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H ANDBOOK OF THE G EOMETRY OF BANACH S PACES

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H ANDBOOK OF THE G EOMETRY OF BANACH S PACES Volume 2 Edited by

W.B. JOHNSON Texas A&M University, College Station, Texas, USA

J. LINDENSTRAUSS The Hebrew University of Jerusalem, Jerusalem, Israel

2003 ELSEVIER Amsterdam • Boston • London • New York • Oxford • Paris San Diego • San Francisco • Singapore • Sydney • Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands © 2003 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science & Technology Rights Department in Oxford, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and ‘Obtaining Permissions’. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2003 Library of Congress Cataloging-in-Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data Handbook of the geometry of Banach spaces Vol. 2 1. Banach spaces I. Johnson, William B., 1944- II. Lindenstrauss, J. 515.7’32 ISBN 0444513051

ISBN: 0-444-51305-1 ∞ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).

Printed in The Netherlands.

Preface 1. Introduction The aim of this Handbook is to present an overview of the main research directions and results in Banach space theory obtained during the last half century. The scope of the theory, having widened considerably over the years, now has deep and close ties with many areas of mathematics, including harmonic analysis, complex analysis, partial differential equations, classical convexity, probability theory, combinatorics, logic, approximation theory, geometric measure theory, operator theory, and others. In choosing a topic for an article in the Handbook we considered both the interest the topic would have for non specialists as well as the importance of the topic for the core of Banach space theory, which is the study of the geometry of infinite dimensional Banach spaces and n-dimensional normed spaces with n finite but large (local theory). Many of the leading experts on the various aspects of Banach space theory have written an exposition of the main results, problems, and methods in areas of their expertise. The enthusiastic response we received from the community was gratifying, and we are deeply appreciative of the considerable time and effort our contributors devoted to the preparation of their articles. Our expectation is that this Handbook will be very useful as a source of information and inspiration to graduate students and young research workers who are entering the subject. The material included will be of special interest to researchers in Banach space theory who may not be aware of many of the beautiful and far reaching facets of the theory. We ourselves were surprised by the new light thrown by the Handbook on directions with which we were already basically familiar. We hope that the Handbook is also valuable for mathematicians in related fields who are interested in learning the new directions, problems, and methods in Banach space theory for the purpose of transferring ideas between Banach space theory and other areas. Our introductory article, “Basic concepts in the geometry of Banach spaces”, is intended to make the Handbook accessible to a wide audience of researchers and students. In this chapter those concepts and results which appear in most aspects of the theory and which go beyond material covered in most textbooks on functional and real analysis are presented and explained. Some of the results are given with an outline of proof; virtually all are proved in the books on Banach space theory referenced in the article. In principle, the basic concepts article contains all the background needed for reading any other chapter in the Handbook. Each article past the basic concepts one is devoted to one specific direction of Banach space theory or its applications. Each article contains a motivated introduction as well as v

vi

Preface

an exposition of the main results, methods, and open problems in its specific direction. Most have an extensive bibliography. Many articles, even the basic concepts one, contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers. The format of the chapters is as varied as the personal scientific styles and tastes of the contributors. In our view this makes the Handbook more lively and attractive. The chapters in Volume 1 were ordered alphabetically according to the first author of the chapter. The chapters in this second volume are ordered by the same principle. Chapters which should have been included in Volume 1 according to this principle but were not ready by the deadline we set for Volume 1 appear instead in the present volume. At the end of this volume there appear a few addenda and corrigenda to articles in Volume 1. Even though this Handbook is quite voluminous it was inevitable that a few aspects of Banach space theory are not covered in it, at least in the depth they deserve. Some of these omissions stem from our planning and others from the fact that a few of the researchers who intended to contribute to this Handbook regretfully were not able to participate. Examples of such aspects are (i) Geometric non-linear functional analysis. (ii) Questions related to parameters (like various approximation numbers and widths) as well as the general area of applications to approximation theory. (iii) The connection between Banach space theory and axiomatic set theory. (iv) Probabilistic inequalities and majorizing measures. Nevertheless, we believe that the Handbook of the Geometry of Banach Spaces presents a reasonably comprehensive and accessible view of the present state of the subject. As for the history of the subject, a short account of the history of local theory is contained in an article by B. Maurey in this volume. A comprehensive history of Banach space theory is now being prepared by A. Pietsch. This volume ends with an author index and a subject index which is common to both volumes.

William B. Johnson and Joram Lindenstrauss

List of Contributors Argyros, S.A., University of Athens, Athens (Ch. 23) Godefroy, G., Université Paris VI, Paris (Ch. 23) Gowers, W.T., Centre for Mathematical Sciences, Cambridge (Ch. 24) Kalton, N., University of Missouri, Columbia, MO (Chs. 25, 26) Ledoux, M., Université Paul-Sabatier, Toulouse (Ch. 27) Mankiewicz, P., Institute of Mathematics, PAN, Warsaw (Ch. 28) Maurey, B., Université Marne la Vallée, Marne la Vallée (Chs. 29, 30) Montgomery-Smith, S., University of Missouri, Columbia, MO (Ch. 26) Odell, E., The University of Texas, Austin, TX (Ch. 31) Pełczyński, A., Institute of Mathematics, PAN, Warsaw (Ch. 32) Pisier, G., Université Paris VI, Paris and Texas A&M University, College Station, TX (Chs. 33, 34) Preiss, D., University College London, London (Ch. 35) Rosenthal, H.P., The University of Texas at Austin, Austin, TX (Chs. 23, 36) Schechtman, G., Weizmann Institute of Science, Rehovot (Ch. 37) Schlumprecht, Th., Texas A&M University, College Station, TX (Ch. 31) Tomczak-Jaegermann, N., University of Alberta, Edmonton (Ch. 28) Tzafriri, L., The Hebrew University of Jerusalem, Jerusalem (Ch. 38) Wojciechowski, M., Institute of Mathematics, PAN, Warsaw (Ch. 32) Wojtaszczyk, P., Warsaw University, Warsaw (Ch. 39) Xu, Q., Université de Franche-Comté, Besançon (Ch. 34) Zinn, J., Texas A&M University, College Station, TX (Ch. 27) Zippin, M., The Hebrew University of Jerusalem, Jerusalem (Ch. 40) Zizler, V., University of Alberta, Edmonton (Ch. 41)

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Contents of Volume 1 Preface List of Contributors

v vii

1. Basic concepts in the geometry of Banach spaces W.B. Johnson and J. Lindenstrauss 2. Positive operators Y.A. Abramovitch and C.D. Aliprantis 3. Lp spaces D. Alspach and E. Odell 4. Convex geometry and functional analysis K. Ball 5. ΛP -sets in analysis: Results, problems and related aspects J. Bourgain 6. Martingales and singular integrals in Banach spaces D.L. Burkholder 7. Approximation properties P.G. Casazza 8. Local operator theory, random matrices and Banach spaces K.R. Davidson and S.J. Szarek 9. Applications to mathematical finance F. Delbaen and W. Schachermayer 10. Perturbed minimization principles and applications R. Deville and N. Ghoussoub 11. Operator ideals J. Diestel, H. Jarchow and A. Pietsch 12. Special Banach lattices and their applications S.J. Dilworth 13. Some aspects of the invariant subspace problem P. Enflo and V. Lomonosov 14. Special bases in function spaces T. Figiel and P. Wojtaszczyk 15. Infinite dimensional convexity V.P. Fonf, J. Lindenstrauss and R.R. Phelps 16. Uniform algebras as Banach spaces T.W. Gamelin and S.V. Kislyakov ix

1 85 123 161 195 233 271 317 367 393 437 497 533 561 599 671

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17. Euclidean structure in finite dimensional normed spaces A.A. Giannopoulos and V.D. Milman 18. Renormings of Banach spaces G. Godefroy 19. Finite dimensional subspaces of Lp W.B. Johnson and G. Schechtman 20. Banach spaces and classical harmonic analysis S.V. Kislyakov 21. Aspects of the isometric theory of Banach spaces A. Koldobsky and H. König 22. Eigenvalues of operators and applications H. König

707

Author Index Subject Index

975 993

781 837 871 899 941

Contents of Volume 2 Preface List of Contributors Contents of Volume 1

v vii ix

23. Descriptive set theory and Banach spaces S.A. Argyros, G. Godefroy and H.P. Rosenthal 24. Ramsey methods in Banach spaces W.T. Gowers 25. Quasi-Banach spaces N. Kalton 26. Interpolation of Banach spaces N. Kalton and S. Montgomery-Smith 27. Probabilistic limit theorems in the setting of Banach spaces M. Ledoux and J. Zinn 28. Quotients of finite-dimensional Banach spaces; random phenomena P. Mankiewicz and N. Tomczak-Jaegermann 29. Banach spaces with few operators B. Maurey 30. Type, cotype and K-convexity B. Maurey 31. Distortion and asymptotic structure E. Odell and Th. Schlumprecht 32. Sobolev spaces A. Pełczyński and M. Wojciechowski 33. Operator spaces G. Pisier 34. Non-commutative Lp -spaces G. Pisier and Q. Xu 35. Geometric measure theory in Banach spaces D. Preiss 36. The Banach spaces C(K) H.P. Rosenthal 37. Concentration, results and applications G. Schechtman xi

1007 1071 1099 1131 1177 1201 1247 1299 1333 1361 1425 1459 1519 1547 1603

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Contents of Volume 2

38. Uniqueness of structure in Banach spaces L. Tzafriri 39. Spaces of analytic functions with integral norm P. Wojtaszczyk 40. Extension of bounded linear operators M. Zippin 41. Nonseparable Banach spaces V. Zizler Addenda and corrigenda to Chapter 7, Approximation properties by Peter G. Casazza Addenda and corrigenda to Chapter 8, Local operator theory, random matrices and Banach spaces by K.R. Davidson and S.J. Szarek Addenda and corrigenda to Chapter 11, Operator ideals by J. Diestel, H. Jarchow and A. Pietsch Addenda and corrigenda to Chapter 15, Infinite dimensional convexity by V.P. Fonf, J. Lindenstrauss and R.R. Phelps

1635 1671 1703 1743

1817 1819 1821 1823

Author Index

1825

Subject Index

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CHAPTER 23

Descriptive Set Theory and Banach Spaces Spiros A. Argyros Department of Mathematics, University of Athens, 15780, Athens, Greece E-mail: [email protected]

Gilles Godefroy Equipe d’Analyse, Université Paris VI, 4 place Jussieu, F-75252, Paris Cedex 05, France E-mail: [email protected]

Haskell P. Rosenthal∗ Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082, USA E-mail: [email protected]

Contents I. I.1. I.2. I.3. I.4. I.5. I.6. II. II.1. II.2. II.3. II.4. II.5. II.6. III. III.1. III.2.

Basic concepts in descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trees and analytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Set derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact sets of first Baire class functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representable Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Descriptive set theoretic complexity of families of Banach spaces . . . . . . . . . . . . . . . . Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The c0 -theorem and Banach space invariants associated with Baire-1 functions . . . . . . . . Connections between Baire-1 functions, D-functions, and Banach spaces containing c0 or 1 The c0 -theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spreading models associated with two special classes of Baire-1 functions . . . . . . . . . . . Transfinite analogs and the Index Theorem for spaces not containing c0 . . . . . . . . . . . . Some open universality problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weakly null sequences and asymptotic p spaces . . . . . . . . . . . . . . . . . . . . . . . . . Compact families of finite subsets of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schreier families and the repeated averages hierarchy . . . . . . . . . . . . . . . . . . . . . . Schreier families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Research partially supported by NSF Grant DMS-0070547.

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2 Edited by William B. Johnson and Joram Lindenstrauss © 2003 Elsevier Science B.V. All rights reserved 1007

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The repeated averages hierarchy (RA-hierarchy) . . . . . . . . . . . . . III.3. Restricted unconditionality and dichotomies for weakly null sequences III.4. Asymptotic p spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tsirelson and mixed Tsirelson norms . . . . . . . . . . . . . . . . . . . III.5. Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract The goal of this chapter is to reveal the power of descriptive set theory in penetrating the structure of Banach spaces. The chapter is divided into three subchapters, each with its own introduction. Subchapters one, two and three were mostly written by the second, third, and first authors, respectively. Space limitations forced us to leave out many fundamental results and on-going research in this exciting interface. We do hope, however, that our article gives a strong flavor of this aspect of Banach space theory.

Descriptive set theory and Banach spaces

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I. Basic concepts in descriptive set theory In this chapter, we introduce the basic concepts in descriptive set theory, and display some of their applications to Banach space theory. Section 1 shows the classical theory of analytic subsets of Polish spaces, the rank theorem for trees, and the countable height of closed well-founded trees in Polish spaces. Set derivations are introduced in Section 2, and first Baire class functions in Section 3, which is devoted to the proof of a dichotomy result on compact sets of first Baire class functions; application is given to Rosenthal’s 1 -theorem. Representable Banach spaces are defined and studied in Section 4, where a classification theorem is provided. Section 5 is devoted to the topological complexity of families of Banach spaces, in relation with universality problems and index theory. N OTATION . If E is a set, we denote by E N (resp. E [N] ) the set of all all sequences (resp. finite sequences) in E. If s ∈ E [N] and n ∈ E. s n stands for the concatenation of s and n, and |s| is the length of s. We denote by ω1 the first uncountable ordinal or equivalently the set of all countable ordinals. The letter α will usually denote a countable ordinal, hence α ∈ ω1 . We let π1 = A × B → A be the first projection: π1 ((x, y)) = x. If M is a topological space, T(M) denotes the collection of all closed subsets of M. I.1. Trees and analytic sets A topological space S is called a Polish space if it is homeomorphic to a separable complete metric space. The space NN consisting of all sequences of natural numbers, equipped with the product of the discrete topologies, is an important example of a Polish space, which is homeomorphic (through continued fractions) to the set [0, 1] \ Q equipped with the topology induced by the real line. Let N[N] be the set of finite sequences of integers. If s ∈ N[N] , we denote by |s| the length of s. If σ ∈ NN , we write s σ if σ “starts with s”, that is, s(i) = σ (i) for all 1 i |s|. Set V s = σ ∈ NN : s σ . Then for any σ ∈ NN , the family {Vs : s σ } is a basis of clopen neighborhoods of σ . A subset T of N[N] is called a tree if t ∈ T and s < t implies that s ∈ T . We denote by [T ] the “boundary” of T , namely: [T ] = σ ∈ NN : s ∈ T for all s σ . It is easily seen that [T ] is a closed subset of NN , and that any closed subset of NN can be obtained in this way. A tree T is well founded if [T ] = ∅. We denote by S the set of all trees of integers (hence S is contained in the power set of N[N] ) and by WF the subset of S consisting of well-founded trees. P ROPOSITION I.1.1. Let S be a Polish space, and A ⊂ S be a subset of S. The following assertions are equivalent:

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(i) There is a continuous map ϕ : NN → S such that A = ϕ(NN ). (ii) There is a Polish space P and a continuous map ψ : P → S such that ψ(P ) = A. (iii) There is a closed subset F of S × NN such that A = π1 (F ). A subset A of S satisfying (i), (ii), (iii) is called an analytic subset of S. Indeed, (ii) implies (i) since any Polish space is a continuous image of NN ; (i) implies (iii) by considering the graph of ϕ, and (iii) implies (i) since a closed subset of a Polish space is Polish. Using the flexibility of the space NN allows us to show through condition (iii) that many sets are analytic. L EMMA I.1.2. Let (An ) be a sequence of analytic subsets of a Polish space S. Then the sets n∈N

An ,

An

n∈N

are analytic. Indeed write An = π1 (Fn ), with Fn closed in S × NN . The set F = (x, σ, n) ∈ S × NN × N: (x, σ ) ∈ Fn is a closed subset of S × NN × N, and π1 (F ) = n∈N An . The result follows since NN × N N N . A similar argument works for the intersection, using this time that (NN )N NN . Since closed sets are trivially analytic, Lemma I.1.2 shows that all sets in the σ -field generated by closed sets (i.e., all Borel sets) are analytic. Note that Proposition I.1.1 shows that continuous functions map analytic sets to analytic sets. As shown by Souslin, it is not so for Borel sets. We now display the link with trees. Let A = π1 (F ) be analytic, with F closed in S × NN . For all x ∈ S, let B(x, δ) = {y ∈ S: d(x, y) < δ}. We set

T (x) = s ∈ N[N] : B x, |s|−1 × Vs ∩ F = ∅ . It is clear that T (x) is a tree, and moreover that x ∈ A ⇔ T (x) ∈ / WF. Let us denote by ω1 = {α; α < ω1 } the set of all countable ordinals. We define the height h(T ) of a tree T as follows. Given T ∈ S, we “trim it” and define T = s ∈ N[N] ; ∃ n ∈ N such that s n ∈ T .


1011

We proceed by transfinite induction to define

T (α+1) = T (α) and if β is a limit ordinal, T (β) =

T (α) .

α 0} and call this the Baire-1 index of f .

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Now let S be a Polish space. We denote by B1 (S) the space of first Baire class functions from S to R, equipped with the topology τρ (S) of pointwise convergence on S. We use derivations for proving T HEOREM I.3.3. Let {fn } be a sequence of continuous functions from S to R. Let K be the closure of {fn } in Rs equipped with the product topology τρ (S). The following are equivalent: (i) Every f ∈ K \ {fn ; n 1} is the pointwise limit of a subsequence of {fn }. (ii) Every f ∈ K is a Borel function. (iii) K ⊆ B1 (S). P ROOF. Only (ii) ⇒ (i) requires a proof. Pick f ∈ K \ {fn : n 1}, and fix a free ultrafilter U on N such that f (x) = lim fn (x) n→U

for all x ∈ S. For (u, v) ∈ Q2 with u < v, we define a derivation Du,v : for A ⊆ S, with A closed, define Du,v (A) by: x ∈ Du,v (A) if for any neighborhood V of x, one has

n 1: inf fn < u < v < sup fn ∈ U. V ∩A

V ∩A

Assume that F = ∅ is such that Du,v (F ) = F . For any ε = ε(k) ∈ {0, 1}[N] , we construct by induction non-empty open sets Vε in F and a sequence {kn : n −1} of integers such that for all s and t, (i) diam(Vs ) < 2−|s| ; (ii) V s ⊂ Vt if s extends t; (iii) if |s| = n, Vs0 ⊂ {fkn < u} and Vs1 ⊂ {fkn > v}. Indeed we may assume that diam(F ) 1. We start with Vφ = F and k−1 = 1. If the construction is done up to |s| = n and fkn−1 , we let As = n 1; inf fn < u < v < sup fn . Vs ∩F

Vs ∩F

Since As ∈ U for all |s| = n, one has

As ∩ {k kn−1 } = ∅.

|s|=n

We pick kn in this set and then construct the open sets Vs (|s| = n + 1) using fkn . For any σ = σ (k) ∈ {0, 1}N , we define

h(σ ) = {Vs : σ extends s}.


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Clearly, h = {0, 1}N → T(S) is continuous. If V is any free ultrafilter on N and g = limn→V (fkn ◦ h), then g(σ ) >

u+v ⇔ σ ∈ V. 2

Since a non-trivial ultrafilter is a non-Borel subset of {0, 1}N , we have a proof that (ii) implies Du,v (F ) = F for any F = ∅ and any u < v. (α) (S) for all α ∈ ω1 . By the above, there is ξ(u, v) = ξ < ω1 We now define Fα = Du,v such that Fξ = ∅, hence S=

{Fα \ Fα+1 ; α < ξ }.

Let {V : 1} be a basis of the topology of S. For any (u, v) ∈ Q2 , there is an α < ξ and an 1 such that at least one of the two sets Au,v ,α = n 1: inf fn u V ∩Fα

and u,v B,α = n 1: sup fn v V ∩Fα

belong to U . We denote by D the countable collection of all subsets of N of the form Au,v ,α u,v or B,α that belong to U , where (u, v) ∈ Q2 , 1 and α < ξ(u, v) are arbitrary. The set D is the basis of a filter H contained in U , and it is easily checked that for all x ∈ S, f (x) = lim fn (x). n→H

It is now easy, using the fact that H has a countable basis, to find a subsequence of {fn } which τρ (S)-converges to f . This shows (i). R EMARK I.3.4. The above proof shows that if {fn } has no pointwise convergent subsequence, there are u < v, a copy of K = h({0, 1}N) of the Cantor set and a subsequence {fkn } such that (fkn ◦ h)(ε) v if ε(n) = 1. In other words, the subsequence {fkn } is equivalent to the sequence of the coordinate functions on {0, 1}N . Note in particular that {fkn } has no pointwise convergent subsequence, and no Borel pointwise cluster point. It is easily seen that the sequence of coordinate functions on {0, 1}N is equivalent in the Banach space C({0, 1}N ) to the natural basis of 1 (N). Hence Theorem I.3.3 and its proof lead to the 1 -theorem ([112]).

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T HEOREM I.3.5. Let (xn )n1 be a bounded sequence in a Banach space X. Then exactly one of the following properties holds true: (a) (xn ) has a subsequence (xkn ) equivalent to the 1 -basis. (b) Every subsequence of (xn ) has a weak Cauchy subsequence. The argument consists in applying Theorem I.3.3 to S = (BX∗ , ω∗ ), and to fn = xn . Note that condition (i) implies the theorem of Odell and Rosenthal [102]: when a separable Banach space X does not contain 1 (N), every x ∗∗ ∈ X∗∗ is the ω∗ -limit of a weak Cauchy sequence in X. (For another exposition of the 1 -theorem, see [62], Theorem 3.3 of Chapter 24 of this Handbook.)

I.4. Representable Banach spaces When a Banach space X is not separable, it is natural to look for a subset of X which witnesses the non-separability, such as an uncountable biorthogonal system. This cannot be done in full generality. However, we show in this section that it can indeed be done when the Banach space is sufficiently regular. We need the following D EFINITION I.4.1. A Banach space X is representable if it is isomorphic to a subspace Y of ∞ (N), such that Y is analytic in RN – equivalently, analytic in (∞ (N), ω∗ ). It is plain that separable spaces are representable. If Y is separable, Y ∗ is representable as a Kσ in RN . If K is a separable compact subset of (B1 (S), τρ ), where S is a Polish space (called a Rosenthal compact in [56]), then C(K) is representable. The following result motivates Definition I.4.1. T HEOREM I.4.2. Let X be a non-separable representable Banach space. Then X has a biorthogonal system of cardinality of the continuum. That is, there exists a bounded subset {(xσ , xσ∗ ): σ ∈ {0, 1}N} of X × X∗ such that xt∗ (xs ) = δst . P ROOF. We still denote by ω∗ the restriction to X of the ω∗ topology of ∞ (N). Let φ : NN → (X, ω∗ ) be a continuous onto map. Fix ε > 0. An easy transfinite induction provides {xα : α < ω1 } ⊆ X and {xα∗ : α < ω2 } ⊆ X∗ such that (i) xα 1. xα∗ < 1 + ε for all α. (ii) If β < α, xα∗ (xβ ) = 0 and xα∗ (xα ) = 1. Pick σα ∈ NN such that φ(σα ) = xα . Since NN is a Polish space, there is an α0 < ω1 such that for all α > α0 , every neighborhood of σα contains uncountably many of {σγ : γ < ω2 }. Let δ0 be some complete metric on NN . For n 1, we construct balls {Bsn : s ∈ {0, 1}n } in NN of δ0 -radius less than 1/n, such that n+1 ⊆ Bsn for s ∈ {0, 1}n and i ∈ {0, 1}; (a) Bsi (b) there exist fsn ∈ 1 such that fsn < 1 + ε, and fsn > 1 − ε on φ(Bsn ), |fsn | 1/n on φ(Bsn ) if s = s.


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For doing so, we first pick α1 > α0 . By ω∗ -approximation of xα∗1 from 1 , there is an f11 ∈ 1 with f11 < 1 + ε and f11 [φ(σα1 )] = 1. Since φ is continuous, we find B11 containing σα1 of radius less than 1 such that f11 [φ(σ )] > 1 − ε for all σ ∈ B11 . Pick now σ2 > β1 > α1 such that σβ1 ∈ B11 and σα2 ∈ B11 . One has

xα∗2 φ(σβ1 ) = 0;

xα∗2 φ(σα2 ) = 1

and again, we find by ω∗ -approximation f12 ∈ 1 with f12 < 1 + ε and

f12 φ(σβ1 ) = 0;

f12 φ(σα2 ) = 1.

Since (f12 ◦ φ) is continuous on NN and every neighborhood of σβ1 contains uncountably many σγ ’s, there is an α3 > α2 such that σα3 ∈ B11 and f12 [φ(σα3 )] < 1/4. We have now

xα∗3 φ(σα3 ) = 1;

xα∗3 φ(σα1 ) = 0.

Approximating xα∗3 , we find f22 ∈ 1 with f22 < 1 + ε and

f22 φ(σα3 ) = 1;

f22 φ(σα1 ) = 0.

We may now find B12 containing σα2 and B22 containing σα3 such that (a) and (b) are satisfied. It should now be clear how to proceed to complete the construction. To finish the proof, we define A : {0, 1}N → NN by A(σ ) =

Bσn|n

n1

and we let xσ = φ(A(σ )]. We pick xσ∗ ∈ X∗ a ω∗ -cluster point in X∗ of the bounded sequence [r(fσn|n )], where r : 1 → X∗ is the canonical map of restriction to X. Conditions (a) and (b) show that the set (xσ , xσ∗ ) works. This result opens the way to a classification theorem, which we state without proof; (i) follows from Theorem I.4.2. T HEOREM I.4.3. Let X be a representable Banach space. Then (i) X is separable if and only if X contains no uncountable biorthogonal system. (ii) X does not contain 1 ({0, 1}N) if and only if X contains no uncountable Markushevich basis, if and only if (BX∗ , ω∗ ) is an angelic compact space. (iii) X contains 1 ({0, 1}N ) if and only if X containsa total uncountable biorthogonal system, i.e., (xt , xt∗ ) such that xs∗ (xt ) = δs,t and t ker(xt∗ ) = {0}.

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I.5. Descriptive set theoretic complexity of families of Banach spaces Let S be a Polish space; recall that F(S) denotes the set of all closed subsets of S. Let S be any metrizable compactification of S. The map F → F from F(S) to F( S) is one-to-one and maps F(S) to F0 ( S) = {F ; F ∩ P = F }. We recall some basic properties (see [36]): S) is a Gδ -subset of F( S), hence a Polish space for the Hausdorff metric of F( S). 1. F0 ( 2. The induced Borel structure (called the Effros–Borel structure) makes F(S) a standard Borel space. 3. This Borel structure is generated by the sets {F ∈ F(P ); F ∩ Vn = ∅}, where (Vn )n is a basis of the topology of S. Hence it does not depend upon the compactification. It is a classical theorem of Banach that every separable Banach space is isometric to a subspace of C({0, 1}N). The subset V of F(C({0, 1}N)) consisting of all vector subspaces is Borel in the Effros–Borel structure, and it is therefore a standard Borel space (i.e., it is Borel-isomorphic to R). This frame allows us to speak about Borel, resp. analytic, resp. coanalytic families of Banach spaces since all these notions can be defined in a standard Borel space. (Cf. also pp. 262–266 of [72] for a treatment of coanalytic families of Banach spaces.) We denote by E ⊆ V 2 the graph of the equivalence relation of linear isomorphism. T HEOREM I.5.1. The set E is analytic non-Borel in V 2 . The equivalence relation of linear isomorphism has no analytic section. In fact, there is a separable space U such that its equivalence class U is not Borel. Also, it follows from the results in [26] that Lp is not Borel for all p = 2 with 1 < p < ∞. This space U is the universal space constructed in [107]. It is not known whether 2 (N) is the only equivalence class which is Borel. Theorem I.5.1 says that the quotient (V/ ) is not a standard Borel space in any natural structure. T HEOREM I.5.2. Let E ⊆ V be an analytic set of separable Banach spaces, stable under linear isomorphism. If E contains all separable reflexive spaces, then there is X ∈ E such that X is universal, i.e., X contains an isomorphic copy of every separable Banach space. Theorem I.5.2 yields the result of Bourgain that a separable Banach space is universal provided every separable reflexive space embeds into it [23]. In turn, the proof of I.5.2 is founded on Bourgain’s arguments. See Section II.1.6 and also Theorem III.4.7 and the paragraph immediately following for further discussion of Bourgain’s work in this connection. Of course (I.5.2) (and in fact the work in [22]) yields that the class SD of Banach spaces with separable dual, is coanalytic non-Borel. The Szlenk index is then a natural rank defined on this coanalytic class. G. Lancien has exhibited a “universal control function” defined on the countable ordinals, showing that the “dentability index” of any space X in SD may be controlled by its Szlenk index [82] (see also p. 805 of [58]). Thus it follows that the slice derivation is not much slower than the Szlenk derivation given in I.2.2(b). A similar theory can be developed for basic sequences, where linear isomorphism is replaced by equivalence between bases. In this frame, the analogous result to Theorem I.5.1


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holds true; in particular, the set of basic sequences up to equivalence is not a standard Borel space in any natural way.

I.6. Notes and remarks 1. The classical theory of analytic sets goes back to Souslin [126] and Lusin [87]. Classical references include the books [79] (§33–39), [36], and [98]. We refer to [73] and references therein for applications of descriptive set theory to harmonic analysis. Transfinite ranks belong to the classical theory [88]. The derivation of Theorem I.1.4 from Theorem I.1.3 is given in ([74], pp. 146–147). Both are special cases of the Kunen–Martin uniform boundedness theorem ([98], p. 101), which involves arbitrary cardinals. An application of Theorem I.1.4 is given in [26]: there exist uncountably many pairwise non-isomorphic nonHilbertian complemented subspaces of Lp , 1 < p < ∞, p = 2. (Such are all Lp spaces; see [2].) It remains an open question if the cardinality of isomorphism types equals the continuum. 2. Derivations were first considered by Cantor in his 1870 solution of the uniqueness problem for trigonometric series; it is well-known that this work led him to the creation of set theory. Cantor’s derivation is Example 2.2(a). Theorem I.2.1 is in particular an extension of Hurewicz’s theorem [67] asserting that the set of countable closed subsets of [0, 1] is coanalytic non-Borel in the Hausdorff topology. It is a special case of a theorem due to Moschovakis (see [98,133]). A link between Theorem I.2.1 and Theorem I.1.4 is obtained via the tree T ⊆ F(K)[N] consisting of all finite sequences (F0 , F1 , . . . , Fn ) such that d(Fi ) = Fi−1 for 1 i n. Finally, we note the following result due to Ghoussoub and Maurey [55]: Every Banach space with the Radon Nikodym Property (or more generally, the Point of Continuity Property) has a subspace isometric to a dual Banach space. The proof of this result involves a transfinite inductive argument and concepts in our chapter. For an exposition, see [117] and [52]; also see [52] for discussion of the RNP and PCP. 3. Theorem I.3.1 is Baire’s main theorem, whose proof goes back to [13] in the case of realvalued functions of a real variable. This is the first time where transfinite arguments were used in studying sequences of functions. Easy modifications of the original proof provide the general result. Finite and transfinite indices which measure the complexity of first Baire class functions have recently been intensively studied ([74,33,32,34,75,76,84,83,119]), in particular in the context of Banach space theory ([123,66,1,42–46,11,77]). They are an operative tool in the proof of the c0 -theorem [118] (given as Theorem II.2.2 below). The study of pointwise compact subsets of the first Baire class originated in [114]; these sets show up in the proof of Rosenthal’s 1 -theorem [112] and Odell–Rosenthal’s characterizations of Banach spaces not containing 1 [102]; see also [116]. Theorem 4.3 is Rosenthal’s dichotomy [115]. Most of the problems left open in [114] were solved in [25]. We refer to [130] for recent deep classification results. An ordinal ranking of these compact sets has been defined in [91], using some results from [56]. Theorem I.3.3 is from [38], where it was used for an effective version of the results; see also [109].

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4. Representable Banach spaces are defined in [61], where it is shown how to adjust Stegall’s proof [127] in order to obtain Theorem I.4.2. The more complete Theorem I.4.3 [50] follows from Theorem I.4.2 and results from [25,37] and [125]. Under an appropriate determinacy axiom, such results extend to subspaces of ∞ (N) which belong to a projective class in the w∗ -topology [60]. 5. This section presents the frame in which the descriptive complexity of families of Banach spaces can be defined and studied, as shown in [17] (see [20]). Theorems I.5.1 and I.5.2 are established in [20]. These results have important antecedents in Bourgain’s work. Bourgain was the first discoverer of the connection between descriptive set theory and universality results. In [22], Bourgain showed in just a few lines, that a separable Banach space X is universal provided C(K) embeds in X for every countable compact metric space K, thus also giving an extension of Szlenk’s theorem [128]. His elegant argument simply exploits the classical fact that the family of closed countable subsets of [0, 1] is coanalytic non-Borel in F([0, 1]]). (See [116] for an exposition of these results of Bourgain.) A primary motivation for [22] is Bourgain’s ordinal-proof, via the Szlenk index for operators and the Kunen–Martin boundedness theorem, of H.P. Rosenthal’s result that a separable Banach space X is universal provided there is a C(K)-space and an operator T : C(K) → X such that T ∗ X∗ is non-separable. See [120] for an exposition of this further application of descriptive set-theoretic methods. We refer to [18,19,21,78,70,71,82,81] and references therein for related work. The note [57] surveys recent applications of the Szlenk index. The book [72] provides updated information on descriptive set theory.

II. The c0 -theorem and Banach space invariants associated with Baire-1 functions Let X be a separable infinite-dimensional Banach space, and let K denote the unit ball of X∗∗ in its weak* topology; let x ∗∗ ∈ X∗∗ . The main purpose of this subchapter is to discuss the connection between the Baire-properties of x ∗∗ |K and the Banach space structure of X. Thus, x ∗∗ is called a Baire-1 element of X∗∗ if x ∗∗ |K is Baire-1, and a D-element if x ∗∗ |K is a difference of bounded semi-continuous functions. Section II.1 gives a fairly self-contained proof of the result that the Baire-1 elements of X∗∗ \ X correspond to nontrivial weak-Cauchy sequences, while the D elements correspond to non-trivial weakly unconditionally summing series (Theorem II.1.2). Then X contains an isomorph of c0 iff X∗∗ \ X has a D-element, while it contains an isomorph of 1 iff X∗∗ has a non-Baire-1 element (Theorem II.1.3). Section II.1 also introduces the classes of (s) and (ss) sequences, fundamental for discussing the c0 -theorem in Section II.2. Some applications of the c0 theorem are reviewed, such as: If X is non-reflexive and every subspace of X has weakly sequentially complete dual, then c0 embeds in X (Corollary II.2.5). The proof of the c0 -theorem involves a fundamental intrinsic characterization of the D-functions on a compact metric space K, through the transfinite oscillations. These are defined, and then a proof is sketched of the c0 -theorem itself, with particular attention to the “real variables – descriptive set-theoretic” part given in Theorem II.2.17. Section II.3 is devoted to the subclasses of Baire-1 functions called B1/2 (K) and B1/4 (K). The fundamental connections here: sequences generating an 1 -spreading model are associated


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with Baire-1 elements of X∗∗ which are not in B1/2 (K), while sequences generating a summing basis spreading model are associated with the ones in B1/4 (K) (Theorems II.3.5 and II.3.6). Section II.4 treats transfinite spreading models and certain transfinite subclasses of B1/4 -functions, and the quite recent Index Theorem for spaces not containing c0 . The subchapter concludes with some open questions in II.5 concerning the possible universality of certain kinds of Banach spaces in terms of their descriptive set-theoretic structure.

II.1. Connections between Baire-1 functions, D-functions, and Banach spaces containing c0 or 1 Let K be a compact metric space. We change our notation slightly from Chapter I and let B1 (K) denote the class of all bounded (real or complex) valued functions on K of the first Baire class. It follows easily that B1 (K) is a Banach algebra in the supremum norm. D EFINITION II.1.1. f : K → C is called a (complex) difference of bounded semicontinuous functions (a D-function) if there exist continuous functions ϕ1 , ϕ2 , . . . on K so that ϕj (k) < ∞ and ϕj converges to f pointwise. (II.1.1) sup k∈K

We then set ϕj (k): (ϕj )is a sequence in C(K) f D = inf sup k∈K

satisfying (II.1.1) .

(II.1.2)

We also let DBSC(K) denote the set of all D-functions on K, and sometimes abbreviate DBSC(K) by D(K). Recall that an extended real valued function f : K → [−∞, ∞] is called upper semicontinuous if f (x) = limy→x f (y) for all x ∈ K; f is called lower semi-continuous if f (x) = limy→x f (y) for all x ∈ K; f is semi-continuous if it is either upper or lower semicontinuous. (Following Bourbaki, we use non-exclusive lim sups and lim infs; thus, e.g., limy→x f (y) = limε↓0 sup{f (y): d(y, x) < ε}, d the metric on K). It then follows from a result of Baire that f ∈ D(K) if and only if there are bounded lower semi-continuous functions u1 , . . . , u4 on K so that f = (u1 − u2 ) + i(u3 − u4 ). D(K) is also a Banach algebra under the D-norm, and in general · D is not equivalent to the sup norm · ∞ on D(K), that is, since obviously D(K) ⊂ B1 (K), in general D(K) is not closed in B1 (K). Now let X be a separable Banach space and let K equal BX∗ in its weak* topology. Let ∗∗ denote the set of all x ∗∗ ∈ X ∗∗ with x ∗∗ |K ∈ B (K), and let X ∗∗ denote the set of all XB 1 D 1 ∗∗ (resp. X ∗∗ ) as the Baire-1 x ∗∗ ∈ X∗∗ with x ∗∗ |K ∈ D(K). We refer to the members of XB D 1 (resp. D-elements) of X∗∗ . The Baire-1 elements of X∗∗ were first introduced in [64].

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The following result shows the fundamental connection between classes of Baire-1 functions and the Banach space structure of X. T HEOREM II.1.2. Let X and K be as above, and let x ∗∗ ∈ X∗∗ . ∗∗ if and only if there is a weak-Cauchy sequence (x ) in X with x → (a) x ∗∗ ∈ XB j j 1 ∗∗ ∗ x ω . Moreover one can then choose (xj ) with xj x ∗∗ for all j . ∗∗ if and only if there exists a sequence (x ) in X so that (b) x ∗∗ ∈ XD x j j is weakly ∗ unconditionally summing (i.e., |x (xj )| < ∞ for all x ∗ ∈ X∗ ) and nj=1 xj → / X, given ε > 0, one can choose (xj ) so that x ∗∗ ω∗ . Moreover if x ∗∗ ∈ (xj ) is equivalent to the c0 -basis

(II.1.3)

and ∞ ∗ x (xj ) < x ∗∗ |K + ε. sup D

x ∗ ∈BX∗ j =1

(II.1.4)

Before giving the proof, we note an immediate consequence of II.1.2 and a result of Odell and Rosenthal [102]. T HEOREM II.1.3. Let X be a separable Banach space. ∗∗ \ X = ∅. (a) c0 → X if and only if XD ∗∗ = ∅. 1 (b) → X if and only if X∗∗ \ XB 1 ((a) follows immediately from II.1.2(b); (b) from [102]; see also the remark following Theorem I.3.5 above.) To prove II.1.2, we give some preliminary notations, also used in the sequel. Given sequences (bj ) and (ej ) in a linear space, (ej ) is called the difference sequence of (bj ) if e1 = b1 and ej = bj − bj −1 for all j . The summing basis refers to the unit vectors basis for SER, the space of all converging series of scalars, i.e., all sequences (cj ) with cj convergent, under the norm (cj )SER = supn | nj=1 cj |. It is easily seen that SER is isomorphic to c0 ; in fact if (ej ) is the usual c0 -basis and bn = ni=1 ei for all n, then n n j =1 cj bj = supk | j =k cj | for all n, so (bj ) is 2-equivalent to the summing basis, and of course (bj ) is also a basis for c0 . D EFINITION II.1.4. Let (bj ) be a given sequence in a Banach space. (a) (bj ) is called non-trivial weak-Cauchy if (bj ) is weak-Cauchy and non-weakly convergent. if (bj ) is a weak-Cauchy basic sequence so that cj (b) (bj ) is called an (s)-sequence converges whenever cj bj converges. In the above, “(s)” stands for “summing”. It follows easily that an (s) sequence is nontrivial weak-Cauchy, hence its closed linear span cannot be weakly sequentially complete. The following result yields universality of (s)-sequences in non-weakly sequentially complete Banach spaces. For a proof, see Proposition 2.2 of [117].


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P ROPOSITION II.1.5. Let (xj ) be a non-trivial weak-Cauchy sequence in a Banach space. Then (xj ) has an (s)-subsequence. We shall also need to use the difference sequences of (s)-sequences, which are (somewhat surprisingly) characterized as follows. D EFINITION II.1.6. A sequence (ej ) in a Banach space is called a (c)-sequence provided it is a semi-normalized basic sequence so that ( nj=1 ej )∞ n=1 is weak-Cauchy. P ROPOSITION II.1.7. Let (bn ) and (ej ) be sequences in a Banach space with (ej ) the difference sequence of (bj ). Then (bj ) is (s) if and only if (ej ) is (c). We sketch the proof; for details, see Proposition 2.1 of [117]. (Throughout, if (xj ) is a sequence in a Banach space, [xj ] denotes its closed linear span. If (xj ) is basic, (xj∗ ) denotes its sequence of biorthogonal functionals in [xj ]∗ ; xj∗ (xi ) = δij for all i and j .) Suppose first (bj ) is an (s)-sequence, and let (Pj ) be its sequence of basis projections: j Pj x = i=1 ci bi if x = ∞ i=1 ci bi . It follows that the ej ’s are linearly independent; an elementary argument yields that if (Qk ) is the sequence of basis projections for (ej ) (just defined on their linear span), then defining en∗ en∗ =

∞

bj∗

(the series converging ω∗ ),

(II.1.5)

j =n

(en∗ ) is biorthogonal to (en ) and Qn = Pn−1 + en∗ ⊗ bn

for all n

(II.1.6)

(where (en∗ ⊗ bn )(α) = en∗ (x)bn for all x ∈ X). Since (bj ) is (s), the sequence (en∗ ) is uniformly bounded, and (II.1.6) then also yields (Qn ) is uniformly bounded, hence (ej ) is basic and so a (c)-sequence. But if (ej ) is a (c)-sequence, (II.1.6) yields that conversely (Pn ) is uniformly bounded, whence (bj ) is (c). We need one last natural Banach space concept. D EFINITION II.1.8. Let (xj ) and (fj ) be sequences in a Banach space. (xj ) is called WUC (Weakly Unconditionally Cauchy) if ∞ ∗ ∗ x (xj ): x ∈ BX∗ < ∞. (xj ) = sup WUC j =1

(fj ) is called DUC (Difference (weakly) Unconditionally Cauchy) if (fj − fj −1 )∞ j =1 is WUC (where f0 = 0); we then set (fj )DUC = (fj − fj −1 )WUC . application of the uniform boundedness principle yields that (xj ) is WUC if A routine |x ∗ (xj )| < ∞ for all x ∗ ∈ X∗ (WUC sequences are also called weakly unconditionally

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summing in the literature). Since the sequence of partial sums of a WUC sequence is weakCauchy, DUC sequences are weak-Cauchy. We need a natural permanence property of DUC sequences. Given (xj ) and (yj ) sequences in a Banach space, (yj ) is called a convex block basis of (xj ) if there exist 0 = n0 < n1 < n2 < · · · and (λj ) non-negative scalars so that for all j ,

λi = 1

and yj =

nj−1 0 for every open neighborhood U of μ}. Of course supp μ is a closed subset of K. Then Pμ and Pa (S) are both ω∗ -dense in P(S) where S = supp μ.

(II.1.10)

Indeed, if W = Pμ or Pa (S), W is convex and f ∞ = supν∈W | f dν| for all f ∈ C(S), so (II.1.10) follows by the Hahn–Banach theorem. Now define G on M(K) by G(μ) =

F (k) dμ(k) for all μ ∈ M(K).

(II.1.11)

K

It follows from the bounded convergence theorem that G|E ∈ B1 (E), hence also H |E ∈ B1 (E), where H = F − G. Suppose the assertion of the lemma is false. Thus H = 0. Since the linear span of P(K) equals M(K), it follows that there is a ν ∈ P(K) with H (ν) = 0; by multiplying H by −1 if necessary, we may assume that Re H (ν) > 0. Let Z denote the space of λ ∈ M(K) with λ absolutely continuous with respect to ν. Applying the Radon–Nikodym and Riesz

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representation theorems, it follows that there is a bounded Borel-measurable function ϕ on K so that ϕ dλ for all λ ∈ Z. (II.1.12) H (λ) = K

Since then Re H (ν) = K Re ϕ dλ > 0, it follows that (Re ϕ)+ dν > 0, where (Re ϕ)+ denotes the positive part of Re ϕ. Thus we may choose a c > 0 so that ν(L) > 0, where L = {k ∈ K: ϕ(k) c}. Thus if λ ∈ P(K) is such that λ(K \ L) = 0, then (II.1.13) Re ϕ dλ = ϕ dλ c. L

Finally, let μ ∈ P(K) be defined by μ(B) =

ν(B ∩ L) ν(L)

for all Borel sets B

and let S = supp μ. Now it follows from (II.1.12) and (II.1.13) that Re H (λ) c

for all λ ∈ Pμ .

(II.1.14)

On the other hand, if k ∈ K and δk denotes the point-mass probability at k, then G(δk ) = F (δk ) and thus H (λ) = 0

for all λ ∈ Pa (S).

(II.1.15)

Now P(S) is a weak*-closed subset of E, yet (II.1.10), (II.1.14), and (II.1.15) yield that H |P(S) has no point of continuity in P(S). This contradicts the fact that H |E ∈ B1 (E), by the Baire characterization theorem (Theorem I.3.1 above). We are finally prepared for the P ROOF OF T HEOREM II.1.2. Let X, K and X∗∗ be as in the statement of Theorem I.1.2. Let Y = C(K); we may regard X ⊂ Y by the Hahn–Banach theorem. (a) Let f = x ∗∗ |K. If there is a weak Cauchy sequence (xj ) converging w∗ to x ∗∗ , then trivially f ∈ B1 (K) since the xj ’s may be regarded as continuous functions on K, and of course xj |K → f pointwise. Suppose conversely that f ∈ B1 (K). Choose (fj ) a sequence of continuous functions on K with fj → f pointwise, where f = x ∗∗ |K, and assume without loss of generality that x ∗∗ = 1. Let τ : C → C be the continuous retraction onto the disc given by τ (z) = z if |z| 1, τ (z) = z/|z| otherwise; then replacing fj by τ ◦ fj for all j , we may assume that fj 1 for all j . Now we claim that fixing n, then

dist X, conv {fj : j n} = 0.

(II.1.16)


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Indeed, if this were false, by the Hahn–Banach separation theorem, we could choose y ∗ ∈ Y ∗ and an r > 0 with y ∗ (x) = 0 for all x ∈ X

and

Re y ∗ (fj ) r

for all j.

(II.1.17)

By the Riesz representation theorem, there is a complex valued Borel measure μ on K representing y ∗ ; we thus have that x(k) dμ(k) = 0 for all x ∈ X and K (II.1.18) Re fj (k) dμ(k) r for all j. But identifying X∗∗ with X⊥⊥ in Y ∗∗ , we also have that then x ∗∗ (μ) = 0, since (II.1.18) yields that μ ∈ X⊥ . Now in fact, if E = BY ∗ , then x ∗∗ |E ∈ B1 (E). Indeed, letting T : X → Y denote the canonical isometric injection, we have that T ∗ (E) = K, and this (fn ◦ (T ∗ |E)) is a sequence of continuous functions on E converging pointwise to x ∗∗ |E. Thus by Lemma I.1.11, ∗∗ x ∗∗ (k) dμ(k) = 0, (II.1.19) x (μ) = K

but

x ∗∗ (k) dμ(k) r > 0

Re

(II.1.20)

K

by (II.1.18) and the bounded convergence theorem. This contradiction yields (II.1.16). But then it follows that there exists a convex block basis (uj ) of (vj ) and a sequence (xj ) in X so that uj − xj → 0

and xj 1

for all j.

(II.1.21)

But it’s clear that still uj → f pointwise, hence also xj → f pointwise on K. Again by the Riesz representation theorem, we obtain that (xj ) is weak-Cauchy with xj → x ∗∗ w∗ . (b) If (xj ) is as in (a), (xj ) is WUC, which implies immediately that f = x ∗∗ |K ∈ ∗∗ . Given ε > 0, we may choose (f ) a seDBSC(K). Suppose conversely, that x ∗∗ ∈ XD j quence in C(K) (with f0 = 0) so that fj → f ∞

pointwise and

(fj − fj −1 )(k) < f D + ε

(II.1.22) for all k ∈ K.

j =1

Again invoking the Riesz representation theorem, we have that (fj ) < f D + ε, DUC

(II.1.23)

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and moreover (fj ) is weak-Cauchy with fj → x ∗∗ ω∗ . By part (a), there exists also a weakCauchy sequence, (xj ) in X with xj → x ∗∗ ω∗ . But then fj − xj → 0 weakly and (fj ) is / X). The conclusion of II.1.10(b) then yields a non-weakly convergent (assuming x ∗∗ ∈ convex block basis (xj ) of (xj ), so that (xj ) is equivalent to the summing basis, and the proof of II.1.10(b) yields that if ε > 0 is given, (xj ) may be chosen so that (II.1.4) holds. We recall finally the following concept. D EFINITION II.1.12 ([107]). A Banach space X has property (u) provided for any weakCauchy sequence (xj ) in X, there exists a DUC sequence (yj ) in X with xj − yj → 0 weakly. The following result is now an immediate consequence of Theorem II.1.2 and Proposition II.1.10. C OROLLARY II.1.13. Let X be a given Banach space. The following are equivalent. (1) X has property (u). (2) Every non-trivial weak-Cauchy sequence in X has a convex block basis equivalent to the summing basis. ∗∗ = X ∗∗ (if X is separable). (3) XB D 1 II.2. The c0 -theorem We first recall the following class of basic sequences introduced in [116]. D EFINITION II.2.1. A sequence (bj ) in a Banach space is called strongly summing (s.s.) weak-Cauchy basicsequence so that whenever (cj ) is a sequence of scalars if (bj ) is a with supn nj=1 cj bj < ∞, cj converges. The following result yields a general subsequence principle characterizing Banach spaces containing c0 , analogous to the 1974 result in [112] characterizing spaces containing 1 . T HEOREM II.2.2 (The c0 -theorem). Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing subsequence or a convex block basis equivalent to the summing basis. The alternatives of this result are mutually exclusive (we indicate why this is so below). We first draw some immediate consequences (throughout, let X denote an infinitedimensional real or complex Banach space). C OROLLARY II.2.3. The following are equivalent. (1) No subspace of X is isomorphic to c0 . (2) Every non-trivial weak-Cauchy sequence in X has an (s.s.) subsequence.


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P ROOF. (1) ⇒ (2) The second alternative of II.1.2 is excluded, since a sequence equivalent to the summing basis spans a space isomorphic to c0 . (2) ⇒ (1) The summing basis is obviously not (s.s.), hence no sequence equivalent to it is (s.s.). Since X has no sequence equivalent to the summing basis, c0 does not embed in X. This corollary in turn yields a dual characterization of spaces containing 1 . C OROLLARY II.2.4. The following are equivalent. (1) No subspace of X is isomorphic to 1 . (2) For every linear subspace Y of X, every non-trivial weak-Cauchy sequence in Y ∗ has an (s.s.) subsequence. P ROOF. (2) ⇒ (1) is trivial, for if Y is isomorphic to 1 , Y ⊂ X, then c0 is isomorphic to a subspace of Y ∗ , hence Y ∗ has a sequence equivalent to the summing basis. (1) ⇒ (2) Suppose to the contrary that (2) failed. By Corollary II.2.3, Y ∗ contains a subspace isomorphic to c0 , for some Y ⊂ X, hence Y contains a subspace isomorphic to 1 by a result of Bessaga and Pełczyński [16]. The next result gives one of the many motivations for the c0 -theorem. C OROLLARY II.2.5. If X is non-reflexive and Y ∗ is weakly sequentially complete for all subspaces Y of X, then c0 embeds in X; moreover X has property (u). To show this, we first note the following fundamental permanence property of (s.s.) sequences. L EMMA II.2.6. Let (bj ) be an (s.s.) sequence in X. Then ( weak-Cauchy sequence.

n

∗ ∞ j =1 bj )n=1

is a non-trivial

P ROOF. Let F ∈ [bj ]∗∗ . Since (bj ) is basic, it follows that n ∗ F bj bj < ∞. sup n

(II.2.1)

j =1

n ∗ ∗ ∞ Hence since (bj ) is (s.s.), ∞ j =1 F (bj ) converges, proving ( j =1 bj )n=1 is weak-Cauchy. ∗ ∗ But of course (bj ) is a semi-normalized basic sequence, hence (bj ) is a (c)-sequence, so ( nj=1 bj∗ )∞ n=1 is an (s)-sequence and thus non-weakly convergent. R EMARK . A stronger permanence property is given in Proposition II.2.10. We pass now to the P ROOF OF C OROLLARY II.2.5. The hypotheses imply that 1 does not embed in X, since c0 embeds in (1 )∗ and c0 is not weakly sequentially complete. Since X is non-reflexive,

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we may choose a bounded sequence in X with no weakly convergent subsequence. But this sequence in turn has a weak-Cauchy subsequence (xj ) by the 1 -theorem. (xj ) has no (s.s.) subsequence by Lemma II.2.6, so since it is non-trivial weak-Cauchy, it has a convex block basis (fj ) equivalent to the summing basis, whence [fj ] is isomorphic to c0 , so c0 embeds in X. Moreover the same argument applies to any non-trivial weak-Cauchy sequence (xj ) in X; letting (fj ) be as above, (xj − fj ) is weakly null and (fj ) is DUC, so X has property (u). A refinement of this argument yields the following equivalence (see Corollary 1.5 of [118]). C OROLLARY II.2.7. The following are equivalent. (1) Y ∗ is weakly sequentially complete for all subspaces Y of X. (2) X has property (u) and 1 does not embed in X. R EMARK . It is known that setting K = ωω + 1, then C(K) fails property (u) (cf. Proposition 5.3 of [66]), hence Corollary II.2.5 yields the existence of a subspace Y of C(K) with Y ∗ non-weakly sequentially complete. It was apparently unknown before the work in [118] if C(K) contained such a subspace Y , for any K countable compact metric. Actually, a particular (s.s.) sequence was discovered in C(ωω + 1) prior to the formulation and proof of Theorem II.1.2, and the study of this example led eventually to the above general results. We may combine the results given in Section II.1 and the above corollaries to obtain the ∗∗ = X ∗∗ \ X ∗∗ ). following result (where we let XND D C OROLLARY II.2.8. The following are equivalent. (1) Neither c0 nor 1 embeds in X. ∗∗ ∩ X ∗∗ = X ∗∗ \ X. (2) XB ND 1 (3) For all non-reflexive subspaces Y of X, there exists a subspace Z of Y so that neither Z nor Z ∗ is weakly sequentially complete. The proof of the c0 -theorem involves the following natural companion notion for (s.s.) sequences. D EFINITION II.2.9. A basic sequence (ej ) in a Banach space is called coefficient converging (c.c.) if (i) ( nj=1 ej )∞ n=1 is a weak-Cauchy sequence and (ii) whenever (cj ) is a sequence of scalars with supn nj=1 cj ej < ∞, the sequence (cj ) converges. Note that (s.s.) sequences are (s) and (c.c.) sequences are (c) (since the conditions (i) and (ii) force (ej ) to be semi-normalized). The following gives some rather satisfying permanence properties relating (s.s.) and (c.c.) sequences (and also implies Lemma II.2.6).


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P ROPOSITION II.2.10. Let (bj ) be a sequence in a Banach space with difference sequence (ej ). The following are equivalent. (1) (bj ) is (s.s.). (2) (ej ) is (c.c.). (3) (bj ) is basic and (bj∗ ) is (c.c.). (4) (ej ) is basic and (ej∗ ) is (s.s.). Moreover if (bj ) is (s.s.), every convex block basis of (bj ) is also (s.s.). (For the proof, see Section 2 of [118].) We next deal with the proof of the c0 -theorem. We first note a “real-variables” formulation. T HEOREM II.2.11. Let K be a compact metric space, f : K → C a bounded discontinuous function, and (fn ) be a uniformly bounded sequence of continuous functions on K with fn → f pointwise. (a) (fn ) has a convex block basis equivalent to the summing basis (in C(K)) if and only if f ∈ D(K). / D(K). (b) (fn ) has an (s.s.) subsequence if and only if f ∈ Let us note that part (a) follows from the results of Section II.1. Indeed, if f ∈ D(K), then by Proposition II.1.10 and Theorem II.1.2, (fn ) has a convex block basis equivalent to the summing basis. But if (gn ) is a convex block basis of (fn ) with (gn ) equivalent to the summing basis, then also gn → f pointwise and of course supk∈K |(gn − gn−1 )(k)| < ∞, so evidently f ∈ D(K). Finally, note that if (fn ) is (s.s.), f ∈ / D(K). Indeed, otherwise (fn ) would have a convex block basis (gn ) which is equivalent to the summing basis; but also (gn ) would be (s.s.) by the last statement of Proposition II.2.10. Thus the summing basis would be (s.s.), a contradiction. If, e.g., we let K equal BX∗ in its weak* topology, we also immediately obtain that the two alternatives of Theorem II.2.2 are mutually exclusive. The hard part of the c0 -theorem is thus the “if” assertion in II.2.11(b). An important ingredient in the proof of this is an intrinsic characterization of functions in DBSC(K) (for K an arbitrary separable metric space), involving the transfinite oscillations of a given scalar valued functions on K. First, if g is an extended real valued function on K, we denote the upper semicontinuous envelope of g by Ug, that is, Ug(x) = limy→x g(y) for all x ∈ X. D EFINITION II.2.12. Given f : K → C and α a countable ordinal, the α-th oscillation of f , oscα f , is defined as follows: osc0 f ≡ 0. If β = α + 1, define

(II.2.2) osc β f (x) = lim f (y) − f (x) + oscα f (y) for all x ∈ K. y→x

If β is a limit ordinal, set osc β f (x) = sup oscα f (x) for all x ∈ K. α 0). L EMMA II.2.16. (fj ) has an (s.s.) subsequence provided, for every ε > 0 and subsequence (gj ) of (fj ), there is a subsequence (bj ) of (gj ) whose difference sequence (ej ) is ε-(c.c.). For the sake of simplicity, let us suppose that we deal only with real scalars; so f is then real-valued. The following real variables result then yields the c0 -theorem. T HEOREM II.2.17. Let α be a countable ordinal and x ∈ K be given with 0 < pα (f )(x) =def λ < ∞. Let U be an open neighborhood of x and 0 < η < 1. There exists (bj ) a subsequence of (fj ) so that letting (ej ) be the difference sequence of (bj ), then given 1 m1 < m2 < · · · an infinite sequence of indices, there exists a t in U and a k so that k

em2j (t) > (1 − η)λ;

(II.2.6)

j =1

em2j (t) > 0 for all 1 j k;

(II.2.7)

ei (t) < ηλ.

(II.2.8)

i ∈{m / 1 ,m2 ,...} im1

We now sketch the proof of the c0 -theorem: let β be the least countable ordinal with oscβ f unbounded (which is well defined by Theorem II.2.13). Then by Proposition II.2.14, either pβ (f ) or pβ (−f ) must be unbounded, so by replacing f by −f and fn by −fn for all n if necessary, we may assume pβ (f ) is unbounded. Now β must be a limit ordinal, since f is bounded, and (II.2.1) easily yields that then oscα+1 f is bounded if oscα f is. But of course then p˜β (f ) is also unbounded, and we have that pα (f ) is bounded for all α < β and sup pα (f )∞ = ∞.

(II.2.9)

α 0, (fn ) has a subsequence (bn ) whose difference sequence is ε-(c.c.). By (II.2.9), we may choose α < β and x ∈ K so that 2 def λ = pα f (x) > . ε

(II.2.10)

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We now claim that if η is sufficiently small and (bi ) satisfies the conclusion of Theorem II.2.17, its difference sequence (ej ) is ε-c.c. Were this false, we could choose a sequence of scalars (cj ) so that n cj ej 1 for all n

(II.2.11)

j =1

and so that there are 1 = m1 < m2 < · · · with cm2j−1 = 0

and cm2j > ε

for all j.

(II.2.12)

(It is obvious that one may always assume c1 = 0.) Now it is also the case that there is a τ < ∞ so that the basis constant of any difference subsequence of a subsequence of (fj ) is at most τ . Now by Theorem II.2.17, choose t in K and k satisfying (II.2.6)–(II.2.8). Thus we obtain m m 2k 2k ci ei ci ei (t) i=1

i=1

ε

k j =1

em2j (t) −

|ci | ei (t)

(by (II.2.7))

i ∈{m / 1 ,m2 ,...}

ε(1 − η)λ − 2τ ηλ

(II.2.13)

(by (II.2.6), (II.2.8), and the basis constant estimate τ ). Of course if η is chosen small m 2k enough, (II.2.10) and (II.2.13) yield that i=1 ci ei > 2, contradicting (II.2.11). We note that the hypothesis of a given open neighborhood U of x is not used in the above deduction; this is used in a crucial way in the proof of Theorem II.2.17, which is achieved by transfinite induction. A remarkable feature of the argument is that only the statement itself at step α, is needed to obtain the step at α + 1 (the main part of this argument).

II.3. Spreading models associated with two special classes of Baire-1 functions We first recall the following two special classes introduced in [66]. Let K be a fixed compact metric space. D EFINITION II.3.1. (a) B1/2(K) denotes the set of all f : K → C so that there is a sequence (fn ) in DBSC(K) with fn → f uniformly. (b) B1/4(K) denotes the set of all f : K → C so that there is a sequence (fn ) in DBSC(K) with fn → f uniformly and supn fn D < ∞.


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Of course B1/2(K) is a Banach algebra under the sup-norm and B1/2 (K) ⊂ B1 (K). B1/4(K) is also a Banach algebra under the norm f 1/4 = inf λ > 0: there exists (fn ) in D(K) with fn → f uniformly and sup fn D λ .

(II.3.1)

n

We next give some intrinsic criteria. (A function f is called a simple D-function if f ∈ D(K) and has only finitely many values.) P ROPOSITION II.3.2. Let f : K → C be given. The following are equivalent. (1) f ∈ B1/2 (K). (2) f is a uniform limit of a sequence of simple D-functions on K. (3) β(f ) ω; i.e., βn (f, ε) < ∞ for all ε > 0. R EMARK . β(f, ε) and β(f ) denote the Baire-1 oscillation indices defined following Example I.3.2. T HEOREM II.3.3. Let f : K → C be given. The following are equivalent. (1) f ∈ B1/4 (K). (2) There exists a sequence (fn ) of simple D-functions with fn → f uniformly and supn fn D < ∞. (3) oscω f ∞ < ∞. R EMARK . The following remarkable identity is obtained by Farmaki in [45]: for f real valued in B1/4(K), f B1/4 = |f | + osc ω f ∞ . We next recall the basic concepts concerning spreading models in Banach spaces. D EFINITION II.3.4. Let (ej ) be a basis for a Banach space E and let (xj ) be a seminormalized basic sequence in a Banach space X. (xj ) is said to generate (ej ) as spreading model if for all ε > 0 and all k, there is an N so that for any N n1 < n2 < · · · < nk , (xn1 , . . . , xnk ) is (1 + ε)-equivalent to (e1 , . . . , ek ).

(II.3.2)

(ej ) is called a spreading model for (xj ) if some subsequence of (xj ) generates (ej ), and finally (ej ) is called a spreading model for X if it is a spreading model for some basic sequence in X. It is easily seen that if (ej ) is a spreading model for X, (ej ) is 1-spreading; that is, (ej ) is isometrically equivalent to its subsequences. Remarkable results of Brunel and Sucheston (based on Ramsey theory), yield that if (xj ) is any semi-normalized basic sequence in a

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Banach space, then some subsequence of (xj ) generates a spreading model (ej ), which is moreover unconditional if (xj ) is weakly null (see [27,28]; also see Theorem 2.2 of [62] for a nice exposition of the first assertion). It then follows from results in Section II.1 above that if (ej ) is a conditional spreading model for (xj ), (xj ) has a non-trivial weak-Cauchy subsequence, and hence an (s)-subsequence. In turn, it follows that then (ej ) is also an (s)-sequence. Moreover if (xj ) is already an (s)-sequence and (ej ) is a spreading model for (xj ), either (ej ) is conditional, or (ej ) is equivalent to the 1 -basis. Now fix X a separable Banach space, and assume K is ω∗ compact with Ext(BX∗ ) ⊂ K ⊂ BX∗ .

(II.3.3)

The following results yield fundamental equivalences connecting spreading models and the Baire-1 classes given in Definition II.3.1. T HEOREM II.3.5. Let (xn ) be a non-trivial weak-Cauchy sequence in X, and let f = x ∗∗ |K where (xn ) converges ω∗ to x ∗∗ . (1) If f ∈ / B1/2(K), then some subsequence of (xn ) generates a spreading model equivalent to the 1 -basis. (2) If every convex block basis of (xn ) has a spreading model equivalent to the 1 -basis, f∈ / B1/2(K). T HEOREM II.3.6. Let (xn ) and f be as in Theorem II.3.5. (1) If f ∈ B1/4 (K), then some convex block basis of (xn ) generates a spreading model equivalent to the summing basis. (2) If (xn ) generates a spreading model equivalent to the summing basis, f ∈ B1/4(K). Let us note that one can find examples of the above result for X = C(K) with K a countable compact metric space. In fact, if K = ωω + 1, then B1 (K) \ B1/2 (K) = ∅; also then the classical Schreier sequence (xj ) is an example of an (s)-sequence with spreading 2 model equivalent to the 1 -basis. If K = ωω + 1, then B1/4(K) \ DBSC(K) = ∅ (see [66]). It follows using the c0 -theorem and Theorem II.3.6 that there exists an (s.s.) sequence (xn ) in C(K) which has a spreading model equivalent to the summing basis. Actually, a Banach space X is constructed in [66] such that c0 does not embed in X, yet X∗ is separable and for some x ∗∗ ∈ X∗∗ \ X, f ∈ B1/4(K) of K = (BX∗ , ω∗ ) and x ∗∗ |K. We conclude this section with a brief discussion of the class of functions of finite Baire index. It follows from the results in Section II.2 that if K is a compact metric space and f : K → C is such that β(f ) < ω, then f ∈ DBSC(K). Actually, f is then a “strong D-function”; that is, there exists a sequence (fn ) of simple D-functions on K with fn − f D → 0 [33]. Of course it follows via Section II.1 that if (fn ) in C(K) is such that (fn ) is uniformly bounded and fn → f pointwise with f discontinuous of finite index, then (fn ) has a convex block basis equivalent to the summing basis.


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II.4. Transfinite analogs and the Index Theorem for spaces not containing c0 Let (bj ) be a fixed normalized basis for a Banach space B. We modify the terminology of Chapter I slightly and say that a finite sequence (x1 , . . . , xn ) in a Banach space X is λ-equivalent to (b1 , . . . , bn ) if n n n 1 cj x j cj bj λ cj x j λ j =1

j =1

for all scalars c1 , . . . , cn . (II.4.1)

j =1

Now suppose X is a separable Banach space containing no subspace isomorphic to B. Given λ 1, set

Tλ = T X, (bj ), λ = (x1 , . . . , xn ): xi ∈ X for all i and

(x1 , . . . , xn ) is λ-equivalent to (b1 , . . . , bn ), n = 1, 2, . . . .

(II.4.2)

It follows that this is a well founded closed tree in X, hence by Theorem I.1.4, it has a height h(Tλ ) < ω1 . Thus also

def h X, (bj ) = sup h(Tλ ) < ω1 .

(II.4.3)

λ1

Now suppose in addition that (bj ) is 1-spreading, and that (xn ) is an infinite seminormalized basic sequence in X so that (bj ) is equivalent to a spreading model for (xn ). Now given λ 1, let

F (xj ), (bj ), λ = F (xj ), λ = (n1 , . . . , nk ): k 1, n1 < n2 < · · · < nk ,

and (xnj )kj =1 is λ-equivalent to (b1 , . . . , bk ) .

(II.4.4)

Then it follows that F ((xj ), λ) is a well founded tree (regarded as a subset of N[N] ) and we have h(F ((xj ), λ)) h(Tλ ). Thus we obtain,

def

ω + 1 h F (xj ) = sup h F (xj ), λ h X, (bj ) < ω1 .

(II.4.5)

1λ

Of course one may replace the infinitely branching tree N[N] by {F ⊂ N: #F < ∞}, identifying finite subsets of N with their increasing enumerations as in (II.4.3). The well founded tree F = (F (xj ), λ) has the additional property that it is hereditary and pointwise closed. That is, given F ∈ F and G ⊂ F , G ∈ F , and finally given (Fn ) in F with (1Fn ) converging pointwise, then there is an F ∈ F with 1Fn → 1F pointwise.

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A bootstrapping method for (rapidly) defining subtrees of the infinitely branching tree, of arbitrarily large height, was formulated by Alspach and Argyros as follows [1]. (We use the convention: if F, G are finite non-empty subsets of N, F < G means max F < min G; also n F means n min F .) D EFINITION II.4.1. For every limit ordinal ξ , fix (ξn ) a sequence of successor ordinals strictly increasing to ξ . We recursively define for ξ < ω1 , a family Sξ of finite subsets of N, as follows: S0 = {n}: n ∈ N ∪ {∅}. If Sξ has been defined, Sξ +1 =

n

Fi : n 1, n F1 < · · · < Fn , and Fi ∈ Fξ for 1 i n ∪ {∅}.

i=1

If ξ is a limit ordinal and Sα has been defined for all α < ξ , Sξ = {F : for some n 1, F ∈ Sξn and n F } ∪ {∅}. Sξ is termed the Schreier family of order ξ . Note that S1 = {F ⊂ N: #F min F if F = ∅}.

(II.4.6)

It is established in [1] that for all ξ < ω1 , Sξ is hereditary, pointwise closed, and a wellfounded tree with h(Sξ ) = ωξ + 1. For further properties, see Section II.3.2. It is easily seen that a Banach space X has a spreading model equivalent to (bj ) if and only if X has a basic sequence (xj ) so that for some λ 1,

F (xj ), λ ⊃ S1 .

(II.4.7)

This forms the basis for the following concept. D EFINITION II.4.2. Let (ej ) be a 1-spreading basis for a Banach space E, let (xj ) be a semi-normalized basic sequence in a Banach space X, and let 1 ξ < ω1 . (xj ) is said to ξ -generate a spreading model equivalent to (ej ) if there is a λ 1 so that Sξ ⊂ F ⊂ N: F is finite and (xj )j ∈F is λ-equivalent to (ej )j ∈F .

(II.4.8)

See Chapter III for several results concerning the special cases where (ej ) is the usual basis for c0 or 1 ; in particular, see Theorem III.3.12 for a duality equivalence. Also see [47] and [48] for interesting dichotomies concerning sequences which ξ -generate spreading models equivalent to the c0 and 1 bases, respectively. We next formulate a certain transfinite analogue of part of Theorem II.3.5.


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T HEOREM II.4.3. Let X be a separable Banach space, let K satisfy (II.3.3), and let (xn ), and f be as in Theorem II.3.5; let 1 ξ < ω1 . If the Baire index β(f ) > ωξ , then some subsequence of (xj ) ξ -generates a spreading model equivalent to the 1 basis. This yields an important boundedness result due to Bourgain [24]. C OROLLARY II.4.4. Let X be a separable Banach space which contains no subspace isomorphic to 1 , and let K satisfy (II.3.3). There exists a countable ordinal α so that

β x ∗∗ |K α

for all x ∗∗ ∈ X∗∗ .

(II.4.9)

N OTE . β(f ) denotes the Baire-1 index of f , defined following Example I.3.2. P ROOF. Let (bj ) be the usual basis for 1 . By the boundedness result cited from Section II.1, we have that there is a countable ordinal α so that

h T X, (bj ), λ α

for all λ 1.

(II.4.10)

Suppose now that β(f ) > ωξ . Then by the results of Section II.1 and Theorem II.4.3, there is a weak-Cauchy sequence (xj ) in X with xj → x ∗∗ ω∗ so that (xj ) ξ -generates a spreading model equivalent to the 1 -basis. Thus by (II.4.5), and the basic properties of Sξ ,

ωξ h F (xj ) α.

(II.4.11)

It follows that β(f ) α · ω.

(II.4.12)

It has recently been discovered jointly by the first author of the present paper and V. Kanellopoulos, that the analogue of Bourgain’s theorem holds in the case of c0 , thus affirming a conjecture of the third author [10]. To formulate this precisely, suppose K is a compact metric space and f : K → C is a bounded function with f ∈ / D(K). It follows from Theorem II.2.13 that there is a least ordinal α < ω1 with oscα f ∞ = ∞. We denote this α by rND (f ), the “non-D index of f ”. T HEOREM II.4.5 (The c0 -index theorem). Let X be a separable Banach space not containing a subspace isomorphic to c0 , and let K = (BX∗ , ω∗ ). Then there exists a countable ordinal α so that rND (x ∗∗ |K) α for all x ∗∗ ∈ X∗∗ . We indicate only some ideas of the intricate proof of this result. Let (bj ) denote the summing basis for SER (the space of converging series). Let K be as above, and suppose f ∈ B1 (K) is discontinuous and satisfies oscξ f ∞ < ∞ for some countable ordinal ξ . Suppose also that (fn ) is a uniformly bounded sequence in C(K) converging pointwise

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to f . Then the “end-goal” of the work in [10], is the proof that there exists a convex block basis (xj ) of (fn ) and a λ > 1 such that

h F (xj ), (bj ), λ ξ. (II.4.13) ∗∗ , then Of course it then follows that assuming f = x ∗∗ |K with x ∗∗ ∈ XB 1

h X, (bj ) ξ.

(II.4.14)

But then were the conclusion of Theorem II.4.5 false, we would obtain that h(X, bj )) = ω1 , which implies that X has an infinite basic sequence equivalent to the summing basis by Theorem I.1.4, a contradiction. To achieve this end-goal, the authors in [10] introduce the concept of the ξ -th variation of a sequence of functions f¯ = (fn ), denoted νξ (f¯ ). First, this requires the novel introduction of a transfinite family of finite subsets of doubleton’s in N, defined as follows, via the notations: given p and q doubleton subsets of N, write p < q if max p < min q; given F a finite non-empty set of such doubletons and n ∈ N , write n F if n min p for all p ∈ F . Now set P0 = {∅}; P1 = {{(n, m)}: n, m ∈ N, n = m}. If Pξ is defined, let Pξ +1 = {p} ∪ F : p < F and F ∈ Pξ ∪ Pξ . If ξ is a countable limit ordinal, choose (ξn ) a strictly increasing sequence with ξ = supn ξn and set Pξ = {F : ∃ n ∈ N with n F and F ∈ Pξn } ∪ {∅}. Now given a sequence of complex-valued functions f¯ = (fn ) defined on a set K and given V ⊂ K, define fi (t) − fj (t). (II.4.15) v˜ξ (f¯, V ) = sup sup F ∈Pξ t ∈V (i,j )∈F

Then define

vξ (f¯, V ) = inf v˜ξ (fm+j )∞ j =1 . m

(II.4.16)

Finally, set vξ (f¯ ) = Vξ (f¯, K). The following basic permanence property for the transfinite variations is then established in [10]. (See Definition II.2.12 for the definition of the transfinite oscillation oscξ f .) T HEOREM II.4.6. Let K, f¯ = (fn ), and f be as above. (a) Given ξ < ω1 and V an open subset of K, sup oscξ f (t) vξ (f¯, V ). t ∈V


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(b) There exists a convex block basis g¯ of f¯ such that vξ (g¯ ) = oscξ f ∞

for all ξ < ω1 .

The end-goal is finally achieved in [10] by showing that given a countable ordinal ξ , if (xn ) is a uniformly bounded point-wise converging basic sequence in C(K), which dominates the summing basis and satisfies v˜ωξ (xn ) < ∞, then for some 1 < λ < ∞, (II.4.13) holds. The proof of II.4.6(a) is achieved following the ideas in the proof of the c0 theorem; in particular, of Theorem II.2.17. The convex block basis in II.4.6(b), termed in [10] the “optimal sequence associated to f ”, is obtained through II.4.6(a) and the following concept and theorem. First, let K and f be as above, and also let g be a bounded positive real valued function on K. Define the relative oscillation of f with respect to g, osc g f , by osc g f (t) = lim f (y) − f (t) + g(y) y→t

for all t ∈ K.

Note that osc α+1 f = osc g f where g = oscα f . The following result in [10], called there the optimal sequences theorem, is then the remaining basic ingredient needed for the proof of Theorem II.4.6, and hence of II.4.5. (The term “optimal sequences” really refers to “optimal convex block basis” in the terminology of this subchapter.) T HEOREM II.4.7. Let K, (fn ), and g be as above, with g upper semi-continuous. Then given ε > 0, there exists a convex block basis (gn ) of (fn ) so that for all n and m, |gn − gm | + g < osc g f ∞ + ε. ∞ We next consider some transfinite Banach algebras of first Baire class functions, and a conjecture, stronger than Theorem II.4.5. Fix K a compact metric space, and for each countable infinite ordinal α let Dα denote the family of all scalar valued bounded functions f with oscα f bounded. It can be seen that Dα is a Banach algebra under the norm f Dα = |f | + oscα f ∞ . One also has that for all 1 ξ and ωξ α < ωξ +1 , Dα = Dωξ ; in particular, this shows K (e.g., K = [0, 1]), it is that rND (f ) = ωξ for some ξ , for any f . For uncountable also known that Dωξ+1 = Dωξ . By Theorem II.2.13, ωα δ for n ∈ G} is a compact family of finite subsets of N with the property that for every convex combination a x with x > δ, there exists a G ∈ F such that x= ∞ n n δ/2 n=1 n∈G an > δ/2. These observations make clear the necessity of studying and understanding the structure of such families. We start with some N OTATION . (i) We denote by [N] (resp. [N] δ then εn an xn C(δ) for all sequences (εn )n∈N ∈ {−1, 1}N . if there exists (iii) The sequence s is said to be Sξ unconditional C > 0 such that for every (an )n∈N , F ∈ Sξ we have that n∈F an xn C ∞ n=1 an xn . In the case of ξ = 1 we shall also use the term C-Schreier unconditional. Concerning these forms of unconditionality we have T HEOREM III.3.2. Let (xn )n∈N be a seminormalized weakly null sequence in a Banach space and ε > 0. There exists a subsequence (xn )n∈M which is: (i) Nearly unconditional. (ii) Convexly unconditional. (iii) (2 + ε) Schreier unconditional. As we have already mentioned after Theorem III.1.2, near and convex unconditionality describe phenomena similar to the content of Theorem III.1.2. Our next proposition highlights this fact and constitutes the main part of the proof of those results. P ROPOSITION III.3.3. Let W be a weakly compact subset of c0 (N) and δ, ε positive reals. Then there exists M ∈ [N] such that the following are fulfilled.

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(1) If F ∈ [M]0 and n∈F φ(n) > δ then there exists φ ∈ W such that n∈F φ (n) > δ − ε and n∈M\F |φ (n)| < ε.

(2) If F ∈ [M] δ then there exists φ ∈ W such

that φ (n) δ − ε and n∈M\F |φ (n)| < ε. Observe that both parts (1) and (2) are indeed analogues of Theorem III.1.2. Their proofs are also obtained by arguments similar to the corresponding proof of Theorem III.1.2. Part (1) concerns near unconditionality and part (2) convex unconditionality. The proof of (2 + ε) Schreier unconditionality uses also similar ideas. Let us point out that we should not expect to obtain Sξ unconditionality for all ξ < ω1 since this would imply that (xn )n∈N must contain an unconditional subsequence. In particular, as was observed by Odell, in the Maurey–Rosenthal examples [93], there is no subsequence which is S2 unconditional. We next present some consequences. For this we need the following. D EFINITION III.3.4. Let s = (xn )n∈N be a sequence in a Banach space. (a) The sequence s is said to be series-bounded if the sequence { nk=1 xk }n∈N of its partial sums is norm bounded. (b) The sequence s is said to be semi-boundedly complete if for every sequence of scalars (an )n∈N such that the sequence (an xn )n∈N is series-bounded we have that limn an = 0. Near unconditionality yields directly the next result, due to Elton (cf. [40,99]). T HEOREM III.3.5 (1st dichotomy). Every seminormalized weakly null sequence either contains a subsequence equivalent to the usual c0 -basis or a semi-boundedly complete subsequence. The next result (cf. [99]), due also to Elton, is another consequence of near unconditionality. P ROPOSITION III.3.6. Let (en )n∈N be a seminormalized nearly unconditional and weakly null Schauder basis for the space X. Assume that no subsequence of (en )n∈N is equivalent to the c0 basis. Then the sequence of biorthogonal functionals (en∗ )n∈N is weakly null. C OROLLARY III.3.7. A Banach space either contains an isomorph of c0 , or a subspace failing the Dunford–Pettis property. Recall that a Banach space X has the Dunford–Pettis property if for every pair (xn )n∈N , of weakly null sequences in X and X∗ respectively we have that xn∗ (xn ) → 0. The following is a generalization of the near unconditionality theorem, involving the repeated averages (cf. [9]).

(xn∗ )n∈N


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T HEOREM III.3.8. Let s = (xn )n∈N be a weakly null sequence and ξ < ω1 . There exists M ∈ [N] such that for every δ ∈ (0, 1] there exists a constant C(δ) > 0 so that the following property is fulfilled: if L ∈ [M], (an )n∈N are scalars in [−1, 1] then ∞ L

L ai ξi · s ai ξi · s C(δ) max δ, i∈F

i=1

for all F ⊂ {n: |an | δ}. This result is used in the proof of Theorem III.3.16 stated below. The appearance of the constant δ in the right-hand side of the above inequality is necessary because we do not have a lower estimate for the quantities ξiL · s. We continue with ξ

ξ

D EFINITION III.3.9. A sequence (xn )n∈N is said to generate an p , (resp. c0 ) spreading model, with ξ a countable ordinal, if (xn ) ξ -generates a spreading model equivalent to the usual basis of the space p (resp. c0 ) as in Definition II.4.2. ξ

ξ

In the sequel we shall restrict our attention to 1 and c0 spreading models. In such a case ξ ξ a bounded sequence (xn )n∈N generates an 1 , (resp. c0 )-spreading model provided there exists a δ > 0 such that for every F ∈ Sξ and all scalar sequences (an )n an xn |an | δ· n∈F

n∈F

resp. an xn δ max |an |: n ∈ F . n∈F ξ

The stability properties of the Schreier families yield that if (xn )n∈N generates an p spreading model then so does every subsequence (xn )n∈M . Furthermore, if ζ < ξ then ξ there exists n0 = n0 (ζ, ξ ) such that (xn )n>n0 is an p spreading model. Bourgain’s ξ p -index, defined through well founded trees, yields that if a sequence (xn )n∈N admits p spreading models for ξ in an unbounded subset of [0, ω1 ) then it contains a subsequence equivalent to the p basis. (See Theorem I.1.4 above and also the discussion in I.2.6, on Section 4.) In particular if (xn )n∈N is a weakly null sequence, then there exists a countable ξ ordinal ξ0 such that for every ξ > ξ0 the sequence (xn )n∈N admits no 1 spreading model. This observation will be used in the second dichotomy (Theorem III.3.11). P ROPOSITION III.3.10. Let s = (xn )n∈N be a seminormalized weakly null sequence and ξ < ω1 such that (xn )n∈N is not ξ -convergent. Then there exists a subsequence s = (xnk )k∈N of (xn )n∈N such that ξ (i) s generates an 1 spreading model.

(ii) s is Sξ -unconditional.

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P ROOF. (i) This is a consequence of the large families lemma (Lemma III.2.10). Indeed Proposition III.2.9 yields that there exists M ∈ [N] and ε > 0 such that for all L ∈ [M] and i ∈ N ξiL s˙ > ε. Set Fε/2 = F ⊂ N: ∃ x ∗ ∈ BX∗ , x ∗ (xn ) ε/2, ∀n ∈ F and observe that Fε/2 is a compact hereditary family which ε/2 norms the family {ξiL : i ∈ N, L ∈ [M]}. From Lemma III.2.10 we obtain that for some L ∈ [M] Sξ (L) ⊂ Fε/2 . Also we may assume that (xn )n∈L is convexly unconditional (Theorem III.3.2). It readily follows ξ that (xn )n∈L generates an 1 spreading model. (ii) This is a consequence of the convex unconditionality of (xn )n∈L obtained in part (i). This proposition and Lemma III.2.1 yield the following result. T HEOREM III.3.11 (2nd dichotomy [11]). Let s = (xn )n∈N be a weakly null sequence and let ξ be a countable ordinal. Then one of the following holds exclusively: (a) The sequence s is ξ -convergent. ξ (b) There exists a subsequence s of s which generates an 1 spreading model and is Sξ unconditional. Moreover there exists a unique ordinal ξ0 such that for all ξ ξ0 the sequence s is ξ -convergent while for all ξ < ξ0 it is not. P ROOF. The fact that one of the two alternatives holds is a direct consequence of Proposition III.2.8 and Proposition III.3.10. The last part of the statement follows from the next C LAIM . If s = (xn )n∈N is ξ -convergent for some ξ , then it is ζ -convergent for all ζ ξ . Indeed, if not, then for some ζ > ξ and some ε > 0 there exists M ∈ [N] such that Sζ (M) ⊂ Fε/2 . Here Fε/2 is the same set as in the proof of Proposition III.3.10. Property 4 of Schreier families yields that for some M ∈ [M] Sξ (M ) ⊂ Fε/2 . Finally Lemma III.2.10 provides L ∈ [M ] such that for every Q ∈ [L] and i ∈ N, ξiQ s˙ > ε/2 hence s is not ξ -convergent which derives a contradiction. For ξ = 1 this theorem yields the well known dichotomy that every seminormalized weakly null sequence either contains a subsequence generating an 1 spreading model or else a Cesaro summable subsequence (cf. [41,113]). For a recent approach of this result see [95]. N OTATION . For a sequence (xn )n∈N in a Banach space X and M ∈ [N] we denote by XM the closed linear span of (xn )n∈M . ξ

ξ

The duality of c0 and 1 spreading models is described in the next result.


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T HEOREM III.3.12 ([9]). Let (xn )n∈N , (xn∗ )n∈N be weakly null sequences in X and X∗ respectively. Assume that for some ε > 0 we have that infn |xn∗ (xn )| > ε. Then, for a countable ordinal ξ , the following are equivalent. ξ (1) For every M ∈ [N] there exists an L ∈ [M] such that (xn )n∈L generates a c0 spreading model. ξ (2) For every M ∈ [N] there exists an L ∈ [M] such that (xn∗ |XM )n∈L generates an 1 spreading model. The implication (1) ⇒ (2) is trivial. (2) ⇒ (1) uses the near unconditionality and the large families lemma. D EFINITION III.3.13. The Banach space X satisfies the ξ -Dunford–Pettis property (ξ -DP) if for every pair (xn )n∈N , (xn∗ )n∈N of weakly null sequences in X and X∗ respectively with (xn∗ )n∈N ξ -convergent, we have that limn xn∗ (xn ) = 0. The space X is said to be hereditarily ξ -DP provided every subspace V of X satisfies the ξ -DP. The next result is a consequence of Theorem III.3.12. C OROLLARY III.3.14. For a Banach space X and 1 ξ < ω1 the following are equivalent. (1) Every seminormalized weakly null sequence in X has a subsequence which generξ ates a c0 -spreading model. (2) X is hereditarily ξ -DP. We pass now to the last dichotomy concerning weakly null sequences. We begin with the following: D EFINITION III.3.15. A seminormalized Schauder basic sequence (xn )n∈N is said to be boundedly convexly complete (b.c.c.) provided the following property holds for all sequences of scalars (an )n∈N such that (an xn )n∈N is series-bounded: Given (Fj )j ∈N , a sequence of consecutive finite subsets of N such that supj n∈Fj |an | < ∞, we have that limj n∈Fj an xn = 0. Let us observe that every b.c.c. sequence (xn )n∈N is also semi-boundedly complete. The converse is not valid. The following extends the 1st dichotomy (Theorem III.3.5) and its proof uses many of the ingredients present in the previous part of this section. T HEOREM III.3.16 (3rd dichotomy [9]). For every seminormalized weakly null sequence s = (xn )n∈N , one of the following alternatives holds exclusively: (a) There exists a boundedly convexly complete subsequence. (b) Every subsequence admits a convex block subsequence equivalent to the unit vector basis of c0 . Moreover (b) is equivalent to: (b ) For all M ∈ [N] there exists N ∈ [M] and ξ < ω1 such that for all L ∈ [N] the convex block subsequence (ξnL · s)n∈N is equivalent to the c0 basis.

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This dichotomy is the analogue, for weakly null sequences, of Rosenthal’s c0 -theorem [118] (Theorem II.2.2 above). However its proof depends on different techniques and it is not known how to obtain one result from the other. We conclude with C OROLLARY III.3.17. Let (xn )n∈N be a seminormalized weakly null sequence with the following property: every subsequence (xn )n∈M admits a further convex block subsequence which is seminormalized and series bounded. Then every subsequence has a convex block subsequence equivalent to the c0 basis.

III.4. Asymptotic p spaces The corner stone of what follows is one of the most important discoveries of the last decades in Banach space theory. That is Tsirelson’s space, invented by Tsirelson (cf. [132]). We recall the definition of the norm of this space which in the sequel will be denoted by T . For x ∈ c00 (N) we set: xT = max x0 ,

sup nE1 0 such that every normalized block basis {xk }nk=1 of (ei )i∈N such that n < supp x1 < · · · < supp xn , is K-equivalent to the usual basis of np (resp. c0n ). Clearly p spaces are asymptotic p spaces. The Tsirelson-space T is an asymptotic 1 space while T ∗ is an asymptotic c0 space. As we will see next, Tsirelson-type norms provide examples of asymptotic p spaces not containing isomorphs of p . Among the properties that one could observe is that any spreading model of an asymptotic p space, generated by a block subsequence of the basis, is isomorphic to p (resp. c0 if p = ∞). An extensive study of asymptotic 1 spaces is presented by Odell, Tomczak and Wagner in [105]. The following interesting result, due to Odell and Schlumprecht, provides a criterion for a Banach space to contain isomorphs of 1 or c0 .


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T HEOREM III.4.2 ([103]). Let X be a Banach space with a Schauder basis (en )n∈N . Assume that every spreading model of X generated by a block subsequence of the basis is isometric to 1 . Then X contains isomorphically 1 . A similar statement holds also for c0 . The proof of this theorem is obtained by transfinite induction. More precisely it is shown ξ ξ that for all ξ < ω1 the space X contains 1 (resp. c0 ) spreading models. As we have mentioned before, this yields that X contains 1 or c0 respectively. Let us point out that the isometric assumption concerning the spreading models is necessary, as Tsirelson space indicates. Further it is not known if a similar statement holds for 1 < p < ∞.

Tsirelson and mixed Tsirelson norms We start with the following: D EFINITION III.4.3. Let M be a compact family of finite subsets of N. (i) A sequence {Ei }ni=1 of successive subsets of N is called M-admissible if there exists G = {mi }ni=1 ∈ M such that m1 E1 < · · · < mn En . A sequence {xi }ni=1 of vectors of c00 (N) is called M-admissible provided that the sequence {supp xi }ni=1 is M-admissible. (ii) A sequence {Ei }ni=1 of finite pairwise disjoint subsets of N is called M-allowable if the family {{min Ei }}ni=1 is M-admissible. A sequence {xi }ni=1 of vectors of c00 (N) is M-allowable provided that the sequence {supp xi }ni=1 is M-allowable. Let us observe that the condition n E1 < · · · < En appearing in the definition of Tsirelson’s norm is equivalent to saying that {Ei }ni=1 is S-admissible where S denotes the Schreier family S1 . The concept of M-admissible families was introduced in two unpublished papers. First it was defined for M = Sξ by the first author and subsequently the general case by Deliyianni jointly with the first author. We are now ready to define Tsirelson and mixed Tsirelson norms. We begin with the first one. (a) Norms of the form Tp (M, ϑ) Let M be a compact family of finite subsets of N, 0 < ϑ < 1 and 1 p < ∞. The space Tp (M, ϑ) is the completion of c00 (N) under the following norm. For x ∈ c00 (N) we set x(p,M,ϑ) = max x0 , sup ϑ

n

1/p p Ei x(p,M,ϑ)

.

i=1

The sup is taken over all M-admissible families {Ei }ni=1 . If p = 1 we shall denote T1 (M, ϑ) by T (M, ϑ). Let us also denote by An the compact family {F ⊂ N: #F n}. Next we list some properties of these spaces.

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T.1 The space T (S, 1/2) is Tsirelson space while Tp (S, 1/2) is what is called the p-convexification of Tsirelson space. Each one of them defines an example of an asymptotic p space not containing p . The dual of T (S, 1/2) is an asymptotic c0 space. Similar properties hold for the spaces Tp (Sξ , ϑ). Moreover every normalized ξ block sequence in Tp (Sξ , ϑ) defines an p spreading model. T.2 For all n ∈ N, 1 p < ∞, ϑ ∈ (0, 1) the space Tp (An , ϑ) is isomorphic to some q , 1 < q < ∞ or to c0 . In particular if p = 1 and ϑ 1/n then T (An , ϑ) is isomorphic to c0 while if ϑ > 1/n then T (An , ϑ) is isomorphic to p where 1/p + 1/q = 1 and 1/n1/q = ϑ. T.3 If M is a compact family with Cantor–Bendixson index i(M) > ω then for all ϑ ∈ (0, 1) the space T (M, ϑ) is a reflexive space not containing any p , 1 < p < ∞. The p-convexification of Tsirelson’s space was defined and studied in [49] and modified versions of it in [69]. The isomorphisms of T (An , ϑ) and p is a result of Bellenot (cf. [14]). Results concerning the relation of Cantor–Bendixson index of M and the structure of the space T (M, ϑ) are contained in [15]. (b) Mixed Tsirelson norms of the form Tp ((Mn )n , (ϑn )n ) Let (Mn )n∈N be a sequence of compact families and (ϑn )n∈N be a null sequence with 0 < ϑn < 1. For 1 p < ∞ we define Tp ((Mn )n , (ϑn )n ) to be the completion of c00 (N) under the norm · ∗ defined as follows: For x ∈ c00 (N) we set,

x∗ = max x0 , sup sup ϑn n

n

1/p p Ei x∗

,

i=1

where the inside sup is taken over all Mn -admissible families {Ei }ni=1 . Such spaces are called mixed Tsirelson spaces. The modified mixed Tsirelson spaces are denoted by TpM ((Mn )n , (ϑn )n ) and their norms are defined by equations similar to the above one where the inside supremum is taken over all Mn -allowable families {Ei }ni=1 . Let us point out that TpM (M, ϑ), the modified Tsirelson-type spaces, are defined in a similar manner. Mixed Tsirelson norms were introduced in [5] and modified mixed Tsirelson norms in [6]. Next we present some properties of mixed Tsirelson spaces as well as some examples. The most interesting families (Mn )n∈N for such examples are the family (An )n∈N and the family (Sn )n which is the first sequence of the Schreier families {Sξ }ξ 1, there exists an equivalent norm | · | on X such that for all infinite-dimensional subspaces Y of X, |x| : x, y ∈ SY λ. sup |y| For a comprehensive discussion of distortion issues, see Chapter 31 in this Handbook by E. Odell and T. Schlumprecht, where their fundamental discovery is reviewed: p is arbitrarily distortable for all 1 < p < ∞ [104]. The reader will also find discussed there the following remarkable result due to Milman and Tomczak-Jaegermann [97]. T HEOREM III.4.5. Let X be a Banach space containing no arbitrarily distortable subspace. Then X contains an asymptotic p space for some 1 p < ∞ or an asymptotic c0 space. It remains an open question as to whether Tsirelson’s space itself, or its dual, could satisfy the hypotheses of III.4.5. On the other hand, it is also open as to whether a space with these properties must be c0 or 1 saturated, implying by a celebrated result of James that it has no distortable subspaces. We continue with the following result due to Tomczak-Jaegermann [131]. T HEOREM III.4.6. Every HI Banach space is arbitrarily distortable. The basic argument in the proof uses transfinite induction and shows that every boundedly distortable Banach space contains Sξ unconditional sequences for all ξ < ω1 . This yields that the space must contain an unconditional basic sequence. Note that this theorem yields that the asymptotic p -space occurring in the conclusion of Theorem III.4.5 may be chosen to have an unconditional basis. We conclude this section with the next result which demonstrates the size and the variety of the class of HI spaces.

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T HEOREM III.4.7 ([4]). Let X be a separable Banach space containing all reflexive HI Banach spaces. Then X is universal, i.e., it contains an isomorphic copy of any separable Banach space. This result extends a previous theorem due to Bourgain [23] (see also [116] and Section I.5 above). Its proof is also related to transfinite constructions. For a Banach space Y with a basis (en )n∈N Bourgain has obtained a long sequence (Rξ (Y ))ξ 0. Must there exist an infinite-dimensional subspace Y and a colour j such that for every vector y in the unit sphere of Y there is a vector z in the unit sphere of Y such that y − z < ε and f (z) = j ? In other words, can Y be found such that every point in the sphere of Y is close to a point of some given colour? The Banach space X in the above problem may either be a general one or something specific such as 2 . In both cases the answer is far from obvious. An equivalent and often more useful formulation is the following. P ROBLEM 4.2. Let X be a Banach space, let f : S(X) → R be a uniformly continuous function and let ε > 0. Must there exist an infinite-dimensional subspace Y such that, given any two vectors x, y ∈ S(Y ), |f (x) − f (y)| < ε? These problems are discussed fully in Chapter 31 on distortion [19], so we shall confine ourselves to a few remarks. First, it is an easy consequence of Milman’s proof of Dvoretzky’s theorem that the problems have positive answers if we ask merely that Y should have arbitrarily large finite dimension. Moreover, using the proof of Krivine’s theorem instead of that of Dvoretzky’s (for all these results see [16]), one can ask for Y to be generated by a block basis of any given basic sequence in X. In particular, if X is c0 or p , Y can be chosen isometric to n∞ or np .

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Second, there are results in Ramsey theory which are encouraging. One of them is a theorem of Hindman [14], which can be formulated as follows. T HEOREM 4.3. Let the finite subsets of N be coloured with finitely many colours. Then there exist finite subsets A1 , A2 , A3 , . . . of N, with max Ai < min Ai+1 for every i, such that all non-empty unions of finitely many of the Ai have the same colour. If one identifies finite subsets of N with 01-sequences in the natural way, then the above theorem says that for any finite colouring of a sort of discrete sequence space there is a monochromatic “infinite-dimensional block subspace”. This seems to be evidence in favour of a positive answer to Problems 4.1 and 4.2. The theorems of Dvoretzky and Krivine show that any counterexample to the problems has to be very definitely infinite-dimensional. Hindman’s theorem suggests that it must also be non-combinatorial in an essential way if it is to defeat known Ramsey-theoretic arguments. Despite these two restrictions, counterexamples are known to exist, but because of the restrictions, they are very mysterious. The first known explicit counterexample was, in the second formulation, a simple renorming, due to Odell, of a small variant of Tsirelson’s space (see [15]). The famous distortion problem concerns the case X = p and was solved in the negative by Odell and Schlumprecht [18]. The ideas they introduced have now been used to show that for almost no space X is there a positive answer: this is in striking contrast with the finite-dimensional situation. This article is concentrating on positive results, so we shall now consider the one case where there is a Ramsey result for Banach spaces and their subspaces, which is when the space is c0 . This space is sufficiently like a space of zeros and ones, and addition of disjointly supported vectors is sufficiently like the union of disjoint sets, that ideas from the proof of Hindman’s theorem can be used successfully. The first step of the argument is to define a sort of combinatorial approximation of c0 . To begin with, we do this just for the positive part of the unit sphere (i.e., the set of vectors with non-negative coordinates). Given ε > 0 and k = k(ε) sufficiently large, let A (for alphabet) be the set {0} ∪ {(1 − ε)r : 0 r k}. Then any vector y in the positive part of the sphere of c0 can be approximated to within ε by a finitely supported vector x all of whose coordinates belong to A. In addition, we can insist that at least one coordinate of x is 1. Let X be the set of all finitely supported vectors with coordinates in A, at least one of which is 1. The advantage of the alphabet A over a more obvious one like {0, k −1 , 2k −1 , . . . , 1} is that it is easy to deal with scalar multiplication. Indeed, if we define a binary operation ∗ on A by letting a ∗ b be ab if ab ∈ A and 0 otherwise, then |ab − a ∗ b| < (1 − ε)k , which we have assumed to be less than ε. Given a ∈ A and x = (x1 , x2 , . . .) ∈ X, let us define a ∗ x to be the vector (a ∗ x1 , a ∗ x2 , . . .). Although a ∗ x does not belong to X (unless a = 1), we still have the property that if x and y are in X and have disjoint supports, then x + a ∗ y and a ∗ x + y belong to X. Because X is closed under the addition of disjointly supported vectors, it has the structure of a partial semigroup. This allows us to use ultrafilter techniques, which also yield a very neat proof of Hindman’s theorem (due to Glazer). Let us begin with a lemma which is fundamental in this kind of argument.

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L EMMA 4.4. Let (S, +) be a compact Hausdorff semigroup such that addition is rightcontinuous, meaning that for every x ∈ S the map y → y + x is continuous. Then S contains an idempotent; that is, an element x such that x + x = x. P ROOF. By an easy Zorn’s lemma argument there is a minimal non-empty compact subsemigroup T of S. Let x ∈ T be arbitrary. Since T is compact and addition is rightcontinuous, the set T + x is compact. Moreover, (T + x) + (T + x) = (T + x + T ) + x ⊂ T + x, so T + x is a subsemigroup. But T + x ⊂ T , so by the minimality of T we must have T + x = T . It follows that there exists z ∈ T such that z + x = x. Now the set of all such z is compact, as it is the inverse image of {x} under the continuous map z → z + x. Moreover, it is closed under addition. Therefore, by minimality again, it is the whole of T . It follows that x + x = x. We shall apply Lemma 4.4 to a certain semigroup of ultrafilters. Let us say that an ultrafilter α on the set X defined above is cofinite if every set of the form Xn = {x ∈ X: x1 = · · · = xn−1 = 0} belongs to α. (This is the appropriate non-triviality condition, and is stronger than saying merely that α is non-principal.) The following notation is incredibly useful, not just to save writing but also as an aid to thought. If P (x) is a statement about vectors x ∈ X, we shall write (αx) P (x) for the statement {x ∈ X: P (x)} ∈ α. One can think of α as a (finitely additive) measure on X and read (αx) as “for α-almost every x”. Thus, α becomes a quantifier. The ultrafilter properties of α convert into the rules (αx) P (x) and (αx) Q(x)

if and only if (αx) P (x) and Q(x)

and not (αx) P (x)

if and only if

(αx) not P (x),

which we shall use repeatedly. If α and β are two cofinite ultrafilters on X, we define their sum to be α + β = A ⊂ X: (αx) (βy) x + y ∈ A . We interpret the statement x + y ∈ A to be false if x and y are not disjointly supported. Notice, however, that because β is cofinite and x is finitely supported we have that (αx) (βy) x and y are disjointly supported so in the ultrafilter world, questions of support are more or less irrelevant. It may make it easier to understand ultrafilter addition if we state a third quantifier rule, which is equivalent to the definition (as may be seen by replacing “P (x)” by “x ∈ A”):

(α + β)x P (x)

if and only if

(αx) (βy) P (x + y).

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It is easy to check (and this is one of the advantages of the quantifier notation) that ultrafilter addition is associative. We shall also define “linear combinations” of cofinite ultrafilters on X in a similar way. Let α and β be two such ultrafilters and let a ∈ A. Then α + a ∗ β = A ⊂ X: (αx) (βy) x + a ∗ y ∈ A and a ∗ α + β = A ⊂ X: (αx) (βy) a ∗ x + y ∈ A . Clearly this definition can be extended to general combinations a1 ∗ α1 + · · · + an ∗ αn , provided that at least one ai equals 1. Once again, there are corresponding quantifier rules. For example,

(α + a ∗ β)x P (x)

if and only if

(αx) (βy) P (x + a ∗ y).

It is possible and useful to interpret a ∗ α as an ultrafilter itself, not on X buton a ∗ X, which is the combinatorial version of the set of points in c0 of norm a. Let Y = a∈A a ∗ X (the combinatorial unit ball rather than unit sphere). Then Y is still a partial semigroup, and if we define a ∗ α to be {A ⊂ a ∗ X: (αx) a ∗ x ∈ A}, then a ∗ α + β is simply the sum of two ultrafilters defined on Y . We state the next lemma without proof, since it is basically an exercise to check everything. L EMMA 4.5. Let S be the set of all cofinite ultrafilters on X. Then S is a compact Hausdorff semigroup, under ultrafilter addition. Moreover, for every a ∈ A and every β ∈ S, the maps α → a ∗ α + β and α → α + a ∗ β are continuous. Notice that Lemmas 4.4 and 4.5 imply the existence of an idempotent ultrafilter in S. The next result is a strengthening of this fact, and is the key to the whole argument. L EMMA 4.6. There exists an ultrafilter α ∈ S such that α + a ∗ α = a ∗ α + α = α for every a ∈ A. P ROOF. Set a = (1 + ε)−1 and for j = 0, 1, 2, . . ., k let us write Xj for a k−j ∗ X and Sj for the set of all cofinite ultrafilters on Xj . Lemma 4.5 tells us that Sj is a compact Hausdorff semigroup and that linear combinations are right continuous. (This is not quite directly true, but follows from the fact that in Lemma 4.5 there is an implicit dependence on k, and there is an isomorphism between Xj and the case k = j of X.) We shall prove by induction that for every j k there is a cofinite ultrafilter α on Xj such that a i ∗ α + α = α + a i ∗ α = α for i = 0, 1, . . . , j . The result is true when j = 0, because X0 consists of finitely supported non-zero vectors all of whose coordinates are either 0 or a k . Therefore, all we need to find is an idempotent ultrafilter. The existence of this follows from Lemmas 4.4 and 4.5. So let us now suppose


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that j > 0, and that we have a cofinite ultrafilter β on Xj −1 such that β + a i ∗ β = a i ∗ β + β = β for i = 0, 1, . . . , j − 1. Let T = {α ∈ Sj : a ∗ α = β}. It is easily checked that the map α → a ∗ α is a continuous map from Sj to Sj −1 . Therefore T is compact, as is the set T + β. This second set is also closed under addition, since if a ∗ γ = a ∗ δ = β, then a ∗ (γ + β + δ) = a ∗ γ + a ∗ β + a ∗ δ = β + a ∗ β + β = β + β = β. (The equality a ∗ β + β = β follows even when j = 1 because then β ∈ S0 and a ∗ β = 0.) By Lemma 4.4 we can find an idempotent in T + β. That is, there is an ultrafilter γ ∈ Sj such that a ∗ γ = β and γ + β + γ + β = γ + β. Now set α = β + γ + β. If 1 i j , then α + a i ∗ α = β + γ + β + a i ∗ β + a i−1 ∗ β + a i ∗ β = β + γ + β = α (when i = j we are using the fact that a j β = 0) while α + α = β + γ + β + β + γ + β = β + (γ + β) + (γ + β) = β + (γ + β) = α. Therefore, the ultrafilter α gives us the inductive step.

By a block basis of X, we shall mean a sequence in X that forms a normalized block basis of c0 . If x1 , x2 , . . . is a block basis of X, then by the subspace generated by x1 , x2 , . . . we shall mean the set of all points in X of the form ni=1 ai ∗ xi . (Notice that ai must equal 1 for at least one i.) A subspace generated by a block basis is a block subspace. The ultrafilter we have just constructed implies the following combinatorial Ramsey result. T HEOREM 4.7. Let X be finitely coloured. Then X contains a monochromatic infinitedimensional block subspace. We do not give the proof here, because it is formally almost identical to the proof of Theorem 4.13 which is more central to our concerns and will be given later. Notice, however, that if k = 1, then Theorem 4.7 is equivalent to Hindman’s theorem. One can use Theorem 4.7 to prove a positive answer to 4.2 in the case of functions f that depend only on the (pointwise) modulus of a vector. (One could think of these as “unconditional” functions.) We now turn to the more general situation. Let Z be the set of all finitely supported vectors in the unit sphere of c0 with coordinates in A ∪ −A, and note that Z is an ε-net of the entire unit sphere of c0 . The definitions of multiplication in A ∪ −A, scalar multiplication in Z, block subspaces and so on are all obvious. The next lemma will allow us to use the method of Lemma 4.6 to find an ultrafilter appropriate for results about Z. One cannot hope for exact results about Z because of the colouring according to the sign of the first non-zero coordinate. Accordingly, even though Z is discrete, we must consider approximations. The alphabet A ∪ −A is a totally ordered set. Let us say that two elements a, b ∈ A are neighbours if they are next to each other in this ordering. Given

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vectors x, y ∈ Z, let us say that they are neighbours if xi is a neighbour of yi for every i. Let the neighbourhood N(x) of x be the set of all neighbours of x. Given a subset E ⊂ Z, let us define the expansion E of E to be the set of all neighbours of elements of E, or equivalently the union of all neighbourhoods of elements of E. L EMMA 4.8. There exists a cofinite ultrafilter α on Z such that, for any set E ∈ α, the set −E also belongs to α. P ROOF. Let β be any cofinite ultrafilter on Z, let a = (1 − ε) and let α = a k ∗ β − a k ∗ β + a k−1 ∗ β − a k−1 ∗ β + · · · + a ∗ β − a ∗ β + β − β + a ∗ β − a ∗ β + a 2 ∗ β − a 2 ∗ β + · · · + a k ∗ β − a k ∗ β. We shall see that α has the required property. For notational convenience, let a1 , a2 , . . . , am be the sequence a k , −a k , . . . , a, −a, 1, −1, . . ., a k , −a k of coefficients used above. Then, using our quantifier rules for linear combinations, the statement that E belongs to α is equivalent to the statement (βx1 ) . . . (βxm )

m

ai ∗ xi ∈ E.

i=1

By relabelling some of the dummy variables, we can say also that

(βx0 ) . . . (βxm−1 )

m−1

ai+1 ∗ xi ∈ E.

i=0

Now the sequence a1 , . . . , am has been chosen with the property that, for every i < m, ai is a neighbour of −ai+1 , and also so that a1 and am are neighbours of 0. Therefore, from the second statement, we can deduce that (βx1 ) . . . (βxm )

m

ai ∗ xi ∈ −E

i=1

which shows that −E ∈ α, as required.

The above trick is very similar to an argument used by Brunel and Sucheston to find an unconditional spreading model in any Banach space, using invariance under spreads. Let us define an approximate ultrafilter on Z to be a filter γ such that, whenever Z = C1 ∪ · · · ∪ Cr , there exists i such that the expansion C i belongs to γ . We shall say that


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a filter γ is sign-invariant if E ∈ γ implies that −E ∈ γ . Notice that sign-invariance is equivalent to the quantifier rule (γ x) P (x)

if and only if (γ x)P (−x).

C OROLLARY 4.9. There exists a cofinite sign-invariant approximate ultrafilter γ on Z. P ROOF. Let α be the ultrafilter constructed in Lemma 4.8. Define γ to be the set of all E such that E ∈ α and −E ∈ α. Then the sign-invariance of γ is immediate, and all we need to do is show that γ is an approximate ultrafilter. This is also easy. If Z = C1 ∪ · · · ∪ Cr , then Ci belongs to α for some i, and by Lemma 4.8 so does −C i . Obviously C i belongs to α as well, so it belongs to γ . The next lemma is another exercise in checking uninteresting facts, so again we state it without proof. L EMMA 4.10. The set S of all cofinite sign-invariant approximate ultrafilters on Z is a compact Hausdorff semigroup under filter addition. Moreover, the maps α → α + a ∗ β and α → a ∗ α + β are continuous for every a ∈ A ∪ −A and every β ∈ S. The next lemma is deduced from Lemma 4.10 in a similar way to how Lemma 4.6 is deduced from Lemma 4.5. However, the proof is not quite identical: instead of defining Zj to be a k−j ∗ Z and letting Sj be the set of cofinite sign-invariant approximate ultrafilters on Zj , one should instead define Sj to be a k−j ∗ S. (If one does not do this, then it is not true that a ∗ Sj = Sj −1 .) In fact, the proof given in [10] is for this reason not quite correct. A correct proof, with full details, can be found in [4] on pages 318–319. L EMMA 4.11. There exists a cofinite sign-invariant approximate ultrafilter α on Z such that α + a ∗ α = a ∗ α + α = α for every a ∈ A. Given a block basis x1 , . . . , xn in Z we shall write x1 , . . . , xn for the (combinatorial) subspace generated by x1 , . . . , xn . C OROLLARY 4.12. There exists an approximate ultrafilter α on Z such that E ∈ α implies that (αx) (αy) x, y ⊂ E. P ROOF. Let α be the filter from Lemma 4.11 and let E ∈ α. For every j ∈ {0, 1, . . . , k} we know that (αx) (αy) x + a j ∗ y ∈ E and (αx) (αy) a j ∗ x + y ∈ E.

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By the quantifier rule for sign-invariance, we can deduce that (αx) (αy) ± x ± a j ∗ y ∈ E and (αx) (αy) ± a j ∗ x ± y ∈ E and the required property of α follows from the quantifier rule for conjuctions of statements. Notice that the property of α in Corollary 4.12 translates into the quantifier rule (αx) P (x)

if and only if

(αx) (αy) ∀z ∈ x, y P (z).

We can now deduce a Ramsey theorem for colourings of Z. As commented earlier, almost exactly the same argument proves Theorem 4.7. T HEOREM 4.13. Let Z be finitely coloured with colours C1 , . . . , Cr . Then there exists i such that C i contains an infinite-dimensional block subspace of Z. P ROOF. Let α be the filter from Corollary 4.12, let i be such that C i ∈ α and let C = C i . We shall apply our quantifier rules over and over again. To begin with we know that (αx) x ∈ C. By the rule stated just before this theorem we can deduce that (αx) (αy) x, y ⊂ C. It follows (after renaming x and y) that there exists x1 such that (αx) x1 , x ⊂ C. Suppose inductively that we can find x1 , . . . , xn such that (αx) x1 , . . . , xn , x ⊂ C. Then by the quantifier rule we find that (αx) (αy) ∀z ∈ x, y x1 , . . . , xn , z ⊂ C which is easily seen to be equivalent to the statement (αx) (αy) x1 , . . . , xn , x, y ⊂ C. Hence, we can find xn+1 to continue the induction. We therefore construct a sequence x1 , x2 , . . . with the property that x1 , . . . , xn ⊂ C for every n, which means that the block subspace generated by x1 , x2 , . . . is a subset of C. It is easy to deduce from this a result about uniformly continuous functions on the sphere of c0 .


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T HEOREM 4.14. Let f be a uniformly continuous function from the unit sphere of c0 to R and let δ > 0. Then there is an infinite-dimensional subspace W of c0 such that |f (v) − f (w)| < δ for every pair of vectors v, w in the unit sphere of W . P ROOF. Without loss of generality, f takes values between 0 and 1. Choose ε and k such that the set Z is an η-net of the unit sphere of c0 , where η is small enough that |f (x) − f (y)| < δ/5 whenever x − y 2η, and partition [0, 1] into sets I1 , . . . , Ir of diameter at most δ/5. For 1 j r let Cj be the set of x ∈ Z such that f (x) ∈ Ij . By Theorem 4.13 there is an infinite-dimensional block subspace (in the combinatorial sense) V of Z contained in C j for some j . It is easy to check that the distance between two neighbours in Z is at most η, so for every v ∈ V there exists x ∈ Z such that v − x 2η and f (x) ∈ Ij . It follows that f (v) lies strictly within δ/5 of a point in Ij . Now let W be the block subspace in c0 generated by V . It is easy to check that V is an η-net of the unit sphere of W . Therefore, for every w in the sphere of W , f (w) lies strictly within 2δ/5 of a point in Ij . It follows easily that W has the required property.

5. Banach-space dichotomies While it is now known that a general Banach space does not need to contain a subspace with nice symmetry properties (see [15]), there is still scope for dichotomy theorems, that is, results that assert the existence of a subspace with one of two extreme properties. Rosenthal’s 1 -theorem is an example of such a theorem. In this section, we shall consider some others. Typically, they involve saying that if a Banach space fails to contain a subspace with some good symmetry property, then it must have a subspace which lacks symmetry in a very extreme way. It is natural to try to use Ramsey theory to obtain such results, colouring appropriate objects according to whether they are good (helping to produce symmetry) or bad (showing no symmetry at all). However, it is not quite so clear what the objects should be, or how to formulate and prove an appropriate result, especially given the counterexamples mentioned at the beginning of the previous section. In order to help us arrive at a good formulation, it will be useful to state a theorem from [11], which was the initial motivation for a more abstract Banach-space Ramsey result. For the definitions in the statement of the theorem, see [15]. All spaces and subspaces in this section will be infinite-dimensional, unless we explicitly say otherwise. T HEOREM 5.1. Every Banach space X has a subspace Y which either has an unconditional basis or is hereditarily indecomposable. Now by standard arguments, we may as well let X be a space of the form (c00 , · ), where c00 is the space of all finitely supported sequences and · is a norm on c00 making the natural basis into a normalized monotone basis (at least after the space is completed, but this technicality is unimportant here). The following lemma, an easy exercise, is stated without proof.

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L EMMA 5.2. A space X = (c00 , · ) contains no unconditional basic sequence if and only if, for every C 1, every block subspace of X contains a sequence y1 < · · · < yn (for some n) of vectors of norm at most 1 such that n n n (−1) yi > C yi . i=1

i=1

Let us call such a sequence y1 < · · · < yn C-conditional. If X contains no unconditional basic sequence, then every block subspace of X contains a finite C-conditional sequence for every C. Encouraged by the proof of Theorem 3.1, let us define a set of finite block sequences in c0 to be large for a block subspace Y of c00 if every block subspace of Y contains one of them. In this terminology, for every C 1 the set of C-conditional sequences in X is large for c00 = X. Can we now give a Banach-spaces version of Theorem 3.1? The answer is no, until we can give a suitable definition of “very large”. The most obvious idea is to say that a collection σ of finite block sequences is very large for X if every block basis of X has an initial segment in σ . However, the distortability results mentioned earlier (and proved in [19]) show that this is too strong a definition, even for sequences of length one. Indeed, let X be a space such that the unit sphere contains asymptotic sets A, B with x − y α > 0 for every x ∈ A and y ∈ B. Let σ be the set of sequences (x) such that x ∈ A. Then σ is large, but every block subspace of X has a point not in A (or even close to A) which can obviously be extended to an infinite block basis. Let us return to our motivating result, Theorem 5.1. We are hoping to use a Ramsey result for finite sequences of blocks, applied to conditional sequences, to obtain a hereditarily indecomposable subspace. The next lemma, also stated without proof as it is an easy exercise and is essentially covered in [15], gives us a clue about what to do. L EMMA 5.3. Let X = (c00 , ·) and let Y be a block subspace of X. Then Y is hereditarily indecomposable if and only if, for every C 1 and any pair Z, W of block subspaces of Y , there is a sequence z1 < w1 < z2 < w2 < · · · < zn < wn (for some n) such that n n (zi + wi ) > C (zi − wi ). i=1

i=1

Notice that the property we require of Y certainly implies that the set of C-conditional sequences is large for every C. (Just let Z = W .) It is not hard to show that it is strictly stronger. Let Σ = Σ(X) be the set of all finite block sequences in X consisting of vectors of norm at most 1. It would now be natural to define a set σ ⊂ Σ to be very large for Y if, for any pair of block subspaces Z, W of Y there was a sequence (z1 , w1 , z2 , w2 , . . . , zn , wn ) in σ such that zi ∈ Z and wi ∈ W for every i. Instead, we shall give what appears to be the strongest definition for which a positive result can be obtained. Given a set σ ⊂ Σ, one can define a two-player game as follows. One player, S (for subspace), chooses a block subspace X1 of X. The other player, V (for vector), chooses a vector x1 ∈ X1 of norm at most 1. Then S chooses a subspace X2 and V chooses x2 ∈ X2


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of norm at most 1. Play continues like this. V wins the game if, at some point, the sequence (x1 , . . . , xn ) resulting from the play belongs to σ . It is an easy but instructive exercise to give an example of a set σ of sequences such that V wins the game, but such that for any n, S can force the game to last for at least n moves. D EFINITION 5.4. A set σ ⊂ Σ of finite block sequences is strategically large, or s-large for Y if V wins the above game when S is restricted to subspaces of Y . As in Section 4 any Ramsey result must be an approximate one, so we now introduce appropriate notation for the approximations. If Δ = (δ1 , δ2 , . . .) is a sequence of non-negative real numbers and σ ⊂ Σ, then σΔ will stand for the set of all sequences (x1 , . . . , xn ) ∈ Σ such that there exists a sequence (y1 , . . . , yn ) ∈ σ with xi − yi δi for every i n. We shall also write σ−Δ for the complement of (σ c )Δ . An equivalent way of defining σ−Δ is as the set of all sequences (x1 , . . . , xn ) ∈ Σ such that (y1 , . . . , yn ) ∈ σ whenever xi − yi δi for every i n. It follows, of course, that (σ−Δ )Δ ⊂ σ . It is also easy to see that (σΔ )−Δ ⊃ σ . We are now ready for our Nash-Williams-type theorem. In order to deal with the approximations, it is necessary for the statement to be a little untidy. However, as we shall show in Corollary 5.6, a tidier version follows easily. T HEOREM 5.5. Let X = (c00 , · ) be a sequence space with basis constant C and let ρ ⊂ Σ(X) be a set of finite block sequences in the unit ball of X. Let Θ = (θ1 , θ2 , . . .) with every θ i > 0 and let Δ = (δ1 , δ2 , . . .) be another sequence of positive real numbers such that C ∞ i=N δi θN for every N . If σ−Θ is large for X, then X has a block subspace Y such that σ2Δ is strategically large for Y . P ROOF. We shall need an important trick (or at least it seems to be important) which was not necessary for the proof of Theorem 3.1. This is to assume that σ is minimal in the following sense. If (x1 , . . . , xn ) is in σ and (y1 , . . . , ym ) is a sequence in Σ such that yi ∈ x1 , . . . , xn for every i, then (y1 , . . . , ym ) is not in σ unless n = m, which implies that yi is a multiple of xi for every i. It is easy to see that if σ is large for X, then σ contains a subset which is minimal in this sense and still large for X, so we lose no generality by making this assumption. As far as possible, the rest of the proof will be organized along similar lines to the proof of Theorem 3.1. Accordingly, given a sequence (x1 , . . . , xn ) ∈ Σ, let us define σ (x1 , . . . , xn ) to be the set of all (possibly null) sequences (y1 , . . . , ym ) such that (x1 , . . . , xn , y1 , . . . , ym ) belongs to σ . Given n ∈ N, let Δn be the sequence (δ1 , . . . , δn , 0, 0, . . .) and let Γn be the sequence (δ1 , . . . , δn , 2δn+1 , 2δn+2 , . . .). We now attempt to construct sequences of vectors x1 < x2 < · · · and block subspaces X = X0 ⊃ X1 ⊃ X2 ⊃ · · · with the following properties, for every n. (i) xn ∈ Xn−1 . (ii) σΔn (x1 , . . . , xn ) is large for Xn . (iii) σΓn (x1 , . . . , xn ) is not strategically large for any subspace of Xn . If the theorem is false, then the induction starts with the null sequence and X0 = X, since σ is large for X and not strategically large for any subspace of X. (We omit the

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word “block” from now on.) As in the proof of Theorem 3.1, let us now suppose that we have constructed x1 , . . . , xn and X1 ⊃ X2 ⊃ · · · but are unable to find xn+1 and Xn+1 to continue the induction. This tells us that, for every y ∈ Xn such that y 1 and xn < y, and for any subspace Y ⊂ Xn , there is a subspace Z of Y such that the following statement holds: either no sequence in σΔn+1 (x1 , . . . , xn , y) is contained in Z, or σΓn+1 (x1 , . . . , xn , y) is strategically large for Z. Let us write P (y, Z) for this statement. Notice that P (y, Z) implies P (y, W ) for every subspace W of Z. Using this fact, we shall construct a sequence z1 < z2 < · · · of unit vectors in Xn and subspaces Xn = Z0 ⊃ Z1 ⊃ Z2 ⊃ · · · with the following properties, for every k. (a) zk ∈ Zk−1 . (b) For every y ∈ z1 , . . . , zk , either no sequence in σΔn (x1 , . . . , xn , y) is contained in Zk or σΓn (x1 , . . . , xn , y) is strategically large for Zk . Note that (b) is not quite the statement ∀y P (y, Zk ) because the approximations are different. The induction starts with the null sequence and Z0 . To see that it can always be continued, let us suppose that we have constructed z1 , . . . , zk and Z1 ⊃ · · · ⊃ Zk . Let zk+1 be any unit vector in Zk such that zk < zk+1 . We need yet another induction to define Zk+1 , but fortunately it is not too complicated. Let w1 , . . . , wN be a δn+1 -net of the unit ball of the subspace z1 , . . . , zk . Now choose subspaces Zk ⊃ W1 ⊃ W2 ⊃ · · · ⊃ WN such that, for each i, the statement P (wi , Wi ) holds. This we can do because for every y ∈ Xn such that y 1 and xn < y and for every subspace Y ⊂ Xn there is a subspace Z ⊂ Y such that P (y, Z) is true, as has already been mentioned. Let Zk+1 = Wn . By the hereditary property of P (y, Z), we find that P (wi , Zk+1 ) for every i n. If we now let y ∈ z1 , . . . , zk be any vector of norm at most one, then statement (b) (with k replaced by k + 1) follows easily on approximating y by some wi with y − wi δn+1 . We are now ready to diagonalize. Let Z be the subspace generated by the block basis z1 , z2 , . . . and let Y be any subspace of Z. Because σΔn (x1 , . . . , xn ) is large but not strategically large for Y (by (i) and (ii)), we know that there must be some y ∈ Y such that Y and hence Z contains a sequence belonging to σΔn (x1 , . . . , xn , y). Let k be minimal such that y ∈ z1 , . . . , zk . It follows that Zk contains a sequence in σΔn (x1 , . . . , xn , y). By (b), we deduce that σΓn (x1 , . . . , xn , y) is strategically large for Zk , and hence for Z. To summarize, we have shown that, for every subspace Y ⊂ Z, there exists y ∈ Y such that σΓn (x1 , . . . , xn , y) is strategically large for Z. But this implies that σΓn (x1 , . . . , xn ) is strategically large for Z, which contradicts (iii). This shows that our original (outer) induction continues for ever. So now let us diagonalize again, letting V be the subspace generated by x1 , x2 , . . . . We claim that no sequence in V belongs to σ−Θ . To see this, let (v1 , . . . , vr ) ∈ Σ consist of vectors in V and choose n such that every vi belongs to the subspace x1 , . . . , xn and at least one xi is not a multiple of a vi . Because σΔn (x1 , . . . , xn ) is large for Xn , we can find a sequence (y1 , . . . , yN ) ∈ σ such that yi − xi δi whenever i n. Let φ be the linear map from x1 , . . . , xn to y1 , . . . , yn that takes xi to yi for every i. Since each vi has norm at most 1, no coefficient can exceed C, so by the triangle inequality and our choice of Δ, we find that φ(vi ) − vi θ i for every i. But because (y1 , . . . , yN ) ∈ σ and σ satisfies the minimality


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condition stated at the beginning of the proof, we know that (φ(v1 ), . . . , φ(vk )) ∈ / σ , which / σ−Θ . implies that (v1 , . . . , vk ) ∈ We have therefore contradicted the assumption that σ−Θ was large, which proves the theorem. C OROLLARY 5.6. Let X = (c00 , · ) be a sequence space, let σ ⊂ Σ(X) be a collection of finite block sequences, let Δ be a sequence of positive real numbers and suppose that σ is large for X. Then there exists a subspace Y of X such that σΔ is strategically large for Y . P ROOF. Let Θ = Δ/2 and let ρ = σΘ . Then ρ−Θ is large for X. Theorem 5.5 implies that, for a sufficiently small sequence Γ , there is a subspace Y such that ρΓ is strategically large for Y . Since this remains true when Γ increases, we can of course take Γ to be Θ. However, ρΘ = σΔ . A further reformulation is possible in terms of colourings: if Σ(X) is coloured with two colours, σ and τ , then there is a subspace Y of X such that either every finite block sequence in the unit ball of Y belongs to τ , or σΔ is strategically large for Y . Put this way, the Ramsey-theoretic nature of the result is emphasized, but also the difference between this result and conventional results in Ramsey theory. The two most important differences are that it is weaker (in that being strategically large is weaker than being very large) and that there is an asymmetry between σ and τ . About the second point we shall have more to say later. Let us now return to our motivating theorem, and see how it follows easily from Corollary 5.6. P ROOF OF T HEOREM 5.1. Let X = (c00 , · ) be a sequence space containing no unconditional basic sequence, and for N ∈ N let σN be the set of all block sequences (x1 , . . . , xn ) of vectors in X of norm at most one such that n n n (−1) xi > N xi i=1

i=1

and such that at least one xi has norm 1. By Lemma 5.2, we know that, for every N , σN is large for X. It is an easy exercise to find, for every N , a positive sequence ΔN such that n n (−1)n yi > (N/2) yi i=1

i=1

for every (y1 , . . . , yn ) ∈ (σN )ΔN . Let us now apply Corollary 5.6 repeatedly, to obtain a nested sequence X1 ⊃ X2 ⊃ · · · of block subspaces of X such that, for every N , (σN )ΔN is strategically large for XN . Let Y be a diagonal subspace, that is, a subspace generated by a block basis y1 , y2 , . . . with yn ∈ Xn for every n. It is easy to see that (σN )ΔN is strategically large for Y , whatever the value of N .

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Now let Z and W be arbitrary block subspaces of Y . If S plays the strategy Z, W, Z, W, . . . , then V can win, whatever the value of N . This shows that the condition of Lemma 5.3 is satisfied, and hence that Y is hereditarily indecomposable. Theorem 5.5 has an extension from sets of finite sequences to analytic sets of infinite sequences. To be precise, let Σ(X) now stand for the set of all infinite block sequences of X, with the blocks of norm at most 1, and give Σ(X) the (metrizable) topology of pointwise convergence. (This is a slight oversimplification, as we need Σ(X) to be complete. One can deal with this technicality by considering sequences (xn , λn )∞ n=1 where the xn form a normalized block basis and the λn belong to the interval [0, 1].) It is obvious how to extend to sets of infinite sequences of the definitions of large, strategically large and so on. T HEOREM 5.7. Let X = (c00 , · ) be a sequence space with basis constant C and let σ be an analytic subset of Σ(X). Let Δ = (δ1 , δ2 , . . .) be a sequence of positive reals. If σ is large for X, then X has a block subspace Y such that σΔ is strategically large for Y . A proof of this theorem can be found in [11]. The slightly weaker statement where Σ(X) consists of normalized block sequences is also proved in [2]. Theorem 5.7 ought to be a very useful tool for proving further Banach-space dichotomies. However, the results so far have been a little disappointing. In particular, no natural theorem is known that uses anything like the full strength of the theorem. However, there is one consequence (which can be deduced from the much simpler special case where σ is assumed to be closed) of some interest, as it provides a dichotomy for Banach spaces with unconditional bases, and thus converts Theorem 5.1 from a dichotomy to a trichotomy. To appreciate the result it is necessary to know of the existence of a space constructed in [12]. This space has an unconditional basis, but is otherwise as free of structure as possible. What does this mean? Well, it is an easy exercise to show that if X is a Banach space ∞ with an unconditional basis (xn )n=1 , then every diagonal map, that is, linear map of the form Λ : an xn → λn an xn , is continuous, provided that it satisfies the trivial necessary condition that the λn are bounded. (Such maps are also called multipliers.) The space XU in [12] has the property that the diagonal maps are essentially the only continuous maps defined on it. To be precise, every map in L(XU ) is the sum of a diagonal map and a strictly singular map. (It would be interesting to know what can be said about maps from subspaces of XU to XU . See Problem 5.13 below for a concrete question.) In particular, the space XU is not isomorphic to any proper subspace of itself. On the other hand, most spaces with unconditional bases have far more structure than this, which suggests that some sort of dichotomy might be true. Let us define a Banach space X to be quasi-minimal if any two subspaces Y , Z of X have further subspaces U , V that are isomorphic. This is true in particular if X is minimal, which means that every subspace Y of X has a further subspace isomorphic to X. However, there are non-minimal examples of quasi-minimal spaces, of which Tsirelson’s space is a famous one. (See [7] for a proof of this.) Thus, in a certain sense, quasi-minimal spaces are rich in isomorphisms between subspaces. The following result concerning quasi-minimal spaces is a consequence of Theorem 5.7 for closed sets σ .


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T HEOREM 5.8. Every Banach space X with an unconditional basis has a subspace Y such that either Y is quasi-minimal or no subspace of Y is isomorphic to any proper subspace of itself. As it stands, the above theorem is not a proper dichotomy, since it is likely that there are quasi-minimal Banach spaces Y such that no subspaces are isomorphic to further proper subspaces. (There is even a suggestion for how to construct such a space, but nobody has checked that it really works.) However, it is a consequence of the following stronger result which is a genuine dichotomy. T HEOREM 5.9. Every Banach space X with an unconditional basis has a block subspace Y such that either Y is quasi-minimal or no two subspaces of Y generated by disjointly supported block bases are isomorphic to each other. The deduction of Theorem 5.8 from Theorem 5.9 uses standard facts about block bases and a criterion of Casazza. Details can be found in [11]. These results suggest further lines of investigation, but it seems that dichotomies as tidy as those of Theorem 5.1 and Theorem 5.9 are hard to come by. For example, if it is true that a quasi-minimal space X may have the property that no subspace of X is isomorphic to further proper subspace, it is tempting to think of such a space X as ‘bad’ and to try to prove a dichotomy along the following lines: every quasi-minimal space has a subspace which is either bad or, in some appropriate sense to be determined, very good. But what should a good space be like? If the previous results are anything to go by, one might expect such a space to be very rich in isomorphisms from subspaces to proper subspaces. Let P be the set of Banach spaces that are isomorphic to some proper subspace. If X has no bad subspace, then it follows immediately that every subspace Y of X has a subspace in P . That is, X is P -saturated. What stronger property could we ask for from a very good subspace? One possibility is that every subspace should belong to P . However, it seems quite likely that this definition does not lead to a dichotomy theorem. Indeed, here is a conjecture that would, if true, place considerable restrictions on any positive statements one might hope to prove. C ONJECTURE 5.10. There exists a quasi-minimal Banach space X with an unconditional basis such that every block subspace Y has a subspace Z generated by a block basis z1 , z2 , . . . such that the shift operator with respect to this basis is an isomorphism, and also has a subspace W which is isomorphic to no proper subspace of itself. Even a weaker statement, where it is assumed only that Z is isomorphic to some proper subspace, would contradict many natural conjectures about P -saturated spaces. Proving the above conjecture would require a breakthrough in the technology for producing counterexamples. The methods so far have tended to be all-or-nothing, in the sense that if one uses them to define a space with certain bad qualities, then those qualities will saturate the space, or at least the part of the space where the methods are used. This is a somewhat vague statement, but the following unsolved problem (which I heard from Haskell Rosenthal) may clarify it.

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P ROBLEM 5.11. Let X be a Banach space such that every subspace of X has a further subspace with an unconditional basis. (Such a space is called unconditionally saturated.) Does it follow that X is decomposable? Equivalently (given Theorem 5.1), does there exist an indecomposable Banach space with no hereditarily indecomposable subspace? To construct such a space, one would have to remove a great deal of structure from the space while somehow leaving subspaces relatively untouched. A positive solution to the following problem would improve the analogy between Theorem 5.1 and Theorem 5.9. P ROBLEM 5.12. Let X be a Banach space with an unconditional basis such that no two subspaces Y and Z generated by disjointly supported block bases are isomorphic. Does it follow that every continuous linear map from a subspace Y of X into X is a strictly singular perturbation of the restriction of a diagonal map on X? Even the following special case of Problem 5.12 is open. P ROBLEM 5.13. Let XU be the Banach space constructed in [12] with an unconditional basis. Is every continuous linear map from a subspace Y of XU into XU a strictly singular perturbation of the restriction of a diagonal map on XU ? A FINAL REMARK . Very recently Problem 5.11 was solved by Argyros and Manoussakis. As one might expect, the answer is no. It will be very interesting to see whether their construction leads to a new generation of Banach spaces with notable “non-hereditary” properties.

References [1] S.A. Argyros, G. Godefroy and H.P. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1007–1069 (this Handbook). [2] J. Bagaria and J. Lopez Abad, Weakly Ramsey sets in Banach spaces, Adv. Math. 160 (2001), 133–174. [3] E. Behrends, New proofs of Rosenthal’s 1 -theorem and the Josefson–Nissenzweig theorem, Bull. Polish Acad. Sci. Math. 43 (1995), 283–295. [4] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Amer. Math. Soc. Colloq. Publ. 48, Amer. Math. Soc., Providence, RI (2000). [5] B. Bollobás, Combinatorics. Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability, Cambridge Univ. Press, Cambridge (1986). [6] A. Brunel and L. Sucheston, On B-convex Banach spaces, Math. Systems Theory 7 (1974), 294–299. [7] P. Casazza and T.J. Shura, Tsirelson’s Space, Lecture Notes in Math. 1363, Springer, New York (1989). [8] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. [9] J. Farahat, Espaces de Banach contenant 1 , d’apreès H.P. Rosenthal, Espaces Lp , Applications Radonifiantes et Géométrie des Espaces de Banach, Exp. No. 26, Centre de Math., École Polytech., Paris (1974). [10] W.T. Gowers, Lipschitz functions on classical spaces, European J. Combin. 13 (1992), 141–151. [11] W.T. Gowers, An infinite Ramsey theorem and some Banach-space dichotomies, Ann. of Math, to appear.


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[12] W.T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997), 543– 568. [13] R.L. Graham, K. Leeb and B.L. Rothschild, Ramsey’s theorem for a class of categories, Adv. in Math. 8 (1972), 417–433. [14] N. Hindman, Finite sums from sequences within cells of a partition of N, J. Combin. Theory Ser. A 17 (1974), 1–11. [15] B. Maurey, Banach spaces with few operators, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1247–1297 (this Handbook). [16] B. Maurey, Type, cotype and K-convexity, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1299–1332 (this Handbook). [17] C.St.J.A. Nash-Williams, On well quasi-ordering transfinite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33–39. [18] E. Odell and Th. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281. [19] E. Odell and Th. Schlumprecht, Distortion and asymptotic structure, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1333–1360 (this Handbook). [20] H.P. Rosenthal, A characterization of Banach spaces containing 1 , Proc. Nat. Acad. Sci. 71 (1974), 241– 243. [21] H.P. Rosenthal, A characterization of Banach spaces containing c0 , J. Amer. Math. Soc. 7 (1994), 707–748.


CHAPTER 25

Quasi-Banach Spaces Nigel Kalton∗ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 3. Linear subspaces and basic sequences . . . . . . . . . . . 4. The three-space problem and minimal extensions . . . . . 5. The Krein–Milman theorem . . . . . . . . . . . . . . . . 6. Operators and the structure of Lp -spaces when 0 < p < 1 7. Lattices and natural spaces . . . . . . . . . . . . . . . . . 8. Analytic functions and applications . . . . . . . . . . . . 9. Tensor products and algebras . . . . . . . . . . . . . . . . 10. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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∗ Supported in part by NSF DMS-9870027.


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1. Introduction The theory of the geometry of Banach spaces has evolved very rapidly over the past fifty years. By contrast the study of quasi-Banach spaces has lagged far behind, even though the first research papers in the subject appeared in the early 1940’s ([18,6]). There are very sound reasons to want to develop understanding of these spaces, but the absence of one of the fundamental tools of functional analysis, the Hahn–Banach theorem, has proved a very significant stumbling block. However, there has been some progress in the non-convex theory and arguably it has contributed to our appreciation of Banach space theory. A systematic study of quasi-Banach spaces only really started in the late 1950’s and early 1960’s with the work of Klee, Peck, Rolewicz, Waelbroeck and Zelazko. The efforts of these researchers tended to go in rather separate directions. The subject was given great impetus by the paper of Duren, Romberg and Shields in 1969 which demonstrated both the possibilities for using quasi-Banach spaces in classical function theory and also highlighted some key problems related to the Hahn–Banach theorem. This opened up many new directions of research. The 1970’s and 1980’s saw a significant increase in activity with a number of authors contributing to the development of a coherent theory. An important breakthrough was the work of Roberts in 1976 [73] and [75] who showed that the Krein–Milman Theorem fails in general quasi-Banach spaces by developing powerful new techniques. Quasi-Banach spaces (Hp -spaces when p < 1) were also used significantly in Alexandrov’s solution of the inner function problem in 1982 [4]. During this period three books on the subject appeared by Turpin [86], Rolewicz [77] (actually an expanded version of a book first published in 1972) and the author, Peck and Roberts [56]. In the 1990’s it seems to the author that while more and more analysts find that quasi-Banach spaces have uses in their research, paradoxically the interest in developing a general theory has subsided somewhat. In this short article we will only give a glimpse of the theory, and we have tried to make the subject accessible for an audience which is primarily interested in and familiar with Banach space theory. There is no attempt to be encyclopaedic. Thus we will look carefully at problems related to the existence of closed subspaces which are very much in the spirit of the recent work of Gowers and Maurey [29] in Banach space theory. We will also consider the problem of characterizing the complemented subspaces of Lp (0, 1) when 0 < p < 1. In the last few sections we consider how the theory of lattices, analytic functions and tensor products alters in the non-convex setting. Although this article is devoted to quasi-Banach spaces, much of the early theory was developed in the context of more general topological vector spaces or sometimes F -spaces (complete metric linear spaces). In some cases (such as the study of the Krein–Milman theorem for compact convex sets, see Section 5) restricting to quasi-Banach spaces loses nothing in terms of generality, and in most cases there is relatively little loss.

2. Preliminaries In this section we will review a few elementary concepts and definitions. Further details can be found in one of the books [78,56].

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Let us recall first that a quasi-norm · on vector space X over the field K = R or C is a map X → [0, ∞) with the properties: • x = 0 if and only if x = 0. • αx = |α|x if α ∈ K, x ∈ X. • There is a constant C 1 so that if x1 , x2 ∈ X we have x1 + x2 C(x1 + x2 ). The constant C is often referred to as the modulus of concavity of the quasi-norm. A very basic and important result is the Aoki–Rolewicz theorem ([6,77]). This result can be interpreted as saying that if 0 < p 1 is given by C = 21/p−1 then there is a constant B so that for any x1 , . . . , xn ∈ X we have n 1/p n p xk B xk . k=1

(2.1)

k=1

It is then possible to replace · by an equivalent p-subadditive quasi-norm ||| · ||| so that

1/p |||x1 + x2 ||| |||x1 |||p + |||x2 |||p . X is said to p-normable if (2.1) holds. We will say that X is p-normed if the quasi-norm on X is p-subadditive. In general it is convenient to assume unless otherwise mention that a quasi-Banach space is p-normed for some p > 0. The quasi-norm · induces a metric topology on X: in fact a metric can be defined by d(x, y) = |||x − y|||p , when the quasi-norm is p-subadditive. X is called a quasi-Banach space if X is complete for this metric. Note that if we assume X is p-normed for some p > 0 then the quasi-norm is a continuous function for the metric topology. It is important to emphasize that the standard basic results of Banach space theory such as the Uniform Boundedness Principle, Open Mapping Theorem and Closed Graph Theorem which depend only on completeness apply to this type of space; however applications of convexity such as the Hahn–Banach theorem are not applicable. If X and Y are quasi-Banach spaces then L(X, Y ) denotes the space of bounded linear operators T : X → Y under the quasi-norm T = sup{T x: x 1}. A special and important case is the dual space X∗ = L(X, K) which is always a Banach space. The best known examples of quasi-Banach spaces are the spaces p and Lp (0, 1), when 0 < p < 1. These spaces are p-normable. It is readily seen that ∗p = ∞ but Lp (0, 1)∗ = {0}. Notice that p has a separating dual while Lp has a trivial dual. This latter result is due to Day [18] in what is arguably the first paper on quasi-Banach spaces. Another important example is the Hardy space Hp , i.e., the closed subspace of dθ Lp (T, 2π ) spanned by the functions {einθ : n 0}. Although Lp has trivial dual, Hp has a separating dual ([22]). If X has a separating dual then we can define an associated norm on X by the formula xc = sup x ∗ (x): x ∗ 1 . It can be easily shown that · c is the largest norm on X dominated by the original quasinorm. The completion of X with this norm Xc is called the Banach envelope of X. In a natural sense Xc and X have the same dual space.

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Many of the standard notions in Banach space theory can be carried through to quasiBanach spaces. For example, the notions of (Rademacher) type and cotype (see [32]) can be defined in exactly the same way. T HEOREM 2.1. Let X be a quasi-Banach space of type p where 0 1 then X is normable (i.e., a Banach space). We remark that there are non-locally convex spaces of type one which are not Banach spaces [40]. The Krivine–Maurey–Pisier theorems on finite representability of np ’s have analogues in this setting (we refer to [36,7] and [9]). In particular Dvoretzky’s theorem always has an appropriate generalization (see [36] and [19]). Let us also mention an important substitute for convexity in complex quasi-Banach spaces. We will say that a quasi-norm (which we assume r-subadditive for some r < 1) is plurisubharmonic if for any x, y ∈ X then 1 x 2π

2π

x + eiθ y dθ.

0

For a discussion of this condition see [17]. It is important to note that the spaces Lp and their subspaces have plurisubharmonic norms. We will discuss this condition further in Section 8.

3. Linear subspaces and basic sequences In this section we will discuss some very fundamental structure problems for quasi-Banach spaces concerning linear subspaces of quasi-Banach spaces. Many of the results and problems in this section are interesting in the category of F -spaces but we will restrict ourselves to quasi-Banach spaces for clarity. A fundamental and still unresolved problem is the following: P ROBLEM 3.1 (The atomic space problem). Does every quasi-Banach space have a proper closed infinite-dimensional subspace? A quasi-Banach space X is called atomic if it has no proper closed infinite-dimensional subspaces. Very little is known about this problem. For a recent contribution in the context of F -spaces see [70]. Although Problem 3.1 remains elusive much progress has been made in understanding the structure of subspaces of quasi-Banach spaces. Before reviewing this progress we discuss the historical context for some of these ideas. Since the failure of the Hahn–Banach theorem is a characteristic of non-locally convex spaces, it is natural that much of the early research in the area was devoted to trying to understand this phenomenon. A Banach space has a very rich dual space and this also

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means that it has a very rich class of closed subspaces (each non-trivial continuous linear functional gives rise to a closed subspace of codimension one). Therefore an associated problem for quasi-Banach spaces is to find (infinite-dimensional) proper closed subspaces. It is clear from Day’s result that it is possible to find a closed subspace E of Lp when 0 < p < 1 and a continuous linear functional e∗ ∈ E ∗ which cannot be extended; indeed E can be taken to be one-dimensional. This construction will work in any space X which fails to have a separating dual. In the space p for 0 < p < 1 more work is required but Peck [68] gave a similar example of the failure of the Hahn–Banach theorem. Later, Duren, Romberg and Shields [23] found a representation of the dual of Hp for 0 < p < 1 and used it to show that the Hahn–Banach theorem also fails in these spaces. Their work led them to formulate a conjecture. They defined a quasi-Banach space X (or more generally an F-space) to have the Hahn–Banach Extension property (HBEP) if whenever e∗ is a continuous linear functional on a closed subspace E of X then e∗ has an extension x ∗ ∈ X∗ . They also defined the notion of a proper closed weakly dense (PCWD) subspace as a proper closed subspace E so that the quotient X/E has trivial dual. They then asked whether a quasiBanach space with (HBEP) is necessarily locally convex and whether a any non-locally convex quasi-Banach space has a PCWD-subspace. It is easy to see that if X has HBEP then it must have a separating dual and every quotient must have HBEP; hence if X has a PCWD-subspace it cannot have HBEP. These two questions had a considerable impact on the theory because they focused attention on the problem of subspaces. In effect HBEP is equivalent to the statement that the weak and norm topologies have the same closed subspaces. After important contributions in [79] and [71] the first of these problems was resolved in [33]: T HEOREM 3.2. A quasi-Banach space X has HBEP if and only if X is locally convex (i.e., a Banach space). The method of proof of Theorem 3.2 relies on the construction of basic sequences. Of course, there is no guarantee that quasi-Banach spaces will contain basic sequences (unlike Banach spaces). In fact an atomic space (if it exists) would be an immediate counterexample; but we will later show how to construct a quasi-Banach space without a basic sequence. However it is natural to start by imitating as far as the possible the classical Bessaga–Pełczyński basic sequence selection techniques. It soon becomes clear that the role of the weak (or weak∗ ) topology can be replaced by any weaker Hausdorff vector topology τ on X so that X has an equivalent τ -lower-semi-continuous quasi-norm. We will call such a topology polar. P ROPOSITION 3.3 (Basic sequence selection criterion). Let X be a quasi-Banach space and suppose (xn ) is a sequence so that lim xn = 0 for some polar vector topology τ but inf xn > 0. Then (xn ) has a subsequence which is basic. We recall that a sequence (xn ) in a quasi-Banach space X is called a Markushevich basis if [xn ] = X and there is a bi-orthogonal sequence (xn∗ ) so that (xn∗ ) separate the points of X.

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We will say that (xn ) is a Markushevich basic sequence if it is a Markushevich basis for its closed linear span. An immediate corollary of this proposition is: P ROPOSITION 3.4 (Markushevich basic sequence selection criterion). Let X be a quasiBanach space and suppose (xn ) is a sequence so that lim xn = 0 for some weaker Hausdorff vector topology τ but inf xn > 0. Then (xn ) has a subsequence (yn ) which is a Markushevich basic sequence and whose bi-orthogonal sequence (yn∗ ) in [yn ]∗ satisfies sup yn∗ < ∞. An alternative approach to this result was given by Drewnowski [21]. We are now a position to indicate a proof of Theorem 3.2: P ROOF OF T HEOREM 3.2. Since X has (HBEP) it is clear that X∗ separates points and therefore the Banach envelope norm , c induces a weaker Hausdorff vector topology on X. We argue that it cannot be a strictly weaker topology than the quasi-norm topology. Indeed, if it is strictly weaker, then using Proposition 3.4 one can find a sequence (xn ) such that xn c < 4−n but xn = 1 for all n and (xn ) is a Markushevich basis for its closed linear span E with bi-orthogonal functionals (xn∗ ) satisfying sup xn∗ < ∞. Then we can ∞ −n ∗ ∗ ∗ define e ∈ E by e = n=1 2 xn∗ . Suppose e∗ can be extended to a bounded linear functional f ∗ ∈ X∗ . Then f ∗ (4n xn ) = 2n but 4n xn c 1 for all n. This contradiction shows that X coincides with its Banach envelope. Once this is established it is not too difficult to prove a companion result for PCWD subspaces [37]: T HEOREM 3.5. Let X be a quasi-Banach space with a separating dual. If X has no PCWD subspace then X is locally convex. Notice however that the hypothesis of a separating dual is required here. We will see later that this hypothesis is necessary: there exist non-locally convex quasi-Banach spaces which do not have any quotient with trivial dual. Let us now return to the discussion of basic sequences. Theorems 3.3 and 3.4 yield some characterizations of spaces with basic sequences: T HEOREM 3.6. Let X be a separable infinite-dimensional quasi-Banach space. Then the following conditions on X are equivalent: (i) X contains a basic sequence. (ii) Thereis descending sequence (Ln ) of infinite-dimensional closed subspaces of X with ∞ n=1 Ln = {0}. (iii) There is a family L of infinite-dimensional closed subspaces such that {L: L ∈ F } is infinite-dimensional for any finite subset F of L but {L: L ∈ L} = {0}. (iv) There is a strictly weaker Hausdorff vector topology on X. These implications are relatively easy. The equivalence of (ii) and (iii) simply follows from the Lindelof property for separable metric spaces. That (i) implies (ii) is trivial. For

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(ii) implies (iv) simply consider the vector topology on X induced by the semi-quasinorms x → d(x, Ln ) for n = 1, 2, . . . . Thus the only implication with any difficulty here is that (iv) implies (i). Let τ be a Hausdorff vector topology on X, which is strictly weaker than the original quasi-norm topology qn. Let τ ∗ be a maximal Hausdorff vector topology on X strictly weaker than qn (such a topology must exist). Let τ ∗∗ be the quasi-norm topology on X defined by taking the τ -closure of the original unit ball as a new unit ball. Then the maximality of τ means that either τ ∗∗ = τ ∗ or τ ∗∗ = qn. But the former case means that the identity i : (X, τ ∗ ) → (X, qn) is almost continuous and a form of the Closed Graph Theorem comes into play: one deduces that τ ∗ = qn a contradiction. It follows that τ ∗∗ = qn and so τ ∗ is a polar topology. One can use the Lindelof property to construct a weaker metrizable Hausdorff vector topology ρ which is still polar. Then an application of Theorem 3.3 completes the proof. The last condition leads to the definition of a minimal space as any quasi-Banach space which does not have any weaker Hausdorff vector topology. A separable quasi-Banach space is minimal if and only if it contains no basic sequence (separability is redundant here, but that requires a little more explanation). Obviously an atomic space must be minimal but we shall see that the converse is false. Let us now illustrate the problem by considering an arbitrary separable Banach space X. Let L be a maximal collection of infinite-dimensional closed subspaces of X with the property that any finite intersection is infinite-dimensional. Let E = {L: L ∈ L}. There are three possibilities: • E = {0}. Then by Theorem 3.6 X is non-minimal. • E is infinite-dimensional. Then E is atomic. • dim E < ∞. In this case X/E contains a basic sequence, but X could still be minimal. The third possibility suggests a way of constructing a minimal space with no atomic subspace. It is even possible to hope for an example where dim E = 1 and X/E is a Banach space. Obviously one needs that X is not a Banach space: this brings into focus a distinct problem which also received a considerable amount of attention in the 1970’s: the three space problem for Banach spaces, which is discussed in the next section. It will turn out that there is a counterexample of this nature and it is closely related to the recent work of Gowers and Maurey [29,28]. T HEOREM 3.7 ([52]). There is a quasi-Banach space which does not contain a basic sequence. We will postpone discussion of this theorem to Section 4. In view of Theorem 3.7 it is possible to ask whether such examples can be created in classical spaces such as Lp when p < 1. In fact there are two positive results which show that every subspace of Lp has a basic sequence. The first result is due to Bastero [8] who show that the theory of Krivine–Maurey stability can be extended to quasi-Banach spaces. This shows that: T HEOREM 3.8. If X is closed subspace of Lp when 0 < p < 1 then X contains a subspace isomorphic to r for some p r 2. The second result of Tam [84] gives a general and important criterion for the existence of basic sequences. Notice that this result also includes the case of subspaces of Lp .

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T HEOREM 3.9. Let X be a complex quasi-Banach space with a plurisubharmonic quasinorm. Then X contains a basic sequence. To conclude this section, we note that in [30] an example is created of a sequence (fn ) contained in Lp when 0 0 and every subsequence (fn )n∈M is fundamental in Lp .

4. The three-space problem and minimal extensions We now turn our attention to a central problems of the area in the 1970’s, the three-space problem for local convexity which asked if there is a non-locally convex quasi-Banach space X with a closed subspace E such that both E and X/E are locally convex. This problem belongs to a family of three-space problems for which we refer to [12]. T HEOREM 4.1. There is a non-locally convex quasi-Banach space X with a subspace E of dimension one so that X/E is isomorphic to 1 . Theorem 4.1 is due independently (and essentially simultaneously) to the author, Ribe and Roberts [38,72] and [75]. However the examples created in each case were very different. Suppose X is a quasi-Banach space. We will say that Y is a minimal extension of X if there is a subspace E of Y with dim E = 1 and Y/E ≈ X. We will say that Y is the trivial extension (or that Y splits) if L is complemented, i.e., Y ≈ L ⊕ X in the natural way. We say ([55]) that X is a K-space if every minimal extension of X is trivial. We now describe a general construction of a minimal extension (first used in [38] and [72]). Let us suppose X is a quasi-Banach space (over the field K). Let X0 be any fixed dense linear subspace of X (of course X0 = X is a possible choice). We say that a map F : X0 → K is quasilinear if: (1) F (αx) = αF (x) for x ∈ X0 and α ∈ K. (2) There is a constant K so that

F (x1 + x2 ) − F (x1 ) − F (x2 ) K x1 + x2 ,

x1 , x2 ∈ X.

We then define a quasi-norm on K ⊕ X0 by (α, x) = α − F (x) + x, F

x ∈ X0 , α ∈ K.

The completion of K ⊕ X0 for this quasi-norm is a minimal extension of X which we denote K ⊕F X. Conversely every minimal extension of X is isomorphic (as an extension) to K ⊕F X for a suitable quasilinear map F (see [37]). It is clear that if F and G are any two quasilinear maps on X0 then F and G define equivalent quasi-norms on K ⊕ X0 if and only if there is a constant C so that |F (x) − G(x)| Cx for every x ∈ X0 . In this case

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we will say that F and G are equivalent. The minimal extension K ⊕F X splits if and only if there is a linear map G : X0 → K equivalent to F , i.e., G satisfies estimate of the form F (x) − G(x) Cx,

x ∈ X0 .

(4.2)

In this way K-spaces are characterized in terms of an approximation property. We refer to [11] for related results. We also note the connection with the concept of Hyers–Ulam functional stability (see [31]); the question is essentially is whether a functional which satisfies a perturbation of the functional equation for linear maps is itself a perturbation of a linear map. Let us note that if X is a Banach space then (4.2) is equivalent to an estimate of the form n n

F (xi ) C xi , x1 , . . . , xn ∈ X0 . (4.3) F (x1 + · · · + xn ) − i=1

i=1

P ROOF OF T HEOREM 4.1. To prove Theorem 4.1 we follow the construction of Ribe [72] of a space now known as the Ribe space. According to the preceding discussion, it is enough to exhibit a function F : c00 → K defined on the dense subspace c00 of all finitely supported sequences in 1 , which is quasilinear and fails to satisfy (4.3). Ribe’s example is the functional ∞ ∞ ∞ F (x) = xk log |xk | − xk log xk k=1

k=1

k=1

(where 0 log 0 := 0). A slight modification yielding an equivalent quasi-norm is the functional Λ(x) =

∞ k=1

xk log

|xk | . x

To see that (4.3) does not hold it is enough to compute F (e1 + · · · + en ) − −n log n.

(4.4) n

k=1 F (ek ) =

The Ribe space immediately produces the necessary example for Theorem 4.1. Also as observed by Roberts [75] it gives an example of a non-locally convex space with no quotient with trivial dual; this gives a counterexample to complement Theorem 3.5. At this point let us mention an important open problem: P ROBLEM 4.2. Classify those Banach spaces which are K-spaces (i.e., so that every minimal extension is trivial). Is it is true that a Banach space X is a K-space if and only if X∗ has non-trivial cotype? There is some body of evidence to support the conjecture in Problem 4.2. The known results are:

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T HEOREM 4.3 ([38]). Suppose X is a Banach space with non-trivial type. Then every minimal extension of X is trivial. T HEOREM 4.4 ([58]). Suppose X is a Banach space which is the quotient of an L∞ -space. Then every minimal extension of X is trivial. It is perhaps worth noting that the latter theorem can be restated in terms of a stability theorem for set functions. T HEOREM 4.5 ([58]). There is a universal constant so that whenever A is an algebra of subsets of some set Ω and F : A → R is a set function satisfying: F (A ∪ B) − F (A) − F (B) 1 if A ∩ B = ∅ then there is an additive set function μ with F (A) − μ(A) K for every A ∈ A. There are some non-locally convex K-spaces: T HEOREM 4.6 ([38]). If 0 < p < 1 then every minimal extension of p or Lp splits. Let us return to the case of 1 . As we have seen it is possible to characterize minimal extensions of 1 via quasilinear maps on c00 . It turns out that it is possible up to equivalence to characterize quasilinear maps in a very convenient form. To understand this let us first + of note that it is only necessary to specify a quasilinear map F on the positive cone c00 c00 since any map obeying the conditions for quasilinearity on the positive cone can be extended by the formula

F (x) = F x + − F (x − ), where x + = max(x, 0) and x − = max(−x, 0). This extension is then unique up to equivalence. Let X be a Banach sequence space, i.e., a space of sequences equipped with a norm · X such that • The basis vectors en ∈ X. • If ξ ∈ X and |ηk | |ξk | for every k then η ∈ X and ηX ξ X . • For every n ∈ N the linear functional η → ηn is continuous. • If ξ is a sequence such that n ∈ N Sn ξ = (ξ1 , . . . , ξn , 0, . . .) ∈ X and sup Sn ξ X < ∞ then ξ ∈ X and ξ X = supn∈N Sn ξ X .

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The last condition here is usually called the Fatou property. We can now define an associ+ ated quasilinear map on c00 by the formula ΦX (x) = sup

∞

ξ X 1 k=1

xk log |ξk |.

(4.5)

This functional was introduced under the name indicator function of X in [51] and later under the name entropy function of X in [66] where it plays an important role in the solution of the distortion problem. The fact that it is quasilinear is first observed in [51]. By way of illustration consider the case X = 1 , when as easy calculation gives that ΦX = Λ where Λ is defined by Eq. (4.4). Note that Φp = p1 Λ and Φ∞ = 0. The entropy functions yield an important source of minimal extensions of 1 . They do not completely classify minimal extensions because each is convex on the positive cone. A complete classification is however obtained in [51]: T HEOREM 4.7. Let F : c00 → K be a quasilinear map (for the 1 -norm). Then there exists a positive α and a Banach sequence space X so that F is equivalent to α(ΦX − ΦX∗ ) where X∗ is the Köthe-dual of X. P ROOF OF T HEOREM 3.7. We return to Theorem 3.7. As suggested in the discussion it is reasonable to hope for an example of a minimal extension of 1 with no basic sequence. If Y is this minimal extension and L is the kernel of the quotient map onto 1 this requires that every infinite-dimensional closed subspace of Y contains L. Clearly this is very much related to the construction of Gowers and Maurey [29] of a Banach space where any two infinite-dimensional subspaces almost intersect. Now if we write Y in the form K ⊕F 1 where F is defined on c00 , then we can translate our requirement to a condition on F . This is that F cannot be equivalent to a linear functional on any infinite-dimensional subspace of c00 . This in turn is equivalent to the requirement that for every infinite-dimensional subspace E of c00 we have: sup F (x): x ∈ E, x 1 = ∞. In this language, this is a type of distortion problem similar to distortion problem for Banach spaces solved in [66]. Clearly Theorem 4.7 suggests we should try use a functional of the type F = ΦX where X is a suitably exotic Banach sequence space. The correct choice is a space used by Gowers [28] which is a modification of the original Gowers–Maurey construction in [29]. The proof that such an example works requires technical calculations similar in spirit to work in [29]; we refer to [52]. One final remark on this example is in order: one can find a dense subspace which has HBEP. Thus in Theorem 3.2 it is necessary to assume that X is complete; it should be noted that in [71] a number of results are proved for metrizable topological vector spaces with HBEP, without the assumption of completeness.

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5. The Krein–Milman theorem A classic problem in non-locally convex spaces asks whether every compact convex subset of a quasi-Banach space has an extreme point. This can be traced back as least as far as [61], and probably much further. Although, at first sight, this problem is unrelated to the three-space problem of the preceding section, in retrospect their negative solutions require very much the same constructions. T HEOREM 5.1. There is a compact convex subset of Lp when 0 < p < 1 which has no extreme points. Theorem 5.1 is due to Roberts [73] and [74]. Roberts’s original construction is contained in [74] and this only gives a compact convex subset of some quasi-Banach space without extreme points, while [73] contains a much simplified approach and the theorem as stated. Since [73] is not readily available, a good reference for this argument is [56]. Roberts then used the key ideas in his proof of Theorem 5.1 to prove Theorem 4.1. In this article we will take the opposite direction, which in hindsight seems the right way to look at things. We now turn to the proof of Theorem 5.1. The approach used by Roberts in both [73] and [75] is through the notion of a needlepoint. If X is a quasi-Banach space we say that x ∈ X is a needlepoint if given ε > 0 there exists a finite set F so that x ∈ co F , v < ε if v ∈ F and for any y ∈ co F there exists 0 α 1 with y − αx < ε. X is called a needlepoint space if every point of X is a needlepoint. The main ingredients of the proof are: P ROPOSITION 5.2. If X is a needlepoint space then X contains a compact convex set K with no extreme points. P ROPOSITION 5.3. The space Lp for 0 < p < 1 is a needlepoint space. In [75] it is simply shown that there is a needlepoint space: the fact that Lp has this property is proved in [73]. For the proof of Proposition 5.2 we refer to [56]. The construction is in fact is a quite logical inductive argument using at each stage that every point of the space is a needlepoint. We will describe an approach to Proposition 5.3 which uses the notion of minimal extensions. In fact, it is very easy to see that if Y is a non-trivial minimal extension of a Banach space X so that X ≈ Y/L where dim L = 1 then every e ∈ L is a needlepoint. Thus we have an easy direct construction of a non-zero needlepoint by using the Ribe space. Of course, the Ribe space is not a needlepoint space; in fact any needlepoint space must have trivial dual. However we next note: T HEOREM 5.4 ([42]). The Ribe space is isomorphic to a subspace of Lp when 0 < p < 1.

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It is in fact instructive to see that the Ribe space is a rather natural subspace of Lp . Let (ξn ) be a sequence of independent random variables each with the Cauchy distribution, i.e., 1 λ(ξn ∈ B) = π

B

dx 1 + x2

so that

eisξn (t ) dt = e−|s| .

Then consider the space E generated by the constant function 1 and the sequence {|ξn |}∞ n=1 . It may be shown that if α0 , α1 , . . . ∈ R is a finitely supported sequence then ∞ ∞

∞ αk |ξk | ∼ α0 − Λ (αk )k=1 + |αk |, α0 + k=1

p

k=1

where Λ is defined in (4.4). For details we refer to [42]. This implies that E is isomorphic to the Ribe space and the constant function 1 is a needlepoint of Lp . To complete the argument we need only note that Lp is a transitive space, i.e., given any f, g ∈ Lp with f = 0 there is a bounded linear operator T : Lp → Lp with Tf = g. As the image of a needlepoint is necessarily a needlepoint this means that Lp is a needlepoint space. This then completes the proof of Proposition 5.3 and hence of Theorem 5.1. There has been some investigation of geometric properties of a compact convex set K which guarantee the existence of extreme points. One way to formulate this idea is to consider a Banach space X and a compact linear operator T : X → Y where Y is a quasiBanach space. Let K = T (BX ). Then K is a symmetric compact convex set in Y . We can then consider geometric conditions on X so that K is affinely homeomorphic to a compact convex subset of a locally convex space; this is equivalent to the existence of a separating family of continuous affine functions on K. Two results of this nature are known: T HEOREM 5.5. Suppose X is a Banach space and T : X → Y is bounded linear operator. Then the collection of continuous affine functions on K = T (BX ) separates the points of K if either of the following conditions hold: (1) ([26]) X contains no subspace isomorphic to 1 . (2) ([58]) T is compact and X is an L∞ -space. Finally an example is created in [50] of a compact convex set which is not affinely homeomorphic to a subset of L0 [0, 1]. It should be pointed out that Proposition 3.2 of [50] has an error: the set K is not convex (see [5]); however the original set K0 or its symmetrization can be used in its place.

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6. Operators and the structure of Lp -spaces when 0 < p < 1 In this section we will treat some results on operators and their representations and discuss the isomorphic structure of the spaces Lp [0, 1]. If X is a quasi-Banach space with trivial dual then the algebra L(X) may, in fact, be rather small, as one does not have the rich class of finite-rank operators. Let us say that a space X is rigid if L(X) = CI . In [57] the following result is proved (see also [90] for a quasi-normed but incomplete example of a rigid space): T HEOREM 6.1. If 0 < p < 1 then Lp has a subspace X so that every quotient of X is rigid. In a recent preprint, Roberts shows there are many rigid spaces by showing: T HEOREM 6.2 ([76]). Every separable p-normable quasi-Banach space with trivial dual is the quotient of a separable rigid p-normable quasi-Banach space. One of the classical results on non-locally convex spaces is a theorem of Williamson [92] which says essentially that the theorem of the Fredholm alternative remains valid. If X has trivial dual so that X∗ reduces to {0} then this implies that any compact operator K : X → X has only zero in its spectrum. Later Pallaschke [67] observed that if X is also transitive then K must itself be the zero operator, since if K = 0 one can find an endomorphism T : X → X so that T K has one in its spectrum. In particular this result applies to Lp when p < 1. This result was generalized by Turpin [86] to a wider class of spaces and in [56] Pallaschke’s result is extended to strictly singular operators. These results suggested a question (due to Pełczyński): is it possible to find a compact endomorphism of a space with trivial dual? In effect, this is equivalent to a more general question: does there exist a quasi-Banach space X with a trivial dual and a non-zero compact operator K : X → Y where Y is any quasi-Banach space? If such an example can be constructed we can suppose K(X) dense in Y so that Y ∗ = {0} and consider the map (x, y) → (0, Kx) on X ⊕ Y . Let us say then that a space X admits compact operators if there is a non-zero compact operator T : X → Y where Y is some quasi-Banach space. Pełczyński’s question was resolved in [59]: T HEOREM 6.3. There is a quasi-Banach space with trivial dual which admits compact operators. The proof used some classical function theory and the earlier results of Duren, Romberg and Shields [23]. We would like however to indicate a slightly different proof based on [60] and the results of Section 3. The key observation is that if X is a quasi-Banach space whose unit ball BX is compact for some (Hausdorff) vector topology τ , then one can mimic the proof of the Banach–Dieudonné theorem to show that the topology τˆ which is defined to be the finest topology on X agreeing with τ on bounded sets is a vector topology. Let us take X = 2 (np ), where 0 < p < 1. Then X has certain features of a reflexive Banach space; in particular, its unit ball is weakly compact. If we use the weak topology w for τ then τˆ

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is the “bounded weak” topology bw. We show next that every closed subspace E of X is bw-closed. In fact if not we can find a sequence xn ∈ BX converging in the bw-topology to a point x ∈ / E. Using Theorem 3.3 or even a simple gliding hump argument we can suppose (xn − x) is a basic sequence equivalent to a block basis of the original basis. But then passing to a further subsequence we can suppose it is equivalent to the canonical basis of 2 . But there is a linear functional ϕ on its closed linear span with ϕ(xn −x) = 1 for all n, which produces a contradiction. Now an application of Theorem 3.5 gives a subspace E so that X/E has trivial dual; in this space the bw-topology factors to a quotient topology for which the unit ball is compact. We remark that this topology by its construction is locally p-convex and so easily provides examples of compact operators into p-normable quasi-Banach spaces. For full details see pp. 140–146 of [56]. Let us note that Sisson [81] showed that there is an example of a rigid space admitting compact operators; this can now also be deduced from Theorem 6.2. The examples indicated above leave open the question of whether specific spaces admit compact operators. Of course the obvious example is the space Lp when 0 < p < 1. This space does not admit compact operators; in fact more is true: T HEOREM 6.4 ([35]). Suppose 0 < p < 1. Let T : Lp → Y be a non-zero operator into some quasi-Banach space Y (or even a topological vector space). Then there is an infinitedimensional subspace H of Lp which is isomorphic to 2 and so that T |H is an isomorphism. This result also holds in a wide class of non-locally convex Orlicz spaces. Let us sketch a very simple proof that there cannot be a compact operator K : Lp → Y where we assume Y has an r-subadditive quasi-norm. Let f ∈ Lp and suppose rn are the Rademacher functions. If K(f rn ) has any cluster point it must be zero since for any subsequence 1 n f (rk1 + · · · + rkn ) converges to zero. Hence limn→∞ K(f rn ) = 0. Let An = {rn = 1} and Bn = {rn = −1}. Then Kf = 2K(f χAn ) − K(f rn ) = K(f rn ) − 2K(f χBn ). Thus

1/r Kf 21−1/r lim inf Kf χAn r + Kf χBn r n→∞

1/p 21−1/p lim inf Kf χAn p + Kf χBn p n→∞

2

1−1/p

Kf p .

This yields K 21−1/p K, i.e., K = 0. It is very possible that Theorem 6.4 is not the best result here. Under special hypotheses one can do much better: T HEOREM 6.5. Suppose 0 < r < p < 1. Then:

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(1) ([39]) Let T : Lp → Lp be a non-zero operator. Then there is a subspace E of Lp so that E ≈ Lp and T |E is an isomorphism. (2) ([41]) Let T : Lp → Lr be a non-zero operator. Then if p < q 2 there is a subspace E of Lp with E ≈ Lq and T |E is an isomorphism. These results follow from representation theorems which we discuss shortly. Let us remark at this point that part (2) extends to operators taking values in natural which we will discuss in Section 7. P ROBLEM 6.6. Suppose 0 < p < 1 and let T : Lp → Y be any non-zero operator. Suppose p < q 2. Does it follow that there is a subspace E ≈ Lq (or even q ) so that T |E is an isomorphism? We will now turn to some basic questions on the structure of the spaces Lp [0, 1] when 0 < p < 1. Some of these questions can be regarded as analogues of similar questions for the Banach spaces Lp when p 1. However, the theory has a different flavor. A simple example is the fact that the quotient of Lp by a one-dimensional subspace is never isomorphic to Lp when p < 1 [55]. In fact this is an immediate consequence of the observation in Theorem 4.6 that every minimal extension of Lp splits. A key result is the concrete representation of operators on Lp [0, 1] given in [39]. This result which was inspired by an earlier similar result for the case p = 0 due to Kwapień [62] has a somewhat similar form also in the case p = 1 [39]. Results of similar type have been studied for operators T : Lp → L0 in [41] and also for operators on general rearrangementinvariant spaces [44] and [47]. T HEOREM 6.7. Suppose T : Lp [0, 1] → Lp [0, 1] is a bounded operator where 0 < p < 1. Then there is a sequence of Borel functions an : [0, 1] → K and Borel maps σn : [0, 1] → [0, 1] so that: (1) |an (s)| |an+1 (s)| for n = 1, 2, . . . and s ∈ [0, 1]. (2) σn (s) = σm (s) when m = n and s ∈ [0, 1]. ∞ p (3) n=1 |an (s)| < ∞ for almost every s ∈ [0, 1]. ∞ an (s)p ds T p λ(B) for every Borel set B ⊂ [0, 1]. (4) −1 n=1 σn B

(5) If f ∈ Lp then Tf (s) =

∞

an (s)f (σn s)

s-a.e.

n=1

Conversely if (an ) and (σn ) are given satisfying (1)–(4) then (5) defines a bounded operator on Lp . This theorem rather easily yields Theorem 6.5(1). It also gives a simple proof that Lp is a primary space, i.e., if Lp = X ⊕ Y then either X or Y must be isomorphic to Lp . (In fact essentially the same proof can be given in the case p = 1; see [39] and [24].)

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However an intriguing question, closely related to a similar question in the case p = 1, is: P ROBLEM 6.8. Is Lp [0, 1] a prime space when 0 < p < 1, i.e., is any infinite-dimensional complemented subspace isomorphic to Lp ? It is well-known that the spaces p are prime when 0 < p < 1. This is due to Stiles [82], by a proof similar to that of Pełczyński for the case p 1. In [39] it is shown that the space L0 [0, 1] is prime by using Kwapień’s representation of operators for this case. The case 0 < p < 1 is however more difficult and remains open. The natural way to attack this problem is to take an arbitrary projection P on Lp and use Theorem 6.7 to represent it. It turns out that one must show that there is a Borel set B of positive measure and n ∈ N so that |an | > 0 on B and σn is one-to-one on B. In the case p = 0 this final step can be completed rather easily but in the case 0 < p < 1 it is not quite so clear. In [42] a detailed study of this and related problems was undertaken and a curious but unsatisfactory result was obtained: T HEOREM 6.9. Suppose 0 < p < 1. Then Lp has at most two non-trivial complemented subspaces up to isomorphism. If the second mysterious complemented subspace were to exist it would have some remarkable properties. For example, there would be an averaging projection of the space of vector-valued functions Lp (Z) = Lp ([0, 1]; Z) onto its subspace of constant functions. P ROBLEM 6.10. Suppose 0 < p < 1 and X is any quasi-Banach space. Is it possible that there is an averaging projection on Lp (X)? P ROBLEM 6.11. Suppose 1 p < ∞ and suppose X is a quasi-Banach space so that there is an averaging projection on Lp (X). Is X locally convex? We remark that in [39] it is shown that there is no averaging projection on Lp (Lp ). Some partial results on Problem 6.11 are given in [45].

7. Lattices and natural spaces We next present some basic facts in the theory of quasi-Banach lattices. It turns out that some interesting complications arise in the theory, associated with problems of convexity. Let us note first that the discussion of pp. 40–41 of [64] of homogeneous functions applies verbatim to quasi-Banach lattices. In this way we can define the notions of p-convexity and q-concavity as in [64] or [32] for quasi-Banach lattices. It is clear, for example, that the fundamental examples Lp when 0 0. These issues are discussed in [46] and [16].

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The issues involved relate to the Maharam submeasure problem, for which we refer to the article [25]. Suppose Ω is a set and Σ is a σ -algebra of subsets of Ω. A set function φ : Σ → [0, ∞) is called a submeasure if we have • φ(∅) = 0, • φ(A) A ⊂ B, ∞ φ(B) if ∞ • φ( n=1 An ) ∞ n=1 φ(An ) for every (An )n=1 ⊂ Σ. φ is called a continuous or Maharam submeasure if An ↓ ∅ implies φ(An ) ↓ 0. The unsolved Maharam submeasure problem asks whether every Maharam submeasure is equivalent to a measure, in the sense that if φ is a Maharam submeasure, then there is a measure μ so that μ(A) = 0 if and only if φ(A) = 0. See [58] for a partial result. A submeasure φ is called pathological if whenever μ is a measure with μ φ then μ = 0. This is equivalent to the fact that given any ε > 0 there exist B1 , . . . , Bn ∈ Σ so that 1 χBk (1 − ε)χΩ n

(7.6)

max φ(Bk ) ε.

(7.7)

n

k=1

and

1kn

Pathological submeasures have been constructed by several authors, but the simplest example seems to be that due to Talagrand [83]. The Maharam problem quoted above is equivalent to asking whether a pathological submeasure can be continuous. Let us suppose then that (Ω, Σ, φ) is a submeasure space. Suppose 0 < r < ∞. Let X be the space of all Σ-measurable functions f so that:

∞

f X =

φ |f | > t r dt

1/p < ∞.

0

It is not hard to show that X is a quasi-Banach lattice. However if φ is pathological then X cannot be p-convex for any p > 0; this follows routinely from (7.6) and (7.7). It is clear that if X has a p-subadditive quasi-norm then we have (p, 1)-convexity, i.e., n n 1/p p |xk | xk , k=1

x1 , . . . , xn ∈ X.

k=1

It is shown in [16] that we then have P ROPOSITION 7.1. Suppose 0 < p < 1 and that X is a p-normable quasi-Banach space (with a p-subadditive quasi-norm). Then if 0 < q < r 1 with 1/q − 1/r = 1/p − 1 we

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have: n 1/r n 1/q r q |xk | xk , k=1

x1 , . . . , xn ∈ X.

k=1

In [46] the notion of lattice-convexity or L-convexity is introduced. A quasi-Banach lattice X lattice-convexity orL-convex if there exists C > 0 so that if u, x1 , . . . , xn ∈ X with maxk |xk | |u| but n1 nk=1 |xk | |u| then u C maxk xk . T HEOREM 7.2 ([46]). Let X be a p-normable quasi-Banach lattice. The following conditions are equivalent: (1) X is lattice-convex. (2) X is r-convex for some r > 0. (3) X is r-convex for every 0 < r < p. There are special situations which guarantees lattice-convexity: T HEOREM 7.3 ( [46,54]). Let X be a quasi-Banach lattice with non-trivial cotype. Then X is lattice-convex. We remark that in [46] this theorem is deduced from a result on the Maharam submeasure problem proved in [58]. In [54] a simpler direct proof is given. T HEOREM 7.4 ([46]). Let X be a quasi-Banach lattice which is isomorphic to a subspace of a lattice-convex quasi-Banach lattice. Then X is lattice-convex. Based on this theorem, we introduce the class of natural spaces. A quasi-Banach space X is called natural if it is linearly isomorphic to a subspace of a lattice-convex quasiBanach lattice. In practice this implies that X is isomorphic to an ∞ -product of spaces of the type Lp (μ) where 0 < p < ∞ is fixed. It may be shown that a p-normable space is natural if and only if it is finitely representable in the space weak Lp or Lp,∞ . The motivation for this definition is that most spaces that arise in analysis are natural. Natural spaces are relatively good spaces to work in. We mention, for example, that natural spaces must always contain a basic sequence. In fact it is not difficult to see that if X is separable and natural then there is a one-one operator T : X → Lp (0, 1). If T is not an isomorphism then X is not minimal and we can apply Theorem 3.6 while if T is an isomorphism then Theorem 3.8 applies. Of course this means that the example in Theorem 3.7 is not natural. There are some simpler examples of non-natural spaces. For example, the spaces Lp (T)/Hp for 0 < p < 1 (essentially proved in [41]) and the Schatten ideals Cp for 0 < p < 1 fail to be natural [49]. The importance of lattice-convexity is that this assumption allows us to use many of the powerful techniques available in the study of Banach lattices. Let us mention the example

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of square-function arguments. It is well-known that in a Banach lattice with non-trivial cotype one has an estimate of the form (see [32] and [64]): n 1/2 n εk xk ≈ |xk |2 E . k=1

k=1

It follows from Theorem 7.3 above and similar arguments that this estimate works in any quasi-Banach lattice with cotype. We also mention the Krivine generalization of Grothendieck’s theorem (see [64], p. 93), which remains valid (with a different constant) even for operators between lattice-convex quasi-Banach lattices. One of the most interesting lines of application of these ideas is in the study of uniqueness of unconditional bases in certain natural quasi-Banach spaces (see [88]). By combining well-established techniques from Banach space theory with the additional information that if X has an unconditional basis (un ) then it simultaneously an unconditional basis in the Banach envelope, it is possible to prove some quite powerful uniqueness results for a wide class of spaces with unconditional bases. This line of research was initiated in [34] where it is shown that the spaces p and many Orlicz sequence spaces for 0 < p < 1 have unique unconditional basis. A general uniqueness criterion was developed in [53] which was improved in a recent paper of [93]: T HEOREM 7.5. Let X be a natural quasi-Banach space with a normalized unconditional basis (un ). Suppose that: (1) X is isomorphic to X ⊕ X. ∞ q (2) There exists q < 1 so that if ∞ n=1 an un converges then n=1 |an | < ∞. Then for any normalized unconditional basis (vn ) of X there is a permutation π of N so that (un ) and (vπ(n) ) are equivalent. A weaker predecessor of this result was used in [53] to show that Hp (Tm ) and Hp (T n ) are isomorphic for 0 < p < 1 if and only if m = n. For further results on uniqueness see [63,1] and [2].

8. Analytic functions and applications Let X be a complex quasi-Banach spaces. Let Ω be an open subset of the complex plane C. Then a function F : Ω → X is called analytic if for every z0 ∈ Ω there exists δ > 0 and xn ∈ X for n 0 so that if |z − z0 | < δ then F (z) =

∞

xn z n .

n=0

This definition of an X-valued analytic function was first employed by Turpin [86]. It is rather easy to see that other possible definitions based on complex differentiability do not work satisfactorily in quasi-Banach spaces. For example, if D is the open unit

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disk with standard area measure then the map F : C → Lp (D) where p < 1 defined by F (z)(w) = (z − w)−1 is actually complex differentiable but does not have a local power series expansion, and indeed is not infinitely differentiable. The basic theory of analytic functions was developed by Turpin [86], who noticed that if K is a compact subset of Ω then the convex hull of F (K) remains bounded in X. This enables one to show that there is a factorization of f through a Banach space. More precisely, if Ω0 is an open relatively compact subset of Ω then there is a Banach space Y , a one-one bounded injection j : Y → X and an analytic function G : Ω0 → Y so that F = j ◦ G. Since the theory of Banach space-valued analytic functions is very well understood one can use this device to prove many of the basic desired properties of analytic functions. For example, if f is analytic on a disk {z: |z − z0 | < r} then f has a (necessarily unique) power-series expansion valid throughout the disk. However, the theory of analytic functions is by no means as clean as for Banach spaces. The first obstacle is the Maximum Modulus Principle. A simple example due to Alexandrov [3] shows what can happen. Consider the function F : D → Lp (T) defined by F (z)(eiθ ) = e−iθ (1 − e−iθ z)−1 . This map is plainly analytic and extends continuously toa function on the closed unit disk D. Its power series expansion is given −i(n+1)θ zn . Consider the subspace H (T) and let Q : L → L /H be by F (z) = ∞ p p p p n=0 e the quotient map. Then Q ◦ F is analytic into Lp /Hp and Q(F (0)) = 1. However on the boundary if |z| = 1 then QF (z) = 0, i.e., F (z) ∈ Hp . To see this rewrite F (z) as −¯z(1 − z¯ eiθ )−1 . Of course if X has an equivalent plurisubharmonic quasi-norm then for any analytic function F : Ω → X the map z → F (z) is subharmonic and so we have a Maximum Modulus Principle. In fact this property essentially characterizes spaces for which a form of Maximum Modulus Principle holds. Let us say that X is A-convex if there is a constant C so that for every X-valued polynomial F (z) = nk=0 xk zk we have F (0) C max F (z). |z|=1

Of course this will imply that the same conclusion holds for any continuous function F on the closed unit disk D which is analytic in the interior, so that the space Lp /Hp is an example of a non-A-convex space. It is shown in [49] that X is A-convex if and only if X has an equivalent plurisubharmonic quasi-norm. Since the spaces Lp and p are A-convex it follows trivially that natural spaces are A-convex. Of course by Theorem 3.9 A-convex spaces always contain basic sequences and so this yields another proof that natural spaces also must contain basic sequences. However, it should be noted that the Schatten classes Cp for 0 < p < 1 are A-convex but fail to be natural (see [49]). The treatment of analytic functions valued in a non-A-convex space requires different techniques, but it turns out that the theory is still quite rich. The key ingredient is an atomic decomposition theorem due to Coifman and Rochberg [15]. In this paper, the authors proved some very general atomic decompositions for certain Bergman spaces. As a by-product they extended the results of [23] and [80] to calculate the p-envelope of Hr when 0 < r < 1 (this is the p-normed analogue of the Banach envelope). Let us denote this

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1121

space Br,p . It turns out that Br,p consists of the space of all analytic functions on D such that 1/p

f (w)p 1 − |w|2 p/r−2 dA(w) < ∞, f r,p = D

where dA is standard area measure dx dy. The key to their proof of this is the following atomic decomposition: T HEOREM 8.1. There is a constant r) so that if ψ ∈ Br,p then there exist zk ∈ D C = pC(p, 1/p Cψ and αk ∈ C for k ∈ N so that ( ∞ r,p and k=1 |αk | ) ψ(w) =

∞

ν+1−σ αk 1 − |zk |2 (1 − wzk )−(ν+2) ,

k=1

where σ = 1/r − 1 and ν = [σ ]. Let us illustrate how this can be used to establish some basic estimates (cf. [48]). Suppose X is a p-normable space and that F : D → X is a polynomial. Suppose F (z) = x0 + x1 z + · · · + xn zn . Fix 0 < r < p and define a linear operator T : Br,p → X by T (f ) =

n

xk

k=0

(ν + 1)! f (k) (0). (ν + k + 1)!

Then

T (1 − wzk )−ν+2 = F (zk ). It follows by applying Theorem 8.1 rather crudely we can get an estimate that T C max F (z). z∈D

However

k!(ν + 1)! xk T wk = (ν + k + 1)! so that xk C

ν +k+1 T wk Br,p . k

If we choose r so that σ ∈ N so that σ = ν then this gives a Cauchy-type estimate which is valid for any function F analytic in the open unit disk, (k) F (0) Ck!k 1/p−1 maxF (z). (8.8) z∈D

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N. Kalton

Once one has these Cauchy estimates certain other basic principles of complex analysis follow. For example, it is clear that Liouville’s theorem holds (i.e., a bounded analytic function is constant). This was first noted in [48] although an earlier weaker version for functions analytic on the Riemann sphere was observed by Turpin [86]. Let us also note there is an annular Maximum Modulus Principle: T HEOREM 8.2 ([49]). For any 0 < r < 1 and any 0 < p 1 there is a constant C = C(r, p) so that if X is a p-normed quasi-Banach space and F : D → X is an analytic function then F (0) C sup F (z). r|z|0

It is easily seen that both these definitions can be given in discrete form, e.g., xθ,p ≈

n∈Z

p 2−θpn K 2n , t

1/p .

(3.2)

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N. Kalton and S. Montgomery-Smith

The functor which takes the couple X to Xθ,p is the (θ, p)-method; this clearly provides an example of an exact interpolation method. The theory of this method is well-developed and understood and we can refer to [5] and [8] for a full discussion of such topics as reiteration and duality. For our purposes it is useful to point out an equivalent definition in terms of the J -functional, first obtained in the fundamental paper of Lions and Peetre [76]. Consider the case 1 p < ∞. Define for x ∈ X0 + X1 , x θ,p

p 1/p k θk = inf max xk 0 , 2 xk 1 : x= 2 xk , k∈Z

(3.3)

k∈Z

where the series converges in X0 + X1 . Then x ∈ Xθ,p if and only if x θ,p < ∞ and the norms x θ,p and xθ,p are equivalent. In (3.3) we have formulated the J -method discretely; it is more usual to use a continuous version. The equivalence of the J -method and the K-method of definition can be obtained from the Fundamental Lemma, which we discuss later (Theorem 6.1). Note that we must have that Δ(X) is dense in the spaces Xθ,p provided 1 p < ∞. Using this, one can show a duality theorem [74]: T HEOREM 3.1. Suppose Δ(X) is dense in both X0 and X1 . Then if 1 p < ∞ and 0 < θ < 1 the dual of (X0 , X1 )θ,p can be identified naturally with (X0∗ , X1∗ )θ,q where 1/p + 1/q = 1. The (θ, p)-methods have proved extremely useful in many branches of analysis including Banach spaces. We conclude this section by discussing the first major application of interpolation in Banach spaces theory, the Davis–Figiel–Johnson–Pełczyński factorization theorem [35]. The basic idea of this theorem is to establish conditions under which certain interpolation spaces are reflexive, although in the initial paper the language of interpolation was not used. Later Beauzamy [3] established the general result. Consider the spaces Zn = (Δ(X), J2n ) where Jt (x) = J (x, t) is a norm on Δ(X). Now we can use (3.3) to define a quotient mapping Q : p (Zn )n∈Z → Xθ,p by

θk 2 ak . Q (ak )k∈Z = k∈Z

The following lemma is an easy gliding hump argument: L EMMA 3.2. Suppose 1 < p < ∞ and 0 < θ < 1. Suppose an = (ank )k∈Z is a bounded sequence in p (Zn ) such that for each k we have limn→∞ ank = 0 weakly in X0 + X1 . Then Qan converges to zero weakly in Xθ,p . P ROOF. It is enough to construct a sequence of convex combinations of (Q(ak ))kn which is weakly null. First by Mazur’s theorem we can take convex combinations and assume limn→∞ ank X0 +X1 = 0 for each k. It follows quickly that limn→∞ ank θ,p = 0 for each k. Indeed we can split ank = bnk + cnk where bnk is bounded in X0 and converges

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1137

to zero in X1 while cnk is bounded in X1 and converges to zero in X0 . Then we use the estimate

1−θ θ θ ank θ,p C bnk 1−θ X0 bnk X1 + cnk X0 cnk X1 . It follows that we can find a sequence Nn → ∞ so that θk Q(an ) − 2 ank |k|Nn

→ 0. θ,p

Let bnk = ank if |k| Nn and 0 otherwise. Standard gliding hump arguments show that bn is weakly null. It is then easy to conclude that an is also weakly null. An immediate consequence due to Beauzamy [3] is that: T HEOREM 3.3. Suppose 1 < p < ∞ and 0 < θ < 1. Then (X0 , X1 )θ,p is reflexive if and only if BΔ(X) is relatively weakly compact in Σ(X). This follows from the preceding lemma and the Eberlein–Smulian theorem. Now the Factorization Theorem of Davis, Figiel, Johnson and Pełczyński is given by: T HEOREM 3.4 ([35]). Suppose X and Y are Banach spaces and T : X → Y is weakly compact. Then there is a reflexive space R and a factorization of T = BA where A : X → R and B : R → Y are bounded. P ROOF. We use the following typical trick. Let K be the closure of T (BX ) and let Y0 be the Banach space generated by taking K as it’s unit ball. Let Y1 = Y and then take R = Yθ,p for some choice of 0 < θ < 1 and 1 < p < ∞. For A we treat T as an operator into R and for B we take the inclusion of R into Y . There is a sense in which the (θ, p)-methods give rise to Banach spaces with relatively simple structure. This is the content of a theorem of Levy [71]. T HEOREM 3.5. Suppose Δ(X) is not closed in Σ(X). Then for 0 < θ < 1, 1 p < ∞ the spaces (X0 , X1 )θ,p contain a complemented copy of p . In order to prove this theorem, we first prove a preliminary lemma: L EMMA 3.6. Suppose Δ(X) is not closed in Σ(X). Then either (1) For every t > s > 0 and ε > 0 there exists x ∈ Δ(X) with τ −1 K(τ, x) (1 − ε)s −1 K(s, x), or

s < τ < t,

1138


(2) For every t > s > 0 and ε > 0 there exists x ∈ Δ(X) with K(τ, x) (1 − ε)K(t, x),

s < τ < t.

P ROOF. Note first that t −1 K(t, x) is decreasing and K(t, x) is increasing. Assume (1) fails. Then there exists ε > 0 and t > s > 0 such that t −1 K(t, x) < (1 − ε)s −1 K(s, x). Then for any x ∈ Δ(X) we have K(t, x) (1 − ε)txX1 . Thus, putting δ = 1 − ε/2 we can find u1 ∈ X0 , v1 ∈ X1 so that x = u1 + v1 and u1 X0 txX1 and v1 X1 δxX1 . Now we can iterate the argument as in the Open Mapping Theorem and write v1 = u2 + v2 where u2 X0 δtxX1 and v2 X1 δ 2 txX1 . Continuing in this way we construct ∞ (un )∞ n=1 in X0 and (vn )n=1 in X1 such that un X0 tδ n−1 xX1 ,

vn X1 tδ n xX1

and x = u1 + · · · + un + vn . Clearly ∞ n=1 un converges in X0 and its sum must be x (by computing in X0 + X1 ). Hence xX0 t (1 −δ)−1 xX1 . Similarly the failure of (1) implies that xX1 CxX0 whenever x ∈ Δ(X) for a suitable constant C. Thus if both (1) and (2) fail then the two norms · X0 and · X1 are equivalent on the intersection, and this implies the intersection is closed in X0 + X1 . We now turn to the proof of Theorem 3.5: P ROOF. Note that (3.2) the space Xθ,p can be regarded as a subspace of the p -sum of the space X0 + X1 with the norms 2−θn K(2n , x) for n ∈ Z. To show it has a complemented subspace isomorphic to p requires only the existence of a normalized sequence (xm )∞ m=1 in Xθ,p so that for each n we have limm→∞ K(2n , xm ) = 0 (i.e., xm converges to 0 in X0 + X1 ). This follows by standard gliding hump techniques. If such a sequence does not exist then there is a constant C so that

p 2−nθp K 2n , x

1/p CK(1, x).

n∈Z

That this is impossible follows from the preceding lemma.

4. The complex method We first define the complex method for a Banach couple X which we now assume consists of complex Banach spaces. We introduce a Banach space F of analytic functions as follows. Let S = {z: 0 < !z < 1} and let F be the space of analytic functions F : S → Σ(X)


1139

such that F extends continuously to the closure S and the functions t → F (j + it) are continuous and bounded in Xj for j = 0, 1. We norm F by F F = max

sup

j =0,1 −∞ 0 one can find a sequences (sn )∞ n∈Z , (cn )n∈Z in (0, ∞) such that K(t, x) a + bt +

cn min(sn , t) (1 + ε)K(t, x),

0 < t < ∞.

(6.3)

n∈Z

Note that if K(t, u) a for all t then by Gagliardo completeness uX0 a and similarly if K(t, u) bt for all t then uX1 b. If K(t, u) c min(s, t) for all t then uX0 cs and uX1 c. Thus if apply Theorem 6.2 to (6.3) we obtain Theorem 6.1 with constant C(1 + ε). Next we consider the converse direction. In this case we find (un ) as in Theorem 6.1 and let ψ(t) =

min un X0 , tun X1 .

n∈Z

Thus K(t, x) ψ(t) CK(t, x),

0 < t < ∞.

= (θj (t))j ∈N Now consider the set Γ of continuous maps θ : (0, ∞) → 1 of the form θ (t) where each θj is non-negative and concave and we have θj (t) ϕj (t) but n∈Z θj (t) C −1 ψ(t) for 0 < t < ∞). It is not difficult to see by the Ascoli–Arzela theorem that Γ is compact for the topology of uniform convergence on compacta, and so has a minimal element σ (t) = (σj (t))n∈Z . It then follows without difficulty that in fact we have j ∈Z

σj (t) = C −1 ψ(t),

0 < t < ∞.


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Indeed if σj > C −1 ψ(t) on some maximal open interval I then it is easy to see that σn must be affine on I ; if I = (α, β) where 0 < α < β < ∞ then the fact that ψ is concave leads to a contradiction, while the other cases when α = 0 and/or β = ∞ can be treated similarly. Now using the definition of ψ and the fact that each σj is concave we see that σj (t) = C −1

δj k min uk X0 , tuk X1 ,

k∈Z

where k∈Z δj k = 1 and δj k 0. (One way to see this is to use the representation 6.2 for ψ and σj .) Let vj =

δ j k uk .

k∈Z

Then

∞

j =1 vj

converges absolutely in X0 + X1 to x and

K(t, vj ) Cσj (t) Cϕj (t). It is clear from the foregoing discussion that if we define γ1 as the infimum of all constants C for which Theorem 6.1 holds and γ2 as the infimum of all constants C for which Theorem 6.2 holds then γ1 = γ2 . Their common value, γ is called the K-divisibility constant. Its exact value is unknown and seems to be a challenging problem. The √ best estimate from above was obtained by Cwikel, Jawerth and Milman [26], γ 3 + 2 2. On the other hand, an example of Kruglyak [70] gives a lower estimate γ > 1.6. Let us now sketch the ideas in the proof of Theorem 6.1, but without attempting to give the most delicate estimates (following [5]); we refer the reader to [26] for these. For fixed x ∈ X0 + X1 let us define t0 = 1 and then construct a sequence (tj )j ∈Z by two-sided induction such that for any j we have that one of the three mutually exclusive possibilities holds: K(tj , x) tj K(tj −1 , x) (1) , =2 min K(tj −1 , x) tj −1 K(tj , x) or (2) tj −1 = 0 and K(tj , x) tj K(t, x) , min < 2, K(t, x) tK(tj , x)

0 < t < tj ,

or (3) tj = ∞ and K(t, x) tK(tj −1 , x) , min < 2, K(tj −1 , x) tj −1 K(t, x)

tj −1 < t < ∞.

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For each j ∈ Z such that 0 < tj < ∞ pick vj ∈ X0 and wj ∈ X1 so that x = vj + wj and vj X0 + twj X1 < 2K(tj , x). If tj = 0 let vj = 0 and wj = x. If tj = ∞, let vj = x and wj = 0. Next let uj = vj − vj −1 . If 0 < tj −1 < tj < ∞ then uj X0 4K(tj , x),

uj X1 2tj−1 −1 K(tj −1 , X).

It follows that if tj −1 t tj then

min uj X0 , tuj X1 8K(t, x). If 0 = tj −1 < tj then uj = vj and uj X0 2K(tj , x). In this case either 1 lim K(t, x) K(tj , x) t →0 2 or tj−1 K(tj , x) lim t −1 K(t, x) 2tj−1 K(tj , x). t →0

In the latter case we use Gagliardo completeness to deduce that xX1 2tj−1 K(tj , x) and hence uj X1 xX1 + wj X1 4tj−1 K(tj , x). In either case we have

min uj X0 , tuj X1 8K(t, x),

0 < t tj .

Similarly if tj −1 < tj = ∞ we obtain


tj −1 t < ∞.

Now if tj −1+r t tj +r where r ∈ Z \ {0} then we may see that K(t, x) 2r−1 K(tj , x),

r > 0,

and t −1 K(t, x) 21−r tj−1 −1 K(tj −1 , x),

r < 0.


1149

Hence

min uj X0 , tuj X1 8 · 2−|r| K(t, x). Hence


0 < t < ∞.

j ∈Z

It is clear that uj converges absolutely in X0 + X1 and it is not difficult to check that its sum must be x in all possible cases. The above argument is clearly quite crude with regards to constants. In [26] a somewhat more delicate analysis is performed, keeping track of the intercepts on the axes of the tangents √ to the concave function t → K(t, x). With this analysis one can achieve the constant 3 + 2 2 + ε for any ε > 0. The main conclusion from the principle of K-divisibility is that K-monotone interpolation spaces coincide exactly with interpolation spaces obtained by the K-method: T HEOREM 6.3 (Brudnyi and Kruglyak [11]). If X is a Gagliardo complete couple then any K-monotone interpolation space is given by a K-method. Suppose Y is a K-monotone interpolation space. The idea of Theorem 6.3 is that one can define a Banach function space E on (0, ∞) by ∞ ∞ yj Y : f (t) K(t, yj ), 0 < t < ∞ . f E = inf j =1

j =1

Now if x ∈ X0 + X1 and K(t, x) ∈ E we can find yj ∈ Y so that K(t, x)

∞

K(t, yj ),

0 < t < ∞,

j =1

and ∞ j =1

yj Y 2K(t, x)E .

Then by K-divisibility (Theorem 6.2) we can decompose x =

∞

j =1 uj

in X0 + X1 so that

K(t, uj ) CK(t, yj ), where C is a universal constant. If Y is K-monotone this implies that uj ∈ Y and that we have an estimate uj Y C1 yj Y for some constant C1 depending on Y . Hence x= ∞ j =1 uj ∈ Y and xY C2 K(t, x)E .

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We say that a Banach couple is a Calderón couple if every interpolation space is K-monotone (or, by Theorem 6.3, every interpolation space is given by a K-method). This terminology is based on the classical Calderón–Mitjagin theorem on interpolation spaces for (L1 , L∞ ). This theorem is in some sense already classical, but we will discuss it below as motivation. Let us first make an equivalent formulation of the problem. Suppose 0 = x ∈ Σ(X). We can define an orbit space for x, Ox namely the space {T x: T ∈ L(X)} with the norm yOx = inf T X : T x = y . Then Ox is an interpolation space. If Ox is monotone then there is a constant C so that for any y satisfying K(t, y) K(t, x) we have T ∈ L(X) with T x = y and T X C. We thus have: P ROPOSITION 6.4. X is a Calderón couple if and only if for every x ∈ Σ(X) there is a constant C = C(x) so that if y ∈ Σ(X) and K(t, y) K(t, x) for 0 < t < ∞ then there exists T ∈ L(X) with T x = y. Somewhat surprisingly it appears to be unknown if the constant C can be chosen independent of x. If this is the case we call X a uniform Calderón couple. We now consider the special case of the pair L = (L1 (R), L∞ (R)) where R is equipped with standard Lebesgue measure. In this case the K-functional is computable and is given by the formula:

t

K(t, f ) =

f ∗ (s) ds,

0

where f ∗ is the decreasing rearrangement of |f |, i.e., the function on (0, ∞) given by f ∗ (s) = sup inf f (u). λ(F )=s u∈F

If we introduce f ∗∗ as usual by setting f ∗∗ (t) =

1 t

t

f ∗ (s) ds

0

then of course K(t, f ) = tf ∗∗ (t). We recall a Banach function space X is symmetric (or a symmetric lattice ideal) if it satisfies the condition that if f ∈ X and g is any measurable function with g ∗ f ∗ it follows that g ∈ X and gX f X . It is not difficult to show any interpolation space for L is a symmetric space. Now for f, g ∈ L∞ + L1 let us say f ≺ g if f ∗∗ (t) g ∗∗ (t) for 0 < t < ∞. In many symmetric Banach function spaces X the property f ∈ X and g ≺ f implies g ∈ X and gX f X ; for example, this holds if X is separable. However it does not hold in


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general, so we shall use the term rearrangement-invariant or r.i. space to mean a symmetric space with this additional property. The following theorem is due to Ryff [103] and Calderón [15]: P ROPOSITION 6.5. If f ≺ g then there is an operator T ∈ L(L) with T L = 1 and Tg = f. T HEOREM 6.6. The interpolation spaces for the couple (L1 , L∞ ) coincide with the r.i. spaces on R and (L1 , L∞ ) is a Calderón couple. Let us remark that this theorem is equally valid for the couple (L1 , L∞ [0, 1]). It is natural then to try to extend this Theorem 6.6 to other function spaces. A major advance was made in this direction by Sparr [106] and [107] (see also [1]): T HEOREM 6.7. The Banach couple (Lp0 (w0 ), Lp1 (w1 )) is a Calderón couple for any choice of 1 pj ∞ and any pair of weight functions wj . On the other hand, Ovchinnikov shows that the pair (L1 + L∞ , L1 ∩ L∞ ) is not a Calderon couple [88]. Indeed Maligranda and Ovchinnikov [78] showed that the Lp ∩ Lp and Lp + Lp are not K-monotone with respect to this couple when 1 < p < ∞, p = p and 1/p + 1/p = 1. However these spaces are complex interpolation spaces for this couple. This raises the general question of classifying pairs of r.i. spaces (X0 , X1 ) on either (0, 1) or (0, ∞) which form Calderón couples. For special examples (certain types of Lorentz spaces and Marcinkiewicz spaces) positive results were obtained by Cwikel [23] and Merucci [81]. In [56] a full study of this problem was undertaken and although the results are not complete, a good description was obtained for sufficiently “separated” pairs of spaces, in a sense to be described. Curiously enough some of the properties which surface in the characterization have a flavor suggestive of Banach space theory. We first need to introduce some standard ideas (see [72], for example). If X is an r.i. space on [0, 1] or [0, ∞) then the dilation operators Da on X are then defined by Da f (t) = f (t/a) (where we regard f as vanishing outside [0, 1] in the former case). We can then define the Boyd indices pX and qX of X by log a a→∞ log Da X

pX = lim and qX = lim

a→0

log a . log Da X

In many texts, the reciprocals of pX and qX are used for the Boyd indices following the original convention of Boyd [10]. The Boyd indices are of course extremely useful in interpolation theory because of the classical Boyd interpolation theorem [10], which we will discuss in Section 7.

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For convenience we restrict our discussion to the case of r.i. spaces over (0, ∞). Let en = χ(2n ,2n+1 ) . Associated to each r.i. space X we can introduce a Banach sequence space SX modelled on Z defined by ξ = (ξn )n∈Z if and only if n∈Z ξn en ∈ X and ξ e ξ SX = n n . X

n∈Z

Now it is clear from consideration of averaging projections that (X, Y ) is a Calderón couple if and only if (SX , SY ) is also a Calderón couple. It turns out we can answer this question under separation conditions on the Boyd indices in terms of some conditions with a Banach space flavor. Let us suppose that E is a Banach sequence space modelled on Z. N We shall say that E has the right shift-property (RSP) if whenever (xn )N n=1 , (yn )n=1 are two sequences satisfying (1) supp x1 < supp y1 < supp x2 < · · · < supp xn < suppyn , (2) yn E xn E , n = 1, 2, . . . , N , then N N yn C xn . n=1

n=1

N Similarly we say E has the left-shift property (LSP) if whenever (xn )N n=1 , (yn )n=1 are two sequences satisfying (1) supp x1 > supp y1 > supp x2 > · · · > supp xn > supp yn , (2) yn E xn E , n = 1, 2, . . . , N , then

N N yn C xn . n=1

n=1

We then say that an r.i. space X on (0, ∞) is stretchable if SX has (RSP) and compressible if SX has (LSP). If X is both stretchable and compressible then X is elastic. The main theorems of [56] then assert the following: T HEOREM 6.8. Let (X, Y ) be a pair of r.i. spaces on (0, ∞) such that pY > qX . Then (X, Y ) is a (uniform) Calderón couple if and only if X is stretchable and Y is compressible. T HEOREM 6.9. Let X be an r.i. space on (0, ∞); then (X, L∞ ) is a Calderón couple if and only if X is stretchable. Of course the condition pY > qX is a quite strong separation condition on the Boyd indices; it asserts that the intervals [pX , qX ] and [pY , qY ] do not intersect. A remarkable feature of the conclusion of Theorem 6.8 is that the condition that X is stretchable (or Y is


1153

compressible) is independent of the normalization of en . To illustrate this note that if X is a Lorentz space with norm

f ∗ (t)p w(t) dt

∞

f X =

1/p

0

for a suitable decreasing weight function w then X is always elastic because SX up to normalization is simply p . The above Theorems 6.8 and 6.9 have similar statements when (0, ∞) is replaced by [0, 1] and for sequence spaces. It is necessary simply to formulate the shift properties on sequence spaces modelled on N or Z \ N. We refer to [56] for details. It is possible to give a rather complicated characterization of stretchable and compressible Orlicz spaces. In fact for Orlicz spaces the conditions are equivalent and any such space is elastic. The following theorem is given in [56] (we specialize to [0, 1] for definiteness). T HEOREM 6.10. Let F be an Orlicz function. Then the following conditions on F are equivalent: (1) LF [0, 1] is elastic (respectively, stretchable, respectively, compressible). (2) (L∞ [0, 1], LF [0, 1]) is a Calderón couple. (3) There is a bounded monotone increasing function w : [1, ∞) → R and a constant C so that if 1 s t and 0 < x 1 we have F (tx) F (sx) C + w(t) − w(s). F (t) F (s) (4) There is a bounded monotone increasing function w : [1, ∞) → R and a constant C so that if 1 s t and 0 < x 1 we have F (tx) F (sx) C + w(t) − w(s). F (s) F (t) These conditions are a little difficult to check. They are related to somewhat similar criteria for Orlicz spaces to coincide with Lorentz spaces [84]. Perhaps the simplest practical condition which follows is the following. T HEOREM 6.11. Let X be an Orlicz space on [0, 1]. Then if (X, L∞ [0, 1]) is a Calderón couple we have pX = qX . This allows the construction of some very easy counter-examples to the conjecture that (LF [0, 1], L∞ [0, 1]) is a Calderón couple for every Orlicz function F . In spite of the difficulty in classifying Calderón couples, there is a form of converse to the theorem of Sparr, obtained by Cwikel and Nilsson [31]. Here we consider all possible changes of density. If X is a Banach function space we define X(w) = {f : f w ∈ X} with f X(w) = f wX , where w is a weight function (a strictly positive measurable function).

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T HEOREM 6.12. Let (X0 , X1 ) be a pair of Banach function spaces on [0, 1] or [0, ∞). Suppose that for every pair of weight functions the pair (X0 (w0 ), X1 (w1 )) is a Calderón couple. Then there exist 1 p0 , p1 ∞ and weight functions v0 , v1 so that X0 = Lp0 (v0 ) and X1 = Lp1 (v1 ) up to equivalence of norm. Finally let us note a problem raised by Cwikel which was solved in [80]. Cwikel asked if a pair (X, Y ) of complex Banach spaces is a Calderón couple if and only if every complex interpolation space is K-monotone. In [80] counter-examples are exhibited even for pairs of r.i. spaces. 7. Interpolation spaces for (Lp , Lq ) Throughout this section we will suppose that our rearrangement invariant spaces are over [0, 1] or [0, ∞). Let us start by noting that the K-functional for (Lp , Lq ) can be approximated in terms of the Hardy operators. To this end, we have the following formula of Holmstedt [47]: t

−1/p

K f, t 1/p−1/q ≈

t 1/p ∞ 1/q 1 1 ∗ p ∗ q f (s) ds + f (s) ds . t 0 t t

This formula, combined with the fact that (Lp , Lq ) is a Calderón couple, can be used to obtain useful results. For example, it is now easy to prove the following interpolation result [46]. Given an Orlicz function Φ, we will say that Φ is p-convex if the map t → Φ(t 1/p ) is convex, and q-concave if the map t → Φ(t 1/q ) is concave (we will say that all functions are ∞-concave). T HEOREM 7.1. Let 1 p < q ∞, and let X be an interpolation space for (Lp , Lq ). Then there is a positive constant c such that the following holds. If f, g are functions such that gΦ f Φ for every function Φ that is p-convex and q-concave, and if f ∈ X, then g ∈ X with gX c f X . In [61], the authors were able to obtain the following characterization of interpolation spaces for (Lp , Lq ). In order to state this result, we need the notion of conditional expectation. On [0, 1], this is standard. On [0, ∞) the same construction works, as long as the σ -field’s atoms all have finite measure. T HEOREM 7.2. Let 1 p < q ∞. A rearrangement invariant space X is an interpolation space for (Lp , Lq ) if and only if there is a positive constant c such that for any function f , and any sub-σ -field M whose atoms have finite measure, we have that p

1/p E |f | |M c f X X and if q < ∞

1/q . f X c E |f |q |M X


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This gives the following collection of sufficient conditions, the fourth of which is the classical Boyd interpolation theorem [10], which is also proved in [72] (see also [86]). T HEOREM 7.3. Let X be a rearrangement invariant space, and 1 p < q ∞ Suppose that any of the following hold: • X is p-convex and q-concave; • X is p-convex and has upper Boyd index less than q; • X is q-concave and has lower Boyd index greater than p; • X has Boyd indices strictly between p and q. Then X is an interpolation space for (Lp , Lq ). Finally we end this section with some results about the span of the Rademacher series in rearrangement invariant spaces. That is, given a rearrangement invariant space L on [0, 1], we can form a sequence space RL which is the space of sequences (an ) whose norm ∞ n=1 an rn L is finite. (Here (rn ) denotes the sequence of Rademacher functions on [0, 1].) It was shown by Rodin and Semenov [102] that RL is isomorphic to the space 2 if and only if L contains the space G, where G is the closure of the simple functions in the Orlicz space derived from an Orlicz function equivalent to exp(x 2 ). They went on to calculate the space RL for some lattices that do not contain G. More recently this work was extended by Astashkin [2]. One of the main results of this paper can be summarized as follows. T HEOREM 7.4. A symmetric sequence space S is naturally isomorphic to RL for some rearrangement invariant space L on [0, 1] if and only if S is an interpolation space for the couple (1 , 2 ).

8. Extensions Let us briefly describe the elements of the theory of extensions (or twisted sums). This discussion overlaps the discussion in [60] but here our emphasis is slightly different. We note that a good reference for the general theory of twisted sums in the context of Banach space theory is [18]. Let X and Y be a Banach spaces (or more generally quasi-Banach spaces). An extension of X by Y is (formally) a short exact sequence 0 → Y → Z → X → 0, where Z is a quasi-Banach space. Less formally we regard the space Z is an extension of X by Y if Z ⊃ Y and Z/Y is isomorphic to X. One can of course restrict extensions to lie in the category of Banach spaces. There is a general construction of extensions via quasi-linear maps. Let V be any vector space containing Y (we may take Y = V but some flexibility is useful here.) A map Ω : X → V is called quasi-linear if

1156


• Ω(λx) = λΩ(x), x ∈ X, λ ∈ K. • There is a constant C so that if x1 , x2 ∈ X then Ω(x1 + x2 ) − Ω(x1) − Ω(x2) ∈ Y and

Ω(x1 + x2 ) − Ω(x1 ) − Ω(x2) C x1 + x2 .

(8.1)

We then can define an extension X ⊕Ω Y to be the subspace of X ⊕ V of all (x, v) such that v − Ωx ∈ Y , and equipped with the quasi-norm (x, v) = x + v − Ωx. In general this is not a norm, but it will be equivalent to a norm if it satisfies an estimate of the form n n n (xj , vj ), x1 , . . . , xn ∈ X, v1 , . . . , vn ∈ V . xj , vj C j =1

j =1

j =1

It follows that X ⊕Ω Y is isomorphic to a Banach space if (and only if) (8.1) is replaced by the stronger inequality: n n n Ωxk − Ω xk C xk . k=1

k=1

(8.2)

k=1

In fact, in the above construction, it is only necessary that Ω be defined on a dense linear subspace; the construction above yields a space whose completion is an extension. Now it is a key fact that every extension can be represented in this form. Indeed if Z is an extension of X we can define two maps F : X → Z and L : X → Z such that qF = qL = IX where q is the quotient map. F is defined to be homogeneous (not necessarily linear) and satisfy F (x) 2x, while L is required to be linear (but not necessarily bounded). If we set Ω(x) = F (x) − L(x) then Ω : X → Y is quasilinear and one can easily set up a natural isomorphism between Z and X ⊕Ω Y . Notice, however, that the choice of Ω depends heavily on certain arbitrary choices (e.g., of the linear map L). We refer to an extension Z of X as trivial if there is a bounded projection of Z onto X. In this case Z splits as a direct sum X ⊕ Y . It is easy to show that X ⊕Ω Y splits if and only if there is a linear map L : X → V so that Ωx − Lx ∈ Y for all x and Ωx − Lx Cx,

x ∈ X.

In [60] we discussed the case of minimal extensions. A minimal extension is an extension by the scalar field K. In this case all Banach extensions are trivial, by the Hahn–Banach theorem. A Banach space X is called a K-space if all minimal extensions of X are trivial. The following proposition is then very useful: P ROPOSITION 8.1 ([51]). X is a K-space if and only if any extension of X by a Banach space is (isomorphic to) a Banach space.


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It is conjectured (see [60]) that X is a K-space if and only if X∗ has non-trivial cotype. It is known that any space with non-trivial type is a K-space. An extension of X by X is called a self-extension; in this case we introduce the notation dΩ X = X ⊕Ω X. We will see shortly that these are intimately related with interpolation theory, but first let us discuss the historical origins of the study of self-extensions.

9. Self-extensions of Hilbert spaces A self-extension of a Hilbert space is called a twisted Hilbert space. The basic question of the existence of a non-trivial twisted Hilbert space was apparently first raised by Palais. It was solved in 1975 by Enflo, Lindenstrauss and Pisier [39] who produced the first nontrivial example of a twisted Hilbert space. A few years later in [62] an alternative example was constructed based on the ideas of Ribe’s construction of a non-trivial minimal extension of 1 (see [99] and [60]). We will discuss this example and some variants in this section. It is interesting that the link between minimal extensions of 1 and self-extensions of 2 is now much better understood than it was in 1979, and we will explain this connection later. Let us recall that Ribe’s space is associated to the quasilinear map Ω : c00 → R given by ∞ ∞ ∞ (9.1) ξn log |ξn | − ξn log ξn . Ωξ = n=1

n=1

n=1

We define a corresponding self-extension of 2 , denoted by Z2 by taking Ω : 2 → ω (ω is the space of all sequences) as |ξn | ∞ . Ωξ = ξn log ξ 2 n=1 Here we interpret 0 log 0 and 0 log 0/0 as 0. Thus Z2 is the space of pairs of sequences ∞ ((ξn )∞ n=1 , (ηn )n=1 ) such that (ξ, η) =

∞ n=1

1/2 |ξn |2

+

2 1/2 ∞ | |ξ n ηn − ξn log < ∞. ξ 2 n=1

This equation defines a quasi-norm; the fact that it is equivalent to a norm (and thus Z2 is a genuine Banach self-extension) follows from Proposition 8.1. The Banach space properties of the space Z2 are of some interest. It is immediate that there is a natural unconditional Schauder decomposition into two-dimensional spaces and it is shown in [62] that it has no unconditional basis; in [50] it is shown to fail local unconditional structure as well. In fact in [17] it is shown that any space with a two-dimensional UFDD (En ) (or even a UFDD with bounded dimensions) with local unconditional structure has an unconditional basis which can be chosen from the subspaces. The main unresolved problems concerning Z2 are:

1158


• Is Z2 prime? • Is Z2 isomorphic to its hyperplanes. It has been widely conjectured that Z2 is not isomorphic to its hyperplanes (in some sense, Z2 is even-dimensional and its hyperplanes are odd-dimensional!). Of course, since the celebrated example of Gowers [43], this problem is less pressing. Let us notice (as in [62]) that this construction can be generalized quite a bit. Let F : R → R be any Lipschitz map, and let E be any Banach sequence space and define ∞ |ξn | ΩF (ξ ) = ξn F log . ξ E n=1 Then ΩF induces a self-extension dΩF E of E. In fact one can go further and consider complex sequence spaces and then allow F : R → C. In [57] this idea was exploited taking E = 2 and F (t) = t 1+iα . This produces a complex Banach space Z2 (α) which is not isomorphic to its conjugate space. The conjugate space of a complex Banach space X is ¯ In this case it is not difficult the space X on which multiplication is defined by λ × x = λx. to see that the conjugate space of Z2 (α) is isomorphic to Z2 (−α) and then it can be shown without undue difficulty that Z2 (α) and Z2 (−α) are not isomorphic as complex Banach spaces; see [57] or [6]. Earlier examples had been constructed by probabilistic methods by Bourgain [9] and Szarek [108].

10. Analytic families of Banach spaces In this section we sketch the origins of the theory of non-linear commutators and analytic families and how it relates to the preceding examples. Let us introduce the idea of an analytic family of Banach spaces. To do this we will abstract the ideas of complex interpolation introduced in Section 4; this has the added convenience of incorporating the description of interpolating families of spaces by Coifman, Cwikel, Rochberg, Sagher and Weiss [20]. Let us suppose U is an open subset of the complex plane conformally equivalent to the open unit disk D; in fact we need only consider the case U = D and U = S := {z: 0 < !z < 1}. Next let W be some complex Banach space (the ambient space) and let F be a Banach space of analytic functions F : U → W . We assume that F is equipped with a norm · F such that: • The evaluation map F → F (z) (F → W ) is bounded for each z ∈ U . • If ϕ : U → D is a conformal equivalence then F ∈ F if and only if ϕF ∈ F and F F = ϕF F . We will call such a space F admissible. Then for z ∈ U and x ∈ W we define xz = inf F F : F (z) = x and let Xz = x ∈ W ; xz < ∞ .


1159

The family of spaces (Xz )z∈U is then called an analytic family of Banach spaces. If W0 is the linear span of the spaces {Xz : z ∈ U} then a linear map T : W0 → W0 will be called interpolating if F → T ◦ F is defined and bounded on F . It then follows that T (Xz ) ⊂ Xz for each z ∈ U and T Xz →Xz T ◦ F F →F . If we take a Banach couple X = (X0 , X1 ) and define F as in Section 4 then one may see that {Xz : z ∈ U} is an analytic family and Xz = X!z where Xθ is the complex interpolation space between X0 and X1 . Thus our definition abstracts the ideas of complex interpolation. Under these assumptions any T ∈ L(X, Y ) is interpolating. The upper method also yields an analytic family at least when X is Gagliardo complete. Let us note that it is possible to describe the ideas of this section in much more generality, by relaxing our assumptions on F , so that real and other methods may be included; we refer to [28] for a fuller discussion, using an annulus in place of the disk. To keep our discussion reasonably crisp we will retain our much stronger conditions. We now invoke ideas of Rochberg and Weiss [101] (which in embryonic form appear in work of Schechter [104]). For each z we define a derived space dXz ⊂ W × W by dXz = {(x1 , x2 ): (x1 , x2 )dXz < ∞} where (x1 , x2 ) (10.1) = inf F F ; F (z) = x1 , F (z) = x2 . dXz Let Y be the subspace of dXz defined by x1 = 0. We claim that Y is an isometric copy of Xz . Indeed let ϕ be a conformal map of U onto D with ϕ(z) = 0. Then if F (z) = 0 we can write F = ϕG where GF = F F . Then F (z) = ϕ (z)G(z) and so −1 (0, x2 ) = ϕ (z) x2 Xz . dXz On the other hand dXz /Y is trivially isometric to Xz so that we have a short exact sequence 0 → Xz → dXz → Xz → 0 and dXz is a self-extension of Xz . Thus we can use the ideas from Section 8 to give a representation of dXz in the form dΩ Xz where Ω : Xz → W is a quasilinear map. It is easy enough to see that an appropriate Ω is given by Ω(x) = F (z) where F is any choice of F ∈ F with F (z) = x and F F Cxz . In many circumstances there is a unique optimal choice of F with F F = xz and in this case one can define Ω in a very natural way. In general, there is some arbitrariness in the definition of Ω but any two such choices differ by a bounded function. Thus we have (x1 , x2 ) ≈ x1 z + x2 − Ωx1z . dXz Rochberg and Weiss used this construction to obtain commutator estimates. If T is an interpolating operator then (x1 , x2 ) → (T x1 , T x2 ) is bounded on dXz and this implies

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P ROPOSITION 10.1. If T is an interpolating operator then there is a constant C so that if x ∈ Xz then [T , Ω]x = T Ωx − ΩT x ∈ Xz and [T , Ω]x Cxz . z To conclude this section, let consider the case of interpolation of p -spaces. Suppose 1 p1 < p0 ∞. We will take W = ω, the space of all complex sequences, as our ambient space. Let us consider the space G of analytic functions on the strip S of the form F (z) = ∞ (fk (z))∞ k=1 where (fk )k=1 are bounded analytic functions. We can then extend each fk a.e. to the boundary of the strip by taking non-tangential limits, i.e., fk (j + it) = lim fk (x + it), x→j

j = 0, 1.

Now define F to be the space of all F ∈ G so that F (j + it) ∈ pj a.e. for j = 0, 1 and F F = max

ess sup F (j + it)p

j =0,1 −∞ 0. Hence differentiating uk =0

uk

ξk − xk 0. xk

Let xk∗ = uk /xk if uk = 0 and 0 otherwise. Then ∞

ξk xk∗

k=1

so that x ∗ X∗ 1.

∞ k=1

uk = 1


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Uniqueness is immediate since if u = xx ∗ = yy ∗ where xX = yX = x ∗ X∗ = y ∗ Y ∗ and x, x ∗ , y, y ∗ 0 then ∞ 1 k=1

4

(xk + yk ) xk∗ + yk∗ 1

and this can only happen if xk = yk and xk∗ = yk∗ whenever uk = 0. + Note that this implies that if u ∈ c00 , ΦX (u) =

∞

uk log |xk |,

k=1

where x ∈ BX is determined by the Lozanovsky factorization of u/u1 . We can exploit to canonically extend ΦX to c00 by defining ΦX (u) =

∞

uk log |xk |,

k=1

where x is given by the Lozanovsky factorization of |u|/u1 . This definition of ΦX is also quasi-linear on c00 . Next we turn to complex interpolation of Banach sequence spaces, using the upper method which may be formulated as described at the end of Section 10. If X0 and X1 are two Banach sequence spaces we define F to be the subset of G of functions such that F F := max ess sup F (j + it)X < ∞ j =0,1 −∞ 1/2 ∗ ∗ ∗ ∗ ∗ say for some x ∈ AN . It follows that the norms |x|N = sup{|x (x)|: x ∈ AN } arbitrarily distort [(yi )]. (b) The technique used to show that X must contain an unconditional basic sequence is indirect. One shows that X contains basic sequences of unbounded order in terms of their unconditional structure. Given K < ∞ let T (X, K) be the set of all normalized finite basic sequences (xi )n1 ⊆ X which are K-unconditional. Then T (X, K) is naturally a tree under (xi )n1 (yi )m 1 if n m and xi = yi for i n. If X does not contain an unconditional basic sequence then T (X, K) is well founded (the tree has no infinite branches). Set T (0) = T (X, K), T (α+1) = {(xi )n1 : there exists xn+1 so that (xi )n+1 ∈ T (α) } and 1 (β) (α) T = α 0, k ∈ N and x ∈ X there exists k0 ∈ N so that if k0 n1 < n2 < · · · < nk then k k αi ei − x + αi xi < ε x + i=1

i=1

whenever xi ∈ SGni and |αi | 1 for i k. Recently a second finite- infinite-dimensional bridge structure was defined ([34,30]). We first examine the simplest version of this notion. Suppose that X has a basis or more n generally an FDD (finite-dimensional decomposition) (Ei )∞ i=1 . For n ∈ N we say (xi )1 ∈ n {X, (Ei )}n , the n-th asymptotic structure of X w.r.t. (Ei ), if (xi )1 is a normalized basic sequence with the property that: ∀ε > 0 ∀k1 ∃y1 ∈ SEi ∞ ∀k2 ∃y2 ∈ SEi ∞ · · · ∀kn ∃yn ∈ SEi ∞ k k k 1

2

n

with (xi )n1 (1 + ε)-equivalent to (yi )n1 . Thus (xi )n1 can be found (up to ε) in X arbitrarily far out and arbitrarily separated w.r.t. n m (Ei ). Note that if (zi )m 1 is a normalized block basis of (xi )1 ∈ {X, (Ei )}n then (zi )1 ∈ {X, (Ei )}m . It follows by the existence of spreading models and Krivine’s theorem that there exists p ∈ [1, ∞] so that the unit vector basis of np belongs to {X, (Ei )}n for n ∈ N. P ROPOSITION 27. Let (Ei ) be an FDD for X and suppose that |{X, (Ei )}2 | = 1. Then there exists p ∈ [1, ∞] so that for all n if (xi )n1 ∈ {X, (Ei )}n then (xi )n1 is 1-equivalent to the unit vector basis of np . Moreover X contains almost isometric copies of p , if 1 p < ∞ or of c0 if p = ∞. The first part is easy given our previous remarks. The “moreover” statement is also not difficult to prove directly (see [30]) but will in fact follows from Theorem 30 below. Asymptotic structure can also be understood in terms of trees on X. Let Tk = {(n1 , . . . , nk ): ni ∈ N}. τ ∈ Tk (X, (Ei )) if τ = {(xn1 ,...,ni ): (n1 , . . . , ni ) ∈ Tk } ⊆ SX , j ∞ (xn )∞ n=1 is a block basis of (Ei ) and for every 1 j < k and (ni )1 ∈ Tk , (xn1 ,...,nj ,n )i=1 is a normalized block basis of (Ei ). We shall say that τ ∈ Tk (X, (Ei )) converges to (xi )k1 if (xi )k1 is a normalized basic sequence and for some εi ↓ 0 for all (n1 , . . . , nk ) ∈ Tk , (xn1 ,...,ni )ki=1 is (1 + εn1 )-equivalent to (xi )k1 . P ROPOSITION 28. Let (Ei ) be an FDD for X. Let k ∈ N. Then (xi )k1 ∈ {X, (Ei )}k iff there exists a tree τ ∈ Tk (X, (Ei )) which converges to (xi )k1 . The proposition follows easily from the relevant definitions. The asymptotic structure of X may also be characterized in terms of trees as follows.

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E. Odell and Th. Schlumprecht

P ROPOSITION 29. {X, (Ei )}k is the smallest class C of normalized bases of length k having the property that if τ ∈ Tk (X, (Ei )) and ε > 0 then some branch of τ is (1 + ε)equivalent to a member of C. This follows from the fact that if τ ∈ Tk (X, (Ei )) then there exists a convergent subtree τ ⊆ τ , τ ∈ Tk (X, (Ei )). This latter fact can be proved using Ramsey theory (see [22]). Another interpretation of asymptotic structure is given by the next theorem. Recall that kn+1 for all n and some sequence of integers (Fi ) is a blocking of (Ei ) if Fn = Ei i=k n +1 0 = k1 < k2 < · · ·. (Fi ) is a skipped blocking of (Ei ) if there exist integers 1 p1 q1 < qn q1 + 1 < p2 q2 < q2 + 1 < p3 · · · so that Fn = Ei i=p for n ∈ N. n T HEOREM 30. Let εi ↓ 0. There exists a blocking (Hi ) of (Ei ) so that for all k if (Fi )k1 is k any skipped blocking of (Hi )∞ k and xi ∈ SFi for i k then (xi )1 is (1 + εk )-equivalent to an element of {X, (Ei )}k . P ROOF. By a diagonal argument it suffices to produce for a fixed k ∈ N and ε > 0 a blocking (Hi ) of (Ei ) so that any normalized block basis (xi )k1 relative to any skipped blocking (Fi )k1 of (Hi )∞ 2 is (1 + ε)-equivalent to an element of {X, (Ei )}k . By a standard compactness argument one need only show the validity of the following sentence. ∃N2 ∀x2 ∈ SEi ∞ · · · ∃Nk ∀xi ∈ SEi ∞ , (xi )k1 is (1 + ε)-equivalent to “∃N1 ∀x1 ∈ SEi ∞ N1 N2 Nk an element in {X, (Ei )}k .” If false by formally negating the sentence one easily constructs a tree τ ∈ Tk (X, (Ei )) so that no branch of τ is (1 + ε)-equivalent to any element of {X, (Ei )}k . Proposition 29 then yields a contradiction. Let (Ei ) be an FDD for X. We shall say that X is asymptotic p w.r.t. (Ei ) if there exists K < ∞ so that for all k and (xi )k1 ∈ {X, (Ei )}k , d(xi k1 , kp ) K. This is formally weaker than assuming (xi )k1 is K-equivalent to the unit vector basis of kp but as observed in [30] the weaker assumption implies the stronger at least if 1 < p < ∞ (the case p = 1 or ∞ remains open). T HEOREM 31. If X is asymptotic p w.r.t. the FDD (Ei ) with 1 < p < ∞ then there exists K < ∞ so that for all k, (xi )k1 is K-equivalent to the unit vector basis of kp for all (xi )k1 ∈ {X, (Ei )}k . Also one has a nice duality result. Let us note that the asymptotic structure we have discussed w.r.t. an FDD can be and is indeed done in a more general context in [30], e.g., with respect to fundamental, total minimal systems and the next theorem is valid in that broader context. T HEOREM 32. Let 1 p ∞ and let X be a reflexive asymptotic p space w.r.t. the FDD(Ei ). Then X∗ is asymptotic q w.r.t. (Ei∗ ) where 1/p + 1/q = 1.

Distortion and asymptotic structure

1355

One can also consider infinite-dimensional spaces which reflect the asymptotic structure of X. A space Y with a normalized basis (yi ) is an asymptotic version of X if (yi )n1 ∈ {X, (Ei )}n for all n. This includes the class of all spreading models of normalized block bases of (Ei ). Moreover one can show [30] that there exists such a Y which satisfies {Y, (yi )}k = {X, (Ei )}k for all k ∈ N. (Y is called a universal asymptotic version of X.) The asymptotic structure of a space can be stabilized. P ROPOSITION 33. Let (Ei ) be an FDD for X. There exists a normalized block basis (yi ) of (Ei ) so that if Y = [(yi )] and Z = [(Hi )] where (Hi ) is any FDD obtained by blocking a block basis of (yi ) then for all k Y, (yi ) k = Z, (Hi ) k . One may ask, how small must this stabilized asymptotic structure be? The answer is not very. T HEOREM 34 ([43]). There exists a normalized monotone basis (ei ) for a reflexive space X with the following property. For all k, all normalized monotone bases (xi )k1 and all Y = [(yi )] where (yi )∞ 1 is a block basis of (ei ), (xi )k1 ∈ Y, (yi ) k . In particular X cannot contain an unconditional basic sequence. The example is technically difficult. We will not present the argument but shall present the norm. This gives the flavor of both the construction and of the possibilities afforded by generalizations of conditional Tsirelson-type norms. It is worth noting that a somewhat simpler example is given in [40]. In this case the basis (ei ) is unconditional and the unit vector basis of kp belongs to {X, (ei )}k for all k and p ∈ [1, ∞]. H ⊆ c00 ∩ B∞ is taken to be a countable set of non-zero vectors with three properties: (i) H is dense in B∞ ∩ c00 w.r.t. · 1 . (ii) ∀a ∈ H and intervals I of integers, Ia ∈ H if I a = 0. n 1 n 1 (iii) If a1 < · · · < an in H then ni=1 ai , f (n) i=1 ai and n i=1 ai all belong to H . ∞ A subsequence M = (Mn )n=1 ⊆ N is taken with M1 = 2 and we let σ : (a1 , . . . , an ): n ∈ N, a1 < · · · < an , ai ∈ H for i n → N be an injection satisfying 4 more properties. (iv) If a1 < · · · < an belong to H and I is an interval in N and [j1 , j2 ] = {i: I ai = 0, i = n} = ∅, then σ (I aj1 , . . . , I aj2 ) σ (a1 , . . . , an ). (v) If a1 < · · · < an belong to H then max supp an1< σ (a1 , . . . , an ). < ∞ where as before f (n) = (vi) If Im (σ ) is the range of σ then n∈Im (σ ) f (n) log2 (n + 1).

1356


¯¯ are the predecessor and successor of m in Im (σ ) then for (vii) If m ∈ Im (σ ) and m, ¯ m ¯¯ ∞) ∈ [1, m] ¯ ∪ [m, ' 3 <m 1 f (m) min(, m) f (m) f () , + + 3f (m) 2 m f () m f () m , > m. &

If · ∈ N and X = (c00 , · ) for m 2 we set m 1 ∗ ∗ ∗ ∗ = ai : ai ∈ H ∩ BX∗ for i m and a1 < · · · < am , m i=1 X ∞ X X m A = AX A AX for m 2 and AX = A . m, m= n m=2

AX m

m2

nm

m ∗ X ={ 1 ∗ ∗ X X Let for m 2, Bm i=1 ai : (a1 , . . . , am ) ⊆ A is (A , M, σ )-admisf (m) sible}. ∗ ) is ( AX , M, σ )-admissible means that a1∗ < The statement that (a1∗ , . . . , am ∗ X ∗ ∗ X ∗ · · · < am , a1 ∈ iMm Ai and ai+1 ∈ A σ (a1 ,...,ai∗ ) for 1 i < m. We take ∞ X X X X ∞ B = n=2 Bn and B = (Bn )n=2 . For m 1 set X Cm

=

m X

1 ∗ ∗ ∗ X ai : a1 , . . . , am ⊆ B is B , M, σ -admissible and f (m) i=1

CX =

∞

CnX .

n=1

Then one uses Proposition 11 to show that there exists · ∈ N so that X = (c00 , · ) satisfies for all x ∈ c00 , (viii) x = max(x∞ , sup{|a ∗ (x)|: a ∗ ∈ C X }. This is the space which yields Theorem 34. Asymptotic structure has been generalized in several ways. Milman and Wagner have extended the notion to operators [35]. Also Wagner [55] has given a higher-order ordinal notion in terms of certain α-games for α < w1 . The definition of {X, (Ei )}k yields this game for α = k. And of course one need not assume that the space X has an FDD. Indeed [30] consider the broader forum where the tail spaces of an FDD are replaced by finite codimensional subspaces in any non-trivial filtration on X. Γ ⊆ cof(X), the set of all finite codimensional subspaces of X, is called a filtration on X if for all Y, Z ∈ Γ there exists W ∈ Γ with W ⊆ Y ∩ Z. One then has (ei )k1 ∈ {X, Γ }k if ∀ε > 0 ∀Y1 ∈ Γ ∃y1 ∈ SY1 · · · ∀Yk ∈ Γ ∃yk ∈ SY1 so that (yi )k1 is the (1 + ε)-equivalent to (ei )k1 .


1357

As noted in [30] this can be expressed in terms of a game where Player I chooses Y ∈ Γ and Player II chooses y ∈ SY with each player making k alternate plays starting with I. Thus (ei )k1 ∈ {X, Γ }k iff for all ε > 0 Player II has a winning strategy for the set of all normalized bases (1 + ε)-equivalent to (ei )k1 . By regarding X ⊆ [En ], some FDD, and Γ = {X ∩ [Ei ]∞ n : n ∈ N} one obtains a relativized notion of asymptotic structure w.r.t. an FDD and the relativized versions of the previous structural results remain valid [22]. Working with this Γ is nice because one has a coordinate system. What happens however for Γ = cof(X)? And is there an infinite version of asymptotic structure? These are addressed in [44] and we now discuss some of the results contained therein. We consider the two-player game where Player I chooses Y1 ∈ cof(X) and then Player II chooses y1 ∈ SY1 and then Player I chooses Y2 ∈ cof(X) and so on. Player I wins the ω ≡ {(x )∞ : x ∈ S for all i} if (y ) ∈ A. One can define in A-game for a given A ⊆ SX i 1 i X i a natural way what it means to say that Player I has a winning strategy for A and we denote this by WI (A). For ε > 0 we let Aε = {(yi ) ⊆ SX : there exists (xi ) ∈ A with ω , given the xi − yi < ε/2i for all i ∈ N} and we let Aε be the closure of this set in SX product topology of the discrete topology on SX . ω . There exists a space Z with an FDD (E ) so that X ⊆ Z and T HEOREM 35. Let A ⊆ SX i such that the following are equivalent. (a) For all ε > 0, (WI (Aε )). ω (b) For all ε > 0 there exists a blocking (Gi ) of (Ei ) and δi ↓ 0 so that: if (xn ) ∈ SX and there exist integers 1 = k0 < k1 < · · · with

Id −P kn −1 (xn ) < δn [G ] j j=k n−1 +1

for all n then (xn ) ∈ Aε . Moreover if X∗ is separable, (En ) can be chosen to be shrinking and if X is reflexive, Z can be chosen to be reflexive. In these cases (a) is equivalent to (c) Every weakly null tree T ∈ Tω (X) has a branch in Aε . The hypothesis on T in (c) means that T = (x(n1 ,...,nk ) : n1 < · · · < nk are positive integers) ⊆ SX and the successors of every node, including φ, form a weakly null sequence. ω : (x ) is K-equivalent to the unit vector This theorem can be applied to A = {(xi ) ∈ SX i basis of p } to yield the following.

T HEOREM 36. Let X be reflexive and let 1 0 there exists a finite codimensional subspace X0 of X which (C 2 + ε)-embeds into ( Fi )p for some sequence (Fi ) of finite-dimensional spaces. This theorem generalizes results of [21]. A similar theorem for the p = ∞ case was proved by Kalton [20]: if X does not contain 1 and for some K < ∞ every weakly null tree T ∈ Tω (X) admits a branch K-equivalent to the unit vector basis of c0 , then X embeds into c0 .

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The proofs of these theorems use Ramsey theory and Martin’s theorem that Borel games are determined [27]. Unlike the finite asymptotic structure case there is in general no smallest closed set A of normalized bases so that every weakly null tree T ∈ Tω (X) has a branch nearly in A. So there is no unique notion of infinite asymptotic structure, but Theorem 35 does allow one to say something useful. Finally what can be said if the asymptotic structure is as small as possible, either in the [30] sense or in the sense of spreading models? Proposition 28 yielded some information but using the above results one can say more. Recall that {X, cof(X)}2 denotes the asymptotic structure (of length 2) w.r.t. filtration of all finite codimensional subspaces of X. T HEOREM 37. Let X be reflexive and let |{X, cof(X)}2 | = 1. Then there exists 1 0, some finite codimensional subspace of X (1 + ε)-embeds into the p sum of finite-dimensional spaces. If X has a basis (xi ) and there is only one spreading model (ei ) that can be obtained as a spreading model of a block basis of (ei ) then, by the proof of Krivine’s theorem, one obtains that (ei ) is 1-equivalent to the unit vector basis of c0 or p for some 1 p < ∞. T HEOREM 38. Let (xi ) be a basis for X and assume that all spreading models of a normalized block basis of (xi ) are 1-equivalent to the unit vector basis of 1 (respectively, c0 ). Then X contains an isomorph of 1 (respectively, c0 ). It is still open if the theorem extends to p (1 < p < ∞). There is no isomorphic version of this theorem. For example, all spreading models of T are 2-equivalent to the unit vector basis of 1 yet T does not contain 1 . Most recently a third notion has been constructed which generalizes spreading models [17]. Spreading models are generated by basic sequences. Suppose that (xin )n,i∈N is a normalized array in a Banach space X so that for some K < ∞ each row (xin )i∈N is j K-basic and for all n i1 < · · · < in , (xij )nj=1 is K-basic. Then given ε ↓ 0 one can pron for some p(1) < p(2) < · · ·, so that for all n, duce a subarray (yin ), given by yin = xp(i) n i1 < · · · < in and n 1 < · · · < n n n j j aj yij − aj yj < εn , j =1

j =1

if (aj )n1 ⊆ [−1, 1]. The proof uses Ramsey’s theorem. It follows that n n j aj xij ≡ a j ej lim · · · lim i1 →∞ in →∞ j =1

j =1

exists. (ej ) is called an asymptotic model of X. We refer the reader to [17] for a discussion on asymptotic models.


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CHAPTER 32

Sobolev Spaces∗ Aleksander Pełczyński and Michał Wojciechowski ´ Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, P.O. Box 21, 00-956 Warsaw, Poland E-mail: [email protected]; [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Classical Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The canonical embedding and the Sobolev projection . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Isomorphisms of Sobolev spaces. Linear extension theorems . . . . . . . . . . . . . . . . . . . . . . 4. Non-isomorphism of non-reflexive Sobolev spaces of several variables with classical Banach spaces 5. Properties of C(Q) spaces shared by C (k) (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Embedding theorems of Sobolev type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Interpolation in Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Anisotropic Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Partially supported by the KBN Grant 2 P03A 03614.


. . . . . . . . . .

. 1363 . 1364 . 1367 . 1372 . 1377 . 1393 . 1397 . 1404 . 1408 . 1419


Sobolev spaces

1363

Introduction Sobolev spaces were introduced by S.L. Sobolev in the late thirties of the 20th century. They and their relatives play an important role in various branches of mathematics: partial differential equations, potential theory, differential geometry, approximation theory, analysis on Euclidean spaces and on Lie groups. For comprehensive information the reader might consult the monographs [62,90–92,30,7,95]. The limited volume of this survey forced us to restrict ourselves to the simplest Sobolev spaces. We do not discuss the important relatives of classical Sobolev spaces like weighted Sobolev spaces, Besov spaces, spaces of Bessel potentials, Sobolev–Liouville spaces, Sobolev spaces of functions with fractional derivatives, etc. The reader is referred to the monographs mentioned above and the monographs [94,1] and [70] for study of these objects. Sobolev spaces are Banach spaces of smooth functions of one and several variables with conditions imposed on a few first (distributional) partial derivatives. In the classical case one requires that the derivatives up to a prescribed order belong to some Lp -space. The aim of this survey is to review the results on Banach space isomorphic properties of the simplest Sobolev spaces. Fixing the order of derivatives and a domain Ω on which the functions are defined, for varying p we get the scale of Sobolev spaces. The important feature of this scale is that it naturally embeds into the scale Lp (Ω, E) of E-valued functions where E is a finite-dimensional Hilbert space. This enables to show that under mild conditions on Ω the Sobolev spaces in question are isomorphic as Banach spaces to classical Lp -spaces for 1 < p < ∞. The situation is different in the limit case p = 1 and p = ∞ for functions in two and more variables. These Sobolev spaces are not isomorphic to classical Banach spaces L1 and C, although they share some properties of the corresponding classical spaces. The proofs of these facts require various analytic tools like theory of Fourier multipliers, Sobolev embedding theorem, Marcinkiewicz interpolation theorem, etc. This is the obstacle which we had to overcome while writing the survey to make it accessible for a reader less familiar with hard analysis. The survey consists of 8 sections. Section 1 contains basic definitions. In Section 2 we study the embedding of a given scale of Sobolev spaces into the scale of finite-dimensional vector-valued Lp -spaces. We analyze analytic properties of the orthogonal projection onto the image of the embedding (for p = 2). We establish an explicit isomorphism of Sobolev spaces on Rn with Lp -spaces for 1 < p < ∞. In Section 3 we discuss isomorphisms of Sobolev spaces defined on open subsets of Rn . Under mild conditions of regularity of the boundary of the domain, for fixed p and fixed order of derivatives, the topological dimension of the domain (= the number of variables) suffices for the isomorphism of Sobolev spaces in question. The pioneering result in this direction is Mityagin’s theorem on the isomorphisms of spaces of k-times continuously differentiable functions. The result of this section depends on the existence of linear extension operators. In Section 4 we present “negative” results that Sobolev spaces in L1 -norm and L∞ -norm are not isomorphic to corresponding classical spaces. The results are based on the idea of S.V. Kislyakov of employing the Sobolev embedding theorem. The Sobolev spaces in L∞ -norm and the spaces of

1364

A. Pełczyński and M. Wojciechowski

k-times continuously differentiable functions although non-isomorphic to spaces of continuous functions share various important properties of the latter spaces. These results mostly due to J. Bourgain are discussed in Section 5. Section 6 is devoted to the Sobolev embedding theorem and its improvement (like embeddings into Lorentz and Besov spaces). Section 7 concerns with interpolation properties of scales of Sobolev spaces. Section 8 reflects some research interest of the authors. We deal with non-classical anisotropic Sobolev spaces. 1. Classical Sobolev spaces ∂ |α| denote the partial derivative corresponding to the multi-index α α ∂x1 1 ∂x2 2 ...∂xnαn n n α = (αj )j =1 ∈ Z+ where Z+ := N ∪ {0}; here |α| = nj=1 αj is the order of the derivative * α ∂ α ; we use the convention x α := nj=1 xj j . Define α 0 β ≡def αj βj for j = 1, 2, . . . , n. If Ω is an open set in Rn then Ω denotes the closure of Ω and bd Ω = Ω \ Ω denotes the

Let ∂ α =

boundary of Ω. The set of scalars is denoted by K, it is, either R – the real numbers, or C – the complex numbers. Let D(Ω) be the space of all infinitely many times differentiable scalar-valued functions f : Ω → K with compact support, supp f = {x: f (x) = 0} ⊂ Ω. A function g : Ω → K is said to be the α-th distributional partial derivative of an f : Ω → K, in symbol g = D αf , provided |α| gφ dx = (−1) f ∂ α φ dx for φ ∈ D(Ω). Ω

Ω

Here and in the sequel . . . dx denotes the integration against the n-dimensional Lebesgue measure λn ; by Lp (Ω) we denote Lp (Ω, λn ). If a partial derivative of f is continuous on Ω then the corresponding distributional derivative of f coincides with the partial derivative. We admit that the (distributional) derivative of order 0 of a function f coincides with f . Let 1 p ∞ and k = 0, 1, . . . . Let us put p L(k) (Ω) = f : Ω → K: D αf exists and D αf ∈ Lp (Ω) for |α| k . p

We equip L(k) (Ω) with the norm f Ω,(k),p =

α

D f (x)p dx 1/p , max|α|k essupx∈Ω D αf (x), |α|k Ω

for 1 p < ∞, for p = ∞.

By C0(k) (Ω) we denote the closure of D(Ω) in the norm · Ω,(k),∞ , and by C (k) (Ω) a subspace of L∞ (k) (Ω) consisting of functions which together with their partial derivatives of orders k are uniformly continuous and vanishes at infinity (for unbounded Ω). Clearly C0(k) (Ω) ⊆ C (k) (Ω) ⊂ L∞ (k) (Ω).

Sobolev spaces

1365

WARNING . Usually spaces in sup norms are defined on closed subsets of Rn . However the Sobolev spaces in Lp -norms are naturally defined on open subsets of Rn . To unify domains of functions we work with, we have defined C (k) (Ω) as spaces of uniformly continuous functions on open Ω; the functions uniquely extend on the closure of Ω. On the other hand we work primarily with separable spaces thus we add the condition of vanishing at infinity. This condition is meaningful only for unbounded domains. Thus in our (k) notation C (k) (Rn ) = C0 (Rn ) and C (k) (I n ) = C (k) (I n ). Here and in the sequel I n = (1/2, n n 1/2) ⊂ R . p

(k)

The spaces L(k) (Ω) (1 p ∞), C0 (Ω), C (k) (Ω) for k = 1, 2, . . . are called classical Sobolev spaces. A routine argument gives: P ROPOSITION 1. (i) The classical Sobolev spaces are Banach spaces; p (k) (ii) C0 (Ω), C (k) (Ω) and L(k) for 1 p < ∞ are separable. For regular domains Ω ⊂ Rn Sobolev spaces in Lp norms (1 p < ∞) can be defined as completion of C ∞ -functions in the corresponding norms. Precisely we have p (k) P ROPOSITION 2. C ∞ (Ω) := ∞ k=1 C (Ω) is dense in L(k) (Ω) in the following cases n (i) Ω = R (cf. [88], Chapter V, §2, Proposition 1); p p (ii) there is a linear extension operator from L(k) (Ω) to L(k) (Rn ) which takes C ∞ (Ω) ∞ n into C (R ); (iii) Ω is bounded and has segment property, i.e., every x ∈ bd Ω has an open neighbourhood Ux in Rn and there exists a non-zero vector yx such that for every z ∈ Ω ∩ Ux one has z + tyx ∈ Ω for 0 < t < 1 (cf. [1], Chapter III, Theorem 3.18). R EMARKS . (1) In Proposition 2 one can replace C ∞ (Ω) by its subspace consisting with functions whose unique continuous extension to Ω has compact support. (2) A theorem of Stein (cf. [88], Chapter VI, §3, Theorem 5) provides for a large class p p of domains in Rn a construction of linear extension operators from L(k) (Ω) to L(k) (Rn ) taking C ∞ (Ω) into C ∞ (Rn ). (3) Proposition 2 extends also on bounded domains with so called cone property (cf. [62], §1.1.9 for definition, and §1.1.6 and §1.1.9 for the proof). (4) Proposition 2 extends to Sobolev spaces on compact manifolds. We outline briefly how to define Sobolev spaces on manifolds. For simplicity let M be an n-dimensional compact Euclidean C k -manifold. Let (φj , Oj )j ∈A be a finite atlas of M into

consisting of open sets Oj and homeomorphisms φj : Oj −→ Rn such that Ωj := φj (Oj ) are open subsets of Rn whose closures are compact. Assume furthermore that φj φi−1 ∈ C (k) (Ωj ∩ Ωi )

((j, i) ∈ A × A).

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For 1 p ∞ we put p p L(k) (M) = f : M → K: f φj−1 ∈ L(k) (Ωj ) for j ∈ A , ⎧ −1 p

1/p ⎨ , for 1 p < ∞, j ∈A f φj Ωj ,(k),p f (φj ,Oj )j∈A ,(k),p = ⎩ max for p = ∞. j ∈A f Ωj ,(k),∞ , The definition of C (k) (M) is analogous. p Note that the topology of L(k) (M) does not depend of a particular choice of an atlas; for different atlases we get equivalent norms. In the case where M is a Lie group or a homogeneous subspace of a Lie group (we assume that the group operation is compatible with the differentiable structure of M, i.e., translation by element of the group are diffeomorp phisms) then the norm of L(k) can be naturally defined in terms of the Haar measure of the group and partial derivatives defined by elements of the Lie algebra of the group. This remark also applied to unimodular locally compact Lie groups. The most useful special models of Sobolev spaces on manifolds are the spaces on the groups Rn and the tori Tn , and the spaces on the Euclidean spheres Sn which are homogeneous spaces of the orthogonal groups. Since the spaces under consideration are translation invariant with respect to the group action, we can use powerful tools of Harmonic Analysis to study their structure. The torus Tn is usually identified with spaces Rn of 1-periodic functions with 1-periodic derivatives with respect to each variable. We end this section by introducing Sobolev spaces of measures – BV(k) (Ω). For a measure μ denote by v(μ) the positive measure being the total variation of μ (cf. [25], Vol. I, Chapter III, §1, Definition 1). M(Ω) stands for the Banach space of all scalar-valued Borel measures μ on Ω with bounded total variation with the norm μM(Ω) = v(μ)(Ω). A measure ν is said to be the distributional derivative of a measure μ corresponding to the multi-index α, in symbols D α μ = ν provided

∂ α ϕ dμ = (−1)α Ω

ϕ dν

for ϕ ∈ D(Ω).

Ω

We admit D 0 (μ) = μ. For an open Ω ⊂ Rn and for k = 1, 2, . . . by BV(k) (Ω) we denote the space of all μ ∈ BV (Ω) such that D α (μ) exists and belongs to M(Ω) for 0 |α| k, equipped with the norm D α μBV(k) (Ω) = . M(Ω) 0|α|k

The elements of BV(k) (Ω) can be regarded as functions on Ω. Precisely, it is not hard to show (cf., e.g., [81]) that if μ ∈ BVk (Ω) then μ and all the distributional derivatives D α (μ) for |α| < k are absolutely continuous with respect to the Lebesgue measure λn ; thus it is natural to identify them with functions in L1 (Ω). The space L1(k) (Ω) can be identified with the subspace of BV(k) (Ω) being the image of the isometric embedding f → f · λn

Sobolev spaces

1367

of L1(k) (Ω) into BV(k) (Ω). The space BV(1) (Ω) is often called the space of functions of bounded variation on Ω. For further information on spaces of functions with bounded variation the reader is referred to the books [27,62,104].

2. The canonical embedding and the Sobolev projection p

Given f ∈ L(k) (Ω) the tuple (D αf )|α|k is called a jet of f . A jet can be regarded as a vector-valued function from Ω into KK(n,k) . where K(n, k) is the number of partial derivatives of order k in n variables. Let |α|k Lp (Ω) be the product of K(n, k) copies of the space Lp (Ω) equipped with the norm (fα )p =

1/p p , |α|k fα Lp (Ω)

max|α|k fα L∞ (Ω) ,

for 1 p < ∞, for p = ∞.

. . The spaces |α|k C(Ω) and |α|k C0 (Ω) are defined similarly. Clearly the space . p (Ω) is naturally isomorphic to the vector-valued Lp -space Lp (Ω; l 2 L |α|k K(n,k)). Sometimes it is more convenient to work with the second model. The canonical embedding is the map p

J = JΩ,(k),p : L(k) (Ω) →

/

Lp (Ω)

|α|k

defined by

J (f ) = D αf |α|k

p

for f ∈ L(k) (Ω).

. Similarly one defines the canonical embedding J : BV(k) (Ω) → |α|k M(Ω). Clearly . p p |α|k L (Ω) is an L space on a measure space which is independent of p; the canonical embedding is an isometrically isomorphic embedding. Thus p

C OROLLARY 3. (a) If 1 < p < ∞ then the space L(k) (Ω) is superreflexive. (k) (k) (b) The spaces L1(k) (Ω), L∞ (k) (Ω), C (Ω), C0 (Ω), BV(k) (Ω) are not reflexive because they contain subspaces isomorphic either to l 1 or c0 spanned by functions with disjoint supports. (c) L∞ (k) (Ω) and BV(k) (Ω) are dual Banach spaces.

P ROOF. The statements (a) and (b) are obvious. We prove (c) for BV(k) (Ω) (cf. [80], p Proposition 6.1); the argument for L∞ (k) (Ω) is similar. Let lN for N = 1, 2, . . . denote the space of scalar-valued sequences x = (xj )N j =1 with the norm |x|p =

N 1

|xj |p

1/p

,

for 1 p < ∞,

max1j N |xj |, for p = ∞.

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1 and l ∞ are in duality, it follows from the Riesz representation theSince the spaces lN N 1 ) can be identified with the dual of the space orem that the vector-valued space M(Ω; lN ∞ ); the latter is the space of all continuous functions on Ω with values in l ∞ which C0 (Ω; lN N 1) vanish at infinity and at the boundary of Ω. It is also convenient to identify M(Ω; lN ∞ (resp. C0 (Ω; lN )) with Cartesian products of N copies of M(Ω) (resp. C0 (Ω)) equipped with the suitable l 1 (resp. l ∞ ) norm. Now let N = K(k, n) be the number of multi-indices corresponding to all partial derivatives in n variables of order k. Let

J (μ) = D α μ 0|α|k

for μ ∈ BV(k) (Ω)

1 ). Then J (BV (Ω)) be an isometrically isomorphic embedding of BV(k) (Ω) into M(Ω; lN (k) 1 ∗ ∞ )-topology of is a subspace of M(Ω; lN ) which is closed in the w -topology (= C0 (Ω; lN 1 )). Indeed let F = (D αf ) M(Ω; lN m m 0|α|k for m = 1, 2, . . . and let F = (να )0|α|k . w∗

∞ ) we have Assume that Fm → F as m → +∞. Then for every N -tuple (f(α) ) ∈ C0 (Ω; lN

lim m

f(α) d D αfm =

0|α|k Ω

f(α) dνα .

(1)

0|α|k Ω

Fix a multi-index β with 0 |β| k and specify (f(α) )0|α|k by putting f(α) = 0 for α = β and f(β) = ϕ with ϕ ∈ D(Ω). The definition of distributional partial derivative combined with (1) gives

ϕ d D βfm

ϕ dνβ = lim m

Ω

Ω

= lim(−1)

|β|

m

∂ ϕ · fm dλn = (−1) β

Ω

|β|

∂ β ϕ dν(0,0,...,0) . Ω

Hence νβ = D β ν(0,0,...,0) for 0 |β| k. Thus (νβ )0|β|k ∈ J (BV(k) (Ω)). Therefore J (BV(k) (Ω)) is w∗ -closed. Thus J (BV(k) (Ω)) can be identified with the dual of the ∞ )/(BV (Ω)) , where quotient C0 (Ω; lN ⊥ (k)

∞ BV(k) (Ω) ⊥ = (fα )0|α|k ∈ C0 Ω, lN :

fα d D αf = 0 for f ∈ BV(k) (Ω) .

0|α|k Ω

The space !

.

2 |α|k L (Ω)

is a Hilbert space with the inner product defined by

" (fα ), (gα ) := fα g¯α dx. |α|k Ω

Sobolev spaces

1369

Thus L2(k) (Ω) is a Hilbert space with the inner product f, g =

D αf D α g dx.

|α|k Ω

The orthogonal projection P = PΩ,(k) :

/

onto L2 (Ω) −→ J L2(k) (Ω)

|α|k

is called the Sobolev projection. For “nice” domains Ω the Sobolev projection has stronger analytic properties than . boundedness in |α|k L2 (Ω). In particular we have T HEOREM 4. Let n = 1, 2, . . . , k = 0, 1, . . . . Then (a) The Sobolev projection PRn ,(k) is of weak type (1, 1); (b) PRn ,(k) is of strong type (p, p) for 1 < p < ∞; (c) if either n = 1 or k = 0 then PRn ,(k) is of strong type (p, p) for 1 p ∞. Recall that a subadditive operator T whose domain contains a dense set Y ⊂ L1 (μ; E) ∩ (here E is a finite-dimensional Hilbert space) and whose range is contained in the space of μ measurable E-valued functions on some set is said to be of strong type (r, s) (resp. of weak type (r, s)) for some r, s with 0 r, s ∞ provided there is K ∈ (0, ∞) such that Tf s Kf r (resp. supc>0 μ({|Tf | c}) ( Kc f r )s ) for f ∈ Y . The l.u.b. of K satisfying the inequality in the parenthesis is called the weak type (r, s) constant of T . . Regarding |α|k L2 (Rn ) as vector-valued L2 (Rn ) we see that the shift operators induced by translations of Rn commute with PRn ,(k) . This allows to use the theory of Fourier multipliers in the proof of Theorem 4. Recall that a measurable m : Rn → C is a multiplier of weak type (r, s) (resp. of strong type (r, s)) provided that the operator Tm defined by Tm (f ) = (mfˆ)∨ (for f in an appropriate subspace of functions on Rn ) is of weak type (r, s) (resp. strong type (r, s)). Here gˆ denotes the Fourier transform of g and g ∨ the inverse Fourier transform (cf. [42], Chapter 7). The operator Tm is called the multiplier transform of m. The Sobolev projection PRn ,(k) can be expressed by multiplier transforms of simple rational functions. Let S(Rn ) denote the Schwartz class of C ∞ -functions on Rn which together with all their derivatives are rapidly decreasing at infinity (cf. [42], 7.1.2). Let . . n ) = {(f ) ∈ 2 (Rn ): f ∈ S(Rn ) for |α| k}. For α, β ∈ Zn with S(R L α α + |α|k |α|k |α| k, |β| k and for ξ ∈ Rn we put L∞ (μ; E)

Q(k) (ξ ) =

|α|k

ξ 2α ;

mα,β = i |α|−|β| ξ α ξ β Q−1 (k) (ξ );

Tα,β = Tmα,β .

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P ROPOSITION 5. One has

PRn ,(k) (F ) =

Tα,β (fβ )

|β|k

|α|k

for F = (fβ )|β|k ∈

/

S Rn .

(2)

|β|k

P ROOF. Denote by R(·) the right-hand side of (2). Taking into account that the Fourier transform and its inverse preserve S(Rn ) and the identities: ∨ fˆ = f ;

α f = i |α| ξ α fˆ ∂0

α ∈ Zn+ ; f ∈ S Rn ,

by simple algebraic calculation we verify for F ∈

.

|α|k S(R

n)

the identities:

!

" ! " R(F ), F = R(F ), R(F ) ; /

S Rn ∩ JRn ,(k) L2(k) Rn ; R(F ) ∈ |α|k

R(F ) = F

for F ∈ JRn ,(k) L2(k) Rn .

Thus R(F ) = PRn ,(k) (F ) for F ∈

.

|α|k S(R

n ).

P ROOF OF T HEOREM 4. (a) Since the Schwartz class is dense in L1 (Rn ), it follows from Proposition 5 that PRn ,(k) is of weak type (1, 1) iff all Tα,β have the same property, equivalently iff all mα,β are of weak type (1, 1) multipliers. One gets the latter property of mα,β routinely verifying that each mα,β satisfies the classical Hörmander–Mikhlin criterion (cf. [42], 7.9.5). (b) Since PRn ,(k) is of strong type (2, 2) and is selfadjoint, (b) follows by combining (a) with the Marcinkiewicz interpolation theorem (cf. [107], XII (4.6)). (c) Case k = 0 is trivial. If n = 1 then the functions mα,β are the Fourier transforms of linear combinations of the delta functions and functions from L1 (R) (cf. [75] for details). Thus the corresponding multiplier transforms are convolutions with these linear combinations. Hence they are of strong types (1, 1) and (∞, ∞). Combining Theorem 4 with an easy fact that a Sobolev space in Lp -norm contains a complemented subspace isomorphic to Lp (R) and using the decomposition method one can prove that whenever the Sobolev projection is of strong type (p, p) then the Sobolev space in question is isomorphic to Lp . However using Fourier multipliers we can construct an explicit isomorphism. T HEOREM 6. The multiplier transform T1/√Q(k) extends to an isomorphism from Lp (Rn ) p

onto L(k) (Rn ) for 1 < p < ∞ (k = 0, 1, . . . , n = 1, 2, . . .).

Sobolev spaces

1371

P ROOF. We use the following easy consequence of the Marcinkiewicz multidimensional multiplier theorem (cf. [90], Chapter IV, §6, Theorem 6) If |α| k then ξ α [Q(k) ]−1/2 is a strong type (p, p) multiplier on Rn for 1 0 does not depend on f . Since S(Rn ) is dense in L(k) (Rn ), Tm p has the unique extension to a bounded operator from Lp (Rn ) into L(k) (Rn ). 2α Conversely, the identity m−1 = implies Tm−1 (f ) = |α|k ξ m −|α| α αf ) . Therefore, by (3), α α i T (D f ). Thus T T (D −1 ξ m p ξ m p m |α|k |α|k T −1 (f ) Cα D αf p C1 f Lp , m p |α|k

(k)

where the constant C1 = C1 (p, k, n) does not depend on f .

Theorem 6 is false in the limit cases p = 1 and p = ∞ for n > 1 (cf. Section 4). However for n = 1 it extends on the limit cases (cf. [75]). In particular remembering that C(I ), where I = (−1/2; 1/2), is isomorphic to its Cartesian product with the field of scalars we get (cf. Borsuk [9]) P ROPOSITION 7. The operator f → f is an isomorphism from the subspace {f ∈ 1/2 C (k) (I ): −1/2 f (x) dx = 0} of codimension one in C (k) (I ) onto C (k−1) (I ). Hence all the spaces C (k) (I ) are isomorphic to C(I ) for k = 1, 2, . . . . R EMARKS . (1) Theorems 4 and 6 have their counterparts for Sobolev spaces on Tn . To adopt the proofs we use the following T RANSFERENCE THEOREM . If m is a continuous multiplier on Rn of strong type (p, p) for some p with 1 p ∞ (resp. of weak type (1, 1)), then the “sequence” (m(2πa))a∈Zn is the multiplier on the group Zn – the dual of Tn ; the corresponding norm (resp. the weak type (1, 1) constant) of the multiplier transform on Lp (Tn ) is dominated by the norm of the multiplier transform on Lp (Rn ) (resp. the weak type (1, 1) constant). The strong type part is due to De Leeuw (cf. [92], Chapter 7, Theorem 3.8); for the weak type part cf. [103,2]. n n .(2) Letp apn(PR ,(k) ) denote the norm of PR ,(k) regarded as an operator on |α|k L (R ). One can show that if k = 1, 2, . . . , n = 2, 3, . . . then there are positive constants A(k, n) and B(k, n) such that A(k, n) max(p, p/(p − 1)) ap (PRn ,(k) )

1372


B(k, n) max(p, p/(p − 1)) (cf., e.g., [72]). Thus in these cases PRn ,(k) is not of strong type (1, 1) and (∞, ∞). . (k) (3) JR,(k) (C0 (Rn )) is a complemented subspace of |α|k C0 (R). This is a consequence of the strong type (∞, ∞) of PR,(k) and the density of S(R) in C0 (R). (4) Theorems 4 and 6 and their counterparts for Tn are in fact results on the scales p p of Sobolev spaces (L(k) (Rn ))1 0. Since Rn \ bI (k) (k) quasi-Euclidean a similar argument as in (III) shows that C0 (bI n ) ∼ C0 (I n ) is isomorphic to a complemented subspace of C (k) (I n ). Now in view of (III) we get (IV) by the decomposition method. . It is convenient to denote by ( a∈Z Xa )c0 the c0 Cartesian product of a family (Xa )a∈Z . (V) C0(k) (I n ) ∼ C0(k) (Rn ). For m = 0, 1, 2, . . . , n put F0 = Rn and bd+ F0 = ∅, Fm = x ∈ Rn : 0 < xj < 1 for j = 1, 2, . . . , m , bd+ Fm = x ∈ F m : xj = 0 for j = 1, 2, . . . , m . For a ∈ Z we put Fma = Fm + a · em and bd+ Fma = bd+ Fm + a · em . Let

(k) a

C+ Fm = f ∈ C (k) Fma :

lim

α D f (x) = 0 for |α| k . a

x→bd+ Fm

Then (k)

C+ (Fm ) ∼

/ a∈Z

(k) a

C+ Fm+1

for m = 0, 1, 2, . . ., n − 1.

(6)

c0

Let φ ∈ D(3I ) satisfy φ(t) = 0 for t ∈ [2/3, 1] and φ(t) = 1 for t ∈ [0, 1/3]. For m = . (k) (k) a 0, 1, . . . , n − 1 define T : ( a∈Z C+ (Fm+1 ))c0 → C+ (Fm ) setting Tf = g where g = . (k) (k) a ))c0 and f ∈ C+ (Fm ) is given by (ga ) ∈ ( a∈Z C+ (Fm+1 k ∂i 1 · i fb−1 (πm+1 x) · (xm+1 − b)i , f (x) = gb (x) + φ(xm+1 − b) · i! ∂xm+1 i=0

1376


where b ∈ Z and π(·) satisfy: b xm+1 < b + 1,

πm+1 (x) = (x1 , . . . , xm , b, xm+2 , . . . , xn ).

. (k) (k) (k) a Let S : C+ (Fm ) → ( a∈Z C+ (Fm+1 ))c0 be defined by Sf = g where f ∈ C+ (Fm ) and . (k) a )) is given by g = (ga ) ∈ ( a∈Z C+ (Fm+1 c0 k ∂i 1 · i f (πm x) · (xm+1 − b)i gb (x) = (R|F b f )(x) − φ(xm+1 − b) · m+1 i! ∂xm+1 i=0 b for x ∈ Fm+1 .

It is not hard to check that T and S are bounded linear operators which satisfy and T ◦ S = IdC (k)(F ) .

S ◦ T = Id(.

(k) a a∈Z C+ (Fm+1 ))c0

+

m

Thus T is an isomorphism and we get (6). Since F0 = Rn we get by induction that / (k) C+ (Fn ) . C (k) (Rn ) ∼ c0

(7)

Similarly as in step (III), we get that C (k) (I n ) is isomorphic to a complemented subspace (k) (k) of C+ (Fn ). On the other hand C+ (Fn ) is isomorphic to a complemented subspace of (k) n C (2I ): the embedding is the formal identity (we extend the function on 2I n \ Fn by 0) and the projection is given by 2I n \F n

Pf = IdC (k) (2I n ) − E(k)

◦ R|2I n \F n .

(k) (Fn ). Combining Thus, by the decomposition method and (IV), we obtain C (k) (I n ) ∼ C+ with (7) we get (V). (VI) C (k) (Ω) ∼ C (k) (Rn ). If Ω ⊂ Rn is an arbitrary (not necessarily bounded) quasiEuclidean open set, then similarly as in step (IV) (replacing I n by Rn ) we show that C (k) (Ω) embeds as a complemented subspace into C (k) (Rn ). Then, combining steps (IV) and (V) and using the decomposition method we get (VI). (VII) C0(k) (Ω) ∼ C (k) (Rn ). If Ω ∈ Rn is such that Rn \ Ω is quasi-Euclidean then (k) C0 (Ω) is naturally identified with a subspace of C (k) (Rn ). This subspace is complemented via the projection Rn \Ω

P = IdC (k)(Rn ) − E(k)

◦ R|Rn \Ω . (k)

On the other hand C (k) (I n ) is isomorphic to a complemented subspace of C0 (Ω) exactly as in step (V). Thus combining steps (IV) and (V) and using the decomposition method we get (VII).

Sobolev spaces

1377

R EMARK . In the same way we prove that C (k) (Rn ) ∼ C (k) (M) for some compact Euclidean C k -manifold M. In particular we have C OROLLARY 12. C (k) (Tn ) ∼ C (k) (Sn ) ∼ C (k) (Rn ) for fixed n and k with n = 1, 2, . . . , k = 0, 1, . . . . Theorem 10 is proved similarly. As in the proof of Theorem 8 the crucial role plays the counterpart of step (I) – the infinite divisibility of L1(k) (I n ). For details cf. [79]. The argument in [79] does not use JET; instead it uses an ‘elementary’ explicit construction of an extension operator from a special domain. p

P ROOF OF T HEOREM 11. The assumptions on Ω imply that L(k) (Ω) is isomorphic to a . complemented subspace of Lp . Either we use Theorem 6 or we use that |α|k Lp (Rn ) p is isomorphic to Lp (cf. [44], Vol. I, pp. 14–15). Next we show that L(k) (Ω) contains a p complemented subspace isomorphic to L for arbitrary non-empty Ω ⊂ Rn . Clearly Ω contains a cube aI n + x for some a > 0 and some x ∈ Ω. Since there is a linear extension p p p operator from L(k) (aI n + x) into L(k) (Ω), the space L(k) (Ω) contains a complemented p subspace isomorphic to L(k) (I n ). Let p E = f ∈ L(k) (I n ): f depends on the first coordinate only . p

p

Clearly E ∼ L(k) (I ). Moreover E is complemented in L(k) (I n ) via the projection f → I n−1 f (·, x2 , x3 , . . . , xn ) dx2 dx3 · · · dxn . An application of the decomposition method completes the proof.

4. Non-isomorphism of non-reflexive Sobolev spaces of several variables with classical Banach spaces To the contrary with the spaces of one variable (cf. Proposition 7 and the preceding comment) the non-reflexive Sobolev spaces of more than one variable are not isomorphic to corresponding L1 and C(K) spaces. The main analytic tool used in the proofs of next two theorems is a special case of the Sobolev embedding theorem (cf. Section 6 for more detailed discussion). We use the following notation: the characters of the group Tn are identified with exponents ea : I n → C defined for a = (aj ) ∈ Zn by ea (x) = exp 2πi nj=1 aj xj for x = (xj ) ∈ I n . We put fˆ(a) = n f (x)ea (−x) dx (a ∈ Zn ; f ∈ L1 (I n )). I

T HEOREM 13. Let k, n (k = 1, 2, . . . , n = 2, 3, . . .) be given. Then L1(k) (Ω) is not a L1 space for every non-empty open Ω ⊂ Rn . P ROOF. Since every non-empty open Ω contains a cube tI n + x for some t > 0 and x ∈ Ω, it follows from JET that L1(k) (Ω) contains a complemented subspace isomorphic to L1(k) (I n ). Thus it is enough to restrict ourselves to Ω = I n .

1378


First we consider the case k = 1, n = 2 to which the general case reduces. Define V1 first for trigonometric polynomials on I 2 (= finite linear combinations of the ea ’s for a = (a1 , a2 ) ∈ Z2 ) by

V1 (f ) =

fˆ(a)ea = f.

a∈Z2

C LAIM . V1 extends to a bounded linear operator from L1(1) (I 2 ) into L2 (I 2 ). We shall denote the extension also by V1 . Assume the claim. Define T1 : L2 (I 2 ) → by

L1(1) (I 2 )

T1 (f ) =

fˆ(a)ea(1),

where ea(1) = #

a∈Z2

ea 1 + (2πa1 )2 + (2πa2)2

.

T1 is bounded because (ea(1) )a∈Z2 is an orthonormal system in L2(1) (I 2 ), hence f 2L2 (I 2 )

2 (1) ˆ = f (a)ea 2

L(1) (I 2 )

a∈Z2

2 T1 (f )L1

(1) (I

2)

.

Thus V1 T1 : L2 (I 2 ) → L2 (I 2 ) is not a Hilbert–Schmidt operator because

−1 V1 T1 (ea )2 2 2 = = +∞. 1 + (2πa1 )2 + (2πa2 )2 L (I ) a∈Z2

a∈Z2

Thus L1(1) (I 2 ) is not an L1 -space because by a result of Grothendieck (cf. [44], Section 10, [38], [22], 4.12) every operator on a Hilbert space which factors through a L1 -space is Hilbert–Schmidt. P ROOF OF CLAIM . We start with the identity 1

2 ∂ ∂ h, h = h22 1R− ×R+ $ ∂x1 ∂x2

for h ∈ S R2 ,

(∗)

where 1R− ×R+ is the indicator function of the set {x ∈ R2 : x1 0, x2 0}, “$” denotes the operation of convolution, and ·, · stands for the usual inner product in L2 (Rn ); by 1R− and 1R+ we denote the indicator functions of the negative and the positive halfline respectively. To verify (∗) note that for g ∈ S(R) one has

1R− $ g (t) =

+∞

g (s) ds; t

(1R+ $ g)(t) =

t −∞

g(s) ds.

Sobolev spaces

Thus (1R− ×R+ $

x2 ∂ ∂x1 h)(x1 , x2 ) = − −∞ h(x1 , s2 ) ds2 .

1379

Hence integrating by parts we get

2 1 x2 ∂ ¯ ∂ ∂ h(x1 , x2 ) dx1 dx2 h, h =− h(x1 , s2 ) ds2 1R− ×R+ $ 2 ∂x1 ∂x2 ∂x 2 R −∞ h(x1 , x2 )2 dx1 dx2 = h2 . = 2 R2

It follows from (∗) and the Hausdorff–Young inequality that h22

∂ 1R− ×R+ ∞ h ∂x 1

∂ ∂ 2 −1 ∂ h 2 . ∂x h + ∂x h 1 2 1 ∂x2 1 1 1

(∗∗)

Thus the same inequality holds for h ∈ L1(1) (R2 ) with partial derivatives replaced by distributional derivatives. Now if Λ : L1(1)(I 2 ) → L1(1) (R2 ) is a linear extension operator then for f ∈ L1(1) (I 2 ) and h = Λ(f ) we get f L2 (I 2 ) h2

#

# 1/2 D (1,0) h1 + D (0,1) h1 1/2Λ f L1

(1) (I

2)

.

Clearly f 22 = a∈Z2 |fˆ(a)|2 . This completes the proof of claim and of Theorem 13 in the case k = 1, n = 2. Next consider the case k 2, n = 2. We define Vk : L1(k) (I 2 ) → L2 (I 2 ) and Tk : 2 L (I 2 ) → L1(k) (I 2 ) by Vk (f ) =

(2πa1)k−1 fˆ(a)ea ;

Tk (f ) =

a∈Z2

fˆ(a)ea(k),

a∈Z2

where ea ea(k) = ea L2

(k) (I

2)

and ea L2

(k)

3 4 l k 4 5 (2πa1 )2r (2πa2 )2(l−r). (I 2 ) = r=0 l=0

The boundedness of Tk is proved similarly as the boundedness of T1 using the orthonor(k) mality in L2(k) (I 2 ) of the system (ea )a∈Z2 . The boundedness of Vk uses inequality (∗∗) for the function

∂k h ∂x1k−1

instead of h. Since

(2πa1 )2(k−1) Vk Tk (ea )2 2 2 = = +∞, L (I ) ea 2 2 2 2 2

a∈Z

a∈Z

L(k) (I )

the operator Vk Tk is not Hilbert–Schmidt, hence L1(k) (I 2 ) is not a L1 -space.

1380


The case k 1, n 3 follows from previous cases because L1(k) (I n ) contains the complemented subspace isomorphic to L1(k) (I 2 ) consisting of functions depending on the first two variables; the averaging operator f→

I n−2

f (·, ·, x3 , . . . , xn ) dx3 · · · dxn

is the desired projection.

The operators Vk constructed above are not 1-summing, because 1-summing operators on a Hilbert space are Hilbert–Schmidt (cf. [44], Section 10). Thus C OROLLARY 14. There are bounded non-1-summing operators from L1(k) (Ω) into a Hilbert space (Ω ⊂ Rn , n = 2, 3, . . . , k = 1, 2, . . .). R EMARKS . (1) The operators Vk are special cases of the Sobolev embeddings (see Section 6). (2) Identify L1(k) (Tn ) with the subspace of L1(k) (I n ) being the closure of 1-periodic trigonometric polynomials. Clearly L2 (I n ) can be identified to L2 (Tn ) and L2 (Tn ) is translation invariant isometrically isomorphic with L2(k) (Tn ) for k = 1, 2, . . . . Note that (k)

Tk (L2 (T2 )) ⊂ L1(k) (T2 ). Let I2,1 : L2(k) (T2 ) → L1(k) (T2 ) be the natural embedding and let k denote the restriction of Vk to L1 (T2 ). Thus (V k , I (k) ) is the invariant factorization V 2,1 (k) (with respect to the action of the group T2 ) of an invariant operator on L2(k) (T2 ) which is not Hilbert–Schmidt. Next we discuss another property of Sobolev spaces in L1 -norms which differenties them from L1 (μ)-spaces. The latter spaces are by the Lebesgue decomposition theorem complemented in their second duals. We identify a Banach space with its canonical image in its second dual. T HEOREM 15 (cf. [80]). If n = 2, 3, . . . ; k = 1, 2, . . . then for every non-empty open Ω ⊃ Rn the space L1(k) (Ω) is uncomplemented in its second dual. Note that L1(k) (Ω) does not contain an isomorphic copy of c0 because L1(k) (Ω) is iso. metric to a subspace of |α|k L1 (Ω) (cf. Section 2) which is obviously isomorphic to an L1 (μ) space for some measure μ. Thus combining several facts on Banach lattices (cf. [60], Vol. II, Propositions 1.c.6, 1.a.11, Theorem 1.b.16) with Theorem 15 we get C OROLLARY 16. If Ω, k and n satisfy the assumption of Theorem 15 then L1(k) (Ω) is not isomorphic to any complemented subspace of a Banach lattice. To avoid technical complications we present the proof of Theorem 15 for k = 1. The argument requires some preparation. First observe that the general case reduces to the case

Sobolev spaces

1381

Ω = Rn because L1(1) (Ω) contains a complemented subspace isomorphic to L1(1) (I n ) ∼ L1 (Rn ) (by Theorem 10 and the argument in the beginning of the proof of Theorem 13) and the fact that the property “X is complemented in X∗∗ ” is inherited by complemented subspaces of X (cf. [59]). Next we introduce some notation. We represent Rn = Rn−1 × R and we write x = (y, xn ) with y ∈ Rn−1 and xn ∈ R. We identify Rn−1 with the hyperplane {x = (y, xn ) ∈ Rn : xn = 0}. We put Rn− = {x = (y, xn ) ∈ Rn : xn < 0} and Rn+ = {x = (y, xn ) ∈ Rn : xn > 0}. By D(Rn− ) we denote the space of scalar-valued infinitely many times differentiable functions on Rn− which together with all their partial derivatives are uniformly continuous on Rn− and whose unique continuous extensions to Rn− have compact supports; we use the same symbol to denote the functions on Rn− and their extensions to Rn− . It is not hard to verify (cf. Section 1, Proposition 2) that D(Rn− ) is dense in L1(1) (Rn− ) in the norm · L1 (Rn ) . The next result is due to Gagliardo (cf. [33]); it belongs to so (1) − called “trace theorems” (cf. Section 6, Theorem 42). P ROPOSITION 17. There exists the unique bounded linear surjection (called the trace)

Tr : L1(1) Rn− → L1 Rn−1 such that Tr(φ) = φ|Rn−1 for φ ∈ D(Rn− ). Proposition 17 is an immediate consequence of the next two lemmas and the density of D(Rn− ) in L1(1) (Rn− ) (cf. [80] for details). L EMMA 18. One has φ|Rn−1 L1 (Rn−1 ) φL(1) (Rn− ) P ROOF. Fix φ ∈ D(Rn− ). Then φ(y, 0) = the absolute value against dy we get

φ ∈ D Rn− . 0

∂ −∞ ∂xn φ(y, xn ) dxn

∂ φ|Rn−1 L1 (Rn−1 ) φ ∂x 1 n φL1(1) (Rn− ) . n L (R− )

for y ∈ Rn−1 . Integrating

L EMMA 19. There exists C > 0 such that given ψ ∈ D(Rn−1 ) there exists φ ∈ D(Rn− ) such that φ|Rn−1 = ψ;

φL1

n (1) (R− )

CψL1 (Rn−1 ) .

P ROOF. Pick a non-negative h ∈ D(R) with h(0) = 1. It is not hard to verify that for t = t (ψ) > 0 small enough the function φ defined by φ(y, xn ) = h( xtn )ψ(y) has the desired 0 property with C = ∞ h(xn ) dxn .

1382


Recall that a right inverse of a bounded linear operator T : X → Y (X, Y Banach spaces) is a bounded linear operator S : Y → X such that T S = IdY . The crucial analytic ingredient used in the proof of Theorem 15 is T HEOREM 20 (Peetre [71]). The trace Tr : L1(1) (Rn− ) → L1 (Rn−1 ) admits no right inverse. We postpone the discussion of Peetre’s Theorem until completing the proof of theorem 15. We also need the following result from the theory of Banach spaces L INDENSTRAUSS L IFTING P RINCIPLE = LLP. If a bounded linear surjection Q : X → Y (X, Y Banach spaces) has the property that ker Q is complemented in (ker Q)∗∗ then for every L1 -space E every linear operator T : E → Y admits a lifting T : E → X, i.e., T = QT. In particular if Y is isomorphic to L1 (0, 1), E = Y and T = IdE then T is a right inverse for Q. For a proof of LLP see [59,47]. P ROOF OF T HEOREM 15. It is enough to show that some complemented subspace of L1(1) (Rn ) is uncomplemented in its second dual. By Lemma 19, Theorem 20 and LLP, ker Tr is uncomplemented in its second dual. We show that ker Tr is isomorphic to a complemented subspace of L1(1) (Rn ). Let o L1(1) (Rn ) denote the subspaces of L1(1) (Rn ) consisting of the functions which are odd with respect to the variable xn . This subspace is complemented in L1(1) (Rn ) via the projection f → o f where o f (y, xn ) = (f (y, xn ) − f (y, −xn ))/2 for (y, xn ) ∈ Rn−1 × R a.e. For f ∈ ker Tr we define f˜ : Rn → R by f˜(y, xn ) =

f (y, xn ), −f (y, −xn ),

for xn 0, for xn > 0.

(8)

To prove that o L1(1) (Rn− ) is isomorphic to ker Tr we show that (i) the formula (8) defines a function in o L1(1) (Rn ); (ii) the operator f → f˜ is a surjection onto o L1(1) (Rn ). Note that (i) holds for f ∈ D(Rn− ) ∩ ker Tr. Thus it is enough to show that D(Rn− ) ∩ ker Tr is dense in ker Tr. Fix ε > 0 and f ∈ ker Tr. Since D(Rn− ) is dense in L1(1) (Rn− ), there is fε ∈ D(Rn− ) such that f − fε L1 (Rn ) < ε. Since Tr(fε ) = Tr(f − fε ), Lemma 18 (1) − yields fε |Rn−1 L1 (Rn−1 ) f − fε L1 (Rn ) < ε. Therefore, by Lemma 19, there exists (1)

−

gε ∈ D(Rn− ) such that gε |Rn−1 = fε |Rn−1 and gε L1 (Rn ) < Cε. Hence (fε − gε )|Rn−1 = 0 (1) − and f − (fε − gε )L1 (Rn ) < f − fε L1 (Rn ) + gε L1 (Rn ) < (C + 1)ε. (1)

−

(1)

−

(1)

−

For (ii) note that the map f → f˜ is an isomorphism because f L1

n (1) (R− )

2f L1

n (1) (R− )

f˜L1

(1) (R

n)

. For Φ ∈ o L1(1) (Rn ) ∩ D(Rn ) one has Φ| Rn− = Φ. Thus it suffices to

show that o L1(1) (Rn ) ∩ D(Rn ) is dense in o L1(1) (Rn ). Fix ε > 0. Since D(Rn ) is dense in L1(1) (Rn ), given F ∈ o L1(1) (Rn ) (hence satisfying o F = F ) there is a Φ ∈

Sobolev spaces

1383

D(Rn ) such that F − ΦL1 (Rn ) < ε. Thus F − o ΦL1 (Rn ) = o (F − Φ)L1 (Rn ) (1) (1) (1) F − ΦL1 (Rn ) < ε. (1)

Next we discuss Peetre’s theorem. It can be reformated as follows: there is no linear extension operator from L1 (Rn−1 ) → L1(1) (Rn− ). In that form an elegant proof was given in [16]. Another proof is contained in [80]. The proofs in [16] and [80] use harmonic analysis. Here we present a simple proof, based on an idea from [81], which uses the following purely Banach space property of L1 (μ). L EMMA 21. Let μ be a non-purely atomic measure. Assume that weakly compact operators Tm : L1 (μ) → L1 (μ) satisfy ∞ Tm (g)

L1 (μ)

< +∞

for every g ∈ L1 (μ).

m=1

Then

∞

m=1 Tm

= IdL1 (μ) .

A proof of Lemma 21 can be obtain modifying the argument of Proposition 1.d.1 in [60], Vol. I, see also [81], Lemma 5.3. The next lemma is an improvement of Lemma 18. Let us put Dn = D (0,...,0,1) and U(a, b) = {x ∈ Rn : a < xn < b}. L EMMA 22. (j) Let f ∈ L1(1) (Rn− ). Let −∞ < c < d c∗ < d ∗ < 0. Let us define h : Rn−1 → C by h(y) = (d − c)−1

d

f (y, xn ) dxn c

−1 − d ∗ − c∗

d∗ c∗

f (y, xn ) dxn

(y a.e.-λn−1 ).

Then hL1 (Rn−1 )

U (c,d ∗ )

|Dn f | dλn .

(jj) If c < d < 0 then for f ∈ L1(1) (Rn− ) Tr(f ) − (d − c)−1

d c

f (·, xn ) dxn

L1 (Rn−1 )

U (c,0)

|Dn f | dλn .

1384


P ROOF. Define e : R → R by ⎧ ⎪ (d − c)−1 (xn − c), ⎪ ⎨ 1,

e(xn ) = ∗ ∗ −1 c ∗ − x , ⎪ n ⎪ ⎩ d −c 0,

for c < xn < d, for d xn c∗ , for c∗ < xn < d ∗ , otherwise.

The derivative e exits at every point xn ∈ R \ {c, d, c∗ , d ∗ } and e L∞ (R) < +∞. Moreover h(y) = e (xn )f (y, xn ) dxn (y a.e.-λn−1 ). R−

Fix ε > 0. Pick ϕ ∈ D(Rn−1 ) so that ϕL∞ (Rn−1 ) = 1 and hL1 (Rn−1 ) − ε
0 define eη : R → R – the regularization of e by

η exp −(t/η)2 e(xn − t) dt, eη = ρ(xn ) · √ (2π) R where ρ ∈ D(R− ) does not depend on η and satisfies 0 ρ 1 and ρ(xn ) ≡ 1 for xn in some open interval containing [c, d ∗ ]. The regularization eη ∈ D(R− ) and it satisfies for sufficiently small η > 0,

ϕ(y)eη (xn )f (y, xn ) dy dxn > hL1 (Rn−1 ) − ε, (9) R−

Rn−1

/ (c, d ∗ )) and (because e is continuous and e(xn ) = 0 for xn ∈ eη (xn ) < ε Dn f

L1 (Rn− )

+1

−1

for xn ∈ / c, d ∗ .

(10)

Put φ(x) = ϕ(y)eη (xn ) for x = (y, xn ) ∈ Rn− . Combining the definition of distributional derivative with (9) and with the Fubini theorem we get ∂ = φ(x)D f (x) dx φ(x)f (x) dx n n ∂xn n R− R− ϕ(y)eη (xn )f (y, xn ) dx = =

Rn−

R−

Rn−1

ϕ(y)eη (xn )f (y, xn ) dy dxn

> hL1 (Rn−1 ) − ε.

Sobolev spaces

1385

On the other hand taking into account that |ϕ| 1 and (10) we infer that |φ(x)| = |ϕ(y)η(xn )| < ε(Dn f L1 (Rn− ) + 1)−1 whenever x ∈ Rn− \ U(c, d ∗ ). Thus

Rn−

φ(x)Dn f (x) dx

U (c,d ∗ )

φ(x)Dn f (x) dx

+

Rn \U (c,d ∗ )

U (c,d ∗ )

φ(x) · Dn f (x) dx

|Dn f | dλn + ε.

Hence U (c,d ∗ ) |Dn f | dλn hL1 (Rn ) − 2ε. Passing ε → 0 we get (j). To prove (jj) specify c, d, c∗ , d ∗ so that lim c∗ = lim d ∗ = 0. Note that if f ∈ D(Rn− ) then d∗ obviously if lim c∗ = lim d ∗ = 0 then lim (d ∗ − c∗ )−1 c∗ g(·, xn ) − Tr(g)L1 (Rn−1 ) = 0; thus by density of D(Rn− ) in L1(1) (Rn− ) the same formula holds for all functions in L1(1) (Rn− ). P ROOF OF T HEOREM 20. Assume to the contrary that there exists a right inverse of Tr : L1(1)(Rn− ) → L1 (Rn−1 ), say S. Let B = y ∈ Rn−1 : |y|2 1 . / 2B. Let Pick Φ ∈ D(Rn−1 ) so that Φ(y) = 1 for y ∈ B and Φ(y) = 0 for y ∈

X = g ∈ L1 Rn−1 : g(y) = 0 for y ∈ /B . Obviously X can be identified with L1 (B, λn−1 |B) = L1 (B) and gX = gL1 (Rn−1 ) for g ∈ X. Define S % : X → L1(1) (Rn− ) for g ∈ X by S % (g)(x) = Φ(y)S(g)(y, xn )

for x = (y, xn ) ∈ Rn λn -a.e.

Clearly Tr ◦S % = IdX . Define the operators Um : X → X by U0 = 0, and for m = 1, 2, . . . by Um g(y) = λ1 (Im )−1 S % (g)(y, xn ) dxn · 1B (y) for y ∈ Rn−1 λn−1 -a.e., Im

where Im = (−2−m , −2−m−1 ) and 1B denotes the indicator function of B. By Lemma 22(jj), for every g ∈ X, if m 1 then −1 % % (I ) S (g)(·, x ) dx − Tr ◦S (g) Um g − gL1 (Rn−1 ) λ n n 1 m Im

U (−2−m ,0)

Dn S % (g) dλn .

L1 (Rn−1 )

1386


Thus lim Um g − gL1 (Rn−1 ) = 0 for g ∈ X.

(11)

m→∞

By Lemma 22(j) for m = 1, 2, . . . , we get −1 λ Um+1 g − Um gL1 (Rn−1 ) (I ) 1 m+1 − λ1 (Im )−1

Im+1

Im

S % (g)(·, xn ) dxn

U (−2−m ,−2−(m+2) )

S % (g)(·, xn ) dxn

L1 (Rn−1 )

Dn S % (g) dλn .

Taking into account that each x ∈ Rn− belongs to at most two of the sets U(−2−m , −2−(m+2) ) we get ∞

Um+1 g − Um gL1 (Rn−1 ) U1 gL1 (Rn−1 ) + 2S % (g)L1

n (1) (R− )

m=0

3S % · gL1 (Rn−1 ) .

(12)

Put q = n/(n − 1). By the Sobolev embedding theorem for L1(1) (Rn− ) (cf. Theorem 33 and remark (1) following its proof), % S (g) q n AS % (g) 1 L (R ) L

n (1) (R− )

−

for g ∈ X,

where A = A(k, n) is an absolute constant. Using the later inequality, the inequality between the first and the q-th norms on a finite interval, and the Fubini theorem, for g ∈ X, we get q Um gLq (Rn−1 )

q −1 % S (g)(y, xn ) dxn dy λ1 (Im ) Rn−1 Im % S (g)(y, xn )q dxn dy λ1 (Im )−q

λ1 (Im )−q

Rn−1 Im

Rn−

q q Cm S % (g)L1

% S (g)(x)q dx

n (1) (R− )

q q q Cm S % gL1 (Rn−1 ) ,

m g = Um g regarded as an element of Lq (B) and let Jq,1 where Cm = A · λ1 (Im )−1 . Put U denote the natural embedding of Lq (B) into L1 (B). It follows from the previous inequality

Sobolev spaces

1387

m : L1 (B) → Lq (B) is bounded; obviously Jq,1 is bounded because the measure that U m admits a factorization through the reflexive space λn−1 |B is finite. Hence Um = Jq,1 ◦ U Lq (B). Thus Um and Um − Um−1 are weakly compact for m = 1, 2, . . . . In view of (11) and (12) the desired contradiction follows from Lemma 21. Recall that a Banach space X has local unconditional structure (cf. [36], [22], p. 345, [44], p. 59) if there is a constant C > 0 such that for every finite-dimensional subspace F of X there are a finite-dimensional Banach space E with a basis with unconditional constant one (cf. [44], p. 14) and operators u : F → E and v : E → X such that u · v C and v ◦ u : F → X is the natural (set theoretical) embedding. One has (cf. [32], [22], Theorem 17.5) (FJT). A Banach space has local unconditional structure iff its second dual is isomorphic to a complemented subspace of a Banach lattice. Thus the following result [81] is an improvement of Corollary 16. T HEOREM 23. If Ω ⊂ Rn is an open non-empty set then for n = 2, 3, . . . and k = 1, 2, . . . the spaces L1(k) (Ω) and BV(k) (Ω) do not have local unconditional structure. The proof of Theorem 23 is lengthy (cf. [81] for details). It starts with (FJT) and uses the method of the proof of Theorem 15 to show that members of a certain net of separable subspaces of BV(k) (Rn ) containing L1(k) (Rn ) are not isomorphic to complemented subspaces of Banach lattices. This allows us to show that BV(k) (Rn ) does not have local unconditional structure. This implies that L1(k) (Rn ) does not have local unconditional structure by the following result: T HEOREM 24 (cf. [80], Proposition 6.2, [81], Proposition 7.3). There exists an isomorphic embedding ðk : BV(k) (Rn ) → [L1(k)(Rn )]∗∗ such that ðk (BV(k) (Rn )) is a complemented subspace of [L1(k) (Rn )]∗∗ and ðk ◦ ιk = κk , where ιk : L1(k) (Rn ) → BV(k) (Rn ) is the isometric embedding defined by ιk (f ) = f ◦ λn and κk is the natural embedding of L1(k) (Rn ) into its second dual. 24. For simplicity we identify BV(k) (Rn ) with O UTLINE OF THE PROOF OF T HEOREM. n n J (BV(k) (R )), where J : BV(k) (R ) → |α|k M(Rn ) is the canonical embedding, and we identify L1(k) (Rn ) with its image via ιk . Thus L1(k) (Rn ) can be regarded as the subspace 1 ) defined by of L1 (Rn ; lN

(fα )0|α|k ∈ L1 Rn ; l 1 : fα = D αf for 0 |α| k and for f ∈ L1(k) Rn , where N = K(k, n) is the number of partial derivatives in n variables of order k. 1 ) can be identified with L∞ (Rn ; l ∞ ). Thus, by the Hahn–Banach The dual of L1 (Rn ; lN N extension principle every z∗ ∈ (L1(k) (Rn ))∗ has a norm preserving extension to some ∗

∞ ). Now let (G ) ∞ 1 n (φα[z ] ) ∈ L∞ (Rn ; lN ε ε>0 be a C -approximate identity of L (R ), for

1388


instance Gε (x) = (ε)−n G(x/ε) for x ∈ Rn where G(x) = (2π)−n/2 exp(−|x|22/2). Let Φε be the operator of convolution with Gε , i.e., Φε (ν)(x) = Rn Gε (x − y)ν(dy) for x λn -a.e. Then Φε (BV(k) (Rn )) ⊂ L1(k) (Rn ) for k = 0, 1, . . . and lim ε

Rn

Φε (ν)(x)f (x) dx =

Rn

f (x) dν

for f ∈ D Rn and ν ∈ M Rn .

Given ν ∈ BV(k) (Rn ) we define ð(k) (ν) by

ð(k) (ν) z∗ = LIM z∗ Φε (ν) ε→0

= LIM ε→0

0|α|k

Rn

∗ Φε D α ν φα[z ] dx

∗ for z∗ ∈ L1(k) ,

where LIMε→0 denotes a generalized (Banach) limit (cf. [25], Chapter II.3 (23)). The desired projection from (L1(k) (Rn ))∗∗ onto ðk (BV(k) (Rn )) is the operator ðk ◦ U ∗ where U ∗ : (L1(k) (Rn ))∗∗ → BV(k) (Rn ) is the adjoint operator to the isometric embedding ∞ )/(BV (Rn )) → (L1 (Rn ))∗ defined as follows. Let g = (g ) U : C0 (Rn ; lN ⊥ α 0|α|k be (k) (k) a representative of a coset [g]. Then U ([g]) ∈ L1(k) (Rn ))∗ is defined by

U [g] (f ) =

n 0|α|k R

gα · D α f dλn

f ∈ L1(k) Rn .

R EMARKS . (1) By Corollary 3(c) BV(k) (Ω) is a dual Banach space hence it is always complemented in its second dual (cf. [24]). Thus Theorem 15 does not extend on BV(k) (Ω), moreover ιk (L1(k) (Ω)) is not complemented in BV(k) (Ω). (2) The isometric embedding ðk is not unique and depends of the choice of a Banach limit. (3) One can extend the operator Vk defined in the proof of Theorem 13 on BVk (I n ); one can show in that way that there are bounded non-absolutely summing operators from BVk (Ω) into a Hilbert space. Next we pass to Sobolev spaces in sup norm. We exhibit a pathological property of these spaces much stronger than the non-isomorphism with L∞ -spaces. Recall that a Banach space has GL (= Gordon–Lewis property) provided that every 1-summing operator from X into a Hilbert space factors through L1 (μ) (cf. [36], [22], p. 350, [44], Section 9). T HEOREM 25. Let k, n (k = 1, 2, . . . , n = 2, 3, . . .) be given. Then for every open nonempty Ω ⊂ Rn the spaces C (k) (Ω) and L∞ (k) (Ω) fail GL. n C OROLLARY 26. C (k) (Ω) and L∞ (k) (Ω) (∅ = Ω ⊂ R , n = 2, 3, . . . , k = 1, 2, . . .) are not isomorphic either to quotients of L∞ -spaces or to Banach spaces with local unconditional structure or all the more to complemented subspaces of Banach lattices.

Sobolev spaces

1389

P ROOF. Each of the above properties implies GL (cf. [36], [22], Chapter 17, [44], Section 9). The proof of Theorem 25 uses the theory of invariant r-summing operators for 0 < r < 1 and the weak type (1, 1) of the Sobolev projections PTn ,(k) (cf. Section 2). Recall that a linear operator T : X → Y is r-summing (0 < r < ∞, X, Y normed spaces) provided T (xj )r C r sup x ∗ (xj )r πr (T ) := inf C: x ∗ 1 j

j

(xj )nj=1 ⊂ X, n = 1, 2, . . .

.

We need (cf. [61], [101], III.F.35) G ROTHENDIECK –M AUREY T HEOREM . Every bounded linear operator from an L1 space to a Hilbert space is r-summing for 0 < r 1. Hence every bounded linear operator from a Banach space to a Hilbert space which factors through an L1 -space is r-summing for 0 < r 1. The next proposition on invariant r-summing operators for 0 < r < 1 is crucial for our proof of Theorem 25. P ROPOSITION 27. Let U : C (k) (Tn ) → L2(k) (Tn ) be an invariant r-summing operator for some 0 < r < 1. Then a∈Zn U (ea(k) )2 2 n < ∞. L(k) (T )

Throughout the proof we write P instead of PTn ,(k) . It is.convenient to idenp n tify canonical images in |α|k L (T ) (resp. . Sobolevn spaces in question.with their . p n n |α|k C(T )) and to regard |α|k L (T ) (resp. |α|k C(T )) as vector-valued spaces Lp (Tn , E) (resp. C(Tn , E)) where E = (E, | · |E ) is a finite-dimensional Hilbert space. By Trig(Tn , E) we denote the linear span of the functions ea · ξ (a ∈ Zn , ξ ∈ E). The orthonormal basis (ea(k))a∈Zn of L2(k) (Tn ) is identified via the canonical embedding (k)

with (ea )a∈Zn where e(k) a

= ea · ξa ,

where ξa =

%

(2πi)|α| a α

|β|k (2π)

2|β| a 2β

∈ E. |α|k

The bold letters are used to denote vector-valued functions. We apply the standard notation, * α a α = nj=1 aj j for a = (aj ) ∈ Zn and α = (αj ) ∈ Zn+ . Note that P(ea · ξ ) = ξ, ξa E · e(k) a for ξ ∈ E and a ∈ Zn . Hence P(Trig(Tn , E)) ⊂ Trig(Tn , E). Moreover for each r with 0 < r < 1 there is Ar ∈ (0, ∞) such that

P(f )(x)r dx Ar f r for f ∈ Trig Tn , E . (13) r 1 E Tn

1390


The latter inequality is a consequence of a general easy fact that similarly as in the scalar case for E-valued functions a weak type (1, 1) operator is of strong type (1, r) for 0 < r < 1. P ROOF OF P ROPOSITION 27. We shall show that there is A ∈ (0, ∞) such that U Pf 2 Af 1

f ∈ Trig Tn , E .

(14)

Clearly (14) implies the existence of a bounded operator U1 : L1 (Tn , E) → L2(k) (Tn , E) such that U P = U1 I∞,1 . Here I∞,1 : C(Tn , E) → L1 (Tn , E) denotes the natural embedding. Similarly as in the scalar case every invariant operator, say U1 , from L1 (Tn , E) into L2 (Tn , E) is given by a convolution with an L2 function; precisely there is a U1 ∈ L2 (Tn , L(E, E)) such that

U1 (f ) (s) =

Tn

U1 (s − t) f (t) dt

for s λn -a.e.

Here L(E, E) denotes the (dim E)2 -dimensional space of linear operators on E. In our case where U1 (L1 (Tn , E)) ⊆ L2(k) (Tn ) there is a scalar sequence (ua ) such that the Fourier coefficients of U1 satisfy 1 (a)(ξ ) = U

Tn

ea (−t) · U1 (t) dt (ξ ) = ua ξ, ξa E · ξa

2 a∈Zn U1 (a)L(E,E) = (k) (k) = U1 I∞,1 implies U (ea ) = ua ea

Clearly U1 ∈ L2 (Tn , L(E, E)) yields other hand the identity U P To prove (14) define f y by f y (·) = f (· + y)

ξ ∈ E, a ∈ Zn .

a∈Zn |ua | for a ∈ Zn .

2

< ∞. On the

f ∈ Trig Tn , E , y ∈ Tn .

The definition of r-summing operator yields Tn

U Pf y r2 dy

r π(U ) sup

x ∗ ∈Z

Tn

∗ x (Pf y )r dy,

(15)

where the supremum extends on an arbitrary subset Z of the unit ball of [C (k) (Tn )]∗ which is weak-star dense in the ball. It is convenient to take as Z the set of those x ∗ ∈ [C (k)(Tn )]∗ that there exists h = (hα )|α|k ∈ Trig(Tn , E) such that h1 1 and ∗

x (g) =

Tn

! " g(x), h(x) E dx

g ∈ C (k) Tn ∩ Trig Tn , E .

(16)

Sobolev spaces

1391

Note that the integrand on the left-hand side of (15) is a constant function. Hence U Pf y r2 dy = U Pf r2 . Next we estimate the right-hand side of (15). For a scalarTn valued trigonometric polynomial h define ho by ho (x) = h(−x). We need the identity: Tn

(Pf y )(x)h(x) dx = P ho ∗ f (y) f ∈ Tn ,

(17)

where the convolution ho ∗ f is taken coordinatewise. (It is enough to verify (17) for h = ea n and f = ξ · eb (a, ∈ E).) Let {ξ (α) : |α| k} be an orthonormal basis for E. Fix b ∈ Z ; ξ (α) ∗ x ∈ Z and h = |α|k hα ξ ∈ Trig(Tn , E). Then (16) and (17) yield 1 r 2 ∗ (α) x (Pf y )r = (x), h (x)ξ dx Pf α y n T

E

|α|k

1 2 = (Pf y )(x)hα (x) dx, ξ (α) n |α|k

T

E

r

r n (Pf y )(x)hα (x) dx T

|α|k

=

|α|k

E

o

P h ∗ f (y) α E

r [Br ]r

P ho ∗ f (y)r , α E

|α|k

where Br is an absolute constant which depends only on r, k and n. Integrating dy, against r o using (13), the Hausdorff–Young inequality, and taking into account that |α|k hα 1 [Cr ]r hr1 for some absolute constant Cr we get Tn

∗ x (Pf y )r dy [Br ]r

n |α|k T

[Br Ar ]r

o

P h ∗ f (y)r dy α E

ho ∗ f r α 1

|α|k

[Ar Br ]r

r ho f r α 1 1

|α|k

[Ar Br Cr ]r hr1 f r1 = [Ar Br Cr ]r f r1 . Therefore (U P)(f )2 πr (U )Ar Br Cr f 1 which proves (14).

k I (k) : P ROOF OF T HEOREM 25. First we show that C (k) (T2 ) fails GL. Let U = V ∞,1 (k) k : C (k) (T2 ) → L1 (T2 ) is the natural embedding and V C (k) (T2 ) → L2 (T2 ) where I (k)

∞,1

(k)

1392


is the operator constructed in remark (2) after Corollary 14. It follows from the construction k that of V 2 U e(k) 2 a L a∈Z2

(k)

(T2 )

=

(2πa1 )2k−2 = +∞. ea 2 2 2 2

a∈Z

L(k) (I )

Thus, by Proposition 27 and the Grothendieck–Maurey theorem, U does not factor through (k) any L1 -space. On the other hand U is 1-summing because I∞,1 is 1-summing and therefor r-summing for 0 < r < 1. Thus C (k) (T2 ) and C (k) (I 2 ) isomorphic with the previous one (by Theorem 9) fail GL. The GL property is inherited by complemented subspaces of a space with GL. Since C (k) (Ω) and C0(k) (Ω) contain a complemented subspace isomorphic to C (k) (T2 ) (same ar(k) gument as in the proof of Theorem 13 with JET replaced by WET ), C (k) (Ω) and C0 (Ω) n fail GL for ∅ = Ω ⊂ R (k = 1, 2, . . . , n = 2, 3, . . .). The proof for L∞ (k) (Ω) is similar. R EMARKS . (1) Theorem 13 and the idea to apply Sobolev embeddings is due to Kislyakov [49]. One can prove directly that L1(k) (I n ) is not an L1 -space using the fact that the n-dimensional Sobolev embedding of L1(1) (I n ) into Lp (I n ) for p = n/(n − 1) (cf. Section 6) is a bounded operator but is not (n, 2)-summing while by Kwapień’s result (cf. [57], [93], Corollary 11.12) every operator from an L1 -space to Lp is (p/(p − 1), 2)-summing for 1 < p 2 (cf. [99] for details). Similarly one can show that C (1) (Tn ) is not isomorphic I∞,n/(n−1)

n/(n−1)

S

to an L∞ -space considering the operator C (1) (Tn ) −→ L(1) (Tn ) −→ Ln (Tn ), where I∞,n/(n−1) – the natural embedding, S – the appropriate Sobolev embedding. Now use [56], Theorem 2.b.8, which says that every operator from an L∞ -space to Lp -space is (p, 2)-summing for p 2. (2) The non-isomorphism of C (1) (Sn ) with C(K)-spaces for n 2 is due to Grothendieck who gave in [37] some indication for the proof. Henkin [40] published the complete proof that C (k) (Sn ) for n 2 is not isomorphic to any C(K)-space; he even showed that C (k) (Sn ) is not uniformly homeomorphic to any C(K)-space (cf. also [3], Theorem 10.8). The Grothendieck–Henkin argument is simpler if one uses the spaces on Tn instead of on Sn . The idea bases upon the following FACT (folklore). Let G be a compact Abelian group, E a finite-dimensional Hilbert space, X a translation invariant L∞ -subspace of C(G, E). Then X is complemented in C(G, E) via the orthogonal projection from L2 (G, E) onto X, i.e., this orthogonal projection is of strong type (∞, ∞). O UTLINE OF THE PROOF. The injectivity of ln∞ and the definition of an L∞ space imply the existence of a net of finite-dimensional operators {Tu : C(G, E) → X: u ∈ Σ} whose restrictions Tu |X tend pointwise to the identity on X. Averaging the Tu ’s with respect to the Haar measure of G we get the net {Tu : u ∈ Σ} of invariant operators which tends to the desired projection.

Sobolev spaces

1393

Now specifying G = Tn and X = C (k) (Tn ) we conclude that C (k) (Tn ) is not an L∞ space for n 2 because the corresponding orthogonal projection, which is PTn ,(k) , is not of strong type (∞, ∞) (cf. remark (2) after Proposition 7). (3) Kislyakov [49] first established that C (k) (Tn ) is not isomorphic to a quotient of a C(K)-space. Theorem 25 was proved independently in [50] and [58]. The proofs of Proposition 27 and Theorem 25 presented here are taken from [58]. Actually the inverse to Proposition 27 also holds and it also characterizes invariant nuclear operators (cf. [58], Proposition 1.3 and Theorem 4.1). (4) The space C (k) (Tn ) is not isomorphic to any complemented subspace of the Disc Algebra for k = 1, 2, . . . , n = 2, 3, . . . (cf. [74]). (5) The main classification problem which remains open is: are the integers k and n linear topological invariants for C (k) (I n ) and L1(k) (I n )? Precisely, does the isomorphism

of the spaces C (k) (I n ) and C (k ) (I n ) imply k = k and n = n (n, n = 2, 3, . . . , k, k = 1, 2, . . .)? The same question remains open for Sobolev spaces in L1 -norms.

5. Properties of C(Q) spaces shared by C (k) (Ω) Denote by M = M(I ) the dual of C(I ). If Q is an uncountable compact metric space then [C(Q)]∗ is isometrically isomorphic to M (cf. [44], Section 4). A dual X∗ of a Banach space X is said to be a separable perturbation of M provided that X∗ is isomorphic to M ⊕ F for some separable space F . We have T HEOREM 28. If open ∅ = Ω ⊂ Rn (resp. Rn \ Ω) is quasi-Euclidean then the first dual of C (k) (Ω) (resp. C0(k) (Ω)) is a separable perturbation of M for k = 1, 2, . . . , n = 1, 2, . . . . The proof of Theorem 28 requires some preparation. Note that for a Banach space being a separable perturbation of M is an isomorphic invariant. Hence, by Theorem 9, it is enough to prove that [C (k) (Tn )]∗ ∼ M ⊕ F . By the Riesz representation theorem M . we identify n . The space [ n )]∗ is with the space of all Borel complex-valued measures on T C(T |α|k . therefore identified with the space |α|k M of all tuples (μα )|α|k of measures in M with the norm (μα ). = |α|k μα ; the duality.is given by f , Υ = |α|k Tn fα dμα for Υ = (μα ) . ∈ |α|k M and f = (fα ) ∈ |α|k C(Tn ). Recall that a closed linear subspace G ⊂ |α|k M is a C(Tn )-module provided that

(μα ) ∈ G and f ∈ C Tn implies (f μα ) ∈ G. Here f μ is defined by (f μ)(A) = A f dμ for Borel A ⊂ Tn . Note that since the exponents are linearly dense in C(Tn ) in the sup norm, in the definition of C(Tn )-module one can replace C(Tn ) by the set {ea : a ∈ Zn }. FACT. If G ⊂

.

|α|k M

is a C(Tn )-module then G is complemented in

.

Fact is a particular case of much more general result (cf. [73], Section 1).

|α|k M.

1394


By the standard duality argument the quotient space fied with [C (k) (Tn )]∗ , where

C

(k)

T

n

⊥

= Υ∈

/

.

|α|k M/[C

(k) (Tn )]⊥

is identi-

(k) n

M: f , Υ = 0 for f ∈ JTn ,(k) C T .

|α|k

The crucial role in the proof of Theorem 28 is played by the concept of Henkin measure. Given a sequence (fm ) ⊂ C (k) (Tn ) we write fm ⇒ 0 provided that (i) supm fm C (k) (Tn ) < ∞; α n (ii) lim. m ∂ fm (x) = 0 (x ∈ T , |α| < k). Call Υ ∈ |α|k M a Henkin measure provided that (iii) limm JTn ,(k) (fm ), Υ = 0 whenever fm ⇒ 0. . Denote by MH = MH (n, k) the set of all Henkin measures in |α|k M. L EMMA 29. ⊥

MH ⊃ C (k) Tn ;

(18)

MH is a normed closed subspace of

/

M;

(19)

|α|k

MH is a C Tn -module.

(20)

P ROOF. (18) is obvious. (19) is routine and easy. To obtain (20) it suffices to show that ea Υ ∈ MH for every exponent ea with a ∈ Zn and for every Υ ∈ MH ; equivalently lim m

n |α|k T

α

∂ fm ea dμα = 0 whenever fm ⇒ 0 Υ = (μα ) ∈ MH , a ∈ Zn .

To verify the latter statement we note that fm ⇒ 0 implies fm ea ⇒ 0 for a ∈ Zn . Now invoking (i)–(iii) we use the Domination Convergence Theorem, combined with the identity

∂ αfm ea = ∂ α (fm ea ) −

n $ β≺α j =1

αj ! (2πi)|α−β| a α−β ∂ βfm ea , βj !(αj − βj )!

(21)

where α ≺ β ≡def α = β and αj βj for j = 1, 2, . . . , n. The identity (21) follows from the Leibniz formula for the derivatives of the product.

Sobolev spaces

1395

P ROOF OF T HEOREM 28. Let δx,α (f ) = fα (x) for |α| k, x ∈ Tn , and f = (fα )|α|k ∈ . n ∗ (k) n ∗ |α|k C(T ). Let δx,α be the corresponding functional on C (T ), i.e., δx,α (f ) = δx,α (JTn ,(k) (f )) = ∂ α f (x) for f ∈ C (k) (Tn ). Let

∗ ∗ F = the closed subspace of C (k) Tn generated by δx,α for x ∈ Tn , |α| < k. Then F is separable;

(22)

⊥

F = MH / C (k) Tn .

(23)

To verify (22) recall that if k 1 then the natural embedding C (k) (Tn ) → C (k−1) (Tn ) is compact (the Ascoli theorem). Therefore operator [C (k−1)(Tn )]∗ → the adjoint (k) n ∗ ∗ [C (T )] is also compact. Thus Z = x∈Tn ;|α| 1 and extended independently by Gagliardo [34] and Nirenberg [66] to p = 1. Here “→” stands for the set theoretical inclusion. T HEOREM 33 (Sobolev embedding theorem). p (i) If 1 p < n/k, then L(k) (Rn ) → Lq (Rn ) for 1/p 1/q 1/p − k/n. p (ii) If p = n/k > 1, then L(k) (Rn ) → 1r n/k and p 1 then L(k) (Rn ) → C0 (Rn ). The reader is referred to [1,7,90,62] for various proofs of Theorem 33. We present here briefly the original Sobolev approach for p > 1 (cf. [90], Chapter V), combined for p = 1 with a “weak type” argument which we learned from P. Hajłasz. O UTLINE OF THE PROOF. We restrict ourselves to part (i). It is enough to prove (i) for the “maximal possible” exponent q = pn/(n − pk). The proof reduces, by simple induction, to the case k = 1 and to real-valued functions. We use the Sobolev identity (cf. [90], Chapter V, §2, (18)) f (x) =

n

1 ωn−1

i=1

Rn

∂f yi (x − y) · n dy, ∂xi |y|

where ωn−1 denotes the surface measure of (n − 1)-dimensional sphere. Consequently we have n f (x) C j =1

∂f (x − y) · |y|−n+1 dy. n ∂x R

j

The right-hand side of the above formula is the convolution of the sum of absolute values of derivatives of f with the Riesz potential y → |y|−n+1 . By the Hardy–Littlewood–Sobolev theorem on fractional integration (cf. [90], Chapter V, §1, Theorem 1), this convolution is an operator of weak type (1, 1 + 1/(n − 1)) and it is a bounded operator from Lp (Rn ) to Lq (Rn ) for 1 1. For p = 1 the previous argument gives only that f is in “weak Lq ”. To show that f ∈ q L (Rn ) observe that the weak Lq estimate depends only on the gradient of f . Let f be a smooth positive function with bounded support (differences of such functions are dense in L1(1) (Rn )). Put

fm = max 0, min f − 2m , 2m One shows that fm ∈ L1(1) (Rn ) and f =

(m ∈ Z).

∞

−∞ fm

with

∞

Am = x : f (x) 2m+1 = x : fm (x) = 2m ,

−∞ fm L1(1) (Rn )

< ∞. Setting

Sobolev spaces

1399

we infer that the support of the gradient satisfies supp ∇fm ⊂ Am−1 \ Am . Therefore, by the weak type (1, q) inequality, we get ∇fm 1 q λn (Am ) C 2m (here λn stands for the usual Lebesgue measure on Rn ). Thus Rn

|f |q =

∞ m=−∞ Am−1 \Am ∞

2

(m+1)q

|f |q

4q C

∞

2(m+1)q λn (Am−1 \ Am )

m=−∞

λn (Am−1 ) 4 C

m=−∞

∞

q

q

∇fm 1

m=−∞

q ∇fm 1

∞

q

= 4q C∇f 1 .

m=−∞

The last equality holds because the functions ∇fm have pairwise disjoint supports.

R EMARKS . (1) Theorem 33 remains valid if we replace Rn either by a domain with suitable regularity, e.g., any domain satisfying JET (cf. Section 3), or by a compact manifold without boundary, in particular by Tn . (2) The weak type argument is implicitly contained in [62], Chapter I. (3) For general domains the theory is more complicated; there are counterexamples (cf. [1,62]). (4) The reader is referred to books [62,7], the survey [55], and the memoir [39] for extensive literature and comprehensive discussion of the subject. Next we discuss embeddings of L1(1) (Rn ) into Lorentz and Besov spaces. Let h∗ : R → R denote the non-increasing rearrangement of the function h : Rn → R (cf. [92] for definition). Put hp,q =

p q

∞

t

0

q dt h (t) t

1/p ∗

1/q for 0 < p, q < ∞.

The Lorentz space Lp,q (Rn ) consists of all functions f for with f p,q < ∞. Since Ln/(n−1),1 (Rn ) → Ln/(n−1) (Rn ), the next result slightly improves Theorem 33. T HEOREM 34. One has L1(1) (Rn ) → Ln/(n−1),1 (Rn ) for n = 2, 3, . . . . Theorem 34 is a simple consequence of the more subtle embedding theorem into suitable Besov spaces which are in contained the Lorentz spaces in question. θ(p,n)

T HEOREM 35. Let n = 1, 2, . . . . Then L1(1) (Rn ) → Bp,1 1/n − 1) and 1 < p n/(n − 1).

(Rn ) where θ (p, n) = n(1/p +

1400


Recall the definition of Besov spaces due to Peetre [70]. n Let Ψ (Rn ) be the family of all partitions of unity ψ = {ψj (x)}∞ j =0 ⊂ D(R ) such that: (i) supp ψ0 ⊂ {|ξ | 2}, supp ψj ⊂ {2j −1 |ξ | 2j +1 } for j = 1, 2, . . . , (ii) supj ψˆ j 1 < ∞, ∞ n (iii) j =0 ψj (ξ ) = 1 for ξ ∈ R . For fixed ψ ∈ Ψ , 1 p < ∞, 1 q ∞, and 0 θ < ∞ we define for f ∈ S(Rn ) the norm · Bp,q θ (ψ) by θ f Bp,q θ (ψ) = f p + Bp,q (f ; ψ),

where θ Bp,q (f ; ψ) =

∞ q jθ 2 ψˆ j ∗ f p

1/q .

j =0

One can prove that the above norms are equivalent for different partitions of unity in Ψ (cf. [70,78]). θ (Rn ) is the completion of S(Rn ) in the norm · The Besov space Bp,q θ (ψ) for some Bp,q ψ ∈ Ψ . We will also need another norm defined only for θ > 0 which involves the concept of p-th modulus of smoothness ωp (f ; ·). For 1 p < ∞ and for f ∈ Lp (Rn ) we put ωp (f, t) = sup fs − f p , |s|t

where fs (x) = f (x + s) for x, s ∈ Rn .

by Next for 1 q < ∞ and for 0 < θ < 1 we define the norm · Bp,q θ f Bp,q θ

θ = f p + Bp,q (f ),

where

θ Bp,q (f ) =

∞

t

0

−θ

q dt ωp (f, t) t

1/q .

L EMMA 36. For 1 p < ∞, 1 q < ∞, 0 < θ < 1 the norms · Bp,q and · Bp,q θ θ (ψ) are equivalent. For the proof see ([94], Sections 2.3.2 and 2.5.1, [78], Proposition 3.1). Now we are ready for O UTLINE OF THE PROOF OF T HEOREM 35. It is enough to restrict ourselves to a dense set in the cone of non-negative functions in S(Rn ) which are “non-flat”, i.e.,

λn supp f \ {f > 0} = 0 and λn {f = c} = 0

for c > 0.

Our argument bases on a decomposition of non-flat functions in a sum of slices,

fα,β = max α, min(f, β) − α

(α < β).

Sobolev spaces

1401

One shows (cf. [78], Theorem 2.1) that for non-flat f there is an increasing sequence (αm )∞ m=1 such that if we put fm = fαm ,αm+1 then

fm (x) = f (x);

m

∇fm (x) = ∇f (x)

(for x λn -a.e.);

(28)

m

fm 1 = f 1 ;

m

∇fm 1 = ∇f 1 ;

(29)

m 1/n

(n−1)/n

fm ∞ fm 1

C · ∇fm 1

(m = 1, 2, . . .).

(30)

The inequality (30) is non-trivial; it follows from Federer–Kronrod coarea formula (cf. [30], Theorem 3.2.12, [62], 1.2.4, [78], pp. 76–77). Here C, C1 , C2 , . . . denote absolute constants. Using the above decomposition, to prove Theorem 35 for p < n/(n − 1) it is enough to establish for a single function g the inequality

∞

t −θ(p,n) ωp (g, t)

0

dt C1 · ∇g1 , t

(31)

assuming that g satisfies (n−1)/n

g1

1/n

g∞ C · ∇g1 .

(32)

Combining two elementary inequalities: ωp (g, t) ω1 (g, t)1/p (2g∞ )(p−1)/p and ω1 (g, t) t∇g1 , we get

(p−1)/p 1/p ∇g1 . ωp (g, t) t 1/p 2g∞ We split the left-hand side integral of (31),

∞ 0

=

(33) s 0

+

∞ s

where

1/n . s = g1 /g∞ Thus putting Kp,n =

s 0

21−1/p (n−1)(1−1/p) ,

by (33) and (30),

(p−1)/p 1/p t n−2−n/p ωp (g, t) dt 2g∞ ∇g1 =

(p−1)/p 1/p Kp,n g∞ ∇g1

s

t n−2−n/p+1/p dt

0

· s (n−1)(1−1/n) 1/p (n−1)/n 1/n 1−1/n = Kp,n ∇g1 g1 g∞ C2 · ∇g1 .

1402


To estimate the second integral we need the trivial inequality ωp (g, t) 2gp and the inequality 1−n(1−1/p)

gp C3 · g1

n(1−1/p)

∇g1

which follows immediately from the Hölder inequality 1−n(1−1/p)

gp g1

n(1−1/p)

gn/(n−1)

and the Sobolev embedding (Theorem 33). Putting b = 1 + n/p − n and Qp,n = 2C3 b−1 we get ∞ ∞ b t n−2−n/p ωp (g, t) dt C3 · ∇g1−b g t n−2−n/p dt 1 1 s

s

−b/n b = Qp,n ∇g1−b 1 g1 · g1 /g∞ 1−1/n 1/n b g1 = Qp,n ∇g1−b g∞ 1 C4 · ∇g1 .

This completes the proof in the case p < n/(n − 1). The limit case p = n/(n − 1) follows from the reciprocal relations between the Besov spaces with different parameters. Precisely we need the following (cf. [78], Lemma 3.3) L EMMA 37. If 0 1/p − 1/r < 1/n, p 1, and n = 1, 2, . . . , then n(1/p−1/r) n

Bp,q

R

n

0 → Br,q R for 1 q ∞.

This ends the outline.

Using an improvement of Lemma 37 (cf. [78], Lemma 3.3), a norm of Besov space induced by an appropriate partition of unity and analysing the proof of Theorem 35 one gets C OROLLARY 38. For every partition of unity ψ ∈ Ψ there is a constant Cψ such that 0 Bn/(n−1),1 (f ; ψ) Cψ ∇f 1 for f ∈ L1(1) (Rn ). Specifying the partition of unity in Corollary 38 and applying the Hausdorff–Young and Hölder inequalities we obtain T HEOREM 39. There are positive constants C(n) for n = 2, 3, . . . such that

fˆ(ξ ) 1 + |ξ | 1−n dξ C(n)∇f 1 for f ∈ L1 Rn ; (1)

Rn

Rn

fˆ(ξ ) 1 + |ξ | k−n dξ < ∞ for f ∈ L1 Rn (k = 1, 2, . . .). (k)

Sobolev spaces

1403

The counterparts of all the results of this section for periodic functions are also valid. They either can be obtained by adopting the proofs or can be derived from the results for functions in Rn using the Poisson summation formula (cf. [92], Chapter VII, §2). In particular the periodic counterpart of the second inequality of Theorem 39 is C OROLLARY 40. If f ∈ L1(k) (Tn ) then 2, 3, . . .). Here |a|2 =

%

n 2 j =1 aj

ˆ

a∈Zn f (a)(1 + |a|2 )

k−n

< ∞ (k = 1, 2, . . . , n =

for a = (aj ) ∈ Zn .

R EMARKS . (1) Using the duality between Lorentz spaces one can immediately get Theorem 34 from a result of Faris [29]. In the present form it was proved by Poornima [83]. (2) The counterpart of Theorem 35 and Theorem 39, in particular Corollary 40, was proved by Bourgain in unpublished preprints [10,13]. The outline of the proof of Theorem 35 presented here is taken from [78]. Our proof was strongly influenced by Bourgain’s technique developed in the preprints [10,13]. Earlier by a different method Theorem 35 was obtained by Kolyada [55] (cf. also [53] and [54]). Kolyada’s method allows to strengthen Theorem 39 and Corollary 40, roughly speaking, by controlling weighted norms of Fourier transforms by Sobolev type norms involving only pure derivatives. In particular one gets (cf. [53]) C OROLLARY 41. If f and its pure distributional derivatives D (2,0) f and D (0,2) f belong to L1 (R2 ) then fˆ belongs to L1 (R2 ). In view of Theorem 49, Corollary 41 improves the first part of Theorem 39. (3) Theorems 33, 34, 35, as well as equivalence of norms in Besov spaces (Lemma 36) extend to functions with values in an arbitrary Banach space (cf. [78]). It does not apply to results involving weighted norms of Fourier transforms. (4) The Banach ideals properties of Sobolev embeddings have been extensively investigated. For the classical Rellich–Kondrachov theorem on compactness of Sobolev embeddings and its generalizations the reader is referred to [1], Chapter VI, and [62], Chapter V, §5.5. The problematic of s-numbers of Sobolev embeddings is well presented in the book [56]. Recently in [99] it has been observed that the Sobolev embedding p L(k) (Rn ) → Ls (Rn ) where 1/s = 1/p − k/n is (v, 1)-summing for v > v0 but is not 2n , p}. (v0 , 1)-summing for v0 = max{ 2k+n (5) For other aspects of Sobolev embeddings the reader is referred to the survey of Schechtman [86], and the memoir [39]. Embeddings theorems admit generalizations to embeddings with respect to other measures than the Lebesgue measure on Rn . In particular if the measure in question is concentrated on a submanifold in Rn of the n-dimensional Lebesgue measure zero we get so called trace theorems. The simplest example is the case of the Lebesgue measure of a lower-dimensional linear manifold in Rn .

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T HEOREM 42. Let H be a m-dimensional linear manifold in Rn (1 k < n). Let 1 p p q mp/(n − kp). There is the unique bounded linear operator TrH : L(k) (Rn ) → Lq (H ) such that TrH (φ) = φ|H for φ ∈ H . The reader is referred to the books [1,62] and the survey [55].

7. Interpolation in Sobolev spaces For the real interpolation method (cf. [4], [44], Section 11), we know the complete dep q scription of the interpolation spaces between L(k) (Rn ) and L(k) (Rn ) as well as between p p L(k1 ) (Rn ) and L(k2 ) (Rn ). Let us recall some basic concepts of the real method. Let (X0 , X1 ) be a compatible couple of Banach spaces and Yi be a closed subspace of Xi , i = 0, 1. The couple (Y0 , Y1 ) is said to be K-closed in (X0 , X1 ) if there is C > 0 such that whenever y ∈ Y0 + Y1 is represented in the form y = x0 + x1 with xi ∈ Xi for i = 0, 1, then there exists another representation y = y0 + y1 , with yi ∈ Yi and yi Xi Cxi Xi for i = 0, 1. Recall that for the a compatible couple (X0 , X1 ) of Banach spaces the K-functional is defined for each x ∈ X0 + X1 and t > 0 by K(x, t; X0 , X1 ) = inf x0 X0 + tx1 X1 : x = x0 + x1 , where the infimum extends over all representations x = x0 + x1 with x0 ∈ X0 and x1 ∈ X1 . The real interpolation space (X0 , X1 )θ,q consists of all x in X0 + X1 for which the functional

1

xθ,q = 0

q dt t K(x, t; X0 , X1 ) t θ

1/q ,

0 < θ < 1, 1 q < ∞,

is finite. p p The real interpolation for couples (L(k1 ) (Rn ), L(k2 ) (Rn )) leads to Besov spaces α (Rn ) mentioned in Section 6 (cf. [4,5] for definition). Here Lp (Rn ) = Lp (Rn ). Bp,q (0) p

T HEOREM 43 ([4,5,94]). If k1 , k2 are integers with 0 k1 < k2 then (L(k1 ) (Rn ), p α (Rn ) where α = (1 − θ )k + θ k (1 p ∞). L(k2 ) (Rn ))θ,q = Bp,q 1 2 p

The main ingredient in the proof is to show that the K-functional K(f, t; L(0) (Rn ), is equivalent to min(1, t)f p + ωp (f, t). We consider next the case of fixed smoothness and the exponent varying. Let E be a suitable finite-dimensional Hilbert space and L(E, E) denote the space of linear operators on E. Notice that the description of the interpolation space for the couple p q p (L(k) (Rn ), L(k) (Rn )) with 1 < p, q < ∞ is trivial because J (L(k) (Rn )) is a complemented subspace of Lp (Rn , E) via the Sobolev projection. The limit case is much more delicate. The result is due to DeVore and Scherer (cf. [20,21]). It is a consequence of the next result. p L(1) (Rn ))

Sobolev spaces

1405

n T HEOREM 44. An interpolation couple (J (L1(k) (Rn )), J (L∞ (k) (R ))) is K-closed in the couple (L1 (Rn , E), L∞ (Rn , E)). Hence

1 n ∞ n

p L(k) R , L(k) R θ,p = L(k) Rn for θ = 1 − 1/p. We present after [52] a sketch of the proof of Theorem 44 based on an idea of Bourgain [14]. It requires some preparation which involves the concept of Calderon–Zygmund integral operators. Recall that the translation invariant linear operator T : L2 (Rn , E) → L2 (Rn , E) is called a Calderon–Zygmund singular integral operator if there exist a measurable function K : Rn → L(E, E) and a constant C such that (i) for every x0 ∈ Rn , r > 0 and f ∈ L2 (Rn , E) such that supp f ⊂ B(x0 , r) (Tf )(x) = K(x − y)f (y) dy for x ∈ B(x0 , Cr), (ii) for every y ∈ Rn we have K(x + y) − K(x) L(E,E) dx C. |x|>C|y|

By the classical theory of Calderon–Zygmund operators (cf., e.g., [90], Chapter II) T , being a priori of strong type (2, 2), can be extended to the operator of strong type (p, p) for 1 < p < ∞ and of weak type (1, 1) operator. Hence for every f ∈ Lp (Rn , E) (1 p < ∞), Tf is a measurable E-valued function well defined up to the set of measure zero. Let Q : L2 (Rn , E) → L2 (Rn , E) be a Calderon–Zygmund projection (i.e., projection which is a Calderon–Zygmund operator). One can prove that then Q is also a projection on Lp (Rn , E) for 1 < p < ∞; moreover QQf = Qf if f, Qf ∈ L1 (Rn , E) (cf. [52], Lemma 1). This allows us to introduce the following spaces:

HpQ = f ∈ Lp Rn , E : Qf = f a.e. , 1 p < ∞,

Q H∞ = f ∈ L∞ : f, gE dλn = 0 whenever g ∈ L1 Rn , E and Q∗ g = 0 (clearly the conjugate Q∗ is also a Calderon–Zygmund projection). It is well known that the multiplier transforms of multipliers satisfying Hörmander– Mikhlin criterion are translation invariant Calderon–Zygmund operators (cf. remark after this proof). Hence, by the proof of Theorem 4(a), the Sobolev projection PRn ,(k) is P

a Calderon–Zygmund projection for every n, k = 1, 2, . . . and Hp R p J (L(k) (Rn )).

n ,(k)

coincides with

Q

Q

O UTLINE OF THE PROOF OF T HEOREM 44. First we establish that the couple (H1 , H2 ) is K-closed in (L1 , L2 ). We have to show that there exists C > 0 such that for f ∈

1406


L1 (Rn , E) + L2 (Rn , E) such that Qf = f , and for every decomposition f = g + h with g ∈ L1 (Rn , E), h ∈ L2 (Rn , E) there exists a decomposition f = g˜ + h˜ with g˜ ∈ H1Q , Q ˜ 2 Ch2 . Let a = g1 , b = h2 . We apply now h˜ ∈ H2 such that g ˜ 1 Cg1 , h the Calderon–Zygmund decomposition of g on the level t = b2 a −1 (cf. [18]), i.e., the decomposition g = g0 + g1 for which there exists a measurable set Ω ⊂ Rn such that |g0 |E C1 · t,

g0 1 C1 a, g1 1 C1 a, |Qg1 |E C1 a λn (Ω) C1 at −1 , Rn \Ω

(here C1 > 0 is a constant independent of g). We put g˜ = Qg1 , h˜ = Q(g0 + h). We claim 1/2 1/2 that f = g˜ + h˜ is the required decomposition. Indeed, since g0 2 g0 ∞ g0 1 ˜ 2 C2 b where C2 = Q2→2 (C1 + 1). Since Qg1 = g1 + (I − Q)(g0 + C1 b, we have h h), (I − Q)(g0 + h) |Qg1 |E + |g1 |E + g ˜ 1= Rn \Ω

Ω

E

Ω

2C1 a + λn (Ω)1/2 · (I − Q)(g0 − h)2 # 2C1 a + C1 b−1 ag0 − h2 # 2C1 a + C1 b−1 a(C1 b + Cb) C3 a, where C3 > 0 does not depend on the functions involved. To prove that the couple Q Q (H2 , H∞ ) is K-closed in (L2 (Rn , E), L∞ (Rn , E)) we use the trick based on the following lemma due to Pisier (cf. [82,52]). P ISIER ’ S LEMMA . Let (X1 , X2 ) be a compatible couple with X1 ∩ X2 dense in X1 ∪ X2 , and Yi be the closed subspace of Xi (i = 1, 2). Then (Y1 , Y2 ) is K-closed in (X1 , X2 ) iff (Y1⊥ , Y2⊥ ) is K-closed in (X1∗ , X2∗ ). Clearly in view of the lemma it is sufficient to show that ((H∞ )⊥ , (H2 )⊥ ) is K-closed Q I −Q∗ in (L1 (Rn , E), L2 (Rn , E)). But (Hp )⊥ = Hp and we can use what we already established, since I − Q∗ is also Calderon–Zygmund projection. To complete the proof we use Wolff type theorem for K-closedness. It says that one can derive K-closedness of the couple (Y1 , Y2 ) in (X1 , X2 ) from the K-closedness of couples (Y1 , F1 ) in (X1 , E1 ) and (F2 , Y2 ) in (E2 , X2 ), where E0 = (X0 , X1 )θ,p , E1 = (X0 , X1 )δ,q , F0 = (Y0 , Y1 )θ,p , F1 = (Y0 , Y1 )δ,q (0 < θ < δ < 1 and 0 < p < q ∞); for details see (cf. [52], Theorem 2). Q

Q

R EMARK . The fact that the multiplier transform of a function kˆ which satisfies Hörmander’s integral condition is a Calderon–Zygmund operator follows, for example, directly

Sobolev spaces

1407

from Hörmander’s proof (cf. [42], Theorem 7.9.5). If we replace w in formula (7.9.17) in [42] by the difference of point masses δy − δ0 and if I (preserving Hörmanders notation) is a cube centered at 0 and containing y, the same argument as in the original proof of Theorem 7.9.5 yields the condition (ii) of the definition of Calderon–Zygmund (scalar-valued) operators. The case of operators in a finite-dimensional space is similar, it is sufficient to consider separately every entry of the multiplier matrix. Our knowledge of the complex interpolation method in the context of Sobolev spaces is still not satisfactory. For the definition of the complex interpolation method see the survey by Johnson and Lindenstrauss, Section 11 in [44], p. 76. Clearly, as in the case of real interpolation, any non-trivial result has to include at least one limit exponent (i.e., 1 or ∞). To state the next result which provides link between the real and the complex interpolation method we need the concept of Fourier type introduced by Peetre [70]. We say that a Banach space X has a Fourier type p for 1 p 2 if the vector-valued

Fourier transform maps Lp (X) into Lp (X) where 1/p + 1/p = 1. Clearly every Banach space has Fourier type 1, and every Hilbert space has Fourier type 2. The following result from the interpolation theory, due to Peetre (cf. [69,63]), provides the link between the real and the complex interpolation method. L EMMA 45. Let (X0 , X1 ) be a Banach couple such that Xj has Fourier type pj , j = 0, 1. Then (X0 , X1 )θ,pθ ⊂ (X0 , X1 )[θ] , where 0 < θ < 1, and 1/pθ = (1 − θ )/p0 + θ/p1 . The above result yields the following T HEOREM 46 (cf. [63]). For the complex interpolation scale one has 1 n p n

p L(k) R , L(k) R [θ] = L(k)θ Rn , 1/pθ = (1 − θ ) + θ/p, 1 < p < ∞. p

P ROOF. The embedding (L1(k) (Rn ), L2(k) (Rn ))[θ] ⊂ L(k)θ (Rn ) for 1/pθ = 1 − θ/2 is obvious. Since L2(k) (Rn ) is a Hilbert space, it has Fourier type 2. Thus by Theorem 44 and Lemma 45 we get

p

p L(k)θ Rn = L1(k) Rn , L2(k) Rn θ,p ⊂ L1(k) Rn , L(k) Rn [θ] . θ

Hence

p (L1(k) (Rn ), L(k) (Rn ))[θ]

p = L(k)θ (Rn ) for p 2. If p > 2, the boundedness of the p/(p−1) n p (L(k) (R ), L(k) (Rn ))[ 1 ] = L2(k) (Rn ). Thus we complete 2

Sobolev projection implies the proof applying the following result of Wolff (cf. [4,102]) which allows us to “glue” together two interpolation scales.

W OLF ’ S THEOREM . Let Xi (i = 0, 1, 2, 3) be the Banach spaces continuously embedded in a suitable topological vector space. Let 0 < θ < η < 1, θ = λμ and η = (1 − μ)θ + μ. If X1 = (X0 , X2 )[λ] and X2 = (X1 , X3 )[μ] then X1 = (X0 , X3 )[μ] and X2 = (X0 , X3 )[η] . It is an open problem, attributed to P. Jones, to describe the complex interpolation scale n (L1(k) (Rn ), L∞ (k) (R ))[θ] .

1408


8. Anisotropic Sobolev spaces An n-dimensional smoothness is a finite non-empty subset S of Zn+ such that if α ∈ S and p β 0 α ∈ Zn+ then β ∈ S. The anisotropic Sobolev space LS (Ω) (1 p ∞) is the space n of all scalar-valued functions f on open Ω ⊂ R having distributional partial derivatives D αf ∈ Lp (Ω) for α ∈ S, with the norm f S,p,Ω =

α

D f (x)p dx 1/p , for 1 p < ∞, maxα∈S essupx∈Rn D αf (x), for p = ∞. α∈S Ω

By C0S (Ω) we denote the closure of D(Ω) in the norm · S,∞ , and by C S (Ω) – the subspace of L∞ S (Ω) consisting of functions which together with their partial derivatives ∂ α with α ∈ S are uniformly continuous on Ω and vanish at infinity. The smoothnesses {Zn+ : |α| k} are called isotropic smoothnesses (k = 0, 1, . . . , n = 1, 2, . . .). Note that every one-dimensional smoothness is isotropic. The classical Sobolev spaces are these which correspond to isotropic smoothnesses; they are also called isotropic Sobolev spaces. The Sobolev spaces on Tn can be defined, for instance, as the closure of trigonometric polynomials with period 1 with respect to each coordinate in the norm · S,p,I n . In general rotations of coordinates do not preserve anisotropic smoothnesses. Thus only for a very few manifolds the anisotropic Sobolev spaces can be reasonably defined. In particular we do not know how to define them on Sn for n 2. A smoothness S is uniquely defined by the set S # of its maximal elements (in the partial order 0); α ∈ S is maximal provided that for every β ∈ S the condition α 0 β implies β = α. It is natural to ask the following question: given a subset S0 of a set S ⊂ Zn+ , not necessarily a smoothness, under what conditions the norms · S,p,Ω and · S ,p,Ω are equivalent? A satisfactory solution for Ω = Rn and p = 1 follows from the next two results. It is convenient to identify Zn+ , hence also S, with an appropriate subset of Rn . T HEOREM 47 ([43]). Let 1 0 such that γ D f C · D αf p p

(34)

α∈S

if and only if γ ∈ conv S. For p = ∞ the condition is more involved. T HEOREM 48 ([8]). Let S be as above and let γ ∈ Zn+ \ S. Then for p = ∞ (34) holds if and only if (∗) there exists an integer k, 0 k n, and a k-dimensional affine subspace Lk ⊂ Rn parallel to some k-dimensional coordinate space, such that γ is an internal point (with respect to Lk ) of conv(Lk ∩ S).

Sobolev spaces

1409

For p = 1 the condition (∗) is known to be necessary. It is not known whether (∗) is sufficient. A partial result is due to Ornstein. T HEOREM 49 ([68]). If γ ∈ Zn+ \ S and all the maximal elements of S (with respect to the partial order “0”) are of the same order as γ then (34) does not hold for p = 1. In particular Theorem 49 yields that if S = {(0, 0), (1, 0), (0, 1), (2, 0), (0, 2)} then the embedding L1(1) (R2 ) → L1S (R2 ) is not a surjection; in other words the L1 -norm of the mixed derivative D (1,1) is not controlled by the L1 -norms of the pure derivatives of the second order. R EMARKS . (1) Using the Transference Theorem (cf. Section 2) we derive from Theorems 47 and 48 their counterparts for Tn . They also imply similar results for domains with the following individual extension property: for each multiindex α and every p ∈ [1, ∞] there is a positive constant Cα,p such that every C ∞ function f on Ω extends to a C ∞ function f˜ on Rn so that ∂ α f˜Lp (Rn ) Cα,p ∂ α f Lp (Ω) . It is easy to see that starlike domains have this property. (2) The reader is referred to [7] for far reaching generalizations of Theorems 47 and 48 involving fractional derivatives and mixed norms. (3) Mityagin [64] after Il’in [43] and prior to Boman [8] constructed elegant examples showing that condition γ ∈ conv S is not sufficient for (34) in the case p = ∞. (4) The reader is referred to [97] for results partially complementing Theorem 49. They are based on Theorem 65 below. Similarly as in the case of isotropic Sobolev . spaces in Section 2 we define the canonp ical embedding J = JΩ,S,p : LS (Ω) → α∈S Lp (Ω) and the Sobolev projection P = . . p PΩ,S : α∈S Lp (Ω) → J (LS (Ω)) where α∈S Lp (Ω) is the space of tuples (fα )α∈S equipped with the norm (fα )p =

α∈S

p

fα Lp (Ω)

1/p

, for 1 p < ∞,

maxα∈S fα L∞ (Ω) ,

for p = ∞.

. . . The spaces α∈S C(Ω) and α∈S C0 (Ω) are defined similarly. The space α∈S Lp (Ω) is naturally isomorphic to Lp (Ω, E), where E is an appropriate finite-dimensional Hilbert space. For a fixed n-dimensional smoothness S we put for ξ ∈ Rn QS (ξ ) =

ξ 2α ;

mα,β (ξ ) = i |α|−|β| ξ α ξ β Q−1 S (ξ );

α∈S

Tmα,β = Tα,β

(α, β ∈ S).

QS is called the fundamental polynomial of S.

1410


The Sobolev projection is given by

PRn ,S (fα ) =

Tα,β (fβ )

α∈S

α∈S

/ (fα ) ∈ S Rn . α∈S

As in the isotropic case the Lp -boundedness and weak type (1, 1) of a Sobolev projection PRn ,S is equivalent to the same properties of multiplier transforms Tmα,β for all α, β ∈ S. This allows us to extend part (b) of Theorem 4 to arbitrary smoothnesses. T HEOREM 50 ([75]). PRn ,S is of strong type (p, p) for 1 < p < ∞. The proof differs from the proof of Theorem 4. Instead of the Hörmander–Mikhlin criterion one uses the Marcinkiewicz multidimensional multiplier theorem (cf. [90], Chapter IV, §6, Theorem 6) to show that each of the multipliers mα,β is of strong type (p, p) for 1 < p < ∞. Note that part (c) of Theorem 4 extends trivially to arbitrary smoothnesses because every one-dimensional smoothness is isotropic. Part (a) of Theorem 4 does not extend to arbitrary smoothnesses. Examples of smoothnesses whose Sobolev projections fail to be of weak type (1, 1) are shown later. There is a transparent characterization of smoothnesses whose Sobolev projections are of strong types (1, 1) and (∞, ∞). T HEOREM 51. For an n-dimensional smoothness S the following are equivalent (i) PRn ,S is of strong type (1, 1); (ii) PRn ,S is of strong type (∞, ∞); (iii) PTn ,S is of strong type (1, 1); (iv) PTn ,S is of strong type (∞, ∞); (v) S has exactly one maximal element. P ROOF. The equivalences (i) ⇔ (ii) and (iii) ⇔ (iv) follow by duality. The implication (i) ⇒ (iii) is a consequence of the Transference Theorem (cf. Section 2). It can be easily verified that (v) implies that S is a Cartesian product of one-dimensional smoothnesses, hence PS is a tensor product of Sobolev projections of these one-dimensional smoothnesses. Since every one-dimensional smoothness is of strong type (1, 1) (cf. Section 2, Theorem 4(c)), its tensor powers have the same property. Thus (v) ⇒ (i). It remains to show that “non(v)” ⇒ “non(iii)”. Recall that PTn ,S is of strong type (1, 1) for some smoothness S iff mα,β is a Fourier transform of a measure from M(Tn ) for every α ∈ S and β ∈ S, where M(Tn ) denotes the space of scalar-valued Borel measures on Tn with finite total variation. We need (for the proof see [48,72]) W IENER CRITERION . If f = μˆ then lim

d(I )→∞ I

f dσ = μ {0} f : Zn → C, μ ∈ M Tn . σ (I )

Sobolev spaces

1411

Here I = I (b; r) := {a ∈ Zn : |aj − bj | rj for j = 1, 2, . . . , n} for r, b ∈ Zn , d(I ) = min1j n |rj |, and σ is the counting measure on Zn . We shall show that “non(v)” implies that mα,α violates Wiener criterion for some α ∈ S. Indeed “non(v)” yields that given α ∈ S there is a β ∈ S such that αj < βj for some (k) j ∈ {1, 2, . . ., n}. For k = 1, 2, . . . define Ik = I (b (k) ; r (k)) by rj = k for j = 1, 2, . . . , n; bj = k for j = j and bj = 4k . Then σ (Ik ) = (2k + 1)n and d(Ik ) = k; if a ∈ Ik then (k)

(k)

a 2α (2k)2|α|(4k + k)2αj and QS (a) (4k − k)2βj . Thus, remembering that mα,α (a) = a 2α /QS (a), we get lim

k→∞ Ik

mα,α (a) (2k)2|α| (4k + k)2αj dσ = lim = 0. k→∞ σ (Ik ) (4k − k)2βj

If mα,α satisfied the Wiener criterion then the limit would not depend on a particular choice of a sequence (Ik ). Since α ∈ S has been taken arbitrarily, we would have lim

d(I )→∞ I

mα,α (a) dσ = 0, σ (I )

α∈S

which contradicts the identity

α∈S mα,α

= 1.

It is an open, probably difficult, problem to characterize these smoothnesses for which the Sobolev projection is of weak type (1, 1). First we enlist some “positive” results. T HEOREM 52 ([6]). Every two-dimensional smoothness is of weak type (1, 1). T HEOREM 53 ([72]). If the fundamental polynomial QS is h-elliptic, in particular if QS is elliptic, then PRn ,S is of weak type (1, 1). Recall that a mixed homogeneity is a vector h = (h1 , h2 , . . . , hn ) ∈ Zn with 1 h1 h2 · · · hn . A polynomial Q : Rn → C is said to be h-elliptic provided that there are | > R where ρh (ξ ) = 0 for C > 0 and R > 0 such that (ρh )degh (Q)(ξ ) C|Q(ξ )| for |ξ ξ = 0 and ρh (ξ ) is the unique positive root of the equation nj=1 ξj2 ρ −2hj = 1 for ξ = (ξj ) ∈ Rn , and degh (Q) is the maximum of h-degrees of monomials appearing in the shortest representation of Q as the sum of monomials; the h-degree of the monomial ξ α is h, α for α ∈ Zn+ . C OROLLARY 54 ([72]). Let S be an n-dimensional smoothness such that the set of maximal elements of S is a set of n pure derivatives. Then PRn ,S is of weak type (1, 1). The proofs of Theorems 52 and 53 are based upon the following (cf. [28], p. 28, [84]).

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T HE FABES –R IVIÈRE CRITERION . Let h be a mixed homogeneity. Let m : Rn → C be a n bounded function. Let s be an integer such that 2s > |h| = j =1 hj . Assume that for every |α| s the function m has continuous in Rn \ {0} partial derivative ∂ α m which satisfies

h,α α 2 R ∂ m(x) R −|h| dx < ∞.

sup R>0

R/2ρh (x)2R

Then m is a weak type (1, 1) multiplier. For h = (1, 1, . . . , 1) we get the classical Hörmander–Mikhlin criterion which we used to prove Theorem 4(a). The (1, 1, . . . , 1)-elliptic polynomials coincide with classical elliptic polynomials. A satisfactory characterization of smoothnesses S with PRn ,S of weak type (1, 1) is also known for smoothnesses of ord S 2, where ord S := max{|α|: α ∈ S}. A smoothness S is non-degenerated if all maximal elements of S are of order 1; it is reducible if ξj divides all symbols of all maximal elements of S for some j ∈ {1, 2, . . ., n}. T HEOREM 55 ([6]). Let S be a non-degenerated smoothness of order 2. Then (i) if ord S = 1 then S is isotropic hence PRn ,S is of weak type (1, 1); (ii) if S is reducible then PRn ,S is of weak type (1, 1); (iii) if S is irreducible and all maximal elements of S are of order 2 then the Sobolev projection is of weak type (1, 1) iff for every i, j ∈ {1, 2, . . . , n}, i = j , either ∂ 2 /∂xj2 ∈ S and ∂ 2 /∂xi2 ∈ S, or ∂ 2 /∂xi ∂xj ∈ S; (iv) if ∂/∂xi are the maximal elements of S for i ∈ A ⊂ {1, 2, . . . , n} and A = ∅ then the Sobolev projection is of weak type (1, 1) iff ∂ 2 /∂xj2 ∈ S for every j ∈ A. The proof of Theorem 55 is complicated and uses a variety of methods (cf. [6]). It follows from Theorem 55(iii) and (iv) that for n 3 there are examples of n-dimensional smoothnesses whose Sobolev projections fail to be of weak type (1, 1). In particular the Sobolev projections are not of weak type (1, 1) for the three-dimensional smoothnesses T3 and S with maximal elements T3# = {(1, 0, 0), (0, 1, 1)} and S # = {(2, 0, 0), (0, 1, 1)}. Yet another example is mentioned in remark (3) below. R EMARKS . (1) Theorem 51 is contained in [75] and (implicitly) in [87]. The arguments there as well as in [51] are rather complicated. The simple argument presented here is taken from [72]; it is based on the idea of J.-P. Kahane to apply Wiener criterion (private communication in 1987). (2) The counterparts of Theorems 52, 53, 55 and Corollary 54 for Tn holds. (3) A simple tool to study Sobolev projections . uses the quantity ap (PS ) = the norm of the projection regarded as an operator on α∈S Lp (Rn ). By duality ap (PS ) = ap (PS ) for p = p/(p − 1). Moreover, by the Marcinkiewicz interpolation theorem (cf. [107], XII (4.6)), if PS is of weak type (1, 1) then ap (PS ) C max(p, p ) for some constant C independent of p. This is not the case for some smoothness. For instance, if S is a fourdimensional smoothness being the Cartesian product of two two-dimensional isotropic smoothnesses, i.e., S # = {(1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1)} then PR4 ,S is the

Sobolev spaces

1413

second tensor power of PR2 ,(1) thus ap (PS ) = [ap (PR2 ,(1))]2 [A(1, 2) max(p, p/(p − 1))]2 (cf. Section 2, remark (2) after Proposition 7). (4) If the n-dimensional smoothness Tn is defined by Tn# = {(1, 0, . . . , 0), (0, 1, . . ., 1)} then ap (PTn ) Cn [max(p, p/(p − 1))]n−1 for some Cn > 0 independent of p (cf. [6], Proposition 2.47). One conjectures that for every n-dimensional smoothness S one has ap (PRn ,S ) CS [max(p, p/(p − 1))]n−1 for some CS > 0 independent of p. We know only that the above estimate is true if we replace n − 1 by n (cf. [100]). (5) Another open problem related to the theory of rational multipliers on Rn is the following. Is it true that for every smoothness S there exists a non-negative integer r such that C[max(p, p/(p − 1))]r ap (PS ) C −1 [max(p, p/(p − 1)]r for some C > 0 independent of p? Theorem 6 (cf. Section 2) extends with almost the same proof to arbitrary smoothnesses. T HEOREM 56. The multiplier transform T1/√QS extends to an isomorphism from Lp (Rn ) p onto LS (Rn ) for 1 < p < ∞ (k = 0, 1, . . . , n = 1, 2, . . .). Applying the Transference Theorem (cf. Section 2) we get the counterpart of Theorem 56 for Tn . We do not know the analogs of the extension theorems WET and JET (cf. Section 3) for anisotropic smoothnesses. We do not know whether C S (Tn ) and L1S (Tn ) are infinitely divisible; in particular we do not know whether C S (Tn ) (resp. L1 (Tn )) is isomorphic to C S (Rn ) (resp. L1S (Rn )) for anisotropic smoothnesses. The next result suffices to prove various properties of C S (Ω) and L1S (Ω) for arbitrary ∅ = Ω ⊂ Rn . p

T HEOREM 57. Let ∅ = Ω ⊂ Rn . Then LS (Tn ) for 1 p ∞ (resp. C S (Tn )) is isomorp phic to a complemented subspace of LS (Ω) for 1 p ∞ (resp. C S (Ω)). P ROOF. All the operators constructed below are well defined for C ∞ functions. Examp ining the formulae we check that the operators are bounded in LS -norms. Thus they can be extended to bounded operators in appropriate Sobolev spaces in p-th norms. We put K n = (−1; 1)n , Qn = (0; 1)n , Hj− = {x = (xj ) ∈ Rn : xj < 0}, Hj0 = {x ∈ Rn : xj = 0}, Kj,n = K n ∩ Hj− , ej – the j -th coordinate versor, A – the closure of a set A ⊂ Rn (j = 1, 2, . . . , n; n = 1, 2, . . .). k r r Step I. Let F (z) = ∞ k=0 ak z be an entire function such that F (2 ) = (−1) for r = ∞ 0, 1, . . . . Let φ be a C non-negative function such that φ(t) = 1 for −1 < t < 0, and φ(t) = 0 for t > 4/3. Put ⎧ (x), for x ∈ Kj,n , ⎪ ⎨f k Ej f (x) = k ak φ(xj )f (x1 , x2 , . . . , xj −1 , −2 xj , xj +1 , . . . , xn ), ⎪ ⎩ for x ∈ K n \ Kj,n .

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Then Ej : C S (Kj,n ) → C S (K n ) is a linear extension operator (cf. [67] for details). The continuity of Ej in the Sobolev norm defined by S follows from the definition of smoothness and the Leibniz formula cα,β ∂ α−β φ∂ β f, ∂ α (φf ) = 00β0α

where cα,β are appropriate “binomial” coefficients. Step II. Put n

S Cj,0 Q = f ∈ C S Qn : lim ∂ α f (y) = 0 for x ∈ Hj0 ∩ Qn ; α∈S

C0S

n

Q

=

n

y→x

n

S Q . Cj,0

j =1

Then (+) C0S (Qn ) is a complemented subspace C S (K n ); (++) C0S (Qn ) is isomorphic to C S (Tn ). To establish (+) it is enough to construct a linear extension operator, say E : C S (K n \ Qn ) → C S (K n ). Then the desired projection is IdC S (K n ) −E; we can regard C0S (Qn ) as a subspace of C S (K n ) extending each function by 0. Let Rj denote the restriction operator to Kj,n for functions defined on a superset of Kj,n . Put Λj := Ej Rj : C S (K n \ Qn ) → C S (K n ). Then the desired extension operator is E=

(−1)k Λjk ◦ Λjk−1 ◦ · · · ◦ Λj1 ;

the sum extends over all sequences of indices n jk > jk−1 > · · · > j1 1 (k = 1, 2, . . . , n). To verify (++) let Zj = Qn + ej and Qj,n = Qn ∪ Zj \ bd Qn ∪ Zj . Denote by Ej : C S (Qn ) → C S (Qj,n ) the linear extension operator which is an obvious modification of Ej defined in Step I. Note that limy→x (∂ α Ej f )(y) = 0 for α ∈ S, x ∈ Qj,n ∩ {Hj0 + 4/3ej }, f ∈ CS (Qn ). Let R|Zj denote the operator of restriction to Zj . Put Nj f (x) =

f (x) + R|Zj Ej f (x + ej ), f (x),

for x ∈ Qn with 0 < xj < 1/3, otherwise.

S (Qn ) onto the subspace of C S (Q ) consisting Then Nj is an isomorphism from Cj,0 n of functions which extend continuously together with their partial derivatives from S to 1-periodic function with respect to j -th coordinate. The inverse of Nj is the operator Mj defined by

f (x) − R|Zj Ej f (x + ej ), for x ∈ Qn with 0 < xj < 1/3, Mj f (x) = f (x), otherwise.

Sobolev spaces

1415

Clearly, C S (Tn ) can be identified with the subspace of C S (Qn ) consisting of functions on Qn which extend continuously on Rn together with their partial derivatives from S to 1-periodic functions with respect to all coordinates. The desired isomorphism from C0S (Qn ) onto C S (Tn ) is defined by N = Nn ◦ Nn−1 ◦ · · · ◦ N1 ; the inverse is defined by M = M1 ◦ M2 ◦ · · · ◦ Mn . This concludes the proof of Step II. Finally observe that by a standard modification of the extension operator constructed in Step I we construct an extension operator from C S (K n ) into C0S (2K n ). Thus C S (K n ) is isomorphic to a complemented subspace of C0S (2K n ). This suffices to get the assertion of the theorem. Our next result is a generalization of Theorems 13 and 25. T HEOREM 58. Let S be an arbitrary n-dimensional smoothness for some n 2. Then the following are equivalent (v) S has exactly one maximal element; (a1 ) L1S (Tn ) (resp. L1S (Rn )) is isomorphic to L1 [0; 1]; (a2 ) L1S (Tn ) (resp. L1S (Rn )) is an L1 -space; (a3 ) every linear operator from L1S (Tn ) (resp. L1S (Rn )) to a Hilbert space is 1-summing; (b1 ) C S (Tn ) (resp. C S (Rn )) is isomorphic to C[0; 1]; (b2 ) C S (Tn ) (resp. C S (Rn )) is a L∞ -space; (b3 ) C S (Tn ) (resp. C S (Rn )) has GL. The proof of Theorem 58 is not straightforward. It bases on the following P ROPOSITION 59 (Solonnikov [89]). Let r, s be positive integers. Then there are positive constants C = C(r, s) and C = C (r, s) such that 2 (r,0) (0,s) ∂ ∂ (A) |ξ1 |r−1 |ξ2 |s−1 fˆ(ξ ) dξ C f (x) dx f (x) dx R2

R2

2

R2

for f ∈ S R ; r s 2 (0,l) r−1 t −1

(k,0) fˆ(a) |a1 | |a2 | ∂ C f L1 (T2 ) ∂ f L1 (T2 ) (B) a∈Z2

for f ∈ Trig T .

k=0 l=0

2

Applying the Plancherel identity to the left-hand sides of (A) and (B) one can view Proposition 59 as a Sobolev embedding type result. The case r = s = 1 is the classical Sobolev embedding for k = 1, n = 2, p = 1, q = 2. The proof of part (A) is easy for both r and s odd. We use the identity 1R− ×R+ $ ∂x∂ 1 h, ∂x∂ 2 g = h, g for h, g ∈ S(R2 ), which is a slight modification of the identity (∗) in Section 4. Apply that identity for h = ∂ (r−1,0)f , g = ∂ (0,s−1)f , and use Plancherel identity and the Hausdorff–Young inequality. For other pairs of positive integers r, s the argument is more sophisticated (cf. [89,75,51]).

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Part (B) follows from (A) routinely. Fix φ ∈ D(Rn ) with φ(x) = 1 for x ∈ I 2 and φ(x) = 0 for x ∈ / 2I 2 . Apply (A) to f · φ where f is here understand as 1-periodic with respect to each coordinate function defined on R2 . Use Leibniz formula for derivatives of the product. To reduce the proof of Theorem 58 to two-dimensional smoothnesses we need L EMMA 60 ([51]). Let n > 2. Then for every n-dimensional smoothness S which has more than one maximal element there exists a two-dimensional smoothness T which has more p than one maximal element such that LS (Rn ) for 1 p ∞ (resp. C S (Rn )) has a comp plemented subspace isomorphic LT (R2 ) (resp. C T (R2 )). The same is true for periodic models. O UTLINE OF THE PROOF. We consider the case of Rn ; the case of Tn can be proved similarly. If S has more than one maximal element then there is an ordering of coordinates and β, γ ∈ S # such that |β| = ord S and β1 < γ1 . Define Φ : Zn → Zn−1 by Φ(α) = (α1 , α2 , . . . , αn−2 , αn−1 + αn ). Then Φ(S) is an (n − 1)-dimensional smoothness with Φ(β) = Φ(γ ) and Φ(β), Φ(γ ) ∈ [Φ(S)]# . Denote by XS (resp. XΦ(S) ) one of the p p spaces LS (Rn ) for 1 p ∞ or C S (Rn ) (resp. LΦ(S) (Rn−1 ) or C Φ(S) (Rn−1 )). Fix a non-negative h ∈ D(R) with h(t) = h(−t) for t ∈ R and R h2 (t) dt = 1. Denote by z◦ the first n − 2 coordinates of a vector z. Define U : XΦ(S) → XS and P : XS → XΦ(S) by

(Ug) x ◦ , xn−1 , xn = g x ◦ , xn−1 + xn h (xn−1 − xn )/2

g ∈ XΦ(S) , x = (xj ) ∈ Rn , ◦

(Pf ) y , yn−1 = f y ◦ , t, yn−1 − t h(yn−1 /2 + t) dt

R

f ∈ XS , y = (yj ) ∈ Rn−1 .

One verifies that U and P are bounded operators satisfying P U = IdXΦ(S) (cf. [75], Lemma 5.3 for details). Now the desired conclusion follows by backward induction. P ROOF OF T HEOREM 58. Same argument as for isotropic spaces shows that L1S (Tn ) (resp. C S (Tn )) has a complemented subspace isomorphic to L1 [0; 1] (resp. to C(0; 1)). Thus the implications (v) ⇒ (a1 ) and (v) ⇒ (b1 ) follow from Theorem 51 by the standard decomposition method. The implications (a1 ) ⇒ (a2 ) ⇒ (a3 ) and (b1 ) ⇒ (b2 ) ⇒ (b3 ) are formal. Thus in view of Theorem 57 and Lemma 60 to complete the proof it is enough to work with periodic models and to establish the implications “non(v)” ⇒ “non(a3)” and “non(v)” ⇒ “non(b3)” under the additional assumption that S is a two-dimensional smoothness. If S is a two-dimensional smoothness with more than one maximal element then there are α and β in S # such that r = β1 − α1 1, s = α2 − β2 1, and the line passing through α and β supports conv S, equivalently rγ2 + sγ1 rα2 + sα1

for γ = (γ1 , γ2 ) ∈ S.

(35)

Sobolev spaces

1417

Apply the inequality (B) of Proposition 59 to f = ∂ (α1 ,β2 ) g for g ∈ Trig T2 . We get 2 g(a) ˆ |a1 |α1 +β1 −1 |a2 |α2 +β2 −1 C g2 1

LS (T2 )

a∈Z2

.

(36)

Define V : L1S (T2 ) → L2S (T2 ) by Vg =

7 g(a) ˆ

a∈Z2

|a1 |(α1 +β1 −1) |a2 |(α2 +β2 −1) ea QS (a)

for g ∈ L1S T2 .

Then using that ( √Qea (a) )a∈Z2 is an orthonormal system in L2S (T2 ) and (36), we infer that S

V is a bounded operator. Thus V I2,1 , where I2,1 : L2S (T2 ) → L1S (T2 ) is the natural embed% α +β −1 α +β −1 1 1 |a2 | 2 2 ding, is not Hilbert–Schmidt. Indeed V (ea ) = |a1 | ea for a ∈ Z2 . RevokQS (a) ing (35) we get |a1 |α1 +β1 −1 |a2 |α2 +β2 −1 #

V I2,1 ea / QS (a) 2 2 2 = = +∞. LS (T ) QS (a) 2 2

a∈Z

a∈Z

Thus V is not 1-summing. The proof of the implication “non(v) ⇒ non(b3)” for two-dimensional smoothnesses is a repetition with a few inessential changes of the proof of Theorem 25. First observe that Proposition 27 extends from isotropic case to the case of two-dimensional smoothnesses, because, by Theorem 52, PS is of weak type (1, 1). Next as in the proof of Theorem 13 we show that U = V I∞,1 (where I∞,1 : CS (T2 ) → L1S (T2 ) is the natural embedding and V : L1S (T2 ) → L2S (T2 ) is just constructed bounded not 1-summing operator) is an example of 1-summing operator which is not L1 -factorable. In the spirit of Theorem 58 is also T HEOREM 61 ([81], Section 8). An n-dimensional smoothness S has exactly one maximal element iff L1S (Rn ) is isomorphic to a complemented subspace of a Banach lattice. Proof of Theorem 61 is based on the same idea as the proof of Theorem 15. It requires a version of Peetre’s Theorem 20 for anisotropic Sobolev spaces spaces. It is natural to ask which anisotropic smoothnesses share properties of spaces of continuous functions discussed in Section 5. For an n-dimensional smoothness S put S∗ := S \ S # ; i.e., S∗ is the set of all nonmaximal elements of S. Clearly if S = {0} then S∗ is a smoothness and the inclusions C S (Ω) → C S∗ (Ω) and C0S (Ω) → C0S∗ (Ω) are bounded for Ω ⊂ Rn . If Ω = Rn then C S (Rn ) → C S∗ (Rn ) is compact for no S = {0}! Call an n-dimensional smoothness S of Ascoli type provided that the inclusion C S (I n ) → C S∗ (I n ) is compact. Clearly isotropic smoothnesses are of Ascoli type and S is of Ascoli type iff for every (equivalently for

1418


some) open bounded ∅ = Ω ⊂ Rn or for Tn the corresponding inclusions are compact. One can show that S = {0} is Ascoli type iff α ∈ S∗ implies α + t (1, 1, . . . , 1) belongs to the interior of conv S for small t > 0. T HEOREM 62. If S is of Ascoli type then the duals of C S (Tn ), C S (Ω) and C0S (Ω) for ∅ = Ω ⊂ Rn are separable perturbations of the dual of C(0; 1). The proof reduces in view of Theorem 57 to the case of C(Rn ). The proof in this case can be found in [73], Theorem A; it is a slight modification of the proof of Theorem 28 (cf. Section 5). Unfortunately in [73] in the formulation of Theorem A it is erroneously stated that “the inclusion C S (Rn ) → C S∗ (Rn ) is compact” instead of “S is of Ascoli type”. Slightly modifying the proof of Lemma 31 we get L EMMA 63.. If an n-dimensional smoothness S is of Ascoli type then J (C S (Tn )) is a rich subspace of α∈S C(Tn ) = C(Tn , E). Thus invoking Theorem 30 we get C OROLLARY 64. If an n-dimensional smoothness S is of Ascoli type then C S (Tn ) has properties (a), (b), (c) stated in Section 5. R EMARKS . (1) Theorem 58 is essentially due to Kislyakov and Sidorenko [51]; in [75] a similar result is proved with (a3 ) replaced by the weaker condition “C S (Rn ) (resp. C S (Tn )) is not isomorphic to a quotient of L∞ -space”. (2) We do not know the characterization of these smoothnesses that the dual of C S (Tn ) is a separable perturbation of an L1 -space. Every two-dimensional smoothness has this property (cf. [73], Proposition 3.2). On the other hand if S and S

are smoothnesses such that S = {0} and S

has more than one maximal element then the dual of C S (Tn ) is not a separable perturbation of an L1 -space where S = S ×S

and n = dim S +dim S

(cf. [73], Proposition 3.1). An interesting question concerning Banach space properties of anisotropic Sobolev spaces is whether L1S (Rn ) contains a complemented infinite-dimensional Hilbertian subspace. The answer is unknown for isotropic spaces. Surprisingly, we are able to construct complemented invariant infinite-dimensional Hilbertian subspaces of some anisotropic Sobolev spaces. Those invariant projections are related to the Paley projections in H 1 (cf. [17], p. 275). For an n-dimensional smoothness S and an m-element set A ⊂ {1, 2, . . . , n} put S|A = pA (S) where pA is the projection defined by pA (x) = (xj )j ∈A for x = (xj )nj=1 ∈ Rn . T HEOREM 65 ([76]). Let S be an n-dimensional smoothness. The space L1S (Tn ) contains an invariant complemented infinite-dimensional Hilbertian subspace iff (2) there is ∅ = A ⊂ {1, 2, . . . , n} such that (S|A )# contains two elements α, β such that |α| − |β| is an odd integer.

Sobolev spaces

1419

The projection is given by the idempotent multiplier which is a characteristic function of some Hadamard lacunary sequence of characters of Tn . We indicate the idea of the proof in the simplest case of the two-dimensional smoothness S = {(2, 0), (1, 0), (0, 1), (0, 0)}. Let (nk ) be sufficiently fast growing sequence of positive integers (for example, nk = k! for k = 1, 2, . . .). Put ak = (nk , n2k ) and let H be the (Hilbertian) subspace of L1S (T2 ) spanned by the characters {eak : k = 1, 2, . . .}. Then the characteristic function of {ak : k = 1, 2, . . .} induced a bounded multiplier transform on * L1S (T2 ). The proof of that statement goes as follows: put Rk (x) = kj =1 (1 + cosx, aj ) and let R be a weak-star limit of Rk . By the classical Riesz products theory, R is a prob k ) = 1 for k = 1, 2, . . . . One can prove (this is the place abilistic measure such that R(a where the property that (2, 0) and (0, 1) have different order modulo 2 is used) that for every f ∈ J (L1S (T2 )) the convolution R ∗ f belongs to the E-valued Hardy space in 2variables-H 1(T2 , E) where E is the four-dimensional Hilbert space. Then we apply the restriction to R ∗ J (L1S (T2 ) of the appropriate Paley projection (cf. [76] for details). If S fails (2), then a complete description of all complemented invariant subspaces of L1S (Tn ) is known, it is the same as for L1 (Tn ) (cf. [85]). T HEOREM 66 ([98]). The following dichotomy holds: for every smoothness S either L1S (Tn ) contains an infinite-dimensional complemented invariant Hilbertian subspace, or every invariant projection in L1S (Tn ) is a convolution with an idempotent measure on Tn . In contrast for the sup-norm we have T HEOREM 67 ([98]). For each n-dimensional smoothness S every invariant projection in C S (Tn ) is a convolution with an idempotent measure on Tn . A consequence of Theorem 65 is C OROLLARY 68 ([77]). An n-dimensional smoothness S satisfies (2) iff there exists an 1-summing invariant surjection from C S (Tn ) onto an infinite-dimensional subspace of L2S (Tn ). Note that the isotropic smoothnesses do not satisfy (2). Hence L1(k) (Tn ) does not have invariant infinite-dimensional Hilbertian subspaces; the invariant projections in the space are only convolutions with idempotent measures (n, k = 1, 2, . . .).

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CHAPTER 33

Operator Spaces Gilles Pisier∗ Équipe d’Analyse, Université Paris VI, Case 186, F-75252 Paris Cedex 05, France Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Minimal tensor product . . . . . . . . . . . . . . . . . . . . . . . 2. Ruan’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Quotient, ultraproduct and interpolation . . . . . . . . . . . . . . 5. Projective tensor product . . . . . . . . . . . . . . . . . . . . . . 6. Haagerup tensor product . . . . . . . . . . . . . . . . . . . . . . 7. Characterizations of operator algebras and modules . . . . . . . 8. The operator Hilbert space OH and non-commutative Lp -spaces 9. Local theory and exactness . . . . . . . . . . . . . . . . . . . . . 10. Applications to tensor products of C ∗ -algebras . . . . . . . . . . 11. Local reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Injective and projective operator spaces . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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0. Introduction The theory of “Operator Spaces” is quite recent. The starting point is the thesis of Ruan [94] who gave an “abstract” characterization of operator spaces. Soon after, Blecher and Paulsen [13] and Effros and Ruan [32] independently discovered that this characterization allows the introduction of a duality in the category of operator spaces and they developed the theory systematically from that point on (cf. [32–36,30,7–9,80]). The notion of operator space is intermediate between that of Banach space and that of C ∗ -algebra. They could also be called “non-commutative Banach spaces” (but the commutative case should be included!) or else “Quantum Banach spaces” (but “quantum” has been used so many times already. . .). An operator space (sometimes in short o.s.) is simply a closed subspace E ⊂ B(H ) of the space B(H ) of all bounded operators on a Hilbert space. This definition is a bit disconcerting: every Banach space E admits (for a suitable H ) an ⊂ B(H ), therefore all Banach spaces can appear as operator spaces. But isometric copy E the novelty is in the morphisms (and the isomorphisms) which are not those of the category of Banach spaces. Instead of bounded linear maps, we use as morphisms the completely bounded (in short c.b.) ones which appeared as a powerful tool in the early 80’s (see [79]) but were already implicit in the pioneering work of Stinespring (1955) and Arveson (1969) on completely positive maps, [4]. The underlying idea is the following: given two operator spaces: E1 ⊂ B(H1 ),

E2 ⊂ B(H2 ),

we want morphisms which respect the realizations of the Banach spaces E1 and E2 as operator spaces. For instance, if there exists a representation π : B(H1 ) → B(H2 ) (i.e., we have π(xy ∗) = π(x)π(y)∗ , π(1) = 1 whence π = 1) such that π(E1 ) ⊂ E2 , then the “restriction” π|E1 : E1 → E2 must clearly be accepted among morphisms, whence a first type. Of course, the drawback is that this class does not form a vector space, but there is also a second type of natural morphisms: suppose given two bounded operators a : H1 → H2 and b : H1 → H2 , and consider the mapping Mab : B(H1 ) → B(H2 ) given by Mab x = axb ∗ . Then again, if Mab (E1 ) ⊂ E2 , it is natural to accept the restriction of Mab to E1 as a morphism. Completely bounded maps can be described as compositions of a morphism of the first type followed by one of the second type. N OTATION . Let E ⊂ B(H ) be an operator space; we denote by Mn (E) the space of n × n matrices with coefficients in E, equipped with the norm: ∀a = (aij ) ∈ Mn (E), 2 1/2 2 h a h ∈ H h 1 . aMn (E) = sup ij j j j i

(0.1)

j

In other words, we view the matrix a as acting on H ⊕ · · · ⊕ H and we compute its usual norm.

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D EFINITION 1. Let E ⊂ B(H ), F ⊂ B(K) be two operator spaces and let u : E → F be a linear map. We say that u is completely bounded if the mappings un : Mn (E) → Mn (F ) defined by un ((aij )) = (u(aij )) are uniformly bounded in the usual sense for the norm defined in (0.1) and we define: ucb = sup un . n1

We will denote by CB(E, F ) the Banach space of all c.b. maps from E into F , equipped with the c.b.-norm. The main interest of the preceding notion is the following fundamental factorization theorem, which appeared independently in the works of Wittstock [106], Haagerup [45] and Paulsen [78], following Arveson’s earlier work [4]. T HEOREM 2. Let E ⊂ B(H ), F ⊂ B(K) (H, K Hilbert), let u : E → F be a linear map and let C 0 be a constant. The following assertions are equivalent: (i) The mapping u is c.b. and satisfies ucb C. (ii) There exist a Hilbert space H1 , a representation π : B(H ) → B(H1 ) and two operators a, b : H1 → K with ab C such that: ∀x ∈ E

u(x) = aπ(x)b∗.

In other words we have: ucb = inf ab ,

(0.2)

where the infimum runs over all factorizations as in (ii) and this infimum is attained. The following extension property is crucial: it is the analog of the Hahn–Banach theorem for operator spaces. C OROLLARY 3. Let H, K be two Hilbert spaces. Consider an operator space F ⊂ B(H ) and a subspace E ⊂ F . Then every c.b. map u : E → B(K) admits a c.b. extension u˜ : F → B(K) such that u ˜ cb = ucb . The corresponding diagram is as follows: F −− −− u˜ − ∪ u→ E −−−−−→ B(H ). In another direction, Theorem 2 implies the decomposability of c.b. maps as linear combinations of completely positive ones. We say that u : E → F is completely positive (c.p. in short) if, with the preceding notation, all the mappings un : Mn (E) → Mn (F ) are positive with respect to the order structures induced by the positive cone of the C ∗ -algebras Mn (B(H )).

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C OROLLARY 4. Every c.b. map u : E → B(K) admits a decomposition u = u1 − u2 + i(u3 − u4 ) with uj c.p. such that: max uj cb max u1 + u2 cb , u3 + u4 cb ucb . j 4

P ROOF (sketch). We note that if a = b in Theorem 2, then u is completely positive. Thus this corollary follows simply from the polarization identity for skewlinear maps. We note that if u is c.p. on a C ∗ -algebra (or on an operator system), we have u = ucb and (in the unital case) u = u(1). We refer the reader to [79] or to [84] for more information on all these results Now that we know the morphisms, the notion of isomorphism is clear: we say that two operator spaces E, F are completely isomorphic (resp. completely isometric) if there exists an isomorphism u : E → F which is completely bounded with a completely bounded inverse (resp. with moreover ucb = u−1 cb = 1). We say that an isometry (not necessarily surjective) u : E → F is a complete isometry if ucb = u−1 |u(E) cb = 1. We say that a mapping u : E → F is a complete contraction if ucb 1. The reader will easily complete (!) this terminology. One of the great advantages of operator spaces over C ∗ -algebras is that they allow the use of finite-dimensional methods in operator algebra theory. More precisely, if E and F are two completely isomorphic operator spaces, we can measure their “degree of isomorphism” by the following distance: dcb (E, F ) = inf ucb u−1 cb | u : E → F ,

(0.3)

where the inf runs over all the complete isomorphisms u from E onto F . This definition is of course modelled on the “Banach–Mazur distance” between two Banach spaces, which is classically defined as: d(E, F ) = inf uu−1 | u : E → F isomorphism .

(0.4)

By convention, we set dcb (E, F ) = ∞ or d(E, F ) = ∞ if E and F are not isomorphic. Consider now a Banach space X. There exists obviously many possible “operator space structures” (in short o.s.s.) on X. By definition, such a structure on X is the data of an isometric embedding j : X → B(H ). We will say that two such structures: j1 : X −→ B(H1 )

and j2 : X −→ B(H2 )

are equivalent if, for any operator space F ⊂ B(K) and any u : X → F , the c.b. norms of u are the same whether we use one of the embeddings j1 or the other j2 . Of course, this boils −1 −1 down to saying that j2 ◦ j1|j and j1 ◦ j2|j are complete isometries, or equivalently 1 (X) 2 (X) that the norms induced respectively by j1 and j2 on Mn (X) are the same for any n 1. Actually, there is no need to distinguish two equivalent operator spaces. In practice, we will always identify them. When it is really necessary, we might want to distinguish the

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“concrete” operator space E ⊂ B(H ) and the “abstract” associated o.s.s., i.e., the associated equivalence class (instead of a “concrete” representative of this class). If X is a C ∗ -algebra, there is of course a natural embedding of X into B(H ) as a ∗ C -subalgebra, by the Gelfand theory. In that case, if j 1 and j 2 are two C ∗ -algebraic embeddings, the preceding equivalence relation is automatic. Indeed, an injective C ∗ -algebra representation j : X → B(H ) is automatically isometric and since jn : Mn (X) → Mn (B(H )) is also an injective representation, it also is isometric, which boils down to saying that j is completely isometric. Therefore if, in the above, j 1 and j 2 are injective C ∗ -representations, then j 1 (j 2 )−1 and j 2 (j 1 )−1 are automatically complete isome|j 2 (X) |j 1 (X) tries. Hence, we may unambiguously speak of the natural structure of a C ∗ -algebra as an operator space. This last observation allows us to change our viewpoint: we can equivalently define an operator space as a subspace of a C ∗ -algebra, since we now know (by Gelfand’s theory) that there is a natural o.s.s. on any C ∗ -algebra. Recall, in particular, that any commutative unital C ∗ -algebra is isomorphic to the space C(T ) of all continuous functions on a compact set T , equipped with the uniform norm. Let B be an arbitrary Banach space. We can associate to it a compact set TB , namely the unit ball of the dual B ∗ equipped with the topology σ (B ∗ , B). We then have an isometric embedding j : B → C(TB ) which allows to equip B with an operator space structure (induced by the C ∗ -algebra C(TB )). We denote by min(B) (following [13]) the resulting operator space. This provides numerous examples. Of course, these examples are not too interesting since they are too “commutative”, but they have the merit of showing how the category of Banach spaces can be viewed as “embedded” into that of operator spaces. Indeed, if B1 , B2 are two Banach spaces, every bounded linear u : B1 → B2 defines a completely bounded map u : min(B1 ) → min(B2 ) with ucb = u. More generally, for any operator space E, every linear map u : E → B defines a c.b. map u : E → min(B) such that u = ucb . In particular, when u = IB , if E = B is equipped with any operator space structure (respecting the norm of B), we have a complete contraction E → min(B). This expresses the “minimality” of min(B). Following [13], we can also introduce the “maximal” structure on B. For that purpose, it is convenient to define first a notion of direct sum in the category. of operator spaces. Let Ei ⊂ B(Hi ) (i ∈ I ) be a collection of operator spaces. We denote i∈I Ei the such that supi∈I xi < ∞ space formed of all families x = (xi )i∈I with xi ∈ Ei , ∀i ∈ I , . and equipped with the norm x = supi∈I xi . The space . .i∈I B(Hi ) is naturally a C ∗ -algebra (that can be seen as embedded into B( i∈I Hi ),. i∈I Hi meaning here the . E → B(H Hilbertian direct sum). Therefore, the isometric embedding i i ) ini∈I i∈I . duces an operator space structure on i∈I Ei . We thus have a notion of direct sum. Let B be an arbitrary Banach space. Let I be the class . of all the mappings u : B → B(Hu ) with u 1. We can define an embedding J : B → u∈I B(Hu ) by setting J (x) =

/

u(x).

u∈I

This embedding allows us to define an operator space structure on B. We denote by max(B) the associated operator space. By construction, we have the following “maximal-

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ity” property: for any operator space E and any u : max(B) → E, u bounded ⇒ u c.b. and u = ucb . In particular, if E = B is equipped with an arbitrary o.s.s. (respecting the norm of B), we have a complete contraction max(B) → E induced by the identity of B. In conclusion, let E = (B, j ) be an arbitrary o.s.s. on B associated to an isometric embedding j : B → B(H ); we then have completely contractive inclusions (= the identity map): max(B) −→ (B, j ) −→ min(B). Thus, maximal operator spaces provide us with another group of examples. Here are two more fundamental examples: we denote R = span[e1j |j 1] ⊂ B(2 ) and C = span[ei1 |i 1] ⊂ B(2 ). One often says C is formed of the column vectors and R of the row vectors in B(2 ). Note that we have: ∀x = (xi ) ∈ 2

xj e1j = |xj |2

1/2

= xi ei1

so that R and C are indistinguishable as Banach spaces: they are both isometric to 2 . In sharp contrast, they are not completely isomorphic and they provide us with two new o.s.s. on the Hilbert space 2 . Thus, at this point we already have four o.s.s. on 2 (which are known to be distinct): min(2 ), max(2 ), R and C. We will soon see that there are actually a whole continuum of such structures! For the moment the typical application of operator space theory is as follows: we have a C ∗ -algebra A equipped with a distinguished system of generators and we consider the operator space E which is the closure of the subspace linearly spanned by these generators (this space E is often isomorphic to a Hilbert space). Then, in many cases, one can “read” on the operator space structure of E several important properties of the C ∗ -algebra which it generates. See [91] for numerous examples illustrating this principle. Although Ruan’s 1988 thesis marks the real “birth” of operator space theory as such, many earlier contributions have had a strong and lasting influence. Among those, the factorization of multilinear completely bounded maps, due to Christensen and Sinclair [21] (and generalized to the operator space setting by Paulsen and Smith [81]) is fundamental (see Section 6). Even earlier, Effros and Haagerup [25] (inspired by Archbold and Batty’s previous work) discovered that operator spaces may fail to be locally reflexive in the c.b. setting in sharp contrast to the Banach space case. Their ideas are closely related to Kirchberg’s spectacular work (see [60,62,105,1]) on exact C ∗ -algebras. In addition, we should recall that “operator spaces” are the descendents of “operator systems”. An operator system is a unital self-adjoint operator space. The theory of operator systems was extensively developed by Arveson [4] and Choi and Effros [19] in the 70’s, using unital completely positive maps as morphisms. Although many of the subsequent ideas appeared already in germs for operator systems, the constant recourse to the order structure stood in the way of a total “linearization” of C ∗ -algebra theory, which operator space theory can now claim to have realized.

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This text is a sort of introduction to the subject. For more information we refer the reader to the books [41,91], or to [84,82] and to the recent proceedings volume [58]. N OTE . The present text is based on the author’s Bourbaki seminar report [87], first translated into English, then expanded and updated.

1. Minimal tensor product Let H1 , H2 be two Hilbert spaces. We denote by H1 ⊗2 H2 their Hilbertian tensor product. Let E1 ⊂ B(H1 ), E2 ⊂ B(H2 ) be two operator spaces. We define a linear (injective) embedding j from the algebraic tensor product, denoted by E1 ⊗ E2 , into B(H1 ⊗2 H2 ) as follows: for x1 ∈ E1 , x2 ∈ E2 we set ∀h1 ∈ H1 , ∀h2 ∈ H2 ,

j (x1 ⊗ x2 )(h1 ⊗ h2 ) = x1 (h1 ) ⊗ x2 (h2 ),

then we extend by linearity. This embedding allows to define an o.s.s. on the completion of E1 ⊗ E2 relative to the induced norm. We denote by E1 ⊗min E2 the resulting operator space. Thus, by definition, we have a complete isometry; E1 ⊗min E2 ⊂ B(H1 ⊗2 H2 ). We denote by · min the norm induced by B(H1 ⊗2 H2 ) on E1 ⊗min E2 . One can then verify that, up to equivalence, the resulting operator space does not depend on the particular realizations of E1 and E2 in B(H1 ) and B(H2 ), but only on their o.s.s. This follows from the next well known observation, very simple but essential for the theory. P ROPOSITION 5. Let E1 , E2 be as above and let F1 ⊂ B(K1 ) and F2 ⊂ B(K2 ) be two other operator spaces. Let u1 ∈ CB(E1 , F2 ) and u2 ∈ CB(E2 , F2 ). Then u1 ⊗ u2 : E1 ⊗ E2 → F1 ⊗ F2 extends to a c.b. map (still denoted by u1 ⊗ u2 ) such that u1 ⊗ u2 CB(E1 ⊗min E2 ,F1 ⊗min F2 ) u1 cb u2 cb . Moreover, the minimal tensor product is injective, meaning that if u1 and u2 are both complete isometries, the same is true for u1 ⊗ u2 : E1 ⊗min E2 → F1 ⊗min F2 . R EMARK 6. Let E1 , E2 be two C ∗ -algebras, then E1 ⊗min E2 is a C ∗ -subalgebra of B(H1 ⊗2 H2 ). Thus we actually have a tensor product for the C ∗ -algebra category, which the preceding extends to operator spaces. By a classical theorem due to Takesaki (see Section 10 below), the norm · min is the smallest C ∗ -norm on the (linear) tensor product of two C ∗ -algebras. In the Banach space category, Grothendieck [44] showed that the injecˇ 2 of two Banach spaces realizes the smallest “reasonable” tensor tive tensor product B1 ⊗B norm on B1 ⊗ B2 . Blecher and Paulsen (cf. [13]) proved an analog of this for E1 ⊗min E2 .

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In passing, let us observe here that for any operator space E and any Banach space B, ˇ B. On the other hand, we have we have an isometric isomorphism E ⊗min min(B) = E ⊗ Mn (E) Mn ⊗min E

isometrically.

In the sequel, we will often identify Mn (E) with Mn ⊗ E. The minimal tensor product is commutative (i.e., E1 ⊗ min E2 E2 ⊗min E1 ) and associative (i.e., for instance (E1 ⊗min E2 ) ⊗min E3 E1 ⊗min (E2 ⊗min E3 )). Therefore, we may unambiguously define (either directly or by iteration) the minimal tensor product E1 ⊗min · · · ⊗min EN of an arbitrary number N of operator spaces, and we again denote by · min the corresponding norm. One easily verifies that ∀xi ∈ Ei

x1 ⊗ · · · ⊗ xN min = x1 · · · xN .

(1.1)

2. Ruan’s theorem Before completion, a Banach space is just a vector space equipped with a norm. Ruan’s fundamental theorem allows to take an analogous viewpoint for operator spaces, but instead of a norm on V , we must consider a sequence of norms · n on Mn (V ) (or a single norm, but on n Mn (E)). Let E be a Banach space, or merely a vector space on C. Suppose given an operator space structure on E. Then, up to equivalence, this is the same as giving ourselves, for each n 1, a norm · n on the space Mn (E) (of n × n matrices with coefficients in E). The problem solved by Ruan’s theorem is the inverse one: which sequences of norms come from an o.s.s. on E? We will first identify two simple necessary conditions. So assume that V is embedded in B(H ) and that · n is the norm induced on Mn (V ) by Mn (B(H )). The following two properties are then easily verified: ∀n 1, ∀a, b ∈ Mn , ∀x ∈ Mn (V ) a · x · bn aMn xMn (V ) bMn , (R1 ) where we denoted a · x · b the matrix product of the matrix x ∈ Mn (V ) by the scalar matrices a and b. ⎧ ⎪ ∀n, m 1, ∀x ∈ Mn (V ), ∀y ∈ Mm (V ), ⎨ x 0 (R2 ) ⎪ = max xn , ym . ⎩ 0 y n+m

We can now state Ruan’s theorem: T HEOREM 7 ([94]). Let V be a complex vector space. For each n 1 we give ourselves a norm · n on Mn (V ). The following assertions are equivalent: (i) There exist a Hilbert space H and a linear injection j : V → B(H ) such that for all n:

IMn ⊗ j : Mn (V ), · n −→ Mn B(H )

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G. Pisier

is an isometry. Equivalently, in other words, the sequence ( · n ) comes from the operator space structure on V associated to j . (ii) The sequence · n satisfies the axioms (R1 ) and (R2 ) above. Let K = K(2 ) be the space of all compact operators on 2 . K can be viewed as a space of bi-infinite matrices, which allows us to consider Mn as “included” in K. We then set: K0 =

Mn .

n1

It is convenient in the preceding theorem to replace the sequence of norms ( · n ) by a single norm on K0 ⊗ E or (after completion) on K ⊗ E. Indeed, the axiom (R2 ) ensures that the embedding (Mn (E), · n ) ⊂ (Mn+1 (E), · n+1 ) is isometric, which allows to define a norm α on K0 ⊗ E as follows: for x ∈ K0 ⊗ E, choose n be such that x ∈ Mn ⊗ E, we then set: α(x) = xn .

(2.1)

Ruan’s theorem establishes a one-to-one correspondence between the set of operator space structures on V (up to equivalence) and the norms α on K0 ⊗ E (or on K ⊗ E) satisfying (R1 ) and (R2 ). R EMARK . Of course, if V is given to us equipped with a norm, we are mostly interested in the o.s.s. on V respecting the norm of V , i.e., such that (x)1 = x for all x ∈ V . It is then easy to check (this is obvious by (1.1) and the preceding theorem) that (R1 ) and (R2 ) imply a ⊗ xn = aMn x for all a in Mn and all x in V . Let α be the norm on K0 ⊗ E associated to this structure as defined in (2.1), and let αmin , αmax be the norms associated respectively to the minimal and maximal structures, as above. We have then: αmin α αmax , which explains the use of the terms “minimal” and “maximal”. I MPORTANT R EMARK . It should be emphasized that the o.s.s. given by Ruan’s theorem are not explicit and, in most of the cases described below (duality, quotient, interpolation), we have no “concrete” description of them. Their existence follows from the Hahn–Banach theorem, cf. the simplified proof of Theorem 7 appearing in [34].

3. Duality Preliminary. Let E, F be two vector spaces. Let u ∈ F ⊗ E ∗ and let u˜ : E → F be the linear map associated to it. When E and F are Banach spaces, we know that u∨ = u ˜ ˇ E ∗ into the and u → u˜ is an isometric embedding of the injective tensor product F ⊗

Operator spaces

1435

space B(E, F ) of all bounded linear maps from E into F . The duality of operator spaces is modeled on this, but the minimal tensor product replaces the injective one and “c.b.” replaces “bounded”. Let E ⊂ B(H ) be an operator space and let E ∗ be its Banach space dual. Then E ∗ can be equipped with a specific o.s.s. characterized by the following property: For any operator space F , the natural map u → u˜ from F ⊗min E ∗ into CB(E, F ) is an isometry.

(3.1)

We have thus an isometric embedding:

∗ E ⊗min F F ⊗min E ∗ −→ CB(E, F ). When dim(F ) < ∞, this embedding is surjective, whence isometric identifications: ∗

E ⊗min F F ⊗min E ∗ CB(E, F ). In the case F = Mn , we have in particular an isometric identification: Mn ⊗min E ∗ CB(E, Mn ).

(3.2)

The basic idea (independently from [13] and [32]) to define this specific o.s.s. is to take the right side of (3.2) to define a sequence of norms on Mn (E ∗ ) and to verify the axioms (R1 ) and (R2 ). Ruan’s theorem then guarantees that there exists a structure on E ∗ verifying (3.2). One then rather easily deduces (3.1) from (3.2). The unicity of the corresponding structure (up to equivalence) is clear since (3.2) determines at most one o.s.s. on E ∗ . Note that for all u : E → F the transposed map t u : F ∗ → E ∗ satisfies ucb = t ucb . More generally, if F is another operator space, we can define an o.s.s. on CB(E, F ) for which, for each n, we have isometrically:

Mn CB(E, F ) CB E, Mn (F ) .

(3.3)

Indeed, there again the norms appearing on the right side of (3.3) satisfy the axioms (R1 ) and (R2 ). Thus, from now on we may consider CB(E, F ) as an operator space (and (3.1), (3.2) then become completely isometric). Examples. The following completely isometric identities can be checked (cf. [13,33]): R ∗ C,

C∗ R

and, for any Banach space B (cf. [13,8]):

min(B)∗ max B ∗ .

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G. Pisier

Let M be a von Neumann algebra with predual M∗ . By duality, the natural structure of M gives us an o.s.s. on M ∗ hence a fortiori on M∗ ⊂ M ∗ . This raises a “coherence” problem, but fortunately everything “ticks”: if we equip M∗ with the preceding o.s.s., its dual is completely isometric (actually equivalent) to M. Thus we still have existence and unicity of the predual of M in the category of operator spaces. In sharp contrast however, this is no longer true for general operator spaces: Le Merdy [67] showed that there exists an o.s.s. on B(H )∗ which is not the dual of any o.s.s. on B(H ).

4. Quotient, ultraproduct and interpolation We will define below some other operations (= functors) on operator spaces. It is worthwhile to emphasize that these operations extend the corresponding ones for Banach spaces. The principle is the same as for the duality: we first work with the underlying Banach spaces to construct the new space (e.g., dual, quotient ultraproduct or interpolation space), and then equip the resulting space with an o.s.s. compatible with the norm and satisfying the “right” functorial properties, specific to each case. For instance, Ruan [94] defined the quotient of two operator spaces E1 , E2 with E2 ⊂ E1 , as follows. We consider the norm · n on Mn (E1 /E2 ) naturally associated to the quotient of normed spaces Mn (E1 )/Mn (E2 ), then we verify (R1 ) and (R2 ). Theorem 7 then ensures that there exists an o.s.s. on E1 /E2 for which we have, for all n 1, an isometric identification: Mn ⊗min (E1 /E2 ) = Mn (E1 )/Mn (E2 ). More generally, we have an isometric identification: K ⊗min (E1 /E2 ) = (K ⊗min E1 )/(K ⊗min E2 ). Thus we now have a notion of “quotient” in the category of operator spaces. Let q : E1 → E1 /E2 be the canonical surjection and let E3 be another o.s. Then a linear map u : E1 /E2 → E3 is c.b. iff uq is c.b. and we have ucb = uqcb . Moreover this notion of quotient satisfies the usual duality rules: we have completely isometric identities (E1 /E2 )∗ = E2⊥

and E2∗ = E1∗ /E2⊥ .

In analogy with the Banach space case, a mapping u : E → F between o.s. is called a complete surjection (resp. a complete metric surjection) if it is onto and if the associated isomorphism E/ ker(u) → F is a complete (resp. completely isometric) isomorphism. Moreover, this is the case iff u∗ : F ∗ → E ∗ is a complete (resp. completely isometric) isomorphism from F ∗ to u∗ (F ∗ ).

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Let U be a non-trivial ultrafilter on a set I . Let (Ei )i∈I be a family of Banach spaces. We denote by the space of all families x = (xi )i∈I with xi ∈ Xi such that supi∈I xi < ∞. We equip this space with the norm x = supi∈I xi . Let nU ⊂ be the subspace formed of all families such that limU xi = 0. The quotient * /nU is called the ultraproduct of the family (Ei )i∈I with respect to U . We denote it by i∈I Ei /U . Now assume that each space Ei is given equipped with an operator space structure. It is very easy to extend the notion of ultraproduct to the operator space setting. We simply define $ $ Mn Xi /U = Mn (Xi )/U. (4.1) i∈I

i∈I

* This identity endows Mn ⊗ i∈I Xi /U * with a norm satisfying Ruan’s axioms (whence also after completion a norm on K ⊗ ( i∈I Xi /U)). Alternatively, we can view an operator space as a subspace of a C ∗ -algebra, then observing that C ∗ -algebras are stable by ultraproducts, we can realize any ultraproduct of operator spaces as a subspace of an ultraproduct of C ∗ -algebras, and we equip it with the induced operator space structure. This alternate route leads to the same operator space structure as (4.1). We now turn to interpolation. Let (E0 , E1 ) be a “compatible” couple of Banach spaces. This means that we are implicitly given two continuous injections E0 → X and E1 → X of E0 , E1 into a common topological vector space X , which allows us to view E0 and E1 as included in X . The typical example is E0 = L∞ , E1 = L1 and X = L0 . Note that actually one can always replace X by a Banach space namely the “sum” E0 + E1 . By this we mean the subspace of X formed of all elements x0 + x1 with x0 ∈ E0 , x1 ∈ E1 and equipped with the norm inf{x0 E0 + x1 E1 }. Similarly, the “intersection” E0 ∩ E1 is equipped with the Banach space norm x = max{xE0 , xE1 }. Furthermore, for each 0 < θ < 1 the complex interpolation method (due to Calderón and Lions independently) associates to each compatible couple (E0 , E1 ) an “interpolation space” denoted by (E0 , E1 )θ . We set Eθ = (E0 , E1 )θ . We thus obtain a “continuous family” (Eθ )0 2, the space n1 = (n∞ )∗ equipped with the o.s.s. dual to the “natural” structure of n∞ , satisfies √ √ n n ∼ √ dS K n1 n, 2 2 n−1 and for OH n we have √ √

1/2 n1/4 / 2 ∼ n/ 2 n − 1 dS K (OH n ) n1/4 .

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1449

Kirchberg observed early on that (OSn , δcb ) is not compact in general (and actually for any n > 2, but the case n = 2 remains open) and he raised the question of the separability of (OSn , δcb ). But again the answer is negative. T HEOREM 21 ([56]). The space (OSn , δcb ) is not separable if n > 2. More precisely, even the subset of all operator spaces isometric to n2 is not separable. The best asymptotic estimate of this “non-separability” is as follows: let δ(n) be the infimum of the numbers δ > 0 for which (OSn , δcb ) admits a countable δ-net in the following sense (we use dcb rather than δcb ): there is a countable subset of D ⊂ OSn such that ∈ D with dcb (E, E) < δ. With this notation, we have, for n = p + 1 with p ∀E ∈ OSn ∃E prime 3 (see [56]): √ n n ∼ √ δ(n). 2 2 n−1 √ On the other hand, (9.1) (or (9.2)) obviously implies that δ(n) n for all n 1. Using √ the (quite delicate) random matrix bounds in [47], one can show that δ(n) n/[2(1 + n)] for all n 1, see [91] for details and an update.

10. Applications to tensor products of C ∗ -algebras Whenever A1 , A2 are C ∗ -algebras, their algebraic tensor product carries a natural structure of a ∗-algebra. By a C ∗ -norm, we mean a norm on A1 ⊗ A2 such that x = x ∗ ,

xy xy and xx ∗ = x2

for all x, y in A1 ⊗ A2 . Since the works of Takesaki (1958) and Guichardet (1965), it has been known that there is a minimal C ∗ -norm · min and a maximal one · max , so that any C ∗ -norm · on A1 ⊗ A2 must satisfy ∀x ∈ A1 ⊗ A2 ,

xmin x xmax .

We denote by A1 ⊗min A2 (resp. A1 ⊗max A2 ) the completion of A1 ⊗ A2 for the norm · min (resp. · max ). A C ∗ -algebra A1 is called nuclear if, for any C ∗ -algebra A2 , we have · min = · max , or equivalently there is a unique C ∗ -norm on A1 ⊗ A2 . For instance, all commutative C ∗ -algebras are nuclear, as well as K(H ) but, by a result due to Wassermann [104], B(H ) is not nuclear when dim H = ∞. From the combined works of Choi and Effros and Connes, a C ∗ -algebra is nuclear iff its bidual A∗∗ is an injective von Neumann algebra. Moreover (Choi and Effros, Kirchberg) this holds iff the identity is approximable pointwise by a net

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G. Pisier

of completely positive maps of finite rank. Kirchberg’s recent work [60] revived the study of pairs of C ∗ -algebras A, B for which A ⊗ B admits a unique C ∗ -norm, i.e., such that A ⊗min B = A ⊗max B

isometrically.

(10.1)

Kirchberg gave the first example of a C ∗ -algebra A for which (10.1) holds when B = Aop (the opposite C ∗ -algebra, i.e., A with the inverse product x · y = yx), but which is not nuclear. Moreover, he showed that (10.1) holds when B = B(H ) and A = C ∗ (F∞ ) the C ∗ -algebra of the free group (denoted by F∞ ) with (say) infinitely many generators, i.e., the C ∗ -algebra generated by the universal representation of the discrete group F∞ . Using operator space theory, it is possible to give a very simple proof of this result (see [86]). Moreover, that same theory also led the authors of [56] to the solution of a long standing problem discussed at length by Kirchberg in [60], as follows T HEOREM 22 ([56]). If dim H = ∞, we have B(H ) ⊗min B(H ) = B(H ) ⊗max B(H ). The paper [56] describes three different approaches to this result with quantitative estimates of variable sharpness. The best estimate (see [100] for more on this) uses the delicate number theoretic results of Lubotzky, Phillips and Sarnak. T HEOREM 23. Let umax λ(n) = sup umin

u ∈ B(H ) ⊗ B(H ), r(u) n ,

where r(u) denotes the rank of u. Then, for any n of the form n = p + 1 with p prime 3, we have √ √ n n ∼ √ λ(n) n. 2 2 n−1 √ R EMARK . The above upper bound λ(n) n is valid for all√n and follows for instance from (9.2). But the delicate point is the proof that λ(n) n(2 n − 1)−1 , which is a consequence of the lower bounds on the number δ(n) appearing in Section 9. We will now merely outline the link between δ(n) and λ(n) to point out which tensors are “responsible” for the large values of λ(n) found in Theorem 23. In order to do that, let n = p + 1 with p prime 3. Let S ⊂ R3 be the Euclidean sphere equipped with its normalized surface measure. We let L20 ⊂ L2 (S, μ) be the subspace of functions with mean zero (i.e., the orthogonal complement of the constant functions) and let ρ : SO(3) → B(L20 ) be the natural representation defined by

ρ(t)f (ω) = f t −1 (ω) .

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. Let ρ = m0 πm be the decomposition of ρ into irreducible components, each finitedimensional since SO(3) is compact. For any subset Ω ⊂ N we set ρΩ =

/

πm

and HΩ =

m∈Ω

/

Hm .

m∈Ω

By [69], for n = p + 1 as above, there are t1 , . . . , tn in SO(3) such that n √ ρ(ti ) = 2 n − 1 1

whence n √ sup πm (ti ) ⊗ πm (ti ) 2 n − 1. m=m

(10.2)

i=1

The construction in [56] shows as a by-product that the n-tuples ρΩ (t1 ), . . . , ρΩ (tn ) are “often” linearly independent. Let EΩ be their linear span, and (assuming EΩ n-dimen1 , . . . , ξ n be the biorthogonal functionals in E ∗ . By the duality of operator sional) let ξΩ Ω Ω spaces (see Section 2 above), we know that there is a Hilbert space HΩ and a specific isometric embedding ∗ ⊂ B(HΩ ). EΩ

i represents the identity map on EΩ , we have necessarily Moreover, since n1 ρΩ (ti ) ⊗ ξΩ by (3.1) and by the injectivity of the minimal tensor product n i ρΩ (ti ) ⊗ ξΩ (10.3) = IEΩ CB(EΩ ,EΩ ) = 1. i=1

min

But, on the other hand, the method used in [56] (see also [102]) shows (here we skip all details) that (10.2) implies that, for any ε > 0, there is an infinite subset Ω ⊂ N such that ρΩ (t1 ), . . . , ρΩ (tn ) are linearly independent and satisfy: n n ρΩ (ti ) ⊗ ξiΩ . (10.4) √ −ε 2 n−1 i=1 B(H )⊗ B(H ) Ω

max

Ω

In conclusion, one obtains Theorem 23 by combining (10.3) and (10.4).

11. Local reflexivity In Banach space theory, the “principle of local reflexivity” says that every Banach space B satisfies B(F, B)∗∗ = B(F, B ∗∗ ) isometrically for any finite-dimensional (normed)

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space F . Consequently, B ∗∗ is always finitely representable in B. This useful principle goes back to Lindenstrauss and Rosenthal with roots in Grothendieck’s and Schatten’s early work (see [24], p. 178 and references there). Similarly, an o.s. E is called “locally reflexive” if we have CB(F, E)∗∗ = CB(F, E ∗∗ ) isometrically for any finite-dimensional o.s. F (and when this holds for all F , it actually holds completely isometrically). This property was “exported” first to C ∗ -algebra theory by Archbold and Batty, then for operator spaces in [25]. In the o.s. setting, it is unclear how this notion relates to the o.s.-finite representability of E ∗∗ into E. More precisely, we will say that an o.s. E is os-finitely representable into another F , if for any ε > 0 and any finite-dimensional subspace E1 ⊂ E there is a subspace F1 ⊂ F such that dcb (E1 , F1 ) 1 + ε. As the reader can guess, not every o.s. is locally reflexive, so the “principle” now fails to be universal: as shown in [25], C ∗ (F∞ ) is not locally reflexive. Local reflexivity passes to subspaces (but not to quotients) and is trivially satisfied by all reflexive o.s. (a puzzling fact since reflexivity is a property of the underlying Banach space only!). It is known that all nuclear C ∗ -algebras are locally reflexive (essentially due to Archbold and Batty, see [25]). More generally, by Kirchberg’s results, exactness ⇒ local reflexivity for C ∗ -algebras (see [59] or [105]), but the converse remains open. Actually, it might be true that exact ⇒ locally reflexive for all o.s. but the converse is certainly false since there are reflexive but non-exact o.s. (such as OH). It was proved recently in [28] that if an operator space X is 1-exact (meaning that dS K (X) = 1) then it is locally reflexive. All this shows that local reflexivity is a rather rare property. Therefore, it came as a big surprise (at least to the author) when, in 1997, Effros, Junge and Ruan [26] managed to prove that every predual of a von Neumann algebra (a fortiori the dual of any C ∗ -algebra) is locally reflexive. This striking result is proved using a non-standard application of Kaplansky’s classical density theorem, together with a careful comparison of the various notions of “integral operators” relevant to o.s. theory (see [53] for an alternate proof). Actually, [26] contains a remarkable strengthening: for any von Neumann algebra M, the dual M ∗ = (M∗ )∗∗ is o.s.-finitely representable in M∗ . This is already non-trivial when M = B(H )! More recently, Ozawa [77] proved that the space of all n-dimensional subspaces of a non-commutative L1 -space (= the predual of a von Neuman algebra) is compact for the dcb -distance. Since the finite representability is always clear in some “weak sense” this compactness contains the preceding statement.

12. Injective and projective operator spaces Injective objects have always played a major role both in Banach space and operator algebra theory. One reason was the quest for generalizations of the Hahn–Banach extension theorem to maps with ranges of dimension more than 1, or infinite. Moreover, in von Neumann algebra theory, injective factors are of crucial importance because of Connes’ landmark paper [23] where he proves (in addition to the equivalence between “injective” “semi-discrete” and “hyperfinite”) that there is only one injective factor on 2 with a finite faithful normal trace (such algebras are called of type II1 ). This can be viewed as a noncommutative analogue of the fact that the Lebesgue interval is the only infinite non-atomic

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countably generated measure space, or equivalently that L∞ ([0, 1]) is the only infinitedimensional L∞ -space over a non-atomic countably generated measure space. A Banach space X is called injective (isometrically) if for any diagram Y ∪ u S −→ X there is an extension u˜ : Y → X with u ˜ = u. An operator space X is called injective if for any such diagram (with Y an o.s. and u c.b.) there is an extension u˜ : Y → X with u ˜ cb = ucb . The basic examples are X = ∞ (Γ ) or X = L∞ (Ω, Σ, μ) with Γ an arbitrary set or (Ω, Σ, μ) an arbitrary measure space. Since any Banach space X embeds isometrically into ∞ (Γ ), it is easy to see that X is injective iff X is the range of a contractive (= of norm 1) projection on ∞ (Γ ). Analogously, any o.s. X embeds completely isometrically into B(H ) for some suitable Hilbert space H . Thus the same reasoning (recall Corollary 3) shows that X is injective iff X is the range of a completely contractive projection on B(H ). In particular, in addition to B(H ), the operator spaces B(C, H ) and B(H, C), or the column and row Hilbert spaces C and R are injective. We note in passing that, by a result due to Tomiyama a contractive linear projection P on a C ∗ -algebra A is automatically completely contractive. However, the examples of C and R (with the projections x → xe11 and x → e11 x) show that the range of P need not be completely isometric to a C ∗ -algebra (see, e.g., [95], p. 97). We will say that a C ∗ -algebra (or a von Neumann algebra) is injective if it is injective as an operator space (with its natural structure). In [95] the following nice characterization of injective operator spaces is given. T HEOREM 24. An operator space X is injective iff there exists an injective C ∗ -algebra A and two projections p, q in A such that X is completely isometric to pAq. Moreover, Roger Smith observed (unpublished) that if X is finite-dimensional, A too can be chosen finite-dimensional. The preceding theorem is closely connected to the important notion of injective envelope of an operator space due to Hamana (see [48] and also an unpublished manuscript). Given an operator space X, we say that an operator space X with X ⊃ X (completely isometrically) is an injective envelope if X is injective and if is the only completely contractive map extending the inclumoreover the identity of X sion X → X. Hamana [48,49] (and Ruan independently) proved that every operator space admits a unique injective envelope. The notion of injectivity also makes sense in the isomorphic setting: a Banach (resp. operator space) X is called λ-injective if for any diagram as before we have an extension u˜ with u ˜ λu (resp. u ˜ cb λucb ). Of course X is then the range of a projection P on ∞ (Γ ) (resp. B(H )) with P λ (resp. P cb λ), but when λ > 1, the structure of these projections can be quite complicated and much less is known. However, there is another notion, “separable injectivity”, on which a lot of work has been done. A separable Banach space (resp. operator space) X is called “separably λ-injective” if for any subspace S ⊂ Y of a separable space Y there is an extension u˜ : Y → X with

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u ˜ λu (resp. u ˜ cb λucb ). We will say that X is separably injective if it is separably λ-injective for some λ < ∞. In 1941, Sobczyk [98] (see also [101]) proved that the Banach space c0 is separably injective as a Banach space, and as a corollary c0 is complemented in any separable superspace. Of course this is also true of any space isomorphic to c0 . Since no other example was found, this raised the question whether actually c0 was the only possible example up to isomorphism. This remained open for a long time until Zippin, in a deep paper [108], proved the converse to Sobczyk’s theorem: X is separably injective only if X is isomorphic to c0 . Since the space K of compact operators on 2 is the natural analog of c0 , it was natural to expect that K should be separably injective as an operator space. However, a very interesting example due to Kirchberg . [59]shows that it is. notthe case. NevertheC ) or ( less, Rosenthal [93] showed that the spaces ( n c 0 n1 n1 Rn )c0 are both separably 2-injective as operator spaces. (Note that, as o.s., these spaces are different, and also distinct from c0 .) In the same paper [93], Rosenthal studied the many possible variants of the extension property. This work was continued with Oikhberg in [75]. Simpler approaches appear in [76] and [3], see also [50]. One of the main results of [75] is that if we restrict the extension property to subspaces S ⊂ Y of a locally reflexive separable operator space Y , and if we assume that u : S → K is a complete isomorphism, then u : S → K admits a c.b. extension u˜ : Y → K. Thus if we restrict to Y locally reflexive we do have the c.b.-analogue of the corollary to Sobczyk’s theorem: K is completely complemented in any locally reflexive separable operator space containing it. Nevertheless, even assuming Y locally reflexive, the c.b.-version of the separable extension property fails for K in general. It remains open whether K is complemented (by a merely bounded projection) in any separable operator space containing it. We now turn to “projective objects”, that is to say spaces satisfying certain lifting properties. We will say that a Banach space (resp. operator space) X is λ-projective if for any ε > 0, any map u : X → Y/S into a quotient of Banach (resp. operator) space admits a lifting u˜ : X → Y with u ˜ (λ + ε)u (resp. u ˜ cb (λ + ε)ucb ). It is an elementary fact that X = 1 (Γ ) satisfies this with λ = 1 as a Banach space. Consequently, X = max(1 (Γ )) satisfies the same as an operator space. It is known ([63]) that these are the only Banach spaces which are λ-projective for some λ. However, in the o.s. setting, there is a larger class of projective spaces. Indeed Blecher [8] proved that although S1 itself is not projective, the direct sum in the sense of 1 of a family of spaces of the form S1ni for some integers {ni | i ∈ I } is 1-projective. He also proved that any operator space is (completely isometric to) a quotient of a space of this form for suitable I and (ni ); he could thus observe that a 1-projective operator space X is 1-projective iff for any ε > 0, X is (1 + ε)-completely isomorphic to a (1 + ε)-completely complemented subspace of a space of this form. Actually, by the more recent results of [28], we can take ε = 0. While there are rather few projective Banach spaces, many more spaces satisfy the “local” version of projectivity (or equivalently a local form of lifting property). The resulting class of Banach spaces is the class of L1 spaces (see [66]) which can be defined in many equivalent ways. One of these is: X is L1 iff X∗∗ is isomorphic to a complemented subspace of an L1 -space. In sharp contrast, the operator space versions of the various de-

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finitions of L1 -spaces (or more generally Lp -spaces) lead to possibly distinct classes of operator spaces, see [39]. This difficulty is of course related to the lack of local reflexivity in general. One of the possible variants is studied in [65] under the name of “λ-local lifting property” (in short λ-LLP): an operator space X has the λ-LLP if for any map u : X → Y/S, any ε > 0 and any finite-dimensional subspace E ⊂ X, the restriction of u to E admits a lifting u˜ : E → Y with u ˜ cb (λ + ε)ucb . It is proved in [65] that X has the λ-LLP iff X∗ is λ-injective. (As a corollary, X λ-projective implies X∗ λ-injective.) See also [32,35] for related results. More recently, in [28] the authors prove that this happens for λ = 1 iff there is an injective von Neumann algebra R and a (self-adjoint) projection p in R such that X∗ (1 − p)Rp

(completely isometrically). ai

It follows that X has the 1-LLP iff there is a net of finite-rank maps of the form X −→ n

bi

S1 i −→ X with ai cb , bi cb 1 which tend pointwise to the identity on X. In another direction, the results of [28] provide an extension to operator spaces of the classical work of Choi, Effros and Connes (see [19]) on nuclear C ∗ -algebras. We will say ai

bi

that an operator space X is λ-nuclear if there is a net of maps of the form X −→ Mni −→ X with ai cb bi cb λ which tends pointwise to the identity on X. It is known (see [19]) that a C ∗ -algebra A is 1-nuclear iff A∗∗ is injective (equivalently is a 1-injective operator space). The o.s. version of this result proved in [28] now reads like this: an operator space X is 1-nuclear iff X is locally reflexive and WEP. We say that X is WEP if the canonical inclusion X → X∗∗ factors completely contractively through B(H ).

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CHAPTER 34

Non-Commutative Lp -Spaces Gilles Pisier∗ Équipe d’Analyse, Université Paris VI, Case 186, F-75252 Paris Cedex 05, France Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Quanhua Xu Laboratoire de Mathématiques, Université de Franche-Comté, UFR des Sciences et Techniques, 16, Route de Gray, 25030 Besançon Cedex, France E-mail: [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. General von Neumann algebras, including type III . . . . . . . . . . . 4. From classic Lp to non-commutative Lp : similarities and differences 5. Uniform convexity (real and complex) and uniform smoothness . . . 6. Non-commutative Khintchine inequalities . . . . . . . . . . . . . . . . 7. Non-commutative martingale inequalities . . . . . . . . . . . . . . . . 8. Non-commutative Hardy spaces . . . . . . . . . . . . . . . . . . . . . 9. Hankel operators and Schur multipliers . . . . . . . . . . . . . . . . . 10. Isomorphism and embedding . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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∗ Partially supported by NSF and Texas Advanced Research Program 010366-163.


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Non-commutative Lp -spaces

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Introduction This survey is devoted to the theory of non-commutative Lp -spaces. This theory (in the tracial case) was laid out in the early 50’s by Segal [181] and Dixmier [47] (see also [110, 183]). Since then the theory has been extensively studied, extended and applied, and by now the strong parallelism between non-commutative and classical Lebesgue integration is well-known. We will see that on the one hand, non-commutative Lp -spaces share many properties with the usual Lp -spaces (to which we will refer as commutative Lp -spaces), and on the other, they are very different from the latter. They provide interesting (often “pathological”) examples which cannot exist among the usual function or sequence spaces. They are also used as fundamental tools in some other directions of mathematics (such as operator algebra theory, non-commutative geometry and non-commutative probability), as well as in mathematical physics. Some tools in the study of the usual commutative Lp -spaces still work in the noncommutative setting. However, most of the time, new techniques must be invented. To illustrate the difficulties one may encounter when studying non-commutative Lp -spaces, we mention here three well-known facts. Let H be a complex Hilbert space, and let B(H ) denote the algebra of all bounded operators on H . The first fact states that the usual triangle inequality for the modulus of complex numbers is no longer valid for the modulus of operators, namely, in general, we do not have |x + y| |x| + |y| for x, y ∈ B(H ), where |x| = (x ∗ x)1/2 is the modulus of x. However, there is a useful substitute, obtained in [1], which reads as follows. For any x, y ∈ B(H ) there are two isometries u and v in B(H ) such that |x + y| u|x|u∗ + v|y|v ∗ . The second fact is about operator monotone functions. Let α be a positive real number. In general, the condition that 0 x y (x, y ∈ B(H )) does not imply x α y α . This implication holds only in the case of α 1. The last fact concerns the convexity of the map x → x α on the positive part B(H )+ of B(H ). For α < 1 this map is concave (actually, the function (x, y) → x α ⊗ y 1−α is concave on B(H )+ × B(H )+ , [112]), but for α 1 convexity holds only if 1 α 2. The reader can find more results of this nature in [17]. Some even worse phenomena may happen. It is well known that composed with the usual trace Tr on B(H ), all the preceding maps have the usual desired properties. For instance, the function x → Tr(x α ) becomes convex for all α 1, as one can expect. Now consider the function

1/α

(x1 , . . . , xn ) → Tr x1α + · · · + xnα on B(H )n+ . In the commutative setting, the convexity of this function for all α 1 and n 1 is extremely useful in many situations. Again, in the non-commutative case, this convexity is not guaranteed, at least for α > 2 (cf. [36]; see also [10] for some related results).

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Despite the difficulty caused by the lack of these elementary properties, we feel the theory has now matured enough for us to be able to present the reader with a rather satisfactory picture. Of course much remains to be done, as shown by the many open problems which we will encounter. We now briefly describe the organization of this survey. After a preliminary section, we discuss the interpolation of non-commutative Lp -spaces (associated with a trace) in Section 2. This is one of the oldest subjects in the field. The main result there allows to reduce all interpolation problems on non-commutative Lp -spaces to the corresponding ones on commutative Lp -spaces. Section 3 can be still considered as a preliminary one. There we introduce the noncommutative Lp -spaces associated with a state or weight. This section also contains two useful results. The first one says that the non-commutative Lp -spaces over the hyperfinite II1 factor are the smallest ones among all those over von Neumann algebras not of type I. The second one is Haagerup’s approximation theorem. In the short Section 4 we discuss very briefly some similarities and differences between the commutative Lp -spaces and their non-commutative counterparts. One remarkable result in the early stage of the non-commutative Lp -space theory is the Gordon–Lewis theorem on local unconditional structure of the Schatten classes. This (negative) result shows that compared with the usual function spaces, the Schatten classes (and so the general non-commutative Lp -spaces) are, in a certain sense, “very non-commutative”. Section 5 discusses the uniform convexities and smoothness, and the related type and cotype properties. Although the problem on the uniform (real) convexity of the noncommutative Lp -spaces goes back to the 50’s, the best constant for the modulus of convexity was found only at the beginning of the 90’s. Two uniform complex convexities (the uniform PL-convexity and Hardy convexity) are also discussed in this section. The central object in Section 6 is the non-commutative Khintchine inequalities, of paramount importance in this theory. Like in the commutative case, they are the key to a large part of non-commutative analysis, including of course the type and cotype properties of non-commutative Lp -spaces, and closely linked to the non-commutative Grothendieck theorem. Section 7 presents some very recent results on non-commutative martingale inequalities. In view of its close relations with quantum (= non-commutative) probability, this direction, which is still at an early stage of development, is likely to get more attention in the near future. Section 8 deals with the non-commutative Hardy spaces. We present there some noncommutative analogues of the classical theorems on the Hardy spaces in the unit disc, such as the boundedness of the Hilbert transformation, Szegö and Riesz factorizations. The first result in Section 9 is Peller’s characterization of the membership of a Hankel operator in a Schatten class. This result is related to Schur multipliers. The rest of this section gives an outline of the recent works by Harcharras on Schur multipliers and noncommutative Λ(p)-sets. The last section concerns the embedding and isomorphism of non-commutative Lp -spaces. Almost all results given there were obtained just in the last few years. This is still a very active direction.


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We end this introductory section by pointing out that we will freely use standard notation and notions from operator algebra theory, for which we refer to [48,104,178,184,185,190].

1. Preliminaries In this section we give some necessary preliminaries on non-commutative Lp -spaces associated with a trace. This requires that the underlying von Neumann algebra be semifinite (see below the definition). In Section 3, we will consider the non-tracial case. M will always denote a von Neumann algebra, and M+ its positive part. We recall that a trace on M is a map τ : M+ → [0, ∞] satisfying (i) τ (x + y) = τ (x) + τ (y), ∀x, y ∈ M+ ; (ii) τ (λx) = λτ (x), ∀λ ∈ [0, ∞), x ∈ M+ ; (iii) τ (u∗ u) = τ (uu∗ ), ∀u ∈ M. τ is said to be normal if supα τ (xα ) = τ (supα xα ) for any bounded increasing net (xα ) in M+ , semifinite if for any non-zero x ∈ M+ there is a non-zero y ∈ M+ such that y x and τ (y) < ∞, and faithful if τ (x) = 0 implies x = 0. If τ (1) < ∞ (1 denoting the identity of M), τ is said to be finite. If τ is finite, we will assume almost systematically that τ is normalized, that is, τ (1) = 1. We often think of τ as a non-commutative (= quantum) probability. A von Neumann algebra M is called semifinite if it admits a normal semifinite faithful (abbreviated as n.s.f.) trace τ , which we assume in the remainder of this section. Then let S+ be the set of all x ∈ M+ such that τ (supp x) < ∞, where supp x denotes the support of x (defined as the least projection p in M such that px = x or equivalently xp = x). Let S be the linear span of S+ . It is easy to check that S is a ∗-subalgebra of M which is w∗ -dense in M, moreover for any 0 < p < ∞, x ∈ S implies |x|p ∈ S+ (and so τ (|x|p ) < ∞), where |x| = (x ∗ x)1/2 is the modulus of x. Now we define

1/p xp = τ |x|p ,

x ∈ S.

One can show that · p is a norm on S if 1 p < ∞, and a quasi-norm (more precisely, a p-norm) if 0 < p < 1. The completion of (S, · p ) is denoted by Lp (M, τ ). This is the non-commutative Lp -space associated with (M, τ ). For convenience, we set L∞ (M, τ ) = M equipped with the operator norm. The trace τ can be extended to a linear functional on S, which will be still denoted by τ . Then τ (x) x1 ,

∀x ∈ S.

Thus τ extends to a continuous functional on L1 (M, τ ). The elements in Lp (M, τ ) can be viewed as closed densely defined operators on H (H being the Hilbert space on which M acts). We recall this briefly. A closed densely defined operator x on H is said to be affiliated with M if xu = ux for any unitary u in the commutant M of M. An affiliated operator x is said to be τ -measurable or simply measurable if

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τ (eλ (|x|)) < ∞ for some λ > 0, where eλ (|x|) denotes the spectral resolution of |x| (corresponding to the indicator function of (λ, ∞)). For any measurable operator x we define the generalized singular numbers by

μt (x) = inf λ > 0: τ eλ |x| t ,

t > 0.

It will be convenient to denote simply by μ(x) the function t → μt (x). Note that μ(x) is a non-increasing function on (0, ∞). This notion is the generalization of the usual singular numbers for compact operators on a Hilbert space (see [72]). It was first introduced in a Bourbaki seminar note by Grothendieck [77]. It was studied in details in [132,62] and [64]. Let L0 (M, τ ) denote the space of all measurable operators in M. Then L0 (M, τ ) is a ∗-algebra, which can be made into a topological ∗-algebra as follows. Let V (ε, δ) = x ∈ L0 (M, τ ): με (x) δ . Then {V (ε, δ): ε, δ > 0} is a system of neighbourhoods at 0 for which L0 (M, τ ) becomes a metrizable topological ∗-algebra. The convergence with respect to this topology is called the convergence in measure. Then M is dense in L0 (M, τ ). We refer to [131] and [191] for more information. The trace τ is extended to a positive tracial functional on the positive part L0+ (M, τ ) of 0 L (M, τ ), still denoted by τ , satisfying

∞

τ (x) =

μt (x) dt, 0

x ∈ L0+ (M, τ ).

Then for 0 < p < ∞,

1/p . Lp (M, τ ) = x ∈ L0 (M, τ ): τ |x|p < ∞ and xp = τ |x|p Also note that x ∈ Lp (M, τ ) iff μ(x) ∈ Lp (0, ∞), and xp = μ(x)Lp (0,∞) . Recall that μ(x) = μ(x ∗ ) = μ(|x|); so x ∈ Lp (M, τ ) iff x ∗ ∈ Lp (M, τ ), and we have xp = x ∗ p . The usual Hölder inequality extends to the non-commutative setting. Let 0 < r, p, q ∞ be such that 1/r = 1/p + 1/q. Then x ∈ Lp (M, τ ), y ∈ Lq (M, τ ) 4⇒ xy ∈ Lr (M, τ ) and xyr xp yq . In particular, if r = 1, τ (xy) xy1 xp yq ,

x ∈ Lp (M, τ ), y ∈ Lq (M, τ ).

This defines a natural duality between Lp (M, τ ) and Lq (M, τ ): x, y = τ (xy). Then for any 1 p < ∞ we have

Lp (M, τ )

∗

= Lq (M, τ ) (isometrically).


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Thus, L1 (M, τ ) is the predual of M, and Lp (M, τ ) is reflexive for 1 < p < ∞. Note that the classical theorem of Day on the dual of Lp for 0 < p < 1 was extended to the non-commutative setting by Saito [176]: the dual of Lp (M, τ ), 0 < p < 1, is trivial iff M has no minimal projection. R EMARK . [114] contains a different construction of non-commutative Lp -spaces via a non-commutative upper integral. Although we will concentrate on non-commutative Lp -spaces in this survey, the more general so-called “symmetric operator spaces” are worth mentioning: let E be a rearrangement invariant (in short r.i.) function space on (0, ∞), the symmetric operator space associated with (M, τ ) and E is defined by and xE(M,τ ) = μ(x)E . E(M, τ ) = x ∈ L0 (M, τ ): μ(x) ∈ E In particular, if E = Lp (0, ∞), we recover Lp (M, τ ). These symmetric operator spaces have been extensively studied, see, e.g., [51,53,54,133,134] and [203] for more information. We end this section by some examples. (i) Commutative Lp -spaces. Let M be an Abelian von Neumann algebra. Then M = ∞ L (Ω, μ) for a measure space (Ω, μ), integration with respect to the measure μ gives us an n.s.f. trace, and Lp (M, τ ) is just the commutative Lp -space Lp (Ω, μ). (ii) Schatten classes. Let M = B(H ), the algebra of all bounded operators on H , and τ = Tr, the usual trace on B(H ). Then the associated Lp -space Lp (M, τ ) is the Schatten class S p (H ). If H is separable and dim H = ∞ (resp. dim H = n), we denote S p (H ) p by S p (resp. Sn ). Note that in our notation S ∞ (H ) is not the ideal of all compact operators on H but B(H ) itself. [72,128] and [182] contain elementary properties of S p (H ). (iii) The hyperfinite II1 factor. Let Mn denote the full algebra of all complex n × n matrices, equipped with the normalized trace σn . Let (R, τ ) =

8

(An , τn ),

(An , τn ) = (M2 , σ2 ), n ∈ N,

n1

be the von Neumann algebra tensor product. Then R is the hyperfinite II1 factor and τ is the (unique) normalized trace on R. There is another useful description of R. Let (εn )n1 be a sequence of self-adjoint unitaries on a Hilbert space, satisfying the following canonical anticommutation relations εi εj + εj εi = 2δij ,

i, j ∈ N.

(CAR)

Let R0 be the C ∗ -algebra generated by the εi ’s. Then R0 admits a unique faithful tracial state, denoted by τ , which is defined as follows. For any finite subset A = {i1 , . . . , in } ⊂ N with i1 < · · · < in we put wA = εi1 · · · εin , and w∅ = 1. Then the trace τ is uniquely determined by its action on the wA ’s: τ (wA ) = 1 (resp. = 0) if A = ∅ (resp. = ∅). Consider R0 as a C ∗ -algebra acting on L2 (τ ) by left multiplication. Then the von Neumann algebra

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generated by R0 in B(L2 (τ )) can be (isomorphically) regarded as the hyperfinite II1 factor R. Note that the family of all linear combinations of the wA ’s are w∗ -dense in R and dense in Lp (R) for all 0 < p < ∞; also note that {wA : A ⊂ N} is an orthonormal basis of L2 (R) (= L2 (τ )). Finally, we mention that the von Neumann subalgebra generated by {ε1 , . . . , ε2n } is isomorphic to M2n , and then the restriction of τ to this subalgebra is just the normalized trace of M2n . We refer to [25] and [158] for more information. (iv) Group algebras. Consider a discrete group Γ . Let vN(Γ ) ⊂ B(2 (Γ )) be the associated von Neumann algebra generated by the left translations. Let τΓ be the canonical trace on vN(Γ ), defined as follows: τΓ (x) = x(δe ), δe for any x ∈ vN(Γ ), where (δg )g∈Γ denotes the canonical basis of l2 (Γ ), and where e is the identity of Γ . This is a normal faithful normalized finite trace on vN(Γ ). A particularly interesting case is when Γ = Fn , the free group on n generators. We refer to [67] and [196] for more on this theme.

2. Interpolation This section is devoted to the interpolation of non-commutative Lp -spaces. It is well known that the non-commutative Lp -spaces associated with a semifinite von Neumann algebra form an interpolation scale with respect to both the real and complex interpolation methods (see (2.1) and (2.2) below). This result not only is useful in applications but also can be taken as a starting point to define non-commutative Lp -spaces associated to a von Neumann algebra of type III (which admits no n.s.f. trace). This is indeed the viewpoint taken by Kosaki [106] (see also [192]). We will discuss this point in the next section. Here we restrict ourselves only to semifinite von Neumann algebras. Thus throughout this section, M will always denote a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ . We refer to [15] for all notions and notation from interpolation theory used below. Let 1 p0 , p1 ∞ and 0 < θ < 1. It is well known that

Lp (M, τ ) = Lp0 (M, τ ), Lp1 (M, τ ) θ (with equal norms),

Lp (M, τ ) = Lp0 (M, τ ), Lp1 (M, τ ) θ,p (with equivalent norms),

(2.1) (2.2)

where 1/p = (1−θ )/p0 +θ/p1 , and where (·, ·)θ , (·, ·)θ,p denote respectively the complex and real interpolation methods. It is not easy to retrace the origin of these interpolation results. Some weaker or particular forms go back to the 50’s (cf., e.g., [47,110,172]). The results in the full generality were achieved by Ovchinikov [133,134] (see also [135] for the real interpolation, and [141] in the case of Schatten classes). (2.1) and (2.2) easily follow from the following result. Recall that μ(x) denotes the generalized singular number of x (see Section 1) and that a map T : X → Y is called contractive (or a contraction) if T 1. T HEOREM 2.1. For any fixed x ∈ L1 (M, τ ) + L∞ (M, τ ) there are linear maps T and S (which may depend on x) satisfying the following properties: (i) T : L1 (M, τ ) + L∞ (M, τ ) → L1 (0, ∞) + L∞ (0, ∞), T is contractive from Lp (M, τ ) to Lp (0, ∞) for p = 1 and p = ∞, and T x = μ(x);


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(ii) S : L1 (0, ∞) + L∞ (0, ∞) → L1 (M, τ ) + L∞ (M, τ ), S is contractive from Lp (0, ∞) to Lp (M, τ ) for p = 1 and p = ∞, and Sμ(x) = x. Although not explicitly stated, Theorem 2.1 is implicit in the literature. It is essentially contained in [4] for Schatten classes, and in some different (weaker) form in [53] for the general case. We will include a proof at the end of the section. R EMARK . In interpolation language, Theorem 2.1 implies that the pair (L1 (M, τ ), L∞ (M, τ )) is a (contractive) partial retract of (L1 (0, ∞), L∞ (0, ∞)). We should emphasize the usefulness of such a result: it reduces all interpolation problems on (L1 (M, τ ), L∞ (M, τ )) to those on (L1 (0, ∞), L∞ (0, ∞)). Recall that (L1 (0, ∞), L∞ (0, ∞)) is one of the best understood pairs in interpolation theory. We now illustrate this by some examples. More applications can be found in [4,53,54] and [133,134]. First let us show how to get (2.1) and (2.2) from their commutative counterparts. P ROOF OF (2.1) AND (2.2). Let x ∈ Lp (M, τ ) (noting that Lp (M, τ ) ⊂ L1 (M, τ ) + L∞ (M, τ )). Let S be the map associated to x given by Theorem 2.1. Then by interpolation

S : L1 (0, ∞), L∞ (0, ∞) θ → L1 (M, τ ), L∞ (M, τ ) θ is a contraction. However, it is classical that 1

L (0, ∞), L∞ (0, ∞) θ = Lp (0, ∞) (with equal norms). Thus we deduce xθ = Sμ(x)θ μ(x)θ = xp ; whence

Lp (M, τ ) ⊂ L1 (M, τ ), L∞ (M, τ ) θ ,

a contractive inclusion.

The inverse inclusion is proved similarly by means of the map T . Therefore, we have shown (2.1). In the same way, we get (2.2). The above argument works in a more general setting as well. C OROLLARY 2.2. Let F be an interpolation functor. Then

F L1 (M, τ ), L∞ (M, τ ) = F L1 (0, ∞), L∞ (0, ∞) (M, τ ). More generally, for any r.i. function spaces E0 , E1 on (0, ∞)

F E0 (M, τ ), E1 (M, τ ) = F (E0 , E1 )(M, τ ).

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This corollary is contained in [53] (and in [4,141] for Schatten classes). R EMARK . As a consequence of Corollary 2.2, we have that for any r.i. function space E the map T (resp. S) in Theorem 2.1 is contractive from E(M, τ ) (resp. E) to E (resp. E(M, τ )). In particular, T and S are contractions between the Lp -spaces in consideration for all 1 p ∞. The following particular case of Corollary 2.2 is worth being mentioned explicitly. Here Kt denotes the usual K-functional from interpolation theory. C OROLLARY 2.3. Let 1 p0 , p1 ∞. Then for any x ∈ Lp0 (M, τ ) + Lp1 (M, τ ) and any t > 0

Kt x; Lp0 (M, τ ), Lp1 (M, τ ) = Kt μ(x); Lp0 (0, ∞), Lp1 (0, ∞) . In particular,

Kt x; L1 (M, τ ), L∞ (M, τ ) =

t

μs (x) ds. 0

R EMARKS . (i) Using a factorization argument, one can easily extend Corollary 2.3 to the case of quasi-Banach spaces, so that p0 , p1 are now allowed to be in (0, ∞]. Then the equality there has to be replaced by an equivalence with relevant constants depending only on p0 , p1 (see also [135]). (ii) As a consequence of the preceding remark, the indices p0 , p1 in (2.2) can vary in (0, ∞]. (iii) (2.1) also extends to the quasi-Banach space case (cf. [201]). P ROOF OF T HEOREM 2.1. Fix an x ∈ L1 (M, τ ) + L∞ (M, τ ). We may assume x 0. Indeed, by polar decomposition, it is easy to reduce the proof to this case. First we suppose x is an elementary operator, i.e., of the form x=

n

a k ek ,

k=1

where for all 1 k n, ak ∈ (0, ∞), and where the ek ’s are disjoint projections with τ (ek ) ∈ (0, ∞). Then we define Py =

n τ (yek ) k=1

τ (ek )

ek ,

y ∈ L1 (M, τ ) + L∞ (M, τ ).

Note that P is the orthogonal projection of L2 (M, τ ) onto its subspace generated by {e1 , . . . , en }. In particular, P is selfadjoint. Let y ∈ L∞ (M, τ ). Then |τ (yek )| y∞ τ (ek ) sup = y∞ . τ (ek ) 1kn τ (ek ) 1kn

P y∞ sup


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Therefore, P is a contraction on L∞ (M, τ ). By duality, P is a contraction on L1 (M, τ ) as well. Let N be the subalgebra generated by {e1 , . . . , en } (the identity of N being e = e1 + n and τ induces a weighted counting measure ν on · · · + en ). Then N is isomorphic to l∞ n l∞ , namely, ν({k}) = τ (ek ) for all 1 k n. It is clear that for any p Lp (N , τ |N ) = lpn (ν) isometrically. n , ν). With this identification, μ(x) is exactly the Thus we can identify (N , τ |N ) with (l∞ usual non-increasing rearrangement of x with respect to the measure ν. On the other hand, it is classical (and easy to prove in our special case) that there are linear maps R and Q satisfying (cf. [33]) n (ν) → L1 (0, ∞) + L∞ (0, ∞), R is contractive from l n (ν) to (i) R : l1n (ν) + l∞ p p L (0, ∞) for p = 1, ∞, and Rx = μ(x); n (ν), Q is contractive from Lp (0, ∞) to (ii) Q : L1 (0, ∞) + L∞ (0, ∞) → l1n (ν) + l∞ n lp (ν) for p = 1, ∞, and Qμ(x) = x. n (ν) Then we set T = RP and S = iQ, where i is the natural inclusion of l1n (ν) + l∞ (= L1 (N , τ |N ) + L∞ (N , τ |N )) into L1 (M, τ ) + L∞ (M, τ ). One easily checks that T and S satisfy all requirements of Theorem 2.1. Therefore, Theorem 2.1 is proved for elementary operators. Before passing to general (positive) operators, we note that T and S constructed above are positive in the sense that y 0 (resp. f 0) implies T y 0 (resp. Sf 0). Now for a positive x ∈ L1 (M, τ ) + L∞ (M, τ ), using the spectral decomposition of x, we may choose an increasing sequence {xn } of elementary positive operators such that xn x for all n 1, limn→∞ μt (xn ) = μt (x) for all t > 0 and limn→∞ xn = x in the topology σ (L1 (M, τ ) + L∞ (M, τ ), L1 (M, τ ) ∩ L∞ (M, τ )). See [64], pp. 277– 278. By the first part of the proof, for each n there are Tn and Sn associated with xn as in Theorem 2.1. Thus (Tn ) is a bounded sequence in B(L∞ (M, τ ), L∞ (0, ∞)). Since B(L∞ (M, τ ), L∞ (0, ∞)) is a dual space with predual L∞ (M, τ ) ⊗∧ L1 (0, ∞), passing to a subsequence if necessary, we may assume that Tn converges to T in B(L∞ (M, τ ), L∞ (0, ∞)) with respect to the w∗ -topology. Thus T is a contraction from L∞ (M, τ ) to L∞ (0, ∞). To show that T also defines a contraction from L1 (M, τ ) to L1 (0, ∞) let y ∈ L1 (M, τ ) ∩ L∞ (M, τ ) and f ∈ L1 (0, ∞) ∩ L∞ (0, ∞). Then

∞

T (y)f = lim

∞

n→∞ 0

0

Tn (y)f ;

whence

∞ 0

T (y)f lim sup Tn (y)1 f ∞ y1 f ∞ , n→∞

which implies that T y ∈ L1 (0, ∞) and T y1 y1 . Hence T extends to a contraction from L1 (M, τ ) into L1 (0, ∞).

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On the other hand, by the positivity of Tn , we have μ(xn ) = Tn xn Tn x. Taking limits, we find μ(x) T x. Hence μ(x) = wT x for some w ∈ L∞ (0, ∞) with w∞ 1. Then one sees that the map T defined by T y = wT y has the property (i) of Theorem 2.1. Similarly, from the sequence (Sn ) we get the desired map S. R EMARK 2.4. The result of interpolation applied to a compatible pair (X0 , X1 ) of Banach spaces depends in general very much on the way in which we view this pair as compatible. There is however an elementary “invariance” property which we will invoke in the sequel, as follows: let (X0 , X1 ) be a compatible pair of Banach spaces. Now let (Y0 , Y1 ) be another compatible pair of Banach spaces and let u0 : Y0 → X0 and u1 : Y1 → X1 be isometric isomorphisms, which coincide on Y0 ∩ Y1 (in that case it is customary in interpolation theory to think of u0 and u1 as the “same” map!). Equivalently, we have an isometric isomorphism u : Y0 + Y1 → X0 + X1 such that the restrictions u0 = u|Y0 and u1 = u|Y1 are isometric isomorphisms respectively from Y0 to X0 and from Y1 to X1 . Then u defines an isometric isomorphism from (Y0 , Y1 )θ to (X0 , X1 )θ , so that (Y0 , Y1 )θ (X0 , X1 )θ

(0 < θ < 1).

This follows from the interpolation property applied separately to u and its inverse. If we assume that the pairs are made compatible with respect to continuous injections J : X0 → X1 and j : Y0 → Y1 . Then to say that u0 and u1 are the “same” map means that u1 j = J u0 . In particular, if X0 = Y0 and if u0 is the identity on X0 = Y0 , then this reduces to u1 j = J .

3. General von Neumann algebras, including type III The construction of non-commutative Lp -spaces based on n.s.f. traces outlined in Section 1 does not apply to von Neumann algebras of type III, which do not admit n.s.f. traces. However, it is known that any von Neumann algebra has an n.s.f. weight (a weight is simply an additive and positively homogeneous functional on the positive cone with values in [0, ∞]). This section is devoted to the non-commutative Lp -spaces associated with a von Neumann algebra equipped with an n.s.f. weight. There are several ways to construct the latter spaces (cf., e.g., [3,78,87,106,113,192]). We will present two of them. The first one is to reduce von Neumann algebras of type III to semifinite von Neumann algebras with the help of crossed products, as proposed by Haagerup [78]. The second way is via the complex interpolation; so it can be considered as a continuation of the results established in the previous section for the semifinite case. This was developed by Kosaki [106] and Terp [192] (see also [88,89] for related results). We begin with the construction via interpolation. Let M be a von Neumann algebra. We know that M is a dual space with a unique predual, denoted by M∗ . We define, as usual, L1 (M) = M∗ and L∞ (M) = M. Now we are confronted with the problem of defining Lp (M) for any 1 < p < ∞. For simplicity and clarity we will consider only the case where M is σ -finite, as in [106]. The reader is referred to [192] for the general case. Fix a


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distinguished normal faithful state ϕ on M. Then we embed M into M∗ by the following left injection j : M → M∗ ,

j (x) = xϕ

(here xϕ(y) = ϕ(yx) ∀y ∈ M).

It is clear that j is a contractive injection with dense range. Thus we obtain a “compatibility” for the pair (M, M∗ ) with respect to which we may consider interpolation spaces between M and M∗ . Now let 1 < p < ∞. Following Kosaki, we introduce the corresponding non-commutative Lp -space as Lp (M, ϕ) = (M, M∗ )1/p . To show the so-defined non-commutative Lp -spaces possess all properties one can expect, one should first note the important fact that L2 (M, ϕ) is a Hilbert space, more precisely, L2 (M, ϕ) = Hϕ , where Hϕ is the Hilbert space in the GNS construction induced by ϕ (obtained after completion of M equipped with the inner product x, y = ϕ(y ∗ x)). The proof of this fact given in [106] uses the modular theory. Here, we would like to point out that it directly follows from a general result in interpolation theory, that we describe as follows. Let X be a complex Banach space. Let X denote the conjugate space of X, i.e., X is just ¯ for any λ ∈ C X itself but equipped with the conjugate complex multiplication: λ · x = λx and x ∈ X. For x ∈ X, x¯ denotes the element x considered as an element in X. Given a linear map v : X → Y , we denote by v¯ : X → Y the same map acting on the “conjugates”. Now suppose that there is a bounded linear map J : X∗ → X which is injective and of dense range. This allows us to consider (X∗ , X) as a compatible pair of Banach spaces. Suppose further that J is positive, i.e., ξ(J (ξ )) 0 for any ξ ∈ X∗ . Then ξ, η = ξ(J (η)) defines ¯ = ξ(x).) Let H be the a scalar product on X∗ . (Note: for ξ ∈ X∗ and x ∈ X we write ξ(x) completion of X∗ with respect to the above scalar product. Note that H contains X∗ as a dense linear subspace. Thus we can define a bounded linear injection v : H → X by simply setting (on an element of X∗ ) v(ξ ) = J (ξ ), and extending by density to the whole of H . Identifying H ∗ with H as well as (X)∗ with X∗ , and denoting by t v : (X)∗ → H ∗ = H the adjoint of v (in the Banach space sense) we see that J = v t v. These facts are well known (and easy to check). The general theorem referred to above is the following T HEOREM 3.1. With the above assumptions, (X∗ , X)1/2 = H with equal norms. R EMARK . This is well-known ([116]) with the additional assumption that X is reflexive. The general form as above was observed in [153], p. 26 (see also [197] and [42] for related results). C OROLLARY 3.2. L2 (M, ϕ) = Hϕ with equal norms. P ROOF. We let X = M∗ , X∗ = M. Recall that the involution on M∗ is defined by ψ ∗ (x) = ψ(x ∗ ) (ψ ∈ M∗ , x ∈ M). Let J : M → M∗ be the map taking x to j (x)∗ = ϕx ∗ and let u1 : M∗ → M∗ be the (linear) isometry taking ψ to ψ ∗ . We have ξ, η =

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ξ(J (η)) = ξ(ϕη∗ ) = ξ(ϕη∗ ) = ϕ(η∗ ξ ), thus we find H = Hϕ with equal norms and we have J = u1 j , so that the result follows by invoking Remark 2.4 (here u0 is simply the identity of M). Using this corollary and the reiteration theorem, we see that the dual space of Lp (M, ϕ)

is (isometrically) equal to Lp (M, ϕ) for any 1 < p < ∞ (1/p + 1/p = 1). The duality is induced by the scalar product of Hϕ , that is, x, y = ϕ(y ∗ x), x, y ∈ M. Corollary 3.2 also yields the Clarkson inequalities in Lp (M, ϕ) for any 1 < p < ∞ (see Theorem 5.1 below). Thus, Lp (M, ϕ) (1 < p < ∞) is uniformly convex. We will see more precise results on this in Section 5. R EMARK . Instead of the left injection considered previously, one could equally take the right injection of M into M∗ , i.e., x → ϕx (here ϕx(y) = ϕ(xy) ∀y ∈ M). Then the resulting interpolation spaces are isometric to those obtained previously. In view of the results in the last section, one is naturally led to consider the real interpolation as well. Set, for 1 < p < ∞ Lp,p (M, ϕ) = (M, M∗ )1/p,p . The problem now is whether Lp,p (M, ϕ) and Lp (M, ϕ) are isomorphic. For the special case of p = 2, the answer is affirmative. Indeed, Theorem 3.1 admits a counterpart for the real interpolation as well (see [116] in the case of reflexive spaces; [205], p. 519 for the general case; see also [42] for more related results). Thus L2,2 (M, ϕ) = L2 (M, ϕ) with equivalent norms. However, this is no longer true for all other values of p, as shown by the following example, due to Junge and the second named author. E XAMPLE 3.3. Let ϕ be the state of B(l 2 ) given by a diagonal operator D of trace 1 and whose diagonal entries are all positive. Then, for any 1 < p = 2 < ∞, the two spaces (B(l 2 ), B(l 2 )∗ )1/p and (B(l 2 ), B(l 2 )∗ )1/p, p do not coincide. Indeed, let R (resp. R∗ ) be the subspace of B(l 2 ) (resp. B(l 2 )∗ ) consisting of matrices whose all rows but the first are zero. It is clear that R = l 2 and R∗ = l 2 (d) isometrically, where d = (dn )n is the sequence of the diagonal entries of D, and where l 2 (d) is the weighted l 2 -space with the norm xl 2 (d) =

1/2 |xn dn |2

.

n

On the other hand, let P : B(l 2 ) → R be the natural projection. P is contractive on B(l 2 ). It is easy to check that under the left injection associated with ϕ, P is also a contractive projection from B(l 2 )∗ onto R∗ . Now assume that for some 1 < p < ∞ the two interpolation spaces (B(l 2 ), B(l 2 )∗ )1/p and (B(l 2 ), B(l 2 )∗ )1/p, p have equivalent norms. Then we deduce that (R, R∗ )1/p and (R, R∗ )1/p, p have equivalent norms too. However, it is wellknown that the first space is still a weighted l 2 -space (and hence a Hilbert space), while the second one is isomorphic to a Hilbert space only when p = 2. Thus we have proved our assertion.


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This construction by interpolation has several disadvantages: there is no natural notion of positive cone, no reasonably handy bimodule action by multiplication of M on Lp (M, ϕ), and finally the case p < 1 is excluded. However, these difficulties disappear in Haagerup’s construction, to which we now turn. Our main reference for Haagerup’s Lp -spaces is [191]. Let M be a von Neumann algebra equipped with a distinguished n.s.f weight ϕ. Let σt = ϕ σt , t ∈ R, denote the one parameter modular automorphism group of R on M associated with ϕ. We consider the crossed product R = M σ R. Recall briefly the definition of R. If M acts on a Hilbert space H , R is a von Neumann algebra acting on L2 (R, H ), generated by the operators π(x), x ∈ M, and the operators λ(s), s ∈ R, defined by the following conditions: for any ξ ∈ L2 (R, H ) and t ∈ R and λ(s)(ξ )(t) = ξ(t − s).

π(x)(ξ )(t) = σ−t (x)ξ(t)

Note that π is a normal faithful representation of M on L2 (R, H ). Thus we may identify M with π(M). Then the modular automorphism group {σt }t ∈R is given by σt (x) = λ(t)xλ(t)∗ ,

x ∈ M, t ∈ R.

There is a dual action {σˆ t }t ∈R of R on R. This is a one parameter automorphism group of R on R, implemented by the unitary representation {W (t)}t ∈R of R on L2 (R, H ): σˆ t (x) = W (t)xW (t)∗ ,

t ∈ R, x ∈ R,

where W (t)(ξ )(s) = e−it s ξ(s),

ξ ∈ L2 (R, H ), t, s ∈ R.

Note that the dual action σˆ t is also uniquely determined by the following conditions σˆ t (x) = x

and σˆ t λ(s) = e−ist λ(s),

∀x ∈ M, s, t ∈ R.

Thus M is invariant under {σˆ t }t ∈R . In fact, M is exactly the space of the fixed points of {σˆ t }t ∈R , namely, M = x ∈ R: σˆ t (x) = x, ∀t ∈ R . Recall that the crossed product R is semifinite. Let τ be its n.s.f. trace satisfying τ ◦ σˆ t = e−t τ,

∀ t ∈ R.

Also recall that any n.s.f. weight ψ on M induces a dual n.s.f. weight ψ˜ on R. Then ψ˜ admits a Radon–Nikodym derivative with respect to τ . In particular, the dual weight ϕ˜ of our distinguished weight ϕ has a Radon–Nikodym derivative D with respect to τ . Then ϕ(x) ˜ = τ (Dx),

x ∈ R+ .

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Recall that D is an invertible positive selfadjoint operator on L2 (R, H ), affiliated with R, and that the regular representation λ(t) above is given by λ(t) = D it ,

t ∈ R.

Now we define the Haagerup non-commutative Lp -spaces (0 < p ∞) by Λp (M, ϕ) = x ∈ L0 (R, τ ): σˆ t (x) = e−t /p x, ∀t ∈ R . (Recall that L0 (R, τ ) denotes the topological ∗-algebra of all operators on L2 (R, H ) measurable with respect to (R, τ ).) It is clear that Λp (M, ϕ) is a vector subspace of L0 (R, τ ), invariant under the ∗-operation. The algebraic structure of Λp (M, ϕ) is inherited from that of L0 (R, τ ). Let x ∈ Λp (M, ϕ) and x = u|x| its polar decomposition. Then u ∈ M and |x| ∈ Λp (M, ϕ). Recall that Λ∞ (M, ϕ) = M and Λ1 (M, ϕ) = M∗ . The latter equality is understood as follows. As mentioned previously, for any ω ∈ M+ ∗, the dual weight ω˜ has a Radon–Nikodym derivative, denoted by hω , with respect to τ : ω(x) ˜ = τ (hω x),

x ∈ R+ .

Then hω ∈ L0 (R, τ )

and σˆ t (hω ) = e−t hω ,

∀t ∈ R.

1 Thus hω ∈ Λ1 (M, ϕ)+ . This correspondence between M+ ∗ and Λ (M, ϕ)+ extends to 1 a bijection between M∗ and Λ (M, ϕ). Then for any ω ∈ M∗ , if ω = u|ω| is its polar decomposition, the corresponding hω ∈ Λ1 (M, ϕ) admits the polar decomposition

hω = u|hω | = uh|ω| . Thus we can define a norm on Λ1 (M, ϕ) by hω 1 = |ω|(1) = ωM∗ ,

ω ∈ M∗ .

In this way, Λ1 (M, ϕ) = M∗ isometrically. Now let 0 < p < ∞. Since x ∈ Λp (M, ϕ) iff |x|p ∈ Λ1 (M, ϕ), we define 1/p xp = |x|p 1 ,

x ∈ Λp (M, ϕ).

Then if 1 p < ∞, ·p is a norm (cf. [78] and [191]), and if 0 < p < 1, ·p is a p-norm (cf. [108]). Equipped with · p , Λp (M, ϕ) becomes a Banach space or a quasi-Banach space, according to 1 p < ∞ or 0 < p < 1. Clearly, xp = x ∗ p = |x|p , x ∈ Λp (M, ϕ).


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R EMARKS . (i) Using [191] Lemma II.5, one easily checks that Λp (M, ϕ) is isometric to a subspace of the non-commutative weak Lp -space Lp,∞ (R, τ ). Also note that in Λp (M, ϕ) the topology defined by · p coincides with the topology induced by that of L0 (R, τ ) (cf. [191]). (ii) One weak point of the Haagerup non-commutative Lp -spaces is the fact that for any p = q the intersection of Λp (M, ϕ) and Λq (M, ϕ) is trivial. In particular, these spaces do not form an interpolation scale. This causes some difficulties in applications (especially when interpolation is used).

As usual, for 1 p < ∞ the dual space of Λp (M, ϕ) is Λp (M, ϕ), 1/p + 1/p = 1. To describe this duality, we need to introduce a distinguished linear functional on Λ1 (M, ϕ), called trace and denoted by tr, which is defined by tr(x) = ωx (1),

x ∈ Λ1 (M, ϕ),

where ωx ∈ M∗ is the unique normal functional associated with x by the above identification between M∗ and Λ1 (M, ϕ). Then tr is a continuous functional on Λ1 (M, ϕ) satisfying

tr(x) tr |x| = x1 ,

x ∈ Λ1 (M, ϕ).

The usual Hölder inequality also holds for these non-commutative Lp -spaces. Let 0 < p, q, r ∞ such that 1/r = 1/p + 1/q. Then x ∈ Λp (M, ϕ) and y ∈ Λq (M, ϕ) 4⇒ xy ∈ Λr (M, ϕ) and xyr xp yq . In particular, for any 1 p ∞ we have tr(xy) xy1 xp yp ,

x ∈ Λp (M, ϕ), y ∈ Λp (M, ϕ).

Thus, (x, y) → tr(xy) defines a duality between Λp (M, ϕ) and Λp (M, ϕ), with respect to which p

∗

Λ (M, ϕ) = Λp (M, ϕ)

isometrically, 1 p < ∞.

This functional tr on Λ1 (M, ϕ) plays the role of a trace. Indeed, it satisfies the following tracial property tr(xy) = tr(yx),

x ∈ Λp (M, ϕ), y ∈ Λp (M, ϕ).

The reader is referred to [191] for more information.

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T HEOREM 3.4. Let M be a von Neumann algebra. (i) Let 0 < p < ∞. If τ is an n.s.f. trace on M, then Lp (M, τ ) (the non-commutative Lp -space described in Section 1) is isometric to Λp (M, ϕ). (ii) Let 0 < p < ∞. Then Λp (M, ϕ) is independent of ϕ, i.e., if ϕ and ψ are two n.s.f. weights on M, then Λp (M, ϕ) and Λp (M, ψ) are isometric. (iii) Let ϕ be a normal faithful state on M and 1 < p < ∞. Then Lp (M, ϕ) and Λp (M, ϕ) are isometric. The first two parts of Theorem 3.4 are due to Haagerup [78] (see also [191]), and the third one to Kosaki [106]. As can be expected, the proof of Theorem 3.4 heavily depends on the modular theory. The preceding statement allows a considerable simplification of the notation, as follows: C ONVENTION . From now on, given a von Neumann algebra M, Lp (M) will denote any one of the non-commutative Lp -spaces associated with M appearing in Theorem 3.4. (The latter shows that these spaces are all “the same”.) However, if M is semifinite, we will always assume that Lp (M) is the Lp -space constructed from an n.s.f. trace as in Section 1. The following basic result is very useful to reduce the failure of certain properties of to the special case of the hyperfinite factor Lp (R). Recall that R denotes the hyperfinite II1 factor (see Section 1).

Lp -spaces

T HEOREM 3.5. Let M be a von Neumann algebra not of type I. Then for any 0 < p ∞ (resp. 1 p ∞) Lp (R) is isometric to a (resp. 1-complemented) subspace of Lp (M). The proof of Theorem 3.5 combines several more or less well-known facts. The key point is that if M is not of type I, then R is isomorphic, as von Neumann algebra, to a w∗ -closed ∗-subalgebra of M which is the range of a normal conditional expectation on M. The reader is referred to [122] for more details and precise references. We end this section with Haagerup’s approximation theorem of an Lp (M) associated with an algebra M of type III by those associated with semifinite von Neumann algebras (cf. [79]). T HEOREM 3.6. Let M be a von Neumann algebra equipped with an n.s.f. weight ϕ. Let Λp (M, ϕ) be the associated Haagerup Lp -space (0 < p < ∞). Then there are a Banach space X (a p-Banach space if 0 < p < 1), a directed family {(Mi , τi )}i∈I of finite von Neumann algebras Mi (with normal faithful finite traces τi ), and a family {ji }i∈I of isometric embeddings ji : Lp (Mi , τi ) → X such that (i) ji (Lp (Mi , τi )) ⊂ ji (Lp (Mi , τi )) for all i, i ∈ I with i i ; (ii) i∈I ji (Lp (Mi , τi )) is dense in X; (iii) Λp (M, ϕ) is isometric to a (complemented for 1 p < ∞) subspace of X.


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4. From classic Lp to non-commutative Lp : similarities and differences A good part of the early theory consisted in extending commutative results over to the non-commutative case; this usually required specific new methods, but without too many surprises. For instance, we have already seen that a non-commutative Lp -space Lp (M) is reflexive for any 1 < p < ∞. Moreover, just like in the commutative case it is easy to check that L1 (M) has the RNP iff M is atomic. Indeed, if M is not atomic, L1 (M) contains a 1-complemented subspace isometric to L1 (0, 1), hence fails the RNP. On the other hand, L1 (M) is weakly sequentially complete for any M. Moreover, there are characterizations of weakly compact subsets in L1 (M), analogous to those in the commutative setting (cf. [190, III.5] and the references given there; see also [140] for more recent results). Moreover, we will see later in Section 5 that any non-commutative Lp -space (0 < p 1) has the analytic RNP. However, the differentiability of the norms of non-commutative Lp -spaces has not been well understood yet. This problem was considered only for the Schatten classes in [194]. It was announced there (with a sketch of proof) that the norm of S p had the same differentiability as that of l p (1 < p < ∞). It seems unclear how to extend this to the general case (or, at least, to the semifinite case). In a different direction, the papers [55,56] are devoted to the problem of characterizing the symmetric spaces of measurable operators for which the absolute-value mapping x → |x| is Lipschitz continuous. In the case of non-commutative L1 -spaces, Kosaki proves in [108] the following useful inequality: for any ϕ and ψ in such a space, we have √

|ϕ| − |ψ| 2 ϕ + ψ1 ϕ − ψ1 1/2 . 1 The passage from the Schatten classes to von Neumann algebras with semifinite traces, i.e., from the discrete to the continuous case, can sometimes be quite substantial. See, for instance, Brown’s extension of Weyl’s classical inequalities: ( |λn (T )|p )1/p T S p (here λn (T ) are the eigenvalues of T repeated according to multiplicity). Brown [26] had to invent a new kind of spectral measure (now called Brown’s measure) to extend this, together with Lidskii’s trace theorem, to the semifinite case. The study of non-commutative Lp -spaces, or more generally, of symmetric operator spaces, goes mainly in two closely related directions: lift topological or geometrical properties from the commutative setting to the non-commutative one, and reduce problems in the non-commutative case to those in the commutative one. We have already seen several examples in both directions. To discuss more illustrations, it is better to place ourselves in the context of symmetric operator spaces. Let M be a semifinite von Neumann algebra equipped with an n.s.f. trace τ , and let E be an r.i. function space on (0, ∞). One naturally expects that properties of E(M, τ ) should be reflected by those of E. Works already done in this direction are too numerous to enumerate. Here we content ourselves with only three examples. The first one concerns the (uniform) Kadets–Klee properties. The lifting of these properties from E to E(M, τ ) has been extensively studied (cf., e.g., [6,37,38,41,50,57]). The second example is about the reduction of weakly compact subsets in E(M, τ ) to those in E. This was

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achieved in [52] and [58] (see also the references there for previous works on this problem). Finally, the geometry of the unit ball of E(M, τ ) was studied in [5] and [40]. More examples and references of this kind will be given in appropriate places in the subsequent sections (see also [49]). Despite the strong analogy between the commutative and non-commutative settings, non-commutative Lp -spaces behave, in some aspects, very differently from their commutative counterparts. One of the most spectacular differences concerns unconditional bases or “unconditional structures”. Already in [111], it was proved that S 1 cannot be embedded into any space with an unconditional basis, in sharp contrast with 1 . But the big surprise came when Gordon and Lewis [74] proved that the Schatten class S p fails to have any unconditional basis when p = 2 in sharp contrast with p or Lp . More generally they proved that S p fails “local unconditional structure” in their sense (abbreviated as GL-l.u.st.; see [94] for the precise definition). This was the first example of a reflexive Banach space which was not isomorphic to any complemented subspace of a Banach lattice. More precisely, let lu(X) denote the GL-l.u.st. constant of a Banach space X (lu(X) is equal to the norm of factorization through a Banach lattice of the identity of X∗∗ ). The following theorem was proved by Gordon and Lewis [74] using a criterion (necessary but not sufficient) for the GL-l.u.st. of a space X: any 1-absolutely summing operator on X must factor through L1 (this is now called the GL-property). More precisely, they obtained the first part of the next statement (the second part comes from [143], see also [180] and [146], 8.d for related results): T HEOREM 4.1. There is a constant C > 0 such that for any 1 p ∞ and any n 1 p

Cn|1/p−1/2| lu Sn n|1/p−1/2|. Consequently, S p does not have the GL-l.u.st. for p = 2. More generally, let X be any Banach lattice of finite cotype (resp. of type > 1), then there is a constant C > 0 such that, if E is any n2 -dimensional subspace (resp. subspace of a quotient) of X, we have p d(Sn , E) C n|1/p−1/2| . Combining Theorem 3.5 and Theorem 4.1, we immediately obtain C OROLLARY 4.2. A non-commutative Lp (M), 1 p < ∞ and p = 2, has the GL-l.u.st. iff M is isomorphic, as Banach space, to L∞ (Ω, μ) for some measure space (Ω, μ). Moreover, this happens iff Lp (M) is isomorphic to a subspace of a commutative Lp -space. Note that M is isomorphic, as Banach space, to a commutative L∞ iff M is the direct sum (∞ sense) of finitely many algebras of the form L∞ (μ; B(H )) (= L∞ (μ) ⊗ B(H )) with dim(H ) < ∞. Another striking divergence from the classical case is provided by the uniform approximation property (UAP in short): by an extremely complicated construction, Szankowski proved that B(2 ) fails the approximation property (AP in short), and moreover ([189])


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that S p (or S p ) fails the UAP for p > 80. It remains a challenging open problem to prove this for any p = 2. We will describe another striking difference in Section 7, that is, a non-commutative Lp -space Lp (M), 0 < p 1, is never an analytic UMD space except when M is isomorphic, as Banach space, to a commutative L∞ -space. Surprisingly, by [122], “stability” provides us with one more sharp contrast. Recall that a Banach space X is stable (in Krivine–Maurey’s sense) if for any bounded sequences {xm }m1 , {yn }n1 in X and any ultrafilters U , V on N lim lim xm + yn = lim lim xm + yn .

m∈U n∈V

n∈V m∈U

It is well known that any commutative Lp -space (1 p < ∞) is stable (cf. [109]). This is no longer true in the non-commutative setting. In fact, we have the following characterization of stable non-commutative Lp -spaces. T HEOREM 4.3. Let 1 p < ∞, p = 2. Then Lp (M) is stable iff M is of type I. The “if” part of Theorem 4.3 was independently proved by Arazy [8] and Raynaud [166]. The “only if” part is due to Marcolino [122]. Marcolino’s proof is divided into two steps. The first one (the proof of which is relatively easy) is that Lp (R), p = 2, is not stable (recalling that R is the hyperfinite II1 factor). The second step is the above Theorem 3.5.

5. Uniform convexity (real and complex) and uniform smoothness The fact that Lp (M), 1 < p < ∞, is uniformly convex and smooth immediately follows from the following Clarkson type inequalities. T HEOREM 5.1. Let 1 < p, p < ∞ with 1/p + 1/p = 1. Then (i) if 1 p 2, &

'1/p 1 p p

x + yp + x − yp 2 p p 1/p xp + yp , x, y ∈ Lp (M);

(5.1)

(ii) if 2 p ∞, &

'1/p 1 p p

x + yp + x − yp 2 p p 1/p xp + yp , x, y ∈ Lp (M).

(5.2)

Inequalities (5.1) and (5.2), of course, have their origin in the classical Clarkson inequalities for commutative Lp -spaces. In the non-commutative setting, some partial or particular

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cases of (5.1) and (5.2) were obtained in [47,128] (see also [49] for additional references). (5.1) and (5.2), as stated above were proved in [78] and [106] (see also [191] and [64]). P ROOF OF T HEOREM 5.1. The proof is almost obvious via the complex interpolation. Indeed, (5.1) (resp. (5.2)) is trivially true for p = 1, 2 (resp. p = 2, ∞). Then the complex interpolation yields (5.1) and (5.2). We also note that (5.1) and (5.2) are dual to each other. Let δX (resp. ρX ) denote the modulus of convexity (resp. smoothness) of a Banach space X. Theorem 5.1 implies the following C OROLLARY 5.2. Let 1 0;

1 p t , p

t > 0.

and ρLp (M) (t)

(ii) for 2 < p < ∞ δLp (M) (ε)

1 p ε , p2p

0 < ε < 2,

and ρLp (M) (t)

The reader can find some applications of the uniform convexity of Lp (M), e.g., in [107,108]. Let us comment on the estimate for the modulus of convexity given by Corollary 5.2 (the same comment, of course, applies to the modulus of smoothness as well). This estimate is best possible only in the case of 2 < p < ∞. We should also point out that in this case the relevant constant 1/(p2p ) is optimal (for it is already so in the commutative case; see [115], p. 63). Keeping in mind the well-known result on the modulus of convexity of commutative Lp -spaces, one would expect that the order of δLp (M) (ε) for 1 < p < 2 be O(ε2 ). This is indeed the case (cf. [193]). In fact, we have a more precise result as follows. T HEOREM 5.3. Let 1 < p < ∞. Then (i) for 1 < p 2

1/2 x2p + (p − 1)y2p '1/p & 1 p p

x + yp + x − yp , 2

∀x, y ∈ Lp (M);

(5.3)

(ii) for 2 < p < ∞, &

'1/p 1 p p

x + yp + x − yp 2

1/2 x2p + (p − 1)y2p ,

∀x, y ∈ Lp (M).

Moreover, the constant p − 1 is optimal in both (5.3) and (5.4).

(5.4)


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This theorem was proved in [14] for Schatten classes. As pointed out by the authors, the arguments there work for semifinite von Neumann algebras as well. Then the general case follows by Theorem 3.6. We should emphasize that the optimality of the constant p − 1 in (5.3) and (5.4) has important applications to hypercontractivity. We will later illustrate this by discussing the Fermionic hypercontractivity. Note that if one does not care about the best constants, one can deduce Theorem 5.3 from the optimal order of δLp (M) (ε) and ρLp (M) (t) obtained in [193] (at least, for Schatten classes). Note also that (5.3) and (5.4) are equivalent by duality. We will include a very simple proof of (5.4) for p = 2n (n ∈ N), and so by interpolation for all 2 0 p−1 2 ε , 8 p−1 2 t , ρLp (M) (t) 2 δLp (M) (ε)

1 < p 2,

and

2 p < ∞.

R EMARK . The constants (p − 1)/8 and (p − 1)/2 in the above estimates are optimal (see [115], p. 63 for the commutative case). Corollaries 5.2 and 5.4 yield the type and cotype of Lp (M) for 1 < p < ∞. C OROLLARY 5.5. Let 1 < p < ∞. Then Lp (M) is of type min(2, p) and cotype max(2, p). The type and cotype of Lp (M) were determined in [193] for Schatten classes, and in [63] for the general case. We will see later that Lp (M) is of cotype 2 for 0 < p 1. Now we turn to the application of the optimality of the constant p − 1 in (5.3) and (5.4) to the Fermionic hypercontractivity. Before starting our discussion, we should point out, however, that in the scalar case (i.e., in the case where M = C) Theorem 5.3 is exactly Nelson’s celebrated hypercontractivity inequality for the two point space (cf. [20] and [130]). This two point hypercontractivity inequality easily yields the optimal hypercontractivity for the classical Ornstein–Uhlenbeck semigroup. Carlen and Lieb used Theorem 5.3 (in the case of Schatten classes) to obtain the optimal Fermionic hypercontractivity, thus solving a problem left open since Gross’ pioneer works in the domain (cf. [75]). Let R be the hyperfinite II1 factor. We recall that R is generated by a sequence (εn )n1 of self-adjoint unitaries satisfying (CAR) (see Section 1). We also recall that {wA : A ⊂ N, A finite} is an orthonormal basis of L2 (R). We define the number operator N by NwA = |A|wA (|A| denoting the cardinality of A). N is an unbounded positive self-adjoint operator on L2 (R). It generates the Fermionic Ornstein–Uhlenbeck semigroup Pt : Pt = e−t N , t 0. One can show that Pt is a contraction on Lp (R) for all 1 p ∞. The optimal Fermionic hypercontractivity is contained in the following

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T HEOREM 5.6. Let 1 < p < q < ∞. Then Pt is a contraction from Lp (R) to Lq (R) iff e−2t (p − 1)/(q − 1). Let us briefly comment on the proof of Theorem 5.6. First, since linear combinations of the wA ’s are dense in Lp (R), it suffices to prove Theorem 5.6 in the finite-dimensional case, that is, when Pt is restricted to the Lp -spaces based on the von Neumann algebra generated by {ε1 , . . . , εn } (n ∈ N). Second, by standard arguments as for the classical Ornstein–Uhlenbeck semigroup, one can reduce Theorem 5.6 to the special case where 2 = p < q. Assuming these reductions, one can use (5.4) to prove Theorem 5.6 by induction on n (noting that the case n = 1 corresponds to Nelson’s two point hypercontractivity). We refer to [35] for the details. R EMARKS . (i) Theorem 5.6 implies, and in fact, is equivalent to the optimal Fermionic logarithmic Sobolev inequality, see [76] and [35]. (ii) Biane [18] obtained the analogue of Theorem 5.6 for the free Ornstein–Uhlenbeck semigroup. Note that this latter semigroup is also ultracontractive (cf. [23,24]). We end the discussion on the uniform convexity and smoothness by providing a simple proof for Theorem 5.3 (except for the best constant). We need only to consider (5.4). We are going to show that for 2 p < ∞ there is a constant Cp , depending only on p, such that &

'1/p 1 p p

x + yp + x − yp 2

1/2 x2p + Cp y2p , ∀x, y ∈ Lp (M).

(5.4p )

To that end, by Theorem 3.6, we can assume that M is semifinite and equipped with a faithful normal semifinite trace τ . The key step in the proof of (5.4p ) is the implication “(5.4p ) ⇒ (5.42p )”. Let us show this. Assume (5.4p ). Let x, y ∈ L2p (M), and set a = x ∗ x + y ∗ y, b = x ∗ y + y ∗ x. Then a, b ∈ Lp (M) and 1 1 2p 2p

p p

x + y2p + x − y2p = a + bp + a − bp 2 2

p/2

by (5.4p ) a2p + Cp b2p

2 p/2

x22p + y22p + 4Cp x22p y22p

p x22p + (2Cp + 1)y22p ; whence (5.42p ) with C2p 2Cp + 1. Therefore, starting with the trivial case p = 2 (noting that C2 = 1), and by iteration, we get C2n 2n − 1 (in fact, C2n = 2n − 1). Thus for these special values of p we obtain the best constant in (5.4). Then for any other value of p, say, 2n < p < 2n+1 , by complex interpolation, we deduce (5.4p ) with

1−θ n+1

θ 2 −1 , Cp 2n − 1 where 1/p = (1 − θ )/2n + θ/2n+1 .


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Now we pass on to the uniform PL-convexity and Hardy convexity of non-commutative Lp -spaces. This time, we admit quasi-normed spaces (so p < 1 is allowed). Let X be a (complex) quasi-Banach space. Let T be the unit circle equipped with normalized Lebesgue measure. For 0 < p < ∞ we denote by Lp (T, X) the usual Lp -space of Bochner measurable functions with values in X. Note that Lp (T, Lp (M)) is just the noncommutative Lp -space based on the von Neumann algebra tensor product L∞ (T) ⊗ M. Let P(X) denote the family of all complex polynomials with coefficients in X: P(X) =

n

xk z : xk ∈ X, 0 k n, n ∈ N . k

k=0

D EFINITION . Let X be a quasi-Banach space. Let 0 0. We define HX (ε) = inf x + zyL1 (T,X) − 1: x = 1, y ε, x, y ∈ X and p hX (ε) = inf f Lp (T,X) − 1: f (0) = 1, f − f (0)Lp (T,X) ε, f ∈ P(X) . p

X is said to be uniformly PL-convex (resp. H p -convex) if HX (ε) > 0 (resp. hX (ε) > 0) for p all ε > 0. HX (ε) (resp. hX (ε)) is called the modulus of PL-convexity (resp. H p -convexity) of X. The uniform PL-convexity was introduced and studied in [45]. It was shown there that in the definition of HX (ε) above, if the L1 -norm is replaced by an Lp -norm, then the resulting modulus is equivalent to HX (ε). The uniform H p -convexity was explicitly introduced in [199]; however, it is already implicit in [80]. It was proved in [202] that if X is uniformly H p -convex for one p ∈ (0, ∞), then so is it for all p ∈ (0, ∞). Thus we say that X is uniformly H-convex if it is uniformly H p -convex for some p. The uniform PLconvexity (resp. H-convexity) is closely related to inequalities satisfied by analytic (resp. Hardy) martingales with values in X. The Enflo–Pisier renorming theorem about the uniform (real) convexity admits analogues for these uniform complex convexities. We refer to [45,199,201,202] and [149] for more information. R EMARKS . (i) For any given 0 0 such that for all quasiBanach spaces X p

HX (ε) αp hX (αp ε),

0 < ε 1.

Consequently, the uniform H-convexity implies the uniform PL-convexity. (ii) If a Banach space X is uniformly convex, it is uniformly H-convex.

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T HEOREM 5.7. Assume 0 < p, q < ∞. Let M be a von Neumann algebra. Then p

hLq (M) (ε) αεr ,

0 < ε 1,

where r = max(2, p, q) and α > 0 is a constant depending only on p, q. R EMARKS . (i) In the case q > 1, Theorem 5.7 easily follows from Corollaries 5.2 and 5.4. Thus the non-trivial part of Theorem 5.7 lies in the case q 1. (ii) Theorem 5.7 implies, of course, that the same estimate holds for the modulus of PL-convexity. (iii) In the case of q = 1, Theorem 5.7 is contained in [80]. In fact, it is this result which motivated the introduction of the uniform H-convexity. Theorem 5.7, as stated above, was proved in [201]. The ingredient of the proof is the Riesz type factorization for Hardy spaces of analytic functions with values in noncommutative Lp -spaces. In Section 8 below we will discuss such a factorization in a more general context. The following corollary completes Corollary 5.5. Thus the non-commutative Lp -spaces have the same type and cotype as the commutative Lp -spaces. C OROLLARY 5.8. Lp (M) is of cotype 2 for any 0 < p 1 and any von Neumann algebra M. This corollary was proved in [193] for p = 1 and in [201] for 0 < p < 1. We recall that a quasi-Banach space X has the analytic Radon–Nikodym property (abbreviated as analytic RNP) if any bounded analytic function F : D → X has a.e. radial limits in X, where D denotes the unit disc (cf. [30,59], and also [32] for additional references). It is known that the uniform H-convexity implies the analytic RNP. Thus we get the C OROLLARY 5.9. Lp (M) has the analytic RNP for any 0 < p 1 and any von Neumann algebra M. The results discussed in this section have all been extended to symmetric operator spaces. We refer to [70,195] for the cotype, uniform convexity, PL-convexity and smoothness in the unitary ideals, and in the general case, to [200] for the uniform convexity and smoothness, to [203,204] for the uniform H-convexity, RNP and analytic RNP (see also [129]). Finally, we mention that [39] contains related results, especially those on the local uniform convexity for symmetric operator spaces.

6. Non-commutative Khintchine inequalities This section is devoted to the non-commutative Khintchine inequalities and the closely related Grothendieck-type factorization theorems. Although all results in this section hold


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for the general non-commutative Lp -spaces, we will restrict ourselves to the semifinite ones, i.e., those constructed from an n.s.f. trace. Letters Ap , Bp , . . . , will denote positive constants depending only on p. Let (εn )n1 be a Rademacher (or Bernoulli) sequence, i.e., a sequence of independent random variables on some probability space (Ω, F , P ) such that P (εn = 1) = P (εn = −1) = 1/2 for all n 1. We first recall the classical Khintchine inequalities. Let 0 < p < ∞. Then for all finite sequences (an ) of complex numbers A−1 a ε a ε B a ε . (6.1) n n n n p n n p Lp (Ω,P )

n1

L2 (Ω,P )

n1

n1

Lp (Ω,P )

(Note that obviously n1 an εn L2 (Ω,P ) = ( n1 |an |2 )1/2 .) These inequalities remain valid (suitably modified) when the coefficients an ’s are vectors from a Banach space X. In that case they are due to Kahane, and are usually called “Khintchine–Kahane inequalities”: for all finite sequences (an ) in X −1 Ap an εn an εn Lp (Ω,P ;X)

n1

L2 (Ω,P ;X)

n1

Bp an εn n1

(6.2)

.

Lp (Ω,P ;X)

In particular, if X is a commutative Lp -space, say X = Lp over (0, 1), (6.2) implies that for all finite sequences (an ) in Lp (0, 1) 1/2 2 A−1 |a | n p n1

Lp

a ε n n n1

L2 (Ω,P ;Lp )

1/2 2 Bp |an | n1

(6.3)

.

Lp

It is (6.3) that we will extend to the non-commutative setting. Now let M be a semifinite von Neumann algebra equipped with an n.s.f. trace τ . Let a = (an ) be a finite sequence in Lp (M) (recalling that by our convention, Lp (M) = Lp (M, τ )). Define 1/2 2 , aLp (M;l 2 ) = |a | n C

n0

p

∗ 2 1/2 . aLp (M;l 2 ) = |a | n R

n0

p

This gives two norms (or quasi-norms if p < 1) on the family of all finite sequences in Lp (M). The corresponding completions (relative to the w∗ -topology for p = ∞) are denoted by Lp (M; lC2 ) and Lp (M; lR2 ), respectively. The reader is referred to [156] for a discussion of these norms.

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Now we can state the non-commutative Khintchine inequalities. T HEOREM 6.1. Let 1 p < ∞, and let M be a semifinite von Neumann algebra. Let a = (an )n0 be a finite sequence in Lp (M). (i) If 2 p < ∞, there is a constant Bp (depending only on p) such that εn an aLp (M;l 2 )∩Lp (M;l 2 ) C R n0

Lp (Ω,P ;Lp (M))

Bp aLp (M;l 2 )∩Lp (M;l 2 ) . C

(6.4)

R

(ii) If 1 p < 2, there is an absolute constant A > 0 (independent of p and a) such that AaLp (M;l 2 )+Lp (M;l 2 ) εn an C

R

n0

Lp (Ω,P ;Lp (M))

aLp (M;l 2 )+Lp (M;l 2 ) . C

(6.5)

R

For the convenience of the reader we recall the norms in Lp (M; lC2 ) ∩ Lp (M; lR2 ) and

Lp (M; lC2 ) + Lp (M; lR2 ):

aLp (M;l 2 )∩Lp (M;l 2 ) = max aLp (M;l 2 ) , aLp (M;l 2 ) C

R

C

R

and aLp (M;l 2 )+Lp (M;l 2 ) = inf bLp (M;l 2 ) + cLp (M;l 2 ) , C

R

C

R

where the infimum runs over all decompositions a = b + c with b ∈ Lp (M; lC2 ) and c ∈ Lp (M; lR2 ). This result was first proved in [117] for 1 < p < ∞ in the case of the Schatten classes. The general statement as above (including p = 1) is contained in [121]. Modulo the classical fact that in all preceding inequalities the sequence (εn ) can be replaced by a lacunary n sequence, say, by (z2 )n1 on the unit circle T, the main ingredient of the proof in [121] is a Riesz type factorization theorem (see Theorem 8.3 below). R EMARKS . (i) Like in the classical Khintchine inequalities (6.1), the constant Bp in (6.4) √ is of order p (the best possible) as p → ∞ (cf. [154, p. 106]). (ii) We have already mentioned that in Theorem 6.1, the sequence (εn ) can be replaced by a lacunary sequence. It is also classical that (εn ) can be replaced by a sequence of independent standard Gaussian variables. (iii) More generally, Theorem 6.1 holds when (εn ) is replaced by certain sequences in a non-commutative Lp -space Lp (N ) and εn an is replaced by εn ⊗ an in Lp (N ⊗ M), for instance, this holds for the generators of a free group, for a free semi-circular system (in


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Voiculescu’s sense [196]) and for a sequence of CAR operators (as in Section 1). Note that in the free cases, (6.4) even holds for p = ∞! The reader is referred to [81,154] for more information, and also to [27,29] for some related results and for the best constants in these inequalities. (iv) Theorem 6.1 also holds for non-commutative Lp -spaces associated with a general von Neumann algebra (cf. [100,101]). Note that [100,101] contains more inequalities related to (6.4) and (6.5). C OROLLARY 6.2. Let (εij ) be an independent collection (indexed by N × N) of mean zero ±1-valued random variables on (Ω, F , P ). For any 2 p < ∞, there is a constant Cp such that for any finitely supported function x : N2 → C, we have |||x|||p εij x(i, j )eij Cp |||x|||p , (6.6) Lp (Ω,P ;S p )

i,j

where 2 p/2 1/p 2 p/2 1/p x(i, j ) x(i, j ) , . |||x|||p = max i

j

j

i

(6.7) A fortiori this implies εij x(i, j )eij

Lp (Ω,P ;S p )

Cp

inf

ε(i,j )=±1

ε(i, j )x(i, j )eij p . S

(6.8)

P ROOF. Take Lp (M) = S p . Let aij = x(i, j )eij . Then ( ij aij∗ aij )1/2 = j λj ejj and ( aij aij∗ )1/2 = μi eii where λj = ( i |x(i, j )|2 )1/2 and μi = ( j |x(i, j )|2)1/2 . Thus (6.6) is a special case of (6.4). R EMARK 6.3. The preceding result remains valid with the same proof when 1 p < 2 provided one changes the definition of |||x|||p to the following one (dual to the other): |||x|||p = inf yp (2 ) + t zp ( ) , 2

where the infimum runs over all possible decompositions of the form x = y + z. R EMARK . [101] contains more inequalities of type (6.6). Here we just mention one of them, which is an extension of (6.6). Let (fij ) be an independent collection of mean zero random variables in Lp (Ω, F , P ) (2 p < ∞). Then 1/p p/2 1/p p 2 e ≈ max f , f , f ij ij p ij p ij 2 p L (Ω,P ;S )

i,j

j

where the equivalence constants depend only on p.

i

i

j

p/2 1/p fij 22 ,

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In the case of 0 < p < 1, it is easy to check that the second inequality of (6.5) still holds. However, this is not clear for the first one. P ROBLEM 6.4. Does the first inequality of (6.5) hold for 0 < p < 1 (with some constant depending on p)? Does Remark 6.3 extend to p < 1? Like in the commutative setting, the non-commutative Khintchine inequalities are closely related to non-commutative Grothendieck type factorization theorems. Indeed, it was shown in [121] that (6.5) in the case of p = 1 is equivalent to the non-commutative little Grothendieck theorem. To go further, we need one more definition. D EFINITION . Let 1 p ∞, 0 < q r < ∞. Let Y be a Banach space, and let u : Lp (M) → Y be an operator (M being a semifinite von Neumann algebra). u is said to be (r, q)-concave if there is a constant C such that for all finite sequences (an ) in Lp (M)

uan r

1/r

q C |an |s

1/q

, p

where |a|s = ((a ∗ a +aa ∗)/2)1/2 denotes the symmetric modulus of an operator a. If q = r, u is simply said to be q-concave. In the case of p = ∞ (then M can be any C ∗ -algebra), the above notion reduces to that of (r, q)-C ∗ -summing operators introduced in [144] and [147]. The following is an easy consequence of the Hahn–Banach theorem (cf. [144] for a proof). P ROPOSITION 6.5. Let M be a semifinite von Neumann algebra. Let 1 q < p ∞ and s = p/q. Then for any operator u : Lp (M) → Y the following assertions are equivalent (i) u is q-concave; (ii) there are a constant C and f ∈ (Ls (M))∗ , f 0, such that q

1/q ua C f |a|s ,

∀a ∈ Lp (M).

The following Grothendieck-type factorization theorem (when Y is a Hilbert space) is equivalent to (6.5) with p in place of p. T HEOREM 6.6. Let 2 < p ∞, and let Y be a Banach space of cotype 2. Then any operator u : Lp (M) → Y is 2-concave, equivalently (via Proposition 6.5), there are a constant C and f ∈ (Lp/2 (M))∗ with f 0 such that

1/2 ua Cu f |a|2s ,

∀a ∈ Lp (M).

Moreover, C can be chosen to depend only on the cotype 2 constant of Y . R EMARK . The basic case p = ∞ (= non-commutative Grothendieck theorem), is proved in [147] (see also [144,146]). In this case, M can actually be any C ∗ -algebra. In the case of


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p < ∞, Theorem 6.6 is essentially the main result in [118]. More generally, [118] proves this for operators u : E(M, τ ) → H , where H is a Hilbert space and E is a 2-convex r.i. space with an additional mild condition. This, together with [121], implies that Theorem 6.1 can be extended to some symmetric operator spaces. The main difficulty in [118] is to obtain Theorem 6.6 with a constant C independent of p, or equivalently which remains bounded when p → ∞. If we ignore this important point, it is very easy to deduce Theorem 6.6 from (6.4), as follows. P ROOF OF T HEOREM 6.6 FOR p < ∞ WITH C = Cp . Since Lp (M) is of type 2, by Kwapień’s theorem (cf. [146], Theorem 3.2), u factors through a Hilbert space. Thus we may assume Y itself is a Hilbert space. Let (an ) be a finite sequence in Lp (M). Then u(an )2

1/2

= u(an )εn

L2 (Ω,P ;Y )

u an εn Cp |an |2s

L2 (Ω,P ;Lp (M)) 1/2

p

by (6.4) .

Therefore, u is 2-concave.

Unfortunately, the preceding proof does not work for p = ∞. The main difficulty in this case is to show that an operator u from M into a space of cotype 2 factors through a Hilbert space. This was done in [147]. The proof given there relies on another result of independent interest, that we state as follows. T HEOREM 6.7. Let 1 < q < ∞. Let u : A → Y be an operator from a C ∗ -algebra A into a Banach space Y . Then the following assertions are equivalent (i) u is (q, 1)-C ∗ -summing; (ii) there are a constant C and a state f on A such that

1/q a1−1/q , ua Cu f |a|s

∀a ∈ A;

(iii) for any 1 r < q there are a constant C and a state f on A such that

1/q ua Cu f |a|rs a1−r/q ,

∀a ∈ A;

(iv) u is (q, r)-C ∗ -summing for any 1 r < q. Thus Theorem 6.7 gives a characterization of (q, r)-C ∗ -summing operators defined on a C ∗ -algebra. (ii) and (iii) above can be reformulated as a Pietsch-type factorization of u through a non-commutative Lorentz space Lq,1 , constructed from the state f via the real interpolation in the spirit of Kosaki’s construction presented in Section 3. The resulting spaces, denoted by Lq,1 (f ), possess properties similar to the usual Lorentz spaces. The reader is referred to [147] for more information.

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R EMARK . There does not seem to be a known characterization similar to that in Theorem 6.7 for (q, r)-concave operators defined on Lp (M) (p < ∞). Let us close this section by an application of Theorem 6.7. T HEOREM 6.8. Let M be any von Neumann algebra and X ⊂ M∗ a reflexive subspace. Then there are a normal state f of M and p > 1 such that X embeds isomorphically into Lp,p (f ), where Lp,p (f ) is the non-commutative Lp -space referred to above. This theorem, proved in [147], is a non-commutative version of a classical theorem due to Rosenthal in the commutative setting. Its proof uses Theorem 6.7 and a previous result in [92] that any reflexive subspace of M∗ is superreflexive. Note that the real interpolation space Lp,p (f ) can be replaced by the corresponding complex interpolation space. R EMARK . Let A be a C ∗ -algebra, and let T : A → 2 be absolutely summing (in the usual sense). If A is commutative, it is well known that T factors as T = T1 T2 , where T2 : A → 2 is bounded and T1 ∈ S 2 . In [161] it is shown that for a general C ∗ -algebra A, one can get a factorization T = T1 T2 , where T2 : A → 2 is bounded and T1 : 2 → 2 belongs to the Schatten class S4 (the exponent 4 is optimal). A fortiori T is compact. In particular, there is no embedding of 2 into a non-commutative L1 -space with absolutely summing adjoint. See [146, p. 68] for background on embeddings of this kind. 7. Non-commutative martingale inequalities This section deals with non-commutative martingale inequalities. The reader is referred to [71] for the classical (= commutative) martingale inequalities. In what follows, M will be a von Neumann algebra equipped with a normal faithful finite normalized trace τ . We begin with some necessary definitions. Let N ⊂ M be a von Neumann subalgebra. The non-commutative Lp -space associated with (N , τ |N ) is naturally identified with a subspace of Lp (M). There is a unique normal faithful conditional expectation E : M → N preserving the trace τ , i.e., τ (E(x)) = τ (x) for all x ∈ M. For any 1 p ∞, E is extended to a contractive projection from Lp (M) onto Lp (N ), still denoted by E. Now let (Mn )n0 be an increasing sequence of von Neumann subalgebras of M such that the union of all the Mn ’s is w∗ -dense in M. Let En be the conditional expectation from M onto Mn . Then as usual, we define a non-commutative martingale (with respect to (Mn )n0 ) as a sequence x = (xn )n0 in L1 (M) such that En (xn+1 ) = xn ,

∀n 0.

If additionally all xn ’s are in Lp (M), x is called an Lp -martingale. Then we set xp = sup xn p . n0

If xp < ∞, x is called a bounded Lp -martingale. The difference sequence of x is defined as dx = (dxn )n0 with dx0 = x0 and dxn = xn − xn−1 for all n 1.


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R EMARK . Let x∞ ∈ Lp (M). Set xn = En (x∞ ) for all n 0. Then x = (xn ) is a bounded Lp -martingale and xp = x∞ p ; moreover, xn converges to x∞ in Lp (M) (relative to the w∗ -topology in the case p = ∞). Conversely, if 1 < p < ∞, every bounded Lp -martingale converges in Lp (M), and so is given by some x∞ ∈ Lp (M) as previously. Thus one can identify the space of all bounded Lp -martingales with Lp (M) itself in the case 1 < p < ∞. The main result of [156] can be stated as follows. Recall that Ap , Bp , . . . , denote constants depending only on p. T HEOREM 7.1. Let M and (Mn )n0 be as above. Let 1 < p < ∞, and let x = (xn )n0 be a finite Lp -martingale with respect to (Mn )n0 . Then A−1 p Sp (x) xp Bp Sp (x),

(7.1)

where for 2 p < ∞, Sp (x) = dxLp (M;l 2 )∩Lp (M;l 2 ) , C

R

and for 1 < p < 2, Sp (x) = inf dyLp (M;l 2 ) + dzLp (M;l 2 ) , C

R

the infimum being taken over all decompositions x = y + z with Lp -martingales y and z. This is the non-commutative Burkholder–Gundy inequalities. Note that in the commutative case, Sp (x) is the Lp -norm of the usual square function of x (so that the above difference between the cases 2 p < ∞ and 1 < p < 2 disappears). The proof of Theorem 7.1 in [156] is rather tortuous, due to the fact that the usual techniques from classical martingale theory, such as maximal functions, stopping times, etc., are no longer available in the non-commutative setting. See [156] and [19] for applications to non-commutative stochastic integrals. For Clifford martingales, some particular cases of Theorem 7.1 also appear in [34]. R EMARK 7.2. The second inequality in (7.1) holds for p = 1 too. This follows from the duality between H1 and BMO, proved in [156]. Like in the commutative case, Theorem 7.1 implies the unconditionality of noncommutative martingale differences. Let us record this explicitly as follows. C OROLLARY 7.3. With the same assumptions as in Theorem 7.1, we have εn dxn Cp dxn , ∀εn = ±1. n0

p

n0

p

(7.2)

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Some rather particular cases of (7.2) also appear in [65,66]. Note that in the case of 2 p < ∞, (7.2) is equivalent to (7.1), modulo the non-commutative Khintchine inequalities. However, in the case of 1 < p < 2, to prove that (7.2) implies (7.1), one needs a noncommutative version of a classical inequality due to Stein. We refer to [156] for more details. R EMARK . If p is an even integer, the second inequality of (7.1) was extended in [155] to sequences more general than martingale difference sequences (the so-called p-orthogonal sequences); moreover, for these values of p, the method of [155] yields that the order of the constant Bp in (7.1) is O(p) (for even integers p), which is optimal as p → ∞. For the convenience of the reader, we recall the optimal order of the constants Ap and Bp in the commutative case (cf., e.g., [31]): Bp is bounded as p → 1 and O(p) as p → ∞; Ap is O((p − 1)−1 ) as p → 1 and O(p1/2 ) as p → ∞. The constants Ap and Bp in (7.1) obtained in [156] are not satisfactory at all (they are of exponential type as p → ∞). Thus finding the optimal order of Ap and Bp in Theorem 7.1 seemed a very interesting question. Very recently, major progress on this was achieved by Randrianantoanina [165], as follows. T HEOREM 7.4 ([165]). There is a constant C such that for any finite non-commutative martingale x in L1 (M) and any sequence (εn ) of signs ε dx C dx n n n . n0

1,∞

n0

1

By interpolation, this implies the optimal order of the constant Cp in (7.2), namely, Cp = O(p) as p → ∞. This, in turn, combined with Theorem 6.1, yields better estimates for Ap , Bp in (7.1), namely Ap is O((p − 1)−2 ) when p → 1 and both Ap and Bp are O(p) when p → ∞ (which for Bp is optimal). It was also shown in [102] that O(p) is the optimal order of Ap as p → ∞. Note that this order is the square of what it is in the commutative case. On the other hand, it was proved in [100] that Bp remains bounded as p → 1. We will now discuss two other inequalities: the Burkholder and Doob inequalities. T HEOREM 7.5. With the same assumptions as in Theorem 7.1, we have A−1 p sp (x) xp Bp sp (x),

(7.3)

where for 2 p < ∞, 1/p

1/2 p 2 , sp (x) = max |dx dxn p , E | n−1 n n0

p

n0

∗ 2 1/2 dx E n−1 n n0

p


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and for 1 < p < 2, 1/p

1/2 p 2 sp (x) = inf dwn p + En−1 |dun | n0

p

n0

∗ 2 1/2 , + En−1 dvn p

n0

where the infimum runs over all decompositions x = w + u + v with Lp -martingales w, u and v. This theorem comes from [100]. It is the non-commutative analogue of the classical Burkholder inequality. Note that in the commutative case ( En−1 (|dxn |2 ))1/2 is the conditioned square function of x. Like in the commutative case, Theorem 7.5 implies a noncommutative analogue of Rosenthal’s inequality concerning independent mean zero random variables; see [100,101] for more details and some applications. T HEOREM 7.6 ([97]). Let M and (Mn ) be as in Theorem 7.1. Let 1 p < ∞. Let (an ) be a finite sequence of positive elements in Lp (M). Then E (a ) C an (7.4) n n p . n0

p

n0

p

Note that in the commutative case, (7.4) is the dual reformulation of Doob’s classical maximal inequality. Although it is clearly impossible to define the maximal function of a non-commutative martingale as in the commutative setting, Junge found in [97] a substitute, consistent with [154], which enables him to formulate a non-commutative analogue of Doob’s inequality itself, which is dual to (7.4). Note that the latter result immediately implies the almost everywhere convergence of bounded non-commutative martingales in Lp (M) for all p > 1. Results of this kind on the almost everywhere convergence of noncommutative martingales go back to Cuculescu [43]. The reader is referred to [44] and [90,91] for more information. R EMARKS . (i) Like the constants in (7.1) the constants in (7.3) and (7.4) obtained in [100, 97] are not satisfactory at all. In fact, they depend on those in (7.1) since the proofs of (7.3) and (7.4) in [97] and [100] use (7.1). The more recent results of [165] imply better estimates for these constants. (ii) It was proved in [102] that the optimal order of the constant Cp in (7.4) is O(p2 ) as p → ∞. This is in strong contrast with the commutative case for, in the commutative case, the optimal order of the corresponding constant is O(p) as p → ∞. The same phenomenon occurs for the optimal order of the best constant in the non-commutative Stein inequality proved in [156], namely, this optimal order is O(p) as p → ∞; again it is the square of what it is in the commutative case. We refer to [102] for more information. (iii) All the preceding results hold in the non-tracial case as well (cf. [100,101,97]).

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In the rest of this section, we briefly discuss the UMD property and the analytic UMD property of non-commutative Lp -spaces, a subject closely related to inequality (7.2). Applying Corollary 7.3 to commutative martingales with values in Lp (M), 1 < p < ∞, we get the unconditionality of commutative martingale differences with values in Lp (M), that is, Lp (M) is a UMD space in Burkholder’s sense (cf. [32] for information on UMD spaces). This is a well-known fact, proved in [21] and [16]. In fact, these authors proved that the Hilbert transform extends to a bounded map on Lp (T; Lp (M)) for any 1 < p < ∞; but this property (called “HT” in short) is equivalent to UMD. We also refer to the next section for discussions on Hilbert type transforms. Together with Theorem 3.6 we obtain the C OROLLARY 7.7. Lp (M) is a UMD space for any 1 < p < ∞ and any von Neumann algebra M. We mention an open problem circulated in the non-commutative world for almost two decades on the UMD property for symmetric operator spaces. P ROBLEM 7.8. Let M be a semifinite von Neumann algebra equipped with an n.s.f. trace τ , and let E be a UMD r.i. space on (0, ∞). Is E(M, τ ) a UMD space? We now turn to the analytic UMD property. Let TN be the infinite torus equipped with the product measure, denoted by dm∞ . Let Ωn be the σ -field generated by the coordinates (z0 , . . . , zn ), n 0. Let X be a quasi-Banach space. By a Hardy martingale in Lp (TN ; X) (0 < p ∞), we mean any sequence f = (fn ) satisfying the following: for any n 0, fn ∈ Lp (TN , Ωn ; X) and fn is analytic in the last variable zn , i.e., fn admits an expansion as follows ϕn,k (z0 , . . . , zn−1 )znk , fn (z0 , . . . , zn−1 , zn ) = k1

where ϕn,k ∈ Lp (TN , Ωn−1 ; X) for n 0, k 0. If in addition, ϕn,k = 0 for all k 2, f is called an analytic martingale. Note that if X is a Banach space and 1 p ∞, any Hardy martingale in Lp (TN ; X) is a martingale in the usual sense. D EFINITION . X is called an analytic UMD space if for some 0 < p < ∞ (or equivalently for all 0 < p < ∞) there is a constant C such that all finite Hardy martingales f in Lp (TN ; X) satisfy ε df C df n n n , ∀εn = ±1. n0

p

n0

p

This notion was introduced in [69]. The apparent weakening obtained by requiring the above inequality be verified only for analytic martingales, is actually an equivalent definition of analytic UMD spaces (cf. [69]). Typical examples of Banach spaces which are analytic UMD but not UMD are commutative L1 -spaces. In fact, all commutative


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Lp -spaces, 0 < p 1, are analytic UMD spaces. We refer to [69] for more information (see also [32]). However, this no longer holds in the non-commutative setting: P ROPOSITION 7.9. Let M be a von Neumann algebra and 0 < p 1. Then Lp (M) is an analytic UMD space iff M is isomorphic, as Banach space, to a commutative L∞ -space. It was proved in [80] that the trace class S 1 is not an analytic UMD space. The ingredient of the proof there is the unboundedness of the triangular projection on S 1 (cf. [111]). (This projection is in fact a non-commutative Riesz projection in the context of the next section.) The same idea also shows that S p is not an analytic UMD space for 0 < p < 1. Noting that the analytic UMD property is “local”, we then deduce the general case from Theorem 3.5.

8. Non-commutative Hardy spaces A classical theorem of Szegö says that if w is a positive function on the unit circle T such that log w ∈ L1 (T), there is an outer function ϕ such that |ϕ| = w a.e. on T. A lot of effort has been made to extend this theorem to operator valued functions, not only for its intrinsic interest, but also because it is the gateway to many useful applications (cf., e.g., [85,86, 46,179,198]). This problem makes sense in the broader context of subdiagonal algebras, introduced by Arveson in the 60’s in order to unify several frequently used non-selfadjoint algebras such as triangular matrices and bounded analytic operator valued functions. In this section we will present the extension to this general context of some classical results for analytic functions in the unit disc, including Szegö’s theorem, boundedness of the Hilbert transform and the Riesz factorization theorem. Throughout this section, unless explicitly indicated otherwise, M will denote a finite von Neumann algebra equipped with a normal faithful finite normalized trace τ . Let D be a von Neumann subalgebra of M. Let E be the (unique) normal faithful conditional expectation of M with respect to D which leaves τ invariant. D EFINITION . A w∗ -closed subalgebra H ∞ (M) of M is called a finite subdiagonal algebra of M with respect to E (or to D) if (i) {x + y ∗ : x, y ∈ H ∞ (M)} is w∗ -dense in M; (ii) E(xy) = E(x)E(y), ∀x, y ∈ H ∞ (M); (iii) {x: x, x ∗ ∈ H ∞ (M)} = D. D is then called the diagonal of H ∞ (M). This notion can be generalized further (see [12]). However, the theory we will give below is, on one hand, satisfactory only for finite subdiagonal algebras as above, and on the other, interesting enough to cover many important cases. R EMARKS . (i) If H ∞ (M) is a finite subdiagonal algebra of M, it is automatically maximal in the sense that it is contained in no proper subdiagonal algebra with respect to E other than itself (see [61]).

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(ii) Consequently, H ∞ (M) admits the following useful characterization (cf. [12]) H ∞ (M) = x ∈ M: τ (xy) = 0, ∀y ∈ H0∞ (M) , where H0∞ (M) = x ∈ H ∞ (M): E(x) = 0 . Here are some examples (see [12] for more). (i) Triangular matrices. Let Mn be the full algebra of all complex n × n matrices equipped with the normalized trace. Let Tn be the algebra of all upper triangular matrices in Mn . Then Tn is a finite subdiagonal algebra of Mn . In this case, the theory we will give below is partly contained in [73]. (ii) Nest algebras. Let P be a totally ordered family of projections in M containing 0 and 1. Let N (P) = {x ∈ M: xe = exe, ∀e ∈ P}. Then N (P) is a finite subdiagonal algebra of M. The above example on triangular matrices is a special case of nest algebras. (iii) Analytic operator valued functions. Let (M, τ ) be a finite von Neumann algebra. Let (L∞ (T), dm) ⊗ (M, τ ) be the von Neumann algebra tensor product (recalling that T is the unit circle equipped with normalized Lebesgue measure dm). Let H ∞ (T, M) be the subalgebra of (L∞ (T), dm) ⊗ (M, τ ) consisting of all functions f such that

τ xf (z) z¯ n dm(z) = 0,

∀x ∈ L1 (M), ∀n ∈ Z, n < 0.

Then H ∞ (T, M) is a finite subdiagonal algebra of (L∞ (T), dm) ⊗ (M, τ ). This is the algebra of “analytic” functions with values in M. More precisely, each element f in H ∞ (T, M) can be extended, using Poisson integrals, to an M-valued function, analytic and bounded in the unit disc admitting f as its (radial or non-tangential) weak-∗ boundary values. The particularly interesting case H ∞ (T, Mn ) or H ∞ (T, B(l2 )) was extensively studied (cf., e.g., [85,46]). Note that B(l2 ) does not fit into our setting; however, for almost all problems we are concerned with, it can be recovered from Mn by approximation. In the remainder of this section, unless specified otherwise, H ∞ (M) will denote a finite subdiagonal algebra of M with diagonal D. For 0 < p < ∞ the corresponding Hardy space H p (M) is defined as the closure of H ∞ (M) in Lp (M). Many results on the classical Hardy spaces in the unit disc have been extended to the present setting. We refer, for instance, to [13,93,127,123,124,126,159,173,175] and [177] for more information and references. We now give some of these extensions. The first one is the Szegö type theorem. T HEOREM 8.1. Suppose w ∈ M and w−1 ∈ L2 (M). Then there are a unitary u ∈ M and ϕ ∈ H ∞ (M) with ϕ −1 ∈ H 2 (M) such that w = uϕ.


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This theorem, proved by Saito [175], improves a previous factorization theorem due to Arveson [12], in which both w and w−1 are supposed to belong to M. Saito’s proof essentially follows the same fashion set out by Arveson, although some extra technical difficulties appear. R EMARKS . (i) The above theorem can be still improved as follows: let 0 < p, q ∞ and w ∈ Lp (M) with w−1 ∈ Lq (M). Then there are a unitary u ∈ M and ϕ ∈ H p (M) with ϕ −1 ∈ H q (M) such that w = uϕ. (ii) In the classical case of analytic functions in the unit disc, for a positive function w on T, the condition log w ∈ L1 (T) is necessary and sufficient for the existence of a factorization w = uϕ, with u ∈ L∞ (T) unimodular and ϕ an outer function. It is an open problem to extend this to the non-commutative setting. Some partial results can be found in [46,86] and [198]. The following is an immediate consequence of Theorem 8.1 (and also of the remark (i) above). C OROLLARY 8.2. Let w ∈ L1 (M) such that w 0 and w−1 ∈ Lp (M) for some 0 < p ∞. Then there is ϕ ∈ H 2 (M) such that w = ϕ ∗ ϕ. By a rather standard argument, one can deduce from Theorem 8.1 the following Riesz factorization theorem, which was proved in [124] (see also [177] for the case where p = q = 2). T HEOREM 8.3. Let 1 p, q, r ∞ with 1/r = 1/p + 1/q. Then any x ∈ H r (M) can be factored as x = yz with y ∈ H p (M) and z ∈ H q (M); moreover, xr = inf yp zq : x = yz, y ∈ H p (M), z ∈ H q (M) . R EMARKS . (i) It seems unclear whether the infimum above is attained. (ii) With the notations in Theorem 8.3, one has the following more precise statement: for any ε > 0 there are y ∈ H p (M) and z ∈ H q (M) such that x = yz and

r/p , μt (y) μt (x) + ε

r/q μt (z) μt (x) + ε ,

∀t > 0.

In particular, if x ∈ H ∞ (M), then y, z ∈ H ∞ (M) and yp zq = xr + o(1) as ε → 0. This allows to partially extend Theorem 8.3 to the case of indices less than 1 (at least, for elements x ∈ H ∞ (M) ⊂ H r (M)). However, it is unknown whether Theorem 8.3, in its full generality, still holds for indices less than 1. The reader can find applications of Theorem 8.3 to Hankel operators in [179,123], to invariant subspaces of Lp (M) in [175], and to the uniform H-convexity in [201] and [203].

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We will now describe the Hilbert transform and Riesz projection. Let x ∈ {a + b∗ : a, b ∈ It is easy to see that x admits a unique decomposition

H ∞ (M)}.

x = a + d + b∗,

with a, b ∈ H0∞ (M), d ∈ D.

Then we define the Hilbert transform H by

H x = −i a − b∗ . Clearly, x + iH x ∈ H ∞ (M); moreover if x is self-adjoint, H x is the unique self-adjoint element in {a + b∗ : a, b ∈ H ∞ (M)} such that x + iH x ∈ H ∞ (M) and E(H x) = 0. Note that

⊥ L2 (M) = H02 (M) ⊕ L2 (D) ⊕ H 2 (M) , where H02 (M) = {x ∈ H 2 (M): E(x) = 0}. One easily checks that H02 (M) (resp. (H 2 (M))⊥ ) is the closure of H0∞ (M) (resp. {x ∗ : x ∈ H0∞ (M)}) in L2 (M). This decomposition of L2 (M) shows that H extends to a contraction on L2 (M), still denoted by H . Now let P be the orthogonal projection of L2 (M) onto H 2 (M) (i.e., P is the “Riesz projection”). Like in the classical case, H and P are linked together as follows 1 1 P = (idL2 (M) + H ) + E. 2 2 Thus, as far as boundedness is concerned, it suffices to consider one of them. T HEOREM 8.4. (i) H extends to a bounded map on Lp (M) for any 1 < p < ∞; more precisely, one has H xp Cp xp ,

∀x = a + b∗ , a, b ∈ H ∞ (M),

where Cp Cp2 /(p − 1) with C a universal constant. (ii) H also extends to a bounded map from L1 (M) into L1,∞ (M) (the non-commutative weak L1 -space). This result was proved in [160]. Of course, (i) above (for the case 1 < p < 2) follows by interpolation from (ii) and the L2 -boundedness of H (and by duality for the case 2 < p < ∞). However, (i) admits a much simpler separate proof (see the discussion below). In the case of triangular matrices, (i) above is often referred to as Matsaev’s theorem (cf. [73]). In this case, the corresponding Riesz projection is the usual triangular projection (see [111] for more results on this projection; see also [208] for related results). Let us discuss another particularly interesting case, that of analytic operator valued functions. Then Theorem 8.4(i) is equivalent to the UMD property of Lp (M) that we already


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saw in the last section. Indeed, considering the finite subdiagonal algebra H ∞ (T, M) of L∞ (T) ⊗ M, one sees that H = H ⊗ idL2 (M) , where H is the usual Hilbert transform on T. Thus the boundedness of H on Lp (L∞ (T) ⊗ M) is equivalent to the fact that H ⊗ idL2 (M) extends to a bounded map on Lp (T; Lp (M)) (noting that Lp (L∞ (T) ⊗ M) = Lp (T; Lp (M))). In other words, Lp (M) has the “HT property”, which is equivalent to the UMD property, as already mentioned in the last section. The main idea of the proof is an old trick due to Cotlar, which still works in the general setting as in Theorem 8.4. The ingredient is the following formula, whose proof is straightforward. L EMMA 8.5. For any x = a + b∗ with a, b ∈ H0∞ (M)

(H x)∗ H x = x ∗ x + H x ∗ H x + (H x)∗ x . Using Lemma 8.5, we easily check that the boundedness of H on Lp (M) implies that on = 2 and iterating, n we deduce that H is bounded on L2 (M) for all integers n 1. Finally, interpolation and duality yield Theorem 8.4(i). We also point out that this argument gives the optimal order of the constant Cp as stated in Theorem 8.4. L2p (M) (see also the proof of Theorem 5.3 above). Then starting from p

R EMARKS . (i) It was shown in [150] that in the case of triangular matrices or analytic operator valued functions, the non-commutative Hardy spaces form an interpolation scale with respect to the real and complex methods. The same arguments work in the general case as well. Thus for any 0 < p0 , p1 ∞ and 0 < θ < 1 p

H 0 (M), H p1 (M) θ,p = H p0 (M), H p1 (M) θ = H p (M), where 1/p = (1 − θ )/p0 + θ/p1 . (ii) The Hilbert transform H enables us to identify the dual of H 1 (M) with the noncommutative analog of the space BMO (for bounded mean oscillation) as in Fefferman’s classical result, namely the space BMO(M) defined as follows: BMO(M) = x + Hy: x, y ∈ L∞ (M) equipped with the norm z = inf x∞ + y∞ : z = x + Hy, x, y ∈ L∞ (M) . We refer to [124,125] for more information. We end this section by an open problem. A famous theorem due to Bourgain states that the quotient space L1 (T)/H 1 (T) is a GT space of cotype 2 (cf., e.g., [146]). It is not clear at all how to extend this theorem to the non-commutative case. P ROBLEM 8.6. Let H ∞ (M) be a finite subdiagonal algebra in M. Is L1 (M)/H 1(M) of cotype 2? or merely of finite cotype?

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In the case of triangular matrices or vector-valued analytic functions, this problem has been circulated in Banach space theory almost since Bourgain’s theorem. By the way, note that any quotient of L1 (M) by a reflexive subspace is of cotype 2. This is the non-commutative version of a theorem due to Kisliakov and Pisier (cf. [146]). It follows from [92] and [145].

9. Hankel operators and Schur multipliers In general it is not so easy to compute (up to equivalence) the S p -norm of an operator x in S p , except when x = (xij ) is a column (or row) matrix and when it is a diagonal one, as follows: xi ei1

xi e1i =

xi eii

=

Sp

Sp

=

|xi |2

1/2

and

Sp

|xi |p

1/p

.

In view of their importance and ubiquity in Analysis, it was natural to wonder about the case when x = (xij ) is a Hankel matrix, i.e., there is a (complex) sequence γ in 2 such that xij = γi+j ,

∀i, j 0.

(9.1)

This case was solved in Peller’s remarkable paper [136] as follows. T HEOREM 9.1. Let x = (xij ) be given by (9.1) and let 1 < p < ∞. Let ϕ(z) = be “its symbol” and let Δ0 ϕ(z) = x00,

Δn ϕ(z) =

z j γj ,

j 0 z

jγ

j

∀n 1 (z ∈ T).

2n−1 j 2 if Λ is assumed Λ(p)cb ). Let (εn ) denote the Rademacher functions on (Ω, F , P ), as in Section 6. Let ER (resp. EΛ ) be the closed subspace of Lp (Ω, F , P ) (resp. Lp (vN(Γ ))) generated by {εn | n 0} (resp. {λ(tn ) | n 0}). Then the linear mappings u and u−1 defined on the linear spans by u(εn ) = λ(tn ) and u−1 (λ(tn )) = εn extend to c.b. maps u : ER → EΛ and u−1 : EΛ → ER with ucb C and u−1 cb Bp . Moreover, the (orthogonal) projection P : Lp (vN(Γ )) → EΛ , defined by P (λ(t)) = λ(t) if t ∈ Λ, and = 0 if t ∈ Λ, is c.b. on Lp (vN(Γ )).


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The preceding results provide non-trivial new examples of c.b. Fourier multipliers on Lp (T). We now turn to Schur multipliers. A linear map T : S p → S p (resp. T : B(2 ) → B(2 )) is called a Schur multiplier if it is of the form

T (x) = ϕ(i, j )xij for some function ϕ : N × N → C. In this case, we write T = Mϕ . The case p = 2 is of course trivial: we have then Mϕ = supi,j |ϕ(i, j )|. In the case p = ∞, it is well known (due essentially to Grothendieck) that bounded Schur multipliers T = Mϕ : B(2 ) → B(2 ) are all of the following special form: there are bounded sequences (xi ) and (yj ) in 2 such that ϕ(i, j ) = xi , yj .

(9.6)

Moreover, we have Mϕ = inf sup xi sup yj , i

j

where the infimum runs over all possible (xi ) and (yj ) satisfying (9.6). This implies in particular (due to Haagerup) that bounded Schur multipliers on B(2 ) (or on S 1 ) are “automatically” c.b. (see [152]). However, the following remains open (we conjecture that the answer is negative): P ROBLEM 9.8. Is every bounded Schur multiplier on S p (1 < p = 2 < ∞) c.b.? Note that it is rather easy to give examples of bounded Fourier multipliers on Lp (G) which are not c.b. when G is any compact infinite commutative group and 1 < p = 2 < ∞ (see [83] or [152], p. 91). P ROBLEM 9.9. Is there a description of c.b. Schur multipliers on S p extending (9.6) to 1 < p = 2 < ∞? It is known ([83], see also [206]) that the space of bounded (or c.b.) Schur multipliers of S p (2 < p < ∞) does not coincide with any interpolation space between the cases p = 2 and p = ∞. D EFINITION 9.10. A subset A ⊂ N × N is called a σ (p)-set (p 2) if {eij | (i, j ) ∈ A} is an unconditional basic sequence in S p . A simple application of Corollary 6.2 shows that this holds iff there is a constant C such that for any finitely supported function x : A → C we have |||x|||p x(i, j )e ij (i,j )∈A

Sp

C|||x|||p ,

(9.7)

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where |||x|p is as defined in (6.7) above. It is easy to see by interpolation that, if 2 p ∞, we have |||x|p xS p for all x in S p . Hence (9.7) implies that the idempotent Schur multiplier corresponding to the indicator function of A is bounded on S p with norm C. For example, any set A for which either one of the two coordinate projections is one to one when restricted to A, is obviously a σ (p)-set. The following result provides much less trivial examples. P ROPOSITION 9.11 ([83]). Let p 2. Let Λ ⊂ Z be a Z(p/2)-set, or more generally a Λ(p)cb -set. Then the set AΛ = {(i, j ) ∈ N2 | i + j ∈ Λ} is a σ (p)-set. P ROOF. This follows from (6.4), applied to the series ϕ(z) = n∈Λ zn ( i+j =n x(i, j )eij ). Indeed for any z we have ϕ(z) p = x(i, j )e = a ij n , S Sp

(i,j )∈AΛ

Sp

n∈Λ

where an = i+j =n x(i, j )eij . By the Λ(p)cb -property of Λ (see Remark 9.4) there is a constant C such that: 1/2 1 n z an p p max an∗ an an an∗ p, L (S ) S C z n an p p .

1/2

Sp

L (S )

But as we just observed we have

an∗ an =

zn an Lp (S p ) =

x(i, j )2 ejj , j

(9.8)

an S p and

an an∗ =

i

x(i, j )2 eii . i

j

Hence (9.8) implies (9.7) with A = AΛ .

R EMARK . In the situation of Proposition 9.11, the same argument shows that if A = AΛ then for any finitely supported function x : A → S p we have Q(x) eij ⊗ x(i, j ) CQ(x), (9.9) S p (2 ⊗2 )

(i,j )∈A

where 1/2 p 1/p ∗ Q(x) = max x(i, j ) x(i, j ) , j

i

Sp

1/2 p 1/p ∗ x(i, j )x(i, j ) . p i

j

S


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A set A ⊂ N2 satisfying (9.9) for some constant C is called a σ (p)cb -set. Equivalently, this means, by (6.4), that {eij | (i, j ) ∈ A} is “completely unconditional” (see Remark 9.6), i.e., for any choice of signs εij = ±1, the transformations

x(i, j )eij →

(i,j )∈A

εij x(i, j )eij

(i,j )∈A

are c.b. on the closure in S p of {eij | (i, j ) ∈ A}. Since the left side of (9.9) remains valid for A = N2 , (9.9) implies that the indicator function of A is a c.b. Schur multiplier on S p . In particular, if x = (x(i, j )) is Hankelian, i.e., x(i, j ) = γ (i + j ) for some finitely supported function γ : Λ → S p , then (9.9) implies q(γ ) γ (n) ⊗ eij i+j =n

n∈Λ

S p (l2 ⊗l2 )

Cq(γ ),

(9.10)

where 1/2 p 1/p ∗ γ (n) γ (n) , q(γ ) = max j

nj

Sp

1/2 p 1/p ∗ . γ (n)γ (n) p j

nj

S

Thus we can “compute” (up to C) the norm of a Hankel operator with “spectrum” in Λ. C OROLLARY 9.12 ([83]). Let p > 2 be an even integer. (i) There are δ > 0 and C such that, for any n, there is a subset An ⊂ [1, . . . , n]2 with |An | δn1+2/p such that {eij | (i, j ) ∈ An } is C-unconditional in S p , i.e., a σ (p)set. (ii) There is an idempotent Schur multiplier T (idempotent means here T 2 = T ) which is bounded on S p but unbounded on S q for any q > p. P ROOF. The proof combines Theorem 9.3 with Rudin’s (combinatorial and number theoretic) construction of a B(p/2) set Λ ⊂ Z such that lim sup sup N −2/p Λ ∩ [a, a + bN] > 0. N→∞ a,b∈N

R EMARKS . (i) It is shown in [84] that, for any p > 2, n1+2/p is the maximal possible order of growth in the first part of Corollary 9.12. (ii) The preceding corollary almost surely remains valid when p > 2 is not an even integer, but no proof is known at the time of this writing. (iii) It is proved in [136] (see also [105] for related estimates on the case p = ∞) that the orthogonal projection from S 2 onto the subspace of all Hankel matrices (i.e., the averaging

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projection) is bounded on S p iff 1 < p < ∞, and for p = 1, it is bounded from S 1 to S 1,2 (a fortiori it is of “weak” type (1, 1)). See [2] for more recent results on (Hankel and Toeplitz) Schur multipliers, in particular for the case S p with p < 1.

10. Isomorphism and embedding In this section we discuss isomorphism and embedding of non-commutative Lp -spaces. Unless explicitly stated otherwise, we will assume all Lp -spaces considered in this section are separable and infinite-dimensional, or equivalently, the underlying von Neumann algebras are infinite-dimensional and act on separable Hilbert spaces. Throughout this section, Lp denotes the classical commutative Lp -space on [0, 1]. The isomorphic classification of commutative Lp -spaces is extremely simple, for there are only two non-isomorphic commutative Lp -spaces: l p and Lp . However, in the noncommutative setting, the situation is far from simple. In fact, it is impossible to list all non-commutative Lp -spaces up to isomorphism. It even seems very hard to classify them according to the different types of the underlying von Neumann algebras. Despite these difficulties, considerable progress has been achieved in the last few years. p Let K p denote the direct sum in the l p -sense of the Sn ’s, i.e., Kp =

/ n1

p

Sn

. p

p p Note that . K is the non-commutative L -space associated with the von Neumann algebra M = n1 Mn , the direct sum of the matrix algebras Mn , n 1. We also recall that if X is a Banach space, Lp (X) stands for the usual Lp -space of Bochner measurable p-integrable functions on [0, 1] with values in X. If X = Lp (M), Lp (X) is just the non-commutative Lp -space associated with L∞ (0, 1) ⊗ M. We should call the reader’s attention to the two different notations for the Schatten classes, equally often used in the literature: S p in our notation is sometimes denoted by C p , and K p by S p ! Recall that R denotes the hyperfinite II1 factor.

T HEOREM 10.1. Let M be a hyperfinite semifinite von Neumann algebra. Let 1 p < ∞, p = 2. Then Lp (M) is isomorphic to precisely one of the following thirteen spaces:

l p , Lp , K p , S p , Lp ⊕ K p , Lp ⊕ S p , Lp K p , S p ⊕ Lp K p ,

Lp S p , Lp (R), S p ⊕ Lp (R), Lp S p ⊕ Lp (R), Lp R ⊗ B l 2 . R EMARKS . (i) The first nine spaces in the above list give precisely all non-commutative Lp -spaces, up to isomorphism, associated with von Neumann algebras of type I. (ii) Theorem 10.1 is proved in [82]. Prior to that, the case of type I was studied in [187]. [82] also contains results on non-commutative Lp -spaces associated with hyperfinite factors of type III and free group von Neumann algebras. More precisely, it is shown there


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that the non-commutative Lp -spaces associated with hyperfinite factors of type IIIλ for all λ ∈ (0, 1] are isomorphic, and the non-commutative Lp -space associated with a free group von Neumann algebra is independent, up to isomorphism, of the number of generators as soon as this number is not less than 2. We refer the interested reader to [82] for more information. The proof of Theorem 10.1 can be reduced to the non-embedding of one noncommutative Lp -space into another. The main general result on this is the following T HEOREM 10.2. Let 0 < p < ∞, p = 2. Let M be a finite von Neumann algebra. Then S p does not embed (isomorphically) into Lp (M). We get immediately the following corollary. C OROLLARY 10.3. Let 0 2, in [82] for 1 p < 2, and in [188] for p < 1. Note that in the special case where M = L∞ (0, 1), Theorem 10.2 was established by McCarthy in the pioneering paper [128]. His result was considerably improved in [74]. In particular, Theorem 4.1 implies that K p does not embed into Lp . In the converse direction, it was proved in [11] that Lp does not embed into S p . In the case of 0 < p < 2, we have the following result, much stronger than Theorem 10.2. T HEOREM 10.4. Let 0 < p < 2. Let M be a finite von Neumann algebra. Let (ui,j )i,j 1 be an infinite matrix of elements in Lp (M) such that supi,j ui,j p < ∞. Suppose that all rows, columns and generalized diagonals of (ui,j )i,j 1 are unconditional. Then one of the following three alternatives holds (i) Some row or column has a subsequence equivalent to the canonical basis of l p ; (ii) There is a constant λ > 0 such that for every integer n some row and some column p contain n elements λ-equivalent to the canonical basis of ln ; (iii) There is a generalized diagonal (uik ,jk )k1 such that n 1 lim 1/p uik ,jk = 0. n→∞ n k=1

p

Here by a generalized diagonal of (ui,j )i,j 1 we mean a sequence (uik ,jk )k1 with i1 < i2 < · · · and j1 < j2 < · · ·. Theorem 10.4 was proved in [82] for 1 p < 2 and in [188] for p < 1. Using Theorem 10.4, we can deduce the following refinement of Theorem 10.1, which comes from [82] for 1 p < 2, and from [188] for p < 1. T HEOREM 10.5. Let M be as in Theorem 10.1, and let 0 < p < 2. If X = Y are listed in the tree in the following figure, then X embeds into Y iff X can be joined to Y through a descending branch

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Several non-embeddings in Theorem 10.5 are already contained in Corollary 10.3 and the discussion just after it. On the other hand, the non-embedding of Lp (K p ) into Lp ⊕ S p was established in [187], and that of Lp (R) into Lp (S p ) in [157]. The proof for the first non-embedding in [187] uses the classical result that Lp contains a subspace isomorphic to l q for all 0 < p < q < 2. This classical result admits a non-commutative version, which is a remarkable result recently obtained by Junge (see Corollary 10.12 below), and which is the main ingredient for the non-embedding of Lp (R) into Lp (S p ). The remaining nonembeddings in Theorem 10.5 can be reduced to the following T HEOREM 10.6. Let 0 < p < 2, and let M and N be finite von Neumann algebras. Let X ⊂ Lp (M) be a closed subspace which contains no subspace isomorphic to l p , and let Y be a quasi-Banach space which contains no subspace isomorphic to X. Then X ⊗p S p does not embed into Y ⊕ Lp (N ), where X ⊗p S p denotes the closure of the algebraic tensor product X ⊗ S p in Lp (M ⊗ B(l 2 )). Theorem 10.6 was proved in [188]. It extends some results in [82]. Like in [82], its proof heavily relies upon Theorem 10.4. Using this theorem and Corollary 10.12 below (and its commutative counterpart, cited above), we deduce that Lp (S p ) (resp. Lp (R ⊗ B(l 2 ))) does not embed in S p ⊕ Lp (R) (resp. Lp (S p ) ⊕ Lp (R)). Subspaces of Lp (M), which have no copy isomorphic to l p , can be characterized as follows. T HEOREM 10.7. Let 0 < p < ∞, p = 2. Let M be a finite von Neumann algebra and X ⊂ Lp (M) a closed subspace. Then the following assertions are equivalent:


(i) (ii) (iii) (iv)

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X contains a subspace isomorphic to l p . For any λ > 1, X contains a subspace λ-isomorphic to l p . p X contains ln ’s uniformly. For any q such that 0 < q < p the norms · q and · p are not equivalent on X.

R EMARK . If one of the preceding assertions holds, then X contains a perturbation of a normalized sequence formed of operators with disjoint support; consequently, if p 1, X contains, for any λ > 1, a subspace λ-isomorphic to l p and λ-complemented in Lp (M). The above theorem is the extension to the non-commutative setting of the classical Kadets–Pełczyński results for commutative Lp -spaces (cf. [103,170]). It was proved in [186] for p > 2, in [82] for 1 p < 2, and in [163] and [188] for 0 2, Theorem 10.7 yields the non-commutative analogue of the following striking dichotomy: C OROLLARY 10.8. Let M and X be as in Theorem 10.7 with 2 < p < ∞. Then either X is isomorphic to a Hilbert space or X contains a subspace isomorphic to l p . R EMARKS . (i) The above corollary is easier for subspaces of S p , and there it holds for all 0 < p < ∞ (cf. [68]). (ii) More generally, Theorem 10.7 was extended in [169] to non-commutative Lp -spaces associated with any von Neumann algebra. (iii) [162,164] and [169] contain more results closely related to Theorem 10.7 and Corollary 10.8. There are many open problems on the subject discussed above. Below we give two of them. Let M and N be two von Neumann algebras of type λ and μ, respectively, where λ, μ ∈ {I, II1, II∞ , III}. Combining Corollary 10.3, Theorem 10.5 and Theorem 3.5, we see that if λ < μ and (λ, μ) = (II∞ , III), then Lp (M) and Lp (N ) are not isomorphic for all 0 < p < ∞, p = 2. It is unknown whether this is still valid for (λ, μ) = (II∞ , III). P ROBLEM 10.9. Let M and N be two von Neumann algebras of type II∞ and III, respectively. Are Lp (M) and Lp (N ) isomorphic for p = 2? Theorem 10.5 solves the embedding problem for all spaces listed there in the case of p < 2. On the other hand, Corollary 10.3 provides some partial solutions in the case of p > 2. However, we do not know whether Theorem 10.5 holds in full generality for p > 2. Below we state three of the most important cases left unsolved in Theorem 10.5. P ROBLEM 10.10. Let p > 2, and let (X, Y ) be one of the three couples (Lp (K p ), S p ⊕ Lp ), (Lp (S p ), S p ⊕ Lp (K p )) and (Lp (R), Lp (S p )). Does X embed into Y ? All previous non-embedding results deal with a couple of non-commutative Lp -spaces with the same index p. However, Junge’s theorem already mentioned above says that S q does embed into Lp (R) for p < q < 2. In fact, Junge [96] proved the following striking result, much stronger than the embedding of S q into Lp (R).

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T HEOREM 10.11. Let 0 < p < q < 2. Then Lq (R ⊗ B(l 2 )) embeds isometrically into Lp (R). As an immediate consequence, we get the C OROLLARY 10.12. Let 0 < p < q < 2. Then both S q and Lq (R) embed isometrically into Lp (R). q

In the commutative case, it is well-known that any ln embeds (uniformly over n) into p some lN . Junge also obtained the non-commutative version of this in [95]. T HEOREM 10.13. Let 0 0, n ∈ N. Then there is N = N(p, q, ε, n) such p q that SN contains a subspace (1 + ε)-isomorphic to Sn . Like in the commutative case, Junge’s arguments for the preceding results are probabilistic. They use non-commutative analogues of p-stable or Poisson processes. The reader is referred to [95,96] for more details and more embedding results. We conclude this section by a few words about the local theory of the non-commutative Lp -spaces, very recently developed in [99], in analogy with the classical Lp -space theory. Actually, it is better (and more convenient in some sense) to develop this theory in the operator space framework. Then the corresponding Lp -spaces are called OLp -spaces in [60]. Many classical results concerning Lp -spaces have been transferred to this noncommutative setting. In particular, any separable OLp -space (with an additional assumption) has a basis. It was also proved that Lp (M) (1 < p < ∞) is an OLp -space when M is injective or the von Neumann algebra of a free group (in the former case, p can be equal to 1). Consequently, these non-commutative Lp -spaces have bases. In the case of p = ∞, it was shown that any separable nuclear C ∗ -algebra has a basis. The interested reader is referred to [99] for more information.

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CHAPTER 35

Geometric Measure Theory in Banach Spaces David Preiss Department of Mathematics, University College London, London WC1E 6BT, UK E-mail: [email protected]

Contents 1. Finite-dimensional geometric measure theory in infinite-dimensional situations 1.1. Rectifiability and density . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Infinite-dimensional geometric measure theory . . . . . . . . . . . . . . . . . . 2.1. Differentiable measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Surface measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Measures and balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Differentiation theorems for Gaussian measures . . . . . . . . . . . . . . 3. Exceptional sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Lipschitz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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Geometric measure theory in Banach spaces

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We will consider the question to what extent the classical relations between measure, derivative and geometry carry over to infinite-dimensional Banach spaces. Such relations may be strangely distorted, like the seemingly simple question of recovering the Radon– Nikodým derivative by the limit of ratios of measures of balls, or answers may not be known, even in basic cases such as existence of common points of Fréchet differentiability of finitely many real-valued Lipschitz functions on a separable Hilbert space. Our goal is to describe some basic notions and results; these notes should be considered as an invitation to the subject and not as a survey of the subject, since many important concepts have necessarily been left out. We first visit two themes which are essentially finite-dimensional even though the surrounding space is infinite-dimensional, the problem of relations between rectifiability and density in general metric spaces and the recently developed theory of currents in metric spaces. In both cases, the structure of a Banach space is not essential for the setting, but it may always be assumed and it was needed to prove some of the deep results. For practically all problems of infinite-dimensional geometric measure theory, the most important difference between finite-dimensional and infinite-dimensional Banach spaces is due to non-existence of a reasonably finite, translationally invariant measure in the latter case: since any ball B(x, r) contains infinitely many disjoint balls of radius r/3, if μ is a non-zero Borel measure on an infinite-dimensional separable Banach space X and the μ-measure of balls depends only on their radii, then every non-empty open set has infinite measure. In fact, if μ is a σ -finite measure on X, the shifted measure is singular with respect to μ for many shifts from X. One way to see this is to assume, as we may, that μ is finite and the norm is square integrable and consider the Cameron–Martin space of μ, H = {x ∈ X; sup{|x ∗ (x)|: x ∗ L2 (μ) } < ∞}. The identity from H to X is compact (in fact, it is much better; for example, if X is Hilbert, it is Hilbert–Schmidt), and it is easy to see that the shift of μ by any vector not belonging to H is singular with respect to μ. Nevertheless, there are measures in infinite-dimensional Banach spaces that have some of the properties normally associated with the Lebesgue measure. In many instances, Gaussian measures have been used as a replacement for the Lebesgue measure. A more general class of such measures is formed by those that are quasi-invariant in a dense set of directions, where a measure is called quasi-invariant in a direction u if its null sets are preserved when shifted by u. We will restrict ourselves to a discussion of the notions of measures differentiable in a direction in 2.1 and to briefly pointing out connections to the possibility of defining surface measures. In 2.3 we visit the amusing results obtained by attempting to understand to what extent exact naïve analogues of finite-dimensional results fail in infinite dimensions. In the short Section 3 we give several notions of exceptional sets. Some readers may find it more convenient to read the definitions from this section only after encountering their use in Section 4, where we treat the problem of existence of Gâteaux or Fréchet derivative of Lipschitz functions and also briefly consider few more exotic derivatives. Out of the directions that have been omitted one should definitely mention the study of various notions of generalized derivatives (or subdifferentials) which often parallels the development of classical real analysis. As an example, see [11]. For many other problems that could fit into this text as well as a number of results relevant to it see [5]. Although from time to time we mention the case of Gaussian measures (defined as those measures whose every one-dimensional image by a continuous linear functional is Gaussian or Dirac), much

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of the extensive research devoted to them has not been mentioned even though it often has deep connections to problems of geometric measure theory. To avoid possible misunderstandings, we will consider only separable Banach spaces unless non-separability is specifically permitted, and only Borel measures. Normally, measures will be positive and finite. There are, however, important exceptions: signed (i.e., realvalued) measures are appearing, out of necessity, in our considerations of differentiability of measures in 2.1 and the Hausdorff measures are positive but often notoriously infinite.

1. Finite-dimensional geometric measure theory in infinite-dimensional situations Many situations from the classical geometric measure theory (including the notions of fractal geometry such as definition of fractal sets via iterated function systems) can be easily transferred to the case when the ambient space is infinite-dimensional. For example, the k-dimensional Hausdorff measure ∞ ∞ 1 k lim inf H (A) = diam (Ai ): A ⊂ Ai , diam(Ai ) < δ α(k) δ→0 k

i=1

i=1

as well as a number of several other k-dimensional measures (spherical measures, packing measures, etc.) have been studied in arbitrary metric spaces. (The constant factor α(k) is chosen so that the k-dimensional Hausdorff measure in Rk coincides with the Lebesgue measure; since we will not need this fact here, for the purpose of these notes we may set α(k) = 1.) The natural setting is often that of metric spaces (and, because these can be embedded into Banach spaces, it usually suffices to consider only these), but with a few notable exceptions, the generality does not bring much new, even though it may help to clarify the assumptions or provide natural proofs. Here we will briefly consider two of the situations in which the interaction with geometry of Banach spaces went much farther.

1.1. Rectifiability and density Much of the development of classical geometric measure theory was driven by attempts to show, under various geometric assumptions on a subset A of Rn of finite k-dimensional measure, that A is k-rectifiable, i.e., that Hk -almost all of A can be covered by a Lipschitz image of a subset of Rk . (The restriction to the Hausdorff measure and sets of finite measure is not necessary, but it is convenient and suffices for this presentation.) Perhaps the most useful rectifiability criterion, the Besicovitch–Federer projection theorem (see [17, Theorem 3.3.12]), did not seem to have any natural counterpart for infinite-dimensional ambient spaces till one was found in the modern setting of currents (see below). Another useful rectifiability criterion (due, in increasing level of generality, to Besicovitch, Marstrand and Mattila) says that (under the above assumptions) A is k-rectifiable if and only if its k-dimensional density Θ k (A, x) = limδ→0 Hk (A ∩ B(x, r))/(α(k)r k ) is equal to one at Hk almost every x ∈ A. This could well be true in every metric space, although


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it was shown only in the cases when either k = 1 or the space in question is a subset of a uniformly convex Banach space. (So the simplest unknown case is the two-dimensional measure in ∞ 3 .) The implication saying that rectifiable sets have density one almost everywhere was proved by Kirchheim [27] in full generality. The key behind this is the existence of metric derivative (see 4.2.6). One of the corollaries of this work is that the area formula remains valid in the general situation; in the simpler case of injective Lipschitz mappings this says that the k-dimensional measure of the image can be calculated as an integral of a suitable Jacobian over the domain, where the Jacobian depends just on the metric derivative of the mapping. 1.2. Currents The development of the theory of currents was motivated by the difficulty to prove existence results for higher-dimensional minimal surfaces by classical methods. The basic ideas behind the (finite-dimensional) theory of currents, as found, for example, in [17], were similar to those behind the introduction of distributions: a k-dimensional oriented surface gives rise to a linear functional on the space of differential k-forms; among such functionals one chooses (and calls currents) those that still form a (weak∗ ) compact set but already have many features of ‘genuine surfaces’. The main notions may be transfered to Hilbert spaces (or even to those Banach spaces in which suitable results on differentiability of Lipschitz mappings hold) without any change, but deeper results appear to be based on concepts that are not available beyond the finite-dimensional situation. We will now describe an important recent development which shows that the theory needs no concept of differentiability and that many strong results remain valid even when the ambient space is infinite-dimensional. It is due to Ambrosio and Kirchheim [2], based on an idea of De Giorgi and, incidentally, does not even use the concept of differential forms. Instead of considering integration over a k-dimensional (smooth) oriented surface S ⊂ Rn as a linear functional on k-dimensional differential forms, we will consider it as a (k + 1)-linear functional on the space Dk+1 (Rn ) of (k + 1)-tuples of Lipschitz functions (f, π1 , π2 , . . . , πk ). So, in the simplest case of S = [a, b] being an interval on the line (so b k = n = 1), the associated current is T (f, π) = [a,b] f dπ = a f (x)π (x) dx. Somewhat more generally, if g is a Lebesgue integrable function on Rn (which can be imagined as the multiplicity of the k = n-dimensional oriented surface S = {x: g(x) = 0}), the associated current is T (f, π1 , π2 , . . . , πk ) = fg dπ1 ∧ dπ2 ∧ · · · ∧ dπk S

= f (x)g(x) det π1 (x), π2 (x), . . . , πk (x) dx, and analogous formulas associate such (k + 1)-linear forms to (smooth) k-dimensional surfaces with real multiplicity also in case when k < n. The key observation is that the functionals T in the above examples have the following three properties.

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(a) T is (k + 1)-linear. (b) The restriction of T to the set of (k + 1)-tuples with |f | 1 and Lip(πi ) 1 is continuous in the topology of uniform convergence of the f ’s and pointwise convergence of the πi ’s. (c) T is local in the sense that T (f, π1 , π2 , . . . , πk ) = 0 whenever some πi is constant on {x: f (x) = 0}. For the currents from our examples, (c) expresses the fact that they depend on the derivatives of the πi ’s. The fact that (b) holds (which is not so obvious) has been recognised (often implicitly) as important in a number of other connections. We can now define a k-dimensional current T in a complete metric space E as a functional on Dk+1 (E) satisfying (a)–(c). The key requirement of locality means that T depends in a weak sense on the derivative of the πi ’s, and it is the main point in showing that currents have the property that transforming the πi ’s by a Lipschitz mapping of Rk multiplies the f by the determinant of the transformation. (In particular, currents are anti-symmetric in the πi ’s.) One may therefore use the more suggestive notation T (f dπ1 ∧ · · · ∧ dπk ) for T (f, π1 , . . . , πk ), although the symbols dπi themselves may have no meaning. Currents behave like measures in the* first variable; more precisely there is a finite measure μ such that T (f, π1 , π2 , . . . , πk ) i Lip(πi ) |f | dμ; the least such measure is called the mass of T and denoted by T . The total mass of T is defined as M(T ) = T (E). Standard operations on currents are defined in a natural way. The existence of mass allows one to extend currents to arbitrary bounded Borel f ; in particular, the restriction of a current to a Borel set B may be defined as (f, π1 , . . . , πk ) → T (f χB , π1 , . . . , πk ), where χB is the indicator function of B. The push-forward of T by a Lipschitz mapping φ : E → F is φ+ T (f, π1 , . . . , πk ) = T (f ◦ φ, π1 ◦ φ, . . . , πk ◦ φ) and the boundary of a (k + 1)-dimensional current S is ∂T (f, π1 , . . . , πk ) = S(1, f, π1 , . . . , πk ). However, boundaries are tricky: ∂T is a functional satisfying (a) and (c), but there is no reason why it should satisfy the continuity requirement (b). Currents for which ∂T satisfies (b) are called normal and are the first of the basic classes of current. Two other concepts arise naturally from the wish to define a notion that should represent generalized surfaces (with integer multiplicity): a k-dimensional current T is rectifiable if its mass is absolutely continuous with respect to the k-dimensional measure on some k-rectifiable set and it is an integer current if the push-forward to Rk of any restriction of T to a Borel set is representable by a Lebesgue integrable integer-valued function. It is not known if every k-dimensional current in Rk (k 3) is rectifiable (this problem is close to that of describing sets of non-differentiability of Lipschitz mappings in Rk ) but normal k-dimensional currents in Rk are necessarily rectifiable and in fact correspond exactly to functions of bounded variation. It follows that, within normal currents in Rn , the new concepts coincide with the standard ones. These classes of currents admit many natural characterizations similar to those obtained for currents in finite-dimensional spaces. In particular, strong rectifiability criteria which are false for sets are valid for currents. In the presence of suitable results on differentiability of Lipschitz mappings these classes of currents may be defined in a more customary way via exterior algebra and integer currents are those whose density is an integer multiple of the corresponding area factor. (The area factor is related to the Jacobians mentioned above


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as well as to the problem of finding optimal rectangles enclosing a given convex body, which was studied, for example, in [39].) The Plateau problem may be stated in the full generality of a complete metric space E: given a k-dimensional integer rectifiable current S with compact support and zero boundary, find a (k + 1)-dimensional integer rectifiable current T having the least total mass such that ∂T = S. In general, there may exist no currents T with ∂T = S, but such currents always exist if E is a Banach space. However, to assure the existence of a suitable minimizing sequence (whose limit would give a solution to the Plateau problem under fairly general assumptions, for example, if E is dual of a separable space) one needs that the following isoperimetric inequality holds: for every k-dimensional integer rectifiable current S with ∂S = 0 there is a (k + 1)-dimensional integer rectifiable current T with ∂T = S and M(T ) c(M(S))(k+1)/ k , where c is a constant depending on E and k only. Whether this holds in every Banach space is an open problem; it has been proved in duals of separable Banach spaces having a weak∗ finite-dimensional decomposition. 2. Infinite-dimensional geometric measure theory The relation between infinite-dimensional measures and geometry of the infinite-dimensional Banach space E is much weaker than in the finite-dimensional case. Consider just the problem of describing the image of a non-degenerated Gaussian measure γ on E by a continuous linear transformation T : even for simple transformations, such as T x = 2x, the image is singular with respect to γ and so no analogue of the classical substitution theorem can hold for such transformations. The same situation occurs for most shifts. The most common setting is therefore not only a Banach space E equipped with a measure μ, but also with a vector space H of the set of directions in which μ behaves invariantly; it is also often assumed that H is (a continuous image of) a Hilbert space. (The basic example is, of course, a Gaussian measure γ in E with H being its Cameron–Martin space.) The (possibly non-linear) transformations of the form x → x + h(x), where h : E → H are the natural candidates for which the substitution theorem may be valid. The role of geometry of E has nearly disappeared and in fact E is usually just assumed to be a locally convex space. Below we comment on the background of the basic concepts of derivative of measures in Banach spaces and briefly indicate some directions of research. Then we discuss results showing that not only covering theorems but even some of their natural corollaries often fail in infinite-dimensional situation even for Gaussian measures. 2.1. Differentiable measures In much of modern analysis in finite-dimensional spaces, the role of pointwise derivative has been completely overshadowed by that of derivative in the sense of distributions. If f : R → R is Lebesgue integrable, its distributional derivative may be defined as a Lebesgue integrable function g : R → R such that the formula for integration by parts

(1) φ (t)f (t) dt = − φ(t)g(t) dt

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holds for every smooth φ : R → R with bounded derivative. However, observing that in (1) the functions f and g are only used as acting on functions by integration, i.e., as measures, we may consider it as defining that the distributional derivative of a signed measure μ is a signed measure ν such that that φ (t) dμ(t) = − φ(t) dν(t) for every smooth bounded φ : R → R with bounded derivative. On the real line, this generality is partly spurious, since it is easy to see that a measure μ on R has this derivative if and only if it is a function of bounded variation. (Somewhat loosely, one says that the measure μ(E) = E f (t) dt is a function, namely, the function f .) However, the derivative may well be a measure which is not a function, for example, the derivative of the function f (t) = signum(t) is the Dirac measure. A similar approach is used in Rn to define distributional partial derivatives; and again their existence means that the measure is a function. In fact, it is again a function of bounded variation, usually by definition (see, for example, [61]). The definition of distributional derivatives of measures admits a direct generalization to Banach spaces (where we have no notion of a measure being a function): the derivative of a (finite Borel) signed measure μ in direction w is a signed measure Dw μ such that

Dw φ(x) dμ(x) = −

φ(x) dDw μ(x)

(2)

for every bounded continuously differentiable φ : X → R with bounded derivative. The definition immediately implies that the set of directions of differentiability of μ is a linear space, the mapping w → Dw μ is linear and that differentiation commutes with convolution, i.e., Dw (ν ∗ μ) = ν ∗ Dw μ provided that Dw μ exists. Directional derivatives of measures may be equivalently defined by more direct formulae: derivative of μ in the direction w in Skorochod’s sense is defined by (see [52, §21] for details)

φ dDw μ = − lim

r→0

φ(x + rw) − φ(x) dμ(x) r

(3)

provided that the limit exists for every bounded continuous φ : X → R; the functional defined by the limit is necessarily an integral with respect to a measure. Another approach which was developed in finite-dimensional spaces by Tonneli needs essentially no modification in infinite-dimensional spaces: we require that μ has a disintegration

φ dμ =

Y R

φ(y + tw)ψy (t) dt dν(y),

where Y is a complement of span{w}, ν is a probability measure on Y and ψy are (right continuous) functions of bounded variation; under these conditions we define

φ dDw μ =

Y R

φ(y + tw) dψy (t) dν(y).

(4)


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It is easy to see that the derivatives of measures defined by (3) or (4) satisfy (2). If (2) holds, we obtain (3) by denoting νr (E) = λ{t ∈ [0, r]: tw ∈ E}/r, inferring from the formula for differentiation of convolution that φ(x + rw) − φ(x) dμ(x) φ dνr ∗ Dw μ = − r first for every bounded continuously differentiable φ : X → R with bounded derivative and then, by approximation, for every bounded continuous φ : X → R, and by letting r → 0. Finally, to obtain (4) from (2), we disintegrate

h dDw μ =

Y R

h(y + tw) dσy (t) dν(y)

and let ψy (t) = σy (−∞, t]. By approximation, it suffices to show that (4) holds for every continuously differentiable function φ : X → R with bounded derivative and with {t t ∈ R: φ(y + tw) = 0 for some y ∈ Y } bounded. For any τ > 0 denote gτ (y + tw) = −∞ φ(y + sw) − φ(y + (s + τ )w) ds and use (2) and integration by parts to infer that

Dw gτ (x) dμ(x) = −

gτ (x) dDw μ(x) = −

=

Y R

Y R

gτ (y + tw) dσy (t) dν(y)

Dw gτ (y + tw)ψy (t) dt dν(y).

Since Dw gτ (x) = φ(x) − φ(x + τ w), (2) follows by letting τ → ∞. Currently the most useful notion of derivative of a measure μ (often called differentiability in the sense of Fomin) is obtained by requiring additionally that Dw μ be absolutely continuous with respect to μ. This is equivalent to validity of (3) for every bounded Borel measurable function or to differentiability at t = 0 of the function assigning to t ∈ R the measure μ shifted by tw when the space of measures is equipped with the usual norm. The Radon–Nikodým derivative of Dw μ with respect to μ is called the logarithmic derivative of μ in direction w; one readily sees that this term is justified in the finite-dimensional situation. All these notions have been treated as a special case of differentiability of mappings of the real line into the space of signed measures equipped with various topologies in [50]; another particular case of this treatment is the notion of differentiability of measures along vector-fields. (Of course, in this generality some of the equivalences mentioned above may fail.) Under very mild assumptions, these authors also prove the key forb mula dμa / dμb = exp( a -t (x) dt), where -t is the logarithmic derivative of t ∈ R → μt . (See [50] for the history of this formula and its applications.) In the setting when H is a subspace of E consisting only of directions of logarithmic differentiability and h : X → H , one can, under appropriate assumptions, compute the logarithmic derivative of t → (id + th)+ μ and the Radon–Nikodým derivative d(id + th)+ μ/ dμ from the derivative of h and directional logarithmic derivatives of μ – the latter gives a substitution theorem mentioned above (see [51]). The assumptions alluded to here are, of necessity, much stronger than those mentioned so far: since the formulas involve either the trace of the derivative of h in

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the direction of H or the determinant of id + th (x). (This also explains why H is supposed to carry a Hilbert space structure.) A Gaussian measure is logarithmically differentiable exactly in the directions of its Cameron–Martin space and the derivatives may be found explicitly. For these measures, the above results form just a beginning of the story; see, for example, [8] for much more. The natural problem of unique determination of a measure by its logarithmic derivative has been answered negatively in [38]. (Prior to it, several authors noted that a positive answer would not only mean that some correspondence between functions and measures survives to the infinite-dimensional situation but would also have interesting applications.)

2.2. Surface measures Several approaches have been suggested to the definition of the surface measure induced by a given measure μ on E. A natural way is to assume that the surface is defined as {x: ϕ(x) = 0} where ϕ : E → R is such that for sufficiently many functions g on E the measures ϕ+ (gμ) have continuous density kg with respect to the Lebesgue measure; the value of the surface integral of such g is then kg (0)/k1 (0). To prove the assumption of continuity of kg , one may use differentiability of μ together with the Malliavin method. (More details may be found in [8].) Uglanov’s method [56] is based on the idea that, if μ has logarithmic derivative in direction w and G is the graph of a smooth function from a complement of Rw to Rw then Dw μ{a + tw: a ∈ A, t 0} should be a measure on G which is absolutely continuous with respect to the corresponding surface measure on G with known Radon–Nikodým derivative. This can be used to define the surface measure of subsets of G provided that it does not depend on the choice of w. This independence is shown under suitable assumptions, which appear less stringent than in other methods. It is natural to imagine that the theory of surface measures (or, more generally, measures on surfaces of finite co-dimension), could be understood also as theory of integration of differential forms and/or currents of finite co-degree. Such possibility has been explored in a series of papers starting from [49].

2.3. Measures and balls Only little seems to be known about the interplay between geometry of an infinitedimensional Banach space X and behaviour of measures on it. A reasonably clear picture showing that the situation is rather complicated has been obtained concerning the questions that developed from the attempts to find valid infinite-dimensional counterparts to the differentiation theorem for measures according to which in finite-dimensional Banach spaces the Radon–Nikodým derivative g of a Borel measure μ with respect to a Borel measure ν is, at almost every x, obtained by the limit of the ratio of their averages on balls around x: g(x) = limr→0 μ(B(x, r))/ν(B(x, r)). It turned out that the differentiation theorem for measures holds for all Borel measures in a Banach space if and only if it is finite-dimensional; a similar statement holds even for complete metric spaces (and for


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other families than balls) with a suitable combinatorial definition of finite dimensionality. (Like most results mentioned in this subsection, this may be found in [36].) Moreover, in a Hilbert space even much weaker statements are false as is shown by the following rather involved example from [43]. (Some numerical constants in the construction in [43] should not be taken too literally; a more accessible construction should eventually appear in [36].) There are a Gaussian measure ν in 2 and a positive function f ∈ L1 (ν) such that B(x,r) f dν/ν(B(x, r)) tends to infinity as r tends to zero not only for every x but even uniformly on 2 , in other words lim inf

r→0 x∈2 B(x,r)

f dν/ν B(x, r) = ∞.

(5)

Among the possible corollaries of the differentiation theorem for measures that are not negated by the above example the most natural one is the determination of measures by balls: the differentiation theorem applied to μ and μ + ν gives that μ = 12 (μ + ν), hence μ = ν whenever μ and ν coincide on all balls. The hope that at least this statement holds in general metric spaces was dashed by Davies in [14]: there is a compact metric space on which two different Borel probability measures coincide on all balls. The basic idea of this beautiful example is the construction, for any given α, β > 0, of two measures μ, ν on a finite metric space M in which the only distances are one and two (so M is easy to imagine as the vertex set of a graph; points joined by an edge have distance one, remaining points have distance two) which coincide on all closed balls with radius one and satisfy μ(M) = α and ν(M) = β. Such a space is obtained as a graph on n + n2 vertices consisting of a complete graph on n ‘inner’ vertices, to each of which n different ‘outer’ vertices are joined. The main observation is that each ball of radius one consists either of one inner and one outer vertex, or of n inner and n outer vertices; then a straightforward calculation gives μ and ν provided that n is large enough. (For example, all inner vertices may have μ measure α/2n and ν measure α/2n − (β − α)/(n2 − n), and all outer vertices may have μ measure α/2n2 and ν measure α/2n2 + (β − α)/(n2 − n).) Replacing recursively points by rescaled copies of such spaces, one finds a compact metric space M0 of diameter one on which two different Borel measures μ0 , ν0 coincide on all balls of radius less than one. The final space and measures are obtained as M0 ∪ M1 , μ0 + ν1 and ν0 + μ1 , where M1 is another copy of M0 (with corresponding measures μ1 , ν1 ) and the distance between points of M0 and M1 is defined as one. The above construction showing that even the determination of measures by balls may be false in general metric spaces clearly leads only to highly non-homogeneous spaces and it cannot produce, for example, a Banach space. Indeed, for Banach spaces such an example does not exist ([45]): two finite Borel measures μ and ν coinciding on all balls in a separable Banach space X necessarily coincide on all Borel subsets of X. The argument blows suitably placed balls to show that μ and ν coincide on many convex cones that factor through a finite-dimensional subspace; a simple consideration of finite-dimensional projections of μ and ν then gives that they coincide on all half-spaces given by linear functionals belonging to a weak∗ dense subset of the dual unit ball. Hence μ and ν have the same Fourier transform, and the statement follows.

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It is an open problem if the determination of measures by balls holds under relatively weak homogeneity assumptions, e.g., in complete separable metric Abelian groups. In particular, except for a few special cases it is not known whether two finite Borel measures in a Banach space coincide provided that they coincide on all balls with radius at most one. (Note that by the above proof this holds for balls of radius at least one.) This motivates the attempts to prove the Banach space result without the use of Fourier transform. One such attempt noted that it would be enough to show that the family of Borel sets in a separable Banach space X is the smallest family of subsets of X containing balls and closed under complements and countable disjoint unions; the latter family is necessarily closed also under countable monotone unions and intersections. It has been recently shown that this statement holds in finite-dimensional spaces (it is not easy; both existing proofs ([20] and [60]) use Besicovitch’s covering theorem), but not in an infinite-dimensional Hilbert space (any non-trivial intersection of two balls supplies a counterexample [25]). The extent of the difference between finite- and infinite-dimensional situation is apparent from the following two amusing statements concerning separable Hilbert spaces H : (A) The statement “whenever μ(B) ν(B) for all balls B with radius 1, then μ ν” holds if and only if H is finite-dimensional. (B) The statement “whenever μ(B) ν(B) for all balls B with radius 1, then μ ν” holds if and only if H is infinite-dimensional. The statements considered in (A) and (B) are sometimes called positivity principles for small and large balls, respectively. The positivity principle for small balls follows from the differentiation theorem for measures, and so it holds in all finite-dimensional Banach spaces. The example behind the other implication of (A) uses the Gaussian measure ν in 2 and positive function f ∈ L1 (ν) such that (5) holds: a simple modification achieves f dν < 1 and B(x,r) f dν > ν(B(x, r)) for r 1; and the example needed for (A) is ob tained with μ(A) = A f dν. This example also answers negatively the question of validity of positivity principle for all balls in general Banach spaces: In the space 2 ⊕∞ R consider ν1 + ν−1 + μ0 and ν0 , where the index r indicates the image measure under the mapping x ∈ 2 → x ⊕ r. The statement (B) is much easier: in the n-dimensional case one may consider for μ the Lebesgue measure on a ball and for ν a small multiple of the Dirac measure in its centre. In the infinite-dimensional case one shows that the characteristic function of each cylinder (set of the form π −1 (B), where π is an orthogonal projection with a finite-dimensional range and B is any ball in the range), can be obtained as a limit of functions from the convex cone generated by characteristic functions of balls with radius 1. This reduces the proof to the finite-dimensional case of small balls. The negative results mentioned above are remarkably unstable. Riss [48] shows that every Banach space may be renormed so that the positivity principle holds for large balls. Only future investigations may reveal the fate of this little corner of geometric measure theory. It is possible that useful connections exist to probability theory on Banach spaces, especially to problems of large deviations; a small indication of this may be given by the use of Chernoff’s theorem in [15] to prove the determination of measures by balls for measures with finite Laplace transform, or by the fact that more information may be sometimes obtained by using large balls instead of small balls.


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2.4. Differentiation theorems for Gaussian measures The importance of Gaussian measures justifies their separate treatment, although from the point of view of the problems considered here the results are rather sparse. It is not difficult to see that Gaussian measures on Banach spaces are determined by their values on balls of radii at most one [9]. Nevertheless, in 2.3 we noted that even for a Gaussian measure ν in a separable Hilbert space the differentiation theorem for L1 functions may fail not only almost everywhere, but even uniformly (see Eq. (5)). The density theorem may fail as well [42]: there are a Gaussian measure ν in 2 and a Borel set E ⊂ 2 of positive ν measure such that limr→0 ν(E ∩ B(x, r))/ν(B(x, r)) = 0 for ν almost every x, although here it is not known if the failure may be uniform. On the other hand, by [55] there are infinite-dimensional Gaussian measures in separable Hilbert spaces for which the differentiation theorem holds for all functions from Lp for p > 1; a sufficient condition is that the eigenvalues σk of their covariance satisfy σj +1 cσj /j α for some α > 5/4. The proof is rather technical and its important ingredient is the dimension-independent estimate of the Hardy–Littlewood maximal operator from [54].

3. Exceptional sets Here we discuss several notions of “null” or “negligible” sets in a Banach space that appear as exceptional sets in various questions of behaviour of mappings between infinitedimensional spaces. With the notable exception of the topological notion of the sets of the first category, these involve metric or linear conditions and sometimes behave in an unexpected way. First category sets are, unfortunately, rarely useful in problems where main point is to capture some of the roles played in finite-dimensional spaces by the Lebesgue null sets. When convenient, we will give the definition for Borel sets only with the understanding that a possibly non-Borel set is null if it is contained in a Borel null set. (However, it should be pointed out that Borel measurability is not a point of pedantry; leaving it out may easily not only change the definitions but render them meaningless.) The most appealing replacement for Lebesgue null sets in infinite-dimensional Banach spaces or even in complete separable metric Abelian groups is due to Christensen [12]. Let G be an Abelian topological group whose topology is metrizable by a complete separable metric. A Borel set E ⊂ G is Haar null if there is a Borel probability measure μ on G such that every translate of G has measure zero. (Sometimes these sets are referred to as Christensen null. They have also been rediscovered under the name of shy sets.) Haar null sets form a σ -ideal since, if En are Haar null and μn are the corresponding measures, the measure obtained as an infinite convolution of suitable portions of μn witnesses that ∞ n=1 En is Haar null. If the group is locally compact, these null sets coincide with those of Haar measure zero. If, however, the group is not locally compact, then every compact set is Haar null; this follows from the following important generalization of Steinhaus’s theorem: if a Borel set A is not Haar null then A − A contains a neighbourhood of the identity in G. Haar null sets have been, deservedly, investigated in their own right. Since measures on G are inner regular with respect to compact sets, any subset of G containing a translate of

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every compact set is not Haar null. This has been used in [53] to show that if G is not locally compact, then it contains an uncountable collection of disjoint closed subsets which are not Haar null. Reflexive Banach spaces have been characterized as those in which every closed convex set with empty interior is Haar null, see [35] and [34]. Relatively simple examples show that a Fubini-type result is false for Haar null sets. A property useful in application to differentiability is that a Haar null set is Lebesgue null on lines parallel and arbitrarily close to any given line. Although the knowledge that an exceptional set is Haar null suffices for many applications (in particular, if we just need to have one non-exceptional point), there are situations in which this is not the case. Several seemingly different stronger notions of negligible sets in a Banach space X were defined (implicitly or explicitly) to improve, in particular, the results on Gâteaux differentiability of Lipschitz functions. Recall that a cube measure in X is N under the mapping t → x + tk xk any image of the product Lebesgue measure on [0, 1] provided that xk < ∞. A cube or Gaussian measure is non-degenerate if every closed hyperplane gets measure zero. A Borel set E is a separable Banach space X is called (a) cube null if it is null for every non-degenerate cube measure on X [32], (b) Gauss null if it is null for every non-degenerate Gaussian measure on X [40], and (c) Aronszajn null if for every sequence un ∈ X with dense span, E can be written as a union of Borel sets En such that the intersection of En with any line in the direction un is of one-dimensional Lebesgue measure zero [3]. Clearly, Aronszajn null sets are Gauss as well as cube null. It is in fact not difficult to see that Gauss and cube null sets coincide, and may be equivalently defined as those Borel sets that are null for every measure with a dense set of directions of differentiability (cf. [6]). A remarkable result of Csörnyei [13] shows that every cube null set is Aronszajn null, and so all these notions coincide. The main difficulty of the proof stems from the requirement that the sets En be Borel; note however that without assuming this the definition (c) would become meaningless since by an observation from [6] the whole space would be Aronszajn null. Clearly, (a) or (b) immediately show that every Aronszajn null set is Haar null. The converse is false because there are compact Aronszajn non-null sets (e.g., cubes) while every compact set K is Haar null. This example also shows that the analogue of Steinhaus’s theorem is no longer valid for Aronszajn null sets. As one would expect, in finite-dimensional spaces Aronszajn null sets coincide with Lebesgue null sets. By an example from [7], Aronszajn null sets are not invariant under C ∞ Lipschitz isomorphisms. It follows that these sets cannot provide full characterization of exceptional sets that are invariant under such mappings (such as sets of non-differentiability). A possible remedy, suggested by Bogachev [7], is to consider sets which are null for all measures differentiable with respect to a spanning sequence of vector-fields un (i.e., such that for every x the vectors un (x) span X). Another possible remedy, from [47], is to consider sets which can be written as a union of Borel sets En such that, for some un ∈ X and εn > 0, the intersection of En with any curve γ : R → X with Lip(t → γ (t) − tun ) < εn is of onedimensional measure zero. Many questions concerning these sets are open. For example, it is not known if a formally stronger notion in the spirit of (c) is equivalent to the one described here or if an analogue of Csörnyei’s result holds in this situation. In fact, it is


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not even known if in finite-dimensional spaces (of dimension at least three) these null sets coincide with Lebesgue null sets. Intriguing invariance problems for the null sets defined above remain open and deserve to be mentioned because of their possible application to the problem of classifying Banach spaces up to Lipschitz isomorphisms. We have already noted that Aronszajn null sets are not invariant under Lipschitz isomorphisms. The same holds for Haar null sets; moreover, in [28] there is an example of a Haar non-null set in 2 which can be transformed by a Lipschitz isomorphism into a hyperplane; note that hyperplanes are not only Aronszajn null, but have to be null for any notion for which we wish to have any reasonable statement on differentiability of Lipschitz functions almost everywhere. For the intended application to the Lipschitz isomorphism problem it would be enough to know that a null set cannot be transformed into a set whose complement is null; as the title of [28] indicates, even this may happen for Haar null sets, but it is unknown for Aronszajn null sets. No pertinent examples are known for the non-linear notions of null sets. We now briefly describe another appealing notion of exceptional sets which is a metric strengthening of the notion of first category sets; unfortunately, for our purposes, it has the disadvantage that it cannot describe non-differentiability sets of Lipschitz functions since these may well be of second category. These sets, however, play an important role in studying more exceptional behaviour (for example, non-differentiability of continuous convex functions). Out of the huge number of non-equivalent notions (cf. [58]), the two most natural ones (in our context) are σ -porous sets and σ -directionally porous sets. It suffices to define the notions of porous and directionally porous sets only, since the prefix ‘σ ’ means ‘the union of countably many of.’ A set E ⊂ X is porous if there is 0 < λ < 1 such that for every x ∈ E there are xn ∈ X converging to x and rn > λxn − x such that the balls B(xn , rn ) are disjoint from E. It is porous in direction u if the xn may be found on the line through x in direction u, and it is directionally porous if there is u such that it is porous in direction u. The same notion of σ -porosity (though a different notion of porosity) is obtained if we define porosity with λ independent of x; for σ -directional porosity we could even allow dependence of u on x. The porosity σ -ideals are Borel: every σ -(directionally) porous sets is contained in a Borel σ -(directionally) porous set. Clearly, σ -directionally porous sets are σ -porous, and σ -porous sets are first category. Every σ -directionally porous set is Aronszajn null: if E is porous in direction u, it is porous in every direction sufficiently close to u; so, if μ has a dense set of directions of differentiability, then E is porous in a direction v of differentiability of μ and the disintegration along this direction shows that μ(E) = 0. In finite-dimensional spaces the notions of σ -porous and σ -directionally porous sets coincide, otherwise they differ: by [46], every infinite-dimensional space is a union of a σ -porous set and of an Aronszajn null set. For super-reflexive spaces a much stronger decomposition statement (using sets that are sometimes called strongly very porous) can be found in [33]. (Both these statements have been used to give examples concerning nondifferentiability, and so we will meet them again.) These constructions are based on the trivial observation that a Borel set which meets every k-dimensional affine subspace (where k is fixed) in a Lebesgue null set is necessarily Aronszajn null. Our next classes of exceptional sets consist of very small sets indeed (at least compared to the previous classes). Their importance stems from the fact that they have been used in the only results in differentiability in which we have complete characterizations (see

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Section 4.1.1). They are the σ -ideals of sets that can be covered by countably many k codimensional Lipschitz, respectively δ-convex, hypersurfaces. The k codimensional Lipschitz, respectively δ-convex, hypersurfaces are defined as sets of the form {w + φ(w): w ∈ W }, where X is a direct sum X = W ⊕ U and φ : W → U is Lipschitz, respectively δ-convex. Recall that a Lipschitz mapping φ : X → Y is δ-convex if for every y ∗ ∈ Y ∗ the composition y ∗ ◦ φ may be expressed as a difference of two continuous convex functions. These σ -ideals are Borel and are properly contained in all the previous σ -ideals; they are becoming smaller with k increasing, and those defined using δ-convex hypersurfaces are properly contained in the σ -ideals defined via Lipschitz hypersurfaces. In case k = dim(X), both notions give just countable sets. The prefix ‘k-codimensional’ will be omitted if k = 1. Finally, we meet a σ -ideal combining measure and category in a useful (and non-trivial way). It has been used in [31] to obtain first (and so far only) infinite-dimensional results on Fréchet differentiability of Lipschitz functions almost everywhere. The basic idea is to consider a suitable completely metrizable space of measures differentiable in direction of a spanning sequence of vector-fields (which may depend on the measure) and define that a set is null if it is null for residually many of these measures. This, however, appears to be technically complicated, and so the definition uses a parametric approach (in which the condition of differentiability of measures is not so apparent). Let Σ = [0, 1]N be endowed with the product topology and the product Lebesgue measure μ, and let Γ (X) denote the space of continuous maps γ : Σ → X having continuous partial derivatives. We equip Γ (X) with the topology of uniform convergence of the maps and their partial derivatives and define a Borel set E ⊂ X to be Γ -null if μ t ∈ Σ: γ (t) ∈ E = 0 for residually many γ ∈ Γ (X). Note that, since Γ (X) is completely metrizable, the family of null sets forms a proper σ -ideal of subsets of X. A simpler but less useful variant of the notion of Γ -null sets may be obtained by replacing Σ by [0, 1]k . These notions of Γ -null sets can be viewed as a special case of a general scheme defining negligibility of a set A in the space by requiring that it is negligible inside all except negligibly many elements of the hyperspace (space of subsets, space of measures). For example, the set P(X) of Borel probability measures on a complete separable metric space X considered as a subset of the dual to the space of bounded continuous functions with the weak∗ topology is completely metrizable, so one can try to consider as negligible those sets A ⊂ X that satisfy μ(A) = 0 for residually many μ ∈ P(X). Another, purely topological, example can be obtained by using the space K(X) of non-empty compact subsets of X equipped with the Hausdorff metric and defining negligibility of a set A ⊂ X by requiring that C ∩ A is of the first category in C for residually many C ∈ K(X). The usefulness of the notions introduced in these two examples is somewhat diminished by the easily seen fact that for Borel sets they are both equivalent to the notion of the sets of the first category. (In this connection, one may note that, without any condition on a set A ⊂ X, A is of the first category if and only if C ∩ A = ∅ for residually many C ∈ K(X). This follows immediately by noting that the union of the compacts belonging to a Gδ subset of K(X) is a Suslin set.) Nevertheless, the negligibility notion from the second example (in possibly non-separable spaces) and its variants have been successfully used in [37] to


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show existence of points of Gâteaux differentiability of continuous convex functions on certain non-separable Banach spaces (where the set of points of differentiability need not be Borel).

4. Differentiability We will consider the question of existence of points of Fréchet and Gâteaux differentiability of continuous convex functions and of Lipschitz mappings between Banach spaces. For definitions, basic information and, in particular, for the concept of Radon–Nikodým property of a Banach space X and the result that all Lipschitz mappings of finite-dimensional spaces into X are differentiable (at least at one point or, equivalently, almost everywhere) see Section 7 of [21].

4.1. Convex functions The study of differentiability problems for continuous convex functions is greatly simplified by several facts (cf. [21]): the sets of points of Gâteaux as well as of Fréchet differentiability are Gδ (the latter even in non-separable spaces), if one term of a sum of such functions is (Gâteaux or Fréchet) non-differentiable at x, then the sum is non-differentiable at x, at every point the one-sided directional derivatives exist and form a convex continuous and positively 1-homogeneous function of the directions, hence Gâteaux differentiability at x is equivalent to the requirement that f (x + th) + f (x − th) − 2f (x) = o(t) as t → 0 and Fréchet differentiability is equivalent to the requirement that f (x + h) + f (x − h) − 2f (x) = o(h) as h → 0, the subdifferential ∂f (x) = {x ∗ ∈ X∗ : x ∗ (u) f (x + u) − f (x) for all u ∈ X} is non-empty and differentiability has a simple description as a property of the subdifferential: f is Gâteaux differentiable at x if and only if its subdifferential at x is a singleton and f is Fréchet differentiable at x if and only if the multi-valued mapping y → ∂f (y) is single valued and norm continuous at x, i.e., for every ε > 0 there is δ > 0 such that ∂f (y) ⊂ B(x ∗ , ε) for all y ∈ B(x, δ) and x ∗ ∈ ∂f (x). A number of results on convex functions has been generalized to statements about monotone operators: a mapping T of a set E ⊂ X to the family of non-empty subsets of X∗ is called monotone if (y ∗ − x ∗ )(y − x) 0 for all x, y ∈ E, x ∗ ∈ T (x), and y ∗ ∈ T (y). (Note that some authors require E = X but allow T (x) to be empty.) Basic properties and references to situations in which they play a significant role may be found in [41]. Important examples of monotone operators are provided by subdifferentials of continuous convex functions. Standard results on convex functions have their counterpart in the theory of monotone operators. For example, the simple but useful fact that continuous convex functions are locally Lipschitz may be obtained as a corollary of the fact that monotone operators on open sets are locally bounded. Another direction in which the subdifferential approach may be understood is via selection theorems. It is easy to see that the mapping T (x) = ∂f (x) is weak∗ upper semicontinuous (i.e., the set {x: T (x) ⊂ G} are open for every weak∗ open G ⊂ X∗ ) and has non-empty weak∗ compact values (we abbreviate both these properties of T by saying

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that T is weak∗ usco). If ξ : X → X∗ is a selection for T (i.e., ξ(x) ∈ T (x) for all x) which is norm-to-weak∗ continuous at x, then f is Gâteaux differentiable at x; if it is even norm-to-norm continuous, then f is Fréchet differentiable at x. Since the T given by the subdifferential is locally bounded, one may obtain differentiability results from purely topological statements on the existence of selections of weak∗ usco mappings of, say, topological spaces having the Baire property into the unit ball of X∗ . This approach gives also results for monotone operators, since maximal monotone operators an open sets are (locally bounded and) weak∗ usco. The price paid for the higher generality of the approach is weaker information about the size of the set of points of differentiability; in the selection approach one may hardly expect stronger exceptional sets than those of the first category. 4.1.1. Gâteaux differentiability of convex functions The remarkable results of [59] give a complete description of the size of sets of points of Gâteaux non-differentiability of continuous convex functions. They considerably strengthen a series of previous infinitedimensional results starting with Mazur as well as more detailed previous results in the finite-dimensional case. The set of points of Gâteaux non-differentiability of an arbitrary continuous convex function on a separable Banach space X can be covered by countably many δ-convex hypersurfaces. Conversely, for every set E contained in countably many δ-convex hypersurfaces there is a continuous convex function on X which is Gâteaux nondifferentiable at every point of E. Note that in case dim(X) = 1 we recover the classical statement that the sets of points of non-differentiability of convex functions on R are exactly countable sets. The natural generalization of the question answered in the previous paragraph is the study of those points at which the subdifferential is large. Again, [59] gives a complete answer, which reduces to the previous statement if k = 1: for an arbitrary continuous convex function on a separable Banach space the set of those x at which the dimension of the affine span of the subdifferential is at least k can be covered by countably many k-codimensional δ-convex hypersurfaces. Conversely, for every set E contained in countably many k-codimensional δ-convex hypersurfaces there is a continuous convex function on X such that for every x ∈ E the dimension of the affine span of ∂f (x) is at least k. To indicate the way in which this is proved, let N denote the set of points x for which the dimension of the affine span of ∂f (x) is at least k. Given a k-dimensional subspace U of X, u∗ ∈ U ∗ , and ε > 0, the set B of those x ∈ X for which there is x ∗ ∈ ∂f (x) extending u∗ such that x ∗ < 1/ε and f (x + h) − f (x) x ∗ (h) + εh for all h ∈ U is a subset of N ; moreover, by separability, N is a countable union of such sets B. It is therefore enough to consider one such set B and show that it is covered by a k-codimensional δ-convex hypersurface (which still needs work). To show the converse it suffices to consider the case of one hypersurface; then one can define the required convex function using the convex functions describing the hypersurface. The above discussion of Gâteaux differentiability may be modified to show that for any monotone operator T on an open subset of a separable Banach space the set of points at which T (x) contains at least k affinely independent elements can be covered by countably many k-codimensional Lipschitz hypersurfaces. Applying this fact to the subdifferential of a convex function gives, however, only a weaker version of the above results.


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Let us briefly mention some points of the non-separable theory; for more information see [16,18] or [41]. X is said to be a weak Asplund space if every continuous convex function on X is Gâteaux differentiable on a residual set and a Gâteaux differentiability space, if every continuous convex function on X is Gâteaux differentiable at least at one point (or, equivalently, on a dense set). An interesting class of weak Asplund spaces which includes all separable spaces as well as spaces admitting a Gâteaux smooth norm (or just a Gâteaux differentiable Lipschitz bump function) is formed by those Banach spaces whose dual unit ball B ∗ satisfies the condition (of nature similar to topological descriptions of Radon–Nikodým property of duals as in 4.1.2(e)) that inthe following ‘fragmentability’ game the second player has a strategy guaranteeing that ∞ k=1 Fk ∩ Gk is a singleton: the first player starts by choosing a non-empty weak∗ closed subset F1 of B ∗ , then the second player chooses a weak∗ open set G1 such that F1 ∩ G1 = ∅, then the first player chooses a weak∗ closed subset F2 of F1 such that F2 ∩ G1 = ∅, then the second player chooses a weak∗ open set G2 such that F2 ∩ G2 = ∅, etc. This implies that B ∗ belongs to the so called Stegall’s class, i.e., has the property that any weak∗ usco mapping of a Hausdorff topological space Z to B ∗ has a selection which is weak∗ continuous on a residual set. With our definition, it can be shown by considering a minimal usco mapping of Z to B ∗ and showing that it is single-valued on a residual set with the help of the Banach–Mazur game in Z. (No use of the fact that B ∗ is a ball has been made; the argument works in any compact Hausdorff space satisfying the fragmentability condition.) Recent papers of Kalenda [23,24] and of Kenderov [26] show that these classes are different and do not coincide with weak Asplund spaces. Very recently Moors and Somasundaram [37] used the hyperspace based notions of negligibility mentioned at the end of Section 3 to answer the key open problem of the theory by producing a Gâteaux differentiability space that is not weak Asplund. 4.1.2. Fréchet differentiability of convex functions We have seen in [21] which separable spaces have the property that every continuous convex function on them is Fréchet differentiable on a dense Gδ set; they are precisely those whose dual is separable. Other characterizations follow from this, and similarly satisfactory results hold in non-separable setting. Banach spaces in which continuous convex functions have points of Fréchet differentiability are called Asplund spaces. They are characterized by any of the following equivalent properties: (a) Every continuous convex function on X has a point of Fréchet differentiability. (b) Every continuous convex function on X is Fréchet differentiable on a dense Gδ set. (c) The dual of every separable subspace of X is separable. (d) X∗ has the RNP. (e) For every non-empty bounded set E ⊂ X∗ and every ε > 0 there is a weak∗ open set S meeting E such that S ∩ E has diameter less than ε. (f) For every non-empty weak∗ compact convex E ⊂ X∗ and every ε > 0 there are u ∈ X and δ > 0 such that the weak∗ slice S(E, u, δ) = {x ∗ ∈ E: x ∗ (u) supy ∗ ∈E y ∗ (u) − δ} has diameter less than ε. The equivalence of (a) and (b) follows from the set of points of Fréchet differentiability being Gδ ; see [21]. From the negation of one of the statements (c)–(e) one may prove the negation of (f); if then E is a non-empty weak∗ compact convex set without small weak∗ slices, the function f (x) = supx ∗ ∈E x ∗ (x) is nowhere Fréchet differentiable, and so is the sum of the original norm with f (x) and f (−x). In this way, we even see that every non-Asplund space admits an equivalent norm satisfying lim suph→0 (x + h + x − h − 2x)/h > ε for

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some ε > 0; such norms are sometimes called ε-rough. To prove, say, (e) ⇒ (b), one may use (e) to deduce that every weak∗ usco mapping of a Hausdorff space Z to B ∗ has a selection norm continuous on a residual set; this is a purely topological statement whose proof and use is similar to what was described in 4.1.1 for Gâteaux differentiability. Another possibility is to prove the statement in the separable case only (see later), and use the method of separable reduction. (For a general approach to this method see [10].) The question of describing the size of the set of points of Fréchet differentiability of continuous convex functions is not yet fully understood even in the separable case. However, we have: the set of points of Fréchet non-differentiability of any continuous convex functions on a separable Asplund space is σ -porous. To see this, let xk∗ be dense in the dual and let Fk,l be the set of those x in whose neighbourhood the Lipschitz constant of f does not exceed l and for which there is x ∗ ∈ ∂f (x) such that xk∗ − x ∗ < 1/ l and there are arbitrarily small h such that f (x + h) − f (x) − x ∗ (x) > 4h/ l. Then for any y ∈ B(x + h, h/ l 2 ) and any y ∗ ∈ ∂f (y) we have y ∗ (h) f (y) − f (y − h) / Fk,l . Hence f (x + h) − f (x) − 2h/ l xk∗ (h) + h/ l; so xk∗ − y ∗ 1/ l and y ∈ B(x + h, h/ l 2 ) ∩ Fk,l = ∅, which shows that Fk,l is porous. The proof is finished by observing that the set of points of Fréchet non-differentiability is covered by the union of Fk,l . The above argument may be modified to show that for every monotone operator on a Banach space with a separable dual there is a porous set outside of which the operator is single-valued and norm-to-norm upper semi-continuous. In both these results the porosity may be strengthened (to so-called σ -cone porosity, in which the holes, instead of balls, may be cones); for details see [41]. Although these notions are still some way from a description of the size of sets of Fréchet non-differentiability of convex functions, there is a (very strong) porosity condition which enables the construction of badly differentiable functions: if X is uniformly convex, rn 5 0 and E ⊂ X is such that for every x ∈ E and every λ > 1 there are zn such that zn − x < λrn and B(zn , rn /λ) ∩ E = ∅, then there is a continuous convex function on X which is Fréchet non-differentiable at every point of E. The basic idea in construction of such a function is (assuming that E ⊂ B(0, 1)) to consider the supremum of all affine function majorized by the restriction of a uniformly convex function to E ∪ (X \ B(0, 2)). A similar construction (together with adding the functions constructed for a sequence of such sets E) was used by Matoušková [33] to show that on every super-reflexive separable space there is an equivalent norm whose set of points of Fréchet differentiability is Aronszajn null. This example, in particular, shows that the results mentioned so far are not strong enough to produce, given a convex continuous function g on a separable Asplund space X and a Lipschitz mapping f of X to an RNP space Y , one point x ∈ X at which g would be Fréchet differentiable and f Gâteaux differentiable. This question was answered by [31]: the set of points of Fréchet non-differentiability of any convex continuous function on a separable Asplund space is Γ -null. (See 4.2.1 for results on Gâteaux differentiability implicitly alluded to here, and 4.2.2 for a more general version of this statement.) Note that the incompatibility of Γ -null sets and σ -porous sets (which probably carries over to σ -cone porous sets) makes it is difficult to conjecture how a characterization of sets of Fréchet non-differentiability may look like.


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4.2. Lipschitz functions One of the problems guiding the development of differentiability results for Lipschitz mappings is the Lipschitz isomorphism problem: if X, Y are Lipschitz isomorphic (i.e., if there is a bijection f : X → Y such that both f and f −1 are Lipschitz) are they linearly isomorphic? Much of the theory described below can be, and has been, successfully used to obtain partial answers. The basic idea is that the derivative of f should provide such an isomorphism. For this to work, one has to assume additional properties of X, Y such as the RNP (or just reflexivity, or even just super-reflexivity), since otherwise f may be nowhere differentiable. However, at the present time any direct use of this program comes to the obstacle caused by the open problem whether the Gâteaux derivative of a Lipschitz isomorphism of 2 onto itself is surjective at least at one point. Another approach to the Lipschitz isomorphism problem is based on the observation that a Gâteaux derivative of a Lipschitz isomorphism f of X to Y is surjective provided that all compositions y ∗ ◦ f are Fréchet differentiable for all y ∗ ∈ Y ∗ . In fact, a dense set of y ∗ suffices, which gives a good reason why we care so much about the problem of finding a common point of Gâteaux differentiability of a Lipschitz mapping and of Fréchet differentiability of countably many real-valued Lipschitz functions. In these arguments Fréchet derivative may be replaced by almost Fréchet derivative (see 4.2.4). The weakening of the concept of Fréchet derivative can be pushed even further, to the so called affine approximation property of [4]; we will not consider these results here, but mention that in the super-reflexive case most applications of almost Fréchet differentiability (see 4.2.4) to the Lipschitz isomorphism problem may be also obtained with the help of the uniform version of this property. We should recall that differentiability results for general Lipschitz functions cannot be obtained using sets of the first category, as there are Lipschitz functions f : R → R which are differentiable only on first category sets. In fact, on the real line these results cannot be obtained by any means weaker than the Lebesgue measure, since for every set N ⊂ R of Lebesgue measure zero there is a Lipschitz f : R → R which in non-differentiable at all points of N ; moreover, by [57] the sets of non-differentiability of Lipschitz functions R → R are characterized as Gδσ sets of Lebesgue measure zero. 4.2.1. Gâteaux differentiability of Lipschitz functions The question how small are the sets of points of Gâteaux non-differentiability of Lipschitz functions does not have a complete answer yet; it is not even known if in Rn (n 3) the σ -ideal generated by the sets of non-differentiability of real-valued Lipschitz functions coincides with the Lebesgue null sets. Nevertheless, the fact that locally Lipschitz mappings of separable spaces into RNP spaces are Gâteaux differentiable outside Haar null sets mentioned in [21] or the following stronger result are sufficient for a number of purposes: every locally Lipschitz mapping of a separable space into an RNP space is Gâteaux differentiable outside an Aronszajn null set. There are several remarkably simple proofs of this statement: the basic idea is that, if f : X → Y and un ∈ X have a dense span, the sets En of those x ∈ X for which the directional derivative f (x, un ) = limt →0 (f (x + tun ) − f (x))/t does not exist are Borel and, by the RNP of Y , the intersection of En with any line in the direction un is of onedimensional Lebesgue measure zero. It remains to show that the set of points of Gâteaux

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non-differentiability at which directional derivatives exist in a spanning set of directions is Aronszajn null; this set is reasonably well understood, since: if f is a locally Lipschitz mapping of a separable space X to Y , then the set of those x ∈ X at which f has the directional derivative in a spanning set of directions but is not Gâteaux differentiable is σ -directionally porous. To see this, one shows, for every u, v ∈ X, y, z ∈ Y and ε, δ > 0, the directional porosity of the set E of those x0 ∈ X such that Lip(f, B(x0 , δ)) < 1/ε, f (x0 + tu) − f (x0 ) − ty + f (x0 + tv) − f (x0 ) − tz ε|t| for |t| < δ and there are arbitrarily small |s| such that f (x0 + s(u + v)) − f (x0 ) − s(y + z) > 4ε|s|: if |s| is small and x − (x0 + su) < ε2 |s|, then f (x + sv) − f (x) − sz f (x0 + s(u + v)) − f (x0 ) − s(y + z) − f (x + sv) − f (x0 + s(u + v)) − f (x0 + su) − f (x0 ) − sy − / E. The set in question is covered by countably many f (x0 + su) − f (x) > ε|s|, so x ∈ such sets E since it suffices to consider u, v from a dense countable subset of X, y, z from a dense countable subset of the span of f (X) and rational ε, δ. Intriguing questions are obtained when one attempts to use these results to answer the Lipschitz isomorphism problem. The Gâteaux derivative of a Lipschitz isomorphism f of a separable Banach space X onto an RNP space Y , whenever it exists, is a linear isomorphism onto a closed subspace of Y (so X has the RNP as well). This subspace is complemented if, e.g., Y is reflexive (see [5]). Nevertheless, the following key problem is still open: if f is a Lipschitz isomorphism of 2 onto itself, is there a point at which its Gâteaux derivative is a linear isomorphism of 2 onto itself? One may hope that the Gâteaux derivative of any Lipschitz isomorphism between RNP spaces is, at least at one point, a linear isomorphism between them; this more general version of the problem is open as well. Since it is easy to see that if a Lipschitz isomorphism f of X onto Y is Gâteaux differentiable at x and f −1 at f (x), then Df (x) is a linear isomorphism of X onto Y , a positive answer would be obtained if the null sets with respect to which one has the differentiability theorem were invariant under Lipschitz isomorphisms. We have, however, pointed out in Section 3 that this is not the case for Haar null nor for Aronszajn null sets. Other problems on invariance of null sets under Lipschitz isomorphisms treated in Section 3 have been also motivated by the Lipschitz isomorphism problem. For any given notion of null sets, the worst examples would be of the situation when a complement of a null set is mapped onto a set of Gâteaux non-differentiability of some Lipschitz function; such examples are not known even for Haar null sets (and so also not for Aronszajn null sets). Curiously enough, if the Lipschitz isomorphism f : X → Y has all one-sided directional derivatives, then the image of the set at which g : X → Z (with RNP Z) is not Gâteaux differentiable is even Aronszajn null: the image of the set of points at which g is non-differentiable at some direction is contained in the set of non-differentiability of g ◦ f −1 , and the remaining part of the set of non-differentiability points of g is σ -directionally porous, so its f image is also σ -directionally porous, since Lipschitz isomorphisms having one-sided directional derivatives map σ -directionally porous sets to σ -directionally porous sets. The Lipschitz isomorphism problem may well require a strengthening of the above results on Gâteaux differentiability. This motivates the quest for finding smaller σ -ideals of sets for which the differentiability statement still holds (and is genuinely stronger than the use of Aronszajn null sets). The non-linear concepts of Aronszajn null sets briefly discussed in Section 3 provide such σ -ideals. From these results (or directly) is is also easy to see that every Lipschitz function f from a separable Banach space X to an RNP space


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Y is Gâteaux differentiable Γ -almost everywhere. Note again that for these σ -ideals the problem of invariance under Lipschitz isomorphisms is open. 4.2.2. Fréchet differentiability of Lipschitz functions Until recently, the only general result on Fréchet differentiability of Lipschitz mappings, except the case of finitedimensional domain where the concepts of Gâteaux and Fréchet differentiability coincide, was that every Lipschitz mapping f of an Asplund space X to R has points of Fréchet differentiability; a small generalization (for X separable) assumes only the weak∗ closure of the set of of Gâteaux derivatives of f norm separable. It is immediate to deduce from this that the set of points of Fréchet differentiability must be uncountable in every non-empty open set, and stronger information on the size of this set can be obtained by use of the mean value estimate (see 4.2.3). The original proof of the Fréchet differentiability result is rather involved [44]; a simpler (but not simple) proof from [30] is based on the following ideas (we assume X separable): denote by Df (E) the set of all Gâteaux derivatives attained at points of E. Let E1 be a ball of radius one, W1 = Df (E1 ) and let u1 ∈ X be such that the slice S(W1 , u1 , δ1 ) has a small diameter. One can show that there is η1 > 0 such that whenever Df (x) ∈ S(W1 , u1 , η1 ), then lim suph→0 |f (x + h) − f (x) − Df (x)(h)|/h is small. Then one defines E2 as the set of those x from the intersection of E1 and a ball with radius 1/2 at which Df (x)(u1 ) is large and the increments in the direction u1 are uniformly controlled (the real difficulty comes at this point; keeping this control is enabled by an involved estimate of behaviour of derivatives in the plane), and we continue in a similar way requiring now that u2 is close to u1 , etc. The limit of the sequence xn ∈ En is the required point. Even from this rough description it should be clear that this approach shows that every slice S(Df (X), u, δ) (u ∈ X) of the set of Gâteaux derivatives of f contains a Fréchet derivative. One of the main difficulties in proving Fréchet differentiability results, say, for mappings of 2 to finite-dimensional spaces is that the analogous slicing statement is false: by a (complicated) example of [46] there is a Lipschitz mapping f = (f1 , f2 , f3 ) : 2 → R3 such that Df1 (e1 ) + Df2 (e2 ) + Df3 (e3 ) = 0 at every point of Fréchet differentiability of f , but not at every point of Gâteaux differentiability. (Except for understandable misprints, this example, by a computer quirk, uses the meaningless symbol “ ” for (π0 z/rm ).) Any attempt to prove Fréchet differentiability almost everywhere (or even existence of a common point of differentiability of finitely many real-valued functions) is greatly hampered by the fact that there may exist slices of the set of Gâteaux derivatives of f containing no Fréchet derivative. This, however, cannot happen for convex functions. The reason behind this is that they are regular in the following sense: a mapping f : X → Y is called regular at a point x if for every v ∈ X for which the directional derivative f (x, v) exists, (x+t u) limt →0 f (x+t (u+v))−f = f (x, v) uniformly in u with u 1. The key statement on t Fréchet differentiability of Lipschitz mappings with respect to Γ -null sets says (see [31]): if L is a norm separable subspace of the space of linear operators between separable Banach spaces X and Y , then every Lipschitz mapping f : X → Y is Fréchet differentiable at Γ -almost every point of the set at which it is regular, Gâteaux differentiable and its Gâteaux derivative belongs to L. The proof is hard and draws on much of what has been done before. The basic new ingredient comes from the classical descriptive set theory: assuming, for simplicity, that f is Gâteaux differentiable Γ almost everywhere, we observe

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that the mapping γ → Df ◦ γ , being a Borel measurable mapping between complete separable metric spaces Γ (X) and L1 (Σ, L), becomes continuous when restricted to a suitable residual set. To return to general Lipschitz mappings, we observe that the sets of points of their irregularity are σ -porous. Hence every Lipschitz mapping of f : X → R is Fréchet differentiable Γ -almost everywhere if and only if every σ -porous set is Γ -null. The condition of σ -porous sets being null does not hold in 2 (as we will see in 4.2.3, not all Lipschitz f : 2 → R are Fréchet differentiable Γ -almost everywhere) but it can be proved in spaces whose structure is similar to c0 (for example, for spaces containing an asymptotically c0 sequence of finite co-dimensional subspaces). The basic method of avoiding porous sets is to modify a given γ ∈ Γ (X) close to a point at which it belongs to a given porous set so that it passes through a hole. Unfortunately, the resulting sequence of so modified γn ∈ Γ (X) may not converge (in the space Γ (X)). However, in the presence of a c0 structure we can make the modification on disjoint sets of coordinates and so achieve the convergence. These argument then give that if X is a subspace of c0 , or a space C(K) with K countable compact, or the Tsirelson space, then all the σ -porous subsets of X are Γ -null; hence all real-valued Lipschitz functions on these spaces are Fréchet differentiable Γ -almost everywhere. In fact, if X is a subspace of c0 , or C(K) with K countable compact, then the space of bounded linear operators from X to any RNP space Y is separable, and so every Lipschitz mapping between such spaces is Fréchet differentiable Γ -almost everywhere. 4.2.3. Mean value estimates One of the important applications of derivatives or their generalizations is their use to estimate the increment of a function. The model statement is Lebesgue’s variant of the fundamental theorem of calculus saying that for a real-valued b Lipschitz function f of one real variable f (b) − f (a) = a f (t) dt and its corollary, the mean value estimate, that for every ε > 0 there is t ∈ [a, b] such that f (t)(b − a) > f (b) − f (a) − ε. For real-valued Lipschitz functions on a Banach space X one cannot expect that a point of differentiability can be found on the segment [a, b], and so the mean value estimate either uses a point of differentiability close to [a, b] or replaces the derivative by its generalization (this approach will not be used here). The mean value estimate for Gâteaux derivatives follows immediately from the fact that every Haar null set is null with respect to linear measure on a dense set of lines. In fact this gives a stronger statement: if X is separable, G ⊂ X is open, N ⊂ X is Haar null, and f : G → R is Lipschitz, then for every segment [a, b] ⊂ G and every ε > 0 there is x ∈ G \ N such that Df (x)(b − a) > f (b) − f (a) − ε. Since no almost everywhere result is known for Fréchet derivatives, the mean value estimate for them is proved by following more carefully the construction of points of differentiability: if X is an Asplund space, G ⊂ X is open, and f : G → R is Lipschitz, then for every segment [a, b] ⊂ G and every ε > 0 there is x ∈ G at which f is Fréchet differentiable such that Df (x)(b − a) > f (b) − f (a) − ε. (As in the existence result, it suffices to assume that the weak∗ closure of the set of Gâteaux derivatives of f is norm separable.) The mean value estimate may be used to show that the set of points of Fréchet differentiability of these mappings cannot be too small: if any of its projections on R were not of full outer Lebesgue measure, we would find a non-constant everywhere differentiable Lipschitz function ϕ on R having derivative zero at the projection of every point of Fréchet differentiability of f ; adding to


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f a large multiple of the composition of ϕ and the projection would produce a function violating the mean value estimate. If X is separable, this shows that any one-dimensional projection of the set of points of Fréchet differentiability is of full Lebesgue measure (since it is measurable). It is not known if an analogous statement holds also for two-dimensional projections. A higher-dimensional version of the mean value statement for Fréchet derivative of a mapping f of X to a finite-dimensional space may be understood as the statement that every slice of the set of Gâteaux derivatives of f contains a Fréchet derivative. This holds for mappings which are Fréchet differentiable Γ -almost everywhere. (Basically, one considers a γ representing a small finite-dimensional parallelepiped on which the mapping is well approximated by a linear mapping belonging to the slice; for a slight modification of γ one gets Fréchet differentiability μ-almost everywhere and, by the divergence theorem, the mean of the derivative changes only as little as we wish.) Because of this and of the example of [46] (which was already mentioned above), we see that Fréchet differentiability Γ -almost everywhere is false for real-valued Lipschitz functions on 2 . 4.2.4. Almost Fréchet derivative It has been already mentioned that for the Lipschitz isomorphism problem notions of derivatives weaker than Fréchet derivative may be pertinent. A function f : X → Y is called almost Fréchet differentiable is for every ε > 0 there are x ∈ X and a bounded linear operator T (both x and T may depend on ε) such that lim supu→0 f (x + u) − f (x) − T (u)/u < ε. In [29] these derivatives were shown to exist for mappings of supper-reflexive spaces to finite-dimensional spaces (by a rather involved proof). This result was extended in [22], with a more transparent proof, to the case of asymptotically uniformly smooth spaces; this paper should be consulted for details and applications. 4.2.5. Weak∗ derivative For a Lipschitz mapping f of a separable Banach space X to the dual of a separable space Y one defines the weak∗ directional derivative of f at x in (x) direction u as the weak∗ limit, as t → 0, of f (x+t u)−f . The weak∗ Gâteaux different tiability of f at x is defined by requiring that this mapping be linear in u. The existence results for Gâteaux derivatives of Lipschitz functions hold also in this setting (and do not need any RNP requirement). Of course, some of the properties of Gâteaux derivatives are lost; in particular, the weak∗ Gâteaux derivative of a Lipschitz isomorphism may well be zero at some points. However, mean value estimates still hold, so these derivatives are not trivial; and, starting from [32] and [19] have been successfully used to study the Lipschitz isomorphism problems for spaces without Radon–Nikodým property. 4.2.6. Metric derivative The standard example of a nowhere differentiable Lipschitz mapping of (0, 1) to L1 (0, 1), given by f : x → χ(0,x) where χE denotes the indicator function of the set E, is an isometry. This is not just a chance, since the one-dimensional case of the following result due to Kirchheim [27] says that every Lipschitz mapping of (0, 1) to a metric space locally (near to a.e. point) multiplies the distance by a constant as if it were differentiable (no RNP type condition on of the range is needed). If f is a Lipschitz mapping of an open subset of Rn to a metric space, then for a.e. x ∈ Rn there is a seminorm · x on Rn such that limt →0 -(x + tu, x + tv)/t = u − vx for all u, v ∈ Rn . (For an

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application see 1.1.) A new proof of this statement, which relates it to differentiability of Banach space valued mappings has been found in [1]. We may assume that the target is the dual of a separable Banach space. Then f is weak∗ differentiable almost everywhere, and it is natural to assume that ux = Df (x)(u) is the required seminorm; this can in fact be shown by decomposing Rk into countably many sets in which the weak∗ derivative does not oscillate much (the oscillation is measured in a metric metrizing the weak∗ topology of a ball in E) and using the density theorem together with mean value estimates.

References [1] L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (3) (2000), 527–555. [2] L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math. 185 (1) (2000), 1–80. [3] N. Aronszajn, Differentiability of Lipschitz mappings between Banach spaces, Studia Math. 57 (1976), 147–190. [4] S. Bates, W.B. Johnson, J. Lindenstrauss, D. Preiss and G. Schechtman, Affine approximation of Lipschitz functions and nonlinear quotients, Geom. Funct. Anal. 19 (1999), 1092–1127. [5] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. I, Amer. Math. Soc. Colloq. Publ. 48, Amer. Math. Soc., Providence, RI (2000). [6] V.I. Bogachev, Negligible sets in locally convex spaces, Mat. Zametki 36 (1984), 51–64. English transl.: Math. Notes 36 (1984), 519–526. [7] V.I. Bogachev, Some results on differentiable measures, Mat. Sb. (N.S.) 127(169) (3) (1985), 336–351 (in Russian). [8] V.I. Bogachev, Gaussian Measures, Math. Surveys Monographs 62, Amer. Math. Soc., Providence, RI (1998). [9] C. Borell, A note on Gauss measures which agree on small balls, Ann. Inst. H. Poincaré 13 (1977), 231–238. [10] J.M. Borwein and W.B. Moors, Separable determination of integrability and minimality of the Clarke subdifferential mapping, Proc. Amer. Math. Soc. 128 (1) (2000), 215–221. [11] J.M. Borwein, W.B. Moors and X. Wang, Generalized subdifferentials: a Baire categorical approach, Trans. Amer. Math. Soc. 353 (10) (2001), 3875–3893. [12] J.P.R. Christensen, On sets of Haar measure zero in Abelian Polish groups, Israel J. Math. 13 (1972), 255– 260. [13] M. Csörnyei, Aronszajn null and Gaussian null sets coincide, Israel J. Math. 111 (1999), 191–202. [14] R.O. Davies, Measures not approximable or not specifiable by means of balls, Mathematika 18 (1971), 157–160. [15] U. Dinger, Measure determining classes of balls in Banach spaces, Math. Scand. 58 (1986), 23–34. [16] M. Fabian, Gâteaux Differentiability of Convex Functions and Topology, Canad. Math. Soc. Monographs, Wiley-Interscience (1997). [17] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York (1969). [18] P. Habala, P. Hájek and V. Zizler, Introduction to Banach Spaces I and II, Univerzita Karlova, Praha (1996). [19] S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), 225–251. [20] S. Jackson and R.D. Mauldin, On the σ -class generated by open balls, Math. Proc. Cambridge Philos. Soc. 127 (1) (1999), 99–108. [21] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. [22] W.B. Johnson, J. Lindenstrauss, D. Preiss and G. Schechtman, Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math. Soc. (3) 84 (2002), 711–746. [23] O. Kalenda, Stegall compact spaces which are not fragmentable, Topology Appl. 96 (2) (1999), 121–132.


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[24] O. Kalenda, A weak Asplund space whose dual is not in Stegall’s class, Proc. Amer. Math. Soc. 130 (7) (2002), 2139–2143. [25] T. Keleti and D. Preiss, The balls do not generate all Borel sets using complements and countable disjoint unions, Math. Proc. Cambridge Philos. Soc. 128 (2000), 539–547. [26] P.S. Kenderov, W.B. Moors and S. Sciffer, A weak Asplund space whose dual is not weak* fragmentable, Proc. Amer. Math. Soc. 129 (12) (2001), 3741–3747. [27] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113–123. [28] J. Lindenstrauss, E. Matoušková and D. Preiss, Lipschitz image of a measure null set can have a null complement, Israel J. Math. 118 (2000), 207–219. [29] J. Lindenstrauss and D. Preiss, Almost Fréchet differentiability of finitely many Lipschitz functions, Mathematika 86 (1996), 393–412. [30] J. Lindenstrauss and D. Preiss, A new proof of Fréchet differentiability of Lipschitz functions, J. Eur. Math. Soc. 2 (2000), 199–216. [31] J. Lindenstrauss and D. Preiss, On Fréchet differentiability of Lipschitz maps between Banach spaces, Ann. Math. 157 (2003), 257–288. [32] P. Mankiewicz, On the differentiability of Lipschitz mappings in Fréchet spaces, Studia Math. 45 (1973), 15–29. [33] E. Matoušková, An almost nowhere Fréchet smooth norm on superreflexive spaces, Studia Math. 133 (1999), 93–99. [34] E. Matoušková, Translating finite sets into convex sets, Bull. London Math. Soc. 33 (6) (2001), 711–714. [35] E. Matoušková and C. Stegall, A characterization of reflexive Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 1083–1090. [36] L. Mejlbro, D. Preiss and J. Tišer, Determination and differentiation of measures, in preparation. [37] W.B. Moors and S. Somasundaram, A Gâteaux differentiability space that is not weak Asplund, submitted. [38] N.V. Norin and O.G. Smolyanov, Some results on logarithmic derivatives of measures on a locally convex space, Mat. Zametki 54 (6) (1993), 135–138. English transl.: Math. Notes 54 (5–6) (1993), 1277–1279. [39] A. Pelczynski and S.J. Szarek, On parallelepipeds of minimal volume containing a convex symmetric body in Rn , Math. Proc. Cambridge Philos. Soc. 109 (1991), 125–148. [40] R.R. Phelps, Gaussian null sets and differentiability of Lipschitz mappings on Banach spaces, Pacific J. Math. 77 (1978), 523–531. [41] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed., Lecture Notes in Math. 1364, Springer, New York (1993). [42] D. Preiss, Gaussian measure and the density theorem, Comment. Math. Univ. Carolin. 22 (1981), 181–193. [43] D. Preiss, Differentiation of measures in infinitely dimensional spaces, Proc. Conf. in Topology and Measure III, Greifswald (1982), 201–207. [44] D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), 312–345. [45] D. Preiss and J. Tišer, Measures on Banach spaces are determined by their values on balls, Mathematika 38 (1991), 391–397. [46] D. Preiss and J. Tišer, Two unexpected examples concerning differentiability of Lipschitz functions on Banach spaces, GAFA Israel Seminar 92–94, V.D. Milman and J. Lindenstrauss, eds, Birkhäuser (1995), 219–238. [47] D. Preiss and L. Zajíˇcek, Directional derivatives of Lipschitz functions, Israel J. Math. 125 (2001), 1–27. [48] E.A. Riss, The positivity principle for equivalent norms, Algebra i Analiz 12 (3) (2000), 146–172. English transl.: St. Petersburg Math. J. 12 (3) (2001), 451–469. [49] O.G. Smolyanov, De Rham currents and the Stokes formula in Hilbert space, Dokl. Akad. Nauk SSSR 286 (3) (1986), 554–558. [50] O.G. Smolyanov and H. von Weizsäcker, Differentiable families of measures, J. Funct. Anal. 118 (2) (1993), 454–476. [51] O.G. Smolyanov and H. von Weizsäcker, Change of measures and their logarithmic derivatives under smooth transformations, C.R. Acad. Sci. Paris Sér. I Math. 321 (1) (1995), 103–108. [52] A.V. Skorochod, Integration in Hilbert Spaces, Nauka, Moscow (1975) (Russian). [53] S. Solecki, On Haar null sets, Fund. Math. 149 (1996), 205–210.

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[54] E.M. Stein and J.O. Strömberg, Behavior of maximal functions in Rn for large n, Ark. Mat. 21 (1983), 259–269. [55] J. Tišer, Differentiation theorem for Gaussian measures on Hilbert space, Trans. Amer. Math. Soc. 308 (1988), 655–665. [56] A.V. Uglanov, Integration on Infinite-Dimensional Surfaces and its Applications, Math. Appl. 496, Kluwer Academic Publishers, Dordrecht (2000). [57] Z. Zahorski, Sur l’ensemble des points de non-derivabilité d’une fonction continue, Bull. Soc. Math. France 74 (1946), 147–178. [58] L. Zajíˇcek, Porosity and σ -porosity, Real Anal. Exchange 13 (1987–88), 314–350. [59] L. Zajíˇcek, On the differentiability of convex functions in finite and infinite dimensional Banach spaces, Czechoslovak Math. J. 29 (1979), 340–348. [60] M. Zelený, The Dynkin system generated by balls in Rd contains all Borel sets, Proc. Amer. Math. Soc. 128 (2) (2000), 433–437. [61] W.P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Math. 120, Springer, New York (1989).

CHAPTER 36

The Banach Spaces C(K) Haskell P. Rosenthal∗ Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082, USA E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. The isomorphic classification of separable C(K)-spaces 3. Some Banach space properties of separable C(K)-spaces 4. Operators on C(K)-spaces . . . . . . . . . . . . . . . . . 5. The complemented subspace problem . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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∗ Research partially supported by NSF Grant DMS-0070547.

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The Banach spaces C(K)

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1. Introduction A C(K)-space is just the space of scalar-valued continuous functions on a compact Hausdorff space K. We focus here mainly on the case where K is metrizable, i.e., the case of separable C(K)-spaces. Our main aim is to present the most striking discoveries about the Banach space structure of C(K)-spaces, and at the same time to describe the beautiful, deep intuitions which underlie these discoveries. At times, we go to some length to describe the form and picture of an argument, without giving the full technical discussion. We have also chosen to present proofs which seem the most illuminating, in favor of more advanced and sophisticated but (to us) less intuitive arguments. The following is a summary of our exposition. Section 2 deals with the by now classical isomorphic classification of the separable C(K)-spaces, dating from the 50’s and 60’s. It begins with Milutin’s remarkable discovery: C(K) is isomorphic to C([0, 1]) if K is an uncountable compact metric space. We give a fully detailed proof, modulo some standard basic facts (summarized in Lemma 2.5), which follows an argument due to Ditor. This yields that every separable C(K)-space is isometric to a contractively complemented subspace of C(D), D the Cantor discontinuum (Theorem 2.4), through a natural inverse limit argument, given in Lemma 2.11 below. The way inverse limits work (in the metrizable setting) is given in Lemma 2.12, and Theorem 2.4 is deduced after this. The isomorphic classification of the C(K)-spaces with separable duals, due to Bessaga and Pełczyński, occupies the balance of this section. Their remarkα able result: the spaces C(ωω +) form a complete set of representatives of the isomorphism classes, over all countable ordinals (Theorem 2.14). We do give a detailed proof that C(K) is isomorphic to one of these spaces, for all countable compact K (of course, we deal only with infinite-dimensional C(K)-spaces here). This is achieved through Theorem 2.24 and Lemma 2.26. We do not give the full proof that these spaces are all isomorphically distinct, although we spend considerable time discussing the fundamental invariant which accomα plishes this, the Szlenk index, and the remarkable result of Samuel: Sz(C(ωω +)) = ωα+1 for all countable ordinals α (Theorem 2.15). We give a variation of Szlenk’s original formulation following 2.15, and show it is essentially the same as his in Proposition 2.17. We then summarize the invariant properties of this index in Proposition 2.18, and give the relα atively easy proof that Sz(C(ωω +)) ωα+1 in Corollary 2.21. We also show in Section 2 how the entire family of spaces C(α+) (up to algebraic isometry) arises from a natural Banach space construction: simply start with c0 , then take the smallest family of commutative C ∗ -algebras containing this, and closed under unitizations and c0 -sums. (This is the family (Yα )1α 0 . Next, fix i, and for f ∈ C( fi (k) = f (k, i)

.n

j =1 Kj ),

if k ∈ Ki ,

define fi on K by (2.12)

fi (k) = 0 if k ∈ Ki .

(2.13)

fi · ϕi

(2.14)

Then is continuous.

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Indeed, since τ maps Ki × {i} homeomorphically into Ki , it follows that fi |Ki is continuous, and so of course (fi · ϕi )|Ki is also continuous. Since ϕi (x) = 0 for all x ∈ Ki and ϕi is continuous on K, it follows that if (xn ) is a sequence in K ∼ Ki such that xn → x with x ∈ Ki , then (fi · ϕi )(xn. ) = 0 for all n and also (fi · ϕi )(x) = 0, proving (2.14). Finally, define T : C( ni=1 Ki ) → C(K) by Tf =

n

fi ϕi

for all f ∈ C

i=1

Then fixing f ∈ C( any k ∈ K

n /

Ki .

(2.15)

i=1

.n

i=1 Ki ),

we have that indeed Tf ∈ C(K) by (2.14). Moreover for

n Tf (k) fi (k)ϕi (k) i=1

maxfi (k) ϕi (k) i

f ∞ .

(2.16)

Thus T = 1 and of course T is linear. Finally, if f ∈ C(K), then for any k, n 0

τ f i (k)ϕi (k) T τ 0 f (k) = i=1

=

f (k)ϕi (k)

= f (k)

n

ϕi (k) = f (k),

(2.17)

i=1

completing the proof.

We need one more tool; inverse limit systems of topological spaces. We just formulate the special case needed here (see Lemma 2 of [23] for the general situation). L EMMA 2.12. Let (Kn )∞ n=1 be a sequence of compact metric spaces, and for each n, let ϕn : Kn+1 → Kn be a given continuous surjection. There exists a compact metrizable space K∞ satisfying the following for all n: There exists a continuous surjection ϕñ : K∞ → Kn ,

(2.18)

ϕn ϕñ+1 = ϕ˜ n .

(2.19)


1557

Letting Yn = ϕ˜ n0 (C(Kn )), then ∞

Yn

is dense in C(K∞ ).

(2.20)

n=1

If moreover ϕn admits a regular averaging operator for each n, then ϕ˜ 1 admits a regular averaging operator. R EMARK . The space K∞ is essentially determined by (2.16) and (2.17) and is called the inverse limit of the system (Kn , ϕn )∞ n=1 . P ROOF. Let K∞ be the subset of (kj ) ∈ K∞

*∞

n=1 Kn

iff kj = ϕj (kj +1 )

defined by for all j.

(2.21)

Of course the axiom of choice yields that K∞ is not only non-empty, but for all n, ϕ˜ n : K∞ → Kn is a surjection, where

ϕ˜ n (kj ) = kn

for any (kj ) ∈ K∞ .

(2.22)

* K∞ is also a closed subset of ∞ n=1 Kn , where the latter is endowed with the Tychonoff topology, it also follows immediately that fixing n, then (2.19) holds. But this implies that Yn ⊂ Yn+1 .

(2.23)

Indeed, say y ∈ Yn and let y = ϕñ0 (f ), for a (unique) f ∈ C(Kn ). But 0

0 ϕ˜ n0 (f ) = f ◦ ϕñ = f ◦ ϕn ◦ ϕñ+1 = ϕ˜ n+1 ϕn f ∈ Yn+1 .

(2.24)

Now it follows that ∞ n=1 Yn is a unital subalgebra of C(K∞ ) separating its points, hence this is dense in C(K∞ ) by the Stone–Weierstrass theorem. Finally, if each ϕn admits a regular averaging operator, then for all n there exists a contractive linear projection Pn : Yn+1 → Yn from Yn+1 onto Yn . It follows that there exists a unique contractive linear projection P : ∞ j =1 Yj → Y1 such that for all n and y ∈ Yn+1 , P (y) = P1 P2 · · · Pn−1 Pn (y).

(2.25)

But then P uniquely extends to a unique contractive projection from C(K∞ ) onto P (Y1 ) by (2.20), completing the proof via Proposition 2.9. Finally, we give the P ROOF OF T HEOREM 2.4. Let K1 = K. We inductively define K2 , K3 , . . . , KN , . . . satisfying the hypotheses of Lemma 2.12.

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H.P. Rosenthal

Step 1. Choose n1 > 1 and W11 , . . . , Wn11 compact subsets of K1 , each with non-empty interior and diameter less than one, such that K=

n1

int Wj1 .

(2.26)

j =1

. Set K1 = nj 1=1 Wj1 ; endow K1 with the metric described in the comments preceding Lemma 2.12, and let ϕ1 : K1 → K be the continuous surjection given by Lemma 2.11. Thus ϕj (Wj1 × {j }) = Wj1 for all j and ϕj admits a regular averaging operator by Lemma 2.11. . Step m. Assume that Km = nj m=1 Wjm has been defined, and fix j , 1 j nm . Thus Wj = Wjm × {j } is a natural clopen subset of Km . Choose kj > 1 and Wj,1 , . . . , Wj,kj compact subsets of Wj , each with non-empty interior, and diameter less than 1/(m + 1), with Wj =

kj

int Wj,i .

(2.27)

i=1

Next, let nm+1 =

nm

j =1 kj

Win+1 = Wψ(i)

and for 1 i nm+1 ,

(2.28)

where ψ : {1, . . . , nm+1 } → {(j, i): 1 i kj , 1 j nm } is a bijection. .n m+1 Set Kn+1 = j m+1 , endow Km+1 with the metric described preceding Lem=1 Wj ma 2.11 and choose ϕm : Km+1 → Km the continuous surjection admitting a regular averaging operator given by Lemma 2.11. Thus in fact ϕm (Wj, × {}) = Wj,i , where ψ() = (j, i) for all j and i and moreover

nm+1

Km =

int Wjm+1 .

(2.29)

j =1

This completes the inductive construction of the Km ’s and ϕm ’s. Lemma 2.12 now yields the existence of a continuous surjection ϕ : K∞ → K admitting a regular averaging operator, where K∞ satisfies the conclusion of Lemma 2.12. It remains only to check that K∞ is perfect and totally disconnected, hence homeomorphic to D. The details for this quite naturally involve the further structure of inverse systems. For each 1 j n, define the map ϕj,n : Kn+1 → Kj by ϕj,n = ϕj ◦ ϕn . Now in our particular construction, we have that for all m and 1 j nm , ϕm maps Wjm × {j } isometrically onto Wjm .

(2.30)

But then it follows that ϕj,m |Wjm ×{j }

is an isometry for all i m and 1 j nm .

(2.31)


1559

Let us endow K∞ with the metric ∞

dj (xj , yj ) , d (xj ), (yj ) = 2j

(2.32)

j =1

where dj is the metric on Kj . Then it follows that for all m and 1 j nm m < diam W j

2 , m

m = ϕ˜ j W m × {j } . where W j j

(2.33)

m . But then there are points x and y in W m × {j } Indeed, suppose (xk ) and (yk ) belong to W j j such that xi = ϕi,m (x) and yi = ϕi,m (y) for all 1 i m, and have by (2.31), m ∞ 1 1 1 1 (x , y ) + + m (2.34) d d(xi , yi ) m i i j j 2 2 m 2 j =1

j =m+1

since diam Wjm < 1/m. m : 1 j nm ; m = 1, 2, . . .} is a It then follows that the family of clopen subsets {W j base for the topology of K∞ . Hence K∞ is totally disconnected. Finally, our insistence that at each stage m we “label” at least 2 sets contained in Wjm × {j } (i.e., kj > 1), insures that m contains at least two points for all m and 1 j nm , whence K∞ is indeed perfect. W j B. C(K) spaces with separable dual via the Szlenk index Of course C(K)∗ is separable if and only if K is infinite countable compact metric. It is a standard result in topology that every such K is homeomorphic to C(α+) for some countable ordinal α ω. We use standard facts about ordinal numbers. An ordinal α denotes the set of ordinals β with β < α; α+ denotes α + 1. Finally, for α a limit ordinal, C0 (α) denotes the space of continuous functions on α vanishing at infinity, which of course can be identified with {f ∈ C(α+): f (α) = 0}. The Banach spaces C(ωα +) arise quite naturally upon applying a natural inductive construction to c = C(ω+) (the space of converging sequences). Indeed, for any locally compact Hausdorff spaces X1 , X2 , . . . , we have that

def C0 (X1 ) ⊕ C0 (X2 ) ⊕ · · · c = Y 0

(2.35)

. is again algebraically isometric to C0 (X) where X = ∞ j =1 Xj , the “direct sum” of the spaces X1 , X2 , . . . . Of course then Y has a unique “unitization” as the space of continuous functions on the one point compactification of X, which we’ll denote by Y ⊕ [1]. The norm here is quite explicitly given as (2.36) y ⊕ c · 1 = sup sup yj (ω) + c, j ω∈Xj

1560

H.P. Rosenthal

where y = (yj ) ∈ Y . Now define families of C(K) spaces (Yα )1α 0. We define a derivation dε on the ω∗ -compact subsets K of X∗ as follows: Let δε (K) denote the set of all x ∗ ∈ K such that there exists a sequence (xn∗ ) in K with xn∗ → x ∗ ω∗ and xn∗ − x ∗ ε for all n. (2.38)


1561

Now define: dε (K) = δε (K)

ω∗

.

(2.39)

Now define a transfinite descending family of sets Kα,ε for 0 α < ω1 as follows. Let K0,ε = K and K1,ε = dε (K). Let γ be a countable ordinal and suppose Kα,ε defined for all α < γ . If γ is a successor, say γ = α + 1, set Kγ ,ε = dε (Kα,ε ).

(2.40)

If γ is a limit ordinal, choose (αn ) ordinals with αn < γ for all n and αn → γ ; set Kγ ,ε =

∞

Kαn,ε .

(2.41)

n=1

We may now define ordinal indices as follows. D EFINITION 2.16. Let K be a ω∗ -compact subset of X∗ , with X a separable Banach space. (a) βε (K) = sup{α ω1 : Kα,ε = ∅}. (b) β(K) = supε>0 βε (K). (c) Sz(X) = β(Ba X∗ ). Now it is easily seen that in fact there is a (least) α < ω1 with Kα,ε = Kα+1,ε . Moreover one has that then Kα,ε = ∅ iff K is norm separable iff βε (K) < ω1 , and then α = βε (K) + 1. Thus one obtains that Sz(X) < ω1 iff X∗ is norm-separable. Szlenk’s index was really only originally defined for Banach spaces with separable dual. In fact, however, one arrives at exactly the same final ordinal indices as he does, assuming that 1 is not isomorphic to a subspace of X, in virtue of the 1 -theorem [58]. The derivation in [67] is given by: K → τε (K) where τε (K) is the set of all x ∗ in K so that there is a sequence (xn∗ ) in K and a weakly null sequence (xn ) in Ba(X) such that limn→∞ |xn∗ (xn )| ε. Now let Pα (ε, K) be the transfinite sequence of sets arising from this derivation as defined in [67]. Let also ηε (K), the “ε-Szlenk index of K,” equal sup{α < ω1 : Pα (ε, K) = ∅} and η(K) = supε>0 ηε (K)}. The following result shows the close connection between Szlenk’s derivation and ours; its routine proof (modulo the 1 -theorem) is omitted. P ROPOSITION 2.17. Let X be a separable Banach space containing no isomorph of 1 , and let K be a ω∗ -compact subset of X∗ . Then for all ε > 0 and countable ordinals α, ε Pα , K ⊃ Kα,ε ⊃ Pα (2ε, K). (2.42) 2 Hence

η

ε , K βε (K) η(2ε, K) 2

and thus η(K) = β(K).

(2.43)

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H.P. Rosenthal

One may now easily deduce the following permanence properties. P ROPOSITION 2.18. Let X, Y be given Banach spaces and K, L weak* compact norm separable subsets of X∗ . (a) L ⊂ K implies β(L) β(K). (b) If T : Y → X is a surjective isomorphism, then β(T ∗ K) = β(K). (c) If Y ⊂ X and π : X∗ → Y ∗ is the canonical quotient map, then β(πK) β(K). In turn, this yields the following isomorphically invariant properties of the Szlenk index. C OROLLARY 2.19. Let X and Y be given Banach spaces with norm-separable duals. Then if Y is isomorphic to a subspace of a quotient space of X, Sz(Y ) Sz(X). R EMARK . Of course this yields that Sz(X) = Sz(Y ) if X and Y are of the same Kolmogorov dimension; i.e., each is isomorphic to a subspace of a quotient space of the other. This reveals at once both the power and the limitation of the Szlenk index. For the next consequence of our permanence properties of the Szlenk index, recall the Cantor–Bendixon index Ca(K) of a compact metrizable space K, defined by the cluster point derivation. For W ⊂ K, let W denote the set of cluster points of W . Then define K (α) , the α-th derived set of K, by K (0) = K, K (α+1) = (K (α)) , and K (β) = α 0, K is “δ-separated,” i.e., k − k δ

for all k = k in K.

(2.45)

Then βδ (K) = β(K) = Ca(K).

(2.46)

(b) Sz(C(ωα + 1)) α for any countable ordinal α. P ROOF. (a) We actually have that for any 0 < ε δ and any closed subset W of K, δε (W ) = dε (W ) = W .

(2.47)


1563

But then for all countable ordinals α, Kα,ε = K (α)

(2.48)

which immediately yields (2.46). As for (b), we have that

Ca ωα + = α.

(2.49)

But ωα + is obviously ω∗ -homeomorphic to a 2-separated subset K of Ba(C(ωα +))∗ . Thus we have

Sz C ωα + = β(K) = β2 (K) = α.

(2.50)

We may now obtain, via Theorem 2.14, the “easier” half of Theorem 2.15. C OROLLARY 2.21. Let 0 α < ω1 . Then α

Sz C ωω + ωα+1 .

(2.51)

P ROOF. For each positive integer n, we have that α

Sz C ωω ·n + ωα · n by 2.20(b). α

(2.52)

α

But C(ωn·ω +) is isomorphic to C(ωω +) by Theorem 2.14, and hence since the Szlenk index is isomorphically invariant (by Corollary 2.19) α

Sz C ωω + n · ωα

for all integers n,

(2.53)

which implies (2.51).

We next deal with (a) of Theorem 2.14. We first give a functional analytical presentaα tion of the spaces C(ωω ), using injective tensor products. We first recall the definitions (see [22], specifically pp. 485–486). D EFINITION 2.22. Given Banach spaces X and Y , the injective tensor norm, · ε , is defined on X ⊗ Y , the algebraic tensor product of X and Y , by n n ∗ ∗ ∗ ∗ ∗ xk ⊗ yk = sup x (xk )y (yk ): x ∈ Ba X , y ∈ Ba Y k=1

ε

(2.54)

k=1

for any n, x1 , . . . , xk in X and y1 , . . . , yk in Y . The completion of X ⊗ Y under this norm ∨

is called the injective tensor product of X and Y , denoted X ⊗ Y .

1564

H.P. Rosenthal ∨

When K and L are compact Hausdorff spaces, then we have that C(K) ⊗ C(L) is canonically isometric to C(K × L), where the elementary tensor x ⊗ y in C(K) ⊗ C(L) is simply identified with the function (x ⊗ y)(k, ) = x(k)y() for all k, ∈ K × L. α We then obtain the following natural tensor product construction of the spaces C(ωω +) (where we use the “unitization” given in (2.36)). Define a family (Xα )α 0, d ∈ D: f (d) − f (d−) > ε

is finite.

(3.4)

It then follows that defining T : B → ∞ (D) by (Tf )(d) =

f (d) − f (d−) 2

for all f ∈ B, d ∈ D

(3.5)

then T is a linear contraction valued in c0 (D). Now for each d ∈ D, define fd ∈ B by fd (k) = 1 if k < d,

fd (k) = −1 if k d.

(3.6)

Now since d ∈ K(1) , it follows easily that dist(fd , C(K)) = 1. In fact, letting π : B → B/C(K) be the quotient map, we have that for any n, k distinct points d1 , . . . , dn , and arbitrary scalars c1 , . . . , cn , cj fdj . (3.7) cj fdj = max |cj | = T π j

This easily yields that in fact T is a quotient map, and moreover if d1 , d2 , . . . is an enumeration of D, then (πfdj ) is isometrically equivalent to the usual c0 basis and [πfdj ] = B/C(K). The next lemma is the crucial tool for our non-complementation results.

1574

H.P. Rosenthal

L EMMA 3.13. Let K be a compact subset of [0, 1] so that K(n) = ∅. If K is countable, let D = K(1) . If K = [0, 1], let D be a countable dense subset of (0, 1) (the open unit interval). Assume for each d ∈ D, there is given gd ∈ C(K). Then given ε > 0, there exist d1 , . . . , dn in D and v in K so that (fd + gd )(ν) > 1 − ε j j

for all 1 j n.

(3.8)

P ROOF. For convenience, we assume real scalars. Note also that in the case K = [0, 1], K(n) = (0, 1). Let ε > 0 be fixed. First choose d1 ∈ K(n) ∩ D. Now choose δ1 > 0 so that letting V1 = (d1 − δ1 , d1 + δ1 ) ∩ K, then gd (d1 ) − gd (x) < ε 1 1

for all x ∈ V1 .

(3.9)

For simplicity in notation, set a = gd1 (d1 ). Now if x ∈ V1 and x > d1 , (fd1 + gd1 )(x) < −1 + a + ε.

(3.10)

If x ∈ V1 , x < d1 , then (fd1 + gd1 )(x) > 1 + a − ε.

(3.11)

max |1 + a − ε|, |−1 + a + ε| = 1 − ε + |a| 1 − ε.

(3.12)

But

Now since d1 is a two-sided cluster point of K(n−1) , it follows that def

V11 = (d1 − δ1 , d1 ) ∩ K(n−1) ∩ D = ∅ and def

V12 = (d1 , d1 + δ1 ) ∩ K(n−1) ∩ D = ∅. 1 = V 1 or V 2 , and then Hence if follows from (3.7)–(3.11) that we may set V 1 1 1 ∩ K(n−1) ∩ D = ∅ V

(3.13)

and (fd + gd )(x) > 1 − ε 1 1

1 . for all x ∈ V

(3.14)

1 ∩ K(n−1) ∩ D, and proceed in exactly the same way as in the first Now choose d2 ∈ V 1 an open neighborhood of d2 so that step. Thus, we first choose V2 ⊂ V gd (d2 ) − gd (x) < ε 2 2

for all x ∈ V2 .

(3.15)


1575

2 an open subset of Since d2 is a right and left cluster point of K(n−2) we again choose V V2 such that (fd + gd )(x) > 1 − ε 2 2

2 for all x ∈ V

(3.16)

and so that 2 ∩ K(n−2) ∩ D = ∅. V

(3.17)

Continuing by induction, we obtain d1 , . . . , dn in D, dn+1 ∈ K, and open sets in K, 0 ⊃ V 1 ⊃ V 2 ⊃ · · · ⊃ V n , so that for all 1 j n + 1, dj ∈ V j and K =V (fd + gd )(x) > 1 − ε j j

j . for all x ∈ V

Evidently then d1 , . . . , dn and v = dn+1 satisfy the conclusion of the lemma.

(3.18)

We are now prepared for our main non-complementation result. T HEOREM 3.14. Let n > 1 and let K and D be as in Lemma 3.13. Set B = rcl(K, D). Then if P is a bounded linear projection of B onto C(K), P n − 1.

(3.19)

Hence if K = [0, 1] or if K is countable and K(n) = ∅ for all n, C(K) is an uncomplemented subspace of rcl(K, D). P ROOF. Let λ = P ; also let π : B → B/C(K) be the quotient map. Then letting Y = kernel P , standard Banach space theory yields that π(Y ) = B/C(K) and y (λ + 1)πy for all y ∈ Y.

(3.20)

Now by Proposition 3.11 and its proof, B/C(K) is isometric to c0 and in fact [π(fd )]d∈D = B/C(K) and (πfd ) is isometrically equivalent to the c0 -basis (for c0 (D)). But it follows from (3.20) that we may then choose (unique) yd ’s in Y so that πyd = πfd

for all d

(3.21)

and cd yd (λ + 1) max |cd |

(3.22)

for any choice of scalars cd with cd = 0 for all but finitely many d. But (3.21) yields that for each d ∈ D there is a gd ∈ C(K) so that yd = fd + gd

for all d.

(3.23)

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H.P. Rosenthal

At last, given ε > 0, we choose d1 , . . . , dn in D and v ∈ K satisfying (3.13), i.e., the conclusion of Lemma 3.13. But then we may choose scalars c1 , . . . , cn with |cj | = 1 for all j , so that cj (fdj + gdj )(v) > 1 − ε

for all 1 j n.

(3.24)

Hence n n cj y d j cj (fdj + gdj ) (v) > n − nε. j =1

(3.25)

j =1

Finally, (3.22) and (3.25) yield that λ + 1 > n − nε.

(3.26)

But ε > 0 was arbitrary, so the conclusion of the theorem follows.

To complete the proof of Amir’s theorem, we only need to exhibit a subset K of [0, 1] with K homeomorphic to ωω + and K(w) = ∅. This is easily done, in the next result. P ROPOSITION 3.15. Let α = ωn + for some 1 n ω. Then there is a subset K of [0, 1] which is homeomorphic to α, so that K (j ) = K(j ) for all j ω. Moreover then rcl(K) is algebraically isometric to C(K) ⊕ C(K). P ROOF. Obviously, we may put K inside [−1, 1] or any particular interval [a, b] instead. For n = 1 let K = {1/n, −1/n, 0: n = 1, 2, . . .}. Then evidently K(1) = K (1) = {0}, K is homeomorphic to ω+, and rcl(K) is clearly algebraically isometric to c ⊕ c = C(K) ⊕ C(K). Suppose 1 n < ∞, α = ωn +, and K = K α has been constructed satisfying the conclusion of the proposition. Let now {Kj : j ∈ Z ∼ {0}} be a family of “copies” of K α , where for each j 1, Kj ⊂

1 1 , j +1 j

(3.27(i))

while if j −1 Kj ⊂

1 1 , j j +1

Finally, let K α+1 =

j ∈Z Kj

(3.27(ii)) ∪ {0} where Z = Z ∼ {0}.


1577

Then for any 1 i n, (i) K α+1,(i) = Kj ∪ {0} j ∈Z

=

Kj,(i) ∪ {0}

j ∈Z α+1 = K(i) .

(3.28)

In particular, for any j , Kj(n) consists of a single point, xj . Thus, α+1 = {xj , x−j , 0: j ∈ N} K α+1,(n) = K(n)

(3.29)

and of course as in the first step K α+1,(n+1) = Kα+1,(n+1) = {0}.

(3.30)

Now letting X = {f ∈ rcl(K α+1 ): f (x) = 0 for all x 0} and Y = {f ∈ rcl(K α+1 ): f (x) = 0 for all x < 0}, then rcl K α+1 = X ⊕ Y

(3.31)

(algebraically and isometrically, ∞ direct sum). But it follows easily from our induction hypothesis that X and Y are both algebraically isometric to C(ωn+1 +), whence the final statement of the proposition holds. This proves the result for all n < ω. Of course, for α = ω, we may now just repeat the entire procedure, this time placing inn 1 1 1 ) and ( −n+1 , −n ), a “copy” of K ω + which we have constructed side each interval ( n1 , n+1 above, thus achieving the proof. C OROLLARY 3.16. For each n > 1, there is a unital subalgebra An of C(ωn · 2+) with An algebraically isometric to C(ωn +), which is not λ-complemented in C(ωn · 2+) for any λ < n − 1. There is also a unital subalgebra B of C(ωω +) which is algebraically isometric to C(ωω +) which is uncomplemented in C(ωω +). from the preceding two results, P ROOF. The first assertion follows immediately . .∞ for the n A ) is uncomplemented in ( final assertion, it follows that B0 =def ( ∞ j =1 j c0 n=1 C(ω · ω 2+))c0 . But the second space is just C0 (ω ), while B0 is also algebraically isometric to C0 (ωω ). Hence just taking the unitizations of each, the result follows. R EMARK 3.17. Of course, since C(ωn +) is isomorphic to c0 , it has the separable extension property. Thus, there exists λn so that for all separable Banach spaces X ⊂ Y and of T to Y with T λn T . Our operators T : X → C(ωn +), there is an extension T argument yields that λn > n − 1. Actually, Amir proves in [8] that λn = 2n + 1 for all n = 1, 2, . . . .

1578

H.P. Rosenthal

We finally deduce Milutin’s result that the Cantor map induces an uncomplemented embedding of C([0, 1]) in C(D). We identify D with {0, 1}N as above, and let ϕ : D → [0, 1] be the Cantor map defined above. P ROPOSITION 3.18. Let D be the set of dyadic rationals in (0, 1); i.e., D = {k/2n : 1 k < 2n , n = 1, 2, . . .}. There exists an algebraic surjective isometry T : C({0, 1}N) → rcl([0, 1], D) such that T (ϕ 0 (C[0, 1])) = C([0, 1]). Thus ϕ 0 (C[0, 1]) is uncomplemented in C({0, 1}N ) by Theorem 3.14. P ROOF. Define a “standard” partial inverse map τ : [0, 1] → D as follows. If x ∈ [0, 1], x∈ / D, there is a unique y ∈ D with ϕ(y) = x, and define τ (x) = y.

(3.32)

If x ∈ D, then there is a unique (ε j ) ∈ D so that for a unique n 1, εj = 0 for all j > n, εn = 1, and ϕ((εj )) = x, i.e., x = nj=1 εj /2j . Now define

τ (x) = (εj ) .

(3.33)

Of course then

ϕ τ (x) = x

for all x ∈ [0, 1].

(3.34)

Now define T by

(Tf )(x) = f τ (x) for all f ∈ C(D), x ∈ [0, 1].

(3.35)

Now it easily follows that T is an algebraic isometry mapping C(D) into ∞ [0, 1]. We now easily check that

T ϕ0f = f

for all f ∈ C [0, 1] .

(3.36)

Moreover, if f ∈ C(D), then Tf is continuous at x for all x ∈ / D.

(3.37)

Finally, let x ∈ D and (εj ) = τ (x) with n as given preceding (3.33). Let (ym ) be a sequence in [0, 1] with ym → x. Suppose first that x < ym

for all m.

Then it follows that for all m,

(m) (m) τ (ym ) = ε1 , . . . , εn , βn+1 , βn+2 , . . .

(3.38)


1579

and in fact then τ (ym ) → τ (x) = (ε1 , . . . , εn , 0, . . .). Hence

(Tf )(ym ) = f τ (ym ) → f τ (x) = (Tf )(x)

(3.39)

by continuity of f . Thus Tf is indeed right continuous at x. Suppose next that ym < x

for all m.

(3.40)

This time, it follows that

(m) (m) τ (ym ) = ε1 , . . . , εn−1 , 0, βn+1 , βn+2 , . . . for all m, and in fact now def

τ (ym ) → (ε1 , ε2 , . . . , εn−1 , 0, 1, 1, 1 . . .) = z. Hence now,

(Tf )(ym ) = f τ (ym ) → f (z)

(3.41)

by continuity of f , showing that Tf has a left-limit. Thus we have indeed proved that

T C(D) ⊂ rcl [0, 1], D . We may check, however that conversely if f ∈ rcl([0, 1], D), then defining f˜ on D by f˜(τ x) = f (x) for all x ∈ [0, 1],

f˜(y) = f ϕ(y) −

(3.42) (3.43)

if y ∈ D ∼ τ ([0, 1]), then f˜ ∈ C(D), and hence finally T satisfies the conclusion of 3.18, completing the proof.

4. Operators on C(K)-spaces Throughout, K denotes a compact Hausdorff space. By an operator on C(K) we mean a bounded linear operator from C(K) to some Banach space X. Of course, a “C(K)-space” is just C(K) for some K. We first recall the classical result of Dunford and Pettis. T HEOREM 4.1 ([25]). An operator on a C(K)-space maps weakly compact sets to compact sets. See [33], p. 62 for a proof. We note the following immediate structural consequence.

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C OROLLARY 4.2. Let T : C(K) → C(K) be a given weakly compact operator. Then T 2 is compact. Hence if T is a projection, its range is finite-dimensional. Evidently the final statement may be equivalently formulated: every reflexive complemented subspace of a C(K)-space is finite-dimensional. We are interested here in non-weakly compact operators. Before focusing on this, we note the following structural result due to the author [57]. T HEOREM 4.3. A reflexive quotient space of a C(K)-space is isomorphic to a quotient space of an Lp (μ)-space for some 2 p < ∞. Let us note that conversely, Lp is isometric to a quotient space of C([0, 1]) for all 2 p < ∞. (Throughout, for all 1 p < ∞, Lp denotes Lp (μ), where μ is Lebesgue measure on [0, 1].) Theorem 4.3 is in reality the dual of the version of the main result in [57]: every reflexive subspace of L1 is isomorphic to a subspace of Lp for some 1 < p 2. We now focus on the main setting of this section – “fixing” properties of various classes of non-weakly compact operators on C(K)-spaces. D EFINITION 4.4. Let X, Y, Z be Banach spaces. An operator T : X → Y fixes Z if there exists a subspace Z of X with Z isomorphic to Z so that T |Z is an isomorphism. We now summarize the main results to be discussed here. The first result is due to Pełczyński [52]. T HEOREM 4.5. A non-weakly compact operator on a C(K)-space fixes c0 . To formulate the next result, we will need the notion of the Szlenk-index of an operator. D EFINITION 4.6. Let X and Y be separable Banach spaces and T : X → Y be a given operator; let ε > 0. The ε-Szlenk index of T , βε (T ), is defined as βε (T ∗ (Ba(X∗ ))), where βε is as in Definition 2.16. Sz(T ), the Szlenk index of T , is defined as:

Sz(T ) = sup βε (T ) = β T ∗ Ba Y ∗ . ε>0

The results in Section 2 show that Sz(C(ωω +)) = ω2 , it was in fact rather easy to obtain that Sz(C(ωω +)) ω2 . It follows easily that if an operator on a separable C(K)-space fixes C(ωω +), its Szlenk index is at least ω2 . The converse to this, is due to Alspach. T HEOREM 4.7 ([1]). Let K be a compact metric space, X be a Banach space, and T : C(K) → X a given operator. The following are equivalent. (1) Sz(T ) ω2 . (2) βε (T ) ω for all ε > 0. (3) T fixes C(ωω +). We next give another characterization of operators fixing C(ωω +), due to Bourgain.


1581

D EFINITION 4.8. Let X, Y be Banach spaces and T : X → Y be a given operator. T is called a Banach–Saks operator if whenever (xj ) is a weakly null sequence in X, there is a subsequence (xj ) of (xj ) so that n1 nj=1 T (xj ) converges in norm to zero. X has the weak Banach–Saks property if IX is a Banach–Saks operator. It is easily seen that c0 has the Banach–Saks property, It is a classical result, due to Schreier, that C(ωω +) fails the weak Banach–Saks property [64]. Thus any operator on a C(K)-space fixing C(ωω +), is not a Banach–Saks operator. Bourgain established the converse to this result in [18]. T HEOREM 4.9. A non-Banach–Saks operator on a C(K)-space fixes C(ωω +). Bourgain also obtains “higher ordinal” generalizations of Theorem 4.7, which we will briefly discuss. The final “fixing” result in this summary is due to the author. T HEOREM 4.10 [56]. Let K be a compact metric space, X be a Banach space, and T : C(K) → X be a given operator. Then if T ∗ (X∗ ) is non-separable, T fixes C([0, 1]). The proofs of these results involve properties of L1 (μ)-spaces, for by the Riesz representation theorem, C(K)∗ may be identified with M(K), the space of scalar-valued regular countably additive set functions on B(K) the Borel subsets of K. A. Operators fixing c0 Theorem 4.5 follows quickly from the following two L1 theorems, which we do not prove here. The first is due to Grothendieck (Théorème 2, p. 146 of [30]). T HEOREM 4.11. Let W be a bounded subset of M(K). Then W is relatively weakly compact if and only if for every sequence O1 , O2 , . . . of pairwise disjoint open subsets of K, μ(Oj ) → 0 as j → ∞, uniformly for all μ in W .

(4.1)

The second is a relative disjointness result due to the author. P ROPOSITION 4.12 ([55]). Let μ1 , μ2 , . . . be a bounded sequence in M(K), and let E1 , E2 , . . . be a sequence of pairwise disjoint Borel subsets of K. Then given ε > 0, there exist n1 < n2 < · · · so that for all j ,

|μnj |(Eni ) < ε.

i=j

(For any sequence (fj ) in a Banach space, [fj ] denotes its closed linear span.)

(4.2)

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P ROOF OF T HEOREM 4.5. Assume X is a Banach space and T : C(K) → X is not weakly compact. Then also T ∗ : X∗ → C(K)∗ = M(K) is non-weakly compact, so in particular

def W = T ∗ Ba X∗ is non-weakly compact.

(4.3)

Of course then W is not relatively weakly compact, since it is closed. Thus by Gothendieck’s theorem, we may choose η > 0, a sequence O1 , O2 , . . . of disjoint open sets in K, and a sequence μ1 , μ2 , . . . in W with μj (Oj ) > η

for all j .

(4.4)

Let then 0 < ε < η. By passing to a subsequence of the Oj ’s and μj ’s, we may also assume by Proposition 4.12 that

|μj |(Oi ) < ε

for all j .

(4.5)

i=j

For each j , by (4.4) we may choose fj ∈ C(K) of norm 1 with 0 fj 1 and fj supported in Oj , so that fj dμj > η.

(4.6)

Set Z = [fj ]. It is immediate that Z is isometric to c0 ; in fact (fj ) is isometrically equivalent to the c0 basis. Thus we have that given n and scalars c1 , . . . , cn , n cj fj T max |cj |. T j

(4.7)

j =1

But for each j n n ∗ ∗ ci fi sup T x ci fi T x ∗ ∈Ba X∗ i=1 i=1 n ci fi dμj i=1 |ci | |fi | d|μj | |cj | fj dμj − i=j

|cj |η − max |ci |

|μj |(Oi )

i=j

|cj |η − max |ci |ε.

(4.8)


1583

But then taking the max over all j , we get that n ci fi (η − ε) max |ci |. T

(4.9)

i=1

(4.6) and (4.9) yield that T |Z is an isomorphism, completing the proof.

B. Operators fixing C(ωω +) We will not prove 4.7. However, we will give the description of the isometric copy of C(ωω +) which is fixed, in Bourgain’s proof of this result. We first indicate yet another important description of the C(α+)-spaces, due to Bourgain, which is fundamental in his approach. D EFINITION 4.13. Let T∞ be the infinitely branching tree consisting of all finite sequences of positive integers. For α, β in T∞ , define α β if α is an initial segment of β; i.e., if α = (j1 , . . . , jk ) and β = (m1 , . . . , m ), then k and ji = mi for all 1 i k. Also, let (α) = k, the length of α. The empty sequence ∅ is the “top” node of T∞ . A set T ⊂ T∞ will be called a tree if whenever β ∈ T and α ∈ T∞ , α β and α = ∅, then α ∈ T . Finally, a tree T is called well-founded if it contains no strictly increasing sequence of elements of T∞ . We now define Banach spaces associated to trees T , denoted XT , as follows. D EFINITION 4.14. Let T be a well founded tree, and let c00 (T ) denote all systems (cα )α∈T of scalars, with finitely many cα ’s non-zero. Define a norm · T on c00 (T ) by (cα ) = max cγ . T α∈T

(4.10)

γ α

Let XT denote the completion of c00 (T ) under · T . P ROPOSITION 4.15. Let T be an infinite well-founded tree. Then there exists a countable limit cardinal α so that XT is either isometric to C0 (α) (if T has infinitely many elements of length 1 and φ ∈ / T ) or to C(α+). Conversely, given any such ordinal α, there exists a tree T with XT isometric to C0 (α) or to C(α+). Moreover, let T be a given infinite tree, and for α ∈ T let bα be the natural element of c00 (T ): (bα )(β) = δαβ . Let τ : N → T be a bijection (i.e., an enumeration) so that if τ (i) < τ (j ), then i < j . Then (bτ (j ) )∞ j =1 is a monotone basis for XT . All of the above assertions and developments are due to Bourgain [18], except for the basis assertion, which is due to Odell. The author is most grateful to Professor Odell for his personal explanations of these results.

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We next indicate the trees Tn corresponding to the spaces C(ωn +) for 0 n ω. For n finite, simply let Tn be all finite sequences of positive integers of length at most n; also let Tn0 = Tn ∼ {∅}. Finally, let Tω0 = ∞ {(n, α): α ∈ Tn−1 }, and let Tω = Tω0 ∪ {∅}. The n=1 reader should have no difficulty in establishing the assertions of 4.15 in this special case. In particular, for all 1 n ω, XTn0 is isometric to C0 (ωn ) and XTn is isometric to C(ωn +). There still remains the intuitive issue: what is the picture for a subspace of C(K) which is isomorphic to C(ωω +) and fixed by an operator T satisfying the hypotheses of 4.7? The following elegant description gives Bourgain’s answer. D EFINITION 4.16. Let F be a family of non-empty clopen subsets of a totally disconnected infinite compact metric space K. F is called regular if (a) any two elements of F are either disjoint or one is contained in the other. (b) F has no infinite totally ordered subsets, under the order A B if A ⊃ B. We again leave the proof of the following motivating result to the reader. P ROPOSITION 4.17. Let F be an infinite regular family of clopen subsets of K. There is a well founded tree T and an order preserving bijection τ : F → T . [F ] is a subalgebra of C(K) which is algebraically isometric to C(α+) or C0 (α) for some countable ordinal α. Moreover, identifying τ (A) with the basis elements bτ (A) of Proposition 4.15, then τ extends to a linear isometry of [F ] with XT . Conversely, given any well-founded tree T , then there exists a regular family F (for a suitable K) with F order isomorphic to T . Let us just indicate pictures for the regular families corresponding to Tn0 and Tn , and thus to C0 (ωn ) and C(ωn +). Of course, a sequence of disjoint clopen sets spans c0 isometrically. T10

◦

◦

◦

◦

···

We get T1 by putting all these inside one clopen set, which actually corresponds then to the function 1 in C(ω+) = c. T1

◦

◦

◦

◦

···

Now we can get T20 by repeating T1 infinitely many times. T1

◦ ◦ ◦ ◦···

◦ ◦ ◦ ◦ ···

◦ ◦ ◦ ◦ ···

···


1585

Of course, we then put all these inside one clopen set, to obtain T2 . Finally, we obtain Tω0 by choosing a sequence of disjoint clopen sets O1 , O2 , . . . and inside On , we put the regular system corresponding to Tn0 . Bourgain proves Theorem 4.7 by establishing the following general result. T HEOREM 4.18. Let K a totally disconnected compact metric space, X a Banach space, and T : C(K) → X a bounded linear operator be given such that for some ε > 0 and countable ordinal α βε (T ) ωα . Then there is a regular system F of clopen subsets of K with Y =def [F ] isometric to C0 (ωω·α ) such that T |Y is an isomorphism. The whole point of our exposition: one must choose a regular family of clopen sets to achieve the desired copy of C(ωω +); this requires the above concepts. R EMARK 4.19. In view of Milutin’s theorem, it follows that for any compact metric space K and operator T : C(K) → X, T fixes C(ωω·α +) provided βε (T ) = ωα . Thus Theorem 4.7 follows, letting α = 1. Actually, Alspach obtains in [1] that if βε (T ) ω for some ε > 0, K arbitrary, then still T fixes some subspace of C(K) isometric to C0 (ωω ). We turn next to the basic connection between the weak Banach–Saks property and C(ωω +). We first give Schreier’s fundamental example showing that C(ωω +) fails the weak Banach–Saks property; i.e., there exists a weakly null sequence in C(ωω +), such that no subsequence has its arithmetic averages tending to zero in norm. P ROPOSITION 4.20. There exists a sequence U1 , U2 , . . . of compact open subsets of ωω such that setting bj = XUj for all j , then (a) no point of ωω belongs to infinitely many of the Uj ’s, (b) for all scalars (cj ) with only finitely many cj ’s non-zero r (4.11) cji : j1 < · · · < jr and r j1 . cj bj = max i=1

Before proving this, we first show that the sequence (bj ) in 4.20 is an “anti-Banach– Saks” sequence. For convenience, we restrict to real scalars. P ROPOSITION 4.21. Let (bj ) be as in 4.20. Then bj → 0 weakly. Define a new norm on the span of the bj ’s by r |cji |: r = j1 and j1 < · · · < jr . cj bj = max i=1

Then x |||x||| 2x for all x ∈ [bj ].

(4.12)

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It follows immediately that given j1 < j2 < · · ·, then norm as r → ∞. In fact, we have for all k that 2k bji i=1

1 r

r

i=1 bji

does not tend to zero in

2k 1 k bji . 2 2

(4.13)

i=1

We also see the fundamental phenomenon: any k-terms of the bj ’s past the k-th are 2-equivalent to the 1k -basis. It also follows, incidentally, that the norm-condition (4.13) alone, insures that bj → 0 weakly. We prefer however, to give the simpler argument which follows from 4.20. Finally, it follows that the sequence (bj ) in 4.20 is an unconditional basic sequence. P ROOF OF 4.21. Since no k belongs to infinitely many of the Kj ’s, it follows that XKj → 0 pointwise on ωω +, which immediately yields that XKj → 0 weakly, by the “baby” version of the Riesz representation theorem. The lower estimate in (4.12) is trivial. But if we fix cj ’s, r 1 and r = j1 < j2 < · · · < jr , then there is a subset F of j1 , . . . , jr , with r 1 c |cji |. i 2 i∈F

(4.14)

i=1

But if we enumerate F as i1 < · · · < ik , then trivially k r j1 i1 , hence r 1 1 ci |cji | |||cj bj |||. cj · bj 2 2 i∈F

(4.15)

i=1

P ROOF OF 4.20. We give yet one more (and last!) conceptualization of the compact countable spaces ωn + and ωω +. We identify their elements with certain finite subsets of N. Let F be a family of finite subsets of N, so that F contains no infinite sequences F1 , F2 , . . . with Fn Fn+1 for all n, and such that F is closed under pointwise convergence (where Fj → F means XFj → XF pointwise on N). It follows easily that F is then a compact metric space. Now first let Fn be the family of all subsets of N of cardinality as most n. It follows that (j ) Fn is homeomorphic to ωn +. In fact, we obtain by induction that Fn = Fn−j , so that (0) finally Fn = {∅}. Now we “naturally” obtain Fω homeomorphic to ωω + as follows. def

Fω = {∅} ∪

∞

{α ∪ n: (α) n and the least element of α n}.

(4.16)

n=1

In other words, Fω consists of all finite sets whose cardinality is at most its least element. It is clear that Fω is indeed compact in the pointwise topology, and moreover it is also clear (n) that Fω = ∅ for all n = 1, 2, . . . . Finally, it is also clear that for each n, the n-th term in


1587 (ω)

the above union is homeomorphic to ωn +, and so we have that Fω = {∅}, whence Fω is homeomorphic to ωω +. Now for each j , define Uj = {α ∈ Fω : j ∈ α}.

(4.17)

It then easily follows that Uj is a clopen subset of Fω , and of course φ ∈ / Uj for any j . It is trivial that no α ∈ Fω belongs to infinitely many Uj ’s since α is a finite set. Finally, let (cj ) be a sequence of scalars with only finitely many non-zero elements; then for any α ∈ Fω , cj : j ∈ α . cj bj (α) =

(4.18)

But if α = {j1 , . . . , jr } with j1 < j2 < · · · < jr , then by definition of Fω , r j1 , and conversely given j1 < · · · jr with r j1 , {j1 , . . . , jr } ∈ Fω . Thus (4.18) yields (4.11), completing the proof. Next we discuss Banach–Saks operators on C(K)-spaces. Actually, Theorem 4.9 is an immediate consequence of Theorem 4.7 and the following remarkable result ([18], Lemma 17) (see Definition 4.8 for the ε-Szlenk index of an operator) T HEOREM 4.22. Let X and Y be Banach spaces with X separable and T : X → Y be a given operator. Then if the ε-Szlenk index of T is finite for all ε > 0, T is a Banach–Saks operator. In particular, if the ε-Szlenk index of X is finite for all ε > 0, X has the weak Banach–Saks property. Just to clarify notation, we first give the P ROOF OF T HEOREM 4.9. Let T : C(K) → Y be a non-Banach–Saks operator. It is trivial that then, without loss of generality, we may assume that C(K) is separable, i.e., that K is compact metric. Then by Theorem 4.22, there exists an ε > 0 such that βε (T ∗ (Ba(Y ∗ ))) ω. But then T fixes C(ωω +) by Theorem 4.7. The initial steps in the proof of Lemma 17 of [18] (given as a lemma there) can be eliminated, using a fundamental dichotomy discovered a few years earlier. Moreover, the details of the proof of Lemma 17 itself do not seem correct. Because of the significance of this result, we give a detailed proof here. The following is the basic dichotomy discovered by the author in [59]; several proofs have been given since, see, e.g., [46]. T HEOREM 4.23 ([59]). Let (bn ) be a weakly null sequence in a Banach space. Then (bn ) has a subsequence (yn ) satisfying one of the following mutually exclusive alternatives: (a) n1 nj=1 yj tends to zero in norm, for all subsequences (yj ) of (yj ).

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(b) (yj ) is a basic sequence so that any k terms past the k-th are uniformly equivalent to the 1k basis. Precisely, there is a δ > 0 so that for all k < j1 < · · · < jk and scalars c1 , . . . , ck , k k ci yji δ |ci |. i=1

(4.19)

i=1

Note that it follows immediately that if (yj ) satisfies (b), there is a constant η > 0 so

that for any subsequence (yj ) of (yj ), 2k j =1 yj ηk for all k, hence no subsequence of (yj ) has averages converging to zero in norm. In modern terminology, (yj ) has a spreading model isomorphic to 1 . Notice that Schreier’s sequence given in Proposition 4.20 is a witness to this general phenomenon. Now we give the P ROOF OF T HEOREM 4.22. Let T : X → Y be a given operator, and suppose T is nonBanach–Saks. We may assume without loss of generality that T = 1. Choose (xn ) a weakly null sequence in X so that the arithmetic averages of T xn do not tend to zero in norm. Assume that xn 1 for all n. Now choose (bn ) a subsequence of (xn ) so that setting yn = T bn for all n, then for some δ > 0 (yn ) satisfies (4.19).

(4.20)

Let K = T ∗ (Ba(Y ∗ )). We shall prove that Pm (δ, K) = ∅ for all m = 1, 2, . . . ,

(4.21)

where the sets Pm (δ, K) are those originally defined by Szlenk in his derivation (as defined above, preceding Proposition 2.17). We need the following fundamental consequence of (4.19). For all m and α = (j1 , . . . , jm ) with m < j1 < j2 < · · · < jm , there exists a yα∗ ∈ Ba(Y ∗ ) with ∗ y (yj ) δ

for all j ∈ α.

(4.22)

Γm, = α ⊂ { + 1, + 2, . . .}: #α = m

(4.23)

α

Given m, 1, we set

(i.e., Γm, is just all m element subsets of N past the -th term). We now prove the following claim by induction on m. C LAIM . For all m, m and families {yα∗ : α ∈ Γm, } with yα∗ satisfying (4.22) for all α ∈ Γm, , there is a weak∗ -cluster point of {T ∗ (yα∗ ): α ∈ Γm, } belonging to Pm (δ, K).


1589

(This is a more delicate version of the apparently incorrect argument in [18]; the author nevertheless greatly admires the ingenuity of Bourgain’s discussion there). The case m = 1 is really immediate, just using the ω∗ -compactness of K. After all, given any n > , then by definition, ∗ ∗

T y (bn ) = y ∗ (yn ) δ (n) (n)

(4.24)

∗ ) lies in P (δ, K) since (b ) is weakly null in Ba(X). hence any ω∗ -cluster point of (T ∗ y(n) 1 n Now suppose the claim is proved for m, let m + 1, and let {yα∗ : α ∈ Γm+1, } be given with yα∗ satisfying (4.22) for all α ∈ Γm+1, . Fix n > , and define y˜α∗ for each α ∈ Γm,n by ∗ . y˜α∗ = yα∪{n}

But then for all α ∈ Γm,n , y˜α∗ satisfies (4.22), hence by our induction hypothesis, there exists a ω∗ -cluster point xn∗ of {T ∗ (y˜α∗ ): α ∈ Γm,n } which belongs to Pm (δ, K). But since α ∪ {n} ∈ Γm+1, , we have that ! ∗ ∗ " ! ∗ T y˜ , bn = y α

α∪{n} , yn

" δ

(4.25)

for all α ∈ Γm,n ; then also ∗ x (bn ) δ. n

(4.26)

∗ ∗ But then if x ∗ is a weak∗ -cluster point of (xn∗ )∞ n=+1 , x ∈ Pm+1 (δ, K), and of course x is ∗ ∗ ∗ indeed a weak -cluster point of {T (yα ): α ∈ Γm+1, }. This completes the induction step of the claim which then shows (4.21), so the proof of Theorem 4.22 is complete.

C. Operators fixing C([0, 1]) We finally treat Theorem 4.10. We shall sketch the main ideas in the proof given in [56]. We first note, however, that there are two other proofs known, both conceptually different from each other and from that in [56]. Weis obtains this result via an integral representation theorem for operators on C(K)-spaces [68]. For extensions of this and further complements in the context of Banach lattices see [29] and [26]. Finally, the general result 4.18 also yields 4.10, as noted by Bourgain in [18]. Indeed, suppose T : C(K) → X are given as in the statement of 4.10, where K is totally disconnected. Then 4.18 yields that T fixes C(L) for every countable subset L of [0, 1]. But the family of all compact subsets L of [0, 1] such that T fixes C(L), forms a Borel subset of the family of all compact subsets L of [0, 1] in the Hausdorff metric; the countable ones, however are not a Borel set by classical descriptive set theory. Hence there is an uncountable compact L ⊂ [0, 1] so that F fixes L, and then C(L) is isomorphic to C([0, 1]) by Milutin’s theorem. We also note that a refinement of the arguments in [56] yields the following generalization of 4.10, due jointly to Lotz and Rosenthal [44]: let E be a separable Banach lattice with E ∗ weakly sequentially complete, X be a Banach space, and T : E → X be an operator with T ∗ (X∗ ) non-separable. Then T fixes C([0, 1]). For extensions of this and further complements in the context of Banach lattices, see [29] and [26].

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We first introduce some (standard) terminology. Let X be a Banach space, Y be a subspace of X, and W be a subset of Ba(X∗ ). We say that W norms Y if there exists a constant λ 1 so that (4.27) y λ sup w(y) for all y ∈ Y . w∈W

If (4.27) holds, we say that W λ-norms Y . Now let K be a compact metric space. Theorem 4.10 then follows immediately from the following stronger statement: T HEOREM 4.24. A non-separable subset of Ba(C(K)∗ ) norms a subspace of C(K) isometric to C(D). Indeed, we may obviously assume that T : C(K) → X has norm one. Then assuming T ∗ X∗ is non-separable, so is W = T ∗ (Ba(X∗ )), and thus 4.24 yields a subspace Y of C(K) with Y isometric to C(D) and T |Y an isomorphism; then T fixes C(D), and so of course C([0, 1]), which is isometric to a subspace of C(D). The proof of 4.24 proceeds by reduction to the following almost isometric result. L EMMA 4.25. Let Z be a subspace of C(K)∗ with Z isometric to L1 . Then for every ε > 0, Ba(Z) (1 + ε)-norms a subspace of C(K) which is isometric to C(D). We shall sketch some of the ideas in the proof of 4.25 later on. We first note that the actual proof of Theorem 4.24 yields the following dividend. C OROLLARY 4.26. Let Z be a non-separable subspace of C(K)∗ . Then for all ε > 0, Ba(Z) (1 + ε)-norms a subspace of C(K) which is isometric to C(D). Of course 4.26 has the following immediate consequence. C OROLLARY 4.27. Let X be a quotient space of C(K) with X∗ non-separable. Then X contains a subspace (1 + ε)-isomorphic to C(D) for all ε > 0. R EMARK 4.28. The conclusion badly fails for subspaces X of C(D) which are themselves isomorphic to C(D). Indeed, it is proved in [42] that for every λ > 1 there exists a Banach space X which is isomorphic to C(D) but contains no subspace λ-isomorphic to C(D); of course X is isometric to a subspace of C(D). We now take up the route which leads to Lemma 4.25. Say that elements μ and ν of C(K)∗ are pairwise disjoint if μ and ν are singular, regarding μ, ν as complex Borel measures on K. The next result is proved by a two-step transfinite induction. L EMMA 4.29. Let L be a convex symmetric non-separable subset of Ba(C(K))∗ . Then there is a δ > 0 so that for all 0 < ε < δ, there exists an uncountable subset {α }α∈Γ of L and a family {μα }α∈Γ of pairwise disjoint elements of Ba(C(K)∗ ) so that for all α, μα − α ε

and μα δ.

Moreover if L is the unit ball of a subspace of C(K)∗ , one can take δ = 1.

(4.28)


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Now of course the family {μα /μα : α ∈ Γ } is isometrically equivalent to the usual 1 (Γ ) basis. But this is also an uncountable subset of a compact metrizable space, Ba(C(K)∗ ) in the ω∗ -topology. So it follows that we may choose α1 , α2 , . . . distinct el∗ ements of Γ with (fn )∞ n=1 ω -dense in itself where fn = μαn /μαn for all n; note that ∞ (fn )n=1 is isometrically equivalent to the usual 1 basis. A variation of an argument of Stegall [66] now yields P ROPOSITION 4.30. Suppose X is a separable Banach space and (fn ) in X∗ is isometrically equivalent to the 1 basis and ω∗ -dense in itself. Then there exists a subspace U of X∗ , isometric and ω-isomorphic to C(D)∗ , such that for all x ∈ X, (4.29) sup u(x) supfn (x). n

u∈Ba(U )

Of course U is obtained as T ∗ (C(D))∗ where T is constructed to be a quotient map of X onto C(D)). For our purposes, we only need that U contains a subspace isometric to L1 . We now complete the proof of Theorem 4.24, using 4.25, 4.29 and 4.30. Let L be as in 4.24. We may assume that L is convex and symmetric, for if L is the closed convex hull of L ∪ −L, |(x)| for all x ∈ C(K). Let δ satisfy the conclusion of Lemma then sup∈L |(x)| = sup∈ L 4.29, and let 0 < ε so that 1−ε−

ε > 0. δ

(4.30)

Now let (α )α∈Γ and (μα )α∈Γ satisfy the conclusion of 4.29. Choose α1 , α2 , . . . distinct ∗ α’s so that (fn )∞ n=1 is ω -dense in itself, where fn = μαn /μαn for all n. Also let yn = αn and δn = μαn for all n. Next, choose Z a subspace of C(K)∗ isometric to L1 such that for all x ∈ C(K), (4.31) sup z(x) supfn (x) n

z∈Ba Z

thanks to Proposition 4.30. Finally, choose X a subspace of C(K) with X isometric to C(D)) so that (4.32) (1 − ε)x sup z(x) for all x ∈ X, z∈Ba Z

by Lemma 4.25. Now by our definition of the yn ’s and δn ’s, we have for all n that (by (4.28)) fn − yn /δn

ε ε . δn δ

Thus finally fixing x ∈ X with x = 1, we have 1 − ε supfn (x) by (4.31) and (4.32) n

(4.33)

(4.34)

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ε 1 supyn (x) + δ n δ

by (4.33).

(4.35)

Hence ε 1−ε− δ supyn (x). δ n

(4.36)

Thus letting λ = ((1 − ε − εδ )δ)−1 , we have proved that L λ-norms X. Finally, if L is the unit ball of a non-separable subspace of C(K)∗ , we may choose δ = 1 by 4.29; but then it follows that since ε may be chose arbitrarily small, λ is arbitrarily close to 1, and this yields Corollary 4.26. We finally treat Lemma 4.25. Let then Z be a subspace of C(K)∗ which is isometric to L1 . Standard results yield that there exists a Borel probability measure μ on K, a Borel measurable function θ with |θ | = 1, a compact subset S of K with μ(S) = 1, and a σ -subalgebra S of the Borel subsets of S such that (S, S, μ|S) is a purely non-atomic measure space and Z = θ · L1 (μ|S). (We adopt the notation: θ · Y = {θy: y ∈ Y }.) The desired isometric copy of C(D) which is (1 + ε)-normed by Z is now obtained through the following construction. L EMMA 4.31. Let μ, E, and S be as above, and let ε > 0. Then there exist sets Fin ∈ S and compact subsets Kin of S satisfying the following properties for all 1 i 2n and n = 0, 1, . . . . (i) Kin ∩ Kin = Fin ∩ Fin = ∅ for any i = i. n+1 n+1 n+1 ∪ K2i and Fin = F2i−1 ∪ F2in+1 . (ii) Kin = K2i−1 n n (iii) Ki ⊂ Fi . (iv) (1 − ε)/2n μ(Kin ) and μ(Fin ) 1/2n . (v) θ |K10 is continuous relative to K10 . (This is Lemma 1 of [56], with condition (v) added as in the correction to [56].) We conclude our discussion with the P ROOF OF L EMMA 4.25. Let F = K10 and let A denote the closure of the linear span of {χKin : 1 i 2n , n = 0, 1, 2, . . .} in C(F ). Then A is a subalgebra of C(F ) algebraically isometric to C(D). Hence also, def Y = θ¯ · A

is a subspace of C(F ) isometric to C(D).

(4.37)

(θ¯ denotes the complex conjugate of θ , in the case of complex scalar.) Now let E : C(F ) → C(K) be an isometric extension operator, as insured by the Bosuk theorem (Lemma 2.5(b) above). Finally, set X = E(Y ). So, evidently X is a subspace of C(K), isometric to C(D). We claim that Ba(Z)

1 − norms X 1 − 2ε

(4.38)


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which yields 4.25. Of course, it suffices to show that for a dense linear subspace X0 of X sup z(x) (1 − 2ε)x for all x ∈ X0 .

(4.39)

z∈Ba Z

We take X0 to be the linear span of the functions E(θ¯ · χKin ). So, fix n and let n

φ = θ¯

2

ci χKin

for scalars c1 , . . . , c2n with maxi |ci | = 1.

(4.40)

i=1

˜ = 1. Of course we identify the elements of Z with the comFinally, let φ˜ = E(φ). So φ plex Borel measures in θ · L1 (μ|S). So choose i with |ci | = 1 and let f = θ χFin /μ(Fin ). Then f L1 (μ) = 1, so f · μ as an element of Z, also has norm 1. We have that f φ˜ dμ

Kin

f φ˜ dμ −

Fin ∼Kin

f φ˜ dμ

f φ˜ dμ

=

μ(Kin ) − μ(Fin )

μ(Kin ) μ(Fin ) − μ(Kin ) − μ(Fin ) μ(Fin )

1 − 2ε

Fin ∼Kin

since φ˜ 1

by Lemma 4.31.

This concludes the proof, and our discussion of Theorem 4.10.

(4.41)

5. The complemented subspace problem In its full generality, this problem, (denoted the CSP), is as follows: let K be a compact Hausdorff space and X be a complemented subspace of C(K). Is X isomorphic to C(L) for some compact Hausdorff space L? We first state a few results which hold in general, although most of them are easily reduced to the separable case anyway. All Banach spaces, subspaces, etc., are taken as infinite-dimensional. T HEOREM 5.1 ([52]). Every complemented subspace of a C(K)-space contains a subspace isomorphic to c0 . This is an immediate consequence of Corollary 4.2 and Theorem 4.5. The next result refines Milutin’s theorem to the non-separable setting. P ROPOSITION 5.2 ([23]). A complemented subspace of a C(K)-space is isomorphic to a complemented subspace of C(L) for some totally disconnected L.

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Let us point out, however: it is unknown if a (non-separable) C(K)-space itself is isomorphic to C(L) for some totally disconnected L. Later on, we shall give results characterizing c0 (or rather c) as the smallest of the complemented subspaces of C(K)-spaces. Of course Theorem 4.10 characterizes C([0, 1]) as the largest separable case. T HEOREM 5.3 ([56]). Let X be a complemented subspace of a separable C(K)-space with X∗ non-separable. Then X is isomorphic to C([0, 1]). P ROOF. Assume then X is complemented in C(K) with C(K) separable, i.e., K is metrizable. Then K is uncountable, since C(K)∗ itself must be non-separable. Thus by Milutin’s theorem, C(K) is isomorphic to C([0, 1]). By Theorem 4.10, X contains a subspace isomorphic to C([0, 1]). By Pełczyński’s weak injectivity result (Theorem 3.1), X contains a subspace Y isomorphic to C([0, 1]) with Y complemented in C(K). Thus by the decomposition method (applying Proposition 1.2 to C(D) instead), X is isomorphic to C([0, 1]). For the next result, recall that a Banach space X is called primary if whenever X is isomorphic to Y ⊕ Z (for some Banach spaces Y and Z), then X is isomorphic to Y or to Z. The following result is due to Lindenstrauss and Pełczyński. C OROLLARY 5.4 ([42]). C([0, 1]) is primary. P ROOF. Suppose C([0, 1]) is isomorphic to X ⊕ Y . Then X∗ or Y ∗ is non-separable, and hence either X or Y is isomorphic to C([0, 1]) by the preceding result. R EMARK 5.5. Actually, the stronger result is obtained in [42]: let X be a subspace of C([0, 1]). Then C([0, 1]) embeds in either X or C([0, 1])/X. Also, it is established in [5] and independently, in [16], that C(K) is primary for all countable compact K. Thus, all separable C(K)-spaces are primary. We next give characterizations of C([0, 1]) which follow from Theorem 5.3 and some rather deep general Banach space principles. We assume K is general, although the result easily reduces to the metrizable case. T HEOREM 5.6. Let X be a complemented subspace of C(K). The following are equivalent. (1) C([0, 1]) embeds in X. (2) 1 embeds in X. (3) L1 embeds in X∗ . (4) X∗ has a sequence which converges weakly but not in norm. P ROOF. The implications (2) ⇒ (3) and (1) ⇒ (3) are due to Pełczyński (for general Banach spaces X) [53]. (Actually, (3) ⇒ (2) is also true for general X, by [53] and a


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refinement due to Hagler [32].) Of course (1) ⇒ (2) is obvious, and so is (3) ⇒ (4), since 2 is isometric to a subspace of L1 . We show (4) ⇒ (2) ⇒ (1) to complete the proof. Let then (xn∗ ) in X∗ tend weakly to zero, yet for some δ > 0, ∗ x > δ n

for all n.

(5.1)

Then for each n, choose xn in Ba(X) with ∗ x (xn ) > δ. n

(5.2)

Now since X is complemented in C(K), X has the Dunford–Pettis property (i.e., X satisfies the conclusion of Theorem 4.1). But then (xn ) has no weak-Cauchy sequence.

(5.3)

Indeed, if a Banach space Y has the Dunford Pettis property, then yn∗ (yn ) → 0 as n → ∞ whenever (yn∗ ) is weakly null in Y and (yn ) is weak-Cauchy in Y ; so (5.3) follows in virtue of (5.2). But then by the 1 -theorem [58], (xn ) has a subsequence equivalent to the 1 -basis, hence (2) holds. (2) ⇒ (1) Let P : C(K) → X be a projection and let Y be a subspace of X isomorphic to 1 . Let Z be the conjugation-closed norm-closed unital subalgebra of C(K) generated by Y . Then by the Gelfand–Naimark theorem (which holds in this situation for real scalars also), Z is isometric to C(L) for some compact metric space L. Let T = P |Z. Since T |Y = I |Y , T ∗ (Z ∗ ) is non-norm-separable. Hence (1) holds by Theorem 4.10. For the remainder of our discussion, we assume the separable situation. Thus, K denotes a compact metric space; a “C(K)-space” refers to C(K) for some K, so it is separable. Now of course Theorem 5.3 reduces the CSP to the case of spaces X complemented in C(K) with X∗ separable. If the CSP has an affirmative answer, such an X must be c0 saturated (see Proposition 3.5). This motivates the following special case of the CSP, raised in the 70’s by the author. P ROBLEM 1. Let X be a complemented subspace of C(K) so that X contains a reflexive subspace. Is X isomorphic to C([0, 1])? Although this remains open, it was solved in such special cases as: 2 embeds in X, by Bourgain, in a remarkable tour-de-force. T HEOREM 5.7 ([19]). Let X be a Banach space and let T : C(K) → X fix a subspace Y ∗ ∗ of C(K) so that Y does not contain ∞ n ’s uniformly. Then T (X ) is non-separable. Of course then T fixes C([0, 1]) by Theorem 4.10, and so we have the C OROLLARY 5.8. Let X be complemented in C(K) and assume X contains a subspace Y which does not contain ∞ n ’s uniformly. Then X is isomorphic to C([0, 1]).

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For the remainder of our discussion, we focus on spaces X with separable dual. The next result is due to Benyamini, and rests in part on a deep lemma due to Zippin ([70,71]) which we will also discuss. T HEOREM 5.9 ([13]). Let X be a complemented subspace of C(K). Then either X is isomorphic to c0 or C(ωω +) embeds in X. The following result is an immediate consequence, in virtue of the decomposition method and weak injectivity of C(ωω +), i.e., Theorem 3.1. C OROLLARY 5.10. A complemented subspace of C(ωω +) is isomorphic to c0 or to C(ωω +). Now of course, Theorem 5.9 implies Zippin’s remarkable characterization of separably injective spaces, since if C(ωω +) embeds in X, it also embeds complementably, and hence X cannot be separably injective by Amir’s theorem [8], obtained via Theorem 3.14 above. In reality, Theorem 5.9 rests fundamentally on the main step in [70],which may be formulated as follows [13]. (Let us call βε (Ba X∗ ) the ε-Szlenk index of X, where βε is given in Definition 1.11.) L EMMA 5.11 ([70]). Let X be a Banach space with X∗ separable, and let 0 < ε < 1/2. There is a δ > 0 so that if W is a ω∗ -compact totally disconnected (1 + δ)-norming subset of Ba(X∗ ) and if γ < ωα+1 with α the δ-Szlenk index of X, then there exists a subspace Y of C(W ) with Y isometric to C(γ +) so that for all x ∈ X, there exists a y ∈ Y with iW x − y (1 + ε)iW x.

(5.4)

(Here, (iW x)(w) = w(x) for all w ∈ W . Also, iW = i if W = Ba X∗ .) Zippin also proved in [70] the interesting result that for any separable Banach space X and δ > 0, there is a (1 + δ)-norming totally disconnected subset of Ba(X∗ ). Benyamini establishes the following remarkable extension of this in the main new discovery in [13]. T HEOREM 5.12. Let X be a separable Banach space and ε > 0. There exists a ω∗ compact (1 + ε)-norming subset W of Ba(X∗ ) and a norm one operator E : C(W ) → C(Ba(X∗ )) so that EiW x − ix εx

for all x ∈ X.

(5.5)

The preceding two rather deep results hold for general Banach spaces X. In particular, the non-linear approximation resulting from Zippin’s Lemma (Lemma 5.11) shows that in a sense, the C(K)-spaces with K countable play an unexpected role in the structure of general X. The next quite simple result, however, needed for Theorem 5.9, bears solely on the structure of complemented subspaces of C(K)-spaces. It yields (for possibly nonseparable) X that if X is isomorphic to a complemented subspace of some C(K)-space, then X is already complemented in C(Ba X∗ ) and moreover, the best possible norm of the projection is found there.


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P ROPOSITION 5.13 ([14]). Let X be given, let L = Ba(X∗ ), and suppose λ 1 is such that for some compact Hausdorff space Ω, there exist operators U : X → C(Ω) and V : C(Ω) → X with IX = V ◦ U

and U V λ.

(5.6)

Then i(X) is λ-complemented in C(L). P ROOF. Without loss of generality, U = 1. Let Ω be regarded as canonically embedded in C(Ω)∗ . Thus letting ϕ = U ∗ |Ω, ϕ maps Ω into L. So of course ϕ ◦ maps C(L) into C(Ω). We now simply check that V ϕ ◦ i(x) = x

for all x ∈ X.

(5.7)

Then it follows that V ϕ ◦ is a projection from C(L) onto iX, and of course ◦ V ϕ V ϕ ◦ λ.

(5.8)

R EMARK 5.14. Theorem 5.12 and the preceding proposition may be applied to C(K)spaces themselves to obtain that C(K) is (1 + ε)-isomorphic to a (1 + ε)-complemented subspace of C(D), for all ε > 0. Thus the main result in [13] yields another proof of Milutin’s theorem. We prefer the exposition in Section 2, however, for the above result “loses” cc the isometric fact that C(K) → C(D) for all K. The next remarkable result actually yields most of the known positive results in our present context. T HEOREM 5.15 ([13]). Let X∗ be separable, with X a Banach space isomorphic to a complemented subspace of some C(K)-space. There exists a δ > 0 so that if α is the δSzlenk index of X and γ < ωα+1 , then X is isomorphic to a quotient space of C(γ +). P ROOF. By the preceding result, i(X) is already complemented in C(L) where L = Ba(X∗ ) in its ω∗ -topology. Let P : C(L) → i(X) be a projection and let λ = P . Now let 0 < ε be such that 1 ελ < . 2

(5.9)

Choose ε > δ > 0 satisfying the conclusion of Zippin’s lemma. Now choose W a (1 + δ)norming totally disconnected ω∗ -compact subset of L and a norm one operator E satisfying the conclusion of Theorem 5.12; in particular, (5.5) holds. Finally let x ∈ X, and choose y ∈ Y satisfying (5.4). Then since E = 1, Ey − EiW x εiW x.

(5.10)

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Then by (5.5)

Ey − ix ε iW x + x 2εx.

(5.11)

Since P ix = ix, we have P Ey − ix 2εP x 2ελx.

(5.12)

Since 2ελ < 1 by (5.6) and of course ix = x, it follows finally by (5.12) that P E|Y maps Y onto X, completing the proof. We now obtain the P ROOF OF T HEOREM 5.9. Suppose first that the ε-Szlenk index of X is finite for all ε > 0. Then by Theorem 5.15, there is a positive integer n so that X is isomorphic to a quotient space of C(ωn +). But in turn, C(ωn +) is isomorphic to c0 , and so X is thus isomorphic to a quotient space of c0 . Finally, every quotient space of c0 is isomorphic to a subspace of c0 by a result of Johnson and Zippin [35]. But X is also a L∞ -space by a result of Lindenstrauss and the author [41], and hence also by the results in [35], X is isomorphic to c0 . If the ε-Szlenk index of X is at least ω for some ε > 0, then X contains a subspace isomorphic to C(ωω +) by Alspach’s result, Theorem 4.7. Recall that a Banach space X is called an L∞ -space if there is a λ > 1 so that for all finite-dimensional E ⊂ X, there exists a finite-dimensional F with E ⊂ F ⊂ X so that

(5.13) d F, ∞ n λ, where n = dim F. If λ works, X is called an L∞,λ -space. Using partitions of unity, it is not hard to see that a C(K)-space is an L∞,1+ -space, i.e., it is an L∞,1+ε -space for all ε > 0. However, a Banach space X is an L∞,1+ -space if and only if it is an L1 (μ)-predual; i.e., X∗ is isometric to L1 (μ) for some μ. We shall discuss these briefly later on. The result of [41] mentioned above: a complemented subspace of an L∞ -space is also an L∞ -space. In general, L∞ -spaces are very far away from C(K)spaces; however the following result due to the author [60], shows that small ones are very nice. (The result extends that of [35] mentioned above.) P ROPOSITION 5.16 ([60]). Let X be a L∞ -space which is isomorphic to a subspace of a space with an unconditional basis. Then X is isomorphic to c0 . (This was subsequently extended in [29] to L∞ -spaces which embed in a σ –σ Banach lattice.) Theorem 5.15 actually yields that if X is as in its statement, there exists a countable compact K so that X and C(K) have the same Szlenk index, with X isomorphic to a quotient space of C(K). Remarkably, Alspach and Benyamini prove in [6] that for any L∞ -space X with X∗ separable, one has that C(K) is isomorphic to a quotient space of X, K as above. So in particular, using also a result from Section 2, we have


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T HEOREM 5.17 ([13,6]). Let X be isomorphic to a complemented subspace of a C(K)space with X∗ separable. Then the Szlenk index of X is ωα+1 for some countable ordinal α α and then X and C(ωω +) are each isomorphic to a quotient space of the other. Despite the many positive results discussed so far, the eventual answer to the CSP seems far from clear. We conclude this general discussion with two more problems on special cases. Let then X be isomorphic to a complemented subspace of a C(K)-space with X∗ separable. P ROBLEM 2. Does X embed in C(α+) for some countable ordinal α? What if Sz(X) = ω2 ? 2

Finally, what is the structure of complemented subspaces of C(ωω +)? Specifically, 2

P ROBLEM 3. Let X be a complemented subspace of C(ωω +) with Sz(X) = ω2 . Is X 2 isomorphic to C(ωω +)? If Sz(X) = ω3 , is X isomorphic to C(ωω +) itself? We next indicate complements to our discussion. Alspach constructs in [1] a quotient space of C(ωω +) which does not embed in C(α+) for any ordinal α; thus Problem 2 cannot be positively solved by just going through quotient maps. The remarkable fixing results Theorems 4.7 and 4.9 cannot be extended without paying some price. Alspach proves in 2 2 [4] that there is actually a surjective operator on C(ωω +) which does not fix C(ωω +). α+1 This result has recently been extended by Gasparis [28] to the spaces C(ωω +) for all ordinals α and an even wider class of counterexamples is given by Alspach in [4]. Thus an affirmative answer even to Problem 3 must eventually use the assumption that one has a projection, not just an operator. We note also results of Wolfe [69], which yield rather complicated necessary and sufficient conditions that an operator on a C(K)-space fixes C(α+). Some of the original motivation for the concept of L∞ -spaces was that these might characterize C(K)-spaces by purely local means. However Benyamini and Lindenstrauss discovered this is not the case even for L∞,1+ -spaces. They construct in [14] a Banach space X with X∗ isometric to 1 , such that X is not isomorphic to a complemented subspace of C([0, 1]). We note in passing, however, that the CSP is open for separable spaces X (in its statement) which are themselves L1 (μ) preduals. It is proved in [36] that separable L1 (μ) preduals X are actually isometric to quotient spaces of C([0, 1]). Hence if X is such a space and X∗ is non-separable, X contains for all ε > 0 a subspace (1 + ε)isomorphic to C([0, 1]), by the results of [56] discussed above. Also, it thus follows by Theorem 4.5 that separable L1 (μ) preduals contain isomorphic copies of c0 . Bourgain and Delbaen prove in [20] that separable L∞ -spaces are not even isomorphic to quotients of C([0, 1]) in general; they exhibit, for example, an L∞ -space such that every subspace contains a further reflexive subspace. For further counterexample L∞ -spaces of a quite general nature, see [21]. Here are some positive results on the structure of separable L∞ -spaces, which of course yield results on complemented subspaces of C(K)-spaces. Results of Lewis and Stegall

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[39] and of Stegall [66] yield that if X is a separable L∞ -space, then X∗ is isomorphic to 1 or to C([0, 1])∗ . Thus in particular, the duals of complemented subspaces of separable C(K)-spaces are classified. It is proved in [34] that every separable L∞ -space X has a basis which is moreover shrinking in case X∗ is separable. A later refinement in [49] yields that the basis (bj ) may be chosen with d([bj ]nj=1 , ∞ n ) λ for all n (for some λ); of course this characterizes L∞ -spaces. We conclude with a brief discussion of the positive solution to the CSP problem in the isometric setting. Let L be a locally compact 2nd countable metrizable space and let X be a contractively complemented subspace of C0 (L). Then X is isomorphic to a C(K)space. The reason for this: such spaces X are characterized isometrically as Cσ -spaces. This is proved for real scalars in [43], and for complex scalars in [27]. For real scalars, X is a Cσ subspace of C(L) provided there exists an involutive homeomorphism σ : L → L such that X = {f ∈ C(L): f (σ x) = −f (x) for all x ∈ X}. See [27] for the complex scalar case. It follows by results of Benyamini in [11] that such spaces are isomorphic to C(K)spaces; in fact it is proved in [11] that separable G-spaces are isomorphic to C(K)-spaces. This family of spaces includes closed sublattices of C(K)-spaces. It then follows (using the known structure of Banach lattices) that if a complemented subspace X of a separable C(K)-space is isomorphic to a Banach lattice, Xis isomorphic to a C(K)-space. On the other hand, it remains an open question, if complemented subspaces of Banach lattices are isomorphic to Banach lattices. We note finally that Benyamini later constructed a counterexample to his result in the non-separable setting, obtaining a non-separable sublattice of a C(K)-space which is not isomorphic to a complemented subspace of C(L) for any compact Hausdorff space L [12]. For further complements on the CSP in the non-separable setting, see [71]; for properties of non-separable C(K)-spaces, see [48] and [72]. We note finally the following complement to the isometric setting [7]. If a Banach space X is (1 + ε)-isomorphic to a (1 + ε)-complemented subspace of a C(K)-space for all ε > 0, then X is contractively complemented in C(L) where L = (Ba(X)∗ , ω∗ ), hence X is a Cσ -space.

References [1] D.E. Alspach, Quotients of C[0, 1] with separable dual, Israel J. Math. 29 (1978), 361–384. [2] D.E. Alspach, A quotient of C(ωω ) which is not isomorphic to a subspace of C(α), α < ω, Israel J. Math. 33 (1980), 49–60. [3] D.E. Alspach, C(K) norming subsets of C[0, 1]∗ , Studia Math. 70 (1981), 27–61. [4] D.E. Alspach, Operators on C(ωα ) which do not preserve C(ωα ), Fund. Math. 153 (1997), 81–98. [5] D.E. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals, Israel J. Math. 27 (1977), 64–92. [6] D.E. Alspach and Y. Benyamini, C(K) quotients of separable L∞ spaces, Israel J. Math. 32 (1979), 145– 160. [7] D.E. Alspach and Y. Benyamini, A geometrical property of C(K) spaces, Israel J. Math. 64 (1988), 179– 194. [8] D. Amir, Projections onto continuous functions spaces, Proc. Amer. Math. Soc. 15 (1964), 396–402. [9] S.A. Argyros and A.D. Arvanitakis, A characterization of regular averaging operators and its consequences, Studia Math. 151 (3) (2002), 207–226.


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[10] S.A. Argyros, G. Godefroy and H.P. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the Geometry of Banach spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003) (this Handbook). [11] Y. Benyamini, Separable G spaces are isomorphic to C(K) spaces, Israel J. Math. 14 (1973), 287–293. [12] Y. Benyamini, An M-space which is not isomorphic to a C(K) space, Israel J. Math. 28 (1–2) (1977), 98–102. [13] Y. Benyamini, An extension theorem for separable Banach spaces, Israel J. Math. 29 (1978), 24–30. [14] Y. Benyamini and J. Lindenstrauss, A predual of 1 which is not isomorphic to a C(K)-space, Israel J. Math. 13 (1972), 246–259. [15] C. Bessaga and A. Pełczyński, Spaces of continuous functions IV (On isomorphic classifications of spaces C(S)), Studia Math. 19 (1960), 53–62. [16] P. Billard, Sur la primarité des espaces C(α), Studia Math. 62 (2) (1978), 143–162 (French). [17] K. Borsuk, Über Isomorphie der Funktionalräume, Bull. Int. Acad. Polon. Sci. A 1/3 (1933), 1–10. [18] J. Bourgain, The Szlenk index and operators on C(K)-spaces, Bull. Soc. Math. Belg. Sér.B 31 (1) (1979), 87–117. [19] J. Bourgain, A result on operators on C[0, 1], J. Operator Theory 3 (1980), 279–289. [20] J. Bourgain and F. Delbaen, A class of special L∞ spaces, Acta Math. 145 (1981), 155–176. [21] J. Bourgain and G. Pisier, A construction of L∞ -spaces and related Banach spaces, Bol. Soc. Brasil Mat. 14 (2) (1983), 109–123. [22] J. Diestel, H. Jarchow and A. Pietsch, Operator ideals, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 437–496. [23] S. Ditor, On a lemma of Milutin concerning operators in continuous function spaces, Trans. Amer. Math. Soc. 149 (1970), 443–452. [24] S. Ditor, Averaging operators in C(S) and lower semicontinuous sections of continuous maps, Trans. Amer. Math. Soc. 175 (1973), 195–208. [25] N. Dunford and B.J. Pettis, Linear operators on summable functions, Trans. Amer. Math. Soc. 47 (1940), 232–392. [26] T. Figiel, N. Ghoussoub and W.B. Johnson, On the structure of non-weakly compact operators on Banach lattices, Math. Ann. 257 (3) (1981), 317–334. [27] Y. Friedman and B. Russo, Contraction Projections on C0 (K), Trans. Amer. Math. Soc. 273 (1982), 57–73. [28] I. Gasparis, A class of 1 -preduals which are isomorphic to quotients of C(ωω ), Studia Math. 133 (1999), 131–143. [29] N. Ghoussoub and W.B. Johnson, Factoring operators through Banach lattices not containing C(0, 1), Math. Z. 194 (2) (1987), 153–171. [30] A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type C(K), Canad. J. Math. 5 (1953), 129–173. [31] J. Hagler, Embedding of L1 -spaces in conjugate Banach spaces, Thesis, University of California at Berkeley (1972). [32] J. Hagler, Some more Banach spaces which contain L1 , Studia Math. 46 (1973), 35–42. [33] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. [34] W.B. Johnson, H.P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. [35] W.B. Johnson and M. Zippin, On subspaces and quotients of ( Gn )p and ( Gn )c0 , Israel J. Math. 13 (1972), 311–316. [36] W.B. Johnson and M. Zippin, Separable L1 preduals are quotients of C(Δ), Israel J. Math. 16 (1973), 198–202. [37] R. Kaufman, A type of extension of Banach spaces, Acta Sci. Math. V 26 (1965), 163–166. [38] E. Kirchberg, On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), 35–63. [39] D.R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to 1 (Γ ), J. Funct. Anal. 12 (1973), 177–187. [40] J. Lindenstrauss and H.P. Rosenthal, Automorphisms in c0 , 1 and m, Israel J. Math. 7 (1969), 227–239. [41] J. Lindenstrauss and H.P. Rosenthal, The Lp -spaces, Israel J. Math. 7 (1969), 325–349.

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[42] J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249. [43] J. Lindenstrauss and D.E. Wulbert, On the classification of the Banach spaces whose duals are L1 spaces, J. Funct. Anal. 4 (1969), 332–349. [44] H.P. Lotz and H.P. Rosenthal, Embeddings of C(Δ) and L1 [0, 1] in Banach lattices, Israel J. Math. 31 (1978), 169–179. [45] S. Mazurkiewicz and W. Sierpinski, Contribution à la topologie des ensembles dénombrables, Fund. Math. 1 (1920), 17–27. [46] S. Mercourakis, On Cesàro summable sequences of continuous functions, Mathematika 42 (1) (1995), 87– 104. [47] A.A. Milutin, Isomorphisms of spaces of continuous functions on compacts of power continuum, Tieoria Func. (Kharkov) 2 (1966), 150–156 (Russian). [48] S. Negrepontis, Banach spaces and topology, Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, eds, Elsevier (1984), Chapter 23, 1045–1142. [49] N.J. Nielsen and P. Wojtaszczyk, A remark on bases in Lp -spaces with an application to complementably universal L∞ -spaces, [50] T. Oikhberg and H.P. Rosenthal, Extension properties for the space of compact operators, J. Funct. Anal. 179 (2) (2001), 251–308. [51] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. [52] A. Pełczyński, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Polish Acad. Sci. Math. Astr. Phys. 10 (1962), 265–270. [53] A. Pełczyński, On Banach spaces containing L1 (M), Studia Math. 30 (1968), 231–246. [54] A. Pełczyński, On C(S)-subspaces of separable Banach spaces, Studia Math. 31 (1968), 231–246. [55] H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36; Correction, ibid., 311–313. [56] H.P. Rosenthal, On factors of C([0, 1]) with non-separable dual, Israel J. Math. 13 (1972), 361–378. [57] H.P. Rosenthal, On subspaces of Lp , Ann. Math. 97 (1973), 344–373. [58] H.P. Rosenthal, A characterization of Banach spaces containing 1 , Proc. Nat. Acad. Sci. USA 71 (1974), 2411–2413. [59] H.P. Rosenthal, Normalized weakly null sequences with no unconditional subsequences, Durham Symposium on the Relations Between Infinite-Dimensional and Finite-Dimensional Convexity, Bull. London Math. Soc. 8 (1976), 22–24. [60] H.P. Rosenthal, A characterization of c0 and some remarks concerning the Grothendieck property, Longhorn Notes, The University of Texas Functional Analysis Seminar (1982–83), 95–108. [61] H.P. Rosenthal, The complete separable extension property, J. Operator Theory 43 (2000), 324–374. [62] H.P. Rosenthal, Banach and operator space structure of C ∗ -algebras, to appear. [63] C. Samuel, Indice de Szlenk des C(K) (K espace topologique compact dé nombrable), Seminar on the Geometry of Banach Spaces, Vols. I, II, Publ. Math. Univ. Paris VII, Paris (1983), 81–91. [64] J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergence, Studia Math. 2 (1930), 58–62. [65] A. Sobczyk, Projection of the space (m) on its subspace (c0 ), Bull. Amer. Math. Soc. 47 (1941), 938–947. [66] C. Stegall, Banach spaces whose duals contain 1 (7), with applications to the study of dual L1 (M)-spaces, Trans. Amer. Math. Soc. 176 (1973), 463–477. [67] W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable Banach spaces, Studia Math. 30 (1968), 53–61. [68] L.W. Weis, The range of an operator in C(X) and its representing stochastic kernel, Arch. Math. 46 (1986), 171–178. [69] J. Wolfe, C(α) preserving operators on C(K) spaces, Trans. Amer. Math. Soc. 273 (1982), 705–719. Elsevier, Amsterdam (to appear). [70] M. Zippin, The separable extension problem, Israel J. Math. 26 (3–4) (1977), 372–387. [71] M. Zippin, Extensions of bounded linear operators, Handbook of Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1703–1736 (this Handbook). [72] V. Zizler, Nonseparable Banach spaces, Handbook of Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1737–1810 (this Handbook).

CHAPTER 37

Concentration, Results and Applications Gideon Schechtman∗ Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel E-mail: [email protected]

Contents 1. Introduction: approximate isoperimetric inequalities and concentration 2. Methods of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Isoperimetric inequalities, Brunn–Minkowski inequality . . . . . . 2.2. Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Product spaces. Induction . . . . . . . . . . . . . . . . . . . . . . . 2.4. Spectral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Bounds on Gaussian processes . . . . . . . . . . . . . . . . . . . . 2.6. Other tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Dvoretzky-like theorems . . . . . . . . . . . . . . . . . . . . . . . 3.2. Fine embeddings of subspaces of Lp in lpn . . . . . . . . . . . . . 3.3. Selecting good substructures . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. 1605 . 1606 . 1606 . 1610 . 1614 . 1619 . 1624 . 1625 . 1625 . 1625 . 1627 . 1631 . 1632

Abstract Concentration inequalities are estimates for the degree of approximation of functions on metric probability spaces around their mean. It turns out that in many natural situations one can give very good such estimates, and that these are extremely useful. We survey here some of the main methods for proving such inequalities and give a few examples to the way these estimates are used.

∗ The author was partially supported by The Israel Science Foundation founded by The Academy of Science and Humanities.



Concentration, results and applications

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1. Introduction: approximate isoperimetric inequalities and concentration Let (Ω, F , μ) be a probability space where F is the Borel σ -field with respect to a metric d on Ω. The isoperimetric problem for the probability metric space (Ω, F , μ, d) is: given 0 < a < 1 and ε > 0, what is inf μ(Aε ); A ∈ F , μ(A) = a ? and for what A is it attained. Here Aε , the ε neighborhood of A, is defined as Aε = {ω ∈ Ω; d(ω, A) < ε}. There are relatively few interesting cases, some of which will be described below, in which the answer to this question is known. However, it turns out that for many applications a solution to a somewhat weaker question is sufficient: instead of finding the actual infimum of the quantity above it is enough to find a good lower bound to μ(Aε ), subject to μ(A) = a. We shall refer to such a lower bound as a solution to the approximate isoperimetric inequality (for the given space and parameters) provided the solution is optimal except for absolute constants in the “right places”. Let us illustrate the above by the example most relevant for us. The space under question will be (S n−1 , F , μ, d). Here S n−1 is the unit sphere in Rn , d the geodesic distance, F the Borel σ -field and μ the normalized Haar measure (the unique probability measure on S n−1 which is invariant under the orthogonal group). Lévy [38] stated and sketched a proof of the isoperimetric inequality for this space. For every a and ε the minimal set is an (arbitrary) cap (i.e., a d-ball) of measure a. For a cap B of measure 1/2, Bε is a cap of radius π/2 + ε. A standard computation then implies that, for a = 1/2, say, and √ 2 any ε μ(Aε ) μ(Bε ) 1 − π/8 e−ε n/2 for any Borel set A ⊂ S n−1 of measure 1/2. 2 Any inequality, μ(Aε ) 1 − e−cε n , holding for all A with μ(A) = 1/2, with c an absolute constant, will be referred to as an approximate isoperimetric inequality (for sets of measure 1/2) in this case. As we shall see below these inequalities are extremely powerful, the value of the constant c is of little importance for the applications we have in mind, and it is much easier to prove the approximate inequality than the isoperimetric one. Moreover, several proofs of the approximate isoperimetric inequality in this case (and there are many of them) can be generalized to other situations in which no isoperimetric inequality is known. The importance of the approximate isoperimetric inequalities stems from the fact that they imply the following concentration phenomenon. In the setup above, if μ(Aε ) 1 − η/2 for all A with μ(A) 1/2 and if f : Ω → R is a function with Lipschitz constant 1, i.e., |f (x) − f (y)| d(x, y) for all x, y ∈ Ω, then μ({x; |f (x) − M| ε}) η. Here M denotes the median of the function f , i.e., is defined by μ({f M}), μ({f M}) 1/2. This is easily seen (and first noticed by Lévy in the setting of S n−1 ) by applying the inequality μ(Aε ) 1 − η/2 once for the set {f M} and once for {f M}. If η is small this is interpreted as “any such f is almost a constant on almost all of Ω”. For example, in the example above we get that any Lipschitz function of constant one, 2 f : S n−1 → R, satisfies μ({x ∈ S n−1 ; |f (x) − M| ε}) 2 e−ε n/2 , which is quite counterintuitive. The median M can be replaced by the expectation of f , Ef = S n−1 f dμ provided we change the constants 2, 1/2 to other absolute constants. Furthermore, each of these two

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concentration inequalities is also equivalent (with a change of constants) to

2 μ × μ (x, y) ∈ S n−1 × S n−1 ; f (x) − f (y) ε C e−cε n . This holds not only in this particular example but in great generality (see, for example, [45], V. 4). The opposite statement to the one in the second to last paragraph also holds. Concentration implies approximate isoperimetric inequality: if μ({x; |f (x) − M| ε}) η for all Lipschitz functions with constant one then μ(Aε ) > 1 − η for all sets A of measure at least 1/2. This follows easily by considering the function f (x) = d(x, A). Milman realized the relevance of Lévy’s concentration inequality to problems in Geometry and Functional Analysis. Using it he found in [43] a new proof of Dvoretzky’s theorem [11] on Euclidean section of convex bodies which was much more accessible than the complicated original proof. Much more importantly, his proof is subject to vast variations and generalizations. See Section 3.1 for this proof. Except for using the idea of concentration in many instances himself, Milman also promoted the search for new concentration inequalities and new applications of them. We refer the interested reader to an expository article [44] written by Milman on the subject. In this article we survey many (but not all) of the methods of proof of concentration and approximate isoperimetric inequalities. We tried to concentrate mostly on methods which are quite general or that we feel were not explored enough and should become more general. There are many different such methods with some overlap as to the inequalities they prove. Section 2 contains this survey. In Section 3 we give a sample of applications of concentration inequalities. There are many more such applications. At some points our presentation is very sketchy since on one hand many of the applications need the introduction of quite a lot of tools not directly connected to the main theme here and on the other hand some of the subjects dealt with in this application section are also dealt with, with more details in other articles in this Handbook. We hope we give enough to wet the reader’s appetite to search for more in the original sources or the other articles of this Handbook. We would like to emphasize that this is far from being a comprehensive survey of the topic of concentration. This author has a soft point for new ideas in proofs and in many instances below preferred to give a glimpse into these ideas by treating a special case or a version of the relevant result which is not necessarily the last word on it rather than to give all the details on the subject. There are two recent books related to the subject matter here. Ledoux’s book [34] is very much in the spirit of Sections 2.3 and 2.4. Chapter 3 12 in Gromov’s book [19] presents a different point of view on the subject of concentration. 2. Methods of proof 2.1. Isoperimetric inequalities, Brunn–Minkowski inequality We start by stating two forms of the classical Brunn–Minkowski inequality. Here | · | denotes Lebesgue measure in Rn and A + B denotes the Minkowski’s addition of sets in Rn ; A + B = {a + b; a ∈ A, b ∈ B}.


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T HEOREM 1. (1) For every n and every two nonempty measurable subsets of Rn A and B, |A + B|1/n |A|1/n + |B|1/n .

(1)

(2) For every n, every two non-empty measurable subsets of Rn A and B and every 0 < λ < 1, λA + (1 − λ)B |A|λ |B|1−λ .

(2)

Equality in either inequality holds if and only if A and B are homothetic. Theorem 1 has many different proofs. We refer to [56] for two of them and for an extensive discussion concerning this theorem. A variation of this theorem was proved by Prékopa and Leindler [50,36]. One possible proof of their theorem is by induction on the dimension (see, e.g., [49]). Theorem 1 is a simple consequence of this theorem. T HEOREM 2. Let f, g, h be integrable non-negative valued functions on Rn and let 0 < λ < 1. Assume

h λx + (1 − λ)y f (x)λ g(y)1−λ ,

for all x, y ∈ Rn ,

(3)

then

Rn

h

λ Rn

f

1−λ Rn

g

.

(4)

Theorem 1 provides a simple proof of the classical isoperimetric inequality in Rn . To avoid restricting ourselves to bodies whose surface area is definable we prefer to state it as: for every 0 < a < ∞ and every ε > 0, among all bodies of volume a in Rn the ones for which the volume of Aε is minimal are exactly balls of volume A. Maurey [42] noticed that Theorem 2 can be used to give a simple proof of the approximate isoperimetric inequality on the sphere (or equivalently for the canonical Gaussian measure on Rn ). Recently, Arias-de-Renya, Ball and Villa [4] discovered an even more direct proof of the approximate isoperimetric inequality on the sphere, using Theorem 1. Their proof actually establishes a far reaching generalization originally due to Gromov and Milman [21]. We refer to [25] for a discussion of the notion of uniform convexity. We only recall the following (equivalent) definition for the modulus of convexity δ of a normed space (X, · ): x + y ; x, y 1, x − y ε . δ(ε) = inf 1 − 2

(5)

Given a norm · on Rn we consider, in the following theorem, the set S = {x ∈ Rn ; x = 1} with the metric d(x, y) = x − y and the Borel probability measure μ(A) = |{tA; 0 t 1}|/|{x; x 1}|.

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T HEOREM 3. Let · be a norm on Rn and let δ be the modulus of convexity of (Rn , · ). Then for any Borel set A ⊂ S and any ε > 0, μ(Aε ) > 1 − 2μ(A)−1 e−2nδ(ε/2) .

(6)

P ROOF. Let K = {x; x 1} and ν the normalized Lebesgue measure on K. By considering the set {tA; 1/2 t 1} it is clearly enough to prove that, for B ⊂ K, ν(Bε ) > 1 − ν(B)−1 e−2nδ(ε) . Put C = {x ∈ K; d(x, B) ε} then, for all x ∈ B, y ∈ C, (x + y)/2 1 − δ(ε), i.e.,

B +C ⊂ 1 − δ(ε) K 2 therefore, by the Brunn–Minkowski inequality,

2n ν(B)ν(C) 1 − δ(ε) e−2nδ(ε) .

Since for the Euclidean norm on Rn , δ(ε) ε2 /8, we get a simple proof of the approximate isoperimetric inequality for the sphere S n−1 (with the Euclidean or geodesic distance and Haar measure) discussed in the introduction. C OROLLARY 4. If A ⊂ S n−1 and ε > 0 then μ(Aε ) > 1 − 2μ(A)−1 e−nε

2 /16

.

Consequently, if f : S n−1 → R is a function with Lipschitz constant 1 then

2 μ x; |f (x) − M| ε 8 e−nε /16 . There are several ways to prove the isoperimetric inequality (as opposed to approximate isoperimetric inequalities) on the sphere. Some of them generalize to give isoperimetric inequalities in other situations. We refer to Appendix I in [45] in which Gromov presents a generalization based on Levy’s original proof and proves an isoperimetric inequality for Riemannian manifolds in term of their Ricci curvature. A particularly useful instance of this generalization is the case of O(n) equipped with its Haar measure and Euclidean metric (i.e., the Hilbert–Schmidt norm). [14] contains a relatively easy and self-contained proof of the isoperimetric inequalities on the sphere by symmetrization. It seems however to be very special to S n−1 . We now sketch very briefly a proof by another method of symmetrization which is not very well known and which we think deserves to be better known. It seems to have the potential to generalize to other situations, see, for example, the last paragraph of this subsection. The method is due to Baernstein II and Taylor [6] and is written in detail with indications towards generalizations in [7]. S KETCH OF PROOF OF L EVY ’ S ISOPERIMETRIC INEQUALITY . Given a hyperplane H through zero in Rn we denote S0 = S n−1 ∩ H and by S+ and S− the two open half spheres


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in the complement of H . Let also σ = σH be the reflection with respect to H . Of course σ is an isometry with respect to the (Euclidean or geodesic) metric on S n−1 , it satisfies σ 2 = identity and preserves the Haar measure. It also satisfies that if x, y ∈ S+ then d(x, y) d(x, σ (y)). Given a set A ⊆ S n−1 we define its two point symmetrization A∗ with respect to the above decomposition as

A∗ = A ∩ (S+ ∪ S0 ) ∪ A ∩ S− ∩ σ (A ∩ S+ ) ∪ σ (A ∩ S− \ σ (A ∩ S+ )) , i.e., we “push up” elements of A ∩ S− into S+ using σ whenever there is space available. The term symmetrization seems a bit misleading since we desymmetrize as far as symmetry with respect to H is concerned. The point of course is that A∗ is closer to cap than A is and in that sense is more symmetric. Note that if A is Borel, μ(A∗ ) = μ(A). It is also easy to prove that for every ε > 0 and for every A ⊆ S n−1 ∗

A ε ⊆ (Aε )∗ . In particular,

μ A∗ ε μ (Aε )∗ μ(Aε ). The definition of the symmetrization procedure and the last property hold for any metric probability space (K, μ) admitting an isometric and measure preserving involution σ and any partition of the complement of K0 = {x; x = σ (x)} into K− , K+ provided this involution and partition satisfy the following properties: K+ = σ (K− ) and d(x, y) d(x, σ (y)) for all x, y ∈ K+ . To prove the isoperimetric inequality we would like to apply the operation A → A∗ with respect to many hyperplanes, reach a set so that no farther application of this operation improves μ(Aε ) and prove that such a set must be a cap. We’ll sketch in a minute how to do that for S n−1 but we would like to emphasize again that this seems plausible in other situations as well and we think it deserves further investigation. Consider the metric space C of all closed subsets of S n−1 with the Hausdorff metric. Fix A ∈ C and consider the set B ⊆ C of all sets B ∈ C satisfying: • For all ε > 0 μ(Bε ) μ(Aε ) and • μ(B) = μ(A). One checks that the set B is closed in C. Fix a point x0 ∈ S n−1 and let C be the closed cap centered at x0 with measure μ(A). It is enough to prove that C ∈ B. For any hyperplane H with x0 ∈ / H we denote by S+ the open half sphere containing x0 . One now proves that B → μ(B ∩ C) is upper semicontinuous on C. Consequently, μ(B ∩ C) attains its maximum on B, say at B. We shall show that B ⊇ C which will prove the claim. If this is not the case then μ(B \ C) = μ(C \ B) > 0. Let x ∈ B \ C and y ∈ C \ B be points of density of the respective sets and let H be the hyperplane perpendicular to the segment [x, y] and crossing it at the midpoint (x + y)/2. Let B(x, r) ⊂ S− , B(y, r) ⊂ S+ be small balls such that μ(B(x, r) ∩ (B \ C)) >

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0.99μ(B(x, r)) and μ(B(y, r) ∩ (C \ B)) > 0.99μ(B(y, r)). Applying the symmetrization B → B ∗ with respect to this hyperplane, most of B(x, r) will be transferred into B(y, r) while no point of C ∩ B is transferred to a point which is not in C. Thus, μ(B ∗ ∩ C) > μ(B ∩ C). Since B ∗ also belongs to B we get a contradiction. With a bit more effort the proof above can be adjusted to show that caps are the only solutions to the isoperimetric problem in S n−1 . The method of proof presented here can be used to prove other isoperimetric-like inequalities. [13] contains an explicit example in which the setting is similar to the one above (i.e., dealing with subsets of S n−1 ) with the difference that one measures the ε-boundary of a set A in a different way: the measure of the set of all pairs (x, y) with x ∈ A, y ∈ /A and the distance between x and y is at most ε. In [13] this is used to solve a (discrete!) problem concerning an efficient algorithm for approximating the maximal cut in a graph.

2.2. Martingales Recall that for f ∈ L1 (Ω, F , P ) and for G, a sub σ -algebra of F , the conditional expectation, E(f |G), of f given G is the unique h ∈ L1 (Ω, G, P|G ) satisfying

h dP = A

f dP

for all A ∈ G.

(7)

A

(h is the Radon–Nikodým derivative of the measure ν(A) = A f dP on G with respect to P|G .) The correspondence f → E(f |G) is a linear positive operator of norm one on all the spaces Lp (Ω, F , P ), 1 p ∞. Some additional properties of this operator are: • If G ⊂ G is a sub σ -algebra then E(E(f |G)|G ) = E(f |G ). • If g ∈ L∞ (Ω, G, P ) then E(fg|G) = gE(f |G). • For the trivial σ -algebra G = {∅, Ω}, E(f |G) = Ef , the expectation of f . Given a finite or infinite increasing sequence of σ -algebras, F0 , F1 , . . . , a sequence of elements of L1 (Ω, F , P ), f0 , f1 , . . . , is said to be a martingale with respect to F0 , F1 , . . . if fi = E(fj |Fi ) for all i j . We shall always assume here that F0 is the trivial σ -algebra {∅, Ω} and that the sequence is finite with the last terms being fn = f and Fn = F . Then, fi = E(f |Fi ), i = 0, 1, . . . , n. We also denote di = fi − fi−1 , i = 1, 2, . . . , n, and call the sequence {di }ni=1 the martingale difference sequence. One set of examples of a martingale is the following: let Xi be a sequence of mean zero independent random variables and put fi = ij =0 Xj , then {fi } is a martingale with respect to {Fi } where Fi is the smallest σ -algebra with respect to which X0 , . . . , Xi are measurable. In a lot of senses a general martingale resembles this particular set of examples. There are many inequalities estimating the probability of the deviation of f = fn from f0 = Ef in terms of the behavior of the sequence {di }. In the next proposition we gather some of them. (1) is due to Azuma [5] or [57], p. 238. (2) and (3) are due to Pisier [47], (2) was first used in [26]. (4) is a generalization to the martingale case of Prokhorov’s inequality. In a somewhat weaker form it first appears in [29]. The form here is from [23].


1611

P ROPOSITION 5. (1) For all t > 0, n

2 2 di ∞ . P ω; |f (ω) − Ef | t 2 exp −t /2

(8)

i=1

(2) For all 1 0, P

q

ω; |f (ω) − Ef | t K exp −δ t/ di ∞ p,∞ ,

(9)

where q −1 + p−1 = 1, K and δ depend only on p and {ai }ni=1 p,∞ = max1j n j 1/p aj∗ with {aj∗ } denoting the decreasing rearrangement of the sequence {|aj |}. (3) For all t > 0, P

ω; |f (ω) − Ef | t K exp − exp δt/ di ∞ 1,∞ ,

(10)

where K and δ are absolute constants. (4) Put M = max1in di ∞ and S 2 = ni=1 E(di2 |Fi−1 )∞ . Then, for all t > 0, P

Mt t · arc sinh ω; |f (ω) − Ef | t 2 exp − . 2M 2S 2

(11)

The proofs of these and similar inequalities are usually quite simple. Let us sketch the proof of (1). If Fi is “rich” enough, extreme points in the set {d ∈ L∞ (Ω, Fi , P ); E(d|Fi−1 ) = 0, |d| a} have constant absolute value equal to a. Consequently for all λ ∈ R,

2 2 E eλdi |Fi−1 cosh λdi ∞ eλ di ∞ /2 .

(12)

Extending Fi (to become rich enough) if necessary, this inequality holds always. It follows that E eλ

n

i=1 di

2 n−1 2 = E E eλ i=1 di |Fn−1 eλ dn ∞ /2 .

(13)

Iterating this (by applying E(·|Fn−2 ), then E(·|Fn−3 ) . . .) we get E eλ(f −Ef ) eλ

2 n d 2 /2 i ∞ i=1

(14)

.

Applying Chebyshev’s inequality we get, for positive λ, P

ω; f (ω) − Ef t P ω; eλ(f (ω)−Ef )−λt 1 e−λt E eλ(f −Ef ) e−λt +λ

2

n

2 i=1 di ∞ /2

.

Minimizing over positive λ and repeating this with negative λ we get the result.

(15)

1612

G. Schechtman

Yurinski [64] was probably the first to use martingale inequalities in the context of Banach space valued random variables. The point is that if Xi are independent Banach space valued random variables and we form the martingale fi = E( nj=1 Xj |Fi ) then the martingale differences satisfy |di | Xi . This can be used to estimate the tail behavior of nj=1 Xj . Maurey [41] noticed that martingale deviation inequalities can be used to prove approximate isoperimetric inequality for the interesting case of the permutation group. We present a somewhat simplified version of his proof with some abstractization ([51,45]). The length of a finite metric space (Ω, d) is defined as the infimum of = ( ni=1 ai2 )1/2 over all sequences a1 , . . . , an of positive numbers satisfying: there exists a sequence {Ωk }nk=0 of partitions of Ω with • Ω0 = {Ω} and Ωn = {{ω}}ω∈Ω . • Ωk refines Ωk−1 , k = 1, . . . , n. • If k = 1, . . . , n, A ∈ Ωk−1 , B, C ⊂ A and B, C ∈ Ωk then there is a one to one map h from B onto C such that d(ω, h(ω)) ak for all ω ∈ B. The two basic examples we shall deal with are the Hamming cube, Hn , and the permutation group, Πn . The Hamming cube is the set {0, 1}n with the metric d((εi )ni=1 , (δi )ni=1 ) = #{i; εi = δi }. Πn is the set of permutations of {1, √ d(π, ϕ) = √ 2, . . . , n} with the metric #{i; π(i) = ϕ(i)}. The length is smaller or equal n in the first case and 2 n − 1 in the second. Let us illustrate this in the second example. Fix 1 k n − 1 and i1 , i2 , . . . , ik distinct elements of {1, 2, . . . , n}. Put Ai1 ,i2 ,...,ik = π ∈ Πn ; π(1) = i1 , . . . , π(k) = ik

(16)

and let Ωk be the partition whose atoms are all the sets Ai1 ,i2 ,...,ik where (i1 , i2 , . . . , ik ) ranges over all n!/(n − k)! possibilities. It is clear that the first two requirements from {Ωk }n−1 k=0 are satisfied (with n − 1 replacing n). To show that the third one is satisfied with ai = 2 for i = 1, . . . , n − 1, let A = Ai1 ,i2 ,...,ik−1 ∈ Ωk−1 and B = Ai1 ,i2 ,...,ik−1 ,r , C = Ai1 ,i2 ,...,ik−1 ,s ∈ Ωk and define h : B → C by h(π) = (r, s) ◦ π (where (r, s) is the transposition of r and s). We are now ready to state the main theorem of this section. T HEOREM 6. Let (Ω, d) be a finite metric space of length at most . Let P be the normalized counting measure on Ω. Then, (1) Let f : Ω → R satisfy |f (x) − f (y)| d(x, y) for all x, y ∈ Ω. Then for all t > 0, P

ω; |f (ω) − Ef | t

2 exp −t 2 /22 .

(17)

(2) Let A ⊂ Ω with P (A) 1/2 then for all t > 0

P (At ) 1 − 2 exp −t 2 /82 .

(18)

S KETCH OF PROOF. Let = ( ni=1 ai2 )1/2 with ai and Ωi , i = 0, . . . , n, as in the definition of length. Let Fi be the field generated by Ωi and form the martingale fi = E(f |Fi ), i = 0, . . . , n. Note that fi is constant on each atom B of Ωi and that this constant is


1613

fi|B = Avex∈B f (x). If B, C are two atoms of Ωi contained in an atom A of Ωi−1 then by the third property of the sequence of partitions, |fi|B − fi|C | = |B|

f (x) − f h(x) ai .

−1

(19)

x∈B

Since fi−1|A is the average of fi|B over all atoms B of Fi which are subsets of A, we get from (19) that |fi−1|A − fi|C | ai and since this holds for all such A and C, di ∞ ai . Now apply 5(1). This proves (1). (2) follows from (1) as explained in the introduction. C OROLLARY 7. Let (Ω, d) be either Hn or Πn . (1) Let f : Ω → R satisfy |f (x) − f (y)| d(x, y) for all x, y ∈ Ω. Then for all t > 0, P

ω; |f (ω) − Ef | t 2 exp −t 2 /8n .

(20)

(2) Let A ⊂ Ω with P (A) 1/2 then for all t > 0

P (At ) 1 − 2 exp −t 2 /32n .

(21)

By considering a ball in the Hamming metric it is easy to see that, except for the choice of the absolute constants involved, the result for Hn is best possible. In this case, the exact solution to the isoperimetric problem is known as well (and, for sets of measure 2k /2n , is a ball) [22,15]. For sets of measure of the form 2k /2n this can also be deduced from the method of two-point symmetrization introduced in the previous section. For Πn the solution to the isoperimetric problem is not known. However, again except for the absolute constants involved, the corollary gives the right result: E XAMPLE 8. Let n be odd and define A ⊂ Π2n by A = π; π(i) n for more than n/2 indices i with 1 i n .

(22)

Then, μ(A) = 1/2 and for all k < n/2,

P (Ack )

1 = (2n)! 1 = 2n

n

[ n2 −k]+1

l=0

n n! n! n! l (n − l)! l!

n2 . l

[ n2 −k]+1

(23)

l=0

For k with k/n bounded away from 0 and 1/2, a short computation shows that this is larger 2 than e−δk /n .

1614

G. Schechtman

It is also not hard to see that, at least for some a and t, balls are not the solution to the isoperimetric problem inf{P (At ); P (A) = a} on Πn . We wonder whether there is an equivalent, with constants independent of n (and hopefully natural), metric on Πn for which one can solve the isoperimetric problem. The advantage of the method described above is in its generality; in principle, whenever we have a metric probability space we can estimate its length by trying different sequences of partitions and get some approximate isoperimetric inequality. In reality it turns out that in most specific problems, and in particular when the space is naturally a product space, one gets better results by other methods.

2.3. Product spaces. Induction In [58] Talagrand introduced a relatively simple but quite powerful method to prove concentration inequalities which works in many situations in which the probability space is a product space with many components. The proofs, as naive as they may look, are by induction on the number of components. The monograph [62] contains many more instances in which variants of this method work. Another feature in Talagrand’s work is the deviation from the traditional way of measuring distances; the “distance” of a point from a set is not always measured by a metric. We start with a small variation on the original theorem of Talagrand taken from [27]. T HEOREM 9. Let Ωi ⊂ Xi , i = 1, . . . , n, be compact subsets of normedspaces with diam(Ωi ) 1. Consider Ω = Ω1 × Ω2 × · · · × Ωn as a subset of the 2 sum ( ni=1 ⊕Xi )2 . Let μi be a probability measure on Ωi , i = 1, . . . , n, and put P = μ1 × μ2 × · · · × μn . For a compact A ⊂ Ω denote the convex hall of A byconv(A) and for x ∈ Ω put ϕ(x, A) = dist(x, conv(A)) (with respect to the metric in ( ni=1 ⊕Xi )2 ). Then eϕ

(1)

2 (x,A)/4

1 . P (A)

(24)

In particular, for all t > 0, P

x; ϕ(x, A) > t

1 2 e−t /4 . P (A)

(2) If f : Ω → R is convex and Lipschitz (with respect to the metric of ( with constant 1 then 2 P x; f (x) − f > t 4 e−ct

(25) n

i=1 ⊕Xi )2 )

(26)

for all t > 0 and some universal c > 0. S KETCH OF PROOF. The proof of the first assertion of (1) is by induction. The second assertion of (1) and also (2) (with a bit more effort) follow as in (15). The other


1615

theorems in this section are proved similarly. We shall illustrate the induction step. 2 Assume that eϕ (x,A)/4 dP (x) 1/P (A) for all compact A ⊂ Ω = Ω1 × · · · × Ωn and let A ⊂ Ω × Ωn+1 . For ω ∈ Ωn+1 put A(ω) = {x ∈ Ω; (x,ω) ∈ A} (where, for x = (x1 , . . . , xn ) ∈ Ω, (x, ω) = (x1 , . . . , xn , ω)). Put also B = ω∈Ωn+1 A(ω). Fix a y = (x, ω) ∈ Ω × Ωn+1 and notice that ϕ(y, A) ϕ(x, A(ω)) provided A(ω) = ∅. Also, ϕ(y, A) ϕ(x, B) + 1. From these two inequalities it is easy to deduce that, for all 0 λ 1, ϕ 2 (y, A) λϕ 2 (x, A(ω)) + (1 − λ)ϕ 2 (x, B) + (1 − λ)2 . Using Hölder’s inequality and the induction hypothesis, one gets, for all ω ∈ Ωn+1 , eϕ

2 ((x,ω),A)/4

Ω

2 e(1−λ) /4 P (A(ω)) −λ . P (B) P (B)

(27)

We now use a numerical inequality (which can serve as a good Calculus exercise). For all 0 p 1, inf p−λ e(1−λ)

0p1

2 /4

2 − p.

Using this inequality with p = P (A(ω))/P (B) and integrating (27) over ω, we get P × μn+1 (A) 1 1 ϕ 2 ((x,ω),A)/4 e 2− . (28) P (B) P (B) P × μn+1 (A) Ωn+1 Ω Note that if Xi = {−1, 1} with the uniform measure for each i then by Corollary 7 the same conclusion as in Theorem 9(2) holds for any (i.e., not necessarily convex) function satisfying |f (x) − f (y)| n−1/2 |x i − yi |. However, for a convex function, Theorem 9 gives a much better result since n−1/2 |xi − yi | ( |xi − yi |)1/2 . The theorem above has the disadvantage that, because of the convexity assumption, it applies only to Ωi ’s which lie in a linear space. This is taken care of in the next theorem from [62] which surprisingly is extremely applicable. Given * (Ωi , Fi , μi ), i = 1, . . . , n, form the product space (Ω, P ) with * probability spaces Ω = ni=1 Ωi and P = μi . For x, y ∈ Ω let U (x, y) be the sequence in {0, 1}n which realizes the Hamming distance between x and y, i.e., has 0 exactly in the coordinates i where xi = yi . For a subset A of Ω and for x ∈ Ω we set U (x, A) to be the subset of {0, 1}n consisting of all sequences U (x, y) for some y ∈ A, i.e., U (x, A) = {εi }ni=1 ∈ {0, 1}n ; for some y ∈ A, yi = xi iff εi = 0 . For x ∈ Ω and A ⊂ Ω let ϕ(x, A) = d(0, conv(U (x, A))). It should be noted that, in general, ϕ(x, A) is not induced by a metric. i.e., there is no metric d on Ω such that ϕ(x, A) = inf{d(x, y); y ∈ A}. This is easily seen to be the case for Ω = {0, 1}n , for example. T HEOREM 10. Let A ⊂ Ω then 1 2 eϕ (x,A)/4 . P (A)

(29)

1616

G. Schechtman

In particular, for all t > 0, P

x; ϕ(x, A) > t

1 2 e−t /4 . P (A)

Using the Hahn–Banach theorem one can show that ϕ(x, A) = sup inf αi ; y ∈ A .

αi2 =1

(30)

(31)

{i;yi =xi }

Notice that, if h denotes the Hamming distance on Ω, i.e., h(x, y) = #{i; yi = xi }, then formula (31) implies that ϕ(x, A) h1/2 (x, A). Using this inequality and the martingale 2 method of Section 2.2 one gets only P (ϕ(x, A) > t 1/2 ) C e−ct /n while Theorem 10 2 gives P (ϕ(x, A) > t 1/2 ) < 4 e−t /4 4 e−t /4n for t in the relevant range, 0 < t < n. This illustrates the possible advantage of this inequality over Corollary 7 for Hn . Theorem 10 has many applications. We refer to [62] for some of them. A variant of Theorem 9 and particularly of (26) was recently proved by Ledoux ([32] or [33]). The difference is that the convexity assumption on f is weakened to convexity of each variable separately but the conclusion is only a one-sided deviation inequality: 2 P x; f (x) − f > t 4 e−ct . (32) It is unknown whether a similar lower deviation inequality also holds. The next result was first proved by Talagrand in [59]. The original proof was very complicated but in [62] Talagrand presented a much simpler*inductive proof * which we shall sketch here. Consider a product probability space (Ω = ni=1 Ωi , P = ni=1 μi ). Given a q ∈ N and q + 1 elements of Ω, x, y 1, . . . , y q , we define the “Hamming distance” of x from the q-tuple y 1 , . . . , y q by

q (33) / yi1 , . . . , yi . h x; y 1, . . . , y q = # i; xi ∈ Given q subsets A1 , . . . , Aq of Ω, we define

h(x; A1, . . . , Aq ) = inf h x; y 1, . . . , y q ; y 1 ∈ A1 , . . . , y q ∈ Aq . T HEOREM 11. q h(x;A1,...,Aq ) *q

1 . j =1 P (Aj )

(34)

(35)

In particular, P

for all k ∈ N.

x; h(x; A1, . . . , Aq ) k

1 q −k P (A ) j j =1

*q

(36)


1617

S KETCH OF PROOF OF THE INDUCTION STEP. For A1 , . . . , Aq ⊂ Ω × Ωn+1 and ω ∈ Ωn+1 put Aj (ω) = y ∈ Ω; (y, ω) ∈ Aj ,

j = 1, . . . , q,

(37)

and

Bj =

Aj (u),

j = 1, . . . , q.

(38)

u∈Ωn+1

Fix ω ∈ Ω and k ∈ {1, . . . , q} and put also Cj =

Bj Ak (ω)

if j = k, if j = k.

(39)

One then shows that

h (x, ω); A1, . . . , Aq min 1 + h(x; B1 , . . . , Bq ), h(x; C1, . . . , Cq ) .

(40)

It then follows from the induction hypothesis that P (Bk ) 1 min q, min . 1kq P (Ak (ω)) j =1 P (Bj )

q h((x,ω);A1,...,Aq ) *q

(41)

If 0 hi 1, i = 1, . . . , q, are functions on a probability space then

−1 q $ −1 min q, min hi . hi 1iq

(42)

i=1

This follows easily from the inequality h−1 ( h)q 1 which holds for every function h satisfying q −1 h 1. Using (42) and integrating (41) over Ωn+1 , we get the assertion for n + 1. We shall see in a minute the big advantage of this theorem over the concentration inequality for the Hamming metric. Although it looks like there is not much difference between h(·; A, A), say, and the Hamming distance of a point from a set (d(·, A) of Section 5), it turns out that the last theorem gives much better concentration when it applies. Theorem 11 is still looking for good applications. As far as we know Theorem 11 has basically one application dealing with the tail behavior of norms of sums of independent Banach space valued random variables. This is the original application which led Talagrand to prove this result (see [59] and [62], Section 13). This particular application also has a different proof [31].

1618

G. Schechtman

To illustrate the advantage of Theorem 11 over the basic inequality for the Hamming metric we define a class of functions and state a corollary which amounts to a deviation inequality for this class of functions. For I ⊂ {1, . . . , n} denote ΩI =

$

Ωi

and Ω ∗ =

i∈I

ΩI

I ⊂{1,...,n}

and let f : Ω ∗ → R+ . We say that f is monotone if I ⊂ J ⊆ {1, , . . . , n}

implies f (xi )i∈I f (xj )j ∈J

(43)

for all (xj )j ∈J ∈ ΩJ . We say that f is subadditive if for all I, J disjoint subsets of {1, . . . , n} and all (xi )i∈I ∪J ∈ ΩI ∪J ,

f (xi )i∈I ∪J f (xi )i∈I + f (xj )j ∈J .

(44)

Here is an example of such afunction: let Ωi be subsets of a normed space (X, · ) and put f ((xi )i∈I ) = Aveεi =±1 i∈I εi xi . For x ∈ ΩI , y ∈ ΩJ we shall denote by h(x, y) the number of coordinates in which xi = yi including coordinates in which one or both of xi , yi are not defined. C OROLLARY 12. Let f : Ω ∗ → R+ be monotone, subadditive and satisfy |f (x)−f (y)| h(x, y) for all x, y ∈ Ω ∗ . Then, for all a > 0, 1 k n and q ∈ N, P

x ∈ Ω; f (x) (q + 1)a + k

P (f a)−q q −k .

(45)

For a being the median of f and q = 2, say, one gets P (f 3a + k) 42−k . If a ) k ) n this is much better than what one gets for a general Lipschitz function from, e.g., 2 the martingale method. There one gets P (f a + k) 2 e−k /4n . Note the resemblance with the situation concerning Theorem 9: in both cases we evaluate the probability of deviation of f from its expectation (or median), a quantity which depends only on the behavior of f on Ω (since the probability measure is supported there). However, by extending f to a larger set (in Theorem 9 the convex hull of Ω, here Ω ∗ ), if possible, using its Lipschitz constant on the larger set and some additional properties of the extended function (there convexity, here monotonicity and subadditivity) we get, in some cases a stronger concentration result than the basic one. P ROOF OF C OROLLARY 12. For 1 i q put Ai = A = {x ∈ Ω; f (x) a}. Then

f (x) (q + 1)a + k ⊆ h(x; A1, . . . , Aq ) k .

Indeed, if h(x; A1, . . . , Aq ) < k, let y 1 , . . . , y q ∈ A be such that, putting I = q {i; xi ∈ / {yi1 , . . . , yi }}, #I < k. The complement of I can be written as kj =1 Jj with


1619

j

Jj ⊆ {1, . . . , n} satisfying xi = yi for i ∈ Jj . Then, assuming I is not empty, f (x) f (x|I ) +

q

f (x|Jj )

j =1

j

1 + f y|Jj f x|I + y|J 1 q

j =1

q 1 j

#I + f y|J1 + f y|Jj j =1 q

j

f y #I + f y 1 + j =1

< k + (q + 1)a.

(46)

The corollary follows now immediately from Theorem 11.

The paper [62] also contains a generalization of the concentration inequality for the permutation group, Corollary 7. The (inductive) proof of this result is a bit harder than the other proofs surveyed in this section and we shall not reproduce it. This result also awaits good applications. Equip the symmetric group Sn with its natural probability measure, μ. For σ ∈ Sn and A ⊆ Sn let n 2 f (σ, A) = inf si ; (s1 , . . . , sn ) ∈ VA (σ ) , (47) i=1

where VA (σ ) is the convex hall of the set (s1 , . . . , sn ) ∈ {0, 1}n ; ∃τ ∈ A s.t. ∀i n, si = 0 ⇒ τ (i) = σ (i) . T HEOREM 13. For every A ⊂ Sn , t > 0

μ σ ; f (σ, A) > t

1 e−t /16. μ(A)

(48)

The manuscript [62] contains many refinements of Theorems 10, 11 and 13 which we do not reproduce here.

2.4. Spectral methods Let (Ω, F , μ) be a probability space, A some set of measurable functions and E : A → R+ some function (which we shall refer to as energy function). For f ∈ L2 (Ω) denote by

1620

G. Schechtman

σ 2 (f ) the variance of f , σ 2 (f ) =

and, for f ∈ L2 log L (i.e.,

f log f dμ − 2

f 2 dμ −

2 f dμ

(49)

f 2 log+ f dμ < ∞), denote by ε(f ) the entropy of f 2 ,

ε(f ) =

(f − Ef )2 dμ =

2

2 f dμ log f dμ 2

(50)

(which is necessarily finite). We say that (A, E) satisfy a Poincaré inequality with constant C if σ 2 (f ) CE(f )

for all f ∈ A.

(51)

We say that (A, E) satisfy a logarithmic Sobolev inequality with constant C if ε(f ) CE(f )

for all f ∈ A.

(52)

The main example of an energy function E is related to the gradient or generalization of it. If d is a metric on Ω (and F the Borel σ -field), define the norm of the gradient at x ∈ Ω by ∇f (x) = lim sup |f (x) − f (y)| . d(x, y) y→x

(53)

Note that ∇f (x) by itself is not defined. The reason for this notation is of course that if (Ω, d) is a Riemannian manifold (in particular if it is Rn with the Euclidean distance) and if f is differentiable at x then |∇f (x)| is the Euclidean norm of the gradient of f at x. Define now 2 (54) E(f ) = ∇f (x) dμ(x). The classical Poincaré (or Rayleigh–Ritz) inequality says that, in the case of a compact Riemannian manifold, (51) is satisfied with C = λ−1 1 , λ1 being the first positive eigenvalue of the Laplacian on L2 (Ω, μ). We shall only deal here with the energy function (54). [33] contains many other examples and a comprehensive treatment of the subject of this section. If A is the set of bounded Lipschitz functions on (Ω, d), the norm of the gradient satisfies the chain rule: if φ ∈ C 1 (R) and f ∈ Ω then φ ◦ f ∈ Ω and

∇(φ ◦ f )(x) ∇f (x)φ f (x)

(55)


1621

and consequently E(φ ◦ f ) f 2Lip

φ f (x) 2 dμ(x),

(56)

where f Lip denotes the Lipschitz constant of f . The next theorem, basically due to Gromov and Milman, shows that Poincaré inequality implies concentration. T HEOREM 14. Let (Ω, F , μ, d) be a probability metric space. Let A be the set of bounded Lipschitz functions on (Ω, d) and let E be defined √by (54). Assume that (A, E) satisfies the Poincaré inequality (51). Then for all |λ| < 2/ C and every bounded f with Lipschitz constant 1 E eλ(f −Ef )

240 . 4 − Cλ2

(57)

In particular

% − C2 t

P |f − Ef | > t 240 e

for all t > 0.

(58)

P ROOF. By (51) and (56)

2

C E eg − E eg/2 CE eg/2 gLip E eg 4 for any g ∈ A. In particular, for any λ, λ 2 Cλ2 λf Ee E eλf − E e 2 f 4 or E eλf

1 1−

Cλ2

λ f 2 Ee2 .

4

Iterating we get for every n, E eλf

n−1 $ k=o

1 1−

2k

Cλ2 4k+1

λ 2n E e 2n f

which tends to ∞ $ k=o

1 1−

Cλ2 4k+1

2k eλEf .

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G. Schechtman

R EMARK 15. (1) A simple limiting argument shows now that the assumption that f is bounded is superfluous. (2) The simple example of the exponential distribution on R shows that (except for the absolute constants involved) one can’t improve the concentration function e−ct . As we shall see below, what looks like a slight change, logarithmic Sobolev inequality instead of 2 Poincaré inequality, changes the behavior of the concentration function from e−ct to e−ct . The next theorem is apparently due to Herbst. T HEOREM 16. Let (Ω, F , μ, d) be a probability metric space. Let A be the set of bounded Lipschitz functions on (Ω, d) and let E be defined by (54). Assume that (A, E) satisfies the logarithmic Sobolev inequality (52) then for all λ ∈ R and every bounded f with Lipschitz constant 1 E eλ(f −Ef ) eCλ

2 /4

(59)

.

In particular

2 P |f − Ef | > t 2 e−t /C

for all t > 0.

(60)

P ROOF. Put h(λ) = E eλf , then

ε eλf/2 = Eλf eλf −E eλf log E eλf = λh (λ) − h(λ) log h(λ) .

(61)

Also, from (56), we get,

λ2 λ2 E eλf/2 E eλf = h(λ). 4 4

(62)

Combining (61), (62) and (52) we get

λ2 C λh (λ) − h(λ) log h(λ) h(λ) 4 or, putting k(λ) = λ−1 log h(λ) (and, by continuity, k(0) = Ef ), k (λ) =

1 1 h (λ) C − 2 log h(λ) , λ h(λ) 4 λ

for all λ ∈ R.

It follows that k(λ) − k(0) Cλ/4 and thus E eλ(f −Ef ) = eλ(k(λ)−k(0)) eCλ

2 /4

.

R EMARK 17. A simple limiting argument shows that here too the assumption that f is bounded is superfluous.


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Both Poincaré inequality and logarithmic Sobolev inequality carry over nicely to product spaces in the following sense: for i = 1, 2, . . . , n, let (Ωi , Fi , μi ) be a probability space, + Ai some set *nof measurable functions on Ωi and Ei : Ai → R some energy function. Put (Ω, P ) = i=1 (Ωi , μi ). Given a function f on Ω we denote by fi the same function considered as a function of the i-th variable only, keeping all other variables fixed. De fine E(f ) = EP ni=1 Ei (fi ). Let A denote the set of all functions f such that (for all x1 , . . . , xn ) and for all i, fi is in Ai . One can prove that σ 2 (f ) EP

n

σ 2 (fi )

and ε(f ) EP

i=1

n

ε(fi ),

(63)

i=1

from which the following proposition easily follows. P ROPOSITION 18. Assume (Ai , Ei ), i = 1, . . . , n, all satisfy Poincaré inequality (resp. logarithmic Sobolev inequality) with a common constant, C. Then (A, E) satisfies Poincaré inequality (resp. logarithmic Sobolev inequality) with the same constant, C. E XAMPLE 19. The symmetric exponential measure on R, i.e., the measure with density 1 e−|t | , satisfies Poincaré inequality with constant 4. Consequently, the same is true for the 2 measure on Rn which is the n-fold product of this measure. The canonical Gaussian measure on R and thus on Rn satisfies logarithmic Sobolev inequality with constant 2. The proof of both statements can be found in [33]. The second one is due to Gross and, in view of Theorem 16, implies the concentration inequality for γn , the Gaussian measure on Rn : if f : Rn → R is Lipschitz with constant one with respect to the Euclidean metric then 2 γn f − f dγn > t C e−ct . From this it is not hard to get the concentration inequality for S n−1 . One uses Lemma 22 below. We would also like to state a theorem first proved by Talagrand [61] which “interpolates” between the last two theorems. See [8] and [33] for a relatively simple proof along the lines of the proofs of the last two theorems. We state it only for a specific probability measure P on Rn , the product of the measures with density 12 e−|t | on R. See [33] for generalizations. T HEOREM 20. Let f : Rn → R be a function satisfying f (x) − f (y) αx − y2

and f (x) − f (y) βx − y1 .

(64)

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G. Schechtman

Then, with the probability introduced above,

P f (x) − Ef > r C exp −c min r/β, r 2 /α 2

(65)

for some absolute positive constants C, c and all r > 0. R EMARK 21. Although it deals with a different probability measure, Theorem 20 also implies the concentration inequality for the Gaussian measure on Rn (and thus, via Lemma 22 below, also for the Haar measure on S n−1 ). This follows from a simple transference of the Gaussian measure to the product of the symmetrized exponential measure discussed above. Thus, Theorem 20 can be considered as a strengthening of these inequalities. We refer to [61] and [33] for that and further discussion. Although the methods in this and the previous section are specialized to product measures, there is a way to transfer such results to some other situations. In particular to the case of unit balls of np spaces equipped with the normalized Lebesgue measure. The basic A;0t 1}| tool is the following simple result: consider the measure μ(A) = |{t |{x;xp 1}| on the surface of the np ball, 0 < p < ∞. Consider also n independent random variables X1 , X2 , . . . , Xn p each with density function cp e−|t | , t ∈ R. (Note that necessarily cp = p/2Γ (1/p).)

L EMMA 22. Put S = ( ni=1 |Xi |p )1/p . Then ( XS1 , XS2 , . . . , XSn ) induces the measure μ on ∂Bpn . Moreover, ( XS1 , XS2 , . . . , XSn ) is independent of S. See [54] for a proof. This lemma is used there to compute the tail behavior of the q norm on the np ball. Recently ([55]) this result was strengthen, in the case p = 1, q = 2, to give a concentration inequality for general Lipschitz functions, with respect to the Euclidean metric, on the n1 ball B1n . The proof combines most of the results of this section and we shall not give it here. T HEOREM 23. There exist positive constants C, c such that if f : ∂B1n → R satisfies |f (x) − f (y)| x − y2 for all x, y, ∈ ∂B1n then, for all t > 0,

μ x; |f (x) − Ef | > t C exp(−ctn).

(66)

2.5. Bounds on Gaussian processes As we shall see below, in the application sections, concentration inequalities are used mostly to find a point ω, in the metric probability space under consideration, in which a big collection of functions {Gt (ω)}t ∈T are each close to its mean. There may be other ways to reach such a conclusion. Assuming the means of all the functions under consideration are zero, it would be enough, for example, to prove that E supt ∈T |Gt | is small (then, for a set of ω’s of measure at least 1/2, supt ∈T |Gt (ω)| is at most 2×small).


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When T is a metric space and Gt a Gaussian process (meaning that any finite linear combination of the Gt ’s has a Gaussian distribution) the evaluation of E supt ∈T |Gt | is an extensively studied subject in Probability (having to do with the existence of a continuous version of the process). See, for example, [35]. There are well studied connections between the quantity E supt ∈T |Gt | and the entropy (or covering) function of the metric space T as well as with other properties of T . A recent achievement in this area is Talagrand’s majorizing measure theorem which relates the boundedness of E supt ∈T |Gt | to the existence of a certain measure (called majorizing measure) on T and gives new ways to estimate this quantity. A recent book treating this subject is [10]. We’ll not get into it any further here; we only remark that the proofs in this area are very much connected with concentration properties of Gaussian variables.

2.6. Other tools We dealt above mostly with geometric and probabilistic tools to prove concentration and approximate isoperimetric inequalities. There are many other methods and results that are not discussed here for lack of space. In particular we didn’t discuss at all combinatorial methods. For example, the (exact) isoperimetric inequality for the Hamming cube (from which Corollary 7 for that case follows) was first proved by Harper [22] (see also [15] for a simpler proof) by combinatorial methods. There are also geometrical and probabilistic methods we didn’t discuss. [48] contains a yet another nice probabilistic proof of Corollary 4 due to Maurey and Pisier. It uses properties very special to Gaussian variables and thus does not seem to generalize much. [53] contains a generalization of Corollary 4 to harmonic measures on S n−1 . The proof is by reduction to the Haar measure. A new probabilistic method for proving concentration inequalities which emerged recently is that of transportation cost, see [40], where it was initiated by Marton, and the followup in [63]. [34] devotes a chapter to this subject. This method seems very much related to Kantorovich’s solution of Monge’s “mass transport” problem although, as far as I know, no concrete relation has been found yet. The “localization lemma” proved by Lovasz and Simonovits in [39] is a way to reduce certain integral inequalities in Rn to integral inequalities involving functions of one variable. It can be used to prove certain approximate isoperimetric inequalities as is explored in [30]. The list above is far from exhausting all the sources on this vast subject.

3. Applications 3.1. Dvoretzky-like theorems The introduction of the method(s) of concentration of measure into Banach Space Theory was initiated by Milman in his proof [43] of Dvoretzky’s theorem concerning spherical sections of convex bodies [11]. Although this topic is extensively reviewed in the article

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G. Schechtman

[16] in this Handbook, I would like to begin the applications section with a statement of the theorem and a brief description of its proof. T HEOREM 24. For all ε > 0 there exists a constant c = c(ε) > 0 such that for any n-dimensional normed space X there exists a subspace Y of dimension k c log n such that the Banach–Mazur distance d(Y, k2 ) 1 + ε. See [25] for the definition of the Banach–Mazur distance. The one-to-one correspondence between n-dimensional normed spaces and n-dimensional symmetric convex bodies (and the fact that every 2n-dimensional ellipsoid has an n-dimensional section which is a multiple of the canonical Euclidean ball) easily shows that the theorem above is equivalent to the following geometrical statement. By a convex body in Rn we mean a compact convex set with non-empty interior. T HEOREM 25. For all ε > 0 there exists a constant c = c(ε) > 0 such that every centrally symmetric convex body K admits a k c log n dimensional central section K0 and a positive number r satisfying rB ⊂ K0 ⊂ (1 + ε)rB, where B is the canonical Euclidean ball in the subspace spanned by K0 . S KETCH OF PROOF. Since the statement of each of the two theorems is invariant under invertible linear transformations, we may assume that the unit ball K of X = (Rn , · ) satisfies B2n ⊂ K and the canonical Euclidean ball B2n in Rn is (the) ellipsoid of maximal volume among all ellipsoids inscribed in K. (It is a theorem of F. John that the maximal volume ellipsoid is uniquely determined but we do not need this fact here.) A relatively easy theorem of Dvoretzky and % Rogers [12] (see also [45], p. 10) implies now that E = E · = S n−1 x dμ(x) > c logn n for some absolute constant c. Denoting by ν the normalized Haar measure on the orthogonal group O(n) and applying Corollary 4 to the function x → x, which is Lipschitz with constant one, we get that, for every fixed x ∈ S n−1 ,

ν u; ux − E > εE = μ x ∈ S n−1 ; x − E > εE < e−cε

2 E2 n

< e−cε

2 log n

.

Fix a k-dimensional subspace V0 ⊂ Rn and an ε net N in V0 ∩ S n−1 of cardinality smaller than (3/ε)k . The existence of such a net follows from an easy volume argument (see [45], 2 p. 7). It then follows that if (3/ε)k e−cε log n < 1, i.e., if k is no larger than a constant depending on ε times log n, then

ν u; ux − E > εE, for some x ∈ N < 1 which implies that there is a u ∈ O(n) such that (1 − ε)E ux (1 + ε)E,

for all x ∈ N .


1627

It now follows from a successive approximation argument that similar inequalities hold for all x ∈ S n−1 which implies the conclusion of the theorem for the subspace uV0 . We next state another application of the concentration inequality on the Euclidean sphere. This lemma of Johnson and Lindenstrauss is much simpler but has a lot of applications including “real life” ones like efficient algorithms for detecting clusters. T HEOREM 26. Let x1 , x2 , . . . , xn be points in some Hilbert space. If k c > 0 an absolute constant), then there are y1 , y2 , . . . , yn ∈ k2 satisfying

c ε2

log n (with

xi − xj yi − yj (1 + ε)xi − xj

(67)

for all 1 i = j n. S KETCH OF PROOF. We may assume that the points xi are in n2 . Fix a k < n and a rank k orthogonal projection P0 on n2 . When u ranges over O(n), P = uP0 u−1 ranges over all rank k orthogonal projections. It √ is not hard to check that, for all x ∈ S n−1 , −1 E = O(n) uP0 u x dν(u) is of the order k/n and thus, for every x ∈ S n−1 ,

2 ν u; uP0 u−1 x − E > εE = μ x; P0 x − E > εE < e−cε k . It follows that, if k

C ε2

log n, there is a u ∈ O(n) for which

xi − xj −1 (1 + ε)E (1 − ε)E uP0 u xi − xj for all i = j . The range of uP0 u−1 is k-dimensional. Take yi =

uP0 u−1 xi (1−ε)E .

3.2. Fine embeddings of subspaces of Lp in lpn When specializing the proof of Theorem 24 to the case of X = nr , one sees quite easily that if 1 r < 2 then for all ε > 0 there exists a constant c = c(r, ε) > 0 such that for all n there exists a subspace Y of nr of dimension k cn whose Banach–Mazur distance to Euclidean space, d(Y, k2 ) 1 + ε. (For 2 < r < ∞ the same holds with k cn2/r .) This subject is extensively reviewed in [16]. Since it is known (and follows from the existence of p-stable random variables, see below) that, for r < p 2, p embeds isometrically into Lr [0, 1], it is natural to ask whether a similar statement holds with 2 replaced by p, i.e., whether, for r < p < 2, kp (1 + ε)embeds into nr for k proportional to n. Noticing that Gaussian variables are very different from p-stable ones for p < 2 (the first decay exponentially while the latter only polynomially), and that the concentration inequality behind the proof of Theorem 24 has very much to do with the exponential decay of Gaussian variables, one’s first guess would be that the

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G. Schechtman

answer to the question above is negative (and probably that k can only be some logarithmic function of n). It turns out, however, that the answer to the question above is positive. It was proved in [26] that for 1 p < 2 and for every n and ε, n1 contains a subspace Y with d(Y, kp ) < 1 + ε where k c(p, ε)n. This was the first result concerning “tight embeddings” that didn’t deal with Euclidean spaces. It was proved using certain approximation of p-stable random variables and concentration inequalities for martingales as discussed in Section 2.2. This result lead to a series of generalizations and results of similar nature. We refer to [28] for a survey of this topic. Here we only deal with two such examples of generalizations. We would first like to mention a result of Pisier [47], generalizing the result above from the side of the containing space, n1 . p Recall that a random variable h whose characteristic function is given by E eit h = e−c|t | , for some positive constant c, is called (symmetric) p-stable. Lévy proved the existence of such variables for 0 2.) A p-stable variable has r-th moment for all r < p but doesn’t have p-th moment. For 1 < p < 2 we’ll denote from now on by h the p-stable variable whose first moment is equal to 1. This defines its distribution uniquely. If h, h1 , . . . , hn are independent and identically distributed then it is easy to see (compute the characteristic function) that ni=1 αi hi also has the same distribution as h as long as ni=1 |αi |p = 1. In particular the span of h1 , . . . , hn in L1 [0, 1] is isometric to np . For 1 < p < 2, the stable type p constant of a Banach space X, denoted STp (X), is the smallest constant C such that, E hi xi Cn1/p sup xi

(68)

1in

definition for all finite sequences {x1 , . . . , xn } of elements of X. (This is an equivalent to the more common one where n1/p sup1in xi is replaced with ( ni=1 xi p )1/p .) Pisier’s result is: T HEOREM 27. For each 1 0 there is a positive constant c = c(p, ε) such that any Banach space X contains a subspace Y satisfying d(Y, kp ) as long as k < cSTp (X)p/(p−1) .

(69)

Since it is easy to see that STp (n1 ) n(p−1)/p , this implies the result of [26] referred to above. A BRIEF SKETCH OF THE PROOF. Pick a finite sequence, x1 , x2 , . . . , xn , of elements of X for which max xi = 1 and E hi xi 12 n1/p STp (X). Let u1 , u2 , . . . be a sequence of independent random variables each uniformly distributed over the set of 2n j elements {±x1, ±x2 , . . . , ±xn }. Put also Γj = i=1 Ai , j = 1, 2, . . . , where the Aj ’s are independent (and independent of the sequence {ui }) canonical exponential variables, i.e., P (Ai > t) = e−t , t > 0. We shall use a representation theorem for p-stable variables, due


1629

to LePage, Woodroofe and Zinn [37] which says in particular that, for some constant cp depending only on p, S=

∞

−1/p

Γj

uj has the same distribution as cp n−1/p

j =1

n

hi xi

(70)

i=1

and in particular, ES 2p STp (X). Note that for any functional x ∗ , x ∗ (S) is a p-stable variable. If S1 , . . . , Sk are independent and all have the same distribution as S then it is easily seen that if ki=1 |αi |p = 1 then ki=1 αi Si has the same distribution as S and in particular E ki=1 αi Si = ES. −1/p The next step is to replace the random coefficients {Γj } with the deterministic se −1/p u and let R , . . . , R be independent and all have quence {j −1/p }. Put R = ∞ j j 1 k j =1 the same distribution as R. A computation using the explicit distribution of Γj shows that c

C = ESi − Ri < ∞ and it follows that, if

k

i=1 |αi |

p

= 1,

k k k αi Si − E αi Ri C |αi | E i=1

i=1

i=1

Ck

(p−1)/p

k

1/p |αi |

p

i=1

< Cc(p−1)/p STp (X)

(71)

by the choice of k. Note that ki=1 |αi |p = 1 implies that {αi j −1/p }p,∞ = 1 and thus Proposition 5(2) implies that for all such {αi } and for all t > 0,

P αi Ri − E αi Ri > t K exp −δt p/(p−1) .

(72)

This last equation is of course the place where the method of concentration enters, which was the main thing we wanted to illustrate here. The rest of the proof goes along similar lines to the end ofthe proof of Theorem 24: note that it follows from (71), that, for c small enough, E αi Ri is of order STp (X). Choose an ε net in the sphere of kp of cardinality smaller than (3/ε)k . Then, with high probability, αi Ri is of order STp (X) for all sequences {αi } in the net. By a successive approximation the same holds now for all sequences {αi } in the sphere of np which completes the proof. Another way to generalize the result of Schechtman and Johnson [26] (that cn p nicely embeds in n1 ) is from the side of the embedded space, np . After some initial work by

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G. Schechtman

Schechtman (mostly [52]) on embedding finite-dimensional subspaces of Lp [0, 1] in lowdimensional nr spaces in which a new class of “random embeddings” (which were not related to p stable variables) were introduced, Bourgain, Lindenstrauss and Milman [9] proved that, for 0 < r < 2, every k-dimensional subspace of Lr [0, 1] (1 + ε)-embeds in nr provided n/k is at least a certain power of log n times a constant (depending only on r and ε). See also [27] for a different proof. All the proofs involved use concentration in one way or another. The result of [9] mentioned above was improved and simplified by Talagrand [60]. Since his proof has to do with bounds on Gaussian processes and is related to Section 2.5, we would like to briefly review it. As we have already advertised, the article [28] has more on that subject. Here we shall deal only with the case r = 1. T HEOREM 28. For every ε there is a constant C(ε) such that for all n, every Cn log n n-dimensional subspace Y of L1 [0, 1] is (1 + ε)-isomorphic to a subspace of 1 . We remark in passing that one of the main open problems in this area is whether the factor log n is needed. Besides concentration inequalities the proof uses some other heavy tools and is discussed in [28]. We shall only touch the idea involving bounds on Gaussian processes. T HE IDEA OF THE PROOF. By crude approximation we may assume that Y is a subspace of m 1 for some finite (but huge) m. We would like to show that a restriction to a “random” subset of cardinality Cn log n of the coordinates is a good isomorphism when restricted to Y . Of course this is wrong in general (for instance if Y has an element which is supported on only one coordinate, this element would most probably be sent to zero by such a restriction). The idea is to first “change the density” and send Y to an isometric subspace whose elements are “spread out” over the m coordinates. The idea that this may work was the point of [52]. It will be dealt with in [28] and will not be discussed here any further. We’ll concentrate in describing how to evaluate the norm of the random restriction on Y and the norm of its inverse assuming Y is already in good position. We do it inductively, restricting first to a random set of about half the coordinates where each coordinate is chosen with probability 1/2 and the different choices are independent. Equivalently, let {εi }m i=1 be independent variables each taking the values −1 and 1 with probability 1/2 each. We would like to evaluate the restriction to the set A = {i; εi = 1}. If we could show that sup

m 2 < ε(n, m) |x | − |x | i i

x∈Y,x1

i∈A

(73)

i=1

with ε(n, m) “very small” when n/m is small, then this would mean that (2 times) the restriction to A is very close to an isometry. Iterating this would lead, depending on the behavior of ε(n, m), to the desired random restriction onto a small set of coordinates. Note that the quantity in (73) is equal to supx∈Y,x1 | m i=1 εi |xi || and in particular is the same


1631

for A and its complement. Since we are interested in only one set A, of cardinality at most m/2 satisfying (73), it is enough to establish m E sup εi |xi | ε(n, m). (74) x∈Y,x1 i=1

This quantity is dominated by a similar one with independent standard Gaussian variables gi ’s replacing the εi ’s. So the problem reduces to estimating m E sup gi |xi |, x∈Y,x1 i=1

i.e., the expectation of the supremum of a specific Gaussian process. This makes the connection with Section 2.5. We shall not go into more details here. Theorem 28 has a nice geometrical interpretation which is obtained by looking at the polar body to the unit ball of Y . C OROLLARY 29. Let K be the (Minkowski) sum of segments in Rn (or limit of such bodies, these are called zonoids). Then, for every ε, there is a body L in Rn which is the sum of at most C(ε)n log n segments and which ε approximates K in the sense that L ⊂ K ⊂ (1 + ε)L.

3.3. Selecting good substructures Given a sequence of independent vectors {x1 , x2 , . . . , xn } in a normed space X and an ε > 0, what is the largest cardinality k such that there are k disjoint blocks y1 , y2 , . . . , yk which are (1 + ε)-unconditional or (1 + ε)-symmetric? Recall that by disjoint blocks we mean vectors of the form yi = j ∈σi aj xj , i = 1, . . . , k, with σ1 , σ2 , . . . , σk disjoint subsets of 1, 2, . . . , n. y1 , y2 , . . . , yk is said to be (1 + ε)-unconditional (resp. (1 + ε)-symmetric) if k k εi bi yi (1 + ε) bi yi i=1

i=1

for all signs {εi } and all coefficients {bi }. (resp. if k k εi bi yπ(i) (1 + ε) bi yi i=1

i=1

for all signs {εi }, all permutations π of 1, 2, . . . , k and all coefficients {bi }.)

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These problems and various variations thereof were treated quite successfully by concentration of measure methods. The point is that, fixing a partition σ1 , σ2 , . . . , σk of {1, 2, . . . , n} and coefficients {{aj }j ∈σi }ki=1 , the norms k k bi ei = Aveε bi εj aj xj i=1

and

u

i=1

j ∈σi

k k bi ei = Aveε,π bi εj aπ(j ) xj i=1

s

i=1

j ∈σi

on Rn are 1-unconditional and 1-symmetric respectively. If we can find signs {{εj }j ∈σi }ki=1 such that, for all {bi }, ki=1 bi j ∈σi εj aj xj / ki=1 bi ei u is appropriately close to one, then we found disjoint blocks of length k which are (1 + ε)-unconditional. A similar statement holds for the symmetric case. For lack of space we shall not review all that is known about this subject. The unconditional case was first treated by Amir and Milman in [2,3]. Gowers improved some of their quantitative estimates ([17,18]) and in some instances got, except for possible log factors, the best possible estimates. The symmetric case was treated by Maurey [41] and was the motivation for proving Corollary 7 (for Πn ). We were dealing here only with applications to functional analysis and convexity. There are many applications to other areas which we shall not expand on. There are applications to graph theory (see, e.g., the construction of expander graphs in [1]), to other combinatorial questions, computer science, mathematical physics and probability (in particular to estimating the tail behavior of random variables of the form εi Xi for independent vector valued random variables {Xi }). [62] contains many applications of the material of Section 2.3. References [1] N. Alon and V.D. Milman, λ1 , isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985), 73–88. [2] D. Amir and V.D. Milman, Unconditional and symmetric sets in n-dimensional normed spaces, Israel J. Math. 37 (1980), 3–20. [3] D. Amir and V.D. Milman, A quantitative finite-dimensional Krivine theorem, Israel J. Math. 50 (1985), 1–12. [4] J. Arias-de-Reyna, K. Ball and R. Villa, Concentration of the distance in finite dimensional normed spaces, Mathematika 45 (1998), 245–252. [5] K. Azuma, Weighted sums of certain dependent random variables, Tôhoku Math. J. 19 (1967), 357–367. [6] A. Baernstein II and B.A. Taylor, Spherical rearrangements, subharmonic functions, and ∗-functions in n-space, Duke Math. J. 43 (1976), 245–268. [7] Y. Benyamini, Two point symmetrization, the isoperimetric inequality on the sphere and some applications, Longhorn Notes, Texas Funct. Anal. Seminar, Univ. of Texas (1983–1984), 53–76. [8] S. Bobkov and M. Ledoux, Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution, Probab. Theory Related Fields 107 (1997), 383–400.


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[9] J. Bourgain, J. Lindenstrauss and V.D. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73–141. [10] R.M. Dudley, Uniform Central Limit Theorems, Cambridge Stud. Adv. Math. 63, Cambridge Univ. Press, Cambridge (1999). [11] A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Sympos. on Linear Spaces (Jerusalem, 1961), 123–160. [12] A. Dvoretzky and C.A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. USA 36 (1950), 192–197. [13] U. Feige and G. Schechtman, On the optimality of the random hyperplane rounding technique for MAX CUT, Random Structures Algorithms 20 (2002), 403–440. [14] T. Figiel, J. Lindenstrauss and V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53–94. [15] P. Frankl and Z. Füredi, A short proof for a theorem of Harper about Hamming-spheres, Discrete Math. 34 (1981), 311–313. [16] A.A. Giannopoulos and V.D. Milman, Euclidean structure in finite-dimensional normed spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 707–779. [17] W.T. Gowers, Symmetric block bases in finite-dimensional normed spaces, Israel J. Math. 68 (1989), 193– 219. [18] W.T. Gowers, Symmetric block bases of sequences with large average growth, Israel J. Math. 69 (1990), 129–151. [19] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics 152, Birkhäuser, Boston (1999). [20] M. Gromov and V.D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), 843–854. [21] M. Gromov and V.D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263–282. [22] L.H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Combin. Theory 1 (1966), 385–393. [23] P. Hitczenko, Best constants in martingale version of Rosenthal’s inequality, Ann. Probab. 18 (1990), 1656– 1668. [24] W.B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Contemp. Math. 26 (1984), 189–206. [25] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. [26] W.B. Johnson and G. Schechtman, lpm into l1n , Acta Math. 149 (1982), 71–85. [27] W.B. Johnson and G. Schechtman, Remarks on Talagrand’s deviation inequality for Rademacher functions, Functional Analysis (Austin, TX, 1987/1989), Lecture Notes in Math. 1470, Springer, Berlin (1991), 72–77. [28] W.B. Johnson and G. Schechtman, Finite dimensional subspaces of Lp , Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 837–870. [29] W.B. Johnson, G. Schechtman and J. Zinn, Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab. 13 (1985), 234–253. [30] R. Kannan, L. Lovász and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), 541–559. [31] S. Kwapień and J. Szulga, Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces, Ann. Probab. 19 (1991), 369–379. [32] M. Ledoux, On Talagrand’s deviation inequalities for product measures, ESAIM Probab. Statist. 1 (1995/97), 63–87. [33] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilitiés XXXIII, Lecture Notes in Math. 1709, Springer, Berlin (1999), 120–216. [34] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monographs 89, Amer. Math. Soc., Providence, RI (2001).

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[35] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Isoperimetry and Processes, Springer, Berlin (1991). [36] L. Leindler, On a certain converse of Hölder’s inequality II, Acta Sci. Math. (Szeged) 33 (1972), 217–223. [37] R. LePage, M. Woodroofe and J. Zinn, Convergence to a stable distribution via order statistics, Ann. Probab. 9 (1981), 624–632. [38] P. Lévy, Problèmes Concrets D’Analyse Fonctionnelle, 2nd ed., Gauthier-Villars, Paris (1951) (French). [39] L. Lovász and M. Simonovits, Random walks in a convex body and an improved volume algorithm, Random Structures Algorithms 4 (1993), 359–412. [40] K. Marton, A measure concentration inequality for contracting Markov chains, Geom. Funct. Anal. 6 (1996), 556–571. [41] B. Maurey, Construction de suites symétriques, C.R. Acad. Sci. Paris Sér. A-B 288 (1979), 679–681 (French). [42] B. Maurey, Some deviation inequalities, Geom. Funct. Anal. 1 (1991), 188–197. [43] V.D. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Funct. Anal. Appl. 5 (1971), 28–37. [44] V.D. Milman, The heritage of P. Lévy in geometrical functional analysis, Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), Astérisque 157–158 (1988), 273–301. [45] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, New York (1986). [46] V.D. Milman and G. Schechtman, An “isomorphic” version of Dvoretzky’s theorem, C.R. Acad. Sci. Paris Sér. I Math. 321 (1995), 541–544. [47] G. Pisier, On the dimension of the lpn -subspaces of Banach spaces, for 1 p < 2, Trans. Amer. Math. Soc. 276 (1983), 201–211. [48] G. Pisier, Probabilistic methods in the geometry of Banach spaces (CIME, Varenna, 1985), Lecture Notes in Math. 1206, Springer (1986). [49] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press (1989). [50] A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. [51] G. Schechtman, Lévy type inequality for a class of finite metric spaces, Martingale Theory in Harmonic Analysis and Banach Spaces (Cleveland, Ohio, 1981), Lecture Notes in Math. 939, Springer, New York (1982), 211–215. [52] G. Schechtman, More on embedding subspaces of Lp in lrn , Compositio Math. 61 (1987), 159–169. [53] G. Schechtman and M. Schmuckenschläger, A concentration inequality for harmonic measures on the sphere, Geometric Aspects of Functional Analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77 (1995), 255–273. [54] G. Schechtman and J. Zinn, On the volume of the intersection of two Lnp balls, Proc. Amer. Math. Soc. 110 (1990), 217–224. [55] G. Schechtman and J. Zinn, Concentration on the lpn ball, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1745, Springer, Berlin (2000), 245–256. [56] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications 44, Cambridge Univ. Press, Cambridge (1993). [57] W.F. Stout, Almost Sure Convergence, Probab. Math. Statist. 24, Academic Press (1974). [58] M. Talagrand, An isoperimetric theorem on the cube and the Kintchine–Kahane inequalities, Proc. Amer. Math. Soc. 104 (1988), 905–909. [59] M. Talagrand, Isoperimetry and integrability of the sum of independent Banach-space valued random variables, Ann. Probab. 17 (1989), 1546–1570. [60] M. Talagrand, Embedding subspaces of L1 into l1N , Proc. Amer. Math. Soc. 108 (1990), 363–369. [61] M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, Geometric Aspects of Functional Analysis (Israel, 1989–90), Lecture Notes in Math. 1469, Springer (1991), 94–124. [62] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 73–205. [63] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal. 6 (1996), 587–600. [64] V.V. Yurinskii, Exponential bounds for large deviations, Theor. Probab. Appl. 19 (1974), 154–155.

CHAPTER 38

Uniqueness of Structure in Banach Spaces Lior Tzafriri Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel E-mail: [email protected]

Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1. Uniqueness of general and unconditional bases . . . . . 2. Uniqueness of symmetric bases . . . . . . . . . . . . . 3. Uniqueness of unconditional bases, up to a permutation 4. Uniqueness in finite-dimensional spaces . . . . . . . . 5. Uniqueness of rearrangement invariant structures . . . 6. Uniqueness of bases in non-Banach spaces . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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. 1637 . 1637 . 1641 . 1644 . 1651 . 1658 . 1665 . 1667


Uniqueness of structure in Banach spaces

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0. Introduction In this chapter, we consider Banach spaces which can be represented as spaces of sequences or functions with some specific properties, and study the natural question whether such a representation is unique, up to a notion of equivalence which can vary from case to case. The typical space of sequences is derived froma Banach space X with a normalized ∞ Schauder basis {en }∞ n=1 an en ∈ X, there corresponds the n=1 , where to each element x = sequence of coefficients (a1 , a2 , . . . , an , . . .). Of particular interest in this context will be the spaces with a normalized unconditional or symmetric basis. The notion of uniqueness mentioned above means a different thing in each of the cases under consideration. A Banach space X will be said to have a unique general or unconditional or symmetric basis, up to equivalence, if, for any two normalized bases of the same type, there exists an automorphism T of X which takes one basis into the other. The existence of a normalized Schauder basis {en }∞ n=1 in a Banach space X allows the representation of X as a space of ordered sequences. In the special case where {en }∞ n=1 is an ∞ is again a basis which need not unconditional basis, any permutation {eπ(n) }∞ of {e } n n=1 n=1 ∞ be equivalent to {en }∞ n=1 for every π , unless {en }n=1 is a symmetric basis. Therefore, in the case of spaces with a normalized unconditional basis (which is not symmetric) we have the option of considering a different type of uniqueness of the normalized unconditional basis, namely that of uniqueness up to a permutation and equivalence. More precisely, a space X has a unique normalized unconditional basis, up to a permutation (and equivalence) whenever, for each pair of normalized unconditional bases of X, there exists an automorphism T of X which takes the first basis into a permutation of the second. Of course, both notions of uniqueness coincide in the case of symmetric bases. In the continuous case, we typically consider Banach lattices which can be represented as spaces of measurable functions on a suitable measure space. Most interesting is the special case of the so-called rearrangement invariant (r.i.) function spaces over a measure space (Ω, Σ, μ). The main property of such a space X of functions is that automorphisms of the underlying measure space transform elements of X into elements of X with the same norm. In the Basic Concepts article such spaces are called symmetric lattices. The notion of uniqueness can be also studied in the context of finite-dimensional spaces, but, in this case, we have to be more careful since for spaces of the same dimension all the structures are obviously unique. However, for families of Banach spaces {Xn }∞ n=1 , with dim Xn = n for all n, it makes perfect sense to inquire whether, for any two normalized bases of Xn , of the same type, there exists an automorphism Tn of Xn which takes the first basis into the second and, most importantly, the quantity Tn · Tn−1 is bounded by a constant independent of n but possibly dependent on the structure constant (by structure constant we mean either the basis constant or unconditional or symmetric constant, according to the case under consideration).

1. Uniqueness of general and unconditional bases It is quite easy to prove that a separable Hilbert space has a unique normalized unconditional basis, up to equivalence. Indeed, if {un }∞ n=1 is a normalized K-unconditional basis

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in 2 then, by the parallelogram identity, N 2 N Ave an σn un = |an |2 σn =±1 n=1

n=1

N

for every choice of N and scalars {an }2n=1 . This of course implies that K

−1

∞

1/2 |an |

2

n=1

∞ ∞ 1/2 2 a n un K |an | , n=1

n=1

∞ for any choice of {an }∞ n=1 , i.e., {un }n=1 is K-equivalent to the unit vector basis of 2 . It turns out that a separable Hilbert space does not have unique basis, up to equivalence. This fact, which is less trivial, was first proved by Babenko [2]. His construction is based on the simple observation that the characters {eint }n∈Z , in the order {1, eit , e−it , e2it , e−2it , . . .}, form a basis in the space Lp (T, W (t)), where W (t) is an integrable weight function on the circle T, if and only if the Riesz projection, defined by

P+

+∞

an e

int

=

n=−∞

∞

an eint

n=0

is bounded on this space. This latter question has been extensively studied in harmonic analysis and a well known necessary and sufficient condition for the boundedness of the Riesz projection (or, as a matter of fact, of the Hilbert transform) in the space Lp (T, W (t)) is the so-called Ap -condition (see, e.g., [22, p. 254])

1 sup μ(I ) I

W (t) dt I

1 μ(I )

I

1 W (t)

p−1

1/(p−1) dt

< ∞,

where μ is the Lebesgue measure and the supremum is taken over all intervals I ⊂ T of positive measure. The weight function considered by Babenko is Wα (t) = |t|2α , where α is a suitable number. In order to ensure the integrability of Wα (t), one has to require that 2α + 1 > 0, i.e., that α > −1/2. Furthermore, it is easily verified that the A2 -condition is satisfied by Wα iff 1 − 2α > 0, i.e., when α < 1/2. It follows that, for −1/2 < α < 1/2, the characters {eint }n∈Z , in the order described above, form a basis of the Hilbert space L2 (T, Wα (t)). Equivalently, the sequence

2α + 1 2π 2α+1

1/2

|t|α eint n∈Z


1639

forms a normalized Schauder basis in L2 (T), for any value of −1/2 < α < 1/2. For α = 0, we recover the (unique) unconditional basis of the separable Hilbert space while, for the remaining values of α, we obtain mutually non-equivalent conditional bases of the Hilbert space. Indeed, for α = β in the interval (−1/2, 1/2) and 0 < λ < π , we can find scalars {an }n∈Z so that f (t) =

+∞

an eint = χ[0,λ) (t)

n=−∞

in L2 (T) and a.e. on T. Then the fact that f |t|α = λα−β ; f |t|β

0 < λ < π,

shows that the bases {|t|α eint }n∈Z and {|t|β eint }n∈Z are not equivalent. Another construction of conditional bases in a separable Hilbert space, due to McCarthy and Schwartz [42], is presented in detail in [37, p. 74]. The discussion above can be summarized in the following proposition. P ROPOSITION 1.1. A separable Hilbert space has, up to equivalence, a unique normalized unconditional basis and uncountably many mutually non-equivalent conditional basis. In fact, the following more general result was proved by Pelczynski and Singer [49]. T HEOREM 1.2. Any Banach space with an infinite Schauder basis has uncountably many mutually non-equivalent bases. It turns out that also the spaces 1 and c0 have a unique unconditional basis, up to equivalence, but this fact, which was proved by Lindenstrauss and Pelczynski [35], is more difficult and requires some consequences of the famous Grothendieck inequality. We refer to well-known corollaries of this inequality that every bounded linear operator T : c0 → 1 is 2-absolutely summing and its 2-summing norm π2 (T ) satisfies the inequality π2 (T ) KG T , and that every bounded linear operator U : 1 → 2 is absolutely summing and its 1-summing norm π1 (U ) satisfies π1 (U ) KG U . In both inequalities above, KG denotes as usual the Grothendieck constant. The precise statement of Grothendieck’s inequality together with a simple proof can be found in the Basic Concepts article. In order to prove, e.g., that 1 has a unique unconditional basis, up to equivalence, let {xn }∞ n=1 be a normalized K-unconditional basis ∞of 1 , for some K 1, and fix a senotice that quence of scalars {an }∞ n=1 such that the series n=1 an xn converges. Then ∞ → 1 , defined by T t = ∞ the operator T : c0 n=1 an tn xn , for t = {tn }n=1 ∈ c0 , is of norm T 2K ∞ n=1 an xn (in the real case, 2K can be replaced by K). Hence, by the aforementioned estimate of the 2-summing norm of T , we conclude that π2 (T )

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2KG K ∞ n=1 an xn . Then the definition of π2 (T ), applied for the unit vectors in c0 , together with the fact that {xn }∞ n=1 is assumed to be normalized imply that

∞

1/2 |an |

=

2

n=1

∞

1/2 T en

2

n=1

∞ 2KG K an xn . n=1

This inequality shows that the operator U : 1 → 2 , defined by U ( ∞ n=1 an xn ) = ∞ {an }∞ n=1 an xn ∈ 1 , is bounded by 2KG K. Then, by usn=1 , for any convergent series ing the estimate, discussed above, ofthe 1-summing norm π1 (U ) of U , it follows that ∞ 2 K. Hence, for any x = π1 (U ) 2KG n=1 an xn ∈ 1 , ∞ n=1

∞ εn an xn εn =±1

∞ U (an xn ) π1 (U ) sup |an | = n=1

2 2 4KG K an xn ,

n=1

n=1

which completes the proof of the fact that {xn }∞ n=1 is equivalent to the unit vector basis of 1 . The case of c0 is proved by using a simple duality argument. The summary of this discussion is P ROPOSITION 1.3. Both spaces 1 and c0 have, up to equivalence, a unique normalized unconditional basis. An alternative proof of Proposition 1.3, which does not use Grothendieck’s inequality, is presented in Section 5 of the Basic Concepts article. It turns out that 2 , 1 and c0 are the only Banach spaces with a unique unconditional basis, up to equivalence. This result was proved by Lindenstrauss and Zippin [39]. T HEOREM 1.4. A Banach space has, up to equivalence, a unique unconditional basis if and only if it is isomorphic to one of the spaces 2 , 1 or c0 . Instead of the original proof from [39], we present a shorter proof based on an argument due to W.B. Johnson. Suppose that a space X has, up to equivalence, a unique normalized ∞ unconditional basis {xn }∞ n=1 . Since, for any choice of εn = ±1, n = 1, 2, . . . , {εn xn }n=1 is ∞ ∞ an unconditional basis of X it must be equivalent to {xn }n=1 and, thus, {xn }n=1 is symmetric. Let now {um }∞ m=1 be an arbitrary normalized block basis with constant coefficients of {xn }∞ and denote by U its span. n=1 If we prove that X is isomorphic to X ⊕ U , then {x1 , u1 , x2 , u2 , . . . , xn , un , . . .}


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forms a normalized unconditional basis of a space isomorphic to X. Hence, by the unique∞ ∞ ∞ ness of {xn }∞ n=1 , {un }n=1 is equivalent to {xn }n=1 , i.e., {xn }n=1 is a perfectly homogeneous basis in the terminology of [58]. Thus, by the well known result of Zippin [58], {xn }∞ n=1 is equivalent to the unit vector basis of p ; 1 p < ∞, or of c0 . In order to complete the proof, recall, e.g., that the Rademacher functions {rn }∞ n=1 span a Hilbert space in Lp (0, 1), for any value of p 1, and their span [rn ]∞ in complemented in Lp (0, 1), whenever n=1 n p > 1. In particular, 2p contains a uniformly complemented copy of n2 , for any n and n p > 1. Hence, p contains a complemented copy of the direct sum ( ∞ n=1 ⊕2 )p and, ∞ n by the decomposition method of Pelczynski, p ≈ ( n=1 ⊕2 )p , for p > 1, which shows that the unit vector basis of p is not the unique unconditional basis of this space, up to equivalence, for p > 1, p = 2. In order to prove that X ≈ X ⊕ U , Johnson suggested to use the decomposition method, as follows: let V be the span of a sequence of normalized blocks with constant coefficients of {xn }∞ n=1 which has the universality property that every possible normalized block with constant coefficients of {xn }∞ n=1 appears infinitely many times in the sequences generating V . Then, since any block basis with constant coefficients spans a complemented subspace in a space with a symmetric basis (see, e.g., [34] or [37, p. 123]), X ⊕ V is isomorphic to a complemented in X, i.e., that X ≈ X ⊕ V ⊕ W, for some space W . Since V ⊕ U ≈ V because of the above universality property, it follows that X ≈ X ⊕ V ⊕ U ⊕ W ≈ X ⊕ U, which completes the argument. R EMARK . It can be shown that any space with an unconditional basis which is not unique, up to equivalence, actually has uncountably many mutually non-equivalent unconditional bases. Suppose now that X is a space with a unique, up to equivalence, normalized Schauder ∞ ∞ basis {xn }∞ n=1 . Then {xn }n=1 is equivalent to {εn xn }n=1 , for every choice of εn = ±1, ∞ n = 1, 2, . . . , i.e., {xn }n=1 , is unconditional and thus equivalent to the unit vector basis of 2 , 1 or c0 . However, in each of these three cases, X also has a conditional basis. Simple examples of conditional bases are {en − en−1 }∞ (where e0 = 0 and {en }∞ n=1 in 1 n=1 ∞ denotes the unit vector basis in 1 ) and the summing basis { ∞ i=n en }n=1 in c, which of course is isomorphic to c0 . These facts together with Proposition 1.1 provide a partial proof of Theorem 1.2. 2. Uniqueness of symmetric bases An immediate consequence of Theorem 1.3 is the fact that 2 , 1 and c0 have also a unique symmetric basis, up to equivalence. It turns out that there are considerably more Banach spaces with a unique, up to equivalence, symmetric basis.

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T HEOREM 2.1. The spaces c0 and p ; 1 p < ∞, have, up to equivalence, a unique symmetric basis. ∗ ∞ P ROOF. Fix p > 1, let {en }∞ n=1 denote the unit vector basis of p , {en }n=1 the biorthogonal ∞ ∞ sequence associated to {en }n=1 and let {xm }m=1 be another normalized symmetric basis of ∞ p . By a simple diagonalization argument, one can find a subsequence {xmi }∞ i=1 of {xm }m=1 ∗ such that limi→∞ en xmi exists for any choice of n. If all these limits are equal to zero then one can find a further subsequence of {xm }∞ m=1 , which is equivalent to a block basis of ∞ . Since any normalized block basis of {en }∞ {en }∞ n=1 n=1 is equivalent to {en }n=1 itself in p ∞ ∞ it follows that {xm }m=1 is equivalent to {en }n=1 . In case there exists a value of n for which limi→∞ en∗ xmi = α = 0 then one can easily find an infinite subsequence of {xm }∞ m=1 which is equivalent to the unit vector basis of 1 , thus completing the proof.

The class of spaces with a unique symmetric basis is considerably larger than that of p -spaces and it contains, e.g., all the Orlicz sequence spaces M for which the limit limt →0 tM (t)/M(t) exists. A concrete such function is M(t) = t p /(1 + |log t|); p 1. Another class of spaces with a unique symmetric basis is that of Lorentz sequence spaces is a non-increasing sequence of positive weights d(w, p) where p 1 and w = {wn }∞ n=1 satisfying w1 = 1, limn→∞ wn = 0 and ∞ n=1 wn = ∞. Recall that d(w, p) in the space of all sequences α = {αn }∞ n=1 of scalars so that

∞ p α α = sup π(n) wn π

1/p < ∞,

n=1

where the supremum is taken over all permutation π of the integers. The proofs that these two classes of spaces do have a unique symmetric basis are not very hard but we omit them. They can be found, e.g., in [37, Section 4]. An example of a space with “many” mutually non-equivalent symmetric bases is the so-called space U1 of Pelczynski [47], which is universal for all unconditional bases in the sense that it has an unconditional basis {un }∞ i=1 , and, quite remarkably, every unconditional ∞ basic sequence {vj }∞ j =1 in any Banach space V is equivalent to a subsequence of {un }n=1 . A simple application of the decomposition method shows that the property defining U1 characterizes it up to isomorphism. A simple way of constructing this space was suggested by Schechtman [52]. To this end, let {fn }∞ n=1 be a sequence of continuous functions which is dense in the space C(0, 1) of all continuous functions on [0, 1] and, for any sequence a = {an }∞ n=1 which is eventually zero, i.e., a ∈ c00 , define ∞ |||a||| = sup εn an fn ; εn = ±1, n = 1, 2, . . . . n=1

C(0,1)

The unit vectors {un }∞ n=1 clearly form an unconditional basis in the completion U1 of c00 relative to the norm ||| · |||. If {vk }∞ k=1 is an unconditional basic sequence in an arbitrary


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Banach space V one can assume without loss of generality that V is a subspace of C(0, 1). ∞ Hence, one can find a subsequence {fnk }∞ k=1 of {fn }n=1 so that vk − fnk C(0,1) → 0, ∞ as k → ∞, “fast enough” as to imply that {vk }k=1 is equivalent to {fnk }∞ k=1 . Therefore, ∞ , which completes the proof. {fnk }∞ is unconditional and thus equivalent to {u } nk k=1 k=1 In order to show that U1 has a symmetric basis, a fact which is not a priori obvious, one needs an interpolation argument of Davis [13]: let {xn }∞ n=1 be a normalized 1-unconditional basis in a Banach space X and, for m > 1 and p 1, define the Orlicz function Mp (t) = t p /(1 + |log t|); t > 0, and the norm

1/2 ; αm,p = inf β2Mp + γ 2p

α = βm−1 + γ m, with β ∈ Mp and γ ∈ p ,

for all α ∈ Mp . Then, whenever {mn }∞ n=1 is a sequence of numbers > 1 satisfying the ∞ −1 condition n=1 mn < ∞, the expression α

(p)

∞ = αmn,p xn , n=1

X

p whose unit vectors form a 1-symmetric basis in the compledefines a norm on a space Y p so that tion Yp of Y Kp−1 αMp α(p) Kp αp , ∞ for all α ∈ p . It turns out that {mn }∞ n=1 can be selected as to ensure that {xn }n=1 is equivalent to a block basis with constant coefficients of the unit vector basis of Yp . This proves that X is isomorphic to a complemented subspace of the space Yp , which of course has a symmetric basis. This assertion can be also proved by using the fact from [34] that every space with an unconditional basis is isomorphic to a complemented subspace of a space with a symmetric basis. The advantage of the present proof is that it shows that X can be complementably embedded in uncountably many spaces with mutually non-equivalent symmetric bases. By applying this argument to U1 , one constructs, for each value of p 1, a space Zp (p) with a symmetric basis {zn }∞ n=1 so that U1 is isomorphic to a complemented subspace of Zp . Since both U1 and Zp are isomorphic to their own square one can apply the decomposition method and conclude that U1 ≈ Zp , for any p 1. This further implies that (p) U1 has, quite surprisingly, a symmetric basis which is equivalent to {zn }∞ n=1 . As we have (p) ∞ (q) ∞ noticed before, for p = q, {zn }n=1 is not equivalent to {zn }n=1 , i.e., U1 has even a continuum of mutually non-equivalent symmetric bases. With some additional effort, one can construct more “natural” spaces, namely Orlicz sequence spaces, which also have a continuum of mutually non-equivalent symmetric bases (cf. [36] or [37, p. 153]). The fact that all the examples of spaces with a symmetric basis

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L. Tzafriri

considered so far have either a unique symmetric basis, up to equivalence, or uncountably many mutually non-equivalent symmetric bases lead inevitably to the question whether this is always the case. It turns out that the answer is negative: Read [50] has constructed, for every n = 1, 2, . . . or n = ℵ0 , a space Xn with a symmetric basis which has precisely n different symmetric bases. It should be added that we are very far form being able to characterize the class of spaces with a unique symmetric basis, up to equivalence. In fact, we do not have even a reasonable conjecture.

3. Uniqueness of unconditional bases, up to a permutation The notion of uniqueness of the unconditional basis of a Banach space X can be interpreted as asserting that X can be represented in a unique manner as a space of sequences a = {an }∞ n=1 ∈ X with the property that ∞ {an }∞ n=1 ∈ X ⇒ {εn an }n=1 ∈ X,

for any choice of εn = ±1; n = 1, 2, . . . . Obviously, in this representation X is considered as a set of ordered sequences. Now, if instead we consider X as a space of unordered sequences with the property described above then uniqueness of the representation has a different meaning: uniqueness up to permutation and equivalence. Recall that, as it was mentioned in the introduction, a Banach space X with a normalized unconditional basis {xn }∞ n=1 is said to have unique unconditional basis, up to equivalence and permutation, if, whenever {yn }∞ n=1 is another ∞ , for some normalized unconditional basis of X, then {yn }∞ is equivalent to {x } π(n) n=1 n=1 permutation π of the integers. It turns out that there are considerably more spaces with a unique unconditional basis, up to permutation, than the three space 2 , 1 and c0 , which are known to have, up to equivalence, a unique unconditional basis. One such class of spaces was found by Èdelšte˘ın and Wojtaszczyk [19], who showed that finite direct sums of 2 , 1 and c0 have the uniqueness property mentioned before. T HEOREM 3.1. Each of the Banach spaces 1 ⊕ 2 , 1 ⊕ c0 , 2 ⊕ c0 and 1 ⊕ 2 ⊕ c0 has, up to permutation and equivalence, a unique unconditional basis. In order to describe the ideas used in the proof, consider, e.g., the case of the space 1 ⊕ 2 and assume that {zn = xn + yn }∞ n=1 , where xn ∈ 1 and yn ∈ 2 for all n, is a normalized K-unconditional basis of this space. Put and N2 = n; xn > 1/2K N1 = n; xn 1/2K and notice that both [zn ]n∈N1 and [zn ]n∈N2 are complemented subspaces in 1 ⊕ 2 whose direct sum is the whole space. In order to study these two complemented subspaces of 1 ⊕ 2 , we need a result which “rotates” any complemented subspace of a “nice” direct


1645

sum into a “correct” position. To this end, recall that an operator T from a Banach space X into a space Y is strictly singular if the restriction of T to any infinite-dimensional subspace of X is not an isomorphism. Compact operators are of course strictly singular but these two notions are very different. Clearly, every bounded linear operator from 1 into 2 or, viceversa, from 2 into 1 is strictly singular. Now, we can state the theorem of Èdelšte˘ın and Wojtaszczyk [19] which provide the “rotation” into a “correct” position. T HEOREM 3.2. Suppose that X and Y are two Banach spaces so that every operator from X into Y is strictly singular, and let P be a bounded projection form X ⊕ Y onto a subspace Z. Then one can find an automorphism S of X ⊕ Y and complemented subspaces X0 of X and Y0 of Y such that SZ = X0 ⊕ Y0 . We omit the proof of the theorem which is based on a good understanding of several facts on Fredholm operators. We return now to the proof of Theorem 3.1 in the case of the direct sum 1 ⊕ 2 . The proof will be completed once we show, e.g., that [zn ]n∈N1 ≈ 2 and [zn ]n∈N2 ≈ 1 since both 1 and 2 have a unique unconditional basis, up to equivalence. Notice that if xn =

∞

an,j zj

and yn =

j =1

∞

bn,j zj

j =1

then an,n + bn,n = 1, for all n. However, for n ∈ N1 , |an,n | Kxn 1/2, and thus |bn,n | > 1/2. Now consider the linear operator U from the subspace [εn yn ]n∈N1 of L∞ (2 ), where εn (t); n = 1, 2, . . . , denote the Rademacher functions, into [zn ]n∈N1 which maps εn yn to zn , for all n ∈ N1 . Since both {εn yn }n∈N1 and {zn }n∈N1 are unconditional bases, a well known diagonal argument shows that the corresponding diagonal operator is bounded by KU . This means that cn bn,n zn KU sup cn ε n y n , n∈N1

εn =±1 n=1,2,...

n∈N1

for any choice of {cn }n∈N1 . By Theorem 3.2, if [zn ]n∈N1 is not isomorphic to 2 then it contains a complemented subspace isomorphic to 1 . Hence, one can find a normalized block basis wj = ∈σj d z ; j = 1, 2, . . . , of {zn }n∈N1 , which is equivalent to the unit vector basis in 1 . By passing to a subsequence if necessary, one can assume that the subspaces [y ]∈σj ; j = 1, 2, . . . , are “almost” supported on mutually orthogonal subspaces of 2 in the sense that the unit

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L. Tzafriri

vector basis {en }∞ n=1 of 2 can be split into disjoint subsets {en }n∈nj , j = 1, 2, . . . , so that, essentially speaking, [y ]∈σj ⊂ [en ]n∈ηj , for all j . Therefore, for any sequence {cj }∞ j =1 2 < ∞ and any choice of ε = ±1, for all n, with ∞ |c | j n j =1 ∞ ∞ ∞ 1/2 2 1/2 2 2 cj d ε y ∼ |cj | d ε y K |cj | . j =1

j =1

∈σj

j =1

∈σj

Hence, the series ∞ j =1 cj ∈σj d ε y converges unconditionally and so does the series ∞ ∞ ∈σj b, d z . This implies the convergence of the series j =1 cj j =1 cj wj , whenever ∞ ∞ 2 j =1 |cj | < ∞, which of course contradicts the fact that {wj }j =1 is equivalent to the unit vector basis of 1 . Consequently {zn }n∈N1 is equivalent to the unit vector basis of 2 . In a similar manner but using also a duality argument, one proves that [zn ]n∈N2 ≈ 1 , and thus that {zn }n∈N2 is equivalent to the unit vector basis of 1 . While the finite direct sums constructed out of the three spaces 1 , 2 and c0 have, up to equivalence and permutation, a unique unconditional basis, this is not always the case for infinite direct sums constructed out of the same set of spaces. On the positive side we have, e.g., the following result from [6]. T HEOREM 3.3. Every normalized unconditional basis of an infinite-dimensional complemented subspace of the direct sum (2 ⊕2 ⊕· · ·⊕2 ⊕· · ·)0 , is equivalent to a permutation of the unit vector basis of one of the following six spaces: 2 , c0 , 2 ⊕ c0 ,

∞

⊕n2

, 2 ⊕

n=1

∞

⊕n2

, (2 ⊕ 2 ⊕ · · · ⊕ 2 ⊕ · · ·)0 .

n=1

0

0

A similar statement holds for complemented subspaces of the dual space (2 ⊕ 2 ⊕ · · · ⊕ 2 ⊕ · · ·)1 . Consequently, the six spaces appearing above and their duals have, up to permutation and equivalence, a unique unconditional basis. The proof of Theorem 3.3 is not extremely difficult but still beyond the scope of this article and therefore we omit it here. Another result from [6] is: T HEOREM 3.4. Every normalized unconditional basis of an infinite-dimensional complemented subspace of the direct sum (1 ⊕ 1 ⊕ · · · ⊕ 1 ⊕ · · ·)0 is equivalent to a permutation of the unit vector basis of one of the following six spaces: c0 , 1 , c0 ⊕ 1 ,

∞ n=1

⊕n1

, 1 ⊕ 0

∞ n=1

⊕n1

, (1 ⊕ 1 ⊕ · · · ⊕ 1 ⊕)0 . 0

Consequently, each of these six spaces has, up to permutation and equivalence, a unique unconditional basis.


1647

Though Theorems 3.3 and 3.4 have a similar formulation, the proof of 3.4 in [6] is considerably harder than that of 3.3. This is reflected also by the fact that the function M = M(K) (so that if {xn }∞ n=1 is a normalized K-unconditional basis of one of the spaces X appearing above, then {xn }∞ n−1 is M(K)-equivalent to a permutation of the unit vector basis of X) behaves differently in the two cases discussed above. While the proof of 3.3 gives M(K) as a power of K, the proof of 3.4 yields M = M(K) as an exponential function of K and examples show that an exponential function is really needed. Very recently, Casazza and Kalton [10] provided a simpler proof of the uniqueness, up to a permutation, of the unit vector basis in the space (1 ⊕ 1 ⊕ · · · ⊕ 1 ⊕ · · ·)0 and its dual. Quite surprisingly, the other infinite direct sums which can be constructed out the the three spaces 1 , 2 and c0 do not have the the uniqueness property exhibited in Theorems 3.3 and 3.4. T HEOREM 3.5 (cf. [6]). The direct sums

∞

⊕n∞

n=1

, c0 ⊕ 2

∞ n=1

⊕n∞

, (c0 ⊕ c0 ⊕ · · · ⊕ c0 ⊕ · · ·)2 2

and their duals fail to have a unique unconditional basis, up to equivalence and permutation. n P ROOF. We shall treat here only the case of the dual direct sum X = ( ∞ n=1 ⊕1 )2 ; the other cases can be easily derived from it without too much difficulty. In order to exhibit a normalized unconditional basis of X which is not permutatively equivalent to the unit vector basis of X, we first fix n and let Fi = {Ai,j }nj=1 ; 1 i n, the independent partitions of the interval [0, 1] into sets of measure equal to 1/n (i.e., for A ∈ Fi , B ∈ Fj and i = j, μ(A ∩ B) = μ(A)μ(B) = 1/n2 ). With {ei }ni=1 denoting the unit vector basis of the space n1 , consider the vector valued functions fi,j (t) = n1/2 χAi,j (t)ei ;

1 i, j n,

which form a normalized 1-unconditional basis in a subspace Yn of the space L2 ([0, 1], n1 ) since n 2 1/2 n n 1 1/2 ai,j fi,j = n ai,j χAi,j (t) dt , 0 i,j =1

i=1 j =1

for any choice of {ai,j }ni,j =1 . Indeed, for each 1 i n, the functions n ai,j χAi,j (t) j =1

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L. Tzafriri

and n |ai,j |χAi,j (t) j =1

are both Fi measurable and have the same independence of Fi , distribution. Hence, the 1 i n, ensures that that the functions | nj=1 ai,j χAi,j (t)| and nj=1 |ai,j |χAi,j (t) have the same norm in the space L2 ([0, 1], n1 ). Now notice that if Ei denotes the conditional expectation operator associated to the partition Fi , i.e., the operator defined by Ei ϕ =

ϕ(t) dt χAi,j ,

n Ai,j

j =1

for ϕ ∈ L1 (0, 1), then one can define a projection Pn from L2 ([0, 1], n1 ) onto Yn by setting Pn =

n (Ei ϕi )ei , i=1

whenever f (t) = ni=1 ϕi (t)ei is an element of L2 ([0, 1], n1 ). Then, by using the independence of the partitions {Fi }ni=1 , one gets that Pn f = 2

1 n 0

=

2 |Ei ϕi |

dt

i=1

n

1

|Ei ϕi |2 dt +

i=1 0

=

n

n i,j =1 i =j

1

|Ei ϕi |2 dt +

i=1 0

1

|Ei ϕi ||Ej ϕj | dt

0

n i,j =1 i=j

1 0

1

|Ei ϕi | dt

|Ej ϕj | dt ,

0

for any f as above. Hence, Pn f 2 2

n

1

|ϕi |2 dt = 2f 2 ,

i=1 0

√ √ i.e., Pn 2, which shows that Yn is 2-complemented in L2 ([0, 1], n1 ), or even in the subspace (n1 ⊕ n1 ⊕ · · · ⊕ n1 )2 (nn factors) of L2 ([0, 1], n1 ). It follows that the space Y =( ∞ ⊕Y n )2 is isomorphic to a complemented n=1 subspace of X. On the other hand, the block basis yi = n−1/2 nj=1 fi,j ; 1 i n, of {fi,j }ni,j =1 is 1-equivalent to the unit vector basis of n1 , and thus its span is complemented in Yn , since


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1 is block injective, i.e., 1-complemented, whenever it is embedded as a block basis of a 1-unconditional basis. Consequently, Y contains a 1-complemented copy of X and thus, by the decomposition method, X ≈ Y . The proof will be completed once we show that the natural basis of Y is not equivalent to a permutation of the unit vector basis in X. To this end, notice that the unit vector basis of X has the property that any of its subsets of finite codimension contains, for each n, a subsequence 1-equivalent to the unit vector basis of n1 . The natural basis of Y has, however, a diametrically opposite behavior. Indeed, for each ε > 0 and each integer k, there exists an integer n = n(ε, k) such that any subset of {fi,j }ni,j =1 of cardinality k is (1 + ε)-equivalent to the unit vector basis of k2 and, moreover, this property remains valid in Y provided that, for any such k, we eliminate a suitable finite set of vector from the basis of Y . Another quite well-known space for which the question of uniqueness, up to a permutation, could be settled is the so-called Tsirelson space, introduced in [54] (see also [20]). This space is the completion of the space of sequences x ∈ c0,0 under the minimal norm · T satisfying the conditions: (i)

xT x0 ,

for all x ∈ c00 , and (ii)

1 xi = xi T , 2 i=1

T

i=1

whenever support x1 < support x2 < · · · < support x ; = 1, 2, . . . . The fact that such a norm exists is proved in [20]. T HEOREM 3.6. Every complemented subspace of T with an unconditional basis has, up to permutation and equivalence, a unique unconditional basis. Actually, this result was preceded by a theorem proved in [6] which asserts that the 2-convexification T (2) of T , as well as its complemented subspaces with an unconditional basis, have a unique unconditional basis, up to equivalence and permutation (recall that 1/2 T (2) is obtained from T by defining the norm in it as follows: xT (2) = |x|2 T , for vectors x having the property that the square of their absolute value belongs to T ). The proof is quite difficult. The proof for T (i.e., of Theorem 3.6), due to Casazza and Kalton [9], is the byproduct of a more general study of the uniqueness property in spaces which do not contain uniformly complemented copies of n2 , for all n. For 1 < p = 2, the p-convexification T (p) of T does not have a unique unconditional basis, up to permutation and equivalence, since, as was pointed out by Kalton, T (p) can be represented a Tsirelson sum of np ’s and, in this sum, the factor np has an unconditional basis containing among its vectors an k2 with k ∼ log n. The Banach spaces with a unique unconditional basis, up to permutation and equivalence, which were considered so far in this section, have the additional property that also

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L. Tzafriri

their complemented subspaces with an unconditional basis share the uniqueness feature. For such spaces one can introduce the notion of genus (cf. [6]). A Banach space with an unconditional basis is said to be of genus n if in all its complemented subspaces with an unconditional basis, the normalized unconditional basis is unique, up to equivalence and permutation, and there are exactly n different isomorphic types of complemented subspaces with an unconditional basis. The spaces 1 , 2 and c0 are clearly of genus 1. It turns out (cf. [6]) that these three spaces are the only ones of genus 1. T HEOREM 3.7. 1 , 2 and c0 are the only spaces with a (unique) unconditional basis of genus 1. The idea of the proof from [6] is the following: if X is a space with a normalized uncon∞ ditional basis {xn }∞ n=1 of genus 1 then every infinite subsequence of {xn }n=1 is equivalent . Then, by using a sort of compactification argument involving to a permutation of {xn }∞ n=1 is, up to permutation, a subspreading models and ultraproducts, one can show that {xn }∞ n=1 itself is subsymmetric. symmetric basis. Therefore, one can assume w.l.o.g. that {xn }∞ n=1 ∞ Since {xn }∞ n=1 is subsymmetric any block basis with constant coefficients of {xn }n=1 spans a complemented subspace U of X. Moreover, by a slightly more complicated argument than that used in the case when {xn }∞ n=1 is symmetric, it can be easily shown that X ≈ X ⊕ U . Hence, any normalized block basis with constant coefficients of {xn }∞ n=1 is , for some permutation π of the integers. Then, by a simple modequivalent to {xπ(n) }∞ n=1 ification of Zippin’s characterization of perfectly homogeneous bases from [58], one concludes that {xn }∞ n=1 is equivalent to the unit vector basis in c0 or p , for some p 1. The cases when 1 < p = 2 can be easily dismissed, as shown ∞ before. n n Theorems 3.3 and 3.4 above show that the sums ( ∞ n=1 ⊕2 )0 , ( n=1 ⊕1 )0 and their duals are spaces of genus 2. However, it is not known if there exist other spaces of genus 2. The spaces 1 ⊕ 2 , 1 ⊕ c0 and 2 ⊕ c0 are of genus 3 but, again, we do not have a complete characterization of this class. It is quite possible that the class of spaces of finite genus coincides with that of spaces which are obtained from Hilbert space by taking repeated finite or infinite direct sums in the sense of c0 or 1 . Casazza and Lammers [12] obtained many results on spaces of finite genus, e.g., that the unconditional basis contains a subsequence equivalent to the unit vector basis in either 1 , or 2 or c0 . There is a feeling that this class will be well understood once spaces of genus 2 are characterized. The Tsirelson space T and its 2-convexification are of infinite genus. The question whether the uniqueness of the unconditional basis, up to permutation and equivalence, is a hereditary property has a negative answer. More precisely, Casazza and Kalton [9] constructed the first example of a Banach space with a unique unconditional basis, up to equivalence and permutation, which has complemented subspaces with an unconditional basis lacking the uniqueness property. Their starting point in the Orlicz sequence space M , where M(t) = t/(1 + |log t|), for t 0. In this, as in any other Orlicz block basis with constant coefficients of the form qj+1 space, a normalized qj+1 e )/ e uj = ( n=q n n=q +1 n ; j = 1, 2, . . . , generates a so-called modular space j−1 +1 qj+1j−1 M [sj ], where sj = 1/ n=qj−1 +1 en , for all j . This subspace, which is clearly comple-


1651

mented in M , can be described as the space of all sequences a = (a1 , a2 , . . . , aj , . . .) so that ∞ j =1 Msj (|aj |) < ∞, where Ms (t) = M(st)/M(s). It can be easily verified that if sj → 0 very fast, as j → ∞, then M [sj ] is isomorphic to 1 , and, on the other hand, if infj sj > 0 then M [sj ] ≈ M . By manipulating between these diametrically opposite situations, one can select a normalized block basis {uj }∞ j =1 of the unit vector basis in M which is permutatively equivalent to its square but M [sj ] is not isomorphic to either 1 or M itself. Casazza and Kalton have shown in [9] that in this case {uj }∞ j =1 , is, up to permutation and equivalence, the unique unconditional basis of M [sj ] and, moreover, M [sj ] contains complemented subspaces with a non-unique unconditional basis. The fact that direct sums in the sense of c0 of spaces such as 2 or 1 , which do have a unique unconditional basis, have also the uniqueness property, led to the question (stated explicitly in [6]) whether, whenever a space X has unique unconditional basis, up to permutation, then (X ⊕ X ⊕ · · · ⊕ X ⊕ · · ·)0 also has a unique unconditional basis. It turns out that the answer to this question is negative: Casazza and Kalton [9] have proved that direct sums of T or T (2) in the c0 -sense do not have a unique unconditional basis, up to permutation, in spite of the fact mentioned above that both T and T (2) have this property. 4. Uniqueness in finite-dimensional spaces Since any two bases of a finite-dimensional space are always equivalent the question of uniqueness of the basis makes no sense in the framework of a single finite-dimensional Banach space but rather in that of families of such spaces. As in the case of infinitedimensional spaces, the most interesting results are obtained for families of spaces with an unconditional or symmetric basis. D EFINITION 4.1. Let F be a family of finite-dimensional spaces each of which has a normalized 1-unconditional basis. We say that the members of F have a unique unconditional basis, up to equivalence (and permutation), if there exists a function ψ : [1, ∞) → [1, ∞) such that, whenever a space E ∈ F has another normalized unconditional basis {ej }nn=1 whose unconditional constant is K, then {ej }nj=1 is ψ(K)-equivalent to (a permutation of) the given 1-unconditional basis of E. By replacing in the above definition the word “unconditional” with “symmetric”, one defines the notion of uniqueness of the symmetric basis for the elements of F . Typical families studied in this section are Fp = np ; n = 1, 2, . . . ; 1 p ∞. The same argument, involving the parallelogram identity, which was used in the case of 2 , shows that the members of F2 have a unique unconditional basis, up to equivalence. Furthermore, by using the same argument involving 1-summing and 2-summing operators, as in the case of 1 and c0 , one can show that also F1 and F∞ have the same uniqueness property as F2 . P ROPOSITION 4.2. The members of the families F1 , F2 and F∞ have a unique unconditional basis, up to equivalence.

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L. Tzafriri

The analogy with the infinite-dimensional case might lead one to believe that these three families above are the only ones having, up to equivalence, a unique unconditional basis. It turns out that this fact is false. In order to produce examples of other families whose member have a unique unconditional basis, up to equivalence, one needs a result of Dor and Starbird [14] asserting that, for p > 2, a normalized unconditional basis of p is either equivalent to the unit vector basis of p or it admits, for each n, a block basis which is 2-equivalent to the unit vector basis of n2 . By using this fact together with the known assertion that Hilbert space is uniformly complemented in Lp -spaces, for p > 2 (cf. [48] or [41]), it is deduced in [27, p. 48] that: P ROPOSITION 4.3. There exist constants C < ∞ and M < ∞ such that every normalized K-unconditional basis of p ; p > 1, with √ # K C max p, p/(p − 1) , for some constant C < ∞, is M-equivalent to the unit vector basis of p . We omit the details of the proof but just mention that the bound for√K, appearing in the right-hand side of the above inequality, is actually equal to γp (2 )/ 2, where γp (2 ) denotes as usual the factorization constant of 2 through Lp -spaces. An immediate consequence of Proposition 4.3 is the fact that, for each sequence pn → ∞, the members of the family F = npn ; n = 1, 2, . . . have a unique unconditional basis, up to equivalence. We pass now to some questions concerning the uniqueness of the symmetric basis. We begin with subspaces of Lp . Among the finite-dimensional subspaces of Lp with a symmetric basis one can find the families Fp and F2 . Other interesting subspaces of Lp are (p) the “diagonals” of np ⊕ n2 generated by vectors of the form {ej + wj ej(2)}nj=1 , where (p)

{ej }nj=1 and {ej(2)}nj=1 denote the unit vectors in np , respectively n2 , and {wj }nj=1 is an arbitrary sequence of scalars. Spaces of this type were studies by Rosenthal [51] who coined for them the name Xp -space. If wj = w; j = 1, 2, . . . , n, then obviously we deal with symmetric Xp -space. It turns out that every symmetric basic sequence in Lp ; p > 2, is equivalent to a symmetric Xp -space. This characterization has been proved in the Memoir [27, p. 34]. T HEOREM 4.4. For every p > 2 and K 1, one can find a constant D = D(p, K) < ∞ so that any normalized basic sequence {xj }nj=1 in Lp (0, 1), whose symmetry constant is (p)

K, is D-equivalent to the symmetric Xp -space generated by {ej + wej(2)}nj=1 , where n √ w= xj / n. j =1


1653

P ROOF. In the first step of the proof, the symmetric basic sequence in Lp is replaced by a sequence of so-called “symmetrically exchangeable” random variables, i.e., a sequence of functions in Lp whose joint distribution in Rn remains invariant under permutation and change of sign. In order to describe this construction, let {xj }nj=1 be a normalized K-symmetric basic sequence in Lp (0, 1); p > 2, and Hn the family of all distinct n!2n pairs {π, (εj )nj=1 }, where π is a permutation of {1, 2, . . . , n} and εj = ±1, for all 1 j n. The elements of Hn can be put in a one-to-one correspondence with unit subintervals of [0, n!2n ] of the form [k, k + 1], with k being an integer. If a pair {π, (εj )nj=1 } is in correspondence with the subinterval I of [0, n!2n ] then we define fj (t) = εj xπ(j )(t);

t ∈ I ; 1 j n.

In order to make the functions {fj }nj=1 , which are defined on the interval [0, n!2n ], into an exchangeable sequence we compress the interval [0, n!2n ] into [0, 1] in the obvious linear way thus obtaining a new sequence {gj }nj=1 which is symmetrically exchangeable and K-equivalent to the original sequence {xj }nj=1 . It is easily seen that there is no loss of generality in assuming that each point of [0, 1] belongs to the support of at least one of the functions {xj }nj=1 . Then, by a corresponding change of density, one can make the expression ( nj=1 |xj |2 )1/2 into a constant C. This n however implies that also the square function ( j =1 |gj |2 )1/2 is a constant equal to C = ( nj=1 |xj |2 )1/2 . Since {gj }nj=1 is a 1-unconditional basic sequence in Lp (0, 1) and the space Lp (0, 1) is of cotype p, for p > 2, it follows that n n 1/p p aj gj |aj | , j =1

j =1

for any choice if {aj }nj=1 . On the other hand, since {gj }nj=1 is an orthogonal sequence in √ L2 (0, 1) with gj 2 = C/ n, for all 1 j n, we get that n n n n 1/2 1/2 C 2 2 2 aj gj aj gj = |aj | gj 2 =√ |aj | , n j =1

j =1

j =1

2

j =1

for all 1 j n. This proves one part of 4.3 since n n aj xj K −1 aj gj j =1

j =1

K

−1

max

n j =1

1/p |aj |

p

n 1/2 C 2 , , √ |aj | n j =1

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for all {aj }nj=1 , and, by Khinchine’s inequality, n xj K j =1

n p 1/p εj xj dε j =1

p 1/p KBp C. = K εj xj dε

The opposite inequality is more difficult and requires some ideas of Rosenthal [51] and a certain averaging procedure. We omit this argument which is described in detail in [27]. The significance of Theorem 4.4 becomes √ clear only after we fully understand the meaning of the expression w = nj=1 xj / n. P ROPOSITION 4.5. For every K 1, there exists a constant M = M(K) such that, whenever {xj }nj=1 is a normalized K-symmetric basis in a finite-dimensional subspace X of L n2 ) is M-equivalent to the expression p (0, 1), then d(X, √ the Banach–Mazur distance √ n n j =1 xj / n if 1 p 2 and to n/ j =1 xj if p > 2. P ROOF. Suppose that p > 2. It follows from 4.4 that n n n 1/2 j =1 xj −1 2 aj xj D(K, p) √ |aj | , n j =1

j =1

for any choice of {aj }nj=1 . Furthermore, by using Khinchine’s inequality in Lp (0, 1) and the 2-convexity of Lp (0, 1), for p > 2, we also get that n aj xj K j =1

p 1/p n aj εj xj dε = K j =1

p 1/p n aj εj xj dε j =1

n 1/2 n 1/2 |aj |2 |xj |2 |αj |2 , KBp KBp j =1

j =1

again for any choice of {aj }nj=1 . Combining these two inequalities, we conclude that √

n . d X, n2 KBp D(K) n xj j =1

Next notice that, for every sequence {aj }nj=1 of scalars, we also get n n 1 max |aj | aj xj K xj max |aj |, 1j n 2K 1j n j =1

j =1


1655

i.e., n

n 2 d X, ∞ 2K xj . j =1

Hence,

n

√ n d X, n2 d X, n∞ 2K 2 xj d X, n2 , j =1

i.e.,

√

−1 n n , d X, n2 2K 2 xj j =1

which completes the proof for p > 2. The case 1 p 2 is treated in a similar manner. Proposition 4.5 shows that, for a given finite-dimensional subspace X of Lp (0, 1); 1 p ∞, with a symmetric basis, the expression nj=1 xj is, up to a constant depending only on the symmetry constant of {xj }nj=1 , an invariant of the space X rather than that of the particular symmetric basis {xj }nj=1 . This fact together with Theorem 4.3 imply the following consequence from [27, p. 39]. C OROLLARY 4.6. For p > 2, let Gp denote the family of all finite-dimensional subspaces of Lp (0, 1) which have a 1-symmetric basis. Then the members of Gp have, for any fixed p, a unique symmetric basis, up to equivalence. A trivial duality argument, together with 4.2, shows that, for 1 p ∞, each member of the family Fp has, up to equivalence, a unique symmetric basis. This means that, for every 1 p ∞, there is a function ψp (K) which corresponds to the family Fp by Definition 4.1. It turns out that these functions can be selected as not to depend on p and the following result from [27, p. 39] is true. T HEOREM 4.7. The members of the family symmetric basis.

1p∞ Fp

have, up to equivalence, a unique

We omit the proof which is given in detail in the Memoir [27]. We already see that the class of families whose members have, up to equivalence, a unique symmetric basis is quite large. In an attempt to discover the largest family of space with a unique symmetric basis, Schütt studied in [53] the family Dα of all finitedimensional Banach spaces X with a normalized 1-symmetric basis {xj }nj=1 which satisfy the condition

d X, n2 nα , and proved the following beautiful result which we quote here without giving its proof.

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T HEOREM 4.8. For any α > 0, each member of the family Dα has, up to equivalence, a unique symmetric basis. The abundance of families of finite-dimensional spaces with a unique symmetric basis leads naturally to the question, which was raised in the Memoir [27], whether this is not always the case. Theorem 4.8, mentioned above, shows that the construction of a counterexample is not an easy matter. Eventually, Gowers produced in [23] ingenious examples of finite-dimensional normed spaces with two (asymptotically) non-equivalent symmetric bases. The precise statement is as follows. T HEOREM 4.9. For each integer k, there exists a Banach space of dimension n = 2k with two 2-symmetric normalized symmetric bases {ei }ni=i and {ei }ni=1 whose constant of equivalence is at least exp(log log n/8 log log log n). We shall describe here only the main ideas of Gowers’ construction; the details can be found in the paper [23]. Fix n and let A : Rn → Rn be a linear map defined by an orthogonal matrix which will be specified later. Let {ei }ni=i denote the unit vector basis of Rn and put ei = Aei , for all 1 i n. These will be the two bases under consideration. The next step is to define a norm · on Rn so that both {ei }ni=i and {ei }ni=1 become 2-symmetric bases. To this end, let Ω be the group of symmetries associated to the first basis {ei }ni=1 , i.e., of the linear maps w of the form w

n

a i ei =

i=1

n

εi ai eπ(i) ,

i=1

where εi = ±1, for all 1 i n, and π is a permutation of the integers {1, 2, . . ., n}. To the second basis {ei }ni=1 , we associate the group Ω = {AwA−1 ; w ∈ Ω}. Then put X0 = {εi ei ; εi = ±1, 1 i n} and define sets {Xj }∞ j =1 by induction, as follows: if j 1 is odd then Xj = w x; x ∈ Xj −1 , w ∈ Ω , and if j 1 is even, then Xj = {wx; x ∈ Xj −1 , w ∈ Ω}. n Once the sets {Xj }∞ j =0 have been introduced, we define the norm of a vector x ∈ R by setting

x = max 2−j x, xj ; xj ∈ Xj , j = 0, 1, 2, . . . .


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The fact that · is indeed a norm on Rn is trivial since the above maximum is always attained on a finite set of indices depending of course on the vector x under consideration. In order to prove that {ei }ni=1 is a 2-symmetric basis, fix a vector x = ni=1 ai ei ∈ Rn and assume that x = 2−j |x, xj |, for some integer j and some xj ∈ Xj . If j is odd then, for any w ∈ Ω, wxj ∈ Xj +1 so that wx 2−(j +1)wx, wxj = 2−(j +1) x, xj = 2−1 x. On the other hand, if j is even then xj = wx ˜ j −1 , for some w˜ ∈ Ω and xj −1 ∈ Xj −1 . Hence, ˜ j −1 ∈ Xj so that for any w ∈ Ω, wxj = (ww)x wx 2−j wx, wxj = 2−j x, xj = x, which proves that {ei }ni=1 is indeed 2-symmetric. The proof of the fact that {ei }n2=1 is also 2-symmetric is done in exactly the same way. The above construction is independent of the particular choice of the orthogonal matrix A. The idea now is to select such a matrix A so that e1 = Ae1 is as small as possible and it turns out that one can construct an orthogonal matrix A for which the corresponding vector e1 satisfies e exp(− log log n/8 log log log n). 1 This will suffice since it is not too difficult to show that {ei /ei }ni=1 and {ei /ei }nu=1 are at best e1 -equivalent; this fact follows by comparing the expectations e11 E ni=1 gi ei 1 and e1 E ni=1 gi ei , where {gi }ni=1 is a sequence of independent identically distributed 1 random variables. The construction of the orthogonal matrix A so that e1 = Ae1 is “small” is done by induction on its size. For k = 0 and therefore n = 20 = 1, we let A 0 = (1) while, for k > 0, A k is defined by A k

=

A k−1 Ik−1

Ik−1 −A k−1

,

where Ik−1 denotes the (2k−1 × 2k−1 )-identity matrix. Once the (2k × 2k )-matrix A k is defined, we put Ak = k −1/2 A k . It is easily seen that the matrices {Ak }∞ k=0 are not only orthogonal but also symmetric. The most difficult part of the proof is to show that, for n = 2k , the matrix A = Ak has the property that e1 exp(− log log n/8 log log log n). This argument is quite technical and it relies on a lemma of Harper [24], Bernstein [3], Hart [25] and Lindsey [40] which gives an estimate from below for the number of edges joining vertices of the k-cube of a given cardinality r. We omit these details which, as we mentioned before, can be found in [23]. We conclude this section with some remarks on two notions of uniqueness which are in the spirit of the so-called “proportional” theory of finite-dimensional spaces. The main definition introduced in [11] is the following.

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D EFINITION 4.10. Let F be a family of finite-dimensional spaces each of which has a normalized 1-unconditional basis. We say that the members of F have an almost (somewhat) unique unconditional basis provided there exists a function ϕ(K, λ), defined for all K 1 and 0 < λ < 1, such that, whenever X ∈ F with the given normalized 1-unconditional basis {xi }ni=1 has also another normalized K-unconditional basis {yi }ni=1 then, for any (some) 0 < λ < 1, there exist a subset σ ⊂ {1, 2, . . . , n} and a one-to-one function π : σ → {1, 2, . . . , n} so that {xi }i∈σ is ϕ(K, λ)-equivalent to {yπ(i)}i∈σ . These notions are obviously an extension of the notion of unique unconditional basis, up to equivalence and permutation, introduced above. A thorough study of almost and somewhat uniqueness of unconditional basis is made in the paper [11]. We quote here, without proof, the following result from [11], which is clearly a generalization of Theorem 4.8. T HEOREM 4.11. For any α > 0, each member of the family Dα has an almost unique unconditional basis.

5. Uniqueness of rearrangement invariant structures While in the preceding sections we focused on the uniqueness question only in the setting of sequence spaces, in the present one we pass to a study of similar problems in the framework of rearrangement invariant (r.i.) function spaces on a non-atomic measure space. The main requirement imposed on an r.i. function space X on a finite or σ -finite nonatomic measure space (Ω, Σ, μ) is that, for any automorphism τ of Ω and every measurable function f ∈ X, the function f (τ −1 ) also belongs to X and has the same norm as f . If the measure space (Ω, Σ, μ) is assumed to be non-atomic and separable (i.e., Σ endowed with the usual metric ρ(τ, η) = μ(σ Δη); σ, η ∈ Σ, is a separable metric space) then it is well known that (Ω, Σ, μ) is isomorphic to a finite or infinite interval endowed with the usual Lebesgue measure. Hence, in principle, we can restrict our attention to the canonical cases Ω = [0, 1] and Ω = [0, ∞), both endowed with the Lebesgue measure. In the Basic Concepts article such spaces are called symmetric lattices. In the case when a function space X is invariant with respect to the automorphisms of Ω then the same is true for X , the subspace of the dual X∗ of X which consists of “integrals”, i.e., of functionals of the form xg∗ (t) =

fg dμ,

f ∈ X.

Ω

In most of the interesting cases that appear in analysis, X is a norming subspace of X∗ . This occurs if and only if 0 fn (w) ↑ f (w) a.e. on Ω implies that limn→∞ fn = f . The proof of this simple assertion can be found in [38, 1.b.18]. For convenience, we shall assume here that this is always the case. The formal definition of the notion of r.i. function space, which will be used in the sequel, is the following.


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D EFINITION 5.1. An r.i. function space X on the interval Ω = [0, 1] or on the interval Ω = [0, ∞) is a Banach space of classes of equivalence of measurable functions on Ω such that: (i) For any automorphism τ of Ω, a function f ∈ X if and only if f (τ −1 ) ∈ X, and if this is the case then f (τ −1 ) = f . (ii) X is a norming subspace of the dual X∗ of X and thus X is order isometric to a subspace of X

. As a subspace of X

, the space X is either minimal (i.e., X is the closure of the simple integrable functions on Ω) or it is maximal (i.e., X = X

). (iii) If Ω = [0, 1] then L∞ (0, 1) ⊂ X ⊂ L1 (0, 1), with the inclusion maps being of norm one, i.e., f 1 f X f ∞ , for all f ∈ L∞ (0, 1). (iii ) If Ω = [0, ∞) then L1 (0, ∞) ∩ L∞ (0, ∞) ⊂ X ⊂ L1 (0, ∞) + L∞ (0, ∞), again with the inclusion maps being of norm one. Recall that the norm of a function f in the space L1 (0, ∞) ∩ L∞ (0, ∞) is defined as

f = max f 1 , f ∞ . The space L1 (0, ∞) + L∞ (0, ∞) is often used in interpolation theory and the norm of a function f in this space is usually defined by the formula f = inf g1 + h∞ ; f = g + h , the infimum being taken over all decompositions f = g + h, with g ∈ L1 (0, ∞) and h ∈ L∞ (0, ∞). It is easily verified that if Y = L1 (0, ∞) + L∞ (0, ∞) then Y = L1 (0, ∞) ∩ L∞ (0, ∞). The norm in the space Y can be alternatively defined with the aid of the notion of decreasing rearrangement of a function f on either Ω = [0, 1] or on Ω = [0, ∞). The decreasing rearrangement f ∗ of a function f 0 is defined as the right continuous inverse of the distribution function df of f , which is defined by df (t) = μ w ∈ Ω; f (w) > t . In other words, f ∗ (x) = inf t > 0; df (t) x ;

0 x < μ(Ω).

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If f is not 0 then f ∗ is defined as the decreasing rearrangement of the absolute value |f | of f . It turns out that the norm of a function f ∈ Y = L1 (0, ∞) + L∞ (0, ∞) is equal to

1

f ∗ (x) dx.

0

Indeed, for any decomposition f = g + h, as above, and any subset σ ⊂ [0, ∞), we have that σ

f (t) dt g1 + h∞ μ(σ ).

Hence,

1

f ∗ (x) dx = sup

0

f (t) dt; μ(σ ) = 1 f .

σ

Conversely, fix f ∈ Y and put λ = f ∗ − f ∗ χ[0,1] ∞ . Then

f = f ∗ f ∗ − min λ, f ∗ 1 + min(λ, f )∞ 1

= f ∗ − λ χ[0,1] 1 + λ = f ∗ (x) dx. 0

As in the case of spaces with a symmetric basis, the question of uniqueness (this time of the r.i. structure) has been studied most extensively in Lp -spaces. A result on r.i. function spaces on [0, 1], which is quite useful in the study of uniqueness in Lp -spaces, was proved in the Memoir [27, p. 41]. T HEOREM 5.2. An r.i. function space X on [0, 1], which is isomorphic to a subspace of Lp (0, 1); p > 2, coincides with either Lp (0, 1) or L2 (0, 1), up to an equivalent renorming. P ROOF. Let X be an r.i. function space on [0, 1] and let T be an isomorphism from X into Lp (0, 1); p > 2. For every n and 1 i 2n , denote by ϕn,i the characteristic funcn tion of the interval [(i − 1)/2n , i/2n ). Since, for every n, the sequence {T ϕn,i }2i=1 is a K-symmetric basic sequence in Lp (0, 1) with K T · T −1 , one can use Theorem 4.4 from the previous section and conclude the existence of a constant D < ∞, depending only n on p and on T , so that, for every choice of scalars {ai }2i=1 , we have 2n 2n 2n 1/p 1/2 ϕn,i D p 2 ai |ai | ,w |ai | , ∼ max ϕn,i X i=1

X

i=1

i=1


1661

where w=

ϕn,i i=1 ϕn,i X X

2n

√ 2n

=

1

√ . ϕn,1 X 2n

Hence, for any simple function f over the field generated by the intervals [(i − 1)/2n , i/2n ); 1 i 2n , we have that D f X ∼ max ϕn,1 X 2n/p f p , f 2 . Taking f ≡ 1 we get that the sequence {ϕn,1 X 2n/p }∞ n=1 is bounded by D and thus, with α = lim inf ϕn,1 X 2n/p , n→∞

one concludes that, D f X ∼ max αf p , f 2 , for any simple function f over the dyadic intervals in [0, 1]. If α = 0 then obviously X = L2 (0, 1) and if α > 0 then, since p > 2, X = Lp (0, 1), both equalities up to an equivalent renorming. Theorem 5.2 has been generalized in [26] to a quite large class of pairs X and Y , where Y is an r.i. function space on [0, ∞) and X is a non-atomic Banach lattice isomorphic to a subspace of Y . Under different conditions on Y , stated mostly in terms of p-convexity and q-concavity-notions which are described in the Basic Concepts article – it is shown in [26] that X is isomorphic to a sublattice of Y . For instance, this is the case when Y is p-convex and q-concave, for some p > 2 and q < ∞, and X is r-convex, for some r > 2. The same type of assumptions imply that if Y is an r.i. function space on [0, 1] then X contains a non-trivial band lattice isomorphic to a sublattice of Y . The paper [26] contains also some results on complemented spaces. For instance, if Y is a separable r.i. function space on either [0, 1] or [0, ∞), which contains no 2 as a complemented sublattice, and X is a p-convex Banach lattice, for some p > 2, which is isomorphic to a complemented subspace of Y then X is even lattice-isomorphic to a complemented sublattice of Y . In exactly the same manner as in the proof of Theorem 5.2, one can prove the following version for [0, ∞) (cf. [27, p. 43]). T HEOREM 5.3. An r.i. function space X on [0, ∞), which is isomorphic to a subspace of Lp (0, ∞); p > 2, coincides with one of the spaces Lp (0, ∞), L2 (0, ∞) or L2 (0, ∞) ∩ Lp (0, ∞), up to an equivalent renorming. The norm of a function f ∈ L2 (0, ∞) ∩ Lp (0, ∞) is of course defined by f = max(f 2 , f p ).

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Theorem 5.2 above implies the uniqueness of the r.i. structure on [0, 1] of Lp (0, 1), for p > 2, and thus, by duality, also of 1 < p < 2. The uniqueness of the r.i. structure of L2 (0, 1) is quite trivial, in view of 1.1 above. The uniqueness of the r.i. structure on [0, 1] of L∞ (0, 1) can be easily reduced to that of L1 (0, 1). In order to prove the uniqueness of the r.i. structure on [0, 1] of the space L1 (0, 1), we need the fact that if X is an r.i. function space on [0, 1], which is isomorphic to L1 (0, 1), then, for every n, the sequence of characteristic functions ϕn,i = χ[(i−1)/2n ,i/2n ) ; 1 i 2n , forms a 1-unconditional basic sequence in X whose span is 1-complemented in X, by the conditional expectation operator relative to the field generated by the intervals [(i − 1)/2n , i/2n ); 1 i 2n (i.e., by the operator defined by n i/2n En f = 2n 2i=1 ( (i−1)/2n f dx)ϕn,i ). Then, the same argument, as the one used in Secn

tion 1 to prove that 1 has a unique unconditional basis, shows that {ϕn,i /ϕn,i X }2i=1 is n equivalent to the unit vector basis of 21 , with a constant of equivalence independent of n. We summarize the above observations in the following theorem which, again, has been proved in [27]. T HEOREM 5.4. The space Lp (0, 1) has a unique structure as an r.i. function space on [0, 1], for any value of 1 p ∞. R EMARK . In the case p = 1, Kalton [29] proved a much stronger result: an r.i. function space on [0, 1] which contains a copy of L1 (0, 1), already coincides with L1 (0, 1), up to an equivalent norm, provided it does not contain uniformly isomorphic copies of n∞ . Contrary to an initial belief, the space Lp (0, ∞) does not have a unique structure as an r.i. function space on [0, ∞), unless p = 1, 2 or ∞. In fact, the following result can be easily deduced from Theorem 5.3. T HEOREM 5.5. The space Lp (0, ∞); 1 2,

Lp (0, ∞)

and L2 (0, ∞) + Lp (0, ∞),

if 1 2, the spaces Lp (0, ∞) and Z = L2 (0, ∞)∩Lp (0, ∞) are isomorphic. To this end, notice that the restriction Z|[0,1] is isomorphic to Lp (0, 1) and thus Z contains a complemented subspace isomorphic to n Lp (0, ∞). Conversely, for any n and m, the sequence {χ[(i−1)/2n,i/2n ) }2i=1 spans in Z a symmetric Xp -space, and thus its span embeds complementably in Lp (0, ∞), in a uniform manner. Hence, by a compactness argument using, for instance, ultraproducts, one can show that Z is isomorphic to a complemented subspace of Lp (0, ∞). Then the decomposition method shows that Z ≈ Lp (0, ∞). Many other classes of spaces which admit a unique representation as an r.i. function space on [0, 1] were exhibited in the Memoirs [27] and [29]. We shall state some of these results without proof.


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T HEOREM 5.6. An r.i. function space X on [0, 1], which is q-concave for some q < 2, has unique structure as an r.i. function space on [0, 1]. The other result that we want to state below involves the notion of the Haar basis. Recall k that the Haar system {χn }∞ n=1 on [0, 1] is defined by χ1 (t) ≡ 1 and, for = 1, 2, . . . , 2 and k = 0, 1, . . . , by ⎧

⎪ if t ∈ (2 − 2)/2k+1, (2 − 1)/2k+1 , ⎨1

χ2k + (t) = −1 if t ∈ (2 − 1) 2k+1 , 2/2k+1 , ⎪ ⎩ 0 otherwise. It is an immediate consequence of basic interpolation theorems that the Haar system forms a (monotone) basis in any separable r.i. function space on [0, 1]. We now present a result on uniqueness which was originally proved in [27] under some additional assumptions and whose definitive form, stated below, is due to Kalton [29]. T HEOREM 5.7. Let X be a separable r.i. function space on [0, 1] such that the Haar basis of X is not equivalent to a sequence of mutually disjoint functions in X whose linear span is complemented in X. Then X has unique structure as an r.i. function space on [0, 1]. For using interpolation between two Lp -spaces other than L1 and L∞ it is useful to consider the so-called Boyd indices. In order to define these indices for an r.i. function space X on [0, 1], we need the dilation operator Ds restricted to [0, 1], which is defined for 0 < s < ∞ and f on [0, 1], by f (t/s), 0 t min(1, s), (Ds f )(t) = 0, s < t 1 (in the case s < 1). The operator Ds is a dilation by the ratio s : 1 in the positive direction of the t-axis followed by restriction to [0, 1]. It is easily verified that, for every choice of r and s, and every r.i. function space X on [0, 1], Drs X Dr X Ds X , which eventually ensures that the following limits exist: pX = lim

log s log s = sup log Ds X s>1 log Ds X

qX = lim

log s log s = sup . log Ds X 0<s 0 and w = (wn )∞ n=1 is a non-summable monotone decreasing sequence of positive numbers and the quasi-norm xw,p of an element x ∈ d(w, p) is defined by xw,p = sup π

∞

1/p |xπ(n)| wn p

< ∞.

n=1

In this definition, the supremum is taken over all permutation π of the integers. The uniqueness of the basis in d(w, p) was studied in [45] and further in [30]. One result in this direction, which is proved in [30] and provides the solution to an open question raised in [45], asserts that the space d(w, p) has, up to equivalence, a unique unconditional basis, whenever 0 < p < 1 and the sequence of weights w satisfies the condition 1 (w1 + · · · + wn )1/p = ∞. n→∞ n lim

Another remarkable result proved in [30] states that also the spaces p (q ) have a unique unconditional basis, up to equivalence and permutation, as long as 0 < p, q < 1. The same is true for the spaces c0 (p ), as long as 0 < p < 1 (cf. [33]). Perhaps, the most interesting class of quasi-Banach spaces is that of the Hardy spaces Hp (T), 0 < p < 1, where T denotes the circle, and their m-dimensional version Hp (Tm ), where Tm denotes the m-dimensional torus. The fact that these quasi-Banach spaces have an unconditional basis is not obvious. For m = 1, this follows from the fact proved in [55], that Hp (T) is isomorphic to the dyadic Hardy space Hp (0, 1), for all 0 < p < 1. The usual Haar basis {hn }∞ p (0, 1), 0 < p < ∞, since the dyadic n=1 forms an unconditional basis in H Hp (0, 1) consists of all distribution of the form f = ∞ n=1 an hn for which the expression f p =

∞ 1 0

p/2 1/p |an hn |

2

< ∞.

n=1

For higher dimensions, one verifies that the m-times tensor product of the Haar basis is an unconditional basis in the dyadic Hardy space Hp (0, 1)m , for every value of 0 < p < 1, and that this space is isomorphic to the usual space Hp (Tn ) (cf. [55]). A remarkable result from [56] asserts that every normalized unconditional basis in Hp (T); 0 < p < 1, is equivalent to the Haar basis in some order, i.e., the space Hp (T) has, up to equivalence and permutation, a unique unconditional basis.


1667

A nice application of the uniqueness result in non-locally convex Hardy spaces is the fact (proved first by Kalton, Leranoz and Wojtaszczyk [30]) that, for 0 < p < 1, the spaces Hp (Tm ), m = 1, 2, . . . , are mutually non-isomorphic. If, for example, Hp (Tm ) were isomorphic to Hp (T), for some 0 < p < 1 and some integer m, then the corresponding normalized unconditional bases of these two spaces would be equivalent, up to a permutation. That this is false is checked directly. One should point out that the result asserting that the spaces H1 (Tm ), m = 1, 2, . . . , are mutually non-isomorphic was proved before by Bourgain ([4,5]). Uniqueness of bases can be also studied in the more general context of locally convex spaces (l.c.s.), i.e., vector spaces which admit a fundamental system of convex neighborhoods of 0. The topology of such a space can be defined with the aid of a family of semi-norms. Of particular interest are the Fréchet spaces (recall that these are l.c.s. whose topology is both metrizable and complete). The main difficulty faced in studying uniqueness in l.c.s. is the fact that the notion of “normalized” basis does not make any sense in these spaces. The only Fréchet spaces which have a unique unconditional basis are ω, the space of all scalar sequences, and its dual ω∗ , the space of all scalar sequences which are eventually zero. The uniqueness assertion for ω and ω∗ is due to Köthe and Töplitz [32]. This is the result that in some sense initiated the research on uniqueness of bases. That ω and ω∗ are the only spaces with this uniqueness property is due to Dragilev (cf. [17]). Less restrictive is the notion of quasi-equivalence. Two unconditional bases {xn }∞ n=1 and {yn }∞ n=1 in l.c.s. X, respectively Y , are said to be quasi-equivalent if there exists a ∞ permutation π of the integers and a sequence λ = (λn )∞ n=1 of scalars such that {λn xπ(n) }n=1 and {yn }∞ are equivalent, i.e., there exists an isomorphism T from X onto Y so that n=1 yn = T (λn xπ(n)), for all n. A remarkable result, also due to Dragilev [15], asserts that the Fréchet space A(D) of all the functions which are analytic on the open disk D (endowed with the topology of uniform convergence on each compact subset of the disk) has the property that all its unconditional bases are quasi-equivalent. This fact is also true in the class of all nuclear power series space (cf. [43]). A systematic study of quasi-equivalence in l.c.s. has been carried out by Dragilev [16], Dubinsky [18], Mityagin [44] and recently by Zahariuta [57].

References [1] Y.A. Abramovich and P. Wojtaszczyk, On the uniqueness of order in the spaces p and Lp [0, 1], Mat. Zametki 18 (1975), 313–325 (Russian). [2] K.I. Babenko, On conjugate functions, Dokl. Akad. Nauk SSSR 62 (1948), 157–160 (Russian). [3] A.J. Bernstein, Maximally connected arrays on the n-cube, Discrete Math. 15 (1967), 1485–1489. [4] J. Bourgain, The non-isomorphism of H 1 -spaces in one and several variables, J. Funct. Anal. 46 (1982), 45–47. [5] J. Bourgain, The non-isomorphism of H 1 -spaces in a different number of variables, Bull. Soc. Math. Belg. Sér. B 35 (1983), 127–136. [6] J. Bourgain, P.G. Casazza, J. Lindenstrauss and L. Tzafriri, Banach Spaces with a Unique Unconditional Basis, up to Permutation, Mem. Amer. Math. Soc. 322 (1985).

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[7] D.B. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245–1254. [8] D.L. Burkholder, Martingales and singular integrals in Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 233–269. [9] P.G. Casazza and N.J. Kalton, Uniqueness of unconditional basis in Banach spaces, Israel J. Math. 103 (1998), 141–176. [10] P.G. Casazza and N.J. Kalton, Uniqueness of unconditional bases in c0 products, Preprint. [11] P.G. Casazza, N.J. Kalton and L. Tzafriri, Uniqueness of unconditional and symmetric structions in finite dimensional spaces, Illinois J. Math. 34 (1990), 793–836. [12] P.G. Casazza and M.C. Lammers, Genus N Banach spaces, Preprint. [13] W.J. Davis, Embedding spaces with unconditional bases, Israel J. Math. 20 (1975), 189–191. [14] L.E. Dor and T. Starbird, Projections of Lp onto subspaces spanned by independent random variables, Compositio Math. 39 (1979), 141–175. [15] M.M. Dragilev, Canonical forms of a basis in the space of analytic function spaces, Uspekhi Mat. Nauk 15 (1960), 181–188. [16] M.M. Dragilev, On regular bases in nuclear spaces, Mat. Sbornik 68 (1965), 153–173 (Russian). [17] M.M. Dragilev, Topological vector spaces with equivalent bases, Mat. Zametki 28 (1980), 947–951. [18] E. Dubinsky, The Structure of Nuclear Fréchet Spaces, Lecture Notes in Math. 720, Springer (1979). [19] I.S. Èdelšte˘ın and P. Wojtaszczyk, On projections and unconditional bases in direct sums of Banach spaces, Studia Math. 56 (1976), 263–276. [20] T. Figiel and W.B. Johnson, A uniformly convex Banach space which contains no p , Compositio Math. 29 (1974), 179–190. [21] T. Figiel and P. Wojtaszczyk, Special bases in functional spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 561–597. [22] J. Garnett, Bounded Analytic Functions, Academic Press, Orlando (1981). [23] W.T. Gowers, A finite-dimensional Banach space with two non-equivalent symmetric bases, Israel J. Math. 87 (1994), 143–151. [24] L.H. Harper, Optimal assignments to number of vertices, SIAM J. Appl. Math. 12 (1964), 131–135. [25] S. Hart, A note on the edges of the n-cube, Discrete Math. 14 (1976), 157–163. [26] F. Hernández and N.J. Kalton, Subspaces of rearrangement invariant spaces, Canad. J. Math. 48 (1996), 794–833. [27] W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric Structures in Banach Spaces, Mem. Amer. Math. Soc. 217 (1979). [28] N.J. Kalton, Orlicz sequence spaces without local convexity, Math. Proc. Cambridge Philos. Soc. 81 (1977), 253–278. [29] N.J. Kalton, Lattice Structures on Banach Spaces, Mem. Amer. Math. Soc. 493 (1993). [30] N.J. Kalton, C. Leranoz and P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, Israel J. Math. 72 (1990), 299–311. [31] N.J. Kalton, N.T. Peck and J.W. Roberts, An F -space Sample, London Math. Soc. Lecture Notes Ser. 89, Cambridge Univ. Press (1984). [32] G. Köthe and O. Töplitz, Theorie der halbfiniten unendlichen Matrizen, J. Reine Angew. Math. 165 (1931), 116–127. [33] C. Leranoz, Uniqueness of unconditional bases of c0 (p ), 0 < p < 1, Studia Math. 102 (1992), 193–207. [34] J. Lindenstrauss, A remark on symmetric bases, Israel J. Math. 13 (1972), 317–320. [35] J. Lindenstrauss and A. Pelczynski, Absolutely summing operators in Lp spaces and their applications, Studia Math. 29 (1968), 275–326. [36] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces III, Israel J. Math. 14 (1973), 368–389. [37] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer, Berlin (1977). [38] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer, Berlin (1979). [39] J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, J. Funct. Anal. 3 (1969), 115–125. [40] J.H. Lindsey, Assignments of numbers to vertices, Amer. Math. Monthly 71 (1964), 508–516. [41] B. Maurey, Théoremes de factorisation pour les operateures a valeurs dans un espace Lp , Asterisque 11 (1974).


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[42] C.A. McCarthy and J. Schwartz, On the norm of a finite Boolean algebra of projections and applications to theorems of Kreiss and Morton, Comm. Pure Appl. Math. 18 (1965), 191–201. [43] B.S. Mityagin, Nuclear Riesz scales, Dokl. Akad. Nauk SSSR 137 (1961), 519–522. [44] B.S. Mityagin, Approximative dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16 (1961), 63– 132 (Russian). [45] M. Nawrocki and A. Ortynski, The Mackey topology and complemented subspaces of Lorentz sequence spaces d(w, p) for 0 < p < 1, Trans. Amer. Math. Soc. 287 (1985), 713–722. [46] R.E. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241–264. [47] A. Pelczynski, Universal bases, Studia Math. 32 (1969), 247–268. [48] A. Pelczynski and H.P. Rosenthal, Localization techniques in Lp -spaces, Studia Math. 52 (1975), 263–289. [49] A. Pelczynski and I. Singer, On non-equivalent bases and conditional bases in Banach spaces, Studia Math. 25 (1964), 5–25. [50] C.J. Read, A Banach space with up to equivalence, precisely two symmetric bases, Israel J. Math. 40 (1981), 33–53. [51] H.P. Rosenthal, On the subspaces of Lp (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303. [52] G. Schechtman, On Pelczynski’s paper, “Universal bases”, Israel J. Math. 22 (1975), 181–184. [53] C. Schütt, On the uniqueness of symmetric bases in finite-dimensional Banach spaces, Israel J. Math. 40 (1981), 97–117. [54] B.S. Tsirelson, Not every Banach space contains p or c0 , Functional Anal. Appl. 8 (1974), 138–141 (translated from Russian). [55] P. Wojtaszczyk, Hp -spaces, p 1, and spline systems, Studia Math. 77 (1984), 289–320. [56] P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces II, Israel J. Math. 97 (1997), 253–280. [57] V.P. Zahariuta, On Isomorphic Classification of F -spaces, Lecture Notes in Math. 1043, Springer (1983). [58] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265–272.


CHAPTER 39

Spaces of Analytic Functions with Integral Norm P. Wojtaszczyk Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland E-mail: [email protected]

Contents 1. 2. 3. 4.

Notation . . . . . . . . . . . Bergman spaces . . . . . . . Hardy spaces . . . . . . . . Special operators . . . . . . 4.1. Coefficient multipliers 4.2. Composition operators 5. Isomorphisms of H1 . . . . 6. Isomorphic structure of H1 . 7. Isometric questions . . . . . References . . . . . . . . . . .

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. 1673 . 1673 . 1677 . 1680 . 1680 . 1683 . 1684 . 1696 . 1699 . 1700

Abstract We discuss properties of Banach spaces of analytic functions with integral type norms, in particular Hardy spaces and Bergman spaces. We present results about the isomorphic structure of those spaces, subspaces and complemented subspaces, special operators like multipliers and composition operators. The basic results about the real variable theory of H1 spaces are discussed and the connections with complex theory are explained. Isometric questions are also presented.



Spaces of analytic functions with integral norm

1673

1. Notation In addition to the standard notations of this Handbook as explained in [21] we will also use the following notations: D = z ∈ C: |z| < 1 , Bn = (z1 , z2 , . . . , zn ) ∈ C : n

n

|zj | < 1 , 2

j =1

Sn = (z1 , z2 , . . . , zn ) ∈ C : n

n

|zj | = 1 , 2

j =1

C+ = {z ∈ C: 'z > 0}. Obviously Dn ⊂ Cn will denote the n-fold Cartesian product of D and Tn ⊂ Cn will denote the n-fold Cartesian product of T. By ν we will denote the natural volume Lebesgue measure on any subset of Cn . On the sets D, Dn or Bn this measure is normalized so the measure of the whole set equals 1. The natural arc length probability measure of T will be denoted by λ and the corresponding product measure on Tn by λn . The natural rotation invariant probability measure on Sn will be denoted by σn . For any open subset D ⊂ Cn by H(D) we will mean the space of all holomorphic functions on D. 2. Bergman spaces Let D ⊂ Cn be an open set. For 0 0 be such that z + rBn ⊂ D. The following mean value formula for f ∈ H(D) easily follows from the Cauchy formula (or from expansion of f into power series) f (w) dν(w). (2) f (z) = r −2n z+rBn

From (2) follows immediately that for p 1

f (z) Cf p dist z, Cn \ D −2n/p .

(3)

From (3) we infer easily that a sequence Cauchy in the norm of Bp (D) is almost uniformly convergent in D, so we get

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P. Wojtaszczyk

T HEOREM 1. The space Bp (D) is a closed subspace of Lp (D, ν), in particular it is a Banach space. From (3) and Montel’s theorem we also infer that for each compact subset K ⊂ D the restriction f → f |K is a compact operator from Bp (D) into C(K). This implies C OROLLARY 2. The space Bp (D) is isomorphic to a subspace of p . Tosee it simply take a sequence ∅ = K0 ⊂ K1 ⊂ K2 ⊂ · · · of compact subsets of D such that ∞ j =1 Kj = D, and consider the isometric embedding Bp (D) −→

∞

Lp (Kj +1 \ Kj )

j =0

p

defined as f → (f | Kj +1 \ Kj )∞ j =0 . Since each restriction is a compact operator we can find a finite partition of Kj +1 \ Kj such that for the averaging projection Pj (with respect to this partition) we have for f ∈ Bp (D) 1/p

Kj+1 \Kj

|f − Pj f |

p

10−1 f p .

This gives that f → (Pj (f | Kj +1 \ Kj ))∞ j =0 ∈ ( ding into a space isometric to p .

∞

j =0 Im Pj )p

is an isomorphic embed-

R EMARK 3. An obvious modification of the above proof gives that Bp (D) is almost isometric to a subspace of p . It follows easily from [13] that for p = 2k with k = 2, 3, . . . it is not isometric to a subspace of p . For p = 2k it is unknown. From (3) we see that for z ∈ D the value at z is a continuous linear functional on Bp (D). Since B2 (D) is a Hilbert space, from the Riesz theorem we get that for each z ∈ D there exists a function Kz (w) ∈ B2 (D) such that the orthogonal projection P from L2 (D, ν) onto B2 (D) is given by the formula f (w)Kz (w) dν(w). (4) (Pf )(z) = D

This projection is called the Bergman projection and the function of two variables KD (z, w) = Kz (w) is called the Bergman kernel. The way to construct the Bergman kernel of a given domain is to take an orthonormal basis (ϕn )∞ n=0 in B2 (D) and observe ∞ (w) = n=0 ϕn (w)ϕn (z). Taking as an orthonormal basis in B2 (D)√the system that Kz√ ϕ(z) = n + 1zn we get the Bergman kernel for D and taking the tensors of n + 1zn as a basis for B2 (Dn ) we get the Bergman kernel for the polydisc. The ball is treated similarly, although technical details are slightly more complicated (cf. [41], 3.1). In general it is quite difficult to calculate the Bergman kernel explicitly but in the above-mentioned most important cases we get:


1675

• the Bergman kernel for the unit ball Bn is (1 z, w)−n−1 , *− n n • the Bergman kernel for the polydisc D is i=1 (1 − zi w¯ i )−2 . Note that the Bergman kernel for the polydisc is the product of Bergman kernels for the discs. It is a general fact for product domains – this follows from the above argument. Using the above explicit representations we obtain (cf. [41]) T HEOREM 4. Let D be either Bn or Dn and let 1 < p < ∞. The Bergman projection is (extends to) a bounded projection from Lp (D, ν) onto Bp (D). This is an optimal result. Since H∞ is not complemented in any L∞ space, the Bergman projection can not be bounded for p = ∞ (cf. [16]). Since it is orthogonal it cannot be bounded for p = 1 either. There are however (non-orthogonal) projections which are bounded for 1 p < ∞. We have the following T HEOREM 5 (Forelli–Rudin, cf. [41], 7.1). Let P1 f (z) = n

Bn

f (w)(1 − |w|2 ) dν(w). (1 − z, w)n+2

(5)

The operator P1 is a bounded projection from Lp (Bn , ν) onto Bp (Bn ) for 1 p < ∞. Using those projections (for n = 1) coordinatewise we can get analogous result for polydiscs. When we compare this with Corollary 2 and use the classical result of Pełczyński we get T HEOREM 6 ([29]). Let D be either Bn or Dn and let 1 p < ∞. The space Bp (D) is isomorphic to p . Note also that Theorems 4 and 5 allow us to describe (isomorphically) duals Bp (D)∗ . For 1 < p < ∞ we can identify (using the natural Hilbert space pairing) Bp (D)∗ with Bq (D), p−1 + q −1 = 1. Theorem 5 allows us to identify (for 1 p < ∞) the space Bp (D)∗ with Im P1∗ ⊂ Lq (D, ν), i.e., with the space of those f (z) analytic in D that |f (z)|(1 − |z|2 ) ∈ Lq (D, ν). P ROBLEM 7. This theorem settles the isomorphic structure of Bergman spaces on balls and polydiscs, however many isometric questions remain open. For example, we do not know the Banach–Mazur distance between Bp (D) and p (except for p = 2). Clearly it goes to infinity as p → ∞ but the exact values or good estimates are unknown. Also we do not know basis constants or unconditional basis constants. It seems likely that (for p = 2) there is no monotone basis (or even basic sequence) in Bp (D) nor even a finite-dimensional (but not one-dimensional) norm one projection. Also the description of extreme points (and more refined extreme structure) of the unit ball in B1 (D) is unknown. The same questions make sense for other Bergman spaces. The problem of boundedness in Lp of Bergman projections and analogues of Riesz projections (Szegö projections) for other domains is very delicate and not solved in general.

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We refer the interested reader to [4] and the references quoted there. Let us simply point out that there are very regular domains D ⊂ Cn for which the Bergman projection is bounded only in some interval around 2, not in (1, ∞) as we have seen in our examples. Also there are domains for which Szegö projections are bounded only when p = 2. The question if Bp (D) is always isomorphic to p seems to be open. The feature that greatly simplifies the study of Taylor expansions of functions in Bergman spaces isn that dyadic blocks sum absolutely. To make it more precise let us take f (z) = ∞ n=0 an z ∈ Bp (D). Writing the integral in polar coordinates as f Bp =

1 π

1 0

iθ p f r e dθ r dr

2π

1/p

0

we see that every multiplier Λ = (λn ) which acts on Hp (D) acts also on Bp (D). If for p 1 we take multipliers εn Λn discussed after Theorem 18 or for p > 1 we take as Λn the sequence χ[2n ,2n+1 ] we infer (either from Theorem 18 or from the Littlewood–Paley n theory) that the series ∞ n=0 Λ f converges to f unconditionally in Bp (D), and so > L =:

D

∞ n Λ f (z)2

∼ f Bp .

dν(z)

n=0

Now we have (remember that Λn f = Lp

?1/p

p/2

∞

2n+2

2n−1 γn z

n)

1−2−n−1 2π

Λn (f ) r eiθ p dθ r dr

−n 0 n=0 1−2 ∞ 2π −n n

c

p Λ (f ) eiθ dθ.

2

On the other hand p f Bp

1 0

0

0

1

∞ 2π

>

1

>

0 ∞ n=0

p

Λ (f ) r e dθ r dr

n

Λn (f ) r eiθ p dθ

0

r

2n−1

Λ (f )Hp

n=0 ∞ 2n−1 n/p q r 2

p n Λ (f )Hp ·

−n

r dr

r dr

?p/q > ·

1 ∞ 0

1/p ?p

?p

n

n=0

2

iθ

n=0 ∞ 2π

n=0 > ∞ 1

0

=

(6)

0

n=0

n=0

∞

2

? p n Λ (f )Hp r dr

−n

n=0

r

q/p

q2n−1 nq/p

2

r dr.

(7)


1677

A relatively straightforward calculation shows that the last integral is finite, so from (6) and (7) we obtain f Bp ∼

∞

2

p n Λ (f )Hp

−n

1/p (8)

.

n=0

The details of the above calculation can be found (in much greater generality) in [32] or [53], Section 4. The relation (8) allows an explicite construction of an unconditional basis in Bp (D) equivalent to the unit vector basis in p , 1 p < ∞ (cf. [32,53]). For 1 < p < ∞ the following simple system works after we normalise it properly: we define n −1 k 0 = 1 and for z for n = 0, 1, 2, . . . . The desired system consists of f−1 fn (z) = 2k=0 n −n k 2 −2πk2 n n = 0, 1, 2, . . . fn (z) = z fn (e z) with k = 0, 1, . . . , 2 − 1.

3. Hardy spaces The general theory of Hardy spaces is presented, from various points of view in many books, e.g., [18,26,41,56,14]. In what follows we will present only those facts from the general theory that have big impact on Banach space properties of those spaces. Let us start with the most important case, of Hardy spaces on the unit disc D. For f ∈ H(D) and every r, 0 < r < 1, and p, 0 < p < ∞, we define Mp (f ; r) =

1 2π

2π

iθ p f r e dθ

1/p .

(9)

0

For each f and p the function Mp (f ; r) is an increasing function of r and we denote Hp (D) = f ∈ H(D): sup Mp (f ; r) < ∞ .

(10)

r 0 by Mp (f, r) = ( −∞ |f (x + ir)| dx)1/p and Hp (C+ ) is the set of all f ∈ H(C+ ) such that supr Mp (f, r) = f p < ∞. For such f ’s the function Mp (f, r) is a decreasing function of r. Like in the case of the disc we can define boundary values of functions in Hp and we get the conclusion that all Hp spaces defined above are subspaces of Lp . The orthogonal projection from L2 onto H2 (after the identification via boundary values) is called a Riesz projection. Like in D it is bounded from Lp onto Hp for 1 < p < ∞ (and unbounded for p = 1 and p = ∞). For Dn we apply the one-dimensional result coordinatewise; for Bd it is more difficult (cf. [41], 6.3). There is a natural equivalence between Hp (D) and Hp (C+ ) which is a reflection of the fact that those domains are conformally equivalent in a very nice way. Let us fix the map ϕ(z) = (i − z)/(i + z) which maps C+ conformally onto D. For a function f holomorphic

1680

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in some neighbourhood of D we see that f ◦ ϕ is holomorphic in some neighbourhood of C+ . Moreover we have p ∞ ∞ 1 f i − x f ◦ ϕ(x)p 1 dx = dx. 2 1+x i+x 1 + x2 −∞ −∞ π Substituting eiθ = (i − x)/(i + x) we see that the above integral equals −π |f (eiθ )|p 21 dθ . This means that we have an isometry between Hp (D) and a weighted Hp space on C+ with respect to the measure 1/(1 + x 2 ) dx. A natural way to deal with this weight is to apply an appropriate multiplication operator. The results are summarised in the following T HEOREM 13. The map Ip defined on Hp (D) as

(Ip f )(z) =

1 π 1/p

(f ◦ ϕ)(z) · (1 − iz)

−2/p

i−z = 1/p f (1 − iz)−2/p i+z π 1

(14)

is an isometry from Hp (D) onto Hp (C+ ) for 0 < p ∞. Note first that (1 − iz)−2/p is a well defined analytic function on C+ . Computing the norm we get i − x p −2 f i + x |1 − ix| dx −∞ π iθ p 1 ∞ i − x p 1 1 f e dθ. f = dx = 2 π −∞ i+x 1+x 2π −π

Ip (f )p = 1 p π

∞

Thus (some technicalities aside) we see that Ip is an isometry from Hp (D) into Hp (C+ ). To see that it is onto it suffices to solve (14) for f . Note that the above theorem allows us to carry many properties (in particular the canonical factorisation) from H1 (D) to H1 (C+ ). Using this substitution ∞ we can also compute the Poisson kernel for C+ and check that if f ∈ H1 (C+ ) then −∞ f (t) dt = 0. 4. Special operators The fact that our spaces consist of analytic functions and that we have natural projections associated with them allows us to consider several natural classes of operators.

4.1. Coefficient multipliers Coefficient multipliers are most naturally considered on or on Dn . For simplicity we will D ∞ consider only functions on D. Then a function f (z) = k=0 ak zk analytic in D is identified


1681

∞ ∞ with a sequence of coefficients (ak )∞ k=0 and we consider an operator (ak )k=0 → (λk ak )k=0 where the range space is either some sequence space or a space of analytic functions on D.

Hardy spaces. We will consider only coefficient multipliers on H1 (D) because for p > 1 those are the same as Fourier multipliers on Lp (T) (this follows from Corollary 11) and thus are more properly treated in the theory of Fourier series. Moreover multipliers of H1 (D) have important Banach space consequences. T HEOREM 14 (Fefferman). A multiplier Λ = (λn )∞ n=0 maps H1 (D) into 1 if and only if sup

∞

(k+1)m

m1 k=1

2 |λn |

< ∞.

(15)

n=km+1

This theorem was obtained by Fefferman but never published. A proof of an analogous result on Rn was published in [47]. In Section 5 we will give a proof which follows unpublished argument from [49]. C OROLLARY 15 (Hardy). If f =

∞

n=0 an z

n

∈ H1 (D) then

∞ |an | πf 1 . n+1

(16)

n=0

Clearly the constant π in the above inequality does not follow from Theorem 14. There is also a full description of multipliers from H1 (D) into 2 (cf. [14, 6.4]), namely T HEOREM 16. A multiplier Λ = (λn )∞ n=0 maps H1 (D) into 2 (or equivalently into H2 (D)) if and only if sup N −2

N (n + 1)2 |λn |2 < ∞.

N1

n=0

C OROLLARY 17 (Paley). If f =

∞

∞

n=0 an z

(17) n

∈ H1 (D) then

1/2 |a2n |

2

Cf 1 .

(18)

n=1

From this corollary in particular follows the following well known version of Khinchin’s inequality for lacunary trigonometric series: ∞ ∞ ∞ i2n θ i2n θ i2n θ C an e an e an e , n=0

2

n=0

1

n=0

2

(19)

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n so we infer that the projection P (called Paley’s projection) defined as P ( ∞ n=0 an z ) = ∞ k 2 k=0 a2k z is a bounded projection on H1 (D) and its range is isomorphic with 2 . This in particular implies that H1 (D) is not isomorphic to a complemented subspace of an L1 (μ) space, because such spaces do not contain complemented Hilbert spaces (cf. [21]). Clearly any multiplier from H1 (D) into H2 (D) is also a multiplier into H1 (D), but obviously it is not all. The important class of multipliers from H1 (D) into H1 (D) is given by the following T HEOREM 18 (Stein, cf. [12]). Let Λ = (λn )∞ n=0 be a bounded sequence such that supn0 (n + 1)|λn+1 − λn | < ∞. Then Λ is a multiplier from H1 (D) into H1 (D). Note that this theorem provides an H1 version of the Littlewood–Paley decomposition. Let us define multipliers Λn = (λnk )∞ k=0 by the formula ⎧ ⎨1 if 2n k 2k+1 , n λk = 0 if k 2n−1 or k 2n+2 , ⎩ linear otherwise, 2n then for each sequence εn = ±1 the multiplier ∞ the assumptions of n=0 εn Λ ∞ satisfies the above Theorem 18, so for each f ∈ H1 (D) the series n=0 Λ2n (f ) converges unconditionally to f . Bergman spaces. Multipliers between Bergman spaces were investigated quite intensively recently. The formula (8) can be very useful to see how they look like (at least for p > 1). Note that this formula shows that the real difficulty in describing multipliers from Bp (D) into sequence spaces or into Bq (D) is the description of Fourier multipliers of polynomials in Lp (T). This is a demanding problem of harmonic analysis. There is a full description of multipliers from Bq (D) into Bp (D) for 0 < p 2 q < ∞ given in T HEOREM 19 ([52]). The sequence (λn )∞ n=0 is a multiplier from Bq (D) into Bp (D), 0 < p 2 q < ∞, if and only if sup 2k n 0 the operator CΦ is bounded on every Hp (D) and on each Hp (D) we have CΦ

1 + |Φ(0)| . 1 − |Φ(0)|

(20)

P ROOF. We write Φ = M ◦ ϕ where M is a Möbius transformation with M(0) = Φ(0) and ϕ(0) = 0. A direct computation gives CM 1+|M(0)| 1−|M(0)| . Now it follows from the Schwarz lemma that ϕ maps each disk |z| < r into itself. For a given f ∈ Hp (D) and r < 1 let h(z) be the harmonic function in |z| r which for |z| = r equals |f (z)|p . Since |f (z)|p is subharmonic we see that |f (ϕ(z))|p h(ϕ(z)). Thus 1 2π

2π 0

2π it

p it

f ϕ r e dt 1 h ϕ r e dt = h ϕ(0) = h(0) 2π 0 2π 2π it p

1 1 f r e dt. = h r eit dt = 2π 0 2π 0

Since CΦ = Cϕ ◦ CM the claim follows.

If we are interested to what operator ideals a composition operator belongs then the results are very nice for ideal of compact operators. Namely we have T HEOREM 21. Let Φ : D → D. The following conditions are equivalent: (1) CΦ is compact on H2 (D), (2) CΦ is compact on Hp (D) for some p, 1 p < ∞, (3) CΦ is compact on Hp (D) for all p ∈ [1, ∞). P ROOF. Fix 0 < p, q < ∞ and suppose that CΦ is compact on Hp (D). Since bounded sequences in Hq (D) are normal families we need to show that CΦ (fn )q → 0 on a subsequence for fn convergent to 0 uniformly on compact subsets of D and fn q 1. We factorise fn = In · Fn into inner and outer parts. Clearly there is a subsequence so that both In and Fn converge almost uniformly on this subsequence to I∞ and F∞ respectively, and q/p at least one of those functions equals 0. Put Gn = Fn . Since CΦ is compact on Hp (D) q/p we see that CΦ (In ) converges to CΦ (I∞ ) and CΦ (Gn ) converges to CΦ (F∞ ) in the q/p norm of Hp (D). If F∞ = 0 then CΦ (fn )q CΦ (Fn )q = CΦ (Gn )p so converges

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P. Wojtaszczyk

to 0. If I∞ = 0 then In (Φ(eit )) → 0 a.e. because one checks that compactness of CΦ imq plies |Φ(eit )| < 1 a.e. We have CΦ (fn )q = T |In (Φ(eit ))|q |Gn (Φ(eit ))|p dλ(t), where the first factor in the integrand converges to 0 a.e. and the second is uniformly integrable because CΦ (Gn ) converges in Hp . So we get CΦ (fn )q → 0. To get the intuition of how the size of CΦ reflects properties of Φ note that if Φ is inner then CΦ is an isometric embedding, while if Φ∞ < 1 then CΦ is nuclear. Also note, what we used in the previous proof, that CΦ compact implies |Φ(eit )| < 1 a.e. Thus the intuition suggests that for small operators the function Φ(eit ) touches T rather rearly. The natural measure here turns out to be the Nevanlinna counting function of Φ defined for z ∈ D \ {Φ(0)} by NΦ (z) =: {− log |w|: f (w) = z}. Using this function we can express when CΦ is compact or in p-Schatten class. T HEOREM 22. (a) ([44]) CΦ is compact on H2 (D) (and so on all Hp (D), p ∈ [1, ∞)) if and only if lim|w|→1− |log |w||−1 NΦ (w) = 0. (b) ([30]) CΦ is in Schatten p-class 0 < p < ∞ (in particular nuclear for p = 1) on H2 (D) if and only if

log |z|−1 NΦ (z) p/2 1 − |z|2 −2 dν(z) < ∞. D

For other, more Banach space operator ideals the situation is less clear. There is a lot of open problems and a good place to start looking at this subject is [19]. What seems to be of special importance here is the notion of order boundedness, i.e., when CΦ ({f ∈ Hp (D): f p 1}) is an order bounded subset of some Lr (T). We say that CΦ is β-order bounded if the above holds with r = βp, 0 < β < ∞. It is known (cf. [19, Theorem 9] that this notion does not depend on p and is equivalent to (1 − |Φ|2 )−1 ∈ Lβ (T). With this notion we can get some fragmented information about absolutely summing operators. Namely (cf. [46,19]) P ROPOSITION 23. The following are equivalent: (1) operator CΦ is 1-order bounded, (2) operator CΦ : Hp (D) → Hp (D) is p-nuclear for some (and then for all) 2 p < ∞, (3) operator CΦ : Hp (D) → Hp (D) is p-absolutely summing for some (and then for all) 2 p < ∞. 5. Isomorphisms of H1 Let us consider f (z) = !f (z) + i'f (z) ∈ H1 (C+ ). Clearly the function f (z) is determined by !f (in general 'f is determined by !f up to a constant but for f ∈ H1 (C+ ) both functions have integral 0 so the constant is also determined). So about the boundary value f (t) we can think as !f (t) + i'f (t) where 'f is a boundary value of a harmonic conjugate of the harmonic extension of !f (t). We can reverse this process. For a realvalued function f ∈ L1 (R) let F (z) denotes the harmonic extension of f to C+ and let G(z) denotes the harmonic conjugate of F (z) in C+ . Let us put f˜(t) = limy→0+ G(t + iy).


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We see from the above that we can naturally identify H1 (C+ ) with the space !H1 (R) of all functions f ∈ L1 (R) such that f˜(t) exists a.e. and is in L1 (R). We equipp the space !H1 (R) with the norm f 1 + f˜1 . Now this identification establishes an isomorphism between !H1 (R) and H1 (C+ ) but considered as Banach spaces over reals. If one insists (as is quite natural) on complex scalars one has to extend the definition of !H1 (R) the space !C H1 (R) of complex-valued functions F = f + ig with f, g ∈ !H1 (R). Then H1 (C+ ) is (via boundary values) a closed subspace of !C H1 (R). An operator P (f + ig) =: 1 ˜ ˜ is a complex-linear bounded projection from !C H1 (R) onto H1 (C+ ) 2 [f + if + i(g + ig)] (remember that f˜˜ = − f ). One checks that ker P = {f ∈ !C H1 (R): f¯ ∈ H1 (C+ )}, so clearly !C H1 (R) = H1 (C+ ) ⊕ H1 (C+ ). In the sequel we will usually deal with the space !C H1 (R) but for simplicity of notation we will drop the subscript C. D EFINITION 24. The function f on R belongs to the space BMO(R) if and only if 1 sup |I I ⊂R |

|f − fI | = f ∗ < ∞,

I

where I ⊂ R is an interval in R and fI =

(21)

1 |I | I

f (t) dt.

The space BMO(R) is a Banach space with the above norm if we identify functions differing by a constant a.e. The classical John–Nierenberg inequality implies that for each p, 1 p < ∞, the expression

1 sup I ⊂R |I |

1/p

|f − fI |

p

(22)

I

defines an equivalent norm on BMO(R). Clearly L∞ (R) ⊂ BMO(R). It is quite easy to check that log |x| ∈ BMO(R). A deep theorem of Fefferman and Stein asserts that (!H1 (R))∗ = BMO(R). This assertion requires certain care. For f ∈ !H1 (R) and g ∈ BMO(R) the product f · g need not be integrable. Thus the way to understand the Fefferman–Stein theorem is to observe that the Schwartz class S is dense in !H1 (R) and ∗ say that for each x ∗ ∈ (!H 1 (R)) there exists a unique element g ∈ BMO(R) such that for ∗ f ∈ S we have x (f ) = R f (t)g(t) dt and conversely every element g ∈ BMO(R) gives by this formula a continuous linear functional on !H1 (R). A very useful reformulation of the Fefferman–Stein theorem is the atomic decomposition. A function a(t) is called a p-atom supported on an interval I ⊂ R if supp a ⊂ I and a(t) dt = 0 and moreover ap |I |(1/p)−1 . Our aim now is to provide an outline of the following theorem which provides a set of characterisations of !H1 (R). T HEOREM 25. The following conditions are equivalent for a real-valued function on R: (1) f ∈ L1 (R) and f˜ ∈ L1 (R), i.e., f ∈ !H1 (R), (2) there is an analytic function F (z) ∈ H1 (C+ ) such that f (x) = !F (x), (3) f ∗ (x) =: sup|x−y| 2ε we use the cancelation property of the atom to get 1/q ε 1 1 1 q 1 − a(x) ˜ = a(t) dt ap − x − t dt x x −t −ε −ε x 2ε ε Cap 2 · ε1/q C 2 x x ∞ so |x|>2ε |a(t)| ˜ dx C ε xε2 dx C. Thus for every p-atom a, both a and a˜ are in L1 (R) with uniformly bounded norms, so the same is true for f which is an absolutely convergent sum of atoms.

ε

An entirely analogous considerations can be applied to the circle. The notion of a harmonic conjugate is valid in the disc D and it allows us to define a complex space !H1 (T) of all those functions on T such that f 1 + f˜1 < ∞. We have !H1 (T) = H1 (T) ⊕ H10 (T) where H10 (T) = {f ∈ H1 (T): f (0) = 0}. The notion of a p-atom has a natural analogue on T but we have to consider also a constant function as an atom. With those conventions we have the following analog of Theorem 25.

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T HEOREM 26. The following conditions are equivalent for a real-valued function on T: (1) f ∈ L1 (T) and f˜ ∈ L1 (T), i.e., f ∈ !H1 (T), (2) there is an analytic function F (z) ∈ H1 (D) such that f (x) = !F (x), (3) f ∗ (θ ) =: supz∈S(θ) |F (z)| ∈ L1 (T) where F (z) is a harmonic extension of f to D and S(θ ) = conv({|z| 1/2} ∪ {eiθ }), (4) there exists a sequence of ∞-atoms aj such that f = j λj aj with j |λj | < ∞, (5) there exists a sequence of p-atoms, 1 < p ∞, aj such that f = j λj , aj with j |λj | < ∞, (6) !H1 (T)∗ = BMO(T) with natural duality. A more detailed presentation of the ideas involved in the above theorems can be found in [26] or in most of modern books on harmonic analysis, e.g., [48]. The main use of atoms is that to check continuity of a linear operator defined on H1 it suffices to check that it is uniformly bounded on atoms. As an interesting example of this method we present the proof of Fefferman’s Theorem 14. In our presentation we follow [49]. P ROOF OF T HEOREM 14. In this proof we will use the standard harmonic analysis notation that the n-th Taylor (or Fourier) coefficient of a function g is denoted by g(n). ˆ First let us check that condition (15) is necessary. We will consider shifted Fejer kernels

Gm (t) =

[3m/2]+m

1−

[3m/2]−m

|n − [3m/2]| int e . m

Clearly Gm ∈ H1 (T) and it is well known that Gm = 1. For f from the disc algebra we put f1 (t) = f (2mt)Gm (t). One checks that f1 ∈ H1 (T) and f1 1 f ∞ Gm 1 = f ∞ . We note that there are no cancelations of Taylor coefficients in the product defining f1 so |fˆ1 (n)| 12 |fˆ(k − 1)| for (2k − 1)m < n 2km and k = 1, 2, . . . . This implies ∞ 1 ˆ f (k − 1) 2 k=1

2km n=(2k−1)m+1

|λn |

∞

2km

λn fˆ1 (n)

k=1 n=(2k−1)m+1 ∞ λn fˆ1 (n) Cf1 1 Cf ∞ . n=0

∞ This means that the sequence ( 2km n=(2k−1)m+1 |λn |)k=1 is a continuous multiplier from the disc algebra into 1 and by the classical theorem of Paley (cf. [39]) is square summable. 2km ∞ 2(k+1)m 2 2 So we obtain ∞ k=1 ( n=(2k−1)m+1 |λn |) < ∞. The sum k=1 ( n=2km+1 |λn |) is estimated analogously, so we get the necessity of condition (15). To prove the sufficiency of (15) we will embed H1 (T) into !H1 (T). Because the multiplier is rotation invariant it suffices to check that it is uniformly bounded on all ∞-atoms


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supported on intervals I centered at 1. For such an atom a(t) we have following estimates a(n) ˆ n|I |

for n > 0,

(27)

∞ 2 a(n) ˆ |I |−1 , n=1 ∞

(28)

2 a(n ˆ + 1) − a(n) ˆ |I |/8.

(29)

n=0

π Since a(n) ˆ = −π a(eit ) e−int dλ(t) the estimate (27) follows by integration by parts. Es 2 a2 . The inequality (29) we prove as foltimate (28) is clear since ∞ ˆ n=1 |a(n)| 2 lows: ∞ ∞ 2 2 a(n a(n ˆ + 1) − a(n) ˆ ˆ + 1) − a(n) ˆ n=−∞

n=0

=

π

−π

a(t) eit −a(t)2 dλ(t)

|I |/2 it 1 e −12 dλ(t) = 2 |I | −|I |/2 |I |/2 t 2 dλ(t) |I |/8. −|I |/2

We will also need the observation that if (λn )∞ n=0 satisfies (15) then sup m−1 m2

m

n|λn | < ∞.

(30)

n=2

This is true because for 2j −1 < m 2j m n=2

n|λn |

j

k

2

n|λn |

k=1 n=2k−1 +1

Now we are ready to estimate write

∞

j

k

2

k=1

ˆ n=2 |λn a(n)|.

2

k

|λn | C

n=2k−1 −1

n=2

2k Cm.

k=1

Fix m such that |I |−1 m < 2|I |−1 and

∞ m ∞ (k+1)m λn a(n) λn a(n) λn a(n) ˆ = ˆ + ˆ . n=2

j

k=1 n=km+1

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From (27) and (30) we get m m λn a(n) ˆ |I | na(n) ˆ |I |Cm C. n=2

n=2

ˆ k )| = maxkm

k=1

(k+1)m ∞ k=1

∞ 2 a(n ˆ k ) C

2 ?1/2 |λn |

n=km+1

1/2 .

(31)

k=1

Since for each l = 1, 2, . . . , m we have (k+1)m a(n a(n ˆ + 1) − a(n) ˆ ˆ + l) + ˆ k ) a(km n=km+1

we get (k+1)m 2 2 2 a(n a(n ˆ + 1) + 2m ˆ + 1) − a(n) ˆ ˆ k ) 2a(km n=km+1

and averaging this over l we obtain 2 2 a(n ˆ k ) m

(k+1)m

(k+1)m 2 2 a(n) a(n ˆ + 2m ˆ + 1) − a(n) ˆ .

n=km+1

n=km+1

Thus we get ∞ ∞ ∞ 2 2 2 2 a(n a(n ˆ k ) a(n) ˆ + 2m ˆ + 1) − a(n) ˆ m k=1

n=m+1

n=m+1

∞ so (28), (29) and the choice of m gives ˆ k )|2 C. From (31) we get the n=1 |a(n claim. Let us note that Corollary 15 follows, apart from the constant, either from the formulation of Theorem 14 or can be easily obtained directly using the estimates we have. For an


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atom a(t) supported on an interval I centered at 1 and for m ∼ |I |−1 we have from (27) and (28) ∞ m ∞ |an | |an | |an | + n+1 n+1 n+1 n=0

n=0

n=m+1

m n|I | + n+1 n=0

∞

1/2 |an |

n=m+1

2

∞

1/2 (n + 1)

−2

n=N+1

(m + 1)|I | + Ca2 (m + 1)−1/2 C. If we want to use atoms to estimate operators from H1 into H1 we need a way to decompose functions into atoms. In general this is embodied into the theory of molecules (cf. [12]) but as an example we will give the following simple fact: ∞ L EMMA 27. If f ∈ L1 (R) and −∞ f (t) dt = 0 and f 2 K1 and |f (x)| K2 |x|−2 for |x| 16 then F H1 C = C(K1 , K2 ). P ROOF. We take In = [−2n , 2n ] and write g n =: (f − fIn )1In . The desired atomic decom position into 2-atoms is given by f = g4 + ∞ n=4 (gn+1 − gn ). Using those ideas it is relatively easy to show that !H1 (R) has an unconditional basis. T HEOREM 28. Let Ψ (x) be a wavelet such that Ψ (x) exists in each point and max |Ψ (x)|, |Ψ (x)| C(1 + |x|2)−1 . Then the wavelet basis Ψj k = 2j/2 Ψ (2j x − k) for k, j ∈ Z is an unconditional basis in !H1 (R). For the definition, most basic properties of wavelets and an argument that wavelets as described in this theorem exist, the reader may consult [15] and for more detailed treatment, e.g., [54]. P ROOF ( SKETCH ). Using the appropriate change of variables it suffices to take an ∞-atom a(t) with supp a ⊂ [−1, 1] and show that H1 -norms of all functions j,k ±a, Ψj k Ψj k are uniformly bounded. First note that from Lemma 27 we get Ψj k H1 C2−j/2 .

(32)

For the proof we split our sum into three sums j 0,k

+

j >0,|k|0,|k|5·2j

±a, Ψj k Ψj k = S1 + S2 + S3 .

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Using the assumption about the derivative of the wavelet we integrate by parts and get |a, Ψj k | 23j/2 (1 + |2j − k|)−2 . From this and (32) we infer that the sum S1 is absolutely convergent in H1 . The sum S3 is also absolutely convergent. We see directly that

a, Ψj k 2j/2 1 + |2j − k| −2 .

(33)

This and (32) yields the absolute convergence. For S2 we can show (after some calculations using (33)) that for |x| > 16 we have the pointwise estimate j 0 |k|>5·2j |a, Ψj k | |Ψj k (x)| C|x|−2 . This allows us to apply Lemma 27. It is interesting to note that H1 (T) not only has an unconditional basis but also has certain universality property in this respect (cf. Proposition 37). We can develop a parallel theory for the Haar wavelet h (cf. [15] or [54, 1.1]). This theory has deep connections with martingale theory. What is important from our point of view can be summarised in the following T HEOREM 29. For a function f ∈ L1 (R) the following conditions are equivalent: (1) R ( j k∈Z |f, hj k hj k |2 )1/2 < ∞, (2) there exists a p, with 1 < p ∞ such that f = j λj aj with j |λj | < ∞ and aj ’s dyadic p-atoms, i.e., p-atoms supported on some dyadic interval, (3) for every p, with 1 < p ∞ we have f = j λj aj with j |λj | < ∞ and aj ’s dyadic p-atoms, i.e., p-atoms supported on some dyadic interval, (4) the series j k f, hj k hj k converges unconditionally in L1 (R) to f . The space of functions satisfying any of the above conditions is denoted as H1 (δ). Clearly the Haar wavelet basis is an unconditional basis in H1 (δ). One can define an analogous space of functions on [0, 1]. It is denoted by H1 [0, 1] and actually equals span{hj k : supp hj k ⊂ [0, 1]}. As spaces of functions spaces !H1 (R) and H1 (δ) are different. The important fact is that they are isomorphic. T HEOREM 30. Let Ψ be a wavelet as in Theorem 28. The map Ψj k → hj k extends by linearity to an isomorphism from !H1 (R) onto H1 (δ). The argument for this theorem is very similar to the proof of Theorem 28. Actually this proof also shows that the map f → j k f, Ψj k hj k is bounded from !H1 (R) into H1 (δ). Conversely for a dyadic ∞-atom a(t) supported on [0, 1] we see that j k a, hj k Ψj k can be estimated exactly like S2 in the above proof. The details of the proof of this theorem can be found in [54]. From this easily follows C OROLLARY 31. The space H1 (D) is isomorphic to the space H1 (δ). The success of the atomic decomposition in !H1 (R) lead to the vast generalisation of the atomic approach. A natural framework are the so called spaces of homogeneous type, cf. [12].


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D EFINITION 32. A space of homogeneous type (X, d, μ) is a set X equipped with the quasi-metric d and a positive Borel measure μ such that there exists a constant C such that for all x ∈ X and r > 0 we have

μ B(x, 2r) Cμ B(x, r) < ∞. Once we have a space of homogeneous type (X, d, μ) we can define atoms as Borel measurable functions a(t) on X, such that supp a(t) ⊂ B(x, r), |a(t)| μ(B(x, r))−1 and X a(t) dμ(t) = 0. We define H1 (X, d, μ) as the space of all functions f such that f = λ a where a ’s are atoms and j j j j |λj | < ∞. The norm f H1 is defined as j inf j |λj | where the inf is taken over all atomic representations of the function f . The following examples are important in our context: (1) X = R, d(x, y) = |x − y| and μ the Lebesgue measure. In this case H1 (R, d, λ) equals !H1 (R). (2) X = T, d is the arc-length distance and μ is the normalised Lebesgue measure on T. In this case H1 (T, d, μ) equals !H1 (T). (3) X is either R or the interval [0, 1), d is the dyadic distance, i.e., if x = ∞ k=−∞ xk · 2−(k+1) where xk = 0 or 1 then we put d(x, y) =

∞

|xk − yk |2−(k+1).

k=−∞

The measure μ is the Lebesgue measure. In this case we get the dyadic Hardy space H1 (δ). (4) X = Sn the unit sphere in Cn , d(z, w) = |1 − z, w| and μ = σn . In this case we get the space of functions H1 (Sn , d, σn ) on Sn . It contains (boundary values of) the space H1 (Bn ) as a closed, complemented subspace. This example is investigated and the facts mentioned here are proven in [12] and [17]. A deep theorem of Paul Müller describes the isomorphic type of spaces H1 (X, d, μ). Before we can formulate it we need to introduce finite-dimensional analogues of Hardy n −1 spaces. By H1n , n = 1, 2, . . . , we denote the space span{hj k }n−1,2 j =0,k=0 ⊂ H1 (δ). From Theorem 30 we see that isomorphically we obtain the same sequence of spaces when we replace the Haar wavelet by any basis discussed in Theorem 28. It is interesting that the deep result of Boˇckariov [8] asserts that the Banach–Mazur distance between H1n and n −1 ⊂ H1 (D) is uniformly bounded. span{zj }2j =1 T HEOREM 33 (Müller [35]). Let (X, d, μ) be a space of homogeneous type and let H1 (X, d, μ) be infinite-dimensional. Then H1 (X, d, μ) is isomorphic to H1 (δ) if and only if μ({x ∈ X: μ({x}) = 0}) > 0. When μ({x ∈ X: μ({x}) = 0}) = 0 then H1 (X, d, μ) is n isomorphic either to 1 or to ( ∞ n=1 H1 )1 . The proof of this theorem is very ingenious and complicated. To give even the sketch is well beyond the scope of this survey. When we apply this theorem to the case described in (4) above we see that H1 (Bn ) is isomorphic to a complemented subspace of H1 (δ). Next we show that H1 (Bn ) contains a

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complemented copy of H1 (δ). The natural way to prove it is to use the inner function in the ball: there exists a bounded function ϕ(z) holomorphic on Bn such that ϕ(0) = 0 and for ζ ∈ Sn we have |ϕ(ζ )| = 1 σn -a.e. (cf. [42]). One checks that span{ϕ n (z)}∞ n=0 ⊂ H1 (Bn ) is isometric to H1 (D) (the isometry is given by ϕ n ↔ zn ). Let Σ1 be the sub σ -algebra of the σ -algebra Σ of all measurable subsets of Sn generated by the function ϕ. One checks that the conditional expectation operator from L1 (Sn , Σ, σn ) onto L1 (Sn , Σ1 , σn ) gives a projection from H1 (Bn ) onto span{ϕ n }∞ n=0 . So the decomposition argument (use also Theorem 39) gives T HEOREM 34. The spaces H1 (Bn ) for n = 1, 2, . . . are all isomorphic to the space H1 (δ) and also to !H1 (R) and H1 (D). R EMARK 35. The first correct proof of this theorem was given by Wolniewicz in [55]. This result can be greatly generalised. The definition of H1 space can be naturally extended to H1 (Ω) where Ω is a bounded strictly pseudoconvex domain in Cn . Then the boundary ∂Ω can be made into a space of homogeneous type in such a way that H1 (Ω) is a complemented subspace of the atomic H1 (∂Ω). Also such spaces H1 (Ω) are isomorphic to H1 (δ) as was shown by entirely different methods in [2] and [35]. The situation for polydiscs is dramatically different. T HEOREM 36 ([9,10]). The spaces H1 (Dn ) and H1 (Dm ) are non-isomorphic for n = m. Let us give the idea of the proof of this theorem when n = 1 and m = 2. Suppose that 2j there is an isomorphism T : H1 (D2 ) → H1 [0, 1]. Let (ϕj k )∞ j =0 k=1 be an unconditional basis in H1 (D) equivalent to the normalised Haar basis in H1 [0, 1]. Then the subspace of H1 (D2 ) l spanned by the functions ϕj k (z)w2 where j = 0, 1, 2, . . . , k = 1, 2, . . . , 2j , l = 1, 2, . . . , is complemented in H1 (D2 ) – this follows from the Paley’s theorem. Now fix N and look l at functions T (ϕj k (z)w2 ) with j = 0, 1, . . . , N and k, l as previously. Since for a fixed l l j, k the w-liml→∞ ϕj k (z)w2 = 0, the same holds for T (ϕj k (z)w2 ). Thus, starting from j = N and k = 2N and going backward, we see that there are integers l = l(j, k) such that p(j,k) l(j,k) (up to a small perturbation) T (ϕj k (z)w2 ) = s=s(j,k),r as,r hs,r where

s N, 2N < p N, 2N < s N, 2N − 1 < p N, 2N − 1 < · · · < s(N, 1)

< p(N, 1) < s N − 1, 2N−1 < p N − 1, 2N−1 < · · · < s(0, 1) < p(0, 1). The point is that (when we look at j, k only) we invert the order; the last function N ϕN,2N (z)wl(N,2 ) is mapped by T to a function whose expansion in H1 [0, 1] is at the beginning, while the first function ϕ0,1(z)wl(0,1) has the expansion starting very far away. Nevertheless the spaces spanned by (ϕj,k (z)w2

l(j,k)

)N j =0

2j k=1

are uniformly in N comple-

mented in H1 (D2 ) and this basis is equivalent to the basis (ϕj,k (z))N j =0

2j k=1

in H1 (D).


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l(j,k)

Thus T (ϕj k (z)w2 ) span uniformly complemented subspace of H1 [0, 1] and this ba2j sis is equivalent to (hj,k )N j =0 k=1 . But this contradicts the fact that T inverts the order, i.e., T (ϕj k (z)w2

l(j,k)

)=

p(j,k)

as,r hs,r .

s=s(j,k),r

This is the main technical argument in the proof. To give the reader an idea why it is so, let us consider the model case (which reflects the general situation although it is by no l(j,k) means easy to see). Let us consider the case when T (ϕj,k (z)w2 ) = hj,k · rm(j,k) where r denotes the Rademacher functions and m(j, k) = 10N − 2j − k. (Clearly the exact value of m(j, k) is not essential – the important thing is that it reverse the order.) We have N 2j 1/2 1 aj k hj,k rm(j,k) ∼ |aj k |2 |hj k |2 |rm(j k) |2 0 j =0 k=1

j,k

H1

=

1 0

1/2 |aj k | |hj k | 2

2

N 2j = aj k hj,k . j =0 k=1

j,k

H1

j

2 This shows (and we already know it) that (hj,k · rm(j,k) )N j =0 k=1 in H1 ([0, 1), δ) are unij

2 formly in N equivalent to (hj,k )N j =0 k=1 . But they are not complemented (uniformly in N ) because in the dual space BMO the equivalence breaks down. Let h∗j,k denote the Haar basis normalised in L∞ , i.e., biorthogonal functionals to hj,k . We have

N aj h∗j,0 j =0

∼ sup |aj |.

(34)

BMO

∗ On the other hand for f = N j =0 aj hj,0 rm(j,0) we use the definition of the dyadic BMO to −N get (for the interval [0, 2 ]) f BMO 2N

2−N 0

1/2 |f − f[0,2−N ] |

2

.

But for j = 0, 1, . . . , N we have m(j, 0) > N so we infer that f[0,2−N ] = 0, and also that the Rademacher functions we use are orthogonal on the interval [0, 2−N ]. Thus we get f BMO 2N

2−N 0

1/2 |f |

2

=

N

1/2 |aj |

2

.

j =0

This together with (34) clearly shows that there is no equivalence in BMO.

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The technical side of the above proof was greatly simplified by Müller [36]. It is easy to see that the space !H1 (R) is not a Banach lattice in the natural order ∞ because lattice operations do not preserve the condition −∞ f (t) dt = 0. Actually !H1 (R) is not isomorphic to any non-atomic Banach lattice [23]. There is however a closely related Banach lattice of functions on C+ introduced and studied in greater generality in [11]. It is the tent space T 1 (C+ ) defined as the space of all functions on C+ such that

∞ −∞

|x−y| 0, so that (35) gives equivalent norms for different α’s. Clearly the space T 1 (C+ ) is a Banach lattice of functions. We can embed !H1 (R) as a complemented subspace of T 1 (C+ ) as follows: for f ∈ !H1 (R) ) let u(y, t) be its Poisson integral and put F (y, t) = t ∂u(y,t ∂t . The map f → F is the desired isomorphic embedding (cf. [11]). A bit more transparent complemented embedding of H1 (δ) into T 1 (C+ ) can be realised as follows: let for j, k ∈ Z squares Aj k ⊂ C+ be defined as [k2j , (k + 1)2j ] ⊗ i[2j , 2j +1 ] and let Fj k = 1Ajk 2−j/2 . One checks (cf. [23]) that hj k → Fj k is an isomorphic embedding onto a complemented subspace of T 1 (C+ ). Actually (and it is easy to believe if one understands the above embedding of H1 (δ) into T 1 (C+ )) the space T 1 (C+ ) is isomorphic to the Hilbert space valued H1 -space H1 (δ, L2 ) k 2 or (what is the same) span{zj w2 }∞ j k=0 in H1 (T ). It follows from the Bourgain’s arguments indicated above in the proof of Theorem 36 that the space H1 (δ, L2 ) is not isomorphic neither to H1 (T) nor to H1 (T2 ).

6. Isomorphic structure of H1 We have seen in the previous sections that many H1 -type spaces are isomorphic. In this section we want to elaborate more fully on the isomorphic structure of this space, which we will generically denote by H1 . H1 is a dual space. Let D be any of our standard domains in Cn . Then H1 (D) is a dual space because it is a ω∗ -closed subspace of C(∂D)∗ . In the special case of H1 (D) we obtain from the F. and M. Riesz theorem that H1 (D) = (C(T)/A0 )∗ . Here by A0 we mean the subspace of the disc algebra of all functions vanishing at 0. The predual C(T)/A0 can be isometrically described as a space of compact Hankel operators on the Hilbert space. For a function f ∈ C(T) we define the Hankel operator with index f , Hf (g) = (I − R)(fg) where R is the Riesz projection. Clearly Hf : H2 (T) → H2 (T)⊥ . If f1 − f2 ∈ A0 then Hf1 = Hf2 and we infer that Hf f C/A0 . Also since f ∈ C(T) we can approximate it by trigonometric polynomials, so we can approximate Hf by finite-dimensional operators thus Hf is compact. On the other hand if f ∈ C(T) with f C/A0 = 1 then there exists h ∈ H1 (T) such that h1 = 1 and f h = 1. Using the canonical factorisation we write h = h1 · h2 where h1 , h2 ∈ H2 (T) and h1 2 = h2 2 = 1. Then we have ! " Hf Hf (h1 ) 2 Hf (h1 ), h2 = f h1 h2 = 1.


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This gives that the map f → Hf is an isometry from C/A0 into the space of compact operators on a Hilbert space. When we use the space BMO as a dual of H1 then we can describe the predual of H1 (but only isomorphically) as a space VMO of functions of vanishing mean oscillation where 1 VMO = f ∈ BMO: lim |f − fI | = 0 . |I |→0 |I | I Subspaces and complemented subspaces. Let us first discuss subspaces of H1 . The next proposition shows that many subspaces of L1 are also subspaces of H1 . P ROPOSITION 37. Let X ⊂ L1 (T) be a closed subspace. Suppose that X either has an unconditional basis or X is reflexive. Then X is isomorphic to a subspace of H1 (T). To see the first part fix an unconditional basis (xn ) in X and assume (perturbing it slightly) that each xn (t) is a trigonometric polynomial and fix a sequence kn such that for some strictly increasing sequence of integers ln we will have Λ2ln (eikn t xn ) = eikn t xn , where Λs are multipliers discussed after Theorem 18. Then using the unconditionality of (xn ) and Theorem 18 we get 2 1/2 2 1/2 ikn t e a a a x ∼ x (t) dt ∼ x (t) dt n n n n n n n

1

T

n

ikn t an e xn ∼

T

n

n

which shows that eikn t xn is a basic sequence in H1 (T) equivalent to (xn ). For the second part recall that by Rosenthal’s theorem a reflexive subspace of L1 is a subspace of Lp for some p > 1, that such Lp is a subspace of L1 [40], and that Lp for p > 1 has an unconditional basis (cf., e.g., [15]). As a corollary of the above we get that a reflexive complemented subspace of H1 is isomorphic to 2 (cf. [27, Corollary 2.1]). It was also shown in [27] that every Hilbertian subspace of H1 contains an infinite-dimensional complemented subspace. To see it take (fn )∞ n=1 , a sequence of functions in H1 (D) equivalent to the unit vector basis in 2 and let V : 2 → H1 (D) be defined as V (en ) = fn . Think that H1 (D) = X∗ where X ⊂ K(2 ) the space of all compact operators on 2 . We have V = U ∗ where U : X → 2 is onto. Take xn ∈ X such that xn C and U (xn ) = en . Then for some subsequence zk = xn2k − xn2k+1 is weakly null. Now we work in K(2 ) and see that zk either contains a subsequence zks equivalent to the unit vector basis in c0 (what in our case leads to a contradiction) or to 2 This gives that P (f ) = s f, zks fn2ks is the desired projection. Now let us consider subspaces of H1 (D) which are invariant under rotation. Such a subspace is described by a subset Λ ⊂ N and equals span{zn : n ∈ Λ}. Also, by invariance, if such a subspace is complemented, it is complemented by a multiplier 1Λ . We have seen examples of such multipliers in Paley’s theorem. Other easily checked examples are arithmetic progressions intersected with N. All the other examples are build from the above ones.

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T HEOREM 38 (Klemes [25]). A subspace span{zn : n ∈ Λ} ⊂ H1 (D) is complemented in H1 (D) iff Λ is a finite Boolean combination of lacunary sets, finite sets and arithmetical progressions intersected with N. The version (much more difficult) of Klemes’ theorem for H1 (R), i.e., a characterisation of translation invariant complemented subspaces of H1 (R) was given by Alspach [1]. No extension of those results to several variables are known. Clearly the existence of unconditional basis gives many projections. In particular the following result easily follows from the form of Haar basis in H1 (δ). T HEOREM 39. The space H1 is isomorphic to ( ∞ n=1 H1 )1 , its infinite 1 sum. Actually the Haar basis in H1 (δ) and the Haar basis in H1 [0, 1] are permutatively equivalent and each of them is permutatively equivalent to its infinite 1 sum. P ROOF. Let us identify the normalised in H1 (δ) Haar function with its support I and denote it by hI . Let O = {hI : I ⊂ [0, 1]} and let Dn = {hI : I ⊂ [1 − 2−n+1 , 1 − 2−n ]} with n = 1, 2, . . . Clearly the basis Dn is isometrically .∞ ∞ equivalent to the Haar basis in 1] and D is isometrically equivalent to ( H1 [0, n=1 n n=1 Dn )1 . One easily observes that O\ ∞ D is a basis equivalent to the unit vector basis in 1 . Since each Dn contains a n=1 n we infer that O is permutatively equivsubsequence equivalent to the unit vector basis in 1 alent to ( O)1 . To treat the case of H1 (δ) we define O = {hI : I ⊂ R} and define Dn = {hI : I ⊂ [2n−1 , 2n ] ∪ [−2n , −2n−1 ]} where n = 1, 2, . . . and D0 = {hI : I ⊂ [−1, 1]}. The argument now is analogous. The main unsolved problem about infinite-dimensional complemented subspaces of H1 (D) is whether there are infinitely many isomorphic types of them. The easy ones are obtained from the above theorem, Paley’s theorem which implies that 2 is complemented in H1 (D) and finite-dimensional spaces H1n spanned by increasing subsets of unconditional basis. A routine n argument n yields 10non-isomorphic n types ofn them as , , ⊕ , ( ) , ( ) ⊕ , ( ) , ( H ) , ( H 1 )1 ⊕ 2 , follows 1 2 1 2 1 1 2 2 1 1 n 2 1 ( H1n )1 ⊕ ( 2 )1 , H1 . Two essentially new examples were obtained by a more refined martingale techniques by Müller and Schechtman [37]. One of them, called Y1 in [37], can 2 1/2 < ∞}. This space is not isomorphically be described as {(αn ): ∞ n=1 min{|αn |, |αn | n} isomorphic to H1 but contains subspaces isomorphic to all p with 1 p 2. The other one is a sum of independent copies of H1n and can be described as span{Gn }∞ n=1 where each Gn is an isometric copy of H1n but different Gn ’s consist of statistically independent functions. To be more explicite let (rn )∞ n=1 be the sequence of Rademacher functions n and let Gn be the span in H1 [0, 1] of all Walsh functions of the form rkn11 · · · · · rkjj where ni = 0, 1 and n2 ki < (n + 1)2 . Clearly one can use those two new spaces to form direct sums with old ones to get a more extensive (but still finite) list of all known complemented subspaces of H1 . One should also note that the Haar basis in H1 (δ) gives only three isomorphic types T HEOREM 40 ([34]). Let Λ ⊂ Z × Z bean infinite subset. Then the subspace span{hλ : λ ∈ n Λ} ⊂ H1 (δ) is isomorphic to 1 or to ( ∞ n=1 H1 )1 or to H1 (δ).


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The proof of this theorem is quite complicated and technical so we will only indicate how different possibilities arise. In order to make things more transparent let us consider H1 [0, 1]. If we take any subset B of Haar functions whose supports are in [0, 1] we define the set σ (B) as {t: t ∈ supp hI for infinitely many hI ∈ B}. If |σ (B)| > 0 then span{hI ∈ B} ∼ H1 . The proof builds a block basic sequence of {hI ∈ B} which is very close in the H1 norm and distribution to the original Haar system. This gives that our space contains complemented H1 so by decomposition we get the claim. −1 If |σ (B)| = 0 but sup{|I | I ∈ B} ∼ hJ ∈B,J ⊂I |J |: hI ∈ B} = ∞ then we have span{h ( H1n )1 . We show that in this case our space contains complemented ( H1n )1 and is contained as a complemented subspace in one. If |σ (B)| = 0 but sup{|I |−1 hJ ∈B,J ⊂I |J |: hI ∈ B} < ∞ then a direct calculation shows that the basis B is equivalent to the unit vector basis in 1 . The condition distinguishing cases in the situation when |σ (B)| = 0 is a martingale version of the Carleson condition (cf. [26]). Note also that there is a close similarity between conditions and conclusion of this theorem and Theorem 33. This is not accidental and actually the methods of proof of Theorem 33 are an outgrowth and elaboration of the methods used to prove Theorem 40. There are also some general results about complemented subspaces of H1 . Let us formulate some of them as one theorem. T HEOREM 41. (a) ([34]) Let X ⊂ H1 be isomorphic to H1 . Then there exists Y ⊂ X complemented in H1 and isomorphic to H1 . (b) ([34]) The space H1 is primary, i.e., whenever H1 = X ⊕ Y then either X or Y is isomorphic to H1 . (c) ([37]) A complemented subspace H1 either contains 2 or is isomorphic to a X of n) . complemented subspace of ( ∞ H n=1 1 1 Approximation property. It is easy to see that H1 (D) has the bounded approximation property. The easiest argument is using Fejer’s kernels but it also trivially follows from the existence of unconditional basis. The analogous problem for BMO(R) is much harder. It was solved by Jones [22] who proved T HEOREM 42. The space BMO(R) and thus also H1 (D) has the uniform bounded approximation property. 7. Isometric questions The most natural isometric question about a Banach space is to describe all isometries of the space. In the case of H1 spaces (and actually for all Hp spaces) this question has a very satisfactory answer. For D it is given in the following theorem, but for other spaces the description is analogous.

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T HEOREM 43. The operator T : H1 (D) → H1 (D) is an isometry into iff T (f )(z) = F (z) · f (ϕ(z)) where ϕ(z) is an analytic map from D onto D, i.e., an inner function, and F (z) ∈ H1 (D) such that for every bounded Borel function h(t) on T we have h(t) dλ(t) = (h ◦ ϕ)F (t) dλ(t). T

T

Such an operator T is onto iff ϕ is a Möbius transformation and F (z) = αϕ (z) with α ∈ C and |α| = 1. Another natural problem is the description of the extreme structure of the unit ball. The study of extreme points of unit the ball in H1 (D) was done in [28]. In particular, the extreme points in the unit ball of H1 (D) are described as outer functions of norm 1. To see this take f = I · F with I a non-constant inner function and adjust the constants so that 2π it it 0 |f (e )|I (e ) dt is purely imaginary. Then the decomposition 1 1 1 2 2 f= F (1 + I ) − F (1 − I ) 2 2 2 shows that f is not an extreme point. Conversely if outer f0 = 12 (f1 + f2 ) with fi = 1 and f1 = f2 , we infer that both functions fj (eit ) have the same argument. But then fj /f0 are bounded analytic functions in D with real boundary values, thus constants. This gives a contradiction so f0 is an extreme point. No analogous characterisation is known for other H1 (D) spaces. Some effort went to describe exposed and strongly exposed points in various H1 spaces, but only various sufficient conditions are known (cf. [24,43,50]). Another related problem of describing extreme points in the unit ball of the space VMO and BMO with the natural p-mean oscillation norm was studied in [3]. Very little is known about norm one projections in H1 spaces. All known such projections are restrictions of norm one projections on L1 which leave H1 invariant. It is not known if there are any others. In particular it is unknown if there is a norm one finitedimensional projection on H1 (D) whose whose range has dimension > 1. Also it is unknown if H1 (D) has a monotone basis. Another classical isometric result about H1 (D), but valid also for other spaces H1 (D) is the fact (cf. [38]) that if fn converges weakly in H1 (D) to f and fn = f for all n then actually fn converges to f in norm. This means that the natural norm on H1 (D) has the Kadec–Klee property. Actually a bit stronger result holds: for each ε > 0 there is a δ > 0 such that for every weak∗ convergent (with respect to the natural (H1 , C/A0 ) duality) sequence fn → f such that fn 1 and f 1 − δ we have lim infn=m fn − fm ε. From this result follows that H1 (D) has so called ω∗ -normal structure. This implies that every non-expansive map defined on a ω∗ -closed bounded convex subset of H1 (D) has a fixed point (cf. [5]). References [1] D.E. Alspach, A characterization of the complemented translation-invariant subspaces of H 1 (R), Trans. Amer. Math. Soc. 323 (1) (1991), 197–207. [2] H. Araki, Degenerate elliptic operators, Hardy spaces and diffusions on strongly pseudoconvex domains, Tohoku Math. J. 46 (4) (1994), 469–498.


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[3] S. Axler and A. Shields, Extreme Points in VMO and BMO, Indiana Univ. Math. J. 31 (1982), 1–6. [4] D. Bekollé and A. Bonami, Estimates for the Bergman and Szegö projections in two symmetric domains in Cn , Colloq. Math. 68 (1) (1995), 81–100. [5] M. Besbes, S.J. Dilworth, P.N. Dowling and C.J. Lennard, New convexity and fixed point properties of Hardy and Lebesgue–Bochner spaces, J. Funct. Anal. 119 (2) (1994), 340–357. [6] O. Blasco, Multipliers on spaces of analytic functions, Canad. J. Math. 47 (1995), 44–64. [7] R.P. Boas, Isomorphism between Hp and Lp , Amer. J. Math. 77 (1955), 655–656. [8] S.V. Boˇckariov, Construction of polynomial bases in finite-dimensional spaces of functions analytic in the disk, Proc. Steklov Inst. of Math. (1985), 55–81. [9] J. Bourgain, Non-isomorphism of H 1 -spaces in one and several variables, J. Funct. Anal. 46 (1982), 45–57. [10] J. Bourgain, The non-isomorphism of H 1 -spaces in different number of variables, Bull. Soc. Math. Belg. Sér. B 35 (1983), 127–136. [11] R.R. Coifman, Y. Meyer and E. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335. [12] R.R. Coifman and G. Weiss, Extensions of Hardy spaces and theory use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645. [13] F. Delbaen, H. Jarchow and A.Pełczyński, Subspaces of Lp isometric to subspaces of p , Positivity 2 (1998), 339–367. [14] P. Duren, Theory of H p Spaces, Academic Press, New York (1970). [15] T. Figiel and P. Wojtaszczyk, Bases in function spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 561–597. [16] T. Gamelin and S.V. Kislyakov, Uniform algebras and spaces of analytic functions in the supremum norm, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 671–706. [17] J.B. Garnett and R.H. Latter, The atomic decomposition for Hardy spaces in several complex variables, Duke Math. J. 45 (1978), 845–915. [18] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs (1962). [19] H. Jarchow, Some functional analytical properties of composition operators, Quaestiones Math. 18 (1995), 229–256. [20] M. Jevtić and M. Pavlović, Coefficient multipliers on spaces of analytic functions, Preprint. [21] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. [22] P.W. Jones, BMO and the Banach space approximation problem, Amer. J. Math. 107 (4) (1985), 853–893. [23] N.J. Kalton and P.Wojtaszczyk, On nonatomic Banach lattices and Hardy spaces, Proc. Amer. Math. Soc. 120 (1994), 731–741. [24] A. Kheifets, Nehari’s interpolation problem and exposed points of the unit ball in the Hardy space H 1 , Israel Math. Conf. Proc. 11 (1997), 145–151. [25] I. Klemes, Idempotent multipliers of H 1 (T), Canad. J. Math. 39 (1987), 1223–1234. [26] P. Koosis, Introduction to Hp Spaces, London Math. Soc. Lecture Notes Ser. 40, Cambridge Univ. Press, Cambridge (1980). [27] S. Kwapień and A. Pełczyński, Some linear topological properties of the Hardy spaces H p , Compositio Math. 33 (1976), 261–288. [28] K. deLeeuw and W. Rudin, Extreme points and extremum problems in H 1 , Pacific J. Math. 8 (1958), 467– 485. [29] J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249. [30] D.H. Luecking, Composition operators belonging to the Schatten ideals, Amer. J. Math. 114 (5) (1992), 1127–1145. [31] T. MacGregor and K. Zhu, Coefficient multipliers between Bergman and Hardy spaces, Mathematika 42 (2) (1995), 413–426. [32] M. Mateljević and M. Pavlović, Lp -behaviour of the integral means of analytic functions, Studia Math. 77 (1984), 219–237.

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[33] P.F.X. Müller, On subsequences of the Haar basis in H1 (δ) and isomorphisms between H 1 -spaces, Studia Math. 85 (1987), 73–90. [34] P.F.X. Müller, On subspaces of H 1 isomorphic to H 1 , Studia Math. 88 (1988), 121–127. [35] P.F.X. Müller, The Banach space H 1 (X, d, μ). II, Math. Ann. 303 (1995), 523–544. [36] P.F.X. Müller, A simplification in the proof of the non-isomorphism between H 1 (δ) and H 1 (δ 2 ), Studia Math. 150 (2002), 13–16. [37] P.F.X. Müller and G. Schechtman, On complemented subspaces of H 1 and VMO, Lecture Notes in Math. 1376, Springer (1989), 113–126. [38] D.J. Newman, Pseudo-uniform convexity in H1 , Proc. Amer. Math. Soc. 14 (1963), 676–679. [39] R.E.A.C. Paley, A note on power series, J. London Math. Soc. 7 (1932), 122–130. [40] H.P. Rosenthal, On subspaces of Lp , Ann. of Math. 97 (1973), 344–373. [41] W. Rudin, Function Theory in the Unit Ball of Cn , Springer, Berlin (1980). [42] W. Rudin, New Constructions of Functions Holomorphic in the Unit Ball of Cn , CBMS Regional Conf. Ser. in Math. 63, Providence (1986). [43] D. Sarason, Exposed points in H 1 , Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl. 48, Birkhäuser (1990), 333–347. [44] J.H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), 375–404. [45] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer (1993). [46] J.H. Shapiro and P.D. Taylor, Compact, nuclear and Hilbert–Schmidt composition operators on H 2 , Indiana Univ. Math. J. 23 (1973/74), 471–496. [47] W.T. Sledd and D.A. Stegenga, An H 1 multiplier theorem, Ark. Mat. 19 (1981), 265–270. [48] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ (1993). [49] S.J. Szarek and T. Wolniewicz, A proof of Fefferman’s theorem on multipliers, Preprint N.209, Institute of Math. Polish Academy of Sciences (1980). [50] D. Temme and J. Wiegerinck, Extremal properties of the unit ball in H 1 , Indag. Math. (N.S.) 3 (1) (1992), 119–127. [51] D. Vukotić, On the coefficient multipliers of Bergman spaces, J. London Math. Soc. 50 (1994), 341–348. [52] P. Wojtaszczyk, On multipliers into Bergman spaces and Nevanlinna class, Canad. Math. Bull. 33 (1990), 151–161. [53] P. Wojtaszczyk, On unconditional polynomial bases in Lp and Bergman spaces, Constr. Approx. 13 (1997), 1–15. [54] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge Univ. Press, Cambridge (1997). [55] T. Wolniewicz, On isomorphisms between Hardy spaces on complex ball, Ark. Mat. 27 (1) (1989), 155–168. [56] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, Inc., New York (1990).

CHAPTER 40

Extension of Bounded Linear Operators M. Zippin Mathematics Department, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) The injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Separably injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (c) The class of compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (d) Lifting of operators and extension of isomorphisms to automorphisms . . . . . . . . (e) Extension into C(K) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (f) Extension of operators from subspaces of a space of type 2 into a space of cotype 2 . 2. The injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Separably injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Extension of compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Lifting of operators and extension of isomorphisms to automorphisms . . . . . . . . . . Remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Extension of operators into C(K) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Extension of operators from subspaces of a space of type 2 into a space of cotype 2 . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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. 1705 . 1705 . 1707 . 1707 . 1708 . 1709 . 1710 . 1711 . 1716 . 1716 . 1723 . 1726 . 1728 . 1729 . 1732 . 1735 . 1739


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1. Introduction In this chapter we discuss various extension problems concerning bounded linear operators in Banach spaces. Let X and Y be Banach spaces and let E be a subspace of X. An operator : X → Y is said to be an extension of an operator T : E → Y if T e = T e for all e ∈ E. T The general problem discussed below is the following: when does every member of a class : X → Y (of the same class)? of operators T : E → Y admit an extension operator T We will introduce here the basic definitions and describe the main extension problems. Each of these problems is discussed, in detail, in one of the next sections. In this chapter an “operator” means a bounded linear operator. The starting point of all extension theories is the following, well-known, perfect extension theorem for linear functionals, T HE H AHN –BANACH THEOREM ([22,5,8]). Let X be a Banach space over the real or complex field F and let E be a subspace of X. Then every bounded linear functional e∗ : E → F can be extended to a linear functional x ∗ : X → F with x ∗ = e∗ . Unfortunately, such perfection is rare; few extension theories which deal with more general operator extension problems can avoid compromises. For example, as shown in Section 6, the Hahn–Banach theorem, valid for operators of rank 1, is false for operators of rank 2 once we replace F by some two-dimensional space F . Any attempt to generalize the Hahn–Banach theorem necessarily requires some restrictions: restrictions on the spaces X and E, restrictions on the range space Y , relaxation of the norm preservation condition T = T or restrictions on the class of operators to be extended. We start with a discussion about conditions on the domain space E which ensure, in full generality, the existence of bounded extensions of operators.

(a) The injective spaces Given a Banach space X, a subspace E of X and λ 1, we say that the pair (E, X) has the λ-extension property (λ-EP, in short) if, for every Banach space Y , every operator T : E → Y admits an extension T : X → Y with T λT . The pair (E, X) is said to have the Extension Property (EP in short) if, for every space Y , every operator T : E → Y : X → Y . admits an extension T These two properties are closely connected to the following: a Banach space E is called λ-injective or a Pλ space, if for every space X containing E there is a projection P of X onto E with P λ. E is called injective (or a P space) if, for every X containing E, there is a projection of X onto E. The relations between the above properties are formulated in the following P ROPOSITION 1.1. Let E be a Banach space and let λ 1. The following three assertions are equivalent. (1.1) E is λ-injective. (1.2) For every space X containing E, (E, X) has the λ-EP.

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(1.3) For every pair of spaces Y ⊃ X, every operator T : X → E admits an extension : Y → E with T λT . T P ROOF. (1.1) follows from either (1.2) or (1.3) by extending the identity T = IE . To prove that (1.1) ⇒ (1.2), let X ⊃ E and T : E → Y . Let P be a projection of X onto E with = T P is the desired extension. It remains to establish (1.1) ⇒ (1.3). Let P λ then T ∗ Γ = Ball(E ) and let j : E → ∞ (Γ ) be the isometric embedding of E into ∞ (Γ ) defined by j (e)(e∗ ) = e∗ (e) for all e ∈ E and e∗ ∈ Ball(E ∗ ). Let Y ⊃ X and let T : X → E be an operator. Let S = j T : X → j (E) ⊂ ∞ (Γ ) and extend S to an operator S : Y → l∞ (Γ ) as follows: For each e∗ ∈ Γ , the functional T ∗ e∗ on X may be extended to a functional ye∗∗ ∈ Y ∗ with ye∗∗ = T ∗ e∗ by the Hahn–Banach theorem. Define S : Y → ∞ (Γ ) by Sy(e∗ ) = ∗ S is linear, S = T and S extends S. Now define the desired extension ye∗ (y) then = j −1 P S, where P is a projection of l∞ (Γ ) onto j (E) with T : Y → E of T by T P λ. Proposition 1.1 shows that E is injective if and only if (E, X) has the EP for every X containing E and this property is equivalent to (1.3) with the condition T λT omitted. By using the third property (1.3) it is easily proved that every injective space is λ-injective for some λ 1. A direct consequence of the proof of Proposition 1.1 is that, for every set Γ , the space ∞ (Γ ) is a P1 space. The spaces L∞ (Ω, μ) = L∗1 (Ω, μ) are 1-injective, too. To show this one uses the following, well-known compactness argument: let ω = (Ω1 , Ω2 , . . . , Ωm ) denote a finite partition of Ω into mutually disjoint measurable sets of positive measure. Let A denote the collection of all such partitions, ordered as follows: (Ω1 , Ω2 , . . . , Ωm ) = ω < γ = (Γ1 , Γ2 , . . . , Γn ) if n > m and each Ωi is a union of members of γ . Clearly, A is directed by < and each subspace Eω = [χΩi ]m i=1 of L∞ (Ω, μ) is isometric to ∞ (1, 2, . . . , m). Hence Eω is a P1 space and, whenever X ⊃ L∞ (Ω, μ) there is a projection Pω of X onto Eω with Pω = 1. It is clear that ω Eω is dense in L∞ (Ω, μ). Regarding L∞ (Ω, μ) as a linear topological space under the ω∗ topology (induced by L1 (Ω, μ)), the unit ball U = Ball(B(X, L∞ (Ω, μ))) of the space of bounded operators from X into L∞ (Ω, μ), under the pointwise ω∗ topology, is compact. Since {Pω } is a net in U (directed by 0, T : 1 (Γ ) → Y may be chosen so that T (1 + ε)T . We will present in Section 5 generalizations of this phenomenon. Equally easy is the following observation which demonstrates the role of 1 as a “universal” separable space with respect to extension of operators into a space Y . P ROPOSITION 1.5. Let W be a subspace of a Banach space X and let Q be an operator from a space Z onto X so that Q = 1 and Q(Ball(Z)) ⊃ δ Ball(X). Let Y be any Banach space and suppose that every operator T : Q−1 (W ) → Y admits an extension T : Z → Y with T λT . Then any S : W → Y admits an extension S : X → Y with S −1 λδ S. P ROOF. Given an operator S : W → Y , consider the operator SQ : Q−1 (W ) → Y . If S : Z → Y extends SQ and S λS, then, since S vanishes on ker Q, S induces an S λδ −1 S. operator S from X ∼ Z/ ker(Q) into Y so that SQ = S and S δ −1 The significance of the above fact is the following: since every separable space X is a quotient space of Z = 1 , for any subspace W of X, the understanding of the extension properties of the pair (Q−1 (W ), 1 ) with respect to operators into a space Y sheds light on the same extension properties of the pair (W, X) regarding operators into Y . This will be useful in Section 6 below. The main problems discussed in Section 5 are the following two: P ROBLEM 1.6. Let X = c0 (X = ∞ , resp.). Let E be a subspace of X and let T be an on X? isomorphism from E into X. When can T be extended to an automorphism T It turns out ([47]) that, in all “reasonable” situations an extending automorphism does exist, thus demonstrating the surprising richness of the class of automorphisms on X. The second problem is, in a sense, dual to Problem 1.6. It concerns the possibility of lifting an isomorphism between quotient spaces of 1 to an automorphism on 1 . P ROBLEM 1.7. Let E and F be infinite-dimensional subspaces of 1 and let ϕ : 1 → 1 /E and ψ : 1 → l/F be quotient maps. Suppose that T is an isomorphism from 1 /E onto 1 /F . Does there exist an automorphism T on 1 so that ψ T = T ϕ? Again, surprisingly, the answer is positive and provides a useful tool for extension of operators from subspaces of 1 .


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(e) Extension into C(K) spaces We will see in Section 4 below that, restricting the range space Y to the class of L1 (Ω, μ) preduals, we can nicely extend any compact operator T : E → Y to a compact operator : X → Y with T = T , whenever E ⊂ X. Does the same restriction on Y make T any operator T : E → Y extendable? The answer is negative in general. However, positive partial results can be obtained in the special case of Y = C(K), the space of continuous functions on a compact Hausdorff space K. For example, Lindenstrauss and Pełczyński proved the following: T HEOREM 1.8 ([45], Theorem 3.1). Let E be a subspace of c0 and let Y = C(K), for some compact Hausdorff space K. Then any operator T : E → Y admits, for every ε > 0, an extension T : c0 → Y with T (1 + ε)T . This result opens the door onto a new area. We start with the following: D EFINITION 1.9. Let X be a Banach space, let E be a subspace of X and let λ 1. We say that the pair (E, X) has the λ-C(K) Extension Property (λ-C(K) EP, for short) if for every compact Hausdorff space K, any operator T : E → C(K) admits an extension : X → C(K) with T λT . The pair (E, X) has the C(K) EP if it has the λ-C(K) T EP for some λ 1. The restriction of the range space to the family of C(K) spaces provides us with a simple but effective tool in the form of the following. E XTENSION C RITERION 1.10. Let X be a Banach space, let E be a subspace of X and let λ 1. The pair (E, X) has the λ-C(K) EP if and only if there is a ω∗ -ω∗ continuous function ϕ : Ball(E ∗ ) → λ Ball(X∗ ) which extends functionals (i.e., ϕ(e∗ )(e) = e∗ (e) for all e ∈ E and e∗ ∈ Ball(E ∗ )). The proof is elementary. To establish sufficiency, let T : E → C(K) be an operator with T = 1. Then the function ψT : K → Ball(E ∗ ) defined by ψT (k)(e) = (T e)(k), k ∈ K, is clearly ω∗ continuous. Hence ψ = ϕ ◦ ψT : K → λ Ball(X∗ ) is ω∗ continuous. De : X → C(K) by the equality (Tx)(k) = ψ(k)(x), then T is linear because, for fine T each k ∈ K, ψ(k) is a linear functional; T λ because ψ(k) ∈ λ Ball(X∗ ) and so extends T because ϕ extends functionals: if e ∈ E then ψ(k) λ and, finally, T (T e)(k) = ψ(k)(e) = (ϕ ◦ ψT (k))(e) = ψT (k)(e) = (T e)(k) for all k ∈ K. Conversely, assume that (E, X) has the λ-C(K) EP and put K = Ball(E ∗ ) under the ω∗ topology. Let T : E → C(K) denote the natural isometric embedding defined by (T e)(e∗ ) = e∗ (e) for all e∗ ∈ Ball(E ∗ ). Let T : X → C(K) denote an extension of T with T λT . Denoting by δe∗ the point evaluation functional on C(K) at e∗ , we define ϕ : Ball(E ∗ ) → λ Ball(X∗ ) by ϕ(e∗ ) = T∗ (δe∗ ). It is easy to check that ϕ is ω∗ -ω∗ continuous and extends functionals. An immediate consequence of the Extension Criterion 1.10 is

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C OROLLARY 1.11. Let 1 < p < ∞ and let E be a subspace of p . Then (E, p ) has the 1-C(K) EP. Indeed, the uniform convexity of the unit ball of q (where 1/p + 1/q = 1) yields the existence of a unique functional ϕ(e∗ ) on p which extends any non-zero functional e∗ ∈ Ball(E ∗ ) with ϕ(e∗ ) = e∗ . It is not hard to check that ϕ : Ball(E ∗ ) → Ball(q ) (with ϕ(0) = 0) is ω∗ -ω∗ continuous and hence, by the Extension Criterion, (E, p ) has the 1-C(K) EP. Very little is known about the C(K) EP. In view of the Extension Criterion, the C(K) EP depends on the behavior of the ω∗ topologies of Ball(E ∗ ) and Ball(X∗ ). On the other hand, as is demonstrated in Theorem 6.6 below, the “local” structure of E and X plays a role in connection with the C(K) EP, although this role is much less decisive than the role it plays in the case of the extension of compact operators.

(f) Extension of operators from subspaces of a space of type 2 into a space of cotype 2 One of the most elegant extension theorems is based on the notions of Gaussian type 2 and cotype 2. Let {ψi (t)}∞ i=1 denote a sequence of independent normalized Gaussian random variables on a probability space (Ω, μ). A Banach space X is said to be of Gaussian type 2 (respectively, cotype 2) if there is a constant M > 0 so that, for every finite sequence {xi } ⊂ X, the following inequality holds: 2 (1.7) xi 2 xi ψi (t) dμ(t) M 2 Ω

(respectively,

xi 2 M 2

2 xi ψi (t) dμ(t)).

(1.8)

Ω

γ

The Gaussian type 2 constant T2 (X) of X is the smallest M for which (1.7) holds. The γ Gaussian cotype 2 constant C2 (X) is the smallest M for which (1.8) holds. The notions of “Gaussian type” and “Gaussian cotype” are equivalent to those of “type” and “cotype”, respectively. These notions are considered in the article “Basic concepts in the geometry of Banach spaces” ([27], Section 8). For more details about these properties the reader is referred to [16], Chapter 11 and pages 249–251. A beautiful by-product of the concepts of Gaussian type 2 and cotype 2 is M AUREY ’ S EXTENSION THEOREM ([53]). Let X be a Banach space of Gaussian type 2 and let Y be a space of Gaussian cotype 2. Then there is a constant C = C(X, Y ) such that, for every subspace E of X, every operator S : E → Y admits an extension S :X → Y with S CS. An immediate corollary of Maurey’s Extension Theorem is the following: let X be a Banach space of type 2 and let E be a subspace of X which is isomorphic to a Hilbert


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space H . Then there is a bounded projection P of X onto E. This corollary is drawn from Maurey’s Extension Theorem by putting Y = E and extending the identity IE . A proof of Maurey’s Extension Theorem is given in Section 7. In most of the results described above, given the Banach spaces E ⊂ X and Y we are interested in extending members T : E → Y of a class of operators to operators T : X → Y . However, in many cases we may be satisfied with an extension of an individual operator T into a larger space. The following result suggests a “canonical” way of doing that. be Banach spaces, assume that X is a subspace of X, and L EMMA 1.12. Let X, Y , and X let T : X → Y be an operator. Then there exists a Banach space Y containing Y such that /Y and there is a norm preserving extension T : X →Y of T . Let q X/X is isometric to Y (resp. q) ˜ be the quotient map of X onto X/X (Y onto Y /Y , resp.), let j (resp. j˜) denote (Y to Y , resp.) then there is an isometry I so that the the natural embedding of X into X following diagram commutes: j q X −−−−→ X˜ −−−−→ X/X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ TA IA TA ˜

j −−−q˜−→ Y /Y Y −−−−→ Y

Moreover, if T is an isomorphism and T x γ x for all x ∈ X then T is an isomor phism with Tx ˜ γ x ˜ for all x˜ ∈ X. The detailed proof of the first part can be found in [16], p. 316. The “moreover” part is proved in [35]. Actually, the argument is very simple: we assume that T = 1, define = (X ⊕ Y )1 /W . We identify (X ⊕ {0})1 ⊕ Y )1 : x ∈ X} and put Y W = {(x, −T x) ∈ (X and Y with its isometric image ({0} ⊕ Y )1 /W . Let f : (X ⊕ Y ) 1 → Y be the quowith X = f |(X⊕{0}) . Since T(x, 0) ∈ (X ⊕ {0}1 /W ⊂ ({0} ⊕ Y )1 /W ∼ Y tient map, then put T 1 /Y so that I q = q˜ T . The norm infor every x ∈ X, there is a unique map I : X/X →Y equalities and the fact that I is an isometry are easy to check. Significant parts of the theory described below have been well presented in various books. When discussing these parts we state the results and refer the reader to the books in which detailed proofs are given. N OTATION . We use standard Banach space theory notation as can be found in [48] and [49]. In particular, the ω∗ topology on a bounded subset of a dual space X∗ is the σ (X∗ , X) topology. A net {xα∗ } ⊂ X∗ ω∗ -converges to x ∗ if and only if xα∗ (x) → x ∗ (x) for all x ∈ X. 2. The injective spaces The problem of characterizing the injective spaces seems to be very hard. However, two special cases are completely understood. The first one is the case of P1 spaces. Before stating the result, let us mention two properties used in the characterization of P1 spaces.

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A compact Hausdorff space K is called extremally disconnected if the closure of every open subset of K is open. As is well-known (see, e.g., Theorem 17 of [18]) K is extremally disconnected if, and only if, every non-void subset of C(K) which has an upper bound (with respect to the natural order of C(K)) has a least upper bound. The second relevant notion is the binary intersection property: a Banach space X has the binary intersection property if every family of mutually intersecting closed balls has a common point. T HEOREM 2.1. Let X be a Banach space over the real numbers then the following statements are equivalent: (a) X = C(K) where K is an extremally disconnected compact Hausdorff space. (b) X has the binary intersection property. (c) X is a P1 space. P ROOF. Let us start with the history of this result. Goodner [19] and Nachbin [56] independently proved the implication (a) ⇒ (c). Both proved also (c) ⇒ (a) under the assumption that Ball(X) has an extreme point. Nachbin established the equivalence (b) ⇔ (c). Finally, Kelley [37] settled (c) ⇒ (a) in full generality. We outline his argument here. (a) ⇒ (b) Let u denote the constant function 1 on K. A closed ball B(x, r) with center x and radius r in C(K) is exactly the order segment [x − ru, x + ru] = {y: x − ru y x + ru}. If A = [xα , yα ]α∈A is a collection of mutually intersecting segments then, for every α, β ∈ A, there is a zα,β such that xα , xβ zα,β yα , yβ . Hence {xα }α∈A is orderbounded from above and, by the preceding remark, x = sup{xα : α ∈ A} exists and is a common point for all segments in A. (b) ⇒ (c) By Zorn’s lemma, it suffices to show that if Z ⊃ Y, dim(Z/Y ) = 1 and : Z → X T : Y → X is an operator with norm T = 1 then T admits an extension T with T = 1. Let z ∈ Z \ Y and consider the family {B(T y, z − y): y ∈ Y } of balls in X. Any two of these balls intersect because, for y1 , y2 ∈ Y, T y1 − T y2 y1 − y2 z − y1 + z − y2 . Therefore there is a point e common to all balls of this family. Define T : Z → X by T(az + y) = ae + T y for all az + y ∈ Z. It is easily checked that T extends T and T = T = 1. (c) ⇒ (a) Let Ω0 denote the set of extreme points of Ball(X∗ ) equipped with the ω∗ ω∗ topology and put Ω = Ω 0 = the ω∗ closure of Ω0 . By Zorn’s lemma, there exists a ω∗ open subset K0 of Ω, maximal with respect to the property that −K0 ∩ K0 = ∅. Let ω∗ K = K 0 ; we claim that K is extremally disconnected and X = C(K). The claim easily follows from the following L EMMA 2.2. Suppose that X is a P1 space and let U and V be two ω∗ open subsets of Ω satisfying the two conditions U ∩ V = ∅ = −(U ∪ V ) ∩ (U ∪ V )

(2.1)

and −(U ∪ V ) ∪ (U ∪ V )

is ω∗ dense in Ω.

(2.2)

Extension of bounded linear operators ω∗

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ω∗

Let G = ({0} × U ) ∪ ({1} × V ) and let J : X → C(G) denote the natural isometric embedding (J (x)(g) = u(x) if g = (0, u) and J (x)(g) = v(x) if g = (1, v)). Finally, let E ⊂ Ball(C(G)∗ ) denote the set of point-evaluation functionals on C(G). Then J is a ω∗ ω∗ surjective isometry and J ∗ maps −E ∪ E ω∗ homeomorphically onto −(U ∪ V ) ∪ ω∗ ω∗ ω∗ ω∗ ω∗ ω∗ ω∗ ω∗ (U ∪ V ). Moreover, U ∩ V = ∅ = −(U ∪ V ) ∩ (U ∪ V ). Let us first show that Lemma 2.2 implies the claim preceding the statement of the lemma. The set −K0 ∪ K0 is weak∗ dense in Ω. Applying Lemma 2.2 to U = K0 and V = ∅ we get that −K ∩ K = ∅ and that K is weak∗ open in Ω. Moreover, the embedding J : X → C(K) is surjective. To see that K is extremally disconnected, pick a ω∗ open subset U of K and ω∗ ω∗ ω∗ ω∗ put V = K \ U . Lemma 2.2 implies that U ∩ V = ∅ hence U is open. P ROOF OF L EMMA 2.2. Clearly, G is a compact Hausdorff space under the topology ω∗ ω∗ induced by U and V . Let e(0, u) ∈ C(G)∗ and e(1, v) ∈ C(G)∗ denote the evaluation ω∗ ω∗ functionals at (0, u) and (1, v), respectively, for every u ∈ U and v ∈ V . Kelley’s argument consists of seven easily checked steps: ω∗ ω∗ (i) J ∗ e(0, u) = u and J ∗ e(1, v) = v for all u ∈ U and v ∈ V . (ii) By a standard extreme point argument, if u0 ∈ U ∩Ω0 and y ∗ is an extreme point of ω∗ (J ∗ )−1 (u0 ) ∩ Ball C(G)∗ then y ∗ = ±e(0, u) for some u ∈ U or y ∗ = ±e(1, v) ω∗ for some v ∈ V . Hence, by (i), u0 = J ∗ y ∗ = ±J ∗ (e(0, u)) = ±u and therefore y ∗ = e(0, u0 ). Similarly, if v0 ∈ V ∩ Ω0 and y ∗ is an extreme point of (J ∗ )−1 (v0 ) ∩ Ball C(G)∗ then y ∗ = e(1, v0 ). (iii) Let B denote Ball(C(G)∗ ). The Krein–Milman theorem and (ii) imply that −1 ∗ −1 (u0 ) ∩ B = e(0, u0 ) and J ∗ (v0 ) ∩ B = e(1, v0 ) J

(2.3)

for all u0 ∈ U ∩ Ω0 and v0 ∈ V ∩ Ω0 . Note that J ∗ |B is a ω∗ continuous function which maps B onto Ball X∗ and, in particular, maps the extreme points e(0, u0 ): u0 ∈ U ∩ Ω0 ∪ e(1, v0 ): v0 ∈ V ∩ Ω0 in a one-to-one fashion onto (U ∪ V ) ∩ Ω0 . The fact that X is a P1 space provides us with a ω∗ continuous inverse as follows: (iv) Since X is a P1 space there exists a projection P of C(G) onto J (X) with P = 1. Let S = J −1 P then S : C(G) → X is a surjective operator with S ∗ (Ball(X∗ )) ⊂ Ball(C(G)∗ ). Moreover, S ∗ is a norm isometry, J ∗ S ∗ = IX∗ and, in view of (2.3), we have that S ∗ u = e(0, u) and S ∗ v = e(1, v)

(2.4)

for all u ∈ U ∩ Ω0 and v ∈ V ∩ Ω0 . (v) S ∗ is a ω∗ homeomorphism of Ball(X∗ ) into Ball(C(G)∗ ) and it maps the ω∗ dense subset [−(U ∪ V ) ∪ (U ∪ V )] ∩ Ω0 of Ω onto a ω∗ dense subset of (−E) ∪ E. Since both Ω and (−E) ∪ E are weak∗ compact, S ∗ (Ω) = (−E) ∪ E.

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(vi) Let u ∈ U ∩ Ω0 and v ∈ V ∩ Ω0 then, by (i) and (2.4), S ∗ J ∗ (e(0, u)) = S ∗ u = e(0, u) and S ∗ J ∗ (e(1, v)) = S ∗ v = e(1, v). It follows that S ∗ J ∗ is the identity on a dense subset of (−E) ∪ E. Consequently, S ∗ is a ω∗ homeomorphism on Ω and J ∗ is, on (−E) ∪ E, the inverse homeomorphism, which maps E onto ω∗ ω∗ ω∗ ω∗ U ∪ V . Since {0} × U and {1} × V and, respectively, E and −E are ∗ ∗ ω ω ω∗ ω∗ ω∗ ω∗ ∗ disjoint ω compact subsets, U ∩ V = ∅ = −(U ∪ V ) ∩ (U ∪ V ). (vii) To see that J is surjective, note that S ∗ (Ball X∗ ) is convex and weak∗ compact and, by (vi), each extreme point of Ball(C(G∗ )) is in S ∗ (Ball X∗ ). Hence S ∗ (Ball X∗ ) = Ball(C(G∗ )) by the Krein–Milman theorem. A standard separation argument shows that J (X) = C(K). This proves Lemma 2.2. The equivalence (a) ⇔ (c) in the complex scalars case was established by Hasumi in [23]. Another case of a Pλ space which has a complete description is that of a finitedimensional Pλ space X with λ close to 1. It turns out that such a space is “close” to X , independently of the dimension. More precisely, dim ∞ T HEOREM 2.3 ([75,76]). There exists a positive function ϕ(λ), defined for 1 < λ < 1.001, such that, for every n 1 and 1 k n and for every projection P on n∞ with rank(P ) = k and P = λ < 1.001, the Banach–Mazur distance d(P (n∞ ), k∞ ) < ϕ(λ). Moreover, limλ→1 ϕ(λ) = 1. Most of the research on Pλ spaces revolves around the following C ONJECTURE 2.4. Every Pλ space is isomorphic to a C(K) space for some extremally disconnected compact Hausdorff space K. For general results on Pλ spaces the reader is referred to an excellent discussion in [48], pp. 190–194 and [49], I, pp. 105–106. The main topics considered there are summarized in T HEOREM 2.5. (a) ([61]) Let X be an infinite-dimensional Pλ space. Then X contains a subspace isomorphic to ∞ . Moreover, if X contains a subspace isomorphic to c0 (Γ ) for some infinite Γ then it contains also a subspace isomorphic to ∞ (Γ ). (b) ([62]) While every conjugate C(K) space X is a Pλ space (because then X is complemented in X∗∗ = L∞ (μ) which is a P1 space) there exist P1 spaces which are not isomorphic to dual spaces. (c) ([3]) Let X be a C(K) space which is also a Pλ space. Then K contains an extremally disconnected dense open subset G. (d) ([2,25]) If a C(K) space X is a Pλ space for λ < 2 then K is extremally disconnected. For more information about compact Hausdorff spaces K for which C(K) is a Pλ space the reader is referred to [2–4] and [70–72]. The research on the injectivity property restricted to the class of Banach lattices has been more successful because of the extra structure at hand.


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D EFINITION 2.6. A Banach lattice X is called lattice injective (L-injective, in short) if it is complemented by a positive projection in every Banach lattice containing it as a sublattice. The lattice X is called (λ-L)-injective if, for every Banach lattice Y containing X as a sublattice, there is a positive projection P of Y onto X with P λ. Lotz started the research on L-injective lattices and showed ([51]) that every L-injective Banach lattice is (λ-L)-injective for some λ 1. He proved also that, in addition to P1 spaces, the class of (1-L)-injective lattices contains the spaces L1 (μ) and is closed under ∞ -direct sums. Cartright [13] characterized the (1-L)-injective lattices by using the following notion: D EFINITION 2.7. A Banach lattice X is said to have the splitting property if for every positive elements x1 , x1 and y and positive numbers r1 and r2 which satisfy the inequalities xi ri and x1 + x2 + y r1 + r2 there exist positive y1 and y2 in X with y1 + y2 = y such that xi + yi ri for i = 1, 2. Cartwright’s characterization is the following T HEOREM 2.8 ([13]). A Banach lattice X is (1-L)-injective if, and only if, the following two conditions hold: (a) there is a positive projection P of X∗∗ onto X with P = 1. (b) X has the splitting property. He used these ideas to prove that every finite-dimensional (1-L)-injective lattice is one m(j ) of the spaces ( nj=1 ⊕1 )∞ . Haydon proved in [24] a general representation theorem for (1-L)-injective lattices which uses vector bundles. Unfortunately these isometric tools are not suitable for the isomorphic classification problem of (λ-L)-injective lattices. That problem was solved in [50] in the finite-dimensional case as follows: T HEOREM 2.9. There is a function ϕ(λ) 216 λ27 such that every finite-dimensional (λ-L)-injective Banach lattice X is order isomorphic to a (1-L)-injective Banach lattice Y with d(X, Y ) ϕ(λ). A Banach lattice is called discrete if it coincides with the band generated by its atoms. Lindenstrauss and Tzafriri extend the finite-dimensional methods and prove in [50] also that every discrete L-injective Banach lattice is isomorphic to a (1-L)-injective one. In particular, T HEOREM 2.10. Let X be a discrete injective Banach lattice with countably many atoms. Then X is order isomorphic to one of the following six lattices ∞ , 1 , ∞ ⊕ 1 ,

∞ n=1

⊕n1

, 1 ⊕ ∞

∞ n=1

⊕n1

,

∞

∞ n=1

⊕1

. ∞

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The case of injective order continuous lattices was considered by Mangheni [52]. He proved T HEOREM 2.11. Let X be an injective order continuous lattice. Then X is order isomorphic to an L1 (μ) space.

Remarks and open problems R EMARK 2.12. A different approach to the equivalence (a) ⇔ (c) of Theorem 2.1 can be found in [36] and [39]. The nature of a finite-dimensional Pλ space is a fascinating mystery. The precise statement of the problem is the following: P ROBLEM 2.13. Does there exist a function ϕ(λ) such that, for every n 1, n > k 1 and every projection P on n∞ with rank(P ) = k and P = λ, d(P (n∞ ), k∞ ) ϕ(λ)? A small indication of a positive solution is the result, independently proved by Bourgain [10] and Johnson and Schechtman [29], which states that there is a C(λ) 1 such that every finite-dimensional Pλ space E contains a subspace F of dim(F ) C(λ)−1 dim(E) ) such that d(F, dim(F ) C(λ). ∞ There are some indications that the problem may be easier to handle if the matrix representing P is an orthogonal projection on n2 . This leads to P ROBLEM 2.14. Does there exist a function ϕ(λ) such that for every n 1 and every projection P of n∞ there is an orthogonal projection Q in some N ∞ space such that Q ϕ(λ) and d(P (n∞ ), Q(N )) ϕ(λ)? ∞ The isomorphic identity of the finite-dimensional Pλ spaces is unknown even in some natural special cases. P ROBLEM 2.15. Does there exist a function ϕ(λ) such that, for every finite Abelian group G and any translation invariant projection P on ∞ (G) with P = λ and rank(P ) = k, d(P (∞ (G)), k∞ ) ϕ(λ)? 3. Separably injective spaces The purpose of this section is to present a solution of the characterization problem of separable separably injective spaces and discuss a generalization of this property to nonseparable spaces. It was proved by Sobczyk [66] in 1941 that c0 is separably injective. An elegant proof of this fact, due to Veech [69], is presented above in the introductory article [27]. We will prove the converse, namely,


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T HEOREM 3.1. Let X be a separable separably injective space. Then X is isomorphic to c0 . The proof of Theorem 3.1 is based on the knowledge accumulated in Banach space theory in the sixties and early seventies. It heavily relies on the following facts which are listed in the order in which they are used in the following argument. FACT (a). Every infinite-dimensional quotient space of c0 is isomorphic to a subspace of c0 ([31]). FACT (b). Every infinite-dimensional complemented subspace of c0 is isomorphic to c0 ([57]). FACT (c). The spaces C[0, 1] and C(ωω ) are not separably injective ([2]). FACT (d). A complemented subspace of C[0, 1] which has a non-separable dual is isomorphic to C[0, 1] ([63]). FACT (e). If a complemented subspace E of C[0, 1] has a separable dual E ∗ then E ∗ is isomorphic to 1 ([40]). FACT (f). Let X be a Banach space with a separable dual. If X∗ has a basis then X has a basis such that its biorthogonal functionals form a basis of X∗ ([28]). FACT (g). Let X be a separable space and let W be a subspace of X which is isomorphic to C(ωω ). Then W contains a subspace W0 which is complemented in X and is isomorphic to C(ωω ) ([59]). We prove the theorem by embedding the given separably injective space X into a C(F ) space in a very special position, which makes X (non-linearly) “close” to a certain subspace of C(F ) which is isomorphic to c0 . Our first step will be to show that the existence of this embedding, described in the next lemma, implies Theorem 3.1. L EMMA 3.2. Let X be a γ -separably injective space and let 0 < ε < (8γ )−1 . Then there exist a compact metric space F , a subspace Y of C(F ) which is isomorphic to c0 and an embedding i of X into C(F ) such that (1 − ε)x i(x) x for all x ∈ X. Moreover, for each x ∈ X there is a y ∈ Y with i(x) − y εi(x). Step 1. Lemma 3.2 implies Theorem 3.1 P ROOF. Since X is γ -separably injective, then, with μ = (1 − ε)−1 , i(X) is μγ -separably injective and hence there is a projection P of C(F ) onto i(X) with P μγ . Let Q = P |Y ; let 0 = x ∈ X and pick y ∈ Y such that i(x) − y εi(x). Then Qy − i(x) = P y − P i(x) P εi(x) εμγ i(x). But μγ ε < 1/2, therefore, clearly, Q maps Y

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M. Zippin

onto i(X). Hence X is isomorphic to a quotient space of c0 , and, by Fact (a), X is isomorphic to a subspace of c0 . Since X is separably injective, it is isomorphic to a complemented subspace of c0 and, by Fact (b), X is isomorphic to c0 , as claimed in Theorem 3.1. It thus remains to prove Lemma 3.2, a long argument which is divided into the next six steps. Step 2. The construction of F The space X, being separably injective, is a complemented subspace of C[0, 1] which, by Fact (c), cannot be isomorphic to C[0, 1]. It then follows from Fact (d) that X∗ is separable, and according to Fact (e), this dual space is isomorphic to 1 . Hence there is a ∗ constant ν > 0 (which depends on X) and a basis {ϕn }∞ n=1 of X such that ν −1

∞ n=1

∞ ∞ |an | a n ϕn |an | n=1

(3.1)

n=1

∞ for all real {an }∞ n=1 . We now use Fact (f) which states that X has a normalized basis {xn }n=1 ∞ with basis constant M, say, and with biorthogonal functionals {θn }n=1 which form a basis of X∗ . Following ([6], Proposition 1), given 0 < δ < εM −1 we define, for every n 1, Mn = 2(n+2)δ −1 M, Cn = {j/Mn : j is an integer, −Mn < j Mn }, Hn = {θ ∈ Hn is ω∗ Ball(X∗ ): θ (xj ) ∈ Cj for all 1 j n} and F = ∞ n=1 Hn . Since each set ∞ ∗ compact so is F . Moreover, F is a δ net in Ball(X ). Indeed, let ψ = i=1 ai θi ∈ (1 − δ/2)Ball(X∗ ) then |ai | = |ψ(xi )| 1 − δ/2 for all i 1. Fix n 1 and choose the integer −Mn < j (n) Mn for which (j (n) − 1)Mn−1 < an = ψ(xn ) j (n)Mn−1 . Then ∞ ∞ ∞

−1 −1 j (n)M θ j (n)M θ − a − a 2M Mn−1 δ/2. n n n n n n n=1

n=1

n=1

∞

Hence θ = n=1 j (n)Mn−1 θn ∈ Ball(X∗ ), θ − ψ δ/2 and, because θ (xn ) = j (n)Mn−1 for all n 1, θ ∈ F . Equipped with the ω∗ topology, F is a compact metric space and, for each x ∈ X (3.2) (1 − δ)x Sup ψ(x): ψ ∈ F x. It follows that the embedding i : X → C(F ), defined by (ix)(ψ) = ψ(x), satisfies the desired inequality (1 − ε)x i(x) x for x ∈ X. (3.3) Step 3. The topological properties of F shown above, {Ci }∞ i=1 is a family of finite sets of numbers and, for each ψ = As ∞ a θ ∈ F, a = ψ(x ) ∈ Ci for all i 1. Suppose that n 1 and ai ∈ Ci for 1 i n. i i i i i=1 Put A(a1 , a2 , . . . , an ) = θ ∈ F : θ (xi ) = ai for all 1 i n .


1719

Then A(a1 , a2 , . . . , an ) is a clopen subset of F . The collection A = {A(a1, a2 , . . . , an ): n 1, ai ∈ Ci for 1 i n} is clearly a base for the ω∗ topology of F and satisfies the following conditions: F=

A(a1 )

and A(a1 , a2 , . . . , an ) =

a1 ∈C1

A(a1 , a2 , . . . , an , an+1 )

an+1 ∈Cn+1

(3.4) and

if an+1 = an+1 are in Cn+1

then

A(a1, a2 , . . . , an , an+1 ) ∩ A a1 , a2 , . . . , an , an+1 = ∅.

(3.5)

This establishes the fact that F is a Cantor set. Step 4. The Szlenk index of F Let 0 < δ < 1. Following [67] we assign to each ordinal α, by transfinite induction, a set Fα (δ) as follows: F0 (δ) = F and Fα+1 (δ) = {γ ∈ X∗ : there exists a sequence {yn }∞ n=1 in Ball(X) and γn ⊂ Fα (δ) with ω∗ -lim γn = γ , ω-lim yn = 0 and limn→∞ γn (yn ) δ}. If α is a limit ordinal then Fα (δ) = β (M + 1)δ n and there is a γn = ∞ i=1 bi θi ∈ A(a1 , . . . , an ) ∩ Fk for which γn − θ 2(M + 1)δ. Because both γn and θ are in A(a1, . . . , an ), bin = ai for all 1 i n. Hence ∞

2(M + 1)δ γn − θ bin − ai θi . ∞

i=n+1

Pick zn = ∈ Ball(X) such that (γn − θ )(zn ) > (3/2)(M + 1)δ, then ∞ n n i=n+1 ci xi M + 1 and, hence, if yn = (M + 1)−1 ∞ i=n+1 ci xi then yn 1 and −1 |(γn −θ )(yn )| = (M + 1) |(γn − θ )(zn )| (3/2)δ. Moreover, because {θi )∞ i=1 is a basis, lim ∞ a θ = 0 and, therefore, if n is large enough then i i i=n+1 ∞ γn (yn ) (γn − θ )(yn ) − θ (yn ) (3/2)δ − ai θi (yn ) δ. n i=1 ci xi

i=n+1

Finally, because θi (yn ) = 0 for n > i, ω-lim yn = 0. We have thus constructed a sequence ∞ ∗ (γn )∞ n=1 ∈ Fk with θ = ω -lim γn and a sequence {yn }n=1 ⊂ Ball(X) with ω-lim yn = 0 such that γn (yy ) δ. This means, by definition, that θ ∈ Fk+1 − a contradiction which proves our Claim. The Claim implies that, for each 0 k N , the set Fk − Fk+1 can be covered by a k union of a sequence {Aki }∞ i=1 of members of A for which diam(Ai ∩ Fk ) 4(M + 1)δ. Moreover, because of the set theoretical relations among the members of A, for every k , Ak ⊂ A k ∈ A with Aki ∈ A with diam(Aki ∩ Fk ) 4(M + 1)δ, there is a maximal set A i i i k and A k , say, are either k ∩ Fk ) 4(M + 1)δ. Any two such maximal sets, A diam(A i i j disjoint or identical and the union of these maximal sets covers Fk − Fk+1 . Hence we may assume, without loss of generality, that for each 0 k N there is a sequence {Aki }∞ i=1 of members of A such that the following conditions hold: ∞

Aki ⊃ Fk − Fk+1 ,

(3.9)

i=1

diam Aki ∩ Fk 4(M + 1)δ

for all i 1,

(3.10)

Aki ∩ Akj = ∅ if i = j

(3.11)

for each i 1, Aki is a maximal member of A which satisfies (3.10).

(3.12)

and


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Since FN = FN − FN+1 is ω∗ compact, we may assume that {AN i } is a finite sequence. We now claim that if h < k and Ahi ∩ Akj = ∅ then Ahi ⊂ Akj .

(3.13)

Indeed, since Fh ⊃ Fk , diam(Ahi ∩ Fk ) diam(Ahi ∩ Fh ) 4(M + 1)δ. Hence Ahi ⊂ Akj because, by (3.12), Akj is maximal. Let us denote Ahi ∩ Fh by Bih for all 1 h N and i ∞ h 1 and define G0 = F, GN+1 = ∅ and, for 1 k N, Gk = Fk − k−1 i=1 Bi . Clearly, h=1 N Gk ⊃ Gk+1 for all 0 k N and F = k=0 (Gk − Gk+1 ). Moreover, since Fh ⊃ Fk whenever h < k we have that Gk = Fk −

k−1 ∞

k−1 ∞

h

Ahi ∩ Fh = Fk − Ai ∩ Fk .

h=1 i=1

h=1 i=1

Because Ahi ∩ Fk are relatively open in Fk , Gk is a closed set. We claim that ∞

Aki ∩ Gk ⊃ Gk − Gk+1

for all 0 k N.

(3.14)

i=1

∞ h k / k−1 Bi . We must show that θ ∈ ∞ Indeed, let θ ⊂ Gk − Gk+1 then θ ∈ i=1 i=1 Ai . h=1 ∞ k k ∞ ∞ k / i=1 Bi and therefore, θ ∈ / Bh. Assuming that θ ∈ / i=1 Ai we know that θ ∈ k ∞h=1 h i=1 i But, by (3.9), θ ∈ Gk − (Fk − Fk+1 ) ⊂ Fk+1 . Hence, θ ∈ Fk+1 − h=1 i=1 Bi = Gk+1 – a contradiction which proves (3.14). Finally, if h < k we have that Ahj ∩ Gk = Ahi ∩ Fh ∩ Gk = Bih ∩ Gk = ∅.

(3.15)

Let B = {Aki ∩ Gk : 0 k N, i 1}. We have just proved that B is a decomposition of F into pairwise disjoint closed sets satisfying the condition diam(Aki ∩ Gk ) 4(M + 1)δ for all i 1 and 0 k N . We define Y = {y ∈ C(F ): y is a constant function on each set Aki ∩ Gk , 0 k N, i 1}. Step 6. i(X) is close to Y In view of Lemma 3.2, we must prove that for every x ∈ X, if i(x) = 1 then there is a y ∈ Y with y − i(x) < ε. In view of (3.10), assuming that 4(M + 1)δ < ε, it suffices to prove that, if f (θ ) ∈ C(F ) satisfies sup oscillationAk ∩Gk f (θ ) < ε, i,k

i

(3.16)

then there is a y ∈ Y with y − f < ε. This statement is a consequence of the Approximation Lemma of [74].

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Step 7. Y is isomorphic to c0 Pick a point ωik from each set Aki ∩ Gk and define ωˆ ik in Y ∗ by ωˆ ik (y) = y(ωik ) for = ωˆ k : ωk ∈ Ω. Because each all y ∈ Y . Now let Ω = {ωik ; 0 k N, i 1} and Ω i i k k y ∈ Y is constant on Ai ∩ Gk , the definition of ωˆ i does not depend on ωik . Clearly y = for all y ∈ Y . We claim that Ω is a closed set in the ω∗ topology sup{|ω(y)|: ˆ ωˆ ∈ Ω} k(n) ∗ ∗ ∗ ∗ of Y . Indeed, if ω lim ωˆ i(n) = y ∈ Y , by passing to a subsequence, we may assume that lim ωi(n) = ω in F , hence, y(ω) = y ∗ (y) for all y ∈ Y . Moreover, in view of (3.14), The space Ω is a countable ω ∈ Aki ∩ Gk for some k and i and therefore y ∗ = ωˆ ik ∈ Ω. ∗ ∗ ω compact subset of Y and the map j : Y → C(Ω) defined by (j (y))(ωˆ ik ) = ωˆ ik (y) is an which separates points and isometric embedding. Moreover, j (Y ) is a subalgebra of C(Ω) In order to prove that C(Ω) ∼ c0 it suffices to show, contains the unit hence j (Y ) = C(Ω). is by the classical Mazurkiewicz–Sierpinski theorem [54], that the (N +1)-derived set of Ω (k) empty. To establish this fact we will prove the following: if ωˆ ∈ Ω (= the k-th derivative then there is a θ ∈ Gk such that ωˆ = θˆ . Since the number of sets AN is finite, it would of Ω) i (N+1) = ∅. The assertion is clearly true for k = 0 because G0 = F . then follow that Ω (k+1) be a limit of a sequence (ωˆ k(n) ) Assume for k and proceed by induction. Let ωˆ ∈ Ω i(n) k(n) (k) of distinct points of Ω . We may assume, by the induction hypothesis, that (ω ) ⊂ Gk k(n)

i(n)

k(n) and, by passing to a subsequence, we may assume that limn ωi(n) = θ ∈ F . Therefore, for every y ∈ Y, y(θ ) = ω(y). ˆ Since Gk is closed, θ ∈ Gk . But θ ∈ / Gk − Gk+1 ; indeed, if θ ∈ Gk − Gk+1 then, by (3.14), θ ∈ Aki ∩ Gk for some i. The set Aki is clopen, hence k(n) k(n) ωi(n) ∈ Aki ∩ Gk for large enough n and so, ωˆ ik = ωˆ i(n) eventually, contradicting the fact k(n)

that the points ωˆ i(n) are distinct. It follows that θ ∈ Gk+1 and the assertion is proved. This completes the proof of Lemma 3.2 and thus Theorem 3.1 is proved. Rosenthal [64] has recently suggested the following generalization of separable injectivity which is very natural, in view of Sobczyk’s approach and Proposition 1.2, and opens the door to new problems. Let λ 1. The space E is said to have the λ-Separable Extension Property (λ-SEP, in short) if it satisfies (1.6). Note that this definition makes sense for non-separable spaces E. An example of a 1-SEP space which is not injective is the following: let Γ be an uncountable set and let E = C ∞ (Γ ) = the subspace of ∞ (Γ ) containing all countably supported functions on Γ . E has the 1-SEP because, if X is separable and T : X → C ∞ (Γ ) is any operator, then, clearly, there is a countable Γ0 ⊂ Γ such that T (X) is supported on Γ0 . Since ∞ (Γ0 ) is 1-injective, it follows that, for every Y ⊃ X, T admits an extension C C T : Y → ∞ (Γ0 ) ⊂ C ∞ (Γ ). To see that ∞ (Γ ) is not injective, note that c0 (Γ ) ⊂ ∞ (Γ ). If P is a projection of ∞ (Γ ) onto E then P |c0 (Γ ) is the identity and hence, by [61] there is a subset Γ ⊂ Γ with the same cardinality so that P |∞ (Γ ) is an isomorphism into E. Obvious density character considerations lead to a contradiction. The following are recent results of Rosenthal [64] which generalize Sobczyk’s theorem on the separable injectivity of c0 .


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T HEOREM 3.5. (a) Let {Ej }∞ and let X and Y be j =1 be a sequence of 1-injective spaces ∞ Banach spaces so that X ⊂ Y and Y/X is separable. Put E = ( 1 ⊕Ej )c0 and let ε > 0. Then every operator T : X → E extends to an operator T : Y → E with T (2 + ε)T . (b) Let Z ⊃ E and Z/E be separable. Then there is a projection P of Z onto E with P 2 + ε. ∞ (c) Let {Fj }∞ j =1 be a sequence of spaces with the λ-SEP. Then F = ( j −1 ⊕Fj )c0 has the (λ2 + λ + ε)-SEP for every ε > 0. R EMARK 3.6. Rosenthal originally proved (c) with λ = 1. The extension to λ > 1 is an observation of Oikhberg (see the remark following the proof of Theorem 1.5 of [64]). 4. Extension of compact operators While extension theories for general bounded operators leave basic questions unanswered, once we restrict our attention to the extension of compact operators we arrive at a remarkably complete theory [42]. This study, inspired by earlier work of Grothendieck ([20,21]), establishes beautiful relations among three fields: (a) Extension of compact operators. (b) The local structure of the domain space (or, respectively, the range space). (c) Intersection properties of balls in the domain space (or, respectively, the range space). Let us start by explaining what we mean by “local structure” and what special kind of local structure is relevant here. The work of Grothendieck in the fifties directed attention to the role that is played in Banach space theory by finite rank operators and finitedimensional subspaces. These ideas, developed in the sixties by Lindenstrauss, Pełczyński and Rosenthal ([44,46]) culminated in the theory of Lp spaces. Lindenstrauss was the pioneer of this approach in Banach space theory and completed Grothendieck’s work on the connection between extension of compact operators and the L∞ spaces. A Banach space X is a L∞,λ space if it is paved by a family of finite-dimensional spaces {Xα }α∈I α) with d(Xα , dim(X ) < λ. It is called a L∞ space if it is a L∞,λ space for some λ > 1. We ∞ have seen the role of the binary intersection property of balls in the theory of P1 spaces. Lindenstrauss showed that weaker intersection properties of balls of a space X are equivalent to the existence of compact norm preserving extensions of compact operators from X (or, respectively, almost norm preserving compact extensions of compact operators into the space X). Recall that a Banach space X is said to have the 4-2 intersection property if every collection of four mutually intersecting balls has a non-empty intersection. The main result of [42] is the following fundamental characterization of compact operators extension properties (Y and Z denote Banach spaces). T HEOREM 4.1. The following statements on a Banach space X are equivalent: (4.1) X∗∗ is a P1 space. (4.2) X∗ is an L1 (μ) space. (4.3) Let Z ⊃ Y and let ε > 0. Then every compact operator T : Y → X admits a compact extension T : Z → X with T < (1 + ε)T .

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(4.4) Let Z ⊃ Y and let dim Y 3, dim(Z/Y ) = 1 and ε > 0. Then every operator T : Y → X admits an extension T : Z → X with T < (1 + ε)T . : Z → X∗∗ (4.5) Let Z ⊃ Y . Then every operator T : Y → X admits an extension T with T = T . (4.6) Let Z ⊃ Y and assume that Ball(Z) = conv{Ball(Y ), F } where F ⊂ Z is a finite set. Then every operator T : Y → X admits an extension T : Z → X with T = T . (4.7) Let Z ⊃ X and let Y be a dual space. Then every operator T : X → Y admits an : Z → Y with T = T . extension T (4.8) Let Z ⊃ X then every compact operator T : X → Y admits a compact extension T : Z → Y with T = T . (4.9) Let Z ⊃ X then every weakly compact operator T : X → Y admits a weakly : Z → Y with T = T . compact extension T (4.10) Let Z ⊃ X and assume that dim(Z/X) = 1 and dim Y 3. Then every operator T : X → Y admits an extension T : Z → Y with T = T . (4.11) The space X has the 4-2 intersection property. (4.12) Every family of mutually intersecting balls {B(xα , rα }α∈A where {xα }α∈A is conditionally norm compact has a non-empty intersection. (4.13) X is a L∞,1+ε space for every ε > 0. The equivalences (4.1) ⇔ (4.2) ⇔ (4.3) ⇔ (4.5) were proved by Grothendieck ([20, 21]). The rest of the equivalences are proved in [42], where the reader may find a detailed study of the solution. A summary can be found in [48], pp. 157–160. The implication (4.1) ⇒ (4.13) is a consequence of the principle of local reflexivity ([16], p. 178) and Theorem 2.1. The converse implication follows by a ω∗ compactness argument (see [42], pp. 12–13) from the fact that, by (4.13), X is paved by a family {Xα }α∈I of subspaces, with n(α) n(α) = dim Xα < ∞ and limα d(Xα , ∞ ) = 1. Hence, whenever X∗∗ ⊂ Y there is a net {Pα } of projections of Y into X∗∗ with limα Pα = 1 which converges to a projection of Y onto X∗∗ . In the isomorphic setting there is a similar characterization of spaces whose second duals are Pλ spaces in terms of extension properties. T HEOREM 4.2. Let X be a Banach space. Then the following seven assertions are equivalent. Moreover, the validity of (4.14) for some λ 1 is equivalent to the statement “X∗ is a L1 space”. (4.14) X∗∗ is a Pλ space. : Z → X∗∗ with (4.15) Let Z ⊃ Y then every operator T : Y → X admits an extension T T λT . (4.16) Let Z ⊃ X and assume that Y is a dual space. Then every operator T : X → Y : Z → Y with T λT . admits an extension T (4.17) Let Z ⊃ Y and let ε > 0. Then every compact operator T : Y → X admits a : Z → X with T (λ + ε)T . compact extension T (4.18) Let Z ⊃ Y, let dim(Z/Y ) < ∞ and let ε > 0. Then every compact operator T : Y → X admits an extension T : Z → X with T (λ + ε)T .


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(4.19) Let Z ⊃ X then every compact operator T : X → Y admits a compact extension : Z → Y with T λT . T (4.20) Let Z ⊃ X then every weakly compact T : X → Y admits a weakly compact extension T : Z → Y with T λT . (4.21) Let Z ⊃ X and let Y be of finite dimension. Then every operator T : X → Y admits an extension T : Z → Y with T λT . The implications (4.14) ⇔ (4.15) ⇔ (4.16), (4.17) ⇒ (4.18) ⇒ (4.14), (4.16) ⇒ (4.19) and (4.16) ⇒ (4.20) are proved in [42], pp. 12–15. The rest of the implications were established by Johnson (private communication). Johnson’s arguments are presented below. The “Moreover” part is a straightforward consequence of Theorems I and III of [46] and the principle of local reflexivity. P ROOF OF (4.14) ⇒ (4.17). Embed X∗∗ in a large enough ∞ (Γ ) space W . Then there is a projection P of W onto X∗∗ with P λ and W has the Bounded Approximation Property (BAP). By local reflexivity, X has the BAP and hence, given a compact T : Y → X and ε > 0, there exists a sequence {Tn }∞ n=1 of operators of finite rank, Tn : Y → X, such that ∞ n : Z → W T = n=1 Tn and Tn < T + 12 ελ−1 . Extend each Tn to an operator T −(n+2) −1 ∗∗ of finite rank with Tn < Tn + ε2 λ . Then P Tn : Z → X extends Tn and has norm P Tn λTn λTn + ε2−(n+2) . The principle of local reflexivity yields the existence of an operator Tn : Z → X of finite rank which extends Tn and satisfies Tn < = ∞ λTn + ε2−(n+1) . The operator T is clearly the desired extension. T n n=1 Clearly, each of the two assertions (4.19) and (4.20) formally implies (4.21). It remains to discuss T HE PROOF OF (4.21) ⇒ (4.14). Let Z denote any C(K) space which contains X isometrically. For any finite-dimensional subspace E of X∗ , let JE : E → X∗ denote the natural embedding and let TE : X → E ∗ be the operator defined by TE (x)(e) = e(x) for all e ∈ E and x ∈ X. Then, clearly, TE∗ = JE . Use (4.21) to obtain an extension TE : Z → E ∗ with TE λTE = λ. Define JE = (TE )∗ then JE is a lifting of JE to the L1 (μ)space Z ∗ . If Q denotes the TE

X Z

TE

∗ E

JE

E

X∗

Q ∗

JE Z

natural quotient map of Z ∗ onto X∗ then JE = QJE . We now extend both JE and JE to (non-linear) maps on the whole space X∗ by putting both JE (x ∗ ) = 0 and JE (x ∗ ) = 0 for all x ∗ ∈ X∗ \ E. Consider the set E = {E ⊂ X∗ : E a finite-dimensional subspace} directed by inclusion. The net {JE }E∈E is pointwise convergent to the identity on X∗ . The standard, well known ω∗ -compactness argument (described in detail in Theorem 5.1 below) yields the existence of a subnet {JE }E∈Δ which converges ω∗ -pointwise to a linear operator T : X∗ → Z ∗ with T λ. Since Q is the dual of the natural embedding of X

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into Z, it is ω∗ continuous. Therefore the equality JE = QJE for all E ∈ E implies that IX∗ = QT . Hence X∗ is λ-complemented in the L1 (μ)-space Z ∗ and therefore X∗∗ is λ-complemented in L∞ (μ) and is thus a Pλ space. R EMARK 4.3. It is shown in [42] that each of (4.14)–(4.20) implies (4.22) Let Z ⊃ X, let dim(Z/X) < ∞ and let ε > 0. Then every compact operator T : X → X admits a compact extension T : Z → X with T (λ + ε)T . The statement (4.22) formally implies (4.23) with X, Z and ε > 0 as above, every uniform limit T : X → X of finite rank operators admits a compact extension T : Z → X with T (λ + ε)T . While it is unknown whether (4.22) implies (4.14), Pisier constructed in Theorem 3.7 of [60] an example of a separable space X which fails the approximation property (and hence does not satisfy (4.14)) but every uniform limit T : X →X of finite rank operators on X ∞ ∗ ∗ admits a nuclear representation T = ∞ x i=1 i ⊗ xi with i=1 xi xi λT . Clearly, ∞ if zi∗ ∈ Z ∗ is a Hahn–Banach extension of xi∗ then T = i=1 zi∗ ⊗ xi is an extension of T , T : Z → X and T λT . Consequently, X satisfies (4.23). R EMARK 4.4. A separable space X whose second dual is a P1 space is a precise quotient space of C(Δ) where Δ = Cantor’s set ([30]) and contains c0 ([73]). Bourgain and Delbaen ([11]) constructed, for every λ > 1, a separable space Y with Y ∗∗ a Pλ space which is not a quotient space of a C(K) space and does not contain c0 . Bourgain and Pisier ([12]) showed that for every λ > 1 and every separable space E there is a separable L∞,λ space X such that X/E has the RNP and the Schur property. Thus a space whose second dual is a Pλ space need not be isomorphic to a space whose second dual is a P1 space.

5. Lifting of operators and extension of isomorphisms to automorphisms We mentioned in the Introduction the “lifting property” of 1 (Γ ): for every pair of Banach spaces X and Y such that there is a surjective operator q : Y → X, every operator T : 1 (Γ ) → X can be “lifted” through Y , i.e., there is an operator T : 1 (Γ ) → Y such = T . We start with a generalization of the lifting property of 1 (Γ ) due to Lindenthat q T strauss [41], who introduced a beautiful idea of using ω∗ compactness and the abundance of “good”, finite rank linear operators to prove the linearity of a certain map, which is originally defined by a composition with a non-linear function. T HEOREM 5.1. Let Y and X be Banach spaces such that there is a surjective operator q : Y → X. Assume that kernel(q) is complemented in its second dual. Let E be any : E → Y such that L1 space. Then every operator T : E → X admits a lifting operator T qT = T . P ROOF. Let {Eα }α∈A denote a net, directed by inclusion, of finite-dimensional subspaces Eα ) = γ0 < ∞. Let Tα = T |Eα then, of E such that α∈A Eα = E and supα d(Eα , dim 1 dim Eα α : Eα → Y such that implies the existence of an operator T the lifting property of 1


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α = Tα and Tα γ T where γ is any number greater than γ0 . Because q is surqT jective, there is a homogeneous mapping ϕ : X → Y (which is not necessarily linear or continuous) such that qϕ(x) = x for all x ∈ X and ϕ(x) μx, where μ depends on q only. Let K = Kernel(q) and, for every r > 0 let B(r) = r Ball(K ∗∗ ). Equipped with the ω∗ topology induced by K ∗ , B(r) is compact and hence, by the Tychonoff theo* rem, the product Π = e∈E B((γ + μ)T e) is compact. This ω∗ compactness implies that the net πα (e) defined, for every α ∈ A and e ∈ E, by πα (e) = Tα (e) − ϕ(T (e)) if e ∈ Eα and πα (e) = 0 otherwise, has a limit point π(e). Let P be the assumed projection is the deof K ∗∗ onto K and define T by Te = P π(e) + ϕ(T (e)). Let us show that T sired operator. For every α ∈ A and e ∈ E, πα (e) = Tα (e) − ϕ(T (e)) ∈ K and therefore πα (e) ∈ K ∩ B((γ + μ)T e). Therefore, the limit point π(e) ∈ B(γ + μ)T e and P π(e) ∈ K ∩ B((γ + μ)P T e). It follows that T (γ + 2μ)P T e for all e ∈ E. Clearly, T is homogeneous and it remains to show that T is additive. Because α∈A Eα is dense in E and the family {Eα } directed by inclusion, it suffices to show that (f ). For every α such that for every e, f in some subspace Eα0 , T(e + f ) = T(e) + T α (e + f ) − Tα (e) − Tα (f ) = 0 and ϕ(T (e + f )) − ϕ((T (e)) − ϕ(T (f )) ∈ K Eα ⊃ Eα0 , T hence, πα (e + f ) − πα (e) − πα (f ) = −[ϕ(T (e + f )) − ϕ(T (e)) − ϕ(T (f ))]. Passing to the limit, we get that

π(e + f ) − π(e) − π(f ) = − ϕ T (e + f ) − ϕ T (e) − ϕ T (f ) ∈ K. Therefore, since P is a projection onto K,

P π(e + f ) − P π(e) − P π(f ) = − ϕ T (e + f ) − ϕ T (e) − ϕ T (f ) which proves the additivity of T.

The following variation of the lifting property for abstract L-spaces which is an easy consequence of [40] is presented in [35]. P ROPOSITION 5.2. Let Y and X be Banach spaces such that there is a surjective operator q : Y → X. Assume that X has the Radon–Nickodym property. Then, for every abstract =T. L-space E, every operator T : E → X admits an operator T : E → Y such that q T The rest of this section is presented in detail in [49], I, pp. 108–111. We will only state the results, which were originally proved in [47]. It turns out, in response to Problem 1.7 of the Introduction, that every isomorphism between two quotient spaces of 1 which are not isomorphic to 1 can be “lifted” to an automorphism of 1 . T HEOREM 5.3. Let: q1 : 1 → X1 and q2 : 1 → X2 be quotient maps and assume that neither X1 nor X2 is isomorphic to 1 . Let T be an isomorphism of X1 onto X2 . Then there on 1 such that q2 T = T q1 . In particular, Kernel(q1 ) is isomorphic is an automorphism T to kernel(q2 ). Problem 1.6 of the Introduction has the following complete solutions, both for c0 and ∞ .

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T HEOREM 5.4. Let Y and Z be subspaces of c0 , both of infinite codimension. Assume that T : Y → Z is a surjective isomorphism. Then T extends to an automorphism of c0 . The analogous statement for the space ∞ uses the well-known terminology of Fredholm operators. An operator T on a Banach space X is called a Fredholm operator if both spaces kernel T and X/T (X) are finite-dimensional. The integer i(T ) = dim(kernel(T )) − dim(X/T (X)) is called the index of T . T HEOREM 5.5. Let Y and Z be subspaces of ∞ of infinite codimension and let T be an isomorphism of Y onto Z. Then (a) T can be extended to an automorphism of ∞ if both ∞ /Y and ∞ /Z are nonreflexive. (b) T cannot be extended to an automorphism of ∞ if exactly one of the spaces ∞ /Y and ∞ /Z is reflexive. (c) If both ∞ /Y and ∞ /Z are reflexive then every extension T of T to ∞ is a Fredholm operator. Its index i(T) is an integer valued invariant i(T ) of T and does not depend on the particular extension. The operator T extends to an automorphism of ∞ if and only if i(T ) = 0. Remarks and open problems R EMARK 5.6. The lifting property characterizes the spaces 1 (Γ ) [38] up to isomorphism. For example, in the case of a separable space E with the lifting property, let X = E, q : 1 → E be a quotient map and let I : E → X be the identity. If I: E → 1 lifts I so that q I = I then, clearly, I is an isomorphism of E into 1 and I q is a projection of 1 onto a subspace isomorphic to E. Hence E is isomorphic to 1 , by [57]. R EMARK 5.7. Lindenstrauss’s argument presented in the proof of Theorem 5.1 was originally used in [41] to construct the first example of a subspace U of 1 which is not complemented in any dual space and which does not have an unconditional basis. Let q : 1 → L1 be the natural quotient mapping which maps u2n +k−1 , the (2n + k − 1)k th natural basis element, onto the indicator function of the interval [ k−1 2n , 2n ] where n n = 0, 1, 2, . . . , k = 1, 2, . . . , 2 . If U = kernel(q) were complemented in a dual space then, by Theorem 5.1, the identity I : L1 → L1 could be lifted through 1 thus leading to the contradiction that L1 is isomorphic to a complemented subspace of 1 . The subspace U of 1 is a L1,2 space. It is not known whether the pair (U, 1 ) has the C(K) EP (see Section 6 below for the definition). R EMARK 5.8. Lemma 6.5 and Theorem 6.4 below provide tools which may replace Lemma 1 of [47] for the purpose of extending Theorem 5.4 to the case of c0 (Γ ) with Γ uncountable. We believe that such an extension is valid but have not checked it. R EMARK 5.9. Theorem 5.4 is false if we replace c0 by p (1 p 2), Lp (1 p ∞) and C(K) (if K is a compact Hausdorff space for which C(K) is not isomorphic to c0 ),


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see [47]. The validity of this statement for 1 is easy to see. Let q1 : 1 → c0 be a quotient map and consider the quotient map q2 : 1 ⊕ 1 → c0 ⊕ 1 defined by q2 = q1 ⊕ I where I denotes the identity on 1 . It follows that there is an isomorphism T of kernel q1 onto a subspace Y of 1 = 1 ⊕ 1 such that 1 /Y ∼ c0 ⊕ 1 . Since c0 is not isomorphic to c0 ⊕ 1 , T cannot be extended to an automorphism of 1 . However, in certain important cases, isomorphisms admit extensions to automorphisms. A stronger version of Theorem 5.3 is true if we know more about X1 and X2 , for example: T HEOREM 5.10 ([43]). For i = 1, 2 let qi : 1 → Xi be a quotient map onto a L1 space Xi . Let Ei = kernel(qi ) and assume that Ei is infinite-dimensional. Then E1 is isomorphic to E2 if and only if X1 is isomorphic to X2 . Moreover every isomorphism of E1 onto E2 extends to an automorphism on 1 . It is unknown if Theorems 5.3 and 5.4 characterize 1 and c0 respectively. P ROBLEM 5.11. Let X be an infinite-dimensional separable Banach space. Suppose that for every separable space Y which is not isomorphic to X and for every pair of surjective operators q1 : X → Y and q2 : X → Y there is an automorphism T on X with q1 = q2 T . Is X isomorphic to either 1 or 2 ? P ROBLEM 5.12. Let X be a separable infinite-dimensional Banach space. Assume that, for every pair of isomorphic subspaces Y and Z of X with infinite codimension there is an automorphism T of X such that T (Y ) = Z. Is X isomorphic to either 2 or c0 ? R EMARK 5.13. Recently Ferenczi ([17]) has constructed an example of a space X and its subspace E such that any isomorphic embedding T of E into X is of the form T = J + S, where J is the natural isometric embedding of E into X and S is strictly singular. It is therefore conceivable that the answers to Problems 5.11 and 5.12 may be negative.

6. Extension of operators into C(K) spaces We have seen the effectiveness of the simple Extension Criterion 1.10 in establishing the fact (Corollary 1.11) that, for every 1 < p < ∞ and every subspace E ⊂ p , the pair (E, p ) has the 1-C(K) EP. Another family of classical pairs of spaces which share this property is {(E, L1 ): E ⊂ L1 and dim(E) < ∞}. E XAMPLE 6.1. Let E be any finite-dimensional subspace of L1 = L1 [0, 1]. Then (E, L1 ) has the 1-C(K) EP. P ROOF. Pick a basis {fj }nj=1 of E and let {fj∗ }∞ j =1 denote the corresponding biorthogonal functionals. Let J : E → L1 be the natural isometric embedding then J ∗ is a ω∗ continuous mapping of Ball(L∞ ) onto Ball(E ∗ ). Let Σ denote the Lebesgue σ -field in

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[0, 1] and put i(A) = 1A for A ∈ Σ then i(Σ) ⊂ Ball L ψ : i(Σ) → Ball(E ∗ ) by ∞n. Define ∗ ψ(1A ) = J (1A ) then, as is easily checked, ψ(1A ) = j =1 ( A fj dμ)fj∗ for all A ∈ Σ. It is proved in [65] that the set function Ψ : ψ(i(Σ)) → (i(Σ) defined by Ψ (e∗ ) = ψ −1 (e∗ ) admits a ω∗ continuous selection ϕ such that ψ(ϕ(e∗ )) = e∗ for all e∗ ∈ ψ(i(Σ)). Now consider the subset E = {2 · 1A − 1: A ∈ Σ} which is nothing but the subset of extreme points of Ball(L∞ ). Let β : E → Ball(E ∗ ) be defined by β = J ∗ |E . Then clearly, 1 β = 2ψ − nj=1 ( 0 fj dμ)fj∗ . It is easily checked that β(E) = Ball(E ∗ ) and hence, the set function β −1 = J ∗(−1) ∩ E admits the obvious ω∗ continuous selection 2ϕ − 1 from Ball E ∗ into E which extends functionals. It follows from the Extension Criterion 1.10 that (E, L1 ) has the 1-C(K) EP. The Extension Criterion 1.10 may create the false impression that “almost” every pair (E, X) has the C(K) EP. Let us discuss some counter examples. It is obvious that if H and K are compact Hausdorff spaces and E is a subspace of C(H ) which is isomorphic to C(K) then (E, C(H )) has C(K) EP if and only if E is complemented in C(H ). An example of subspaces E of C[0, 1] which are isomorphic to C[0, 1] but uncomplemented was constructed in [2]. One can use Proposition 1.5 to construct a subspace F of 1 such that (F, 1 ) does not have the C(K) EP. Indeed, with the notations of Proposition 1.5, let X = C[0, 1], let E be the above mentioned example of Amir and denote by Q a quotient mapping of 1 onto X. Let F = Q−1 (E), then, by Proposition 1.5, the pair (F, 1 ) does not have the C(K) EP. E XAMPLE 6.2. There is a two-dimensional subspace E of C[−1, 1] such that (E, C[−1, 1]) does not have the 1-C(K) EP. P ROOF. Let E be the subspace of C[−1, 1] spanned by the functions f (t) = t 2 if −1 t 0 and f (t) = t if 0 t 1 and g(t) = −t if −1 t 0 and g(t) = t 2 if 0 t 1. If (E, C[−1, 1]) has the 1-C(K) EP then, by the Extension Criterion 1.10, there is a ω∗ continuous function ϕ : Ball(E ∗ ) → Ball(C[−1, 1]∗ ) which extends functionals. Let δγ denote the point evaluation functional on C[−1, 1] at γ ∈ [−1, 1] and let δγ0 denote its restriction to E. It is easy to check that ϕ(δγ0 ) must be δγ for all γ close to 1 and 0 0 all γ close to −1. But δ−1 = δ10 , δγ0 → δ10 , and δγ0 −→ δ−1 hence δ1 = limγ →1 δγ = γ →1

γ →−1

limγ →1 ϕ(δγ0 ) = limγ →−1 ϕ(δγ0 ) = limγ →−1 δγ = δ−1 a contradiction. Note that this argument shows that for every subspace F ⊃ E of C[−1, 1], if δ1 |F = δ−1 |F then (E, F ) does not have the 1-C(K) EP. Moreover, it follows that the identity I : E → E ⊂ C(Ball E ∗ ) does not admit a norm preserving extension T : C[−1, 1] → E because, if such extension T existed, then ϕ = T ∗ |Ball(E ∗ ) would be a functional extending ω∗ continuous function into Ball(C[−1, 1]∗ ). Let us now take a look at Theorem 1.8 of the Introduction from the point of view of the Extension Criterion. We know that for every ε > 0 and every subspace E of c0 , the pair (E, c0 ) has the (1 + ε)-C(K) EP. This means that there exists a ω∗ continuous, functional extending function ϕε : Ball(E ∗ ) → (1 + ε)Ball(1 ). It turns out that the following stronger statement is true.


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T HEOREM 6.3 ([77]). Let ε > 0 and let E be a subspace of c0 . Then there exists a ω∗ continuous functional extending function ϕε : Ball(E ∗ ) → (1 + ε)Ball(1 ) such that ϕ(e∗ ) (1 + ε)e∗ for all e∗ ∈ Ball(E ∗ ). The proof of Theorem 6.3 is based on Michael’s Continuous Selection Theorem [55]. The main part of the proof is the following: it is shown that for every subspace E ⊂ c0 , the carrier Φ of Ball(E ∗ ) into the convex subsets of (1 + ε)Ball(1 ) defined by Φ(0) = {0} and, for e∗ = 0, Φ(e∗ ) = {x ∗ ∈ 1 : x ∗ extends e∗ and x ∗ (1 + ε)e∗ } is ω∗ l.s.c. Michael’s theorem ensures the existence of a ω∗ continuous selection ϕ of Φ, which is the desired function. This completes the proof of Theorem 6.3. Theorem 6.3 is an essential tool in the proof of the following generalization of Theorem 1.8. T HEOREM 6.4 ([32]). Let Γ be an uncountable set, let ε > 0 and let E be a subspace of c0 (Γ ). Then the pair (E, c0 (Γ )) has the (1 + ε)-C(K) EP. P ROOF. The first step is a decomposition lemma (which remains true in any space with an extended shrinking basis). L EMMA 6.5 ([32, Lemma 2]). Let Γ be an uncountable set and let E be a subspace of c0 (Γ ). Then Γ can be decomposed into a family {Γα }α∈A of pairwise disjoint countable sets such that if α ∈ A and Eα = {x ∈ E: support(x) ⊂ Γα } then, for every x ∈ E, the restriction x|Γα of x to Γα is in Eα . To prove Theorem 6.4, let {Γα }α∈A and {Eα }α∈A be the decomposition guaranteed by Lemma 6.5. The dual space E ∗ is identical with the 1 (A)-direct sum of {Eα∗ }α∈A . Since each Eα is a subspace of c0 (Γα ) = c0 , by Theorem 6.3, there exists a ω∗ continuous function ϕα : Ball(Eα∗ ) → (1 + ε)Ball(1 (Γα )) which extends functionals and satisfies the inequality ϕα (eα∗ ) (1 + ε)eα∗ for every eα∗ ∈ Ball(Eα∗ ). Define ϕ : Ball(E ∗ ) → ∗ (1 + ε)Ball(1 (Γ )) by ϕ(e ) = α∈A ϕα (e∗ |Eα ) where e∗ ∈ Ball(E ∗ ). It is easily checked that ϕ is ω∗ continuous and extends functionals. The Extension Criterion 1.10 now gives the desired conclusion. The special role of 1 in extension of operators into C(K) spaces has been explained in Proposition 1.5. This result demonstrates the importance of examining those subspaces E of 1 for which (E, 1 ) has the C(K) EP. At the moment, the most general class of subspaces of 1 which are known to share this property is the family of ω∗ closed subspaces of 1 . This statement is a special case of the following ∞ T HEOREM 6.6 ([33]). Let {X n }n=1 be finite-dimensional spaces let ε > 0 and let E be a ∗ ω closed subspace of X = ( Xn )1 , regarded as the dual of ( Xn∗ )c0 . Then (E, X) has the (3 + ε)-C(K) EP. Moreover, if E has the approximation property, then (E, X) has the (1 + ε)-C(K) EP.

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Remarks and open problems Because so little is known about the C(K) Extension Property, there are many open problems. P ROBLEM 6.7. Let W be a subspace of a reflexive space F . Does (W, F ) have the C(K) EP? What if F is superreflexive? What if F is Lp , 1 < p = 2 < ∞? P ROBLEM 6.8. Let E be a reflexive subspace of a separable space X. Does (E, X) have the C(K) EP? What if E is only isomorphic to a conjugate space? What if E = 1 ? (The separability assumption is needed here because the Dunford–Petis property of ∞ and [61] imply that if E is a separable reflexive subspace of ∞ and J : E → C([0, 1]) is an isometric embedding then J cannot be extended to an operator from ∞ into C([0, 1])). If E is a subspace of c0 , then (E, c0 ) has the (1 + ε)-C(K) EP for every ε > 0 ([45]) but need not have the 1-C(K) EP ([32]). We do not know if this phenomenon can occur in the setting of “nice” spaces: P ROBLEM 6.9. If X is a reflexive smooth space and (E, X) has the (1 + ε)-C(K) EP. For every ε > 0, does (E, X) have the 1-C(K) EP? The following observation gives an affirmative answer to Problem 6.9 in a special case. P ROPOSITION 6.10 ([33]). If X is uniformly smooth and (E, X) has the (1 + ε)-C(K) EP for every ε > 0, then (E, X) has the 1-C(K) EP. P ROOF. Since X is uniformly smooth, given ε > 0 there exists δ > 0 so that if x ∗ , y ∗ in X∗ and x in X satisfy x ∗ = x = 1 = x ∗ , x = y ∗ , x with y ∗ < 1 + δ, then x ∗ − y ∗ < ε. Letting φn : Ball(E ∗ ) → (1 + n−1 )Ball(X∗ ) be a weakly continuous extension mapping and letting f : Sphere E ∗ → Sphere X∗ be the (uniquely defined, by smoothness) Hahn–Banach extension mapping, we conclude that

lim sup φn x ∗ − f x ∗ : x ∗ ∈ Sphere E ∗ = 0.

n→∞

∗ That is, {φn |Sphere E ∗ }∞ n=1 is uniformly convergent to f |Sphere E . Since each φn is weakly continuous, so is f |Sphere E ∗ . If E is finite-dimensional, then clearly the positively homogeneous extension of f to a mapping from Ball E ∗ into Ball X∗ is a weakly continuous extension mapping. So assume that E has infinite dimension. But then Sphere E ∗ is weakly dense in Ball E ∗ , so by the weak continuity of the φn ’s and the weak lower semicontinuity of the norm, we have

sup φn x ∗ − φm x ∗ : x ∗ ∈ Ball E ∗

= sup φn x ∗ − φm x ∗ : x ∗ ∈ Sphere E ∗ ,


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which we saw tends to zero as n, m tend to infinity. That is {φn }∞ n=1 is a uniformly Cauchy sequence of weakly continuous functions and hence its limit is also weakly continuous. It is apparent from the proof of Proposition 6.10 that the 1-C(K) EP is fairly easy to study in a smooth reflexive space X because every extension mapping from Ball E ∗ to Ball X∗ is, on the unit sphere of E ∗ , the unique Hahn–Banach extension mapping. Let us examine this situation a bit more in the general case. Suppose E is a subspace of X and let A(E) be the collection of all norm one functionals in E ∗ which attain their norm at a point of Ball E. The Bishop–Phelps theorem [7,15] says that A(E) is norm dense in Sphere E ∗ , hence, if E has infinite dimension, A(E) is weak∗ -dense in Ball E ∗ . Therefore (E, X) has the 1-EP if and only if there is a weak∗ -continuous Hahn–Banach selection mapping φ : A(E) → Ball X∗ which has a weak∗ -continuous extension to a mapping φ ω∗ from A(E) = Ball E ∗ to Ball X∗ , since clearly φ will then be an extension mapping. The existence of φ is equivalent to saying that whenever {xα∗ } is a net in A(E) which weak∗ converges in E ∗ , then {φxα∗ } weak∗ converges in X∗ (see, for example, [9], I.8.5). Now, when X is smooth, there is only one mapping φ to consider, and in this case the above discussion yields the next proposition when dim E = ∞ (when dim E < ∞ one extends ω∗ from Sphere E ∗ = A(E) to Ball E ∗ by homogeneity). P ROPOSITION 6.11 ([33]). Let E be a subspace of the smooth space X. The pair (E, X) fails the 1-C(K) EP if and only if there are nets {xα∗ }, {yα∗ } of functionals in Sphere X∗ which attain their norm at points of Sphere E and which weak∗ converge to distinct points x ∗ and y ∗ , respectively, which satisfy x ∗ |E = y ∗ |E . An immediate, but surprising to us, corollary to Proposition 6.11 is: C OROLLARY 6.12 ([33]). Let E be a subspace of the smooth space X. If the pair (E, X) fails the 1-C(K) EP, then there is a subspace F of X of codimension one which contains E so that (F, X) fails the 1-C(K) EP. P ROOF. Get x ∗ , y ∗ from Proposition 6.11 and set F = span E ∪ (ker x ∗ ∩ ker y ∗ ).

P ROBLEM 6.13. Is Corollary 6.12 true for a general space X? In contrast to Corollary 1.11 we have the following P ROPOSITION 6.14 ([33]). For 1 < p = 2 < ∞, Lp has a subspace E for which (E, Lp ) fails the 1-C(K) EP. It is stated in [45] that, for every subspace E of c0 , not only does (E, c0 ) have the (1 + ε)-C(K) EP but, in addition, if Y is an L1 (μ)-predual, then every operator T : E → Y : c0 → Y with T (1 + ε)T . The proof of Theorem 6.6 points in an extends to a T analogous direction: if E is a ω∗ closed subspace of 1 then, in addition to (E, 1 ) having the (3 + ε)-C(K) EP, for every L∞,λ space Y , every operator T : E → Y extends to an operator T : 1 → Y with T| λ(3 + ε)T .

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P ROBLEM 6.15. Let X be a Banach space, E ⊂ X and let (E, X) have the C(K) EP. Let Y be a L∞ space. Does every operator T : E → Y extend to T : X → Y ?

9

A different point of view on certain operator extension problems is that of splitting twisted sums. Recall that a Banach space Z is called a twisted sum of spaces Y and X (denoted by Z = Y X) if j

q

0→Y → Z → X→0

9

is a short exact sequence, i.e., j is an isometric embedding, q is a quotient map onto X and j (Y ) = kernel(q). We will identify j (Y ) with Y below. The twisted sum Y X is said to split if there is an operator T : X → Y X such that qT = IX . Note that in this case P = I − T q is a projection of Y X onto j (Y ). Conversely, if P is a projection of Y X onto j (Y ) define the operator T : X → Y X for all x ∈ X by T x = z − P z where z ∈ Y X is any element for which q(z) = x (T is well defined because, if q(w) = 0 then P w = w). 9

9

9

9

9

9

P ROPOSITION 6.16 ([35]). Let F be a Banach space and let ϕ : 1 (Γ ) → F be a quotient map. Let E = kernel(ϕ). Then, for every Banach space Y , the following two assertions are equivalent (a) Every bounded operator S : E → Y extends to a bounded operator S : 1 (Γ ) → Y . (b) Every twisted sum Y F splits.

9

9

P ROOF. (b) ⇒ (a) By Lemma 1.12, given an operator S : E → Y , there is a twisted sum Y F such that S extends to an operator S1 : 1 (Γ ) → Y F . Let q be the quotient map of Y F onto F then, since (b) implies that Y F splits, there exists an operator T : Y → Y F such that qT = IF . Let P = I − T q, be the above mentioned projection of Y F onto Y . Then S = P S1 is the desired extension of S. (a) ⇒ (b) Let Y F be any twisted sum and let q : Y ∈ F → F be the quotient map with kernel(q) = Y . Given μ > 1, the lifting property of 1 (Γ ) implies the existence of an operator ψ : 1 (Γ ) → Y F so that qψ = ϕ where ψ < μ. Because E ∈ kernel(ϕ), ψ(E) ∈ kernel(q) = Y . Let ψ0 = ψ|E : E → Y and use (a) to extend ψ0 to an operator ψˆ 0 : 1 (Γ ) → Y . The operator ψ − ψˆ 0 maps 1 (Γ ) into Y F . Since kernel (ψ − ψˆ 0 ) ⊃ E = kernel(ϕ), we may define the operator u : F → Y F by u(e) = (ψ − ψˆ 0 )(x) if e = ϕ(x) for some x ∈ 1 (Γ ). It follows that uϕ = ψ − ψˆ 0 and qu = 0. Hence, quϕ = qψ = ϕ. But ϕ(1 (Γ )) = F , therefore qu = IF and Y F splits. 9

9

9

9

9

9

9 9

9

In the terminology of twisted sums, the first part of Problem 6.8 has a positive solution if so does the following

9

P ROBLEM 6.17. Let F be a reflexive space and Y = C(K) for some compact Hausdorff space. Does every twisted sum Y F split?


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9

Indeed let F be a reflexive space, let W ⊂ F and let Γ be so large that there is a quotient map Q : 1 (Γ ) → F . Put E = Q−1 (W ). By Proposition 1.5, (W, F ) has the C(K) EP if (E, 1 (Γ )) does. But, by Proposition 6.16, if every twisted sum Y F splits then (E, 1 (Γ )) has the C(K) EP. The above mentioned algebraic point of view was a useful tool in [35] to show that if q : L1 → Y is a quotient mapping and c0 ⊂ Y then kernel(q) is not a L1 space. Johnson [26] went deeper into the algebra involved in [35] and showed that kernel(q) does not have the GL-lust. R EMARK 6.18. Recently Kalton [34] proved the following partial inverse of Theorem 6.6: let E be a subspace of 1 such that (E, 1 ) has the C(K) EP and 1 /E has an unconditional finite-dimensional decomposition. Then there is an automorphism T on 1 such that T (E) is ω∗ closed.

7. Extension of operators from subspaces of a space of type 2 into a space of cotype 2 The purpose of this section is to prove Maurey’s Extension Theorem stated in Section 1 (f). The proof presented here is based on Maurey’s argument [53] and the approach of [68]. There is a conceptual difference between this extension theorem and the extension theorems of Sections 4 and 6. In the above sections the desired extension is into a space Y which is paved by a family of finite-dimensional subspaces {Yα } directed by inclusion, d(α) where α Yα = Y , and each Yα is ∞ with d(α) = dim Yα . Since each Yα is a P1 space, the extension of an operator into Yα is trivial. The difficulty is in the passage from the d(α) finite-dimensional ∞ to the infinite-dimensional Y . On the other hand, in the present case, the finite-dimensional construction is where most of the action is while the passage to the infinite-dimensional case is achieved by an ultraproduct argument. This is demonstrated in the final part of the proof of the following L EMMA 7.1. Let X and Y be a Banach spaces and let E ⊂ X. Let c > 0 and let T : E → Y be an operator satisfying the following condition: n ∗ 2 ∗ 2 ⊂ E and {xi }ni=1 ⊂ X, if m (7.1) for any finite sets {ei }m i=1 |x (ei )| i=1 |x (xi )| i=1 m n ∗ ∗ 2 1/2 2 1/2 for every x ∈ X then ( i=1 T ei ) cT ( j =1 xj ) . Then there is a Hilbert space H and an operator S : X → H such that S c and T e Se for all e ∈ E. P ROOF. Pick a finite-dimensional subspace F of X and let M = Ball(F ∗ ). Let m n ∗ ∗ 2 ∗ f (xi )2 , K = ϕ:M → R | ϕ f = f (ei ) − i=1

where m i=1

{ei }m i=1

⊂ E ∩ F,

T ei 2 > c2

n i=1

i=1

{xi }ni=1

xi 2 .

⊂ F and

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M. Zippin

It is easy to verify that K is a convex subset of C(M) and, in view of (7.1), each ϕ ∈ K attains a positive maximum in M. Hence if G = {ψ ∈ C(M): ψ(f ∗ ) < 0 for all f ∗ ∈ C(M)} then G ∩ K = ∅ and, by the standard separation theorem, there is a non-trivial separating measure μF ∈ C(M)∗ and α ∈ R such that μF (ψ) < α μF (ϕ) for all ψ ∈ G and ϕ ∈ K. Since G is the negative cone, α 0 and hence μF is a positive measure on M. It follows that 0 < β = sup{ M |f ∗ (f )|2 dμF (f ∗ ): f ∈ Ball(F )} μF (M). Put νF = β −1 c2 μF then ∗ 2

f (f ) dνF f ∗ : f ∈ Ball(F ) . (7.2) c2 = sup M

Now define an operator SF : F → L2 (M, νF ) by (SF f )(f ∗ ) = f ∗ (f ). It follows from (7.2) that SF c. Moreover, suppose that f ∈ E ∩ F and Tf > c and let x ∈ Ball F . Then the function ϕ(f ∗) = |f ∗ (f )|2 − |f ∗ (x)|2 belongs to K and, by the separating property of νF , SF f 2 − M |f ∗ (x)|2 dνF (f ∗ ) = M ϕ(f ∗ ) dνf (f ∗ ) > 0. Passing to the sup on Ball F we get that SF f 2 sup{ |f ∗ (x)|2 dνF (f ∗ ): x ∈ Ball F } = c2 . It follows that SF (f ) c whenever Tf > c and hence Tf SF f cf

for all f ∈ F ∩ E.

(7.3)

We turn now to the ultra product argument. We use here only basic facts which can be found in Section 9 of [27]. Let I = {F ⊂ X: F a finite-dimensional subspace of X} and let U be an ultrafilter on*I which contains, for each finite-dimensional F ⊂ X the set {Z ∈ I : Z ⊃ F }. Let H = ( i L2 (νF ))U be the ultra product of the L2 (νF ) spaces constructed above. As is well-known, H = L2 (μ) for some measure μ, by the stability of Lp spaces under ultraproducts (see, e.g., [16], Chapter 8). Let SF : X → L2 (νF ) be defined by SF x = SF x if x ∈ F and SF x = 0 otherwise, and define S : X → H by Sx = { SF x}F ∈I . Then, clearly, S is a linear operator and Sx cx for all x ∈ X. Moreover, because T x SF x whenever x ∈ F ∩ E, we get that ce Se for all e ∈ E. C OROLLARY 7.2. Let X and Y be Banach spaces and let E be a subspace of X. Let c > 0 and suppose that T : E → Y is an operator satisfying (7.1). Then there is a Hilbert space H and operators S : X → H and U : H → Y such that U S : X → Y extends T , U 1 and S c. P ROOF. By Lemma 7.1 there is a Hilbert space H and an operator S : X → H such that Se T e for all e ∈ E. Let P be the orthogonal projection of H onto S(E), define U0 : S(E) → Y by U0 Se = T e for all e ∈ E and put U = U0 P . Then clearly U 1 and U Se = T e whenever e ∈ E. Before we proceed to the last step of the proof of Maurey’s theorem let us remind the reader of some special operator ideal norms and relations between them. For more details the reader is referred to Section 10 of [27], [16] or [68]. An operator T : X → Y is called 2-summing if π2 (T ) =def sup{( ni=1 T xi 2 )1/2 : n = 1, 2, . . .} < ∞ where the sup is taken over all finite sets {xi }ni=1 ⊂ X for which sup{( ni=1 |x ∗ , xi |2 )1/2 : x ∗ ∈ Ball X∗ } 1.


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It is easy to see that π2 (T ) is an ideal norm, i.e., a norm which satisfies the inequality π2 (U T V ) U π2 (T )V for every pair of spaces X0 and Y0 and operators U : Y → Y0 and V : X0 → X. The second idealnorm we need is the -norm (see [68], p. 80) defined as follows: for T : n2 → Y, (T ) = ( R n T x2 dγ (x))1/2 where γ is the standard Gaussian measure on R n , the density of which is given by (2π)−n/2 exp(− ni=1 21 |ti |2 ). It is easy n to see that, for every U : m 2 → 2 , (T U ) U (T ). For a general operator S : X → n Y, (S) = sup{(SU ) : U ∈ L(2 , X), U = 1, n = 1, 2, . . .}. Again, it is easily checked that (S) is an ideal norm. The relations between the -norm and the notions of Gaussian type and cotype are the consequences of the identity n 2 1/2 ψi (t)T ui dγ (t) Rn

(T ) =

i=1

for every T : n2 → Y , where {ui }ni=1 is the unit vector basis of n2 . This identity follows from the fact that ψi (t) = t, ui , i = 1, 2, . . . , n, are independent standard Gaussian variables on R n . We treat here Banach spaces over the real numbers but, with the corresponding definitions for complex numbers, all of the following results hold in the complex case. We need the following basic facts about the π2 and norms. L EMMA 7.3. (a) Let X and Y be Banach spaces and assume that either X or Y is finitedimensional. Then, for every operator S : X → Y and ε > 0, there is an integer m, opm m m erators U : X → m ∞ and V : 2 → Y , and a diagonal operator Δ : ∞ → 2 such that S = V ΔU and V ΔU < (1 + ε)π2 (S). (b) Let V : n2 → X be the operator defined by V ui = xi , 1 i n, where {ui }ni=1 de notes the unit vector basis of n2 . Then π2 (V ∗ ) ( n1 xi 2 )1/2 . The proof of (a) is based on the Pietch factorization theorem ([27], Section 10) and the fact that X isometrically embeds into a C(K) space, where K is a compact Cantor space. This C(K) space is paved by a family of m ∞ spaces, each spanned by m disjointly m supported functions {fi }m i=1 . The formal identity I : C(K) → L2 (K, μ) maps {fi }i=1 onto numerical multiples of an orthonormal basis where μ is the measure provided by Pietch’s theorem. Part (b) is obtained by a straightforward computation of π2 (V ∗ ). L EMMA 7.4. Let X be a Banach space of Gaussian type 2 and let S : n2 → X be an γ operator. Then (S) T2 (X)π2 (S ∗ ). P ROOF. By Lemma 7.3(a), given ε > 0 there is an integer m, operators U : X∗ → m n m m ∗ m ∞ , V : 2 → 2 and a diagonal operator Δ : ∞ → 2 such that S = V ΔV and ∗ V ΔU (1 + ε)π2 (S ). Denoting by JX the natural embedding of X into X∗∗ m ∗ we get that JX S = U ∗ Δ∗ V ∗ . Because Δ∗ : m 2 → 1 is a diagonal operator, Δ = m γ γ ( i=1 Δ∗ ui 2 )1/2 . The principle of local reflexivity implies that T2 (X∗∗ ) = T2 (X). These two facts, together with the fact that is an ideal norm, imply the following in-

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equality:

(S) = (JX S) V ∗ U ∗ Δ∗ 2 1/2 m ∗ ψi (t)U ∗ Δui dγ (t) = V R n i=1 1/2 n ∗ ∗ 2 ∗ γ ∗∗ U Δ ui V T X 2

i=1

n 1/2 ∗ ∗ γ 2 V U T2 (X) Δui i=1

γ γ = V ∗ U ∗ ΔT2 (X) (1 + ε)T2 (X)π2 S ∗ . Since ε > 0 is arbitrary, (S) T2 (X)π2 (S ∗ ) as claimed. γ

P ROOF OF M AUREY ’ S EXTENSION THEOREM . Let X and Y be Banach spaces of Gaussian type 2 and cotype 2, respectively, and let E be a subspace of X. In view of Corollary 7.2 it suffices to show that every operator T : E → Y satisfies condition (7.1) with γ γ n c = T2 (X)C2 (Y ). To do that choose any two finite sets {ei }m i=1 ⊂ E and {xj }j =1 ⊂ X. Such that m !

n " ! ∗ " x ∗ , ei x , xj for all x ∗ ∈ X∗ .

(7.4)

j =1

i=1

We may assume, without loss of generality, that m = n because adding a few zero’s to one of the sets will affect neither (7.1) nor (7.3). Define the operators U : n2 → E and V : n2 → X by U ui = ei and V ui = xi for 1 i n. Denoting by J the natural embedding of E into X we see that, by (7.3), U ∗ J ∗ x ∗ 2 = sup{x ∗ (J U (u)): u ∈ Ball(n2 )}2 = n n ∗ 2 ∗ 2 ∗ ∗ 2 ∗ ∗ i=1 |x , ei | i=1 |x , xi | = V x for every x ∈ X . Therefore there is an n n ∗ ∗ ∗ operator W : 2 → 2 with W 1 such that W V = U J . It follows from the definition of cotype and the fact that (T ) is an ideal norm that

n

1/2 2

T ei

=

i=1

n

1/2 2

T J U ui i=1 γ γ C2 (Y )(T J U ) T C2 (Y )(J U ).

(7.5)

Using Lemma 7.4, the fact that π2 is an ideal norm and Lemma 7.3(b) we get that

γ γ (J U ) T2 (X)π2 U ∗ J ∗ = T2 (X)π2 W V ∗ 1/2 n ∗

γ γ 2 xj . T2 (X)π2 V T2 (X) j =1

(7.6)


1739

Combining (7.5) and (7.6) we get that

n i=1

1/2 T ei

2

γ γ T C2 (Y )T2 (X)

n

1/2 xj

2

.

j =1

This establishes condition (7.1) and, in view of Corollary 7.2, completes the proof of Maurey’s extension theorem. The following partial converse of Maurey’s extension theorem has been recently proved in [14]. T HEOREM 7.5. Let X be a Banach space which satisfies one of the following properties (i) there is a constant c so that, for every n 1 and every operator T ∈ B(n2 , X), (T ) cπ1 (T ∗ ). (ii) X has the Gordon–Lewis property (in particular, X may be a Banach lattice). (iii) X is isomorphic to a subspace of a Banach lattice of finite cotype. If X satisfies the conclusion of Maurey’s extension theorem then X is of type 2.

References [1] D.E. Alspach, Quotients of C[0, 1] with separable duals, Israel J. Math. 29 (1978), 361–384. [2] D. Amir, Continuous function spaces with the bounded extension property, Bull. Reg. Council Israel, Sect. F 10 (1962), 133–138. [3] D. Amir, Projections onto continuous function spaces, Proc. Amer. Math. Soc. 15 (1964), 396–402. [4] D. Amir, Continuous function spaces with small projection constants, Proc. Sympos. on Functional Analysis, Hiroshima University (1965). [5] S. Banach, Sur les fonctionnelles lineaires I, II, Studia Math. 1 (1929), 211–216 and 223–239. [6] Y. Benyamini, An extension theorem for separable Banach spaces, Israel J. Math 29 (1978), 24–30. [7] E. Bishop and R.R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. [8] F. Bohnenblust and A. Sobczyk, Extensions of functionals on complex linear spaces, Bull. Amer. Math. Soc. 44 (1938), 91–93. [9] N. Bourbaki, General Topology, Part 1, Addison-Wesley (1966). [10] J. Bourgain, A remark on finite dimensional Pλ -spaces, Studia Math. LXXII (1982), 285–289. [11] J. Bourgain and F. Delbaen, A class of special L∞ spaces, Acta Math. 145 (1980), 155–176. [12] J. Bourgain and G. Pisier, A construction of L∞ spaces and related Banach spaces, Bol. Soc. Brasil Mat. 14 (1983), 109–123. [13] D.I. Cartwright, Extensions of Positive Operators between Banach Lattices, Mem. Amer. Math. Soc. 164 (1975). [14] P.G. Casazza and N.J. Nielsen, The solution to the Maurey extension problem for Banach spaces with the Gordon–Lewis property and related structures, Studia Math., to appear. [15] J. Diestel, Geometry of Banach Spaces – Selected Topics, Lecture Notes in Math. 485, Springer (1975). [16] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. in Adv. Math. 43 (1995). [17] V. Ferenczi, Qulques proprietes des espaces de Banach hereditairement indecomposables, Séminaire Initiation à l’Analyse 36o Année, 1996–1997, no 13. [18] L. Gillman and M. Jerison, Rings of Continuous Functions, Princeton Univ. Press, Princeton, NJ (1960). [19] D.A. Goodner, Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89–108.

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[20] A. Grothendieck, Une caratérisatin vectorielle métrique des espaces L1 , Canad. J. Math. 7 (1955), 552– 561. [21] A. Grothendieck, Résumé de la théorie métrique des produit tensoriels topologiques, Bol. Soc. Math. Sao Paulo 8 (1956), 1–79. [22] H. Hahn, Über lineare Gleichungssysteme in linearen Räumen, J. Reine Angew. Math. 157 (1927), 214– 229. [23] M. Hasumi, The extension property of complex Banach spaces, Tôhoku Math. J. 10 (1958), 135–142. [24] R.G. Haydon, Injective Banach lattices, Math. Z. 156 (1977), 19–47. [25] J.R. Isbell and Z. Semadeni, Projection constants and spaces of continuous functions, Trans. Amer. Math. Soc. 107 (1963), 38–48. [26] W.B. Johnson, Extensions of c0 , Positivity 1 (1997), 55–74. [27] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. [28] W.B. Johnson, H.P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. [29] W.B. Johnson and G. Schechtman, On subspaces of L1 with maximal distances to Euclidean space, Proceedings of Research Workshop on Banach Space Theory (Iowa City, Iowa, 1981), Univ. Iowa, Iowa City, IA (1982), 83–96. [30] W.B. Johnson and M. Zippin, Separable L1 preduals are quotients of C(Δ), Israel J. Math. 16 (1973), 198–202. [31] W.B. Johnson and M. Zippin, Subspaces and quotient spaces of ( Gn )p and ( Gn )0 , Israel J. Math. 17 (1974), 50–55. [32] W.B. Johnson and M. Zippin, Extension of operators from subspaces of c0 (Γ ) into C(K) space, Proc. Amer. Math. Soc. 107 (1989), 751–754. [33] W.B. Johnson and M. Zippin, Extension of operators from ω∗ closed subspaces of 1 into C(K) spaces, Studia Math. 117 (1995), 43–55. [34] N.J. Kalton, On the extension of operators into C(K) spaces, to appear. [35] N.J. Kalton and A. Pełczyński, Kernels of surjections from L1 -spaces with an application to Sidon sets, Math. Ann. 309 (1997), 135–158. [36] R. Kaufman, A type of extension of Banach spaces, Acta Sci. Math. (Szeged) 27 (1966), 163–166. [37] J.L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1954), 323–326. [38] G. Köthe, Topological Vector Spaces I, Springer, New York (1969). [39] H.E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, New York (1974). [40] D.R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to 1 (Γ ), J. Funct. Anal. 12 (1973), 177–187. [41] J. Lindenstrauss, On a certain subspace of 1 , Bull. Acad. Pol. Sci. 12 (9) (1964), 539–542. [42] J. Lindenstrauss, Extension of Compact Operators, Mem. Amer. Math. Soc. 48 (1964). [43] J. Lindenstrauss, A remark on 1 spaces, Israel J. Math. 8 (1970), 80–82. [44] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in Lp spaces and their applications, Studia Math. 29 (1968), 275–326. [45] J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249. [46] J. Lindenstrauss and H.P. Rosenthal, The Lp spaces, Israel J. Math. 7 (1969), 325–349. [47] J. Lindenstrauss and H.P. Rosenthal, Automorphisms in c0 , 1 and m, Israel J. Math. 7 (1969), 227–239. [48] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math., Springer, New York (1972). [49] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, II, Springer, New York (1977). [50] J. Lindenstrauss and L. Tzafriri, On the isomorphic classification of injective Banach lattices, Mathematical Analysis and Applications, Part B, Advances in Mathematics Supplementary Studies 7B, 489–498. [51] H.P. Lotz, Extensions and liftings of positive linear operators on Banach lattices, Trans. Amer. Math. Soc. 211 (1975), 85–100. [52] P.J. Mangheni, The classification of injective Banach lattices, Israel J. Math. 48 (1984), 341–347. [53] B. Maurey, Un théorème de prolongement, C.R. Acad. Sci. Paris A 279 (1974), 329–332.


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[54] S. Mazurkiewicz and W. Sierpinski, Contribution á la topologie des ensembles demonbrables, Fund. Math. 1 (1920), 17–27. [55] E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361–382. [56] L. Nachbin, A theorem of Hahn–Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28–46. [57] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. [58] A. Pełczyński, Strictly singular and strictly nonsingular operators in C(S) spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 31–36. [59] A. Pełczyński, On C(S) subspaces of separable Banach spaces, Studia Math. 31 (1968), 513–522. [60] G. Pisier, Counter examples to a conjecture of Grothendieck, Ann. of Math. 151 (1983), 181–208. [61] H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. [62] H.P. Rosenthal, On injective Banach spaces and the spaces L∞ (μ) for finite measure μ, Acta. Math. 124 (1970), 205–248. [63] H.P. Rosenthal, On factors of C[0, 1] with non-separable dual, Israel J. Math. 13 (1972), 361–378. [64] H.P. Rosenthal, The complete separable extension property, J. Operator Theory 43 (2000), 329–374. [65] D. Samet, Continuous selections for vector measures, Math. Oper. Res. 9 (1984), 471–474. [66] A. Sobczyk, Projections of the space m on its subspace c0 , Bull. Amer. Math. Soc. 47 (1941), 938–947. [67] W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53–61. [68] N. Tomczak-Jaegermann, Banach–Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific and Technical, Essex (1989). [69] W.A. Veech, Short proof of Zobczyk’s theorem, Proc. Amer. Math. Soc. 28 (1971), 627–628. [70] J.E. Wolfe, Injective Banach spaces of type C(T ), Thesis, Berkeley, CA (1971). [71] J.E. Wolfe, Injective Banach spaces of type C(T ), Israel J. Math. 17 (1974), 133–140. [72] J.E. Wolfe, Injective Banach spaces of continuous functions, Trans. Amer. Math. Soc. 235 (1978), 115–139. [73] M. Zippin, On some subspaces of Banach spaces whose duals are L1 spaces, Proc. Amer. Math. Soc. 23 (1969), 378–385. [74] M. Zippin, The separable extension problem, Israel J. Math. 26 (1977), 372–387. [75] M. Zippin, The finite dimensional Pλ space for small λ, Israel J. Math. (1981), 359–364. [76] M. Zippin, Correction to the finite dimensional Pλ spaces with small λ, Israel J. Math. 48 (1984), 255–256. [77] M. Zippin, Applications of E. Michael’s continuous selection theorem to operator extension problems, Proc. Amer. Math. Soc. 127 (1999).


CHAPTER 41

Nonseparable Banach Spaces∗ Václav Zizler Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1, Edmonton, Canada E-mail: [email protected]

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic concepts and examples . . . . . . . . . . . . . . . . . . . Weak compact generating and projectional resolutions . . . . . Biorthogonal systems and quasicomplements . . . . . . . . . . Gâteaux smooth and rotund norms . . . . . . . . . . . . . . . . Uniformly Gâteaux smooth norms . . . . . . . . . . . . . . . . Fréchet smooth and locally uniformly rotund norms . . . . . . C k -smooth norms for k > 1 . . . . . . . . . . . . . . . . . . . Open problems and concluding remarks . . . . . . . . . . . . . A. More on special compact spaces . . . . . . . . . . . . . . . B. More on the weak topology of nonseparable Banach spaces C. More on fragmentability and σ -fragmentability . . . . . . . D. Fundamental biorthogonal systems and Mazur’s intersection E. Uniform homeomorphisms . . . . . . . . . . . . . . . . . . F. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7926-02 (Canada). HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2 Edited by William B. Johnson and Joram Lindenstrauss © 2003 Elsevier Science B.V. All rights reserved 1743


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1. Introduction The theory of nonseparable Banach spaces is a large field, closely related to general topology, differential calculus, descriptive set theory and infinite combinatorics. In this article, we will focus on the interplay of weak topologies, smoothness and rotundity of norms, biorthogonal systems and projectional resolutions of the identity operator in nonseparable Banach spaces. We will discuss some applications in questions on special compact spaces and in smooth approximations and smooth partitions of unity on nonseparable Banach spaces. Many open problems will be mentioned. Gâteaux differentiability of norms is behind sequential and metrizability properties of weak topologies on some sets and is useful in the study of special compact spaces (Sections 2, 5). Fréchet differentiability of norms generates continuous duality mappings from Banach spaces X onto dense sets in their duals X∗ (Section 7). Local uniform rotundity of norms is related to covering properties of weak topologies in Banach spaces and is useful in questions on continuity of the identity map from the weak to norm topologies on sets in Banach spaces (Section 7). Uniform Gâteaux differentiability of norms is instrumental in embedding compact spaces into Hilbert spaces in their weak topology (Section 6). The concept of Markushevich bases provides for a good insight into problems on the Corson–Lindelöf properties of weak topologies, injections into c0 (Γ ) and quasicomplements (Section 4). Markushevich bases are related to two fundamental concepts in nonseparable Banach spaces, namely to the concept of weak compact generating and to the concept of projectional resolutions of the identity operator (Sections 2, 3, 4). In Section 8, higher-order differentiable norms on spaces of continuous functions on uncountable scattered compacts are discussed, together with applications in smooth partitions of unity on such Banach spaces. The article is finished with remarks and open problems (Section 9). An ample list of references is included (Section 10). In the end of each section, the results presented are tested on examples of Banach spaces listed in the end of Section 2. The proofs in this article will be outlined only. In order to make it short, we will say “Proof” and mean the key idea in the proof. The list of survey texts, where the reader can find more information in this area include [6,10,20,31,25,34,57,61,60,65,64,66,67,73,80,105,126,141,142,157,177,185,195, 190,197,198,208,206,219,222,243,253,269,280,284,296,304,306,327] and [328]. Many open problems in this area are discussed, for example, in [20,57,73,80,206] and [222]. The exercises in [61,80] and [327] are accompanied with hints for their solution and discuss many “folklore” techniques as well as examples and counterexamples. We will consider real Banach spaces only and keep the standard notation as it is, e.g., in [168]. In particular, N and R will denote the set of all positive integers and reals respectively. The symbol BX will denote the unit ball of a Banach space X, i.e., BX = {x ∈ X; x 1} and SX will denote the unit sphere of a Banach space X, i.e., SX = {x ∈ X; x = 1}, where · is the norm of X. Unless stated otherwise, dual spaces X∗ are considered in their canonical dual norm f = sup{|f (x)|; x ∈ BX } for f ∈ X∗ . Compact topological spaces are assumed to be Hausdorff. If K is a compact space, C(K) will denote the Banach space of all real-valued continuous functions on K with the supremum norm f = sup{|f (t)|; t ∈ K}. The symbol p (c0 ) denotes the space p (N) (c0 (N)). Similarly, p (c) stands for p (Γ ), where card Γ is the cardinality of the

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continuum c, i.e., the cardinality of the set R. The least infinite (respectively uncountable) ordinal number is denoted by ω0 (respectively ω1 ). When we think of these ordinals as cardinal numbers, we will denote them by ℵ0 and ℵ1 respectively. The Continuum Hypothesis then reads ℵ1 = c. A set S is called countable if card S ℵ0 . The cardinality of an ordinal number μ is denoted by |μ|. A Banach space X is identified with its canonical image in its second dual X∗∗ . If X is a Banach space, x ∈ X and f is in the dual space X∗ , then both f (x) and x(f ) will denote the value of f at x. The norm on a Banach space and its dual norm on X∗ will be denoted by the same symbol · . Often, we will say a space and mean a Banach space and a norm on a Banach space and mean an equivalent norm. By a subspace of a Banach space we will mean a norm closed subspace. The words differentiable and smooth have the same meaning in this article. If we say that the norm of a Banach spaces is differentiable, we mean that it is differentiable away from the origin. We will say that a real valued function f is Fréchet C 1 -smooth on a Banach space X, if the operator x → f (x), from X into X∗ , is norm to norm continuous, where f (x) is the w Fréchet derivative of f at x (see Definition 1). For a set A in a Banach space X, A and A mean the closure of A in the norm topology, respectively in the weak topology of X. If X∗ w∗ is a dual space, A is the closure of A in the weak star topology of X∗ . If K is a subset of a Banach space X, then span K denotes the closed linear hull of K in X. If K ⊂ X∗ , ∗ then span w K denotes the weak star closed linear hull of K in X∗ , while span · K denotes the norm closed linear hull of K. Similarly, conv K is the closed convex hull of K in X. The symbol χA will denote the characteristic function of the set A in the topological space T . We will say, typically, that a set K in a Banach space X is weakly compact if it is compact in the relative topology inherited from the weak topology of X. The density character or density (dens T ) of a topological space T is the minimal cardinality of a dense set in T . Unless stated otherwise, for a Banach space X, dens X is the density of X in the norm topology. A bump function on a Banach space X is a real-valued function on X with bounded non-empty support.

2. Basic concepts and examples In this section we will discuss basic concepts in the interplay of weak topologies and Gâteaux smoothness of norms in nonseparable Banach spaces. We will define projectional resolutions of identity, several types of compact spaces that will be discussed in this article and list Banach spaces that will serve as examples for testing the results presented in this article. D EFINITION 2.1. A real-valued function ϕ on a Banach space X is Gâteaux (Fréchet) differentiable at x ∈ X if there is an f ∈ X∗ such that limt →0 1t (ϕ(x + th) − ϕ(x)) = f (h) for every h ∈ SX (uniformly in h ∈ SX ). Such f is then called the Gâteaux (Fréchet) derivative of ϕ at x and is denoted by ϕ (x), i.e., f (h) = ϕ (x)(h) for every h ∈ X. If · is a norm on a Banach space X and x ∈ SX , we say that · is Gâteaux (Fréchet) differentiable at x if the real-valued function ϕ defined for y ∈ X by ϕ(y) = y is Gâteaux (Fréchet) differentiable at x. In this case we write ϕ (x) = x . If the norm · is Gâteaux


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(respectively Fréchet) differentiable at each x ∈ SX , we say that · is a Gâteaux (respectively Fréchet) differentiable norm on X and call it a G (respectively an F) norm on X. The norm · on X is Gâteaux (Fréchet) differentiable at x ∈ SX if and only if limt →0 1t (x + th + x − th − 2) = 0 for every h ∈ SX (uniformly in h ∈ SX ). If the norm · is Gâteaux differentiable at x ∈ SX , then the functional x is supporting BX at x, i.e., x ∈ SX∗ and x (x) = 1 (cf., e.g., [57, p. 2], [80, Chapter 8]). Every Banach space X in its weak topology is a completely regular space (cf. e.g. [80, Chapter 3, Example] or [73, p. 56]). The following results are from [17,41] and [265]. For more in this direction we refer to Section 3 of this article and references therein. T HEOREM 2.2. Let X be a Banach space. Then X in its weak topology is a Lindelöf space if and only if it is a normal space if and only if it is a paracompact space. While infinite-dimensional Banach spaces are never metrizable in their weak or weak star topologies (cf., e.g., [80, Chapter 3]), BX in its weak topology is metrizable if and only if X∗ is separable and BX∗ in its weak star topology is metrizable if and only if X is separable (cf., e.g., [80, Chapter 3]). We will often use the fact that given a compact space K, C(K) is norm separable if and only if K is metrizable (cf., e.g., [80, Chapter 3]). If X∗ is weak star separable and K is a weakly compact set in X, then K in its weak topology is metrizable by the metric ρ(x, y) = 2−i fi −1 |fi (x − y)| for x, y ∈ K, where {fi } is weak star dense in X∗ (cf., e.g., [80, Chapter 3]). In this case, K in its weak topology is separable and span K is norm separable (cf. Mazur’s theorem (e.g., [80, Chapter 3]) and so is K. A set K in a Banach space X is compact in the weak topology of X if and only if K is weakly sequentially compact in X (i.e., every sequence in K has a subsequence converging in K in its weak topology) (Eberlein, Šmulyan) (cf., e.g., [80, Chapter 4]). A topological space T is called pseudocompact if every continuous real-valued function f on T is bounded. Note that then f attains its supremum on T . The following theorem is a combination of the results of James, Krein, Preiss and Simon (cf. [10, Chapter IV.5], [99] and [98] (where a new proof is given), [80, Chapter 3], [168, 258]). T HEOREM 2.3. A subset K of a Banach space X is weakly compact if K is either weakly pseudocompact or K is closed convex and each f ∈ X∗ attains its supremum on K. The latter happens if K is a closed convex hull of a weakly compact set in X. P ROOF. Let C be a weakly pseudocompact set in a Banach space X. Then K := conv C is a weakly compact set in X by James’ weak compactness theorem as every bounded linear functional attains its supremum on K (= its supremum on C) (cf., e.g., [80, Chapter 3], w [168,99]). Assume a ∈ C \ C. Preiss and Simon proved in [258] that there is a sequence w w (Un ) of nonempty sets in C open in the relative weak topology of C and such that w (Un ) converges to a, i.e., for every neighborhood U of a in C , there is n0 ∈ N such that Un ⊂ U for every n n0 . Let fn be continuous functions on C such that the support of fn is in Un and fn (xn ) = n for some point xn ∈ Un ∩ C for each n. Then consider f := fn .

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It is a continuous function on C as it is locally a finite sum (a ∈ / C). The function f is unbounded on C. This is a contradiction with the pseudocompactness of C, showing that C is weakly closed in K and C is thus weakly compact. For more information in this direction see [10, Chapter IV.5], [305] and [80, Chapter 12] and references therein. For non-linear versions on the James theorem we refer to [15], where it is proved, among other things that if X∗ is infinite-dimensional and separable, then there is a Fréchet C 1 -bump function b on X such that {b (x), x ∈ X} = X∗ . The proof of the separable version of James’ weak compactness theorem (cf., e.g., [57, p. 17], [80, Chapter 3] or [231]) gives that if X is a Banach space and B is a norm separable set in BX∗ such that conv · B = BX∗ , then there is x ∈ SX such that f (x) < 1 for all f ∈ B (Godefroy, Rodé, Simons). This is in contrast with the case of the norm nonseparable set B = {±δt , t ∈ [0, 1]} ⊂ BC[0,1]∗ , where δt is the Dirac measure corresponding to the point t ∈ [0, 1] ([80, Chapter 3]). For a compact space K, a norm bounded set A in C(K) is weakly compact in C(K) if and only if it is a compact set in the topology of pointwise convergence in C(K). This is the Grothendieck theorem (cf., e.g., [60, p. 156]) or [80, Chapter 12]. Goldstine’s theorem asserts that for a Banach space X, BX is weak star dense in BX∗∗ (cf., e.g., [80, Chapter 3]). From the proof of Theorem 2.3, we obtain that if K is a weakly compact set in a Banach w space, A ⊂ K and a ∈ A , then there is a sequence (an ) ⊂ A that weakly converges to a, i.e., every weakly compact set K in a Banach space in its weak topology is an angelic compact. However, if {en } is the sequence of the standard unit vectors in 2 , then one checks w √ directly (cf.,√e.g., [6, p. 110] or [80, Chapter 3, Example]) that 0 ∈ { nen } and no subsequence of { nen } converges weakly to 0 (by the Banach–Steinhaus uniform boundedness principle (cf., e.g., [80, Chapter 3])). Every Banach space in its weak topology has countable tightness, i.e., if A is a subset w w in a Banach space X and a ∈ A , then there is a countable C ⊂ A such that a ∈ C (cf., e.g., [80, Chapter 4]). From the countability of the supports of elements of c0 (Γ ), we can see that B∞ (Γ ) in its weak star topology does not have countable tightness if Γ is uncountable. w Like for every infinite-dimensional Banach space, 0 ∈ S1 . However, no sequence in in S1 weakly converges to 0 (Schur, cf., e.g., [80, Chapter 5]). The dual ball B∗∗ 1 its weak star topology does not have countable tightness ([41]). On the other hand, if a separable Banach space X does not contain any isomorphic copy of 1 , then BX∗∗ in its weak star topology is an angelic compact ([272,273,30]). Note that this implies that if a separable Banach space does not contain any isomorphic copy of 1 , then every element of BX∗∗ is a weak star limit of a sequence from BX , BX∗∗ is weak star sequentially compact and for every bounded sequence {xn } in X, there is a subsequence {xni } such that {f (xni )} is convergent for every f ∈ X∗ . Finally, in this case, card X∗∗ = card X. The latter properties characterize spaces not containing any isomorphic copy of 1 among separable spaces. These are the results of Odell and Rosenthal ([228]) and Rosenthal ([272], cf., e.g., [197, p. 101], [57, p. 115]). If X∗ is separable, then all these properties follow immediately as


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BX∗∗ in its weak star topology is a metrizable compact space. One of the examples of separable spaces X with nonseparable dual such that X does not contain any isomorphic copy of 1 is the space J T discussed below in this section. The following result is the Josefson–Nissenzweig theorem ([171,225], cf., e.g., [61, p. 219], [18]). T HEOREM 2.4. Let X be an infinite-dimensional Banach space and f ∈ BX∗ . Then there is a sequence fn ∈ SX∗ such that fn → f in the weak star topology of X∗ . It is proved in [26] that the statement in Theorem 2.4 is equivalent with the statement that for every infinite-dimensional Banach space X there is a convex continuous function on X that is somewhere Gâteaux but not Fréchet differentiable. Related to Theorem 2.4 is also the following remark: let X be an infinite-dimensional Banach space, fn ∈ X∗ , fn = 1/n for each n. Write each fn = limk fn,k , in the weak star topology, fn,k = n for each n, k ∈ N. Then by the Banach–Steinhaus uniform boundedness principle, no sequence in {fn,k } converges weakly to 0. Compare this with the fact that the uniform limit of a sequence of Baire functions is a Baire one function (i.e., a pointwise limit of a sequence of continuous functions, see [221, Chapter XV.1]). For more applications of Theorem 2.4 in non-linear analysis, we refer to [27] and references therein. In particular, as a corollary of Theorem 2.4, we get that in every infinitedimensional Banach space X, there is a continuous convex function defined on X that is unbounded on BX (cf., e.g., [80, Chapter 8, Example]). The Rainwater–Simons theorem reads as follows: assume that X is a Banach space and B is a subset of BX∗ such that for every x ∈ SX , there is b ∈ B such that b(x) = 1. Then a bounded sequence {xn } ⊂ X converges weakly to x ∈ X whenever b(xn − x) → 0 for every b ∈ B (cf., e.g., [99], [80, Chapter 3]). Note that we can take the set of all extreme points of BX∗ for B by the Krein–Milman theorem (cf., e.g., [80, Chapter 3]). It is the result in [236] (an extension of a related result of Borwein) that a separable Banach space X does not contain an isomorphic copy of 1 if and only if each sequence {fn } in X∗ converges in the norm topology to 0 whenever it converges to zero uniformly on all weakly compact subsets of X (cf., e.g., [26]). Note that if we allow nets instead of sequences above, we get the reflexivity of X by the Mackey–Arens–Katˇetov theorem (cf., e.g., [80, Chapter 4]). The following is Šmulyan’s classical result (cf., e.g., [57, p. 3] or [80, Chapter 8]). We will call it Šmulyan’s lemma. T HEOREM 2.5. The norm · of a Banach space X is Gâteaux (Fréchet) differentiable at x ∈ SX if and only if fn − gn → 0 in the weak star topology (norm topology) of X∗ whenever fn , gn ∈ SX∗ are such that fn (x) → 1 and gn (x) → 1. It follows from Theorem 2.5 that · is Gâteaux differentiable at x ∈ SX if and only if there is a unique supporting functional to BX at x, i.e., a unique f ∈ BX∗ with f (x) = 1, namely f = x . Also, if · is a Fréchet differentiable norm, then the map x → x from SX into SX∗ is norm-to-norm continuous. This map is usually called the duality mapping. Finally, it follows from Theorem 2.5 that Gâteaux and Fréchet differentiability of norms coincide in finite-dimensional spaces.

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Assume that the norm · of X is Gâteaux differentiable at x ∈ SX . As x is continuous on BX∗ in its weak star topology and attains its supremum on BX∗ exactly at x , we get that x is a Gδ point of BX∗ in its weak star topology. This simple fact is behind many sequential properties of weak star and weak topologies on Banach spaces. A point p in a topological space T is called a Gδ point of T whenever p is the intersection of a countable collection of open sets in T . If T is a compact space, this means that T has a countable neighborhood base at p. The following statement is from [75]. It shows the interplay of smoothness, the Bishop– Phelps theorem and the Josefson–Nissenzweig theorem. The Bishop–Phelps theorem asserts that for a Banach space X, those elements of X∗ that attain its supremum on BX form a norm dense set in X∗ (cf., e.g., [168], [57, p. 13] or [80, Chapter 3]). If f ∈ X∗ attains its supremum on BX , we say f attains its norm. T HEOREM 2.6. Assume that the norm of a Banach space X is Gâteaux differentiable and ∗ that S ⊂ BX∗ is such that conv w S = BX∗ . If f ∈ BX∗ , then there is a countable C ⊂ S ∗ w such that f ∈ C . P ROOF. If x ∈ SX and f := x , there is {fn } ⊂ S such that fn (x) → 1. From Theorem 2.5, we have fn → f in the weak star topology of X∗ . Hence the statement holds for f = x . By the Bishop–Phelps theorem, the statement holds for every f ∈ SX∗ . Theorem 2.4 can be used to finish the proof of Theorem 2.6. One of the non-linear versions of the Bishop-Phelps theorem is the following statement. If b is a non-negative Gâteaux smooth continuous function on a Banach space X with bounded non-empty support, then the cone generated by the set {b (x); x ∈ X} is norm dense in X∗ (see, e.g., the text preceding Theorem 7.2). Related to the notion of countable tightness is the following result of Pol ([251], cf., e.g., [80, Chapter 12]). T HEOREM 2.7. Let X be a Banach space. Then the following conditions (i) and (ii) are equivalent. ∗ w∗ (i) If A ⊂ BX∗ and f ∈ A , then there is a countable C ⊂ A such that f ∈ conv w C. (ii) If a family A of convex closed sets in X has empty intersection, then some countable subfamily B of A has empty intersection. D EFINITION 2.8. If the conditions in Theorem 2.7 are satisfied, we say that X has property C. Assuming the Continuum Hypothesis, there is a compact space K that has countable tightness and C(K) does not have property C (Haydon, Pol, see [251]). A compact space K has countable tightness if C(K) is Lindelöf in its pointwise topology ([41]). From applications of property C we mention the following result of Pol [251] that is an extension of the former result of Grothendieck (cf. [195]).


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T HEOREM 2.9. Assume that K is a compact space such that C(K) has property C. Then every finite positive regular measure on K has separable support. The support of a measure is the complement of the set of points that have neighborhoods of measure 0. P ROOF. Assume that the measure μ is supported exactly on K and let us show that K is separable. Fix i ∈ N. For x ∈ K put Cx = {f ∈ C(K); K f dμ 1/i and f (x) = 0}. If f ∈ C , then f = 0 identically on K and thus x∈K x K f = 0, a contradiction. Hence C = ∅. As C is closed and convex for every x ∈ K and C(K) has property C, x x∈K x x∈Ai Cx = ∅ for some countable set Ai ⊂ K. We claim that K = i Ai . Indeed, assume that there is f ∈ C(K) such that f = 0 on i Ai and K f dμ > 0. Find i ∈ N such that x∈Ai Cx , a contradiction. K f dμ > 1/i. Then f ∈ In [251] (cf., e.g., [80, Chapter 12]), Pol proved the following result. T HEOREM 2.10. Property C is a three space property, i.e., a Banach space X has property C, whenever there is a subspace Y of X such that both Y and X/Y have property C. The Banach–Dieudonné theorem asserts that a subspace D in a dual space X∗ is weak star closed if D ∩ BX∗ is weak star closed (cf., e.g., [80, Chapter 4]). The second part of the following result is the Corson–Lindenstrauss result from [43], the first part is in [217] and [21]. For a simple proof of a version of this statement we refer to [83]. T HEOREM 2.11. Let K be a weakly compact set in a Banach space X. Consider K in its weak topology. Then K contains a subset S that is Gδ dense in K and such that the weak and norm topologies on K coincide at each point of S. The set S in its topology from K is metrizable by a complete metric. Thus, in particular, the Gδ points of K form a dense set in K. P ROOF. Namioka proved the first part by applying Baire category arguments to the identity map from the weak topology into the norm topology on K ([217], see Section 9). Earlier, Corson and Lindenstrauss used renorming theory and Gâteaux smoothness of norms to prove the last part of the statement ([43]). The set S is metrizable by a complete metric as it is Gδ in K in its norm topology ([275, p. 164]). Each point of S is clearly a Gδ point of K as S is metrizable and dense in K. Let us illustrate the statement in Theorem 2.11 on the case K := B2 (Γ ) , where Γ is uncountable. In this case, we can put S = S2 (Γ ) , as on the unit sphere of a Hilbert space the norm and weak topologies coincide (see, e.g., Section 7). Moreover, S2 (Γ ) is a Gδ set in B2 (Γ ) in its weak topology as S2 (Γ ) = n (B2 (Γ ) \ (1 − n1 )B2 (Γ ) ). The set S2 (Γ ) is dense in B2 (Γ ) in its weak topology (cf., e.g., [80, Chapter 3, Example]). The Gδ points of B2 (Γ ) in its weak topology are exactly the points of S2 (Γ ) ([195, p. 255]).

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There is a weakly compact set K in a Banach space, such that K in its weak topology is non-metrizable and every closed metrizable subset of K is Gδ in K ([21]). There exists a weakly compact set K in a Banach space such that K in its weak topology is nonmetrizable, convex and all points of K are Gδ points of K ([195, p. 269]). However, if K is a nonseparable, weakly compact convex and symmetric set in a Banach space in its weak topology, then this cannot happen as K contains a one point compactification of an uncountable discrete set ([195], cf., e.g., the proof of Theorem 4.2). We can find by a standard argument that the ball B∞ (Γ ) in its weak star topology has no Gδ points if Γ is uncountable. Next result in this direction is the following theorem, which is a combination of the results in [40,257] and [56] (cf., e.g., [73, Chapter II.2]). Before stating the result, we note that Lipschitz Gâteaux differentiable bump functions can easily be constructed from Gâteaux differentiable norms. T HEOREM 2.12. Let K be a compact space such that C(K) admits a Lipschitz bump function that is Gâteaux differentiable. Then K is sequentially compact and K contains a dense Gδ set that is metrizable by a complete metric. P ROOF. The compact space K is sequentially compact by Theorem 2.13 below. Assuming that a Banach space X admits a Gâteaux differentiable norm, it is proved in [257] that every continuous convex function on X is Gâteaux differentiable on a Gδ dense set in X (cf., e.g., [73, p. 72], i.e., X is then a weak Asplund space. The same conclusion holds if X admits a Lipschitz Gâteaux differentiable bump function ([56,100]). In particular, under the assumptions in Theorem 2.12, the supremum norm of C(K) is Gâteaux differentiable on a dense Gδ set in C(K). It is proved in [40] (cf., e.g., [73, p. 45] that then the conclusion in Theorem 2.12 follows. For further results in this direction see Theorem 7.17, the text preceding Theorem 2.15, the text following Theorem 5.2 and Section 9. In separable Banach spaces X, the set of all points of Gâteaux differentiability of a convex continuous function on X is Gδ dense in X (Mazur, cf., e.g., [80, Chapter 8], [243, p. 12]). We will see in Section 5 that the standard norm of 1 (Γ ) is nowhere Gâteaux differentiable if Γ is uncountable. The set of all points of Gâteaux differentiability of the supremum norm of the space D (see below in this section) is a dense but not a residual set in D (cf., e.g., [73, p. 49]). Assuming the Continuum Hypothesis, Argyros and Mercourakis showed in [13] that 1 (c) admits an equivalent norm, the Gâteaux differentiability points of which form a set that is dense but not residual in 1 (c). It is shown in [149] that in a nonseparable Hilbert space H there exists a continuous convex function on H , the set of all Gâteaux differentiability points of which is not Gδ (even not Borel) in H , though it is residual in H (cf., e.g., [57, Chapter I] or [243, Chapter II]). This is in contrast with the points of Fréchet differentiability (cf., e.g., [80, Chapter 8], [168] or [243, p. 14]). The following result follows from the smooth variational principle (cf., e.g., [57, p. 9] or [53]) and from the results in [130,174] and [289] (cf., e.g., [80, Chapter 10]).


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T HEOREM 2.13. Let X admit a Lipschitz Gâteaux differentiable bump function. Then BX∗ in its weak star topology is sequentially compact and norm-dens X∗ card X. If X admits a Lipschitz Fréchet C 1 -smooth bump function, then norm-dens X∗ = dens X. w∗ P ROOF. Let {fn } be a sequence in BX∗ . For n ∈ N put An = {fn }j n and A = n An . Define the function p on X by p(x) = sup{f (x); f ∈ A}. From the smooth variational principle (cf., e.g., [57, p. 9], [80, Chapter 10] or [53]) it follows that p is Gâteaux differentiable at some x0 ∈ X. From the Gâteaux differentiability of p at x0 , it then follows that p (x0 ) is the weak star limit of a subsequence of {fn } (Theorem 2.5) (cf., e.g., [73, p. 38], [80, Chapter 8] or [57, Chapter II]). If the norm of X is Gâteaux differentiable, then the mapping x → x maps SX onto a norm dense set in SX∗ by the Bishop–Phelps theorem. This gives the second part in the statement. If the norm of X is Fréchet differentiable, then by Theorem 2.5, the mapping x → x is norm to norm continuous. Thus dens X∗ dens X. The reverse inequality holds true for all Banach spaces (cf., e.g., [80, Chapter 3]). The following argument provides for an alternative way of proving the first ˇ part of the statement if we assume the Continuum Hypothesis. First, Cech and Pospíšil showed in [36] that the cardinality of every compact space that is not sequentially compact is greater than or equal to 2ℵ1 . Thus by using the second part of the statement, assuming that the density character of X is c and that BX∗ is not weak star sequentially compact, we get card X∗ (dens X∗ )ℵ0 (card X)ℵ0 ((dens X)ℵ0 )ℵ0 = (cℵ0 )ℵ0 = cℵ0 = c = ℵ1 < 2ℵ1 card BX∗ card X∗ , a contradiction. Theorem 2.13 implies that some spaces, the dual ball of which is not weak star sequentially compact, do not admit Lipschitz Gâteaux differentiable bump functions. This applies for instance to ∞ , as we cannot extract a weak star convergent subsequence from the sequence {fn } ⊂ B∗∞ defined for {ai } ∈ ∞ by fn ({ai }) = an . Indeed, if n1 < n2 . . . , define a ∈ ∞ by ani = (−1)i and aj = 0 otherwise. Then fni (a) = (−1)i . Alternatively, one can show that ∞ does not admit any equivalent Gâteaux differentiable norm as follows: as 1 (c) is isometric to a subspace of C[0, 1]∗ , 1 (c) is isometric to a subspace of ∞ , since every space dual to a separable space is isometric to a subspace of ∞ . Hence dens ∗∞ dens 1 (c)∗ = dens ∞ (c) = 2c > c = card ∞ . Haydon showed that c∞ (ℵ1 ) does not admit any continuous (not necessarily equivalent) Gâteaux differentiable norm (cf., e.g., [57, p. 89]). The non-equivalent norm defined on ∞ by |||x|||2 = 2−i xi2 is continuous and Fréchet differentiable on ∞ . For the use of nonequivalent smooth norms in analysis on Banach spaces we refer, e.g., to [16] and [15] and references therein. We will now define several types of compacts that will be discussed in this article. D EFINITION 2.14. Let K be a compact space. Then (i) K is an Eberlein compact if it is homeomorphic to a weakly compact set in c0 (Γ ) considered in its weak topology for some set Γ . (ii) K is a uniform Eberlein compact if K is homeomorphic to a weakly compact set in 2 (Γ ) considered in its weak topology for some set Γ .

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(iii) K is a Corson compact if for some set Γ , K is homeomorphic to a subset S of [−1, +1]Γ considered in its product topology such that all points of S are countably supported (i.e., if f ∈ S, then card{α ∈ Γ ; f (α) = 0} ℵ0 ). The compact space K is scattered if each subset of K has a relatively isolated point. A tree is a partially ordered set (T , ) such that for every t ∈ T , the set {s ∈ T ; s t} is well ordered. We introduce two elements 0 and ∞ which are not in T , such that 0 < t < ∞ for every t ∈ T . If s, t ∈ T then (s, t] = {u ∈ T ; s < u t}, (0, s] = {u ∈ T ; u s}. For each t ∈ T , r(t) is the unique ordinal which has the same order type as (0, t). The height h(T ) is defined by h(T ) = sup{r(t) + 1; t ∈ T }. We will assume that T is Hausdorff, i.e., if (0, t) = (0, t ) and r(t) = r(t ) is a limit ordinal, then t = t . If t ∈ T , then T + is the set of immediate successors of t, i.e., t + = {u ∈ T ; s < u if and only if s t}. We equip T with the weakest topology τ for which all intervals (0, t] are open and closed. We will identify T = T ∪ {∞} with the Alexandrov compactification of T and denote by C0 (T ) the space of continuous functions f on T such that f (∞) = 0. The space T is a compact scattered space. The full uncountably branching tree of height ω1 is α sup αi , then χ ∈ Kαi . On the other hand, assume [0,β] 2 that for some x ∈ C0 [0, ω1 ], we have x ∈ α 0, (ii) Pα Pβ = Pβ Pα = Pmin{α,β} , P0 = 0, Pμ = Identity, (iii) the density character of Pα X is less than or equal to max{ℵ0 , |α|} for all α, and


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(iv) the map α → Pα x is continuous from the ordinal segment [0, μ] in its order topology into X in its norm topology for every x ∈ X. If the density character of X is ℵ1 , then (Pα+1 − Pα )(X) is separable for all α < μ. This makes the situation easier. For many spaces, we can decompose (Pα+1 − Pα )(X) inductively and achieve that they are separable. The price we pay for this is that we lose norm 1 on projections. This is shown in [308], cf., e.g., [57, Chapter VII]. If {Pα ; α < μ} is a PRI on X, then Pβ x ∈ span{(Pα+1 −Pα )x; α < β} for each x ∈ X and each β μ. If {Pα ; α ω1 } is a PRI for X, then by transfinite induction, each x ∈ X lies in some Pα X for α < ω1 . Indeed, given x ∈ X, the function α → (I − Pα )x is continuous all α some β, where β < ω1 . Note and equals zero at ω1 . Thus this function is zero for that (iv) is usually achieved by ensuring that Pβ X = α 0, the set Γ can be decomposed into Γ = ∞ i=1 Γi so that for every f ∈ BX ∗ and every i, card γ ∈ Γiε ; f (xγ ) > ε < ∞. Any M-basis in a subspace of a WCG space has this property.

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N (iii) A Banach space X is WCD if and only if for some a Marku Σ ⊂ N , X admits shevich basis {xα , fα }α∈Γ such that {xα }α∈Γ ∪ {0} = {LK ; K ∈ K(Σ )}, where LK is weakly compact in X and K1 ⊂ K2 implies LK1 ⊂ LK2 . Here K(Σ ) denotes the collection of all compact subsets of Σ . (iv) A Banach space X is WLD if and only if it admits a weakly Lindelöf M-basis. If X is WLD and {xα , fα } is an M-basis for X, then {xα , fα } is a weakly Lindelöf M-basis and every f ∈ X∗ is countably supported on {xα }. (v) Let X be a Banach space. Then X∗ is WL if and only if X∗ is WLD. (vi) Assume that the density of a Banach space X is ℵ1 . Then X is WLD if and only if X admits a PRI in every equivalent norm. (vii) Assume that the density of a WLD Banach space X is ℵ1 and that {Pα }, α ω1 is a PRI for X. Then α α, β < ω1 such that Pβ = Qβ . (viii) If K is a Corson compact, then C(K) admits a PRI formed by projections that are pointwise continuous and admits an M-basis that is Lindelöf in the pointwise topology. (ix) For separable X, X∗ has property C if and only if X does not contain an isomorphic copy of 1 . (x) A Banach space X is WCD if and only if there is an M-basis {xγ , fγ }γ ∈Γ for X such that Γ can be split into Γ = ∞ i=1 Γn in such a way that given γ ∈ Γ , given f ∈ BX ∗ and given ε > 0, there is n such that γ ∈ Γn and

card γ ∈ Γn ; f (xγ ) > ε < ∞. P ROOF. (i) Let a weakly compact convex and symmetric set K generate a nonseparable WCG space X. We construct a PRI of X such that Pα (K) ⊂ K. Then we use the proof of Markushevich’s result that every separable Banach space admits an M-basis ([197, p. 43]), the construction of appropriate M-bases in (Pα+1 − Pα )X and transfinite induction to get that every WCG space X admits an M-basis that is contained in 2K (cf., e.g., [80, Chapter 11]). The rest of (i) is in [46]. (ii) follows from (iv) and from the results in [93]. For a short proof we refer to [83]. (iii) was proved in [324], based on [205]. (iv) ([233,319,324]) Assume that {xα , fα } is a weakly Lindelöf M-basis for X. Let f ∈ X∗ and n ∈ N be given. Let Un = {x ∈ {xα } ∪ {0}; |f (x)| < 1/n} and Uα = {x ∈ {xα } ∪ {0}, fα (x) > 0}. Then {Un } ∪ {Uα } is an open cover of {xα } ∪ {0} in its weak topology. / Uβ if α = β, From the Lindelöf property, this cover has a countable subcover. Since xα ∈ all but countably many xα are in Un . Hence f is countably supported on {xα }. Thus X is WLD. If X is WLD, then X is WL and has a PRI (Theorem 3.8). From this we have that X has an M-basis {xα , fα }. As X is WL, its weakly closed subset {xα } ∪ {0} is weakly Lindelöf. (v) If X∗ is weakly Lindelöf, then X is Asplund. Indeed, otherwise X∗ contains an uncountable weakly discrete set by Stegall’s result ([288], see [65]). Hence X∗ admits an M-basis by Theorem 7.13. Thus X∗ is WLD by Theorem 4.4 below. (vi) (Kalenda [177]) If dens X = ℵ1 , X admits a PRI for every equivalent norm and BX∗ is not Corson in its weak star topology, then it is proven in [54] that BX∗ in its weak star topology contains a copy D of the segment [0, ω1 ]. It follows from the result in [75]


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(which originated in [95,97] and [124]) that X admits an equivalent norm · such that D ⊂ conv · E, where E is formed by supporting functionals to the ball of · in X at all points of Gâteaux smoothness of · . Since X in · admits a PRI, similarly as in the proof of Theorem 2.19 we get that the dual ball is Corson in its weak star topology, a contradiction. (vii) The first part follows from the fact that BX∗ in its weak star topology is angelic. For the second part we refer to [245]. (viii) We refer to, e.g., [57, p. 254] for a proof, originally in [14] and [316,319] where ideas independently obtained also by Gul’ko and Plichko can be found. (ix) If X contains an isomorphic copy of 1 , then X∗ contains an isomorphic copy of 1 (c) by the result of Pełczyński in [241]. As 1 (c) does not have property C, neither does X∗ . If X does not contain an isomorphic copy of 1 , BX∗∗ in its weak star topology is an angelic space ([30]) and thus X∗ has property C by Theorem 2.7. (x) This uses Sokolov’s characterization of WCD spaces ([286], see also [73, p. 130]) and is proved in [85]. For spaces with unconditional bases there are the following results of Johnson (see [271]) and Argyros and Mercourakis [13]. T HEOREM 4.3. Assume that a Banach space X admits an unconditional basis. Then (i) If X is WCG, then every unconditional basis in X is σ -weakly compact. (ii) X is WLD if and only if it does not contain an isomorphic copy of 1 (ℵ1 ). P ROOF. (i) We refer to [271]. (ii) The space 1 (ℵ1 ) is not WLD (Section 3) and thus it cannot have a copy in X if X is WLD by Theorem 3.8. If X is not WLD, then X contains an isomorphic copy of 1 (ℵ1 ) (the proof is similar to that a separable space X with unconditional basis contains a copy of 1 if X∗ is nonseparable. (James, see, e.g., [197, p. 21] or [73, p. 49]). The following result can be found in [233] and [324]. T HEOREM 4.4. Let a Banach space X admit an M-basis. Then X has property C if and only if X is WL if and only if X is WLD. P ROOF. Obviously, WL implies property C and WLD implies WL by Theorem 3.8. It remains to show that in our case, property C implies X is WLD. To this end, let S denote the collection of all elements of X∗ that are countably supported on {xα }. From Theorem 2.7 it follows that S ∩ BX∗ is weak star closed. Then by the Banach–Dieudonné theorem, S is weak star closed in X∗ as S is a subspace of X∗ . Since S contains all fα , we have S = X∗ . It is easy to see that then BX∗ in its weak star topology is a Corson compact (by considering the evaluation map of the elements of BX∗ on {xα }). If X and Y are separable infinite-dimensional Banach spaces, then there is a bounded linear one-to-one operator from X onto a dense set in Y . Indeed, if {xi , fi } and {yi , gi } are Markushevich bases in X and Y respectively with {fi } and {yi } bounded (cf., e.g., [197,

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−i p. 43]), put for x ∈ X, T x = ∞ i=1 2 fi (x)yi . Then T is a bounded linear operator from X into Y and if T x = 0, then for each i, 0 = gi (T x) = 2−i fi (x) and thus x = 0, as {xi , fi } is a Markushevich basis in X. Moreover, for each i, T (xi ) = 2−i yi and thus T (X) is dense in Y . For nonseparable Banach spaces, the existence of linear injections has a profound impact on the structure of spaces involved. For example, from the proof of Pitt’s theorem (cf., e.g., [197, p. 76]) we obtain that if Γ is uncountable, then there is no bounded linear one-to-one operator from p (Γ ) into q (Γ ) if q < p. Also, if Γ is uncountable, then there is no bounded linear one-to-one operator T from c0 (Γ ) into any reflexive Banach space X. Indeed, otherwise, T ∗ (BX∗ ) would be a weakly compact set in c0 (Γ )∗ , which would be norm compact by Schur’s theorem. Thus T ∗ (X∗ ) would be norm separable and c0 (Γ )∗ would be weak star separable as T ∗ maps X∗ onto a weak star dense set in c0 (Γ )∗ . However, c0 (Γ )∗ is not weak star separable if Γ is uncountable. If Γ is uncountable, then there is no bounded linear operator from c0 (Γ ) into ∞ . Indeed, as ∗∞ is weak star separable, we would otherwise obtain that so is c0 (Γ )∗ , which is not the case. Thus the following theorem is a useful result in this direction. T HEOREM 4.5. Assume that a Banach space X admits an M-basis. Then there is a bounded linear one-to-one operator from X into c0 (Γ ) for some Γ . If X is WCG, then there is a bounded linear one-to-one operator from X∗ into c0 (Γ ) for some Γ that is weak star to weak continuous. P ROOF. If {xα , fα }α∈Γ with {fα } bounded is an M-basis for X, define a map T from X into ∞ (Γ ) by T x(α) = fα (x) for α ∈ Γ . Then T maps span {xα } one-to-one into c0 (Γ ) and from the continuity of T and the closedness of c0 (Γ ) in ∞ (Γ ) we get that T maps X into c0 (Γ ). If {xα , fα }α∈Γ is a weakly compact M-basis for X, define a bounded linear operator T from X∗ into ∞ (Γ ) by Tf (α) = f (xα ). Let {yn } be a sequence of distinct points in {xα }. Then fα (yn ) → 0 for every α and since {xα } ∪ {0} is weakly compact, we have yn → 0 weakly in X. From this it follows that T maps X∗ into c0 (Γ ). The operator T is oneto-one and it is weak star to weak continuous on BX∗ as on bounded sets in c0 (Γ ) the weak and pointwise topologies coincide. Hence T is weak star to weak continuous by the Banach–Dieudonné theorem. T HEOREM 4.6. Every weakly compact set in a Banach space is an Eberlein compact. If X is a WCG Banach space, then BX∗ in its weak star topology is an Eberlein compact. P ROOF. If K is a weakly compact set in a Banach space X, then Z := span K is WCG, Z admits an M-basis and thus there is a bounded linear operator T from Z into some c0 (Γ ) (Theorems 4.2, 4.5). Then T is a weak homeomorphism of K onto a weakly compact subset of c0 (Γ ). Similarly one can prove the second part of the statement by using Theorem 4.2(i). By using Theorem 4.6, Theorem 3.3 and [80, Chapter 12], we can state


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T HEOREM 4.7. Let K be a compact space. Then the following are equivalent. (i) K is an Eberlein compact. (ii) K is homeomorphic to a weakly compact set in a Banach space in its weak topology. (iii) K is homeomorphic to a weakly compact set in a reflexive Banach space in its weak topology. (iv) K is homeomorphic to a pointwise compact space in C(L) for some compact space L in its pointwise topology. The following result can be found, e.g., in [80, Chapter 11]). T HEOREM 4.8. A compact set K is an Eberlein compact if and only if C(K) is WCG if and only if C(K) contains a weakly compact set that separates points of K. that a weakly compact set K ⊂ 12 Bc0 (Γ ) . Let A be the family of all finite P ROOF. Assume *n products i=1 x(γi ), x ∈ K, γi ∈ Γ , joined with the function 1. Then every sequence of distinct elements of A converges pointwise and thus weakly to zero (Grothendieck’s theorem). Thus A ∪ {0} is a weakly compact set in C(K) and span A is an algebra in C(K) that separates points of K. Thus span A = C(K) by the Stone–Weierstrass theorem. Hence C(K) is WCG if K is an Eberlein compact. If C(K) is WCG, then BC ∗ (K) is an Eberlein compact in the weak star topology (Theorem 4.6) and thus K ⊂ BC ∗ (K) is an Eberlein compact. We refer to [80, Chapter 11] for the rest of the proof. T HEOREM 4.9. (i) A Banach space X is a subspace of a WCG Banach space if and only if BX∗ in its weak star topology is an Eberlein compact if and only if C(BX∗ ) is WCG, where BX∗ is considered in its weak star topology. (ii) A Banach space X is WCD if and only if C(BX∗ ) is WCD, where BX∗ is considered in its weak star topology. P ROOF. (i) If X is a subspace of WCG space, then BX∗ is a continuous image of an Eberlein compact and thus BX∗ in its weak star topology is an Eberlein compact by Theorem 2.15. If BX∗ in its weak star topology is an Eberlein compact, then X is a subspace of the WCG space C(BX∗ ) (Theorem 4.8), where BX∗ is considered in its weak star topology. Thus (i) follows by Theorem 4.8. (ii) We refer to [73, pp. 121, 123] for the proof. Note that the operator T x = {2−i xi } maps ∞ one-to-one into c0 and yet, ∞ does not admit any M-basis. Indeed, we have the following results of Johnson in [165] and Plichko in [246] (cf., e.g., [80, Chapter 6]). T HEOREM 4.10. The space ∞ admits no M-basis but it is a complemented subspace of a Banach space Z with an M-basis. P ROOF. We will prove the first part of the statement. Assume that {xα , fα } is an M-basis in X = ∞ . Then put Y := span · {fα }. Let {yn } be a sequence in Y ∩ BX∗ . As all yn have

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countable support on {xα } we can extract a subsequence {ynk } such that ynk → y ∈ BX∗ in the weak star topology. As X∗ is a Grothendieck space ([61, p. 103]), we have ynk → y weakly and thus y ∈ Y . Hence Y is a reflexive space. Consequently Y ∩ BX∗ is weakly compact and thus weak star closed. Hence Y is weak star closed by the Banach–Dieudonné theorem. As Y contains {fα } which is weak star dense in X∗ , Y = X∗ and X is reflexive, a contradiction. Let Γ be a set and Σ be a subset of the space NN . The Banach space c1 (Σ × Γ ) is defined as the subspace of ∞ (Σ × Γ ) formed by all functions whose restrictions to K × Γ belong to c0 (K × Γ ) for all compact sets K of Σ . T HEOREM 4.11. (i) A Banach space X is WCG if and only if there is a bounded linear one-to-one operator from X∗ into some c0 (Γ ), that is weak star to weak continuous. (ii) A Banach space X is WCD if and only if there is a set Σ ⊂ NN , a set Γ and a bounded linear one-to-one operator of X∗ into c1 (Σ × Γ ) that is weak star to pointwise continuous. (iii) X is WLD if and only if there is a bounded linear one-to-one operator from X∗ into some c∞ (Γ ) that is weak star to pointwise continuous. P ROOF. (i) If X is WCG, then the statement follows from Theorem 4.5. If such T exists, then T ∗ (B1 (Γ ) ) ⊂ X is weakly compact and generates X. (ii) was proved in [205]. (iii) If such T exists, then, clearly, BX∗ in its weak star topology is a Corson compact. On the other hand, let {xα , fα }α∈Γ be a weakly Lindelöf M-basis for X (Theorem 4.2). By Theorem 4.2(ii), each element of X∗ is countably supported on {xα }α∈Γ . Assuming that {xα } is bounded, the operator T from X∗ into c∞ (Γ ) defined by Tf (α) = f (xα ), α ∈ Γ , satisfies the requirements. We will now show the missing parts in the proof of Theorem 3.8. First of all, if X is WCD, then it has PRI and an M-basis ([325]). By the proof of Theorem 3.8, X is WL. Thus it follows from Theorem 4.4 that WCD implies WLD. We will now show that X is DENS if X is WLD. For simplicity, let us prove that X is separable if X is WLD and X∗ is weak star separable. Let a countable S ⊂ X∗ be weak star dense in X∗ . Let {xα , fα }α∈Γ ∗ be an M-basis for X. By Theorem 4.2(ii), each element of X is ∗countably supported on {xα }. We note that Γ = f ∈S {α ∈ Γ ; f (xα ) = 0}. As each f ∈ X is countably supported on {xα } and S is countable, we get that Γ is countable. Thus X is separable. Before proceeding, we recall that by a subspace of a Banach space we mean a closed subspace. D EFINITION 4.12. A subspace Y of a Banach space X is said to be quasicomplemented in X if there is a subspace Z of X such that Y ∩ Z = {0} and Y + Z = X. The subspace Z is called a quasicomplement of Y in X. Murray showed in [216] that any subspace of a reflexive separable Banach space X is quasicomplemented in X. This result was extended to all separable spaces X by Mackey


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in [199]. The following result in [319] and [323] is a variant of the result of Lindenstrauss in [193]. T HEOREM 4.13. Assume that X is a WLD space and Y is a subspace of X. Then Y is quasicomplemented in X. P ROOF. If X is separable and {xi , fi } is a Markushevich basis of Y , then fi can be extended to X and {zi } ∈ X and gi ∈ X∗ added such that {xi , zi , fi , gi } is a Markushevich basis of X ([128], cf., e.g., [80, Chapter 11]). Then Z := span{zi } is a quasicomplement of Y in X. Indeed, if w ∈ Y ∩ Z, then fi (w) = 0 = gi (w) for all i. Thus w = 0 as {fi } ∪ {gi } separate the points of X. Having this proved, we proceed by transfinite induction, using projectional resolution of identity {Pα } of X such that Pα Y ⊂ Y . Indeed, if Zα is a quasicomplement of (Pα+1 − Pα )Y in (Pα+1 − Pα )X, then span( Zα ) is a quasicomplement of Y in X. If Z is a Banach space, a sequence (yn ) ⊂ Z ∗ is called w∗ -basic provided that ∗ there exists (xn ) ⊂ Z biorthogonal to (yn ) such that for each y ∈ span w {yn }, y = n ∗ limn i=1 y(xi )yi in the weak star topology of Z ([197, p. 10]). The following is a result of Lindenstrauss and Rosenthal (cf. [166]). T HEOREM 4.14. Assume that Y is a subspace of a Banach space X such that Y ∗ is weak star separable and X/Y has an infinite-dimensional separable quotient. Then Y is quasicomplemented in X. P ROOF. Since X/Y has a separable quotient, there exists a biorthogonal sequence (xn , xn∗ ) in X with (xn∗ ) ⊂ Y ⊥ , (xn∗ ) w∗ -basic and such that xn = 1 for each n ([197, p. 11]). As Y ∗ is w∗ -separable, a biorthogonalization argument gives that there exists a biorthogonal sequence (yn , yn∗ ) for Y with (yn∗ ) ⊂ X∗ , Y ∩ (y ∗ )⊥ = {0} and normalized so that yn∗ = 1 for every n ([197, p. 43]). Define an operator T from X into X by T x = ∞ −n−1 y ∗ (x)x . Then T 1/2 and hence I + T is an isomorphism on X. Thus n n n=1 2 (I + T )∗ is a weak star isomorphism on X∗ . Therefore (xn∗ + T ∗ xn∗ ) is a weak star basic sequence weak star equivalent to (xn∗ ). We have T ∗ xn∗ = 2−n−1 yn∗ for every n. We claim that (xn∗ + 2−n−1 yn∗ )⊥ is a quasicomplement of Y in X. In order to see this, let ∗ x ∗ ∈ Y ⊥ ∩ span w {xn∗ + 2−n−1 yn∗ }. Then x ∗ = limn ( ni=1 αi xi∗ + ni=1 2−n−1 αi yi∗ ) in the weak star topology, for some sequence (αi ) of scalars. As x ∗ ∈ Y ⊥ , for each n, x ∗ (yn ) = 2−n−1 αn = 0. Thus x ∗ = 0, showing that Y + (xn∗ + 2−n−1 yn∗ )⊥ is dense in X. Let now y ∈ Y ∩ (xn∗ + 2−n−1 yn∗ )⊥ . Then xn∗ (y) = 0 for each n as y ∈ Y . Hence yn∗ (y) = 0 for each n. This shows y ∈ (yn∗ )⊥ ∩ Y = {0}. As we mentioned in Section 4, c0 is not complemented in ∞ . This should be compared with the following result of Rosenthal [270]. T HEOREM 4.15. The space c0 is quasicomplemented in ∞ .

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ˇ P ROOF. Let βN denote the Stone–Cech compactification of the discrete space N. Then βN \ N is a perfect compact space. Hence there is a continuous map ϕ from βN \ N onto [0, 1] (cf., e.g., [185, p. 29]). From this it follows ([270]) that L1 [0, 1] is isometric to a subset of C(βN \ N)∗ = c0⊥ ⊂ ∗∞ . Hence c0⊥ contains an isomorphic copy of 2 by Kintchine’s inequality (see, e.g., [80, Chapter 6]). As this copy is weak star closed (2 is reflexive), we get that 2 is isomorphic to a quotient of ∞ /c0 . Hence Theorem 4.15 follows from Theorem 4.14. The following result extends Johnson’s result ([166]) and is from [161]. T HEOREM 4.16. Let X be a WCG Asplund space and Y ⊂ X∗ be a WCG subspace of X∗ . Then Y has a weak star closed quasicomplement in X∗ . The following result of Lindenstrauss is from [194]. T HEOREM 4.17. If Γ is uncountable, then c0 (Γ ) is not quasicomplemented in ∞ (Γ ). P ROOF. Assume that Z is a quasicomplement of c0 (Γ ) in ∞ (Γ ). Let π be the quotient map of ∞ (Γ ) onto ∞ (Γ )/Z. Consider the restriction of π to c0 (Γ ) and call it T . Then T (c0 (Γ )) is dense in ∞ (Γ )/Z and thus ∞ (Γ )/Z is WCG. Hence B(∞ /Z)∗ in its weak star topology is sequentially compact (Theorem 4.6 and the Eberlein–Šmulyan theorem). If yn := π ∗ (xn ), xn ∈ B(∞ (Γ )/Z)∗ , let {xnk } be a weak star convergent subsequence of {xn }. Then {ynk } is weak star convergent in ∞ (Γ )∗ . As ∞ (Γ ) has the Grothendieck property, {ynk } is weakly convergent. Hence π ∗ is a weakly compact operator and so is π . The same is true for T . Hence T ∗ : (∞ (Γ )/Z)∗ → 1 (Γ ) is a weakly compact operator and thus norm compact operator as 1 (Γ ) has the Schur property. Thus T ∗ (∞ (Γ )/Z)∗ is a norm separable subset of 1 (Γ ). As T is one-to-one (use the definition of the quasicomplement), T ∗ maps ∞ (Γ )/Z onto a weak star dense set in c0 (Γ ). This means that c0 (Γ )∗ is weak star separable, which is not the case. The standard unit vector basis of c0 (Γ ) is a shrinking M-basis for c0 (Γ ) for every Γ . From the PRI in C[0, ω1 ] discussed in Section 2 we easily construct an M-basis for this space. None of the spaces D, JL0 and JL2 contain a nonseparable subspace with an M-basis. Indeed, such subspace Y then has property C and thus is WLD (Theorem 4.4). Therefore its dual Y ∗ is not weak star separable (Theorem 3.8). However, Y ∗ is a quotient of one of the weak star separable spaces D ∗ , JL∗0 , or JL∗2 , a contradiction. The spaces JL0 and JL2 admit C ∞ smooth norms (Section 8). However, they do not contain any of p (Γ ), p > 1, or c0 (Γ ) for Γ uncountable. Thus Deville’s theorem on containment of p or c0 in C ∞ -smooth spaces ([50], cf., e.g., [57, Chapter V] or [111]) has no nonseparable analogue. The space JL0 , being a subspace of ∞ , is a subspace of a space with an M-basis (Theorem 4.10). Kunen’s C(K) space does not admit any Mbasis (Section 2). The space JT ∗ does not admit any M-basis as it has property C and is not WLD since it is nonseparable with weak star separable dual. Ciesielski–Pol space CP admits no M-basis as it does not inject into any c0 (Γ ) (Theorem 4.5). The subspace c0


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is quasicomplemented in JL2 and also in JL0 , by Theorem 4.14 as both c0 (c) and 2 (c) contain infinite-dimensional separable complemented subspaces. The subspace C[0, 1] is quasicomplemented in D for the same reason. It follows from the recent results in [172] that if X is a nonseparable WCG subspace of ∞ (Γ ) such that X does not contain an isomorphic copy of 1 , then X is not quasicomplemented in ∞ (Γ ).

5. Gâteaux smooth and rotund norms D EFINITION 5.1. The norm · of a Banach space X is strictly convex or rotund (R) if x = y for every x, y ∈ X such that 2x2 + 2y2 − x + y2 = 0. Thus · is R if and only if its unit sphere contains no non-degenerate line segments. Every separable Banach space admits Gâteaux differentiable and rotund norms (see Theorems 5.2, 5.3 below). This is no longer the case for nonseparable spaces. Most of the renormings by Gâteaux differentiable norms are done by renorming the dual spaces by dual rotund norms. This is because majority of C 1 smooth renormings use the fact that the sum of two convex bodies in a finite-dimensional space is C 1 smooth if one of them is. In infinite-dimensional spaces such sums need not be closed. It is the strict convexity of the dual norm that removes this problem and gives a Gâteaux differentiable norm on spaces. Indeed, let B1 and B2 be the unit balls of the norms · 1 and · 2 on X. Assume that the dual norm of · 2 is rotund. Let · 3 be the norm the unit ball of which is B1 + B2 . Then the dual norm of · 3 is the sum of the dual norms of · 1 and · 2 as f 3 = sup{f (x); x ∈ B1 + B2 } = sup{f (x); x ∈ B1 } + sup{f (x); x ∈ B2 } = f 1 + f 2 . It is easy to see that the sum of two norms is strictly convex if one of them is (cf. Theorem 5.2 below). Thus the dual norm · 3 is then strictly convex, though the algebraic sum of B1 and B2 need not in general be closed. We use the fact that the norm is Gâteaux smooth if the dual norm is strictly convex (cf., e.g., [168]). Indeed, using the remark following Theorem 2.5, if x ∈ SX and f, g ∈ SX∗ are such that f (x) = g(x) = 1, then 2 f + g (f + g)(x) = 2 and from the rotundity of the dual norm we get f = g. Before proceeding, we note that in renorming theory, a new norm ||| · ||| for a Banach space X with the original norm · is constructed such that for some constant C > 0, Cx |||x||| C1 x for every x ∈ X. Many constructions are such that the constant C can be chosen arbitrarily close to 1. We will not explicitly mention this fact in the constructions in this article. On the other hand, please see Problem 5 in Section 9. The following is Mercourakis’ result in [205], cf., e.g., [57, p. 288]. T HEOREM 5.2. Let X be a WCD space. Then X admits an equivalent norm whose dual norm is rotund. In particular, every WCD space admits an equivalent Gâteaux differentiable norm.

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P ROOF. First, we will describe a strictly convex norm on c0 (Γ ) for any Γ . An example of such norm is Day’s norm ([47], cf., e.g., [57, p. 69]) defined for x = {xγ } ∈ c0 (Γ ) by x = sup

n

1/2 2

x (γk )/4

k

,

k=1

where the supremum is taken over all n ∈ N and all ordered n-tuples (γ1 , . . . , γn ) of distinct elements of Γ . We need to show that Day’s norm is rotund. To this end, we first make the following simple observation: > n and |a| < |b|, then a 2 /n2 + b 2 /m2 < b2 /n2 + ∞ 2 if jm1/2 2 2 a /m . Thus x = ( j =1 xγj /4 ) , where γj are distinct and such that |xγ1 | |xγ2 | · · ·. Calling such a sequence {γj } an appropriate sequence for x, we have that if x + y = x + y, x = y = 1 and {γj } is an appropriate sequence for x + y, then 2 = x + y =

∞ (x + y)2 γj

0. Thus it is not true that limt →0 1t (x + teα + x − teα − 2) = 0, which means that the norm is not Gâteaux differentiable at x in the direction eα . From this and from the smooth variational principle ([58], cf., e.g., [57, p. 9], [80, Chapter 10], [53]) it follows that if Γ is uncountable, then 1 (Γ ) admits no Lipschitz Gâteaux smooth bump function. The same conclusion holds for ∞ and ∞ /c0 as the lim sup function produces a convex function that is nowhere Gâteaux differentiable (see, e.g., [57, Chapter I] or [80, Chapter 8, Example]). There is a WLD space X such that the dual norm of X∗ is rotund and yet, X is not WCD ([14]). Note that in the proof of Theorem 5.2 we proved the following result, which goes back to Clarkson [39] and Day [47], cf., e.g., [57, p. 46]. T HEOREM 5.3. Let X and Y be Banach spaces such that there is a bounded linear oneto-one operator from X into Y . Assume that the norm of Y is rotund. Then X admits an equivalent rotund norm. In particular, X admits an equivalent rotund norm whenever there is a bounded linear one-to-one operator from X into c0 (Γ ) for some Γ . The latter happens if X∗ is weak star separable or if X has an M-basis. P ROOF. If {fn } is weak star dense in X∗ , then the bounded linear operator T from X 1 into c0 defined by T x(n) = 2n f fn (x) is one-to-one. If X has an M-basis, we use Theon rem 4.5. The first example of a Banach space X that admits a rotund norm but does not admit any bounded linear one-to-one operator into any c0 (Γ ) was constructed in [44]. One of the spaces of this type constructed in [44] consists of the subspace of ∞ [0, 1] formed by all functions f in ∞ [0, 1] such that for every ε > 0, the second derived set of the set {t ∈ [0, 1]; |f (t)| ε} is empty. The part (i) of the following result was proved by Talagrand in [295]. The part (ii) is due to Partington [238] (cf., e.g., [57, Chapter II.7]). Both results belong to the area of the so called distorted norms, see, e.g., [197, p. 97], [230].

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T HEOREM 5.4. (i) Let ||| · ||| be an equivalent norm on ∞ and · be the standard supremum norm on ∞ . Then there is δ > 0 such that for every n ∈ N, there is a subspace Xn of ∞ isomorphic to ∞ and such that on Xn ,

δ − 2−n · ||| · ||| δ + 2−n · . (ii) If Γ is uncountable, then ∞ (Γ ) in any equivalent norm contains an isometric copy of ∞ in the supremum norm. Thus ∞ (Γ ) admits no strictly convex norm if Γ is uncountable. In particular, there is no bounded linear one-to-one operator from ∞ (Γ ) into c0 (Γ ) if Γ is uncountable. Bourgain showed in [29] that ∞ /c0 admits no equivalent R-norm. We will see in Theorem 8.3 that C[0, ω1 ] has an equivalent C ∞ -smooth norm. However, Talagrand showed in [297] (cf., e.g., [57, p. 313]) the following result. T HEOREM 5.5. There is no equivalent norm on C[0, ω1 ] such that its dual is a rotund norm. P ROOF. Assume that ||| · ||| is a dual rotund norm on C[0, ω1 ]∗ . For α ∈ [0, ω1 ], let δα be the Dirac measure corresponding to α. Then the function α → |||δα ||| is lower semicontinuous on [0, ω1 ]. Thus it is constant, equal to, say, a on a closed cofinal subset A of [0, ω1 ). For α ∈ A, let α be the successor of α in A. The map α → |||(δα + δα )/2||| is lower semicontinuous on A and thus equal to say b on a closed cofinal subset B of A. Let (αn ) be a strictly increasing sequence in B such that α = lim αn . Using the facts that δα is a weak star limit of (δαn + δαn )/2 and that ||| · ||| is weak star lower semicontinuous, we get that lim |||(δαn + δαn )/2||| |||δα |||. Hence b a. On the other hand, |||(δα + δα )/2||| |||δα |||/2 + |||δα |||/2 = |||δα |||. Hence a = b and for every α ∈ B,

(δα + δα )/2 = |||δα ||| + |||δα ||| /2 = a. The following result of Haydon ([143]) contains a key argument in some questions on renorming spaces of continuous functions on trees by smooth norms. The result says that there is no “too flat” Gâteaux smooth norm on C0 [0, ω1 ]. This is then used in Theorem 5.7. T HEOREM 5.6. Let · be an equivalent norm on C0 [0, ω1 ] that satisfies (∗) x + λχ(β,γ ] = x whenever supp x ⊂ [0, β], β < γ < ω1 and 0 λ x(β). Then · is not a Gâteaux differentiable norm on C0 [0, ω1 ]. P ROOF. In fact, assuming that a Gâteaux differentiable norm · · ∞ on C0 [0, ω1 ] satisfies (∗), we construct, by transfinite induction, xα , α < ω1 in C0 [0, ω1 ] such that supp xα ⊂ [0, α]; if β α < γ , then xγ (β) = xα (β); x∞ = xα (α); (xα ∞ ) is strictly increasing and xα − x0 12 (xα ∞ − x0 ∞ ). This is a contradiction as there is no strictly increasing function on [0, ω1 ). Theorem 5.6 can be compared with the result in [79] that C0 [0, ω1 ] admits no lattice Gâteaux renorming and with the result in [81] that C[0, ω1 ] admits no equivalent Gâteaux


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differentiable norm that would be lower semicontinuous in the topology of pointwise convergence on [0, ω1 ). The following is Haydon’s result in [143]. T HEOREM 5.7. Let T be a full uncountably branching tree of the height ω1 and let · be an equivalent norm on C0 (T ). Then there is a subspace of C0 (T ) which is isometric to C0 [0, ω1 ] equipped with a norm that satisfies property (∗) from Theorem 5.6 (cf., e.g., [57, p. 323]). Thus C0 (T ) does not admit any equivalent Gâteaux smooth norm though C0 (T ) is an Asplund space. For a similar reason, the space C0 (T ) does not admit any equivalent rotund norm either. P ROOF. Any norm on C0 [0, ω1 ] that satisfies (∗) is not rotund.

Preiss proved in [256] the following result. T HEOREM 5.8. Every Lipschitz function is Gâteaux (Fréchet) differentiable on a dense set in X if X admits a Gâteaux (Fréchet) differentiable norm. For every Γ , the space c0 (Γ ), C[0, ω1 ], JL0 and JL2 all admit C ∞ smooth norms. This will be discussed in Section 8. However, the space C[0, ω1 ] admits no equivalent Gâteaux differentiable norm that would be pointwise lower semicontinuous for points in [0, ω1 ) ([81]). The space D = C(K) does not admit Lipschitz Gâteaux differentiable bump function, in particular it admits no Gâteaux differentiable norm ([297], cf., e.g., [57, p. 303], [73, p. 46]). This follows from the fact that, otherwise, by Theorem 2.12, the compact space K would contain a dense Gδ completely metrizable set, which is not the case (see Section 2). The space c0 (Γ ) admits a rotund norm by the proof of Theorem 5.2. The spaces JL0 , JL2 and D all admit rotund norms by Theorem 5.3 as their duals are weak star separable. The space C[0, ω1 ] admits a rotund norm by Theorem 7.3 below. The space JT ∗ admits a dual Gâteaux differentiable norm ([133]). The paper [133] contains the study on separable spaces that admit norms whose second dual norm is rotund. Note that X is reflexive if the fourth dual norm of X is rotund (Dixmier). In fact, X is reflexive if the third dual norm of X is Gâteaux differentiable (Giles, Kadets, Phelps, cf., e.g., [80, Chapter 8, Example]). For information on spaces that admit norms whose third conjugate is rotund we refer to, e.g., [283] and [285]. It was proved in [82] that a WLD space X of density ℵ1 is WCG if and only if there is a bounded set S in X with span S = X and there is an equivalent Gâteaux differentiable norm · on X such that the derivative of · at each point of the unit sphere is uniform in the directions in S. 6. Uniformly Gâteaux smooth norms D EFINITION 6.1. The norm · of a Banach space X is uniformly Gâteaux differentiable (UG) if for each h ∈ SX , lim

t →0

1 x + th + x − th − 2 = 0 t

uniformly in x ∈ SX .

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The following result can be found, e.g., in [57, p. 65]. T HEOREM 6.2. If a Banach space Y has a uniformly Gâteaux differentiable norm and a bounded linear operator T maps Y onto a dense set in a Banach space X, then X admits a uniformly Gâteaux differentiable norm. P ROOF. The result follows from Šmulyan’s characterization of UG norms which says that · of X is UG if and only if its dual norm is weak star uniformly rotund (W ∗ UR), i.e., whenever fn , gn ∈ X∗ are such that {fn } is bounded and 2fn 2 + 2gn 2 − fn + gn 2 → 0, then fn − gn → 0 in the weak star topology of X∗ (cf., e.g., [57, p. 63]). Indeed, define the norm on X∗ by |||f |||2 = f 2 + T ∗ f 21 , where · is the original norm of X∗ and · 1 is a weak star uniformly rotund norm on Y ∗ . Like in the proof of Theorem 5.2, we get that ||| · ||| is W∗ UR. If X is separable and {xn } is dense in SX , then the operator T : X∗ → 2 defined for f ∈ X∗ by Tf (i) = 2−i f (xi ) is dual to an operator that maps 2 onto a dense set in X as T is one-to-one. Thus we get Šmulyan’s classical result: C OROLLARY 6.3. Every separable Banach space admits an equivalent UG norm. The space c0 (Γ ) admits an equivalent UG norm for any Γ (use the “identity”operator of 2 (Γ ) into c0 (Γ ) and the fact that the norm of Hilbert space is UG (Theorem 6.2)). The following theorem is the result of Troyanski in [311]. T HEOREM 6.4. Assume that {eα , fα }α∈Γ is a normalized unconditional basis for a Banach space X. Then X admits an equivalent UG norm if and only if for every ε > 0, the set Γ can be decomposed into Γ = Γiε such that for each i and for distinct {γj }ij =1 ⊂ Γiε , we have ij =1 eγj εi. The following result is in [76]. T HEOREM 6.5. Let X be a Banach space with an equivalent uniformly Gâteaux smooth norm. Then X is a Kσ δ subset of (X∗∗ , w∗ ), i.e., X = n1 m1 Km,n , where Km,n are some weak∗ compact sets in X∗∗ . In particular, X is then a WCD space. P ROOF. Assume that · is an arbitrary equivalent norm on X. Pick any G ∈ X∗∗ \X. Let H = G−1 (0) be the subspace of X∗ consisting of all the elements of X∗ that vanish at G. The space H is a norming subspace of X∗ , that is, there is δ > 0 such that for all x ∈ SX , sup{|f (x)|; f ∈ H, f 1} δ (cf., e.g., [80, Chapter 3, Example]). Using the idea in [311] and [209], for any equivalent norm · on X, we define for all n, p ∈ N the subsets Sn,p ( · ) in X as follows:

Sn,p · = x ∈ X; f (x) − g(x) 1/p whenever f, g ∈ X∗ , f 1, g 1 and f + g > 2 − 2/n .


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We now assume that · is an equivalent uniformly Gâteaux smooth norm on X. Thus for any p ∈ N, one has

Sn,p · = X.

(1)

n1

We will show that X=

∗ Sn,p · .

p1 n1

It follows from (1) that it suffices to prove that for any G ∈ X∗∗ \X there is a p ∈ N such that G∈ /

∗ Sn,p · .

(2)

n1

We set H = G−1 (0), and define an equivalent norm q on X by the formula q(x) = sup f (x); f ∈ H, f 1 . We claim that

Sn,p · ⊂ Sn,p (q). In order to prove this claim, we observe the bipolar theorem (cf., e.g., [80, Chapter 4]) implies that the dual unit ball Bq ∗ satisfies ∗ Bq ∗ = f ∈ X∗ ; q ∗ (f ) 1 = f ∈ H, f 1 .

(3)

Therefore if q ∗ (f ) 1, q ∗ (g) 1 and q ∗ (f + g) > 2 − 2/n, there are nets (fα ) and (gα ) in H , weak* convergent to f and g respectively, such that fα 1 and gα 1 for all α. Since the norm q ∗ is weak*-lower semicontinuous, one has q ∗ (fα + gα ) > 2 − 2/n when α is large enough. As · and q ∗ coincide on H , fα + gα > 2 − 2/n for large α. If now x ∈ Sn,p ( · ), we have |fα (x) − gα (x)| 1/p for α large enough. It thus follows that x ∈ Sn,p (q). This shows our claim. In order to prove (2), it therefore suffices to show that one has G∈ /

Sn,p (q)

∗

n1

for p ∈ N large enough. To this end, choose p ∈ N such that p > 1/q ∗∗ (G). Fix n ∈ N and ∗ set for simplicity Sn,p (q) = S. We need to show that G ∈ /S .

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Pick f ∈ Bq ∗ such that G(f ) > 1/p, and let x ∈ X be such that q(x) 1 and f (x) > 1 − 1/n. By (3), there is g ∈ H with q ∗ (g) 1 and g(x) > 1 − 1/n. We have then q ∗ (f + g) (f + g)(x) > 2 − 2/n. ∗

From the definition of S, for all z ∈ S one has (f − g)(z) 1/p. Thus if G ∈ S , then G(f − g) 1/p. This contradicts the fact that G(f − g) = G(f ) > 1/p. ∗ For every p ∈ N, let the family {Kp,q }q be equal to the reindexed family {S n,p ∩ ∗∗ mBX∗∗ }n,m . The family {Kp,q } can be used to verify that X is Kσ δ in X in its weak∗ topology. Theorem 6.5 is proved. The proof of Theorem 6.5 gives the following result ([77]): Assume that M is a bounded set in a Banach space X whose norm satisfies the following property: supx∈M (fn − gn )(x) → 0, whenever fn , gn ∈ SX∗ are such that fn + gn → 2. Then M is weakly relatively compact. UG norms are related to uniform Eberlein compacts. We summarize some of the results in this direction in the following theorem. T HEOREM 6.6. Let X be a Banach space. Then the following are equivalent. (i) The space X admits an equivalent UG norm. (ii) The dual ball BX∗ in its weak star topology is a uniform Eberlein compact. (iii) There is a set Γ and a bounded linear operator from 2 (Γ ) onto a dense set in C(BX∗ ), where BX∗ is considered in its weak star topology. (iv) There is a Markushevich basis {xα , fα }α∈Γ of X such that for every ε > 0, there is a partition Γ = i∈N Γiε and there are integers mεi such that for every i ∈ N and every f ∈ BX∗ , one has card γ ∈ Γiε ; f (xγ ) > ε mεi . P ROOF. The implication (i) ⇒ (ii) was proved in [78] under the additional assumption that X has a PRI in its UG norm. This assumption is redundant due to Theorem 6.5 and Theorem 3.8. The implication (ii) ⇒ (iii) is proved in [21] (cf., e.g., [80, Chapter 12]). The implication (iii) ⇒ (i) follows from Theorem 6.1. (ii) ⇒ (iv) This follows from Theorem 4.2 and Theorem 2.9 in [93]. (iv) ⇒ (ii) This follows from Theorem 2.9 in [93]. The following result is in [76]. C OROLLARY 6.7. Let K be a compact space. Then K is a uniform Eberlein compact if and only if C(K) admits an equivalent UG norm. P ROOF. Let K be a uniform Eberlein compact. Then BC(K)∗ in its weak star topology is a uniform Eberlein compact [21]. Then C(K) admits an equivalent UG norm by Theorem 6.6.


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On the other hand, if C(K) admits a UG norm, then BC(K)∗ in its weak star topology is a uniform Eberlein compact by Theorem 6.6 and so is its closed subspace K. Note that Corollary 6.7 gives a short proof to the result that a continuous image of a uniform Eberlein compact is a uniform Eberlein compact (see Theorem 2.15). It also gives an alternative proof to the result in [184] on the existence of reflexive spaces with no equivalent UG norm. For further results in this area we refer to, e.g., [77]. It is proved there the following result: for a Banach space X of density ℵ1 , the following are equivalent: (1) X admits an equivalent norm that is UG uniformly on a set of directions h in a bounded set M with span M = X. (2) There is a bounded linear weak star to weak continuous operator T from X∗ into some c0 (Γ ) such that for every ε > 0 there is a natural number k such that card{γ ∈ Γ ; |Tf (γ )| > ε} k for every f ∈ BX∗ . There also a smoothness characterization is given of spaces X such that 2 (Γ ) can be mapped onto a dense subset in X (Hilbert generated spaces). The following is Hájek’s result [133], which solved the problem of Troyanski posed in one Frolík’s Winter School in the Czech Republic in the 70’s. For a simple proof of it we refer to [77]. C OROLLARY 6.8. If the norm of a Banach space X is weakly uniformly rotund, then X is an Asplund space. A norm · on X is weakly uniformly rotund if xn − yn → 0 weakly in X whenever xn , yn ∈ SX are such that xn + yn → 2. P ROOF OF C OROLLARY 6.8. If X is a separable space with weakly uniformly rotund norm, then X∗ has a UG norm by the Šmulyan lemma. Then X∗ is a subspace of a WCG space by Theorem 6.6 and Theorem 4.9. Thus X∗ is a DENS space by Theorem 3.8. This means that X∗ is separable. The norm · of a Banach space X is uniformly rotund in every direction (URED) if xn − yn → 0 whenever {xn }, {yn } are bounded sequences in X, 2xn 2 + 2yn 2 − xn + yn 2 → 0 and for some z ∈ SX and λn ∈ R, xn − yn = λn z. The standard norm of every Hilbert space is URED. The proof of Theorem 5.2 gives that X admits an equivalent URED norm if there is a bounded linear one-to-one operator from X into a Hilbert space. For necessary and sufficient conditions for renorming spaces with unconditional bases by URED norms we refer to [311]. If Γ is uncountable, then c0 (Γ ) does not admit any equivalent URED norm. Indeed, let | · | be an equivalent norm on c0 (Γ ), denote by · the supremum norm of c0 (Γ ) and put M = sup{|x|; x 1}. Let un 1 be such that |un | → M. Choose z = 1 such that its support is disjoint from the union of the supports of all un . Then put xn := un + 12 z, yn := un − 12 z. Then xn 1 and yn 1 for all n and thus |xn | M and |yn | M for all n. As | 12 (xn + yn )| → M, we get |xn | → M and |yn | → M. This gives that | · | cannot be URED. The URED norms have been used in the fixed point theory for non-expansive mappings and in the study of uniform Eberlein compacts. We refer to [57, p. 67], [12] and references therein for more information in this

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direction. For applications of URED norms in geometry of spaces with symmetric bases we refer to [310]. Rychtáˇr showed in [276] that a Banach space X with unconditional basis admits an equivalent UG norm if X∗ admits an equivalent URED norm. However, he showed in [277] that for example JL2 admits an equivalent norm whose dual is URED. The following examples due to Hájek, Kutzarova and Troyanski can be found in [133, 131,77] and [184]. E XAMPLES . (i) There is a reflexive Banach space that does not admit any equivalent UG norm. (ii) There is a reflexive Banach space X with a UG norm such that there is no bounded linear operator from any Hilbert space onto a dense subset of X. (iii) There is an equivalent norm on JT space the second dual of which is URED, though there is no UG norm on JT ∗ . (iv) There is a non-reflexive separable Banach space with a norm the bidual norm of which is UG. I do not know of any Banach space X the dual of which would admit a Gâteaux differentiable norm but admits no dual Gâteaux differentiable norm. Unlike the situation with Gâteaux or Fréchet differentiable bumps (see Section 8), any space that admits a uniformly Gâteaux differentiable bump function admits an equivalent UG norm ([299], cf., e.g., [91]).

7. Fréchet smooth and locally uniformly rotund norms D EFINITION 7.1. (i) The norm · of a Banach space X is locally uniformly rotund (LUR) if xn − x → 0 whenever xn , x ∈ X are such that 2x2 + 2xn 2 − xn + x2 → 0. (ii) The norm · of a Banach space X is a Kadets–Klee norm or has the Kadets–Klee property if the norm and weak topologies coincide on the unit sphere of · . If · is LUR, then it is a Kadets–Klee norm. Indeed, if xα , x ∈ SX , xα → x weakly and f ∈ SX∗ is such that f (x) = 1, then 2 lim sup xα + x lim inf xα + x lim inf f (xα + x) = 2. Thus xα − x → 0 as · is LUR. It is not difficult to show that on the unit sphere of the standard norm of 1 (Γ ), the weak star and norm topology coincide (cf., e.g., [57, p. 72]). Šmulyan’s lemma (Theorem 2.5) implies that the norm is Fréchet differentiable if its dual norm is LUR. Indeed, let the dual norm be LUR and x ∈ SX be given. Let fn , gn ∈ SX∗ and f0 ∈ SX∗ be such that f0 (x) = 1 and fn (x) → 1, gn (x) → 1. Then 2 fn + f0 (fn + f0 )(x) → 2 and from LUR we get fn − f0 → 0. Similarly we get gn − f0 → 0. Thus fn − gn → 0, which, by Theorem 2.5, means that the norm of X is Fréchet differentiable at x. If X is a separable Banach space, then X admits an equivalent locally uniformly rotund norm. If X∗ is separable, then X admits an equivalent norm the dual of which is locally uniformly rotund. Both of these fundamental results are due to Kadets (cf., e.g., [111], [57, p. 48], [80, Chapter 8]). If X is separable, then X admits a Fréchet smooth norm if and only if X∗ is separable (cf., e.g., [111], [57, p. 51], [80, Chapter 8]). This follows from


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Kadets’ result above and from the Bishop–Phelps theorem. We will assume these results in the sequel. Therefore we will not discuss the proofs of some separable versions of the results, whenever no misunderstanding can occur. If · is a Fréchet differentiable norm on X, then the map Φ : x → x is Fréchet 1 C -smooth on SX by Šmulyan’s lemma. By the Bishop–Phelps theorem, Φ maps SX onto a dense set in SX∗ . A similar result is obtained by using Fréchet C 1 -smooth bump functions. Instead of the Bishop–Phelps theorem one can use here the following argument. Let b be a C 1 -smooth bump function on X and ϕ = b−2 , where b = 0 and ϕ = +∞ elsewhere. If f ∈ X∗ , then ψ = ϕ − f satisfies the assumptions of the smooth variational principle (cf., e.g., [57, Chapter I], [53]) and thus there is Fréchet C 1 -smooth function h on X and a point x0 in X such that ψ − h attains its minimum on X and the norm of the first derivative of h at all points of X is smaller than a given number ε. Then (ϕ − f − h )(x0 ) = 0 and hence we get that {ϕ (x); x ∈ X} is norm dense in X∗ . Thus we obtain that X is an Asplund space if X admits a Lipschitz Fréchet C 1 -smooth bump function (cf., e.g., [243, p. 66]). Indeed, if X admits a Lipschitz Fréchet C 1 -smooth bump function b, we can suppose that b(0) = 0 and then, given a separable subspace Y of X, the restriction of b to Y is a Lipschitz Fréchet C 1 -smooth bump function on Y and thus {ϕ (xn )} is norm dense in Y ∗ whenever ϕ is constructed as above from b and {xn } is a dense sequence in Y . In fact, the Lipschitz property of b is not needed in showing that X is Asplund, namely it follows that X is an Asplund space if X admits a Fréchet differentiable bump function (cf., e.g., [57, p. 58]). No example of an Asplund space is known that does not admit a (Lipschitz) Fréchet C 1 -smooth bump function (Problem 1 below, see, e.g., [57, p. 89]). The following is the result in [259] (cf., e.g., [57, p. 69]). T HEOREM 7.2. Day’s norm on c0 (Γ ) (defined in the proof of Theorem 5.2) is LUR for every Γ . P ROOF. The proof consists of a qualitative variant of the argument that we used in the proof of Theorem 5.2 (cf., e.g., [57, p. 69]). The following is a variant of the result of Troyanski in [308] and is from [330] (cf., e.g., [57, p. 284].) T HEOREM 7.3. If a Banach space X admits a PRI {Pα }, α μ, and each (Pα+1 − Pα )(X), α < μ, admits a LUR norm, then so does X. P ROOF. We will outline the key idea in the proof of this result for a Banach space X with a transfinite Schauder basis {eα , fα }, i.e., if X admits a PRI {Pα }, α μ, such that dim(Pα+1 − Pα )(X) = 1 for all α < μ. Put Γ = [0, μ). Let An be the family of all finite subsets of Γ with no more thann elements. For x ∈ X and A ∈ An , let EnA (x) = dist(x, span{eα }α∈A ) and FnA (x) = α∈A |fα (x)|. Put Gn (x) = sup{E A (x) + nF A (x); A ∈ An } for n ∈ N and x ∈ X. Finally, put G0 (x) = x0 for x ∈ X, where · 0 is the original norm of X. Let Δ = {0, −1, −2, . . .} ∪ Γ and define Φ : X → c0 (Δ) by

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Φ(x)(−n) = 2−n Gn (x) for −n ∈ {0, −1, −2, . . .}, Φ(x)(α) = |fα (x)| for α ∈ Γ . Define an equivalent norm x on X by x = Φ(x)D , where · is Day’s norm on c0 (Δ). Let us briefly outline the main idea of the proof that · is a LUR norm on X. Let xn , x ∈ X be such that 2xn 2 + 2x2 − x + xn 2 → 0. Let ε > 0. Find n ∈ N and A ∈ An such

that EnA (x) < ε. Assume without loss of generality that / A < min fα (x); α ∈ A . sup fα (x); α ∈

(∗)

Due to the term n in the definition of G n s, if m is big enough and A ∈ Am is so chosen A (x) + mF A (x)) < ε with A ∈ A , then, necessarily, A ⊃ A and thus that Gm (x) − (Em m m A the LUR property of Day’s norm on Em (x) < ε. This is because {fα (x)} ∈ c0 (Γ ). From c0 (Δ) it follows that FnA (xk ) → FnA (x) for all A ∈ An and that Gn (xk ) → Gn (x) for all n. As the topology of the coordinatewise convergence in X is Hausdorff, in order to prove that xk − x → 0 it suffices to show that {xk } is relatively norm compact in X. The A (x ) G (x ) − mF A (x ) G (x) + ε − latter is seen from the fact that for large k, Em k m k m m k A A mFm (x) + ε Em (x) + 3ε 4ε. The fact that the set A above is enlarged to ensure that the relation in (∗) holds true can be avoided by adding more parameters ([330]). Overall, the main thing with the above result is that the set A in the definition of EnA was chosen so that EnA (x) is such that FnA (x) = 0, and the supremum in the definition of Gn is “uniquely located” (see the “rigidity” condition in [145]). This Troyanski’s phenomenon explicitly or implicitly appears again in many results in this area, including the results on smooth partitions of unity or recent results of Haydon, Talagrand and others on higher-order smoothness. In particular, a Banach space X admits a LUR norm if it has a Markushevich basis {xα , fα } such that x ∈ span{xα ; fα (x) = 0} for every x ∈ X. This is a special condition on a Markushevich basis. Every separable Banach space has such bM-basis ([300–302], see also [326]). T HEOREM 7.4. Any WLD space admits a LUR norm. If K is a Corson compact, then C(K) admits an LUR norm that is pointwise lower semicontinuous. Any Banach space with a shrinking M-basis admits a norm whose dual norm is LUR. There is a Banach space with a Markushevich basis that admits no LUR norm. P ROOF. We use Theorems 7.3 and 3.8 to get that every WLD admits a LUR norm ([13, 316]). We refer to, e.g., [57, p. 286] for a proof of the result in [14] and [316] that C(K) admits a pointwise lower semicontinuous LUR norm if K is a Corson compact. We refer to [309] for the rest of the result. As ∞ does not admit any LUR norm (see the text following Theorem 7.19), the space Z from Theorem 4.10 admits no LUR norm. Similarly, like we used the attachment of the term nFnA in the construction of an LUR norm above, we can prove the following result from [122] and [158] (cf., e.g., [57, p. 299]).


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T HEOREM 7.5. (i) Let Y be a subspace of a Banach space X such that X/Y admits an equivalent LUR norm. Assume that Y admits an equivalent LUR (respectively R) norm. Then X admits an equivalent LUR (respectively R) norm. (ii) Let Y be a subspace of a Banach space X. Assume that both Y and X/Y admit norms whose dual norms are LUR. Then X admits an equivalent norm whose dual is LUR. P ROOF. As in Theorem 7.3, we attach the balls in the cosets, centered by using the the Bartle–Graves selection, to the ball of the quotient. If T is a bounded linear operator from a Banach space X onto a Banach space Y , then the Bartle–Graves selector is a continuous map from Y into X such that for y ∈ Y , T (b(y)) = y (cf., e.g., [57, p. 299]). There is a subspace Y of a Banach space X such that Y admits a LUR norm, X/Y admits an R-norm and X does not admit any R-norm ([145]). This can happen because of the fact that we no longer can properly use the continuity of the Bartle–Graves selector. If X∗ is WCG, then we attach to the map of X∗∗ into c0 (Γ ) finite-dimensional subspaces of X (using the angelicity of the second dual ball) to obtain that if X∗ is WCG, then X admits a LUR norm. A different approach for getting this result is in [121] (cf., e.g., [57, p. 296]). The following result is in [86,121], cf., e.g., [57, p. 296]. T HEOREM 7.6. A Banach space X admits a LUR norm if X∗ is WCD. Another method of constructing locally uniformly rotund norms is Godefroy’s “transfer” method ([108,120,71], cf., e.g., [57, pp. 44, 289] or [80, Chapter 11]). This method, based on weak star compactness and a geometric argument, allows to construct dual LUR norms in some dual spaces that do not admit a PRI formed by dual projections. The following is Fabian’s result in [71]. It uses Godefroy’s transfer method. T HEOREM 7.7. Assume that X∗ is WCD. Then X admits an equivalent norm whose dual norm is LUR. P ROOF. We will present here Fabian’s proof of this result in the case X∗ is WCG ([120]). There is a set Γ and a weak star to weak continuous bounded linear operator T from c0∗ (Γ ) onto a norm dense set in X∗ . Indeed, let U be a weak star to weak continuous oneto-one operator from X∗∗ into c0 (Γ ) for some Γ and put T = U ∗ . Then T maps c0∗ (Γ ) onto a norm dense set in X∗ as U is one-to-one and weak star to weak continuous. We construct a dual LUR norm on X∗ as follows. Let · be the original norm of X∗ and let | · | be an equivalent dual LUR norm on c0∗ (Γ ). Such norm exists as the standard basis of c0 (Γ ) is shrinking (Theorem 7.4). For n ∈ N and f ∈ X∗ put |f |2n = inf{f − T g2 + ∞ 1 2 ∗ 2 −n 2 ∗ n=1 2 |f |n . This is an equivalent norm on X which n |g| ; g ∈ c0 (Γ )} and |||f ||| = ∗ is weak star lower semicontinuous and thus it is a dual norm. As T (c0 (Γ )) is norm dense in X∗ , it follows that |f |n → 0 for each f ∈ X∗ . Let us now show that ||| · ||| is LUR. To this end assume that f, fj ∈ X∗ are such that 2|||fj |||2 + 2|||f |||2 − |||f + fj |||2 → 0. Then for every n, 2|f |2n + |fj |2n − |f + fj |2n → 0 with j → ∞. Find g, gj ∈ c0∗ (Γ ) such that |f |2n = f − g2 + n1 |g|2 , |fj |2n = fj − T gj 2 + n1 |gj |2 .

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Then 2|f |2n + 2|fj |2n − |f + fj |2n 2 2 2f − T g2 + |y|2 + 2fj − T gj 2 + |gj |2 n n 2 1 − f + fj − T (g + gj ) − |g + gj |2 n

2 1 f − T g − fj − T gj + 2g2 + 2|gj |2 − |g + gj |2 . n This implies that fj − T gj → f − T g and 2|g|2 + 2|gj |2 − |g + gj |2 → 0. As | · | is LUR, we have |g − gj | → 0. Thus lim sup f − fj lim sup(f − T g + T (g − gj ) + fj − T gj ) = 2f − T g 2|f |n . As this holds for each n and |f |n → 0, we get f − fj → 0 and thus ||| · ||| is LUR. The following is an alternative proof that every WCG space admits a LUR norm (which fact follows from Theorem 7.4). Let the dual norm on 1 (Γ ) be defined by |||x|||2 = x21 + x22 , where xi is the norm of i (Γ ) for i = 1, 2. Let T be a bounded linear one-to-one operator of X∗ into c0 (Γ ) that is weak star to weak continuous (Theorem 4.5). The norm ||| · ||| is LUR (cf., e.g., [57, p. 72]) and we can use T ∗ to apply the proof of Theorem 7.7. There is a Banach space that does not admit any Gâteaux differentiable or rotund norms and whose dual is WLD ([145]). We will now discuss how to combine norms with various smoothness and rotundity properties. This procedure is called the Asplund averaging procedure. One method in this direction is the following approach ([89], cf., e.g., [57, p. 52]). T HEOREM 7.8. Assume that X is a Banach space that admits an equivalent LUR norm. Then the set of all equivalent LUR norms on X is residual in the space P of all equivalent norms on X with the metric of uniform convergence on the original ball of X. The metric space P is a Baire space, i.e., such a topological space T that the intersection of any countable family of open dense sets in T is dense in T . P ROOF OF T HEOREM 7.8. Let r0 be a LUR norm on X. For p ∈ P and j ∈ N, put G(p, j ) = {q ∈ P ; sup{|p2 (x) + j −1 r02 (x) − q 2 (x)|; x ∈ BX } < j −2 }. For k ∈ N, put Gk = {G(p, j ); p ∈ P ; j k} and finally put G = ∞ k=1 Gk . Then Gk is open and dense in P for all k and to finish the proof, by the Baire category theorem, it is enough to show that each element of G is LUR. This is done by using the numbers j −1 and j −2 in the definition of G(p, j ). Similarly, the set of all equivalent norms on a given Banach space X such that their dual norm is LUR is residual in P , provided there is at least one such norm on a space.


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Assume that a Banach space X admits an equivalent LUR norm and also admits an equivalent norm whose dual is LUR. By the Baire category theorem, the set of norms that have both these properties is residual in P . From Theorems 7.4, 7.7 and 7.8, we thus obtain the following result. T HEOREM 7.9. A Banach space X admits a LUR norm whose dual is LUR if X∗ is WCD. The method works for some other types of rotundity and smoothness of the first order. If X∗ is WCD, to get a LUR Fréchet differentiable norm on X we can proceed as follows ([164]). Let · be a LUR norm on X and let · n be a sequence of norms whose dual norms are LUR and so that · n → · uniformly on bounded sets. This can be 2done by using the proofs of Theorems 7.6 and 7.7. Then put for x ∈ X, |||x|||2 = ∞ n=1 xn . By using the standard differentiability rules we can see that ||| · ||| is a Fréchet smooth norm. Assuming that 2|||xn |||2 + 2|||xn |||2 − |||xn + x|||2 → 0, we get the same convergence for each · n and thus the same convergence for · by Osgood’s uniform convergence theorem. Thus xn − x → 0 as · is LUR. Bossard, Godefroy and Kaufman proved in [28] that the set of all Fréchet differentiable norms on every infinite-dimensional separable Asplund Banach space is not a Borel set in P (P is defined in Theorem 7.8). This compares with the result of Mazurkiewicz [203] that the set of all differentiable real valued functions on [0, 1] is not Borel in C[0, 1]. The same holds for the set of all LUR norms on every infinite-dimensional separable Banach space ([28]). The norm ||| · ||| on 1 defined by |||x||| = x1 + x2 is strictly convex and a dual norm. Moreover, on its unit sphere the weak star and norm topologies coincide. Its predual norm on c0 is thus Fréchet differentiable (Theorem 2.5). The norm ||| · ||| is not LUR on 1 (by inspection). There are spaces that admit Kadets–Klee norms and do not admit any strictly convex norms ([145]). However, the following result of Troyanski in [313,314] holds true (cf., e.g., [57, Chapter IV]). T HEOREM 7.10. Assume that a Banach space X admits a norm that is a Kadets–Klee norm and that X admits a norm that is strictly convex. Then X admits an equivalent LUR norm. P ROOF (Raja [261]). There is a norm on X that shares both properties in question (the sum of the norms works). Then each point of the new unit sphere is an extreme point of the new unit ball of BX∗∗ (Lin, Troyanski, cf., e.g., [80, Chapter 3, Example]). Thus the slices form a neighborhood system of any element of the unit sphere of X in the weak topology and thus in the norm topology (cf., e.g., [80, Chapter 3, Example]). A slice is an intersection of BX with a halfspace in X. For m ∈ N, put Am = {x ∈ BX ; diam(BX ∩ H ) 1/m for every halfspace H containing x}. Then Am is closed convex and symmetric and 0 ∈ Int Am . Let · m be the Minkowski functional of the set Am . Let am > 0 be such that am x2m 2−m x2 for all x ∈ X, where · is the original norm of X. Put for x ∈ X,

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am x2m + x2 . We will show that | · | is LUR. To this end let x, xk ∈ X be such 2|x|2 + 2|xk |2 − |x + xk |2 → 0.

(∗)

Assume without loss of generality that x = xk = 1. Given ε > 0, fix m ∈ N with m > 2/ε and x ∈ / Am . As Am is closed, xm > 1. From (∗), 12 (x + xk )m > 1 for large k. 1 Hence 2 (x + xk ) ∈ / Am for large k and thus for every large k there is a halfspace H such 1 that 2 (x + xk ) ∈ H and diam(BX ∩ H ) 1/m < ε/2. Now, either x or xk is in H and since the diameter of BX ∩ H is less then ε we get that x − xk ε. This means that xk → x in the norm and the proof is finished. Assume that a Banach space X has a PRI {Pα } in its Fréchet smooth norm. Then {Pα∗ } is automatically a PRI for X∗ . For simplicity, let us show this for the case that the dual norm of X∗ is locally uniformly rotund. To this end, observe that if the dual norm is LUR, then the weak star and norm topologies of BX∗ coincide at each point of SX∗ . If f ∈ SX∗ , then Pα∗ f → Pβ∗ f in the weak star topology if α → β. As Pα∗ f = Pα∗ (Pβ f ) Pβ∗ f we have from the coincidence of the weak star and norm topologies on SX∗ that Pα∗ f → Pβ∗ f in the norm. In the case of Fréchet differentiable norms we use the Bishop–Phelps theorem. By transfinite induction, this gives that if a WLD Banach space X admits an equivalent Fréchet differentiable norm, then X and all its subspaces admit a shrinking M-basis. Thus the property of admitting a shrinking M-basis is a hereditary property on Banach spaces ([160]). In particular, any subspace of c0 (Γ ) is WCG for any Γ . Moreover, by “forced” inclusion of the derivatives of a Fréchet C 1 -smooth function derived from the C 1 smooth bump into the ranges of the dual projections (cf. the subspace F in Lemma 3.11), we can show, similarly, that a WCG space that admits a Lipschitz, Fréchet C 1 -smooth bump function necessarily admits a shrinking M-basis. The duality mapping of a Banach space is a multivalued map D from X into the subsets of X∗ defined by D(x) = {f ∈ X∗ ; x2 = f 2 = f (x)}. The following is a result in [156]. T HEOREM 7.11. If X is an Asplund space, then the duality map D admits a selector that is a pointwise limit in the norm of X∗ of a sequence of norm-to-norm continuous mappings from X into X∗ , i.e., there is a (Jayne–Rogers) selector J of D and a sequence of normto-norm continuous mappings Jn from X into X∗ such that Jn x − J x → 0 for every x ∈ X. The fact that J is a selector of D means that J (x) ∈ D(x) for x ∈ X. If X is WLD, then by “forced” inclusion of ranges of Jn into the ranges of the dual projections in Valdivia’s construction in [319] and by using Simons’ inequality (cf., e.g., [57, Chapter I] or [80, Chapter 3]) (instead of the Bishop–Phelps theorem) we get the following result in [319,309]. For WCD spaces, this result was proved earlier in [70] and [309]. T HEOREM 7.12. Let X be an Asplund space. Then X admits a shrinking M-basis if and only if it is WLD.


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Hence every WCG Asplund space is hereditarily WCG. An example of a nonseparable hereditarily WCG space that is not Asplund is 2 (c) ⊕ 1 (cf., e.g., [35]). If we assume only that X is an Asplund space, then a PRI on X∗ in its canonical supremum norm can be constructed by using the Jayne–Rogers selector (the projections cannot in general be dual maps). Thus we obtain the result in [74] (cf., e.g., [57, Chapter VI], [73, p. 150]). T HEOREM 7.13. X∗ admits a PRI (non-dual projections in general), an equivalent LUR norm (non-dual in general) and an M-basis if X is an Asplund space. Theorem 7.13 solves for dual spaces the longstanding problem (still open in its full generality) whether every space with the Radon–Nikodým property admits a LUR norm. For the definition, see, e.g., [73, p. 33]. One of the pioneering results in this direction was Tacon’s paper [291]. This paper motivated [160] and many other papers in this area in this period of time. The following result is a combination of the results of Deville, Haydon and Rogers ([49, 146], cf., e.g., [57, p. 311]). T HEOREM 7.14. Let K be a compact space such that the Cantor derived set K (ω1 ) = ∅. Then C(K) admits an equivalent LUR norm whose dual norm LUR. The following results characterize Asplund spaces in terms of some topological properties ([233,67,31]): T HEOREM 7.15. A Banach space X is an Asplund space if and only if X∗ is Lindelöf in the topology of uniform convergence on all separable bounded sets in X, if and only if each separable subset of BX in its weak topology is metrizable. Orihuela’s proof in [233] of the first part of the statement gives also the proof of Alster– Gul’ko–Pol result that X is WL if BX∗ is Corson in the weak star topology (see Section 4, cf., e.g., [80, Chapter 12]). For further results in this direction we refer to [219] and references therein. The following Preiss’ version [255] of the separable reduction argument can be used in the proof that X is an Asplund space if and only if every continuous convex function on X is Fréchet differentiable on a dense set in X (cf., e.g., [57, Chapter I] or [243, Chapter 2]). T HEOREM 7.16. Assume that f is a continuous function on a Banach space X. Then for every separable subspace Z of X, there is a separable subspace W of X such that Z ⊂ W and f is Fréchet differentiable as a function on X at every point of W at which the restriction of f to W is Fréchet differentiable. The following result is related to Theorem 2.12 and in a more general form can be found in Ribarska’s theorem from [266] (presented in [73, p. 88]) together with the results in [57, p. 26] or [73, p. 90].

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T HEOREM 7.17. Assume that X is an Asplund space and K is a weak star compact set in X∗ . Then there is a subset S of K that is dense Gδ in K in its weak star topology and such that the weak star and the norm topologies on K coincide at each point of S. The set S is thus metrizable by a complete metric and all the points of S are Gδ points of K in its weak star topology. The following theorem is a counterpart of the results of Kadets in the separable case that we mentioned in the beginning of this section. It summarizes some results in this section. T HEOREM 7.18. Assume that a Banach space X is WLD. Then the following are equivalent. (i) X admits an equivalent norm whose dual norm is LUR. (ii) X admits an equivalent Fréchet smooth norm. (iii) X admits a Fréchet differentiable bump function. (iv) X is an Asplund space. P ROOF. (i) implies (ii): see the text in the beginning of this section. (ii) implies (iii) is standard. (iii) implies (iv): see the text in the beginning of this section. (v) implies (i): see Theorem 7.12 and Theorem 7.4.

We will now discuss the interplay of covering properties of Banach spaces and renormings by LUR and Kadets–Klee norms. Much progress in this area has been recently achieved ([209,211–213,261,262,260,263]). The following result is in [65]. T HEOREM 7.19. Assume that the norm of a Banach space X has the Kadets–Klee property. Then the norm and weak Borel sets coincide in X and X is a Borel set in X∗∗ in its weak star topology. P ROOF (Schachermayer [65]). If we consider on SX and X \ {0} the weak topology, then the map (t, x) → tx of (0, ∞) ⊕ SX into X \ {0} is a Borel homeomorphism, i.e., together with its inverse, it maps Borel sets onto Borel sets. This follows from the fact that in the inverse function y → (y, y/y), the first coordinate is weak lower semicontinuous and thus Borel and the second coordinate is a composition of x → (x, x/x), which is Borel and a continuous function (x, x/x) → x/x. Moreover, as SX in its weak topology is a completely metrizable space if the norm has the Kadets–Klee property, SX is Gδ in its closure which contains SX∗∗ by Goldstine’s theorem. Thus X in its weak topology is then a Borel set in X∗∗ in its weak star topology. It is known that the norm and weak Borel structures in ∞ do not coincide ([293]). As a corollary of Theorem 7.19, we thus have that ∞ does not admit any Kadets–Klee norm, in particular any LUR norm ([195,309], cf., e.g., [57, p. 74]). In as much as the “method of covers” has been essential in characterizing metrizability in topological spaces, it is quite natural that this method has been helpful in characterizing Banach spaces that admit equivalent LUR or Kadets–Klee norms.


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The first characterization in this direction is in [312] (see also [311] and [314]), where martingales were used. Several properties of coverings related to Kadets–Klee norms were studied in [141,153,152,186] and [187]. The main contribution in [210] consists of replacing topological and probabilistic conditions by linear topological properties of spaces. The following notion is in [262] and originated in [64,65] and [292]. Let Σ1 and Σ2 be families of subsets of a given set S. We say that S has property P (Σ1 , Σ2 ), if there is a sequence (An ) of subsets of S such that for every x ∈ S and every V ∈ Σ1 with x ∈ V there are n ∈ N and U ∈ Σ2 such that x ∈ An ∩ U ⊂ V . If Σ1 is a topology, this definition means that the family of sets {An ∩ Σ2 } is a network for Σ1 . For a Banach space X property P(norm topology, weak topology) (P( · , w) for short) is used in the following result in [262]. T HEOREM 7.20. Let X be a Banach space. Property P( · , w) is equivalent to the existence of a symmetric homogeneous weakly lower semicontinuous function F on X with · F 3 · such that the norm and the weak topologies coincide on the set S = {x ∈ X; F (x) = 1}. It is not known if P( · , w) characterizes the Kadets–Klee renormings. For spaces of continuous functions on trees this is true [145]. On the other hand, P( · , w) implies that the Banach space X is a Borel subset of X∗∗ in its weak star topology ([141,262,232]) and thus X isεthen σ -fragmentable ([153]), i.e., for every ε > 0, there isε a decomposition X= ∞ n=1 Xn such that for every n ∈ N and every non-empty A ⊂ Xn , there is a weak open set U such that A ∩ U = ∅ and diam(A ∩ U ) < ε. For LUR renormings we have the following result from [210]. T HEOREM 7.21. A Banach space X admits LUR norm if and only if for an equivalent ε such that for every n ∈ N and every every ε > 0 there is a decomposition X = ∞ X n=1 n x ∈ Xnε , there is an open halfspace H such that x ∈ H and diam (H ∩ Xnε ) < ε. An open halfspace in X is f −1 (a, +∞) for some f ∈ X∗ \ {0} and a ∈ R. A new transfer technique has been developed by using the covering techniques. A bounded linear one-to-one operator T from a Banach space X into a Banach space Y is called an SLD map if X has property P( · , T −1 (norm-open sets in Y)). The main result here is the following theorem. T HEOREM 7.22. Let T be an SLD map from a Banach space X into a Banach space Y . If Y admits an LUR norm, then X has an equivalent LUR norm. A bounded linear operator T is SLD whenever T −1 is a pointwise limit point of a sequence of norm to norm continuous functions, in particular when T −1 is a Baire 1 map (i.e., a pointwise limit of a sequence of continuous maps). This is the case for instance when the dual operator has norm dense range [210]. In this way we obtain, as a particular case, that X admits an equivalent LUR norm if there is a bounded linear operator T that maps X into c0 (Γ ) for some Γ and is such that T ∗ has a norm dense range in X∗ ([121], cf., e.g., [57, Chapter VII]).

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The only point, where the linearity of T is used is transferring the slices from Y into X. For the covering notions, the only condition needed is the weak to weak continuity, so they are invariant for weak homeomorphisms ([141,232,211,220,148]). Indeed, a weak homeomorphism is Baire one in both directions for the norms ([287]). As an application of this method, let us mention the following result in [211] and [212]. T HEOREM 7.23. Let X be a Banach space that is weakly locally uniformly rotund. Then X has an equivalent LUR norm. The norm is weakly locally uniformly rotund if lim(xn − x) = 0 in the weak topology of X whenever xn , x ∈ X are such that lim(2xn 2 + 2x2 − x + xn 2 ) = 0. In the case of dual norms the following is the result in [261]. T HEOREM 7.24. Let the weak and weak star topologies coincide on the dual sphere of X∗ . Then X∗ admits an equivalent dual LUR norm. In many results in this area of coverings, the fundamental construction of the norm is the one in the proof of Theorem 7.10. D EFINITION 7.25. A compact space K has the Namioka property if for every Baire space E and every continuous map ϕ from E into C(K) endowed with the pointwise topology, there is a dense Gδ subset Ω of E such that ϕ : E → (C(K), · ∞ ) is continuous at every point of Ω. The following result from [54] can be found, e.g., in [57, p. 329]. T HEOREM 7.26. Assume that K is a compact space such that C(K) admits an equivalent LUR norm · that is pointwise lower semicontinuous. Then K has the Namioka property. In particular, any Corson compact has the Namioka property. Any scattered compact space K such hat C(K) admits an equivalent LUR norm has the Namioka property. P ROOF. Let B and S denote the unit ball and the unit sphere of · respectively. It follows from the LUR property and from the pointwise lower semicontinuity of · that the identity map I from B endowed with the pointwise topology into B endowed with the norm topology is continuous at every point of S. Now, let E be a Baire space and let ϕ be a continuous map from E into C(K) in its pointwise topology. The map ψ(x) := x is pointwise lower semicontinuous. Hence there is a dense Gδ subset Ω of E such that ψ is continuous at every point of Ω. As I is continuous at every point of S, we get that any point of Ω is a point of continuity of ϕ : E → (C(K), · ). We can use Theorem 7.4 to finish the second part of the proof. If K is scattered, then the norm closed linear hull of the Dirac measures in C(K)∗ equals to C(K)∗ by Rudin’s theorem (cf., e.g., [80, Chapter 12]). Thus any equivalent norm on C(K) is pointwise lower semicontinuous if K is a scattered compact. An example of a compact set that does not have the Namioka property is B∗∞ in its weak do not have the star topology ([49]). There are trees T such that their compactifications T


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Namioka property ([132]). It is an open problem if there is a Baire space E, a compact set K and a separately continuous function f : E × K → R with no points of joint continuity. Recall that a family F of subsets of a topological space T is σ discrete if F = n Fn , where each Fn is a discrete family, i.e., for each n, each point of T has a neighborhood that meets at most one member of Fn . By the Nagata–Smirnov theorem (cf., e.g., [69, Chapter 4]), every metrizable space has a σ -discrete basis of its topology. It is apparently an open problem if every Banach space admits a σ -discrete basis for its norm topology formed by convex sets. The following result can be found in [260]. T HEOREM 7.27. Assume that a Banach space X admits an equivalent LUR norm. Then the norm topology of X has a σ -discrete basis formed by convex sets. P ROOF. Given ε > 0, define by transfinite induction a family of convex sets {Bα } as follows: B0 = BX , Bα+1 = Bα \ x ∈ X; xα∗ (x) > aα , where xα∗ ∈ SX∗ and aα ∈ R are such that diam BX ∩ x ∈ X; xα∗ (x) > aα < ε. The process ends when Bγ is in the open unit ball of X. For δ > 0 then define convex sets C(α, ε, δ) = Bα ∩ x ∈ X; xα∗ (x) aα + δ . The sets C(α, ε, δ) + B(0, 1/n) and their rational multiples are then used to produce the proof of the result. A family {Hγ ; γ ∈ Γ } issaid to be isolated if for every γ0 ∈ Γ , Hγ0 ∩ γ =γ0 Hγ = ∅. If Γ can be split into Γ = Γn with each family {Hγ ; γ ∈ Γn } being isolated, we say that the family {Hγ ; γ ∈ Γ } is σ -isolated. A family A of subsets of a topological space T is a network in T , if every open subset in T is a union of some members of A. A compact space is called descriptive if its topology has a σ -isolated network. A norm · on X∗ is weak star locally uniformly rotund if fn − f → 0 in the weak star topology whenever fn = f = 1 and fn + f → 2. Raja proved in [263] the following theorem. T HEOREM 7.28. If X is a Banach space, then X∗ admits an equivalent dual weak star locally uniformly rotund norm if and only if BX∗ in its weak star topology is descriptive. In the same paper Raja showed that this gives the following result. C OROLLARY 7.29. If X is an Asplund space, then X∗ admits an equivalent dual LUR norm if and only if BX∗ in its weak star topology is a descriptive compact.

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Raja also showed in [263] that there are non-WCD Banach spaces X such that BX∗ is Corson and at the same time descriptive. From Theorem 7.5(i) we get that the spaces JL0 , JL2 , D and C[0, ω1 ] all admit LUR norms and from Theorem 7.5(ii) we get that JL0 and JL2 admit norms whose duals are LUR. Kunen’s C(K) space admits no Kadets–Klee norm. Indeed, such a unit sphere would then be norm separable as C(K) is hereditarily weakly Lindelöf. Hence C(K) would be separable, a contradiction ([220]). Thus this C(K) space is an Asplund space that admits no LUR norm. The space C[0, ω1 ] admits a LUR norm as it admits a PRI with separable ranges. The space CP of Ciesielski and Pol admits an equivalent LUR norm by Theorem 7.5. The first example of a space with a rotund norm that cannot be mapped into c0 (Γ ) by a bounded linear one-to-one operator was constructed in [44]. The space D = C(K) space for the two arrows space K (Section 2) admits an equivalent LUR norm that is pointwise lower semicontinuous ([152]). Hence this K has the Namioka property by Theorem 7.27. The latter result was first proved in [49].

8. C k -smooth norms for k > 1 We will discuss higher-order differentiability in the Fréchet sense only. Thus we will say that a function f is, for example, C 2 -smooth if the map Φ : x → f (x) from X into X∗ in its norm topology, is Fréchet C 1 -smooth. If we algebraically add the epigraphs of the real functions tχ[0,∞] (t) and t 2 , we get the epigraph of a convex function that is not twice differentiable at the origin. Thus we cannot use, in general, sums of convex bodies to produce C k smooth bodies if k > 1. Recall that for a real valued function f on a Banach space X, the epigraph of f is defined by epi f = {(x, r) ∈ X × R; r f (x)}. D EFINITION 8.1. We say that a norm · on a Banach space X locally depends on finitely many coordinates if for every x ∈ X \ {0} there is a neighborhood U of x, f1 , f1 , . . . , fn ∈ X∗ and a continuous function ϕ on Rn such that z = ϕ(f1 (z), f2 (z), . . . , fn (z)) for all z ∈ U. A typical example of such norm is the supremum norm of c0 (Γ ), for any Γ . Let a bounded linear operator T from C0 [0, ω1 ] into c0 [0, ω1 ] be defined by T x(α) = x(α + 1) − x(α) if α < ω1 and T x(ω1 ) = 0. Given x ∈ C0 [0, ω1 ] with x = 1, choose β = sup{α; |x(α)| = 1}. Then T x(α) = 0. This is an example of what is now called a Talagrand operator (it originated in [297], see [144,145]). For a compact set K, it is a bounded linear operator T from a subspace X of C(K) into c0 (K) such that for every x ∈ X of supremum norm one there is k ∈ K such that |x(k)| = 1 and T x(k) = 0. If T is a Talagrand operator on X ⊂ C(K), then the norm · 1 defined for x ∈ X by x1 = supα∈K {|x(α)| + |T x(α)|} is easily seen to depend locally on finitely many coordinates. Hence C0 [0, ω1 ] admits an equivalent norm that locally depends on finitely many coordinates. As C[0, ω1 ] is isomorphic to C0 [0, ω1 ] we thus have that C[0, ω1 ] admits a norm that locally depends on finitely many coordinates.


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Norms that locally depend on finitely many coordinates are often as good as smooth norms (sometimes even better). And, last but not least, they are often much easier to construct than the smooth norms. For their connection with smooth norms see Problem 3 in Section 9. If a Banach space X admits a continuous bump function that locally depends on finitely many coordinates, then X is an Asplund space and contains an isomorphic copy of c0 ([239,90], [57, Chapter V]). For a connection of such norms to polyhedral spaces see, e.g., [132] and [99]. A Banach space X is called polyhedral if BF is the convex hull of a finite set for every finite-dimensional subspace F of X. For norms that locally depend on countably many coordinates we refer to [92]. The following result can be found in [239] (cf., e.g., [57, p. 189]). T HEOREM 8.2. For any set Γ , the space c0 (Γ ) admits an equivalent norm that is at the same time C 1 -smooth and LUR and is a limit, uniform on bounded sets, of C ∞ -smooth norms that locally depend on finitely many coordinates. P ROOF. For n ∈ N, let ϕn be even, C ∞ smooth real valued function on the reals such on (1/(n + 1), ∞). For that ϕn = 0 on [−1/(n + 1), 1/(n + 1] and ϕ > 0 and ϕ

> 0 n ∈ N, define the function Φn on c0 (Γ ) for x = (xα ) by Φn (x) = α∈Γ ϕn (xα ). Note that Φn is well defined and locally C ∞ smoothly depend on finitely many coordinates. Given n, m ∈ N, put Qn,m = {x ∈ c0 (Γ ); Φn (x) m}. Let · n,m denote Minkowski’s functional of Qn,m . Finally, define the norm on c0 (Γ ) for x ∈ c0 (Γ ) by x2 = ∞ −n−m x2 . Then the norm · has the required properties. Indeed, we can n,m n,m=1 2 see that · is LUR (use the properties of the supports of ϕn ) and is C 1 as it is a sum of terms that have first derivatives bounded. This is not the case with the higher-order deriva tives, where we can only say that · n,m are C ∞ smooth. The following result of Haydon [144] extends the result of Talagrand [297], where C 1 -smoothness was studied. T HEOREM 8.3. The space C[0, μ] admits a C ∞ smooth norm for every ordinal μ. The following result is in [118] (cf., e.g., [57, p. 194]). T HEOREM 8.4. If K is a compact space such that the Cantor derived set K (ω0 ) = ∅, then C(K) admits a C ∞ -smooth norm. As K (ω0 ) = ∅ means K (n) = ∅ for some n ∈ N, Theorem 8.4 follows by induction by using the following result in [118] (cf., e.g., [57, p. 194]). T HEOREM 8.5. Assume that k ∈ N∪{+∞}. Let X be a Banach space and Y be a subspace of X such that Y is isomorphic to c0 (Γ ) for some Γ and that X/Y admits an equivalent C k -smooth norm. Then X admits an equivalent C k -smooth norm. Theorem 8.5 can be proved by a variant of the construction in the proof of Theorem 8.2.

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Hájek recently showed in [138] that C(K) admits a C ∞ smooth norm that locally depends on finitely many coordinates if K (ω1 ) = ∅. Later on, Hájek and Haydon showed [139] that C(K) admits an equivalent C ∞ smooth norm if C(K)∗ admits an equivalent dual locally uniformly rotund norm. The following result is due to Haydon [145]. be the one point compactification of a tree T . Then C(T) admits a T HEOREM 8.6. Let T ∞ C -smooth bump function. Haydon has examples of trees that create many counterexamples for renormings: there are trees T such that C(T) then admits a Gâteaux differentiable norm but no strictly convex norm. The full dyadic tree of the height ω1 , i.e., T = α 0 and P is a subset of T , we will say that P is fragmented by ρ down to ε if whenever A is a non-empty subset of P there is a non-empty relative τ -open subset B of A such that ρ-diameter of B less than ε. We will say that a topological space (T , τ ) is fragmented by ρ if T is fragmented by ρ down to ε for each ε > 0. We will say that a topological space is fragmented if it is fragmented by some metric. Ribarska proved in [266] that if a compact space (T , τ ) is fragmented, then T is fragmented by a complete metric that is stronger than τ . Namioka proved in [218] that if a compact space K is fragmented by a lower semicontinuous metric, then K is homeomorphic to a weak star compact set in the dual of an Asplund space. If this happens for a Corson compact K, then K is necessarily an Eberlein compact ([234,290,87], Rezniˇcenko, cf., e.g., [73, p. 155]). As we already discussed in Section 7, a topological space (T , τ ) is called σ -fragmented by ρ if for every ε > 0, T can be decomposed as T = ∞ n=1 Tn , where (Tn , τ ) is fragmented by ρ down to ε. If X is a separable Banach space, {xi } is dense in X and ε > 0, then X = (xi + εBX ), showing that X in its weak topology is σ -fragmented by the metric given by the norm of X. On the other hand, the Banach space c0 in its weak topology is not a countable union of sets fragmented by the norm as it follows from the Baire category theorem and from the fact that each relatively weakly open set in Bc0 has diameter > 1 ([154]) (cf.,

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e.g., [80, Chapter 12]). Each weakly compact set in its weak topology in a Banach space X is fragmented by the norm of X (Troyanski, cf., e.g., [219] and references therein). If X admits a Gâteaux differentiable norm, then X∗ in its weak star topology is fragmented (Ribarska, cf., e.g., [73, p. 81]). For an extension of this result for Lipschitz Gâteaux differentiable bump functions see [100]. Fragmentability of X∗ in the weak star topology (which is equivalent with BX∗ being fragmented in the weak star topology (see, e.g., [73, p. 86])) in turn implies that X is a weak Asplund space. The converse implication does not hold in general ([181]). BX∗ in its weak star topology is fragmented by the norm of X∗ if and only if X∗ in its weak star topology is σ -fragmented by the norm of X∗ if and only if X is an Asplund space (Namioka, Phelps, cf., e.g., [57, Chapter I], [219]). If X is WCG, then X in its weak topology is σ -fragmented by the norm and all Xi in the definition of σ -fragmentability can be taken weakly closed (Jayne, Namioka, Rogers [152]). If X admits a Kadets–Klee norm, then X in its weak topology is σ -fragmented by the norm and all the sets Xi can be differences of weakly closed sets (Jayne, Namioka, Rogers [152]). Let X be ∞ equipped with its weak topology. The X is not σ -fragmented by its norm. However, X is fragmented by the lower semicontinuous metric τ (x, y) = 2−i min{1, |xi − yi |}. If Γ is uncountable, then the space c∞ (Γ ) equipped with its weak topology is not fragmented by any metric nor is it σ -fragmented by any lower semicontinuous metric (all of this is in [154]). For more information on these topics we refer to [73] and [219] and references therein.

D. Fundamental biorthogonal systems and Mazur’s intersection property A biorthogonal system {xα , fα } in a Banach space X is said to be a fundamental biorthogonal system for X if span{xα } = X. Davis, Johnson and Godun ([45,123] proved that if a Banach space X has a WCG quotient space of the same density character as X, then X has a fundamental biorthogonal system. This is the case with ∞ , as 2 (c) is isomorphic to a quotient of ∞ ([270]). On the other hand, if card Γ > c, then c∞ (Γ ) does not admit any fundamental biorthogonal system (Godun, Kadets, Plichko, cf., e.g., [129, p. 238]). Kunen’s space C(K) does not have any nonseparable subspace with a fundamental biorthogonal system (Section 2). A Markushevich basis {xα , fα }α∈Γ for a Banach space X is called a bounded Markushevich basis if supα∈Γ {xα ·fα } < ∞. For bounded Markushevich bases we refer to [197, p. 44], [244,247]. A Banach space X is said to have Mazur’s intersection property if every closed bounded convex set in X is an intersection of a family of balls in X. By the results of Mazur and Phelps, this property is shared by all spaces with Fréchet differentiable norms (cf., e.g., [57, p. 55] or [80, Chapter 8]). We refer to [158], where it is proved that the non-Asplund space 1 ⊕ 2 (c) can be renormed to possess Mazur’s intersection property and that Kunen’s C(K) space in turn does not admit any norm with Mazur’s intersection property. The tool used in [158] is the concept of biorthogonal systems {xα , fα } in X such that span · {fα } = X∗ . It is shown in [158] that every space that admits such a system admits a norm with Mazur’s intersection property and that this is the case if X = C(K) for the compactification K of any tree. I do not know of any Banach space that would admit a C 1 smooth bump function and would not have Mazur’s intersection property at the same time. Related to


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Mazur’s intersection property is the result that every weakly compact set in a Banach space is the intersection of a family of finite unions of balls. This result was proved by Corson and Lindenstrauss in [43] for weakly compact sets in reflexive spaces (cf., e.g., [80, Chapter 8]) and later on, in full generality, by Godefroy and Kalton in [113]. Every WLD space X can be so renormed that in the new norm each weakly compact convex set is the intersection of a family of balls ([331]). E. Uniform homeomorphisms We have discussed only a little on nonlinear classifications of Banach spaces. We refer the reader to [20,112,114–116,169] and more references given below for this topics. F. Concluding remarks We have seen in this paper that in many cases, the separable and nonseparable theories of Banach spaces differ. Let us finish this article with mentioning a few more instances when this happens. First, it is well known that separable Lp spaces admit unconditional bases if 1 < p < ∞ (Paley). This is no longer true if Lp is nonseparable and p = 2 ([68, 101]). James showed that any non-reflexive separable Banach space with unconditional basis contains either c0 or 1 (see, e.g., [197, p. 23] or [80, p. 185]). However, there exists a nonseparable Banach space X with symmetric basis that does not contain any subspace isomorphic to c0 (Γ ) for uncountable Γ while every infinite-dimensional subspace of X contains a subspace isomorphic to c0 ([310]). This space is thus nonseparable nonreflexive with unconditional basis and does not contain an isomorphic copy of c0 (Γ ) or 1 (Γ ) for uncountable Γ . Lindenstrauss proved that every separable Banach space with unconditional basis is isomorphic to a complemented subspace of a space with a symmetric basis ([197, p. 123]). Troyanski showed in [310] that c0 (Γ ) × 1 (Γ ) is not isomorphic to any subspace of a space with a symmetric basis. An example of the use of [310] in the separable theory is, e.g., in [140]. Acknowledgements I would like to thank Marián Fabian, Gilles Godefroy and Kamil John for their long term collaboration with me in nonseparable Banach spaces. This chapter was prepared when I held the position at the Mathematical Institute of the Czech Academy of Sciences in Prague during the years 1998–2001. I thank this institute for providing me with excellent working conditions that enabled me to work on this chapter. I am grateful to Marián Fabian, Gilles Godefroy, Petr Hájek, William Johnson, Ondˇrej Kalenda, Joram Lindenstrauss, Anibal Molto, Matthias Neufang, José Orihuela, Jan Pelant, Jan Rychtáˇr, Stanimir Troyanski, and Vicente Montesinos, who contributed by their help, advice and/or suggestions to this chapter. Above all, I am indebted to my wife Jarmila for her creating continuing excellent conditions for my life long work in Banach spaces.

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[257] D. Preiss, R.R. Phelps and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings, Israel J. Math. 72 (1990), 257–279. [258] D. Preiss and P. Simon, A weakly pseudocompact subspace of a Banach space is weakly compact, Comment. Math. Univ. Carolin. 15 (1974), 603–610. [259] J. Rainwater, Day’s norm on c0 (Γ ), Proc. Amer. Math. Soc. 22 (1969), 335–339. [260] M. Raja, Measurabilité de Borel et renormages dans les espaces de Banach, Ph.D. thesis, Université de Bourdeaux (1998). [261] M. Raja, On locally uniformly rotund norms, Mathematika 46 (1999), 343–358. [262] M. Raja, Kadets norms an Borel sets in Banach spaces, Studia Math. 136 (1999), 1–16. [263] M. Raja, Weak∗ locally uniformly rotund norms and descriptive compact spaces, to appear. [264] J. Reif, A note on Markushevich bases in weakly compactly generated Banach spaces, Comment. Math. Univ. Carolin. 15 (1974), 335–340. [265] E.A. Rezniˇcenko, Normality and collectionwise normality of function spaces, Vestnik Mosk. Univ. Ser. Mat. (1990), 56–58. [266] N.K. Ribarska, Internal characterization of fragmentable spaces, Mathematika 34 (1987), 243–257. [267] N.K. Ribarska, On having a countable cover by sets of small local diameter, Studia Math. 140 (2000), 99–116. [268] G. Rodé, Superkonvexitat und schwache Kompaktheit, Arch. Math. 36 (1981), 62–72. [269] C.A. Rogers and J.E. Jayne, K-analytic Sets, Academic Press (1980). [270] H.P. Rosenthal, On quasicomplemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp (μ) to Lr (ν), J. Funct. Anal. 4 (1969), 176–214. [271] H. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Comp. Math. 28 (1974), 83–111. [272] H.P. Rosenthal, Pointwise compact subsets of the first Baire class, Amer. J. Math. 99 (1977), 362–378. [273] H.P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 803–831. [274] H.P. Rosenthal, Weak∗ -Polish Banach spaces, J. Funct. Anal. 76 (1988), 267–316. [275] H.L. Royden, Real Analysis, 3rd ed., Macmillan (1988). [276] J. Rychtáˇr, Uniformly Gâteaux differentiable norms in spaces with unconditional basis, Serdica Math. J. 26 (2000), 353–358. [277] J. Rychtáˇr, Uniformly rotund norms in every direction in dual spaces, Proc. Amer. Math. Soc., to appear. [278] W. Schachermayer, Some more remarkable properties of the James tree space, Contemp. Math. 85 (1987), 465–496. [279] Z. Semadeni, Banach spaces non-isomorphic to their Cartesian squares. II, Bull. Acad. Polon. Ser. Sci. Math. Astronom. Phys. 8 (1960), 81–84. [280] Z. Semadeni, Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warsaw (1971). [281] S. Shelah and J. Stepr¯ans, A Banach space on which there are few operators, Proc. Amer. Math. Soc. 104 (1988), 101–105. [282] P. Simon, On continuous images of Eberlein compacts, Comment. Math. Univ. Carolin. 17 (1976), 179– 194. [283] I. Singer, On the problem of nonsmoothness of nonreflexive second conjugate spaces, Bull. Austral. Math. Soc. 12 (1975), 407–416. [284] I. Singer, Bases in Banach Spaces II, Springer (1981). [285] M. Smith, Rotundity and smoothness in conjugate spaces, Proc. Amer. Math. Soc. 61 (1976), 232–234. [286] G.A. Sokolov, On some class of compact spaces lying in Σ products, Comment. Math. Univ. Carolin. 25 (1984), 219–231. [287] V.V. Srivatsa, Baire class 1 selectors for upper semicontinuous set valued maps, Trans. Amer. Math. Soc. 337 (1993), 609–624. [288] C. Stegall, The Radon–Nikodym property in conjugate spaces, Trans. Amer. Math. Soc. 206 (1975), 213– 223. [289] C. Stegall, The Radon–Nikodym property in conjugate Banach spaces II, Trans. Amer. Math. Soc. 264 (1981), 507–519. [290] C. Stegall, More facts about conjugate Banach spaces with the Radon–Nikodým property II, Acta Univ. Carolin. Math. Phys. 32 (1991), 47–54.


1815

[291] D.G. Tacon, The conjugate of a smooth Banach space, Bull. Austral. Math. Soc. 2 (1970), 415–425. [292] M. Talagrand, Sur la structure borelienne des espaces analytiques, Bull. Sci. Math. 101 (1977), 415–422. [293] M. Talagrand, Comparison des boréliens d’un espace de Banach pour topologies faibles et fortes, Indiana Math. J. 27 (1978), 1001–1004. [294] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math. 119 (1979), 407–438. [295] M. Talagrand, Sur les espaces de Banach contenant 1 (τ ), Israel J. Math. 40 (1981), 324–330. [296] M. Talagrand, Pettis Integral and Measure Theory, Mem. Amer. Math. Soc. 307 (1984). [297] M. Talagrand, Renormages de quelques C(K), Israel J. Math. 54 (1986), 327–334. [298] W.K. Tang, On the extension of rotund norms, C.R. Acad. Sci. Paris Sér. I 323 (1996), 487–490. [299] W.K. Tang, Uniformly differentiable bump functions, Arch. Math. 68 (1997), 55–59. [300] P. Terenzi, Every norming M-basis of a separable Banach space has a block perturbation which is norming strong M-basis, Extracta Math. (1990), 161–169. [301] P. Terenzi, Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis, Studia Math. 111 (1994), 207–222. [302] P. Terenzi, A positive answer to the basis problem, Israel J. Math. 104 (1998), 51–124. [303] S. Todorˇcević, Trees and linearly ordered sets, Handbook of Set Theoretic Topology, K. Kunen and J. Vaughan, eds, North-Holland (1984). [304] S. Todorˇcević, Topics in Topology, Lecture Notes in Math. 1652, Springer, Berlin (1997). [305] S. Todorˇcević, Compact subsets of the first Baire class, J. Amer. Math. Soc. 12 (1999), 1179–1212. [306] N. Tomczak-Jaegermann, Banach–Mazur Distances and Finite-dimensional Operator Ideals, Pitman Monographs Surveys Pure Appl. Math. 38 (1989). [307] H. Torunczyk, Smooth partitions of unity on some nonseparable Banach spaces, Studia Math. 46 (1973), 43–51. [308] S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173–180. [309] S. Troyanski, On equivalent norms and minimal systems in nonseparable Banach spaces, Studia Math. 43 (1972), 125–138. [310] S. Troyanski, On nonseparable Banach spaces with a symmetric basis, Studia Math. 53 (1975), 253–263. [311] S. Troyanski, On uniform rotundity and smoothness in every direction in nonseparable Banach spaces with an unconditional basis, C.R. Acad. Bulgare Sci. 30 (1977), 1243–1246. [312] S. Troyanski, Locally uniformly convex norms, C.R. Acad. Bulgare Sci. 32 (1979), 1167–1169. [313] S. Troyanski, On a property of the norm which is close to local uniform convexity, Math. Ann. 271 (1985), 305–313. [314] S. Troyanski, Construction of equivalent norms for certain local characteristics with rotundity and smoothness by means of martingales, Proc. 14 Spring Conference of the Union of Bulgarian Mathematicians (1985), 129–156. [315] M. Valdivia, Some more results on weak compactness, J. Funct. Anal. 24 (1977), 1–10. [316] M. Valdivia, Resolutions of the identity in certain Banach spaces, Collect. Math. 39 (1988), 127–140. [317] M. Valdivia, Some properties of weakly countably determined Banach spaces, Studia Math. 93 (1989), 137–144. [318] M. Valdivia, Projective resolutions of identity in C(K) spaces, Arch. Math. 54 (1990), 493–498. [319] M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collect. Math. 42 (1991), 265–284. [320] M. Valdivia, On certain classes of Markushevich basis, Arch. Math. 62 (1994), 445–458. [321] M. Valdivia, On certain topological spaces, Revista Mat. 10 (1997), 81–84. [322] J. Vanderwerff, Smooth approximations in Banach spaces, Proc. Amer. Math. Soc. 115 (1992), 113–120. [323] J. Vanderwerff, Extensions of Markuševiˇc bases, Math. Z. 219 (1995), 21–30. [324] J. Vanderwerff, J.H.M. Whitfield and V. Zizler, Markuševiˇc bases and Corson compacta in duality, Canad. J. Math. 46 (1994), 200–211. [325] L. Vašák, On a generalization of weakly compactly generated Banach spaces, Studia Math. 70 (1981), 11–19. [326] R. Vershynin, On constructions of strong a uniformly minimal M-bases in Banach spaces, Arch. Math. 74 (2000), 50–60.

1816 [327] [328] [329] [330]

V. Zizler

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Stud. Adv. Math. 25 (1991). D. Yost, Asplund spaces for beginners, Acta Univ. Carolin. 34 (1993), 159–177. D. Yost, The Johnson–Lindenstrauss space, Extracta Math. 12 (1997), 185–192. V. Zizler, Locally uniformly rotund renorming and decomposition of Banach spaces, Bull. Austral. Math. Soc. 29 (1984), 259–265. [331] V. Zizler, Renormings concerning the Mazur intersection property of balls for weakly compact convex sets, Math. Ann. 276 (1986), 61–66. [332] V. Zizler, Smooth extension of norms and complementability of subspaces, Arch. Math. 53 (1989), 585– 589.

Addenda and Corrigenda to Chapter 7, Approximation Properties by Peter G. Casazza 1. p. 309, Proposition 8.8 contains an example of a Banach space with the approximation property (AP) but failing the bounded compact AP. The first such example was due to Reinov [4a]. 2. p. 281, Theorem 2.5. Recently [1a] the classification of Banach spaces with the approximation property has been extended to coverings of compact sets in X by operator ranges from a universal Banach space with quite strong properties. T HEOREM 0.1 [1a]. For a Banach space X the following properties are equivalent: (i) X has the AP. (ii) There exists a reflexive Banach space R with basis and with unconditional finitedimensional decomposition such that for each compact K ⊂ BX and for each ε > 0 there is a compact one-to-one operator T : R → X with T (BR ) ⊃ K and T 1 + ε. (iii) For any compact set K ⊂ X there is an M-basic sequence {xi } in X (with biorthogonal functionals {xi∗ }) such that x = xi∗ (x)xi for each x ∈ K. The key element in Theorem 0.1(ii) is that T is one-to-one. Indeed, every compact set in every Banach space may be covered by an operator range of ( n ⊕n1 )2 . 3. Several surprising results on the stochastic approximation property have just appeared [4a]. Given a Radon probability measure μ on a Banach space X, we say that X has the μ-approximation property (μ-AP, in short) provided there is a sequence {Bn } of finite-rank operators on X so that x − Bn x → 0 for μ almost every x in X. We say that X has the stochastic AP provided X has the μ-AP for every Radon probability measure μ on X. If X is separable, we say that X has the μ-basis property (μ-BP, in short) if there is an M-basis {xn , xn∗ } for X for which ∗ xn (x)xn = 1. μ x ∈ X: x = n

We say that X has the stochastic BP provided X has the μ-BP for every Radon probability measure μ on X. As we have seen in Chapter 7, there is a whole sequence of distinct properties for a Banach space which lie between the AP and the basis property. However, in [4a] it is shown that for any Radon probability measure μ on a separable Banach space X, X has 1817

1818

Addenda and corrigenda to Chapter 7

the μ-AP if and only if it has the μ-BP. Therefore, the stochastic AP and the stochastic BP are equivalent properties for separable Banach spaces. It is further shown in [4a] that there are Banach spaces failing stochastic AP. Finally, another strong result in [4a] is that stochastic AP implies AP for Banach spaces with non-trivial type. 4. In [3a] connections are made between the bounded approximation property (BAP, in short) and the non-linear theory of Banach spaces: T HEOREM 0.2 [3a]. If X and Y are Lipschitz-isomorphic Banach spaces such that X has the BAP then Y also has the BAP. There are no known examples of separable X and Y which are Lipschitz-isomorphic and not linearly isomorphic. However, Theorem 0.2 also applies to nonseparable Banach spaces where counterexamples are known. D EFINITION 0.3. Let X be a Banach space and λ 1. We say that X has the λ-Lipschitz bounded approximation property (λ-Lip BAP, in short) if for every compact set K ⊂ X and every ε > 0 there exists a Lipschitz map F : X → X with finite-dimensional range such that F Lip λ and F (x) − x ε for all x ∈ K. T HEOREM 0.4 [3a]. Let X be an arbitrary Banach space. The following are equivalent: (i) X has the λ-BAP. (ii) X has the λ-Lip BAP.

New references [1a] V.P. Fonf, W.B. Johnson, A.M. Plichko and V.V. Shevchyk, Covering a compact set in a Banach space by an operator range of a Banach space with basis, Trans. Amer. Math. Soc., to appear. [2a] V.P. Fonf, W.B. Johnson, G. Pisier and D. Preiss, Stochastic approximation properties in Banach spaces, Studia Math. (Special issue in honor of A. Pełczyński on the occasion of his seventieth birthday), submitted. [3a] G. Godefroy and N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. (Special issue in honor of A. Pełczyński on the occasion of his seventieth birthday), submitted. [4a] O.I. Reinov, How bad may be a Banach space with the approximation property?, Mat. Zametki 33 (6) (1983), 833–846 (in Russian).

Addenda and Corrigenda to Chapter 8, Local Operator Theory, Random Matrices and Banach Spaces by K.R. Davidson and S.J. Szarek 1. p. 346, the Added in proof section: (i) The paper [1a], which is a revised version of [105], has been circulated in the meantime; it contains additionally some concentration results for not-necessarily-extreme eigenvalues. (ii) More precise (but still presumably far from optimal) results in the same direction as [1a] were obtained in [7a]. 2. p. 346, inequality (4): a factor 1/L in the middle expression is missing. It should read

P(F M + t) 1 − Φ(t/L) < exp −t 2 /2L2 . 3. p. 349, Theorem 2.8: more quantitative results (i.e., estimates valid for any dimension rather than in the limit) were obtained √ in [2a] and [6a]. 4. p. 352, inequality (11): a factor n in the middle expression is missing. It should read

√

P(F 2 + σ t) < 1 − Φ t n < exp −nt 2 /2 . √ 5. p. 353, Theorem 2.13: a factor n in the middle expression in the second displayed formula is missing. It should read # #

max P s1 (Γ ) 1 + β + t , P sm (Γ ) 1 − β − t

√

< 1 − Φ t n < exp −nt 2 /2 . 6. p. 354, Problem 2.14: the existence of the limit was proved in [3a]. 7. p. 357, Problem 2.18: solved in the affirmative in [4a]. 8. The book [5a], and particularly its Section 8.5, overlaps and complements the material presented in Section 2 of the chapter.

New references [1a] N. Alon, M. Krivelevich and V.H. Vu, On the concentration of eigenvalues of random symmetric matrices, Israel J. Math., to appear. [2a] G. Aubrun, A small deviation inequality for the largest eigenvalue of a random matrix, Preprint (2002). 1819

1820

Addenda and corrigenda to Chapter 8

[3a] F. Guerra and F.L. Toninelli, The thermodynamic limit in mean field spin glass models, Comm. Math. Phys. 230 (1) (2002), 71–79. [4a] U. Haagerup and S. Thorbjørnsen, A new application of random matrices: Ext(Cr∗ (F2 )) is not a group, Preprint (2002), available at http://arXiv.org/abs/math.OA/0212265 [5a] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monographs 89, Amer. Math. Soc., Providence, RI (2001). [6a] M. Ledoux, A remark on hypercontractivity and tail estimates for the largest eigenvalues of random matrices, Preprint (2002). [7a] M. Meckes, Concentration of norms and eigenvalues of random matrices, Preprint (2002), available at http://arXiv.org/abs/math.PR/0211192

Addenda and Corrigenda to Chapter 11, Operator Ideals by J. Diestel, H. Jarchow and A. Pietsch The diagram on page 490 should be corrected as follows:

ν1

π2dual

α adj

π2

axis of symmetry

lift

(π1dual )ext

α

π1

dual dual π1 π1 π2 ◦ π2

dual π π2 2

λ1 λ2 λ∞

sur inj λ λ1 ∞

·

ν1 α

α dual ·

axis of symmetry

Since we are in the finite-dimensional setting the 1-nuclear norm ν 1 coincides with the 1-integral norm ι1 .

1821


Addenda and Corrigenda to Chapter 15, Infinite Dimensional Convexity by V.P. Fonf, J. Lindenstrauss and R.R. Phelps 1. p. 610, l. 15. In the mean time there appeared a revised (also somewhat expanded) version of [136]. The right reference at this point is [3a, Section 15]. 2. p. 641, Theorem 5.7 and p. 644, Theorem 5.14 ((1) ⇒ (4)). A more streamlined proof of these assertions appears in [2a]. This paper contains some other related results. For example, if X is separable and non-reflexive and its unit sphere is covered by a union ∗ of caps {Dn }∞ n=1 of radius a < 1 then for every sequence εn → 0 there is an f ∈ X with f = 1 and such that sup{f (x): x ∈ Dn } 1 − εn for every n. 3. p. 662, Proposition 7.8. In [36] it is only proved that any covering of a reflexive space by CCB sets cannot be locally finite. The stronger statement made in Proposition 7.8 is mentioned in [36] as a remark without proof. We do not know at present whether Proposition 7.8 is really true. 4. p. 663, l. 11. Erase the sentence starting with “Subsequent to this. . . ”, and replace it by the following: A survey of more recent results in this direction is given in [1a]. However, in spite of the many results mentioned in this reference, the problem of convexity of Chebyshev sets in Hilbert space is still open.

New references [1a] V.S. Balaganskii and L.P. Vlasov, The problem of convexity of Chebyshev sets, Russian Math. Surveys 51 (1996), 1127–1190. [2a] V.P. Fonf and J. Lindenstrauss, Boundaries and generation of convex sets, Israel J. Math., to appear. [3a] R.R. Phelps, Lectures on Choquet’s Theorem, 2nd ed., Lecture Notes in Math. 1757, Springer (2001).

1823


Author Index Roman numbers refer to pages on which the author (or his/her work) is mentioned. Italic numbers refer to reference pages. Numbers between brackets are the reference numbers. No distinction is made between first and co-author(s).

Aarts, J. 1802, 1806 [1] Abramovich, Y.A. 87, 88, 90–94, 99–105, 108, 110, 111, 116, 117 [1–9]; 118 [10–21]; 535, 558 [1]; 1665, 1667 [1] Adams, R. 1363, 1365, 1398, 1399, 1403, 1404, 1419 [1] Aharoni, I. 829, 830 [1,2]; 906, 935 [1]; 1757, 1796, 1806 [2] Aizenman, M. 354, 360 [1] Akemann, C.A. 1461, 1510 [1] Akilov, G.P. 106, 107, 120 [74] Al-Husaini, A.L. 905, 935 [2] Albiac, F. 1119, 1127 [1,2] Aldous, D.J. 136, 156 [1]; 237, 265 [1]; 515, 526, 527 [1] Alencar, R. 812, 830 [3] Alesker, S. 714, 731–735, 772 [1–5] Alexander, H. 674, 675, 704 [1] Alexandrov, A.B. 1101, 1120, 1127 [3,4]; 1506, 1510 [2] Alexandrov, A.D. 727, 731, 772 [6,7] Alexandrov, A.G. 1800, 1806 [3] Alexopoulos, J. 515, 528 [2,3] Alfsen, E.M. 310, 313 [1]; 611–614, 620, 621, 627, 665 [1–3] Aliprantis, C.D. 21–24, 83 [1]; 87–90, 92–97, 99–105, 108, 110, 111, 113, 116, 117 [4–9]; 118 [10–14,22–29]; 535, 558 [1] Allekhverdiev, D.E. 453, 490 [1] Allen, G.D. 527, 528 [4] Allexandrov, G. 304, 313 [2] Alon, N. 358, 360 [2]; 770, 772 [8]; 1632, 1632 [1]; 1819, 1819 [1a] Alonso, J. 793, 830 [4] Alspach, D.E. 59, 83 [22]; 133, 135, 147, 151, 154, 156, 156 [2–8]; 279, 304, 313 [3,4]; 581, 595 [1]; 839, 862, 868, 868 [1]; 875, 896 [1]; 1019, 1038,

1051, 1052, 1064, 1065 [1,2]; 1351, 1359 [1]; 1560, 1569, 1580, 1585, 1594, 1598–1600, 1600 [1–7]; 1698, 1700 [1]; 1719, 1739 [1] Alster, K. 1754, 1762, 1806 [4,5] Altshuler, Z. 134, 156 [9]; 511, 527, 528 [5,6] Amann, H. 250, 265 [2] Ambrosio, L. 1523, 1544, 1544 [1,2] Amemiya, I. 89, 118 [30]; 440, 490 [2] Amir, D. 744, 745, 770, 772 [9,10]; 793, 820, 822, 830 [5,6]; 1572, 1577, 1596, 1600 [8]; 1632, 1632 [2,3]; 1714, 1717, 1730, 1739 [2–4]; 1745, 1748, 1755, 1762, 1763, 1806 [6,7] Anantharaman-Delaroche, C. 1431, 1455 [1] Andersen, N.T. 341, 360 [3] Anderson, J. 332–335, 360 [4–7]; 1461, 1510 [1] Andô, T. 110, 118 [31]; 130, 147, 156 [10]; 255, 265 [3]; 904, 935 [3] Androulakis, G. 1065 [3]; 1345, 1352, 1359 [2–4] Angelos, J.R. 320, 360 [8] Ansari, S. 545, 547, 558 [2] Ansel, J.P. 385, 390 [1] Antipa, A. 257, 265 [4] Aoki, T. 1101, 1102, 1127 [6] Apostol, C. 330, 332, 360 [9–11] Arai, H. 704, 704 [2] Araki, H. 1470, 1510 [3]; 1694, 1700 [2] Araujo, A. 1191, 1198 [1–3] Arazy, J. 1151, 1172 [1]; 1461, 1467, 1468, 1477–1479, 1507, 1510 [4–10]; 1511 [11] Archangel’skii, A.V. 1745, 1747, 1748, 1763, 1806 [8–10] Arenson, E.L. 87, 118 [15] Argyros, S. 139, 154, 157 [11,12]; 822, 830 [7]; 1019, 1038–1041, 1050, 1051, 1053, 1056, 1058, 1059, 1062–1065, 1065 [1]; 1066 [4–12]; 1081, 1096 [1]; 1253–1256, 1266, 1272, 1275, 1280, 1281, 1295 [1–3]; 1351, 1352, 1359 [1,5]; 1554, 1560, 1600 [9]; 1601 [10]; 1752, 1754, 1756, 1759, 1761,

1825

1826

Author Index

1763, 1767, 1774, 1775, 1781, 1784, 1800–1802, 1806 [11–14] Arias, A. 1440, 1454, 1455 [2,3] Arias-de-Reyna, J. 716, 772 [11]; 1607, 1632 [4] Arnold, L. 343, 360 [12] Aron, R.M. 555, 558 [3]; 676, 704 [3]; 812, 830 [3] Aronszajn, N. 535, 558 [4]; 1532, 1544 [3] Arvanitakis, A.D. 1554, 1600 [9] Arveson, W.B. 339, 340, 360 [13,14]; 535, 558 [5,6]; 1427, 1428, 1431, 1455 [4]; 1495–1497, 1511 [12,13] Ash, M. 215, 230 [1,2] Asimov, L. 615, 618, 620, 621, 626, 627, 665 [4] Asmar, N.H. 249, 265 [5,6]; 1371, 1419 [2] Asplund, E. 663, 665 [5]; 792, 798, 805, 828, 830 [8,9] Astashkin, S.V. 1155, 1172 [2] Aubin, J.P. 433 [1]; 798, 830 [10] Aubrun, G. 1819, 1819 [2a] Axler, S. 1700, 1701 [3] Azagra, D. 409, 422, 433 [2,3]; 799, 830 [11]; 1748, 1753, 1806 [15,16] Azoff, E. 328, 360 [15] Azuma, K. 1610, 1632 [5] Babenko, K.I. 1638, 1667 [2] Bachelier, L. 369, 390 [2] Bachelis, G.F. 204, 230 [3]; 884, 896 [2] Bachir, M. 411, 416, 418, 433 [4,5] Baernstein, A. 259, 265 [7]; 444, 490 [3] Baernstein II, A. 1608, 1632 [6] Bagaria, J. 1094, 1096 [2] Bai, Z.D. 344, 348, 353, 358, 360 [16–19]; 361 [20]; 366 [187] Baire, R. 1019, 1066 [13] Bakhtin, I.A. 88, 118 [32] Bakry, D. 350, 361 [21] Balaganskii, V.S. 1823, 1823 [1a] Ball, K.M. 163, 165, 171, 177, 183, 185, 187, 192, 193 [1–5]; 716, 718, 722, 724, 725, 728, 772, 772 [11,12]; 773 [13–18]; 901, 918, 935 [4,5]; 1223, 1244 [1]; 1481, 1511 [14]; 1607, 1632 [4] Banach, S. 273, 313 [5]; 444, 490 [4]; 524, 528 [7]; 1249, 1295 [4]; 1705, 1739 [5] Banaszczyk, W. 767, 773 [19] Bang, T. 183, 193 [6] Bañuelos, R. 259, 265 [8–10] Bapat, R.B. 98, 113, 118 [33] Barany, I. 175, 193 [7] Barles, G. 426, 431, 433 [6] Barthe, F. 164, 171, 173, 178, 193 [8,9]; 718, 773 [20]; 921, 935 [6] Bastero, J. 519, 528 [8,9]; 769, 773 [21]; 1103, 1106, 1127, 1127 [7–10]

Bates, S. 1539, 1544 [4] Baturov, D.P. 1747, 1806 [17] Bauer, H. 621, 665 [6] Beauzamy, B. 6, 7, 52, 57, 83 [2]; 444, 474, 479, 480, 490 [5,6]; 491 [7]; 549, 550, 555, 558 [3,7–9]; 792, 804, 830 [12]; 1136, 1137, 1172 [3]; 1340, 1352, 1359 [6] Beck, A. 1181, 1198 [4]; 1303, 1330 [1] Becker, R. 613, 665 [7] Beckner, W. 481, 491 [8] Behrends, E. 1079, 1096 [3]; 1749, 1806 [18] Bekollé, D. 1676, 1701 [4] Bell, M. 1796, 1806 [19] Bellenot, S. 1062, 1066 [14] Bellow, A. 260, 265 [11] Ben Arous, G. 345, 346, 361 [22] Benedek, A. 107, 118 [34] Benitez, C. 793, 830 [4,13] Benjamini, Y. 1392, 1419 [3] Bennett, C. 505, 515, 528 [10]; 1133, 1136, 1147, 1172 [4,5]; 1404, 1407, 1420 [4] Bennett, G. 147, 157 [13]; 230 [4]; 342, 361 [23]; 463, 491 [9,10]; 748, 773 [23]; 866, 867, 868 [2]; 962, 973 [1] Benveniste, E.J. 331, 359, 361 [24] Benyamini, Y. 35, 36, 38, 42, 48, 83 [3]; 279, 313 [6]; 342, 354, 361 [25]; 630, 634, 665 [8]; 766, 773 [22]; 822, 829, 830 [14,15]; 906, 935 [7]; 1087, 1096 [4]; 1158, 1172 [6]; 1251, 1260, 1295 [5]; 1310, 1330 [2]; 1348, 1359 [7]; 1521, 1540, 1544 [5]; 1560, 1569, 1594, 1596–1600, 1600 [5–7]; 1601 [11–14]; 1608, 1632 [7]; 1718, 1739 [6]; 1745, 1751, 1752, 1754–1756, 1780, 1797, 1805, 1806 [20–22] Bercovici, H. 341, 361 [26] Berg, I.D. 323–325, 328, 329, 331, 361 [27–29] Bergh, J. 74, 76–78, 80, 83 [4]; 505, 528 [11]; 577, 595 [2]; 692, 704 [4]; 1133, 1136, 1139, 1172 [7,8]; 1280, 1295 [6]; 1404, 1420 [5]; 1437, 1455 [5,6]; 1466, 1511 [15] Berkson, E. 237, 249, 250, 265 [5,12,13]; 1371, 1411–1413, 1419 [2]; 1420 [6]; 1494, 1511 [16] Berman, A. 98, 113, 118 [35] Berman, K. 336, 361 [30]; 859, 868 [3] Bernstein, A.J. 1657, 1667 [3] Bernstein, A.R. 535, 558 [10] Bernstein, S. 470, 491 [11] Bernués, J. 769, 773 [21]; 1062, 1066 [15]; 1127, 1127 [10] Besbes, M. 1700, 1701 [5] Besov, O.V. 1363, 1398, 1399, 1409, 1420 [7] Bessaga, C. 603, 648, 665 [9,10]; 792, 799, 830 [16,17]; 1029, 1045, 1066 [16]; 1249, 1295 [7];

Author Index 1560, 1571, 1601 [15]; 1745, 1759, 1796, 1800, 1806 [23–25] Bhatia, R. 327, 329, 347, 361 [31–36]; 1461, 1511 [17] Biane, P. 360, 361 [37]; 1482, 1491, 1511 [18,19] Billard, P. 569, 595 [3]; 1594, 1601 [16] Bishop, E. 608, 610, 640, 641, 665 [11–13]; 1733, 1739 [7] Björk, T. 374, 390 [3] Black, F. 369, 371, 390 [4] Blasco, O. 250, 265 [14,15]; 894, 896 [3]; 1682, 1701 [6] Blecher, D. 1427, 1430, 1432, 1435, 1438, 1442, 1444, 1445, 1454, 1455 [7–12]; 1456 [13–16] Blower, G. 264, 265 [16] Boas, R.P. 156, 157 [14]; 1678, 1701 [7] Bobkov, S.G. 350, 358, 361 [38]; 1623, 1632 [8] Bochner, S. 265 [17]; 480, 491 [12] Boˇckariov, S.V. 569, 575, 595 [4–6]; 1693, 1701 [8] Bogachev, V.I. 1528, 1532, 1544 [6–8] Bohnenblust, F. 140, 157 [15]; 465, 491 [13]; 1705, 1739 [8] Bolker, E.D. 525, 528 [12]; 768, 773 [24]; 911, 921, 924, 935 [8] Bollobás, B. 1077, 1078, 1096 [5] Boman, J. 1408, 1409, 1420 [8] Bombieri, E. 207, 230 [5]; 555, 558 [9] Bomze, I.M. 112, 119 [43] Bonami, A. 1481, 1511 [20]; 1676, 1701 [4] Bonic, R. 413, 433 [7]; 799, 830 [18] Bonsall, F.F. 608, 665 [14] Border, K.C. 87, 100, 118 [22] Borell, C. 346, 361 [39]; 717, 773 [25]; 1531, 1544 [9] Borovikov, V. 614, 665 [15] Borsuk, K. 1371, 1420 [9]; 1553, 1601 [17] Borwein, J.M. 396, 399, 418, 433 [8,9]; 664, 665 [16]; 798, 820, 830 [19]; 1521, 1538, 1544 [10,11]; 1749, 1806 [26,27] Bossard, B. 793, 805, 828, 830 [20–22]; 1020, 1042, 1044, 1066 [17–21]; 1787, 1806 [28] Bourbaki, N. 1733, 1739 [9] Bourgain, J. 139, 147, 150, 154, 157 [16–21]; 175, 193 [12]; 201, 204, 206, 209, 213, 214, 219–221, 223, 226, 229, 230, 230 [6–11]; 231 [12–14]; 237, 245, 247, 250, 265 [18–22]; 336, 337, 358, 359, 361 [40–44]; 445, 465, 468, 482, 491 [14–17]; 576, 580, 581, 590, 592, 595 [7–11]; 633, 634, 636, 658, 665 [17–20]; 675, 686, 687, 699, 703, 704, 704 [5–11]; 718, 724, 736, 742, 755, 759, 763, 766–772, 773 [26–40]; 805, 830 [23]; 844, 845, 853, 854, 859, 860, 862–867, 868 [4–13]; 884, 885, 891, 892, 895, 896, 896 [4,5]; 897 [6–8]; 918, 925, 927, 935 [9–12]; 1018–1020, 1039, 1046, 1064, 1066 [22–26]; 1158, 1172 [9]; 1221, 1231, 1235,

1827

1240, 1244 [2–4]; 1254, 1295 [8]; 1305, 1330 [3]; 1351, 1359 [8–10]; 1371, 1396, 1397, 1403, 1405, 1411–1413, 1419 [2]; 1420 [6,10–14]; 1494, 1502, 1511 [21,22]; 1581, 1583, 1587, 1589, 1595, 1599, 1601 [18–21]; 1630, 1633 [9]; 1646, 1647, 1649–1651, 1667, 1667 [4–6]; 1694, 1701 [9,10]; 1716, 1726, 1739 [10–12]; 1748, 1758, 1767, 1776, 1802, 1806 [29,30] Bourgin, R.D. 260, 265 [23]; 633, 634, 665 [21,22]; 793, 830 [24]; 1745, 1789, 1807 [31] Boutet de Monvel, A. 349, 361 [45] Bouziad, A. 1807 [32] Boyd, D.W. 514, 528 [13]; 1151, 1155, 1172 [10]; 1663, 1668 [7] Bo˙zejko, M. 1482, 1511 [23,24] Brascamp, H.J. 164, 193 [10]; 718, 773 [41] Bratteli, O. 620, 665 [23]; 1466, 1511 [25] Braverman, M.Sh. 523, 528 [14,15] Brenier, Y. 173, 193 [11]; 713, 773 [42] Bretagnolle, J. 140, 157 [22]; 524, 527, 528 [16,17]; 855, 868 [14]; 906, 935 [13] Bronk, B.V. 344, 361 [46] Brown, D.J. 113, 116, 118 [23,24] Brown, L.G. 323, 331, 361 [47,48]; 1477, 1511 [26] Brown, S.W. 341, 362 [49]; 543, 558 [11] Brudnyi, Yu.A. 74, 83 [5]; 1108, 1127 [11]; 1133, 1142, 1145, 1149, 1172 [11–13]; 1373, 1383, 1420 [15,16] Brunel, A. 1036, 1066 [27,28]; 1074, 1096 [6]; 1304, 1306, 1307, 1310, 1330 [4,5] Bu, S.Q. 139, 157 [23]; 263, 265 [24] Buchholz, A. 1487, 1511 [27–29] Bukhvalov, A.V. 87, 89, 96, 106, 108, 118 [37,38]; 119 [39–41]; 263, 265 [25–27]; 638, 665 [24]; 1484, 1511 [30] Burago, Y.D. 710, 712, 726, 774 [43] Burger, M. 935 [14] Bürger, R. 112, 119 [42,43] Burkholder, D.L. 10, 83 [23]; 126, 128, 157 [24,25]; 237, 238, 241–246, 248–251, 253, 254, 256–259, 262, 266 [28–44]; 477, 482, 491 [18–22]; 523, 528 [18]; 590, 595 [12]; 895, 897 [9]; 901, 906, 935 [15]; 1484, 1492, 1494, 1495, 1511 [31,32]; 1664, 1668 [8] Burkinshaw, O. 21–24, 83 [1]; 87–90, 92–97, 99–105, 108, 110, 111, 113, 117 [4–9]; 118 [10–13,23,25–29]; 535, 558 [1] Busemann, H. 177, 193 [13]; 918, 935 [16] Cabello Sanchez, F. 805, 830 [25] Caffarelli, L.A. 714, 774 [44] Calderón, A.P. 245, 266 [45,46]; 1139–1141, 1143, 1151, 1165, 1172 [14,15]; 1469, 1511 [33]

1828

Author Index

Calkin, J.W. 439, 491 [23] ˇ Canturija, Z.A. 579, 596 [18] Capon, M. 1048, 1066 [29] Caradus, S.R. 558 [12] Carathéodory, C. 712, 774 [45] Carl, B. 175, 193 [14]; 452, 463, 491 [24,25]; 854, 868 [15]; 958, 959, 962, 973, 973 [2–4]; 1305, 1330 [6] Carlen, E.A. 1461, 1481, 1482, 1491, 1511 [14,34–36] Carleson, L. 223, 231 [15] Carne, T.K. 1443, 1456 [17] Carothers, N.L. 135, 147, 156 [7]; 304, 313 [3]; 504, 505, 511, 523, 525–527, 528 [19–32]; 901, 935 [17] Carro, M.J. 1161, 1172 [16] Cartier, P. 610, 665 [25] Cartwright, D.I. 92, 119 [44]; 1715, 1739 [13] Casazza, P.G. 12, 14, 58, 60, 83 [24]; 133, 136, 140, 157 [26–28]; 276, 279, 285, 286, 291–294, 297–299, 302, 304, 309, 311, 313, 313 [7–18]; 511, 527, 528 [6]; 529 [33,34]; 812, 821, 830 [26,27]; 1063, 1065, 1066 [30,31]; 1094, 1096 [7]; 1157, 1172 [17]; 1255, 1295 [9]; 1418, 1420 [17]; 1646, 1647, 1649–1651, 1657, 1658, 1667 [6]; 1668 [9–12]; 1739, 1739 [14] Cascales, B. 1807 [33] Caselles, V. 110, 119 [45,46] Castillo, J.M.F. 1107, 1127 [12]; 1155, 1172 [18]; 1745, 1789, 1799, 1803, 1807 [34,35] Cauty, R. 1127, 1127 [13,14] ˇ Cech, E. 1753, 1807 [36] Cepedello, M. 813, 828, 830 [28,29]; 831 [30,31]; 1807 [37] Cerda, J. 1161, 1172 [16] Ceretelli, O.D. 1170, 1172 [19] Chaatit, F. 1019, 1036, 1045, 1046, 1066 [32,33]; 1067 [34]; 1348, 1359 [11] Chang, K.-C. 433 [10] Chatterji, S.D. 260, 266 [47,48] Chen, S.T. 515, 529 [35–37] Chen, Z.L. 97, 119 [47] Cheridito, P. 383, 390 [5] Chevet, S. 354, 362 [50]; 461, 491 [26] Chilin, V.I. 1477, 1478, 1484, 1511 [37–40]; 1512 [41] Cho, C. 310, 313 [19] Choi, C. 259, 266 [49,50] Choi, M.D. 101, 119 [48]; 323, 362 [51]; 1431, 1455, 1456 [18,19] Choquet, G. 605, 608, 611, 613, 614, 629, 665 [26–33]; 793, 831 [32]; 1025, 1047, 1048, 1067 [35] Choulli, M. 433 [11] Choulli, T. 389, 390 [6,7] Christensen, E. 340, 362 [52]; 1431, 1440–1442, 1444, 1445, 1456 [20,21]

Christensen, J.P.R. 624, 666 [34]; 1018, 1019, 1048, 1067 [36]; 1531, 1544 [12] Ciesielski, K. 1758, 1807 [38] Ciesielski, Z. 575, 583, 585–587, 595 [13–17]; 973, 973 [5] Clarke, F. 412, 423, 425, 433 [12,13]; 434 [14–16] Clarkson, J.A. 128, 157 [29]; 484, 491 [27]; 1775, 1807 [39] Clément, P. 245, 266 [51] ˇ Coban, M.M. 1752, 1807 [40] Cobos, F. 266 [52]; 1471, 1472, 1512 [42] Coifman, R.R. 1120, 1128 [15]; 1158, 1172 [20]; 1406, 1420 [18]; 1682, 1691–1693, 1696, 1701 [11,12] Cole, B.J. 676, 682, 704 [3]; 705 [12] Colin de Verdiere, Y. 1456 [22] Connes, A. 1452, 1456 [23] Connes, B. 215, 231 [16] Contreras, M.D. 642, 666 [35] Cooke, R. 214, 231 [17] Corson, H.H. 624, 629, 662, 666 [32,36,37]; 1747, 1748, 1750, 1751, 1756, 1757, 1760, 1775, 1800, 1803, 1805, 1807 [41–43] Corvellec, J.-N. 425, 434 [17] Coulhon, T. 250, 266 [53]; 1363, 1423 [95] Cowen, C.C. 320, 360 [8]; 471, 491 [28,29] Crandall, M.G. 420, 421, 426, 431, 434 [18–20] Creekmore, J. 505, 529 [38] Csörnyei, M. 1532, 1544 [13] Cuartero, B. 1116, 1117, 1124, 1128 [16] Cuculescu, I. 1493, 1512 [43,44] Cwikel, M. 505, 529 [39]; 1140–1143, 1145, 1147, 1149, 1151, 1153, 1158, 1159, 1161, 1172 [1,20]; 1173 [21–33] Dacunha-Castelle, D. 140, 157 [22,30]; 455, 491 [30]; 524, 527, 528 [16,17]; 855, 868 [14]; 906, 935 [13]; 1304, 1307, 1330 [7] Dadarlat, M. 332, 362 [53] Dalang, R.C. 374, 390 [8] Dancer, E.N. 112, 119 [49] Danilevich, A.A. 263, 265 [27]; 638, 665 [24]; 1484, 1511 [30] Dar, S. 714, 724, 731, 732, 763, 772 [5]; 774 [46–48] Dashiell, F.K. 1775, 1794, 1807 [44] Daubechies, I. 566, 578, 596 [19] David, G. 250, 266 [54]; 591, 596 [20]; 702, 705 [13] Davidson, K.R. 320, 323–325, 328, 331–333, 339–341, 361 [29]; 362 [54–67]; 859, 868 [16]; 969, 973 [6] Davie, A.M. 283, 313 [20,21]; 673, 705 [14] Davies, R.O. 1529, 1544 [14] Davis, B.J. 258, 266 [43]; 523, 528 [18]; 1170, 1173 [34]

Author Index Davis, C. 327–329, 339, 347, 360 [15]; 361 [34,35]; 362 [68] Davis, W.B. 1020, 1067 [37] Davis, W.J. 96, 119 [50]; 265 [21]; 266 [55]; 443, 491 [31]; 646, 666 [38]; 743, 766, 774 [49]; 792, 793, 821, 831 [33–35]; 850, 868 [17]; 1103, 1128 [17]; 1133, 1136, 1137, 1173 [35]; 1254, 1280, 1295 [10]; 1483, 1512 [45]; 1643, 1668 [13]; 1761, 1766, 1803, 1804, 1807 [45,46] Day, M.M. 792, 798, 821, 831 [36,37]; 1101, 1102, 1128 [18]; 1396, 1420 [19]; 1774, 1775, 1807 [47] de Acosta, A. 1191, 1198 [1] de Branges, L. 105, 118 [36]; 541, 558 [13]; 603, 666 [39] de Figueiredo, D.G. 434 [21] de Leeuw, K. 184, 193 [15]; 608, 610, 665 [12]; 965, 974 [19]; 1700, 1701 [28] de Pagter, B. 110, 111, 121 [106]; 245, 266 [51]; 1465, 1467, 1468, 1477, 1478, 1512 [51–56] de Valk, V. 515, 529 [50] Debs, G. 793, 831 [38]; 1019, 1067 [38] Deddens, J.A. 339, 362 [69] Defant, A. 466, 467, 484, 488, 490, 491 [32]; 1513 [98] Defant, M. 590, 596 [21] Degiovanni, M. 425, 434 [17] Deift, P. 349, 362 [70] Delbaen, F. 374, 377–386, 389, 390 [9–16]; 465, 491 [16]; 658, 665 [18]; 901, 905, 911–915, 935 [18]; 1599, 1601 [20]; 1674, 1701 [13]; 1726, 1739 [11] Deliyanni, I. 1062, 1063, 1065, 1066 [5–7,15]; 1253, 1255, 1256, 1275, 1295 [1]; 1352, 1359 [5] Dellacherie, C. 374–376, 390 [17] Deville, R. 33, 34, 57, 83 [6,25]; 406, 408, 409, 413, 415, 418–422, 433 [2,3,11]; 434 [22–31]; 476, 491 [33]; 644, 659, 666 [40,41]; 792, 793, 795, 798, 799, 805, 812–814, 821, 831 [39–47]; 1745, 1747–1750, 1752–1755, 1757–1767, 1772–1778, 1781–1792, 1794–1799, 1801, 1804, 1806 [15]; 1807 [48–59] Devinatz, A. 1495–1497, 1512 [46] DeVore, R.A. 575, 596 [22]; 1404, 1420 [20,21] Diestel, J. 13, 18, 35, 36, 38–40, 47, 55, 60, 65, 66, 72, 83 [7–9]; 106, 107, 119 [51]; 259, 260, 263, 266 [56]; 445, 455, 458, 459, 464, 466, 471, 474, 479, 480, 483, 484, 491 [17,34–37]; 518, 529 [40]; 558 [14]; 675, 681, 690, 705 [15]; 792, 793, 806, 831 [48–50]; 867, 868 [18]; 879, 881, 882, 887, 897 [10]; 929, 935 [19]; 943, 950, 951, 974 [7]; 1222, 1244 [5]; 1378, 1387–1389, 1396, 1420 [22,23]; 1452, 1456 [24]; 1563, 1601 [22]; 1710, 1711, 1724, 1733, 1736, 1739 [15,16]; 1745, 1748, 1749, 1762, 1770, 1803, 1808 [60,61]

1829

Dilworth, S.J. 504, 511, 518, 519, 523–527, 528 [22–29]; 529 [41–49]; 769, 770, 774 [50,55]; 906, 935 [20]; 1103, 1126, 1128 [19,20]; 1142, 1173 [36]; 1700, 1701 [5] Dineen, S. 676, 705 [16]; 812, 830 [3] Dinger, U. 1530, 1544 [15] Ditor, S. 1552, 1554, 1556, 1593, 1601 [23,24] Dixmier, J. 1388, 1420 [24]; 1461, 1463, 1466, 1480, 1512 [47,48] Dixon, A.C. 451, 491 [38] Dixon, P.G. 286, 310, 313 [22] Dmitrovski˘ı, V.A. 347, 362 [71] Dobrowolski, T. 799, 830 [11]; 1753, 1806 [16] Dodds, P.G. 94, 119 [52]; 1465, 1467, 1468, 1477, 1478, 1480, 1511 [37,38]; 1512 [49–58] Dodds, T.K. 1465, 1467, 1468, 1477, 1478, 1480, 1512 [49–57] Doléans-Dade, C. 389, 390 [18] Domenig, T. 471, 492 [39] Doob, J.L. 260, 267 [57] Dor, L.E. 131, 147, 155, 157 [13,31,32]; 230 [4]; 255, 256, 267 [58]; 565, 596 [23]; 748, 768, 773 [23]; 774 [51]; 862, 866, 867, 868 [2,19]; 906, 935 [21]; 1652, 1668 [14] Dore, G. 250, 267 [59–62] Douglas, R.G. 323, 331, 361 [47,48] Doust, I. 256, 267 [63] Dow, A. 1808 [62] Dowling, P.M. 1477, 1512 [50] Dowling, P.N. 263, 267 [64,65]; 1700, 1701 [5] Downarowicz, T. 617, 666 [43] Dragilev, M.M. 1667, 1668 [15–17] Drewnowski, L. 1105, 1128 [21] Driouich, A. 267 [66] Drnovšek, R. 102, 119 [53] Duan, Y. 515, 529 [36] Dubinsky, E. 473, 492 [40]; 1302, 1303, 1330 [8]; 1667, 1668 [18] Dudley, R.M. 1188, 1190, 1193, 1194, 1198 [5,6]; 1625, 1633 [10] Duffie, D. 374, 390 [19] Dugundji, J. 603, 666 [44]; 1756, 1808 [63] Dunford, N. 106, 107, 119 [54,55]; 439, 444, 492 [41,42]; 1366, 1388, 1395, 1420 [25]; 1579, 1601 [25] Dupire, B. 371, 390 [20] Duren, P.L. 675, 694, 705 [17]; 1102, 1104, 1113, 1120, 1128 [22,23]; 1677, 1681, 1701 [14] Durier, R. 650, 655, 666 [45] Durrett, R. 5, 6, 83 [10] Dvoretzky, A. 136, 157 [33]; 458, 475, 492 [43,44]; 720, 735, 774 [52–54]; 1303, 1315, 1330 [9–11]; 1606, 1625, 1626, 1633 [11,12]

1830

Author Index

Dykema, K.J. 345, 357, 358, 362 [72]; 366 [178]; 1171, 1173 [37,38]; 1445, 1458 [103]; 1466, 1487, 1516 [196] Dynkin, E.B. 616, 666 [42] Eaton, M. 910, 935 [22] Ebenstein, S.E. 204, 230 [3]; 884, 896 [2] Èdelšte˘ın, I.S. 149, 157 [34]; 1644, 1645, 1668 [19] Edelstein, M. 664, 666 [46] Edgar, G.A. 260, 263, 267 [65,67–69]; 631, 633, 636, 639, 665 [22]; 666 [47–49]; 1484, 1512 [59]; 1745, 1758, 1764, 1766, 1789–1791, 1802, 1803, 1808 [64–67] Edmunds, D.E. 969, 974 [8] Edwards, D.A. 607, 615, 620, 666 [50–52] Edwards, R.E. 538, 558 [15] Effros, E.G. 310, 313 [1]; 621, 622, 626, 666 [53,54]; 1427, 1431, 1432, 1434, 1435, 1438–1447, 1452, 1454, 1455, 1456 [18–20,25–43]; 1510, 1512 [60] Egghe, L. 260, 267 [70] Einstein, A. 369, 390 [21–23] Ekeland, I. 395, 409, 434 [32,33]; 798, 830 [10]; 831 [51] El Haddad, E.M. 418, 420, 421, 434 [27,28] El Karoui, N. 382, 390 [24] El-Gebeily, M.A. 156, 157 [35] El-Mennaoui, O. 267 [66] Ellentuck, E. 1049, 1067 [39]; 1077, 1096 [8] Ellis, A.J. 615, 618, 620, 621, 626, 627, 665 [4] Elton, J. 1046, 1049, 1050, 1056, 1064, 1067 [40] Emery, M. 378, 379, 390 [25] Enflo, P. 98, 119 [56]; 129, 133, 134, 156 [2]; 157 [36,37]; 235, 237, 267 [71]; 273, 279, 280, 283, 285, 313 [4,23]; 446, 451, 479, 489, 492 [45,46]; 543, 545, 547, 549, 550, 555, 558 [2,3,9,16–19]; 766, 772, 774 [56,57]; 804, 805, 829, 831 [52–54]; 1115, 1128 [24]; 1157, 1173 [39]; 1304, 1330 [12]; 1348, 1359 [12]; 1805, 1808 [68] Engelking, R. 1756, 1793, 1802, 1808 [69] Engle, P. 1397, 1420 [26] Erdös, P. 359, 362 [73]; 444, 492 [47]; 1058, 1067 [41] Evans, L. 1367, 1420 [27] Evans, W.D. 969, 974 [8] Exel, R. 326, 362 [74]; 1495, 1512 [61] Fabes, E.B. 1411, 1420 [28] Fabian, M. 413, 418, 434 [34–37]; 792, 793, 798, 812, 814, 820–822, 828, 831 [41,55–59]; 832 [60–65]; 1537, 1544 [16]; 1745, 1747–1759, 1761–1769, 1771, 1772, 1774–1782, 1785–1789, 1792, 1795, 1797–1799, 1801–1805, 1806 [26]; 1808 [70–91]; 1809 [92]

Fack, T. 1464, 1469, 1480, 1481, 1512 [62–64] Fakhoury, H. 445, 492 [48] Falconer, K. 220, 231 [18] Fang, G. 425, 434 [38–41]; 435 [42] Farahat, J. 1079, 1096 [9] Faris, W.G. 1403, 1420 [29] Farmaki, V. 822, 830 [7]; 1019, 1035, 1038, 1041, 1042, 1045, 1046, 1067 [42–48]; 1754, 1766, 1780, 1781, 1806 [12]; 1809 [93] Feder, M. 304, 313 [24] Federer, H. 1363, 1401, 1420 [30]; 1522, 1523, 1544 [17] Fefferman, C. 220, 223, 231 [19,20]; 267 [72] Feige, U. 1610, 1633 [13] Feldman, J. 535, 558 [5]; 615, 666 [55] Fell, J.M. 610, 665 [25] Feller, W. 606, 617, 666 [56]; 804, 832 [66] Felouzis, V. 1064, 1066 [8]; 1254, 1266, 1272, 1280, 1281, 1295 [2] Fenchel, W. 727, 774 [58] Ferenczi, V. 1252–1254, 1265, 1266, 1295 [11–15]; 1729, 1739 [17] Ferguson, T.S. 906, 935 [23] Ferleger, S.V. 1492, 1512 [65,66] Fernandez, D.L. 250, 267 [73] Fernique, X. 347, 350, 362 [75,76] Fetter, H. 1758, 1809 [94] Fichtenholz, G.M. 1373, 1420 [31] Figá-Talamanca, A. 1466, 1512 [67] Figiel, T. 96, 119 [50,57]; 208, 231 [21]; 249, 267 [74,75]; 276, 283, 289, 309, 313 [25–28]; 443, 475, 491 [31]; 492 [49]; 511, 529 [51]; 583, 585–587, 591–594, 595 [15–17]; 596 [24,25]; 715, 716, 726, 735, 737, 747, 748, 751, 768, 770, 772, 773 [36]; 774 [59–61]; 792, 832 [67]; 853, 854, 858, 863, 868 [20]; 869 [21–23]; 916, 925, 935 [24]; 973, 973 [5]; 1060, 1062, 1067 [49]; 1133, 1136, 1137, 1171, 1173 [35,37]; 1222, 1233, 1234, 1244 [6,7]; 1254–1256, 1280, 1295 [10,16,17]; 1305, 1306, 1330 [13,14]; 1342, 1359 [13]; 1387, 1420 [32]; 1589, 1601 [26]; 1608, 1633 [14]; 1649, 1664, 1668 [20,21]; 1691, 1692, 1697, 1701 [15]; 1761, 1766, 1803, 1807 [46] Fillmore, P.A. 323, 331, 339, 361 [47,48]; 362 [69] Finet, C. 434 [26]; 805, 832 [68]; 1020, 1067 [50]; 1757, 1758, 1767, 1809 [95–97] Fleming, R.J. 901, 935 [25] Flinn, P. 279, 313 [6]; 314 [29]; 511, 528 [30] Floret, K. 466, 467, 484, 488, 490, 491 [32] Foia¸s, C. 330, 332, 360 [9,10] Föllmer, H. 387, 388, 390 [26,27] Fonf, V.P. 34, 35, 83 [26]; 611, 637, 641, 644, 646–650, 653–659, 661–665, 666 [40,57–62]; 667 [63–71]; 668 [96]; 793, 813, 831 [42,43];

Author Index 832 [69]; 1019, 1046, 1067 [51,52]; 1747, 1749, 1795, 1798, 1799, 1807 [51,52]; 1809 [98,99]; 1817, 1818 [1a,2a]; 1823, 1823 [2a] Force, G. 147, 151, 154, 157 [38] Forrester, P.J. 345, 362 [77] Fosgerau, M. 1752, 1804, 1809 [100] Fourie, J. 484, 491 [34] Fradelizi, M. 921, 935 [6] Frampton, J. 413, 433 [7]; 799, 830 [18] Frankiewicz, R. 1805, 1809 [101] Frankl, P. 1613, 1625, 1633 [15] Franklin, Ph. 575, 596 [26] Frankowska, H. 433 [1] Frazier, M. 1141, 1173 [40] Fréchet, M. 906, 935 [26] Fremlin, D.H. 89, 92, 94, 119 [52,58,59]; 1019, 1020, 1066 [25]; 1117, 1128 [25]; 1748, 1758, 1767, 1802, 1806 [30] Friedman, Y. 1509, 1510 [9]; 1512 [68]; 1600, 1601 [27] Friis, P. 325, 362 [78]; 363 [79] Frobenius, G. 98, 119 [60] Frolík, Z. 1802, 1809 [102] Frontisi, J. 828, 832 [70]; 1798, 1809 [103,104] Fry, R. 813, 828, 832 [71] Fuhr, R. 626, 627, 667 [72] Füredi, Z. 175, 193 [7]; 1613, 1625, 1633 [15] Gagliardo, E. 1381, 1398, 1420 [33,34] Gamboa de Buen, B. 1758, 1809 [94] Gamelin, T.W. 60, 84 [27]; 675–677, 682, 704 [3]; 705 [12,18–20]; 879, 895, 897 [11]; 1675, 1701 [16] Gamlen, J.L.B. 129, 157 [39] Gantmacher, V.R. 443, 492 [50] García, C.L. 285, 313 [10] García del Amo, A. 527, 529 [52] Garcia-Cuerva, J. 505, 529 [53] Gardner, R.J. 177, 193 [16–18]; 918, 919, 935 [27–29] Garling, D.J.H. 137, 139, 157 [40]; 246, 263, 264, 266 [55]; 267 [76–78]; 463, 492 [51]; 516, 519, 527, 529 [54,55]; 681, 705 [21]; 793, 831 [33]; 905, 936 [30]; 952, 974 [9]; 1103, 1128 [17]; 1142, 1173 [41]; 1483, 1484, 1494, 1495, 1512 [45,69,70] Garnett, J.B. 674, 675, 705 [22]; 1638, 1668 [22]; 1693, 1701 [17] Garsia, A.M. 209, 231 [22]; 1490, 1512 [71] Gasparis, I. 1051, 1056, 1059, 1064, 1065, 1066 [9]; 1067 [53,54]; 1599, 1601 [28] Gaudet, R.J. 129, 157 [39] Gaudin, M. 345, 364 [122] Geiss, S. 267 [79] Gelfand, I.M. 902, 908, 919, 920, 936 [31,32]

1831

Geman, S. 344, 353, 363 [80] Georgiev, P. 828, 832 [72] Gevorkian, G.G. 565, 573, 596 [27,28] Geyler, V.A. 87, 89, 90, 118 [16]; 119 [40] Ghoussoub, N. 34, 83 [25]; 97, 119 [61–63]; 263, 267 [80–82]; 402–405, 425, 434 [41]; 435 [42–50]; 445, 483, 492 [53]; 635, 637–639, 667 [73–75]; 795, 798, 799, 821, 831 [34,44]; 832 [73]; 1019, 1067 [55]; 1589, 1598, 1601 [26,29]; 1752, 1753, 1775, 1783, 1807 [53] Giannopoulos, A.A. 47, 84 [28]; 164, 169, 177, 180, 193 [19–22]; 342, 358, 363 [81]; 719, 722, 725, 726, 729, 737, 754, 755, 766, 774 [62–70]; 844, 859, 867, 869 [24]; 918, 936 [33]; 1203, 1221, 1224, 1225, 1244 [8]; 1626, 1627, 1633 [16] Giesy, D.P. 1303, 1330 [15,16] Giga, M. 250, 267 [84] Giga, Y. 250, 267 [83,84] Giles, J.R. 1745, 1809 [105] Gillespie, T.A. 237, 249, 250, 265 [5,12,13]; 1133, 1162, 1173 [42]; 1347, 1359 [14]; 1494, 1511 [16] Gillman, L. 1712, 1739 [18] Giné, E. 1181, 1186, 1190, 1191, 1194, 1195, 1198 [1–3,7,8]; 1199 [9,10] Ginibre, J. 219, 231 [23] Girardi, M. 443, 492 [52]; 518, 529 [44] Girko, V.L. 344, 363 [82] Glasner, E. 619, 620, 667 [76] Gleit, A. 658, 667 [77] Glimm, J. 542, 558 [20] Gluskin, E.D. 175, 193 [23]; 454, 492 [54]; 743, 765, 766, 774 [71]; 775 [72–74]; 854, 869 [25]; 946, 968, 974 [10]; 1208, 1212, 1222, 1224, 1244 [9,10]; 1245 [11]; 1254, 1296 [18,19] Godefroy, G. 33, 34, 57, 83 [6]; 84 [29]; 97, 119 [64]; 154, 156, 157 [11,41]; 235, 268 [85]; 285, 295, 310, 314 [30,31]; 406, 408, 409, 415, 434 [24,25]; 435 [51]; 476, 491 [33]; 644, 645, 666 [41]; 667 [78]; 792, 793, 798, 799, 805, 812–814, 821, 822, 828, 829, 830 [22]; 831 [38,45,46]; 832 [60,73–84]; 1016, 1018–1020, 1046, 1067 [50,56–61]; 1081, 1096 [1]; 1112, 1128 [26]; 1560, 1601 [10]; 1745, 1747–1750, 1752–1755, 1757–1767, 1772–1778, 1780–1792, 1795–1802, 1804, 1805, 1806 [28]; 1807 [54–58]; 1808 [74–77]; 1809 [96,106–122]; 1818, 1818 [3a] Godement, R. 616, 667 [79] Godun, B.V. 1767, 1804, 1809 [123]; 1810 [124] Goethals, J.P. 916, 936 [34] Gohberg, I.C. 439, 492 [55]; 1464, 1465, 1496, 1498, 1512 [72]; 1513 [73] Gol’dstein, V.M. 1372, 1420 [35] Gonzalez, M. 812, 832 [85]; 1107, 1127 [12]; 1155, 1172 [18]; 1745, 1789, 1799, 1803, 1807 [34,35]

1832

Author Index

Gonzalo, R. 413, 434 [30]; 812, 813, 831 [47]; 832 [85] Goodey, P. 911, 936 [35] Goodman, V. 147, 157 [13]; 230 [4]; 342, 361 [23]; 463, 491 [10]; 748, 773 [23]; 866, 867, 868 [2]; 1185, 1190, 1199 [11] Goodner, D.A. 1712, 1739 [19] Gordon, Y. 278, 314 [32]; 342, 352, 354, 361 [25]; 363 [83–85]; 454, 463, 466, 492 [51,54,56]; 735, 740, 749, 750, 759, 766, 769, 773 [22]; 775 [75–79]; 858, 869 [26]; 896, 897 [12]; 925, 936 [36]; 946, 952, 968, 974 [9,10]; 1127, 1128 [27]; 1222, 1245 [12]; 1387–1389, 1420 [36]; 1478, 1507, 1513 [74] Gorelik, E. 829, 832 [86] Gorin, E.A. 905, 936 [37] Götze, F. 350, 358, 361 [38] Goullet de Rugy, M. 622, 667 [80] Gowers, W.T. 18, 84 [30]; 136, 157 [42]; 222, 231 [24]; 277, 304, 310, 314 [33]; 770, 775 [80,81]; 812, 813, 832 [87,88]; 1016, 1036, 1051, 1064, 1067 [62,63]; 1087, 1089, 1094–1096, 1096 [10,11]; 1097 [12]; 1101, 1106, 1110, 1128 [28,29]; 1158, 1173 [43]; 1250–1255, 1260, 1261, 1263, 1265–1268, 1271, 1273, 1274, 1276, 1283, 1288, 1296 [20–27]; 1310, 1330 [17]; 1344, 1349, 1359 [15,16]; 1632, 1633 [17,18]; 1656, 1657, 1668 [23] Graham, C.C. 874, 897 [13] Graham, R.L. 1074, 1097 [13] Granas, A. 603, 666 [44] Grandits, P. 389, 390 [28] Granville, A. 207, 230 [5] Greenleaf, F.P. 876, 897 [14] Gripenberg, G. 573, 596 [29] Gripey, R. 1367, 1420 [27] Grobler, J.J. 110, 111, 119 [65,66] Gromov, M. 347, 363 [86]; 714, 717, 744, 745, 775 [82–86]; 1606, 1607, 1633 [19–21] Gronbaek, N. 286, 314 [34] Gross, L. 1301, 1330 [18,19]; 1481, 1482, 1513 [75,76] Grothendieck, A. 273, 281, 282, 288, 289, 309, 314 [35]; 440, 444, 449–452, 457, 459, 466, 467, 483, 486, 489, 490, 492 [57–62]; 911, 936 [38]; 964, 974 [11]; 1021, 1044, 1067 [64]; 1301, 1302, 1330 [20]; 1378, 1392, 1420 [37]; 1421 [38]; 1432, 1438, 1439, 1443, 1446, 1457 [44]; 1464, 1513 [77]; 1581, 1601 [30]; 1723, 1724, 1740 [20,21] Gruenhage, G. 1810 [125] Grünbaum, B. 930, 936 [39] Grz¸as´lewicz, R. 515, 529 [56] Grzech, M. 1805, 1809 [101]

Guédon, O. 718, 749, 775 [78,87,88] Guerra, F. 1819, 1820 [3] Guerre-Delabrière, S. 134–136, 139, 158 [43,44]; 519, 529 [57]; 813, 833 [89]; 1745, 1810 [126] Guionnet, A. 345, 346, 361 [22]; 363 [87] Gul’ko, S.P. 1755, 1762, 1810 [127] Gundy, R.F. 128, 157 [25]; 242, 257, 258, 266 [42–44]; 268 [86]; 523, 528 [18]; 880, 897 [15] Gurarii, P.I. 581, 596 [30] Gurarii, V.I. 581, 596 [30]; 626, 667 [81]; 1771, 1810 [128] Gutiérrez, J.A. 237, 268 [87]; 676, 705 [23] Gutman, A.E. 87, 106, 108, 119 [41] Haagerup, U. 358, 363 [88,89]; 1305, 1330 [21]; 1428, 1431, 1440, 1449, 1452, 1456 [25]; 1457 [45–47]; 1470, 1474, 1476, 1480, 1483, 1484, 1487, 1495, 1506–1509, 1513 [78–82]; 1819, 1820 [4a] Habala, P. 6, 7, 20, 34, 36, 41, 42, 63, 83 [11,12]; 792, 832 [62]; 833 [90]; 1253, 1269, 1295 [15]; 1296 [28]; 1537, 1544 [18]; 1745, 1747–1758, 1761–1766, 1768, 1769, 1771, 1772, 1774, 1775, 1777–1780, 1782, 1785, 1787–1789, 1792, 1797, 1802, 1804, 1805, 1808 [80]; 1810 [129] Hadwiger, H. 733, 775 [89] Hadwin, D. 329, 363 [90] Hagler, J. 1569, 1595, 1601 [31,32]; 1752, 1810 [130] Hahn, H. 1705, 1740 [22] Hájek, P. 6, 7, 20, 34, 36, 41, 42, 63, 83 [11,12]; 434 [36]; 659, 666 [40]; 667 [85]; 792, 799, 813, 821, 828, 831 [31,42,43]; 832 [61,62]; 833 [90–92]; 1537, 1544 [18]; 1745, 1747–1758, 1761–1766, 1768, 1769, 1771, 1772, 1774–1782, 1785, 1787–1789, 1792, 1793, 1795–1799, 1802, 1804, 1805, 1807 [37,51,52]; 1808 [77–80]; 1810 [129,131–140] Hajłasz, P. 1399, 1403, 1421 [39] Halbeisen, L. 1358, 1359 [17] Halberstam, H. 200, 231 [25] Halmos, P.R. 106, 119 [67]; 535, 558 [21] Halperin, I. 505, 526, 529 [58] Halpern, H. 336, 361 [30]; 859, 868 [3] Hamana, M. 1453, 1457 [48,49] Hammand, P. 1046, 1067 [65] Hansell, R.W. 1745, 1791, 1792, 1810 [141] Harcharras, A. 1501–1505, 1513 [83,84] Hardin, C.D., Jr. 901, 902, 905, 936 [40–42] Hardy, G.H. 529 [59]; 956, 974 [12]; 1173 [44] Hare, D. 799, 831 [45] Harmand, P. 310, 314 [36]; 829, 833 [93]; 1745, 1810 [142] Harper, L.H. 745, 775 [90]; 1613, 1625, 1633 [22]; 1657, 1668 [24]

Author Index Harrison, J.M. 374, 376, 377, 391 [29,30] Hart, S. 1657, 1668 [25] Hasumi, M. 1714, 1740 [23] Hayakawa, K. 1142, 1173 [45] Haydon, R.G. 140, 158 [45]; 408, 409, 435 [52–54]; 526, 528 [32]; 611, 615, 644, 667 [82–84]; 799, 813, 833 [94,95]; 901, 935 [17]; 1019, 1030, 1034, 1036, 1045, 1046, 1068 [66]; 1339, 1359 [18]; 1715, 1740 [24]; 1776, 1777, 1784–1787, 1789, 1791, 1794–1796, 1798, 1799, 1810 [139,143–147] Heath-Brown, D.R. 229, 231 [26] Heinrich, S. 305–308, 314 [37]; 443, 444, 455, 492 [63–65]; 793, 829, 833 [96]; 1543, 1544 [19] Helson, H. 1495–1497, 1513 [85,86] Henkin, G.M. 1392, 1396, 1421 [40] Hensgen, W. 250, 263, 268 [88,89] Hensley, D. 175, 176, 193 [24,25]; 724, 775 [91] Henson, C.W. 1107, 1128 [30] Hernández, F.L. 518, 527, 529 [52,60,61]; 530 [62–66]; 1661, 1668 [26] Herrero, D.A. 329–333, 359, 362 [62]; 363 [91–95] Hervé, M. 608, 667 [86] Herz, C. 906, 936 [43] Hestenes, M.R. 1373, 1421 [41] Hewitt, E. 616, 667 [87] Hilbert, D. 439, 441, 492 [66] Hille, E. 442, 492 [67]; 968, 974 [13] Hilsum, M. 1470, 1513 [87] Hindman, N. 1082, 1097 [14] Hirsberg, B. 626, 667 [88] Hitczenko, P. 257, 268 [90]; 1154, 1173 [46]; 1610, 1633 [23] Hoeffding, W. 519, 530 [67] Hoffman, A.J. 328, 363 [96] Hoffman, K. 675, 678, 705 [24]; 875, 878, 887, 897 [16]; 1677, 1701 [18] Hoffman-Jørgensen, J. 473, 492 [68]; 769, 775 [92]; 1180, 1181, 1183–1185, 1188, 1191, 1199 [12–15]; 1302, 1303, 1330 [22,23] Holbrook, J. 327, 361 [36] Holický, P. 798, 833 [97]; 1752, 1758, 1792, 1810 [148,149] Holmstedt, T. 1154, 1173 [47] Holub, J.R. 960, 974 [14] Hörmander, L. 223, 231 [27]; 731, 775 [93]; 1369, 1370, 1373, 1407, 1421 [42] Horn, A. 447, 492 [69] Hsu, Y.-P. 526, 529 [45] Huang, C.-F. 374, 390 [19] Hudzik, H. 515, 527, 529 [37,56]; 530 [68] Huff, R.E. 633, 667 [89,90] Huijsmans, C.B. 96, 119 [68] Hunt, R.A. 505, 530 [69]; 1173 [48] Hurewicz, W. 1019, 1068 [67]

1833

Hustad, O. 626, 627, 667 [91,92] Hutton, C.V. 441, 493 [70] Hyers, D.H. 1108, 1128 [31] Il’in, V.P. 1363, 1398, 1399, 1408, 1409, 1420 [7]; 1421 [43] Ioffe, A.D. 418, 435 [55]; 798, 833 [98] Ionescu Tulcea, A. 260, 268 [91] Ionescu Tulcea, C. 260, 268 [91] Isac, G. 1108, 1128 [31] Isbell, J.R. 1714, 1740 [25] Ishii, H. 420, 421, 431, 434 [18] Ivanov, M. 419, 434 [29] Iwaniec, T. 259, 268 [92,93] Izumi, H. 1470, 1513 [88,89] Jacka, S.D. 385, 391 [31] Jackson, S. 1530, 1544 [20] Jacobson, C. 214, 231 [28] Jacod, J. 376, 391 [32] Jahandideh, M.T. 102, 119 [69] Jain, N.C. 1188, 1190, 1199 [16,17] Jajte, R. 1493, 1513 [90,91] James, R.C. 235, 268 [94]; 275, 314 [38]; 477, 479, 493 [71,72]; 581, 596 [31]; 643, 667 [93]; 792, 804, 833 [99,100]; 1052, 1068 [68]; 1250, 1252, 1256, 1257, 1263, 1296 [29–31]; 1303, 1304, 1306, 1307, 1330 [24–27]; 1331 [28]; 1335, 1336, 1359 [19]; 1758, 1810 [150] Jameson, G.J.O. 87, 119 [70]; 458, 459, 493 [73] Jamison, J.E. 901, 935 [25] Janovsky, L.P. 92, 118 [17] Janssen, G. 1307, 1331 [29] Jaramillo, J.A. 413, 434 [30]; 676, 705 [23]; 812, 813, 831 [47]; 832 [85] Jarchow, H. 47, 55, 60, 65, 66, 72, 83 [9]; 286, 309, 313 [11]; 455, 458, 459, 464, 466, 471, 473, 474, 480, 491 [35]; 493 [74–76]; 518, 529 [40]; 675, 681, 690, 705 [15]; 867, 868 [18]; 879, 881, 882, 887, 897 [10]; 901, 905, 911–915, 929, 935 [18,19]; 943, 950, 951, 974 [7]; 1222, 1244 [5]; 1378, 1387–1389, 1420 [22]; 1452, 1456 [24]; 1490, 1500, 1513 [92]; 1563, 1601 [22]; 1674, 1684, 1701 [13,19]; 1710, 1711, 1724, 1736, 1739 [16] Jarosz, K. 702, 703, 705 [25,26] Jawerth, B. 1141, 1145, 1147, 1149, 1161, 1173 [26,40,49] Jayne, J.E. 631, 668 [94]; 1745, 1761, 1788, 1791, 1794, 1803, 1804, 1810 [147,151–155]; 1811 [156,157]; 1814 [269] Jensen, R. 420, 435 [56] Jerison, M. 1712, 1739 [18] Jevtić, M. 1682, 1701 [20]

1834

Author Index

Ji, G. 1496, 1513 [93] Jiménez Sevilla, M. 664, 665 [16]; 799, 833 [101]; 1758, 1784, 1804, 1811 [158] Johansson, K. 346, 363 [97] John, F. 463, 493 [77]; 718, 775 [94] John, K. 489, 493 [78]; 805, 833 [102]; 1763, 1772, 1787–1789, 1798, 1799, 1811 [159–164] Johnson, W.B. 88–91, 96, 97, 105, 108, 119 [50,62,63,71,72]; 125, 126, 129, 131, 134–136, 140–143, 145–149, 154, 156, 156 [8]; 157 [13]; 158 [46–57]; 164, 190, 193 [26]; 230 [4]; 257, 262, 268 [95,96]; 273–280, 285, 288–296, 298–300, 305, 306, 309, 310, 312, 313 [10,12,19,27]; 314 [28,39–51]; 336, 338, 363 [98]; 443, 445, 459, 483, 491 [31]; 492 [52,53]; 493 [79,80]; 511, 521–523, 529 [51]; 530 [70–72]; 563, 573, 581, 588, 596 [32,33]; 630, 634, 636, 640, 641, 655, 668 [95]; 744, 748, 755, 769, 770, 773 [23]; 775 [95–97]; 784, 792, 793, 805, 821, 829, 831 [35]; 832 [67]; 833 [103–108]; 839–845, 850, 851, 853–855, 858–860, 863, 864, 866, 867, 868 [2,20]; 869 [21,22,27–34]; 906, 925, 929, 936 [44–46]; 950, 951, 953, 957, 961, 964, 965, 974 [15,16]; 1020, 1060, 1062, 1067 [37,49]; 1068 [69]; 1103, 1116, 1119, 1128 [32]; 1133, 1136, 1137, 1157, 1173 [35,50]; 1181, 1192, 1199 [18,19]; 1207, 1208, 1212, 1222, 1224, 1233, 1234, 1242, 1244 [6]; 1245 [13–17]; 1254–1256, 1280, 1295 [10]; 1296 [17]; 1304, 1306, 1331 [30,31]; 1342, 1359 [13]; 1374, 1375, 1377, 1378, 1380, 1387–1389, 1393, 1396, 1404, 1407, 1420 [32]; 1421 [44]; 1454, 1457 [50]; 1478, 1513 [94]; 1535, 1537, 1539, 1543, 1544 [4,21,22]; 1552, 1572, 1579, 1589, 1598–1600, 1601 [26,29,33–36]; 1607, 1610, 1614, 1626, 1628–1630, 1633 [24–29]; 1649, 1652, 1654–1656, 1660–1664, 1668 [20,27]; 1673, 1682, 1701 [21]; 1710, 1716, 1717, 1726, 1731–1733, 1735–1737, 1740 [26–33]; 1745, 1747, 1750, 1752, 1756–1758, 1761, 1765, 1766, 1769, 1771–1773, 1796, 1797, 1800, 1803–1805, 1807 [45,46]; 1811 [165–170]; 1817, 1818 [1a,2a] Jones, L. 148, 158 [47]; 842, 843, 869 [27] Jones, P. 305, 314 [52]; 1373, 1397, 1421 [45,46]; 1699, 1701 [22] Jordan, P. 906, 936 [47] Josefson, B. 1749, 1773, 1811 [171,172] Journé, J.L. 250, 266 [54]; 591, 596 [20] Juhász, I. 1811 [173] Junge, M. 590, 596 [21]; 724, 775 [98,99]; 1447–1452, 1456 [26]; 1457 [51–57]; 1487, 1492, 1493, 1509, 1510, 1513 [95–102] Junilla, H. 1808 [62]

Kadets, M.I. 142, 147, 158 [58]; 463, 493 [81]; 510, 530 [73]; 580, 596 [35]; 611, 647, 667 [68]; 668 [96]; 792, 793, 798, 833 [109–112]; 883, 897 [18]; 929, 936 [48]; 1304, 1331 [32]; 1509, 1514 [103]; 1752, 1771, 1810 [128]; 1811 [174] Kadison, R.V. 333, 334, 363 [99]; 859, 869 [35]; 1463, 1514 [104] Kaftal, V. 336, 361 [30]; 859, 868 [3] Kahan, W.M. 339, 362 [68] Kahane, J.P. 184, 193 [15]; 363 [100]; 644, 645, 668 [97]; 873, 897 [17]; 965, 974 [19]; 1181, 1199 [20]; 1307, 1331 [33] Kakosyan, A.V. 905, 939 [134] Kakutani, S. 443, 465, 493 [82–84] Kalenda, O. 822, 833 [113]; 1537, 1544 [23]; 1545 [24]; 1745, 1756, 1766, 1801, 1811 [175–178] Kalton, N.J. 74, 81, 84 [31]; 95, 120 [73]; 143, 156, 157 [41]; 158 [59,60]; 278, 279, 291–295, 297, 301, 302, 310, 311, 313 [13–16]; 314 [30,53–55]; 403, 435 [57]; 505, 511, 518, 527, 530 [74–77]; 569, 596 [34]; 697, 705 [27]; 769, 775 [77]; 792, 793, 821, 829, 830 [26]; 832 [77–81]; 833 [114]; 843, 854, 868 [7]; 869 [36]; 886, 897 [19]; 901, 936 [49]; 1101, 1103–1127, 1127 [2,11]; 1128 [26,27,30,33–50]; 1129 [51–60]; 1133, 1143, 1144, 1151–1159, 1161, 1162, 1164–1168, 1170, 1171, 1172 [17]; 1173 [27,28,38,51–53]; 1174 [54–63]; 1253, 1255, 1296 [32,33]; 1357, 1359 [20,21]; 1382, 1421 [47]; 1647, 1649–1651, 1657, 1658, 1661–1667, 1668 [9–11,26,28–31]; 1696, 1701 [23]; 1711, 1727, 1734, 1735, 1740 [34,35]; 1796, 1805, 1809 [113–116]; 1818, 1818 [3a] Kamińska, A. 515, 527, 529 [37]; 530 [68,78–81] Kamont, A. 565, 596 [27] Kanellopoulos, V. 1039–1041, 1053, 1066 [10] Kannan, R. 1625, 1633 [30] Kanter, M. 177, 193 [27]; 524, 530 [82] Kantorovich, L.V. 87, 91, 92, 106, 107, 120 [74–76] Karadzov, G.E. 1140, 1174 [64] Karatzas, I. 374, 391 [33] Kashin, B.S. 336, 338, 342, 358, 363 [101–103]; 575, 596 [36]; 749, 762, 775 [100]; 1222, 1245 [18]; 1305, 1331 [34] Katavolos, A. 1432, 1445, 1457 [58] Kato, T. 445, 493 [85] Katz, N. 222, 231 [29,30] Katznelson, Y. 184, 193 [15]; 569, 596 [37]; 965, 974 [19]; 1410, 1421 [48] Kaufman, R. 793, 828, 830 [22]; 833 [115]; 1020, 1068 [70,71]; 1601 [37]; 1716, 1740 [36]; 1787, 1806 [28]

Author Index Kazarian, K.S. 505, 529 [53] Kazhdan, J.L. 621, 622, 666 [54] Kechris, A.S. 1013, 1018–1020, 1032, 1045, 1068 [72–74]; 1811 [179] Keleti, T. 1530, 1545 [25] Keller, O.-H. 603, 668 [98] Kelley, J.L. 1712, 1740 [37] Kelly, B.P. 249, 265 [6] Kemperman, J.H.B. 914, 936 [50] Kendall, D.G. 613, 614, 668 [99] Kenderov, P. 435 [58]; 1537, 1545 [26]; 1752, 1804, 1807 [40]; 1811 [180,181] Kesten, H. 358, 363 [104] Ketonen, T. 859, 869 [37] Kheifets, A. 1700, 1701 [24] Khovanskii, A.G. 732, 776 [102] Kirchberg, E. 1431, 1440, 1448, 1450, 1452, 1454, 1457 [59–62]; 1551, 1601 [38] Kirchheim, B. 1523, 1543, 1544, 1544 [1,2]; 1545 [27] Kiriakouli, P. 1019, 1046, 1068 [75–77] Kishimoto, A. 1440, 1456 [27] Kislyakov, S.V. 60, 64, 84 [27,32]; 468, 493 [86]; 675, 679, 690, 691, 695, 697, 699–701, 705 [28–32]; 879, 882, 895, 897 [11,20,21]; 1133, 1174 [65–68]; 1392, 1393, 1405, 1406, 1412, 1415, 1416, 1418, 1421 [49–52]; 1505, 1514 [105]; 1675, 1701 [16] Kitover, A.K. 87, 100, 118 [15]; 120 [77] Klain, D. 733, 775 [101] Klebanov, L.B. 905, 939 [134] Klee, V.L. 602, 603, 629, 650, 652, 661–663, 666 [32]; 668 [100–107]; 793, 798, 833 [116]; 1111, 1129 [61] Klemes, I. 875, 897 [22]; 1698, 1701 [25] Knaust, H. 821, 833 [117]; 1354, 1357, 1359 [22] Knöthe, H. 713, 776 [103] Koëthe, G. 1454, 1457 [63] Koldobsky, A. 71, 84 [33]; 177, 193 [18,28]; 524, 529 [46]; 853, 869 [38]; 905, 906, 909–911, 918, 919, 921–923, 935 [20,29]; 936 [37,51–59]; 937 [60–62] Kolmogoroff, A.N. 370, 391 [34] Kolyada, V.I. 1399, 1403, 1404, 1421 [53–55] Komisarski, A. 1020, 1068 [78] Komorowski, R. 279, 314 [56]; 315 [57]; 1252, 1296 [34,35]; 1805, 1809 [101] König, H. 71, 84 [33]; 268 [97,98]; 452, 463, 469, 470, 493 [88–90]; 724, 770, 776 [104,105]; 853, 869 [38]; 915–917, 930–934, 937 [63–69]; 944, 953, 954, 957, 959, 961, 962, 964–966, 968, 970, 971, 973, 974 [15,17,18]; 1234, 1245 [19]; 1392, 1403, 1421 [56]

1835

Koosis, P. 327, 361 [34]; 1677, 1678, 1688, 1699, 1701 [26] Korotkov, V.B. 87, 106, 108, 119 [41]; 120 [78] Kosaki, H. 1464, 1466, 1469–1471, 1474, 1476, 1477, 1480, 1512 [64]; 1514 [106–108] Koskela, P. 1399, 1403, 1421 [39] Köthe, G. 457, 493 [91]; 570, 596 [38]; 1667, 1668 [32]; 1728, 1740 [38] Krasnoselsky, M.A. 87, 88, 118 [32]; 120 [79,80] Kraus, J. 1440, 1457 [46] Krawczyk, A. 1811 [182] Krawczyk, L. 389, 390 [6,7,28] Krée, P. 1491, 1511 [34] Krein, M.G. 88, 99, 120 [81,82]; 439, 492 [55]; 1464, 1465, 1496, 1498, 1512 [72]; 1513 [73] Krein, S. 74, 78, 81, 83 [13]; 87, 120 [83]; 1133, 1174 [69] Krengel, U. 87, 120 [84] Kreps, D.M. 374, 376, 391 [29,35] Kriecherbauer, T. 349, 362 [70] Krieger, H.J. 110, 120 [85] Krishnaiah, P.R. 353, 366 [187] Krivelevich, M. 346, 363 [105]; 1819, 1819 [1a] Krivine, J.L. 136, 138, 139, 158 [61,62]; 284, 315 [58]; 455, 491 [30]; 515, 524, 528 [16]; 530 [83]; 812, 833 [118]; 906, 935 [13]; 937 [70]; 1303–1305, 1307, 1314, 1330 [7]; 1331 [35,36]; 1339, 1359 [23]; 1479, 1514 [109] Kruglyak, N.Ya. 74, 83 [5]; 1133, 1142, 1143, 1145, 1147, 1149, 1172 [11,12]; 1173 [29]; 1174 [70] Krygin, A.W. 1478, 1484, 1511 [39,40] Kuelbs, J. 1179, 1182, 1185, 1190, 1199 [11,21–23] Kunen, K. 268 [99] Kunze, R. 1461, 1466, 1514 [110] Kuratowski, K. 1019, 1068 [79] Kurzweil, J. 798, 813, 833 [119,120]; 1811 [183] Kusraev, A.G. 87, 106, 108, 119 [41] Kutateladze, S.S. 87, 106, 108, 119 [41] Kutzarova, D. 304, 313 [2]; 821, 834 [122]; 1062, 1063, 1065, 1066 [6]; 1068 [80]; 1781, 1782, 1811 [184] Kwapień, S. 140, 158 [63,64]; 254, 257, 268 [100]; 275, 315 [59]; 338, 347, 363 [106,107]; 465, 466, 473, 481, 493 [92–94]; 527, 530 [84,85]; 770, 776 [106]; 820, 834 [121]; 855, 856, 858, 869 [23,39]; 875, 879, 880, 886, 887, 889, 897 [23]; 934, 937 [71]; 1115, 1129 [62]; 1181, 1191, 1199 [24,25]; 1222, 1244 [7]; 1304, 1331 [37]; 1392, 1393, 1421 [57,58]; 1446, 1457 [64]; 1478, 1495, 1498, 1514 [111]; 1617, 1633 [31]; 1697, 1701 [27] Kye, S.-H. 1447, 1455, 1457 [65] Kyriazis, G. 579, 596 [39]

1836

Author Index

Laba, I. 222, 231 [29] Lacey, H.E. 465, 493 [95]; 862, 869 [40]; 1716, 1740 [39]; 1745, 1772, 1811 [185] Lamb, C.W. 260, 268 [101] Lamberton, D. 250, 266 [53]; 374, 391 [36] Lammers, M.C. 1650, 1668 [12] Lamperti, J. 905, 937 [72] Lance, E.C. 339, 364 [108] Lancien, G. 792, 793, 804, 805, 821, 829, 832 [78,79]; 834 [123,124]; 1018, 1020, 1068 [81,82]; 1791, 1796, 1805, 1809 [115,116]; 1811 [186,187] Landes, T. 515, 530 [86] Lapeyre, B. 374, 391 [36] Lapresté, J.-T. 1340, 1352, 1359 [6] Larman, D.G. 177, 193 [29]; 735, 776 [107]; 918, 937 [73] Larman, R.R. 1812 [188] Larson, D.R. 341, 364 [109] Latała, R. 460, 493 [96]; 718, 776 [108] Latter, R.H. 1693, 1701 [17] Lazar, A.J. 614, 622, 624–626, 646, 658, 667 [88]; 668 [108–113] Le Hoang Tri 1112, 1127 [5] Le Merdy, C. 1436, 1444, 1452, 1455 [11]; 1457 [53,67,68] Leach, E.B. 798, 834 [125]; 1812 [189] Lebourg, G. 798, 831 [51] Lebowitz, J.L. 354, 360 [1] Ledoux, M. 52, 84 [34]; 338, 350, 361 [21]; 364 [110–112]; 472, 493 [97]; 523, 530 [87]; 740, 756, 776 [109]; 848, 849, 869 [41]; 1179, 1180, 1183, 1185, 1186, 1188, 1190–1196, 1199 [26–32]; 1606, 1616, 1620, 1623–1625, 1632 [8]; 1633 [32–34]; 1634 [35]; 1819, 1820 [5,6] Leduc, M. 407, 409, 435 [59]; 812, 834 [126] Ledyaev, Y. 412, 423, 433 [13]; 434 [14–16] Lee, J.M. 238, 268 [102,103] Lee, P.Y. 1019, 1068 [83] Leeb, K. 1074, 1097 [13] Lehto, O. 259, 268 [104] Leindler, L. 1607, 1634 [36] Leinert, M. 1465, 1470, 1514 [113,114] Lemberg, H. 1314, 1331 [38]; 1339, 1359 [24] Lennard, C.J. 526, 528 [27,28]; 529 [47]; 1477, 1512 [50]; 1700, 1701 [5] LePage, R. 1192, 1199 [33]; 1629, 1634 [37] Leranoz, C. 569, 596 [34]; 1119, 1127 [1,2]; 1129 [53,63]; 1666, 1667, 1668 [30,33] Leung, D.H. 504, 505, 530 [88–92]; 658, 668 [114]; 1019, 1068 [84] Levental, S. 384, 391 [37] Levy, M. 511, 530 [93]; 813, 833 [89]; 1137, 1174 [71]

Lévy, P. 739, 744, 776 [110]; 906, 907, 937 [74]; 1605, 1634 [38] Lewandowski, M. 905, 937 [75] Lewis, D.R. 129, 158 [65]; 278, 279, 313 [6]; 314 [32]; 466, 492 [56]; 493 [98]; 726, 752, 776 [111]; 840, 858, 869 [26,42,43]; 930, 937 [66]; 1222, 1245 [12,20]; 1387–1389, 1420 [36]; 1478, 1507, 1513 [74]; 1600, 1601 [39]; 1717, 1727, 1740 [40] Li, D. 156, 157 [41]; 829, 832 [80,82]; 1046, 1067 [59] Lidski˘ı, V.B. 451, 493 [99]; 968, 974 [20] Lieb, E.H. 164, 173, 193 [10,30]; 718, 773 [41]; 1461, 1481, 1482, 1511 [14,35,36]; 1514 [112] Lifshits, E.A. 87, 120 [80] Lima, Å. 310, 314 [36]; 626, 668 [115,116] Lin, B.L. 133, 157 [27]; 511, 527, 528 [6]; 529 [33,34] Lin, H. 325, 364 [113] Lin, P.K. 471, 493 [100]; 526, 528 [32]; 901, 930, 935 [17]; 937 [66]; 1065, 1068 [80]; 1461, 1510 [10] Linde, W. 475, 484, 493 [101,102]; 905, 937 [78–82] Lindeman, A. 259, 265 [8] Lindenstrauss, J. 7, 10–14, 18, 21–27, 30, 33–36, 38, 42, 48, 50, 51, 78–80, 83 [3,14,15,26]; 84 [35]; 87–91, 105, 108, 119 [72]; 120 [86,87]; 125, 126, 129, 132–136, 140–143, 145, 146, 154, 156, 158 [48,49,66–70]; 159 [71,72]; 164, 175, 190, 193 [12,26]; 208, 231 [21]; 251, 257, 258, 262, 263, 267 [81]; 268 [95,105–107]; 273, 275, 278, 281, 282, 284, 289, 295, 299–301, 305–307, 309, 311, 312, 314 [46,47]; 315 [60–67]; 403, 435 [45]; 440, 443, 459, 465, 466, 475, 492 [49]; 493 [80,103]; 494 [104,105]; 505, 511, 514, 515, 518, 530 [94]; 531 [95–99]; 563, 573, 574, 577, 580, 581, 588, 596 [32,40–43]; 602, 618, 624–626, 630, 634–636, 638–641, 645, 646, 651, 653–655, 658, 661–663, 665 [8]; 666 [37,38]; 667 [69,73]; 668 [95,113,117–123]; 681, 705 [33]; 715, 716, 735–737, 742, 744, 747, 748, 766, 768–770, 773 [30–32,37–39]; 774 [61]; 775 [95]; 776 [112,114,115]; 784, 792, 793, 798, 805, 821, 822, 829, 830 [1,2,6,14]; 831 [34,54]; 832 [69]; 833 [104,106,107]; 834 [127–130]; 839–842, 844, 845, 850, 855, 858, 860, 865, 866, 868 [8]; 869 [28]; 906, 916, 925, 927–929, 935 [7,10–12,24]; 936 [44]; 937 [76,77]; 950, 951, 953, 963–965, 974 [16,21–23]; 1019, 1067 [52]; 1087, 1096 [4]; 1103, 1116, 1119, 1128 [32]; 1129 [64]; 1151, 1155, 1157, 1158, 1167, 1172 [6]; 1173 [39,50]; 1174 [72]; 1181, 1199 [18]; 1207, 1208, 1212, 1222, 1233, 1245 [14,21]; 1249, 1251–1253, 1256, 1260, 1263, 1264, 1285, 1295 [5]; 1296 [36–39]; 1301, 1304–1306, 1310, 1330 [2,13]; 1331 [31,39,40]; 1336, 1348,

Author Index 1359 [7,25]; 1374, 1375, 1377, 1378, 1380–1383, 1387–1389, 1392, 1393, 1396, 1404, 1407, 1419 [3]; 1421 [44,59,60]; 1454, 1457 [66]; 1478, 1480, 1481, 1507, 1511 [11]; 1513 [94]; 1514 [115]; 1521, 1533–1535, 1537–1541, 1543, 1544 [4,5,21,22]; 1545 [28–31]; 1552, 1572, 1579, 1590, 1594, 1597–1600, 1601 [14,33,40,41]; 1602 [42,43]; 1607, 1608, 1626, 1630, 1633 [9,14,24,25]; 1639–1643, 1646, 1647, 1649–1651, 1658, 1664, 1667 [6]; 1668 [34–39]; 1673, 1675, 1682, 1701 [21,29]; 1707–1711, 1714–1716, 1723–1729, 1732, 1733, 1736, 1737, 1740 [27,41–50]; 1745, 1747–1752, 1754, 1756–1760, 1762–1768, 1771–1773, 1775, 1790, 1794–1798, 1800, 1801, 1803–1805, 1806 [2,7,20]; 1807 [42–44]; 1809 [98,99]; 1811 [167–169]; 1812 [190–198]; 1823, 1823 [2a] Lindsey, J.H. 1657, 1668 [40] Linhart, J. 769, 776 [113] Lions, J.L. 1136, 1139, 1140, 1174 [73–76]; 1471, 1472, 1514 [116] Lions, P.L. 420, 421, 426, 431, 434 [18–20] Lisitsky, A. 909, 937 [83] Littlewood, J.E. 956, 974 [12]; 1173 [44] Litvak, A. 736, 737, 746, 767, 769, 773 [19]; 776 [116–118]; 1224, 1245 [11] Litvinov, G.L. 536, 558 [22] Llavona, J.L. 676, 705 [23] Löfström, J. 74, 76–78, 80, 83 [4]; 505, 528 [11]; 577, 595 [2]; 692, 704 [4]; 1133, 1136, 1172 [8]; 1280, 1295 [6]; 1404, 1420 [5]; 1437, 1455 [6]; 1466, 1511 [15] Lomonosov, V.I. 102, 105, 120 [88,89]; 536, 538, 543, 545, 558 [22–26]; 641, 668 [124]; 1397, 1420 [26] Lonke, Y. 910, 911, 937 [62,84] Loomis, L.H. 668 [125] Lopachev, V.A. 905, 910, 937 [85–87] López, G. 1020, 1046, 1066 [21]; 1068 [85,86] Lopez, J.M. 886, 897 [24] Lopez Abad, J. 1094, 1096 [2] Lorentz, G.G. 505, 524, 526, 531 [100–102]; 575, 596 [22] Lorentz, R.A. 578, 596 [44] Loring, T.A. 325, 326, 362 [74]; 364 [114,115] Lotz, H.P. 92, 119 [44]; 455, 456, 484, 494 [106]; 531 [103]; 1589, 1602 [44]; 1715, 1740 [51] Louveau, A. 1019, 1020, 1032, 1045, 1067 [60]; 1068 [73,74]; 1809 [117] Lovaglia, A.R. 792, 834 [131] Lovász, L. 1625, 1633 [30]; 1634 [39] Lowdenslager, D. 1495, 1497, 1513 [86] Lozanovsky, G.Ya. 87, 89, 106, 107, 119 [39]; 120 [90,91]; 1162, 1174 [77]; 1347, 1359 [26] Lubotzky, A. 338, 359, 364 [116]; 1451, 1457 [69]

1837

Lucchetti, R. 435 [58] Luecking, D. 471, 494 [107]; 1684, 1701 [30] Lukacs, E. 937 [88] Lusin, N.N. 1019, 1068 [87,88] Lusky, W. 273, 302, 303, 315 [68–70]; 581, 596 [45–47]; 626, 668 [126,127]; 905, 937 [89] Lust-Piquard, F. 192, 193 [31]; 1486, 1488, 1489, 1514 [117–121] Lutwak, E. 918, 937 [90] Lutzer, D. 1802, 1806 [1] Luxemburg, W.A.J. 87, 96, 119 [68]; 120 [92–94] Lyubich, Y. 914, 916, 917, 937 [91,92]; 938 [93] Maaden, A. 435 [60] MacCluer, B.D. 471, 491 [29] MacGregor, T. 1682, 1701 [31] Mackey, G. 1771, 1812 [199] Magajna, B. 1444, 1457 [70] Magidor, M. 444, 492 [47]; 1058, 1067 [41] Magill, M. 115, 120 [95] Maiorov, V.E. 973, 974 [24] Makai, E. 959, 974 [25] Makarov, B.M. 87, 106, 108, 119 [41] Maleev, R. 814, 834 [132] Maligranda, L. 515, 531 [104]; 1151, 1174 [78] Mandelbrot, B.B. 371, 391 [38] Mandrekar, V. 1191, 1198 [3] Mangheni, P.J. 1716, 1740 [52] Mani, P. 735, 768, 776 [107,119] Mankiewicz, P. 47, 84 [36]; 277, 304, 315 [71–73]; 358, 364 [117]; 633, 668 [128]; 766, 776 [120,121]; 793, 829, 833 [96]; 1216, 1219, 1220, 1222–1226, 1230–1233, 1235–1237, 1239, 1240, 1242–1244, 1245 [22–35]; 1254, 1255, 1296 [40]; 1532, 1543, 1544 [19]; 1545 [32] Manoussakis, A. 1062, 1063, 1065, 1066 [6,7]; 1068 [89,90] Marˇcenko, V.A. 343, 344, 353, 357, 364 [118] Marcinkiewicz, J. 250, 268 [108]; 522, 531 [105] Marciszewski, W. 1019, 1068 [91]; 1796, 1806 [19]; 1812 [200–202] Marcolino, J. 1476, 1479, 1514 [122] Marcus, M.B. 523, 531 [106]; 1190, 1192, 1199 [17,34] Margulis, G.A. 359, 364 [119,120] Marsalli, M. 1496, 1497, 1499, 1514 [123–126] Martin, D.A. 144, 159 [73]; 1358, 1359 [27] Martin, G. 259, 268 [93] Martín, M. 1046, 1068 [86] Marton, K. 1625, 1634 [40] Masani, P. 1495, 1497, 1516 [198] Mascioni, V. 277, 300, 306, 315 [74,75]; 1019, 1036, 1046, 1066 [33]

1838

Author Index

Mastyło, M. 527, 530 [68]; 1143, 1154, 1173 [29]; 1174 [79,80] Masuda, T. 1470, 1510 [3] Mateljević, M. 1677, 1701 [32] Matheron, E. 1797, 1807 [59] Matheson, A. 875, 896 [1] Matoušek, J. 798, 834 [133]; 925, 928, 938 [94] Matoušková, E. 798, 834 [133,134]; 1532, 1533, 1538, 1545 [28,33–35] Mattner, L. 905, 938 [95] Mauldin, R.D. 458, 494 [108]; 1530, 1544 [20] Maurey, B. 11, 51, 53, 84 [37,38]; 125, 133, 134, 136, 139, 140, 148, 158 [45,50,62]; 159 [74–77]; 237, 242, 251, 263, 267 [80–82]; 268 [109]; 277, 284, 304, 310, 314 [33]; 315 [76]; 402–405, 435 [45–49]; 459, 473, 474, 477, 494 [109–112]; 511, 515, 523, 530 [70,83]; 635, 637–639, 667 [73–75]; 745, 770, 771, 776 [122–125]; 799, 812, 821, 829, 832 [73]; 833 [118]; 834 [135,136]; 841, 842, 845, 855, 857, 858, 867, 869 [29,44–46]; 883, 884, 897 [25]; 906, 921, 935 [1,6]; 936 [46]; 953, 957, 961, 964, 965, 974 [15,26]; 1019, 1051, 1055, 1056, 1065, 1067 [55,63]; 1068 [92,93]; 1076, 1081, 1082, 1089, 1090, 1094, 1096, 1097 [12,15,16]; 1101, 1106, 1110, 1128 [29]; 1181, 1191, 1199 [35,37]; 1250–1255, 1260, 1261, 1263, 1265, 1267, 1268, 1271, 1273, 1274, 1276, 1283, 1288, 1296 [26,27,41]; 1301–1306, 1320, 1331 [41–45]; 1344, 1349–1358, 1359 [16,28–31]; 1389, 1421 [61]; 1479, 1514 [109]; 1607, 1612, 1632, 1634 [41,42]; 1652, 1654–1656, 1660–1664, 1668 [27,41]; 1710, 1735, 1740 [53] Maynard, H.B. 634, 668 [129] Mazurkiewicz, S. 1560, 1602 [45]; 1722, 1741 [54]; 1787, 1812 [203] Mazya, V.G. 1363, 1365, 1367, 1373, 1398, 1399, 1401, 1403, 1404, 1421 [62] McAsey, M. 1496, 1514 [127] McCann, R.J. 173, 193 [32]; 713, 776 [126] McCarthy, Ch.A. 568, 596 [48]; 1465, 1480, 1507, 1514 [128]; 1639, 1669 [42] McConnell, T.R. 250, 264, 268 [110–112] McGehee, O.C. 576, 577, 596 [49]; 874, 897 [13] McGuigan, R. 658, 667 [77] McIntosh, A. 327, 329, 347, 361 [35] McLaughlin, D. 1797, 1812 [204] McLaughlin, K.T.-R. 349, 362 [70] McMullen, P. 732, 733, 776 [127,128] McWilliams, R.D. 1045, 1068 [94] Meckes, M. 1819, 1820 [7] Medzhitov, A. 515, 531 [107] Mehta, M.L. 342, 344, 345, 364 [121,122] Mejlbro, L. 1529, 1545 [36] Memin, J. 382, 391 [39]

Mendelson, S. 663, 669 [130] Mercourakis, S. 1019, 1046, 1050, 1058, 1064, 1065, 1066 [11]; 1068 [95,96]; 1587, 1602 [46]; 1745, 1752, 1756, 1761, 1763, 1766, 1767, 1770, 1773–1775, 1784, 1800–1802, 1806 [13,14]; 1812 [205,206] Merton, R.C. 369, 371, 391 [40] Merucci, C. 1151, 1174 [81] Meyer, M. 179, 194 [33]; 724, 728, 749, 759, 775 [78,79]; 776 [105,129]; 777 [130,131] Meyer, P.A. 374–376, 389, 390 [17,18]; 610, 614, 616, 665 [25]; 666 [33]; 669 [131] Meyer, Y. 566, 577, 578, 596 [50]; 1696, 1701 [11] Meyer-Nieberg, P. 87, 89, 92, 96, 120 [96] Mézard, M. 354, 364 [123] Miao, B. 348, 360 [17] Michael, E. 596 [51]; 1731, 1741 [55]; 1755, 1812 [207] Michaels, A.J. 103, 120 [97] Milman, D.P. 792, 834 [137] Milman, M. 1141, 1145, 1147, 1149, 1159, 1161, 1173 [26,28,30]; 1174 [82]; 1407, 1421 [63] Milman, V.D. 47, 48, 51, 53, 83 [16]; 84 [28]; 169, 175, 193 [12,21]; 208, 231 [21,31]; 342, 347, 358, 363 [81,86]; 364 [124]; 471, 475, 492 [49]; 494 [113]; 523, 531 [108]; 710, 713–719, 722–724, 726, 729, 731, 732, 735–750, 752, 754, 755, 758, 759, 762, 766–772, 772 [5,8–10]; 773 [33,34,36–40]; 774 [49,61,64–68]; 775 [85,86]; 776 [114,117,118]; 777 [132–154]; 792, 821, 834 [138]; 844, 845, 848, 850, 852, 859, 865–867, 868 [8,17]; 869 [24,47]; 884, 892, 897 [8,26]; 914–916, 924, 925, 927, 935 [12,24]; 938 [96,97]; 1063, 1068 [97]; 1127, 1129 [65]; 1203, 1221, 1224, 1225, 1230, 1231, 1234, 1243, 1244 [8]; 1245 [36,37]; 1246 [38,39]; 1250, 1296 [42,43]; 1305, 1306, 1310, 1315, 1317, 1325, 1326, 1330 [3,13]; 1331 [46–49]; 1335, 1338, 1350, 1353–1358, 1359 [30,32]; 1360 [33–35]; 1606–1608, 1612, 1625–1627, 1630, 1632, 1632 [1–3]; 1633 [9,14,16,20,21]; 1634 [43–46]; 1745, 1812 [208] Milne, H. 676, 705 [34] Milutin, A.A. 1551, 1572, 1602 [47] Minc, H. 98, 120 [98] Mirsky, L. 329, 364 [125] Misiewicz, J. 906, 909, 911, 938 [98–101] Mityagin, B.S. 459, 494 [114]; 906, 935 [1]; 1174 [83]; 1373, 1409, 1421 [64,65]; 1667, 1669 [43,44] Molto, A. 792, 834 [139]; 1778, 1790–1792, 1812 [209–214] Monat, P. 389, 390 [10] Monniaux, S. 250, 268 [113]

Author Index Montesinos, V. 832 [62]; 1745, 1747–1758, 1761–1769, 1771, 1772, 1774–1780, 1782, 1785, 1787–1790, 1792, 1797, 1802, 1804, 1805, 1808 [80–85]; 1812 [209] Montgomery, H.L. 198, 224–226, 231 [32]; 345, 364 [126]; 555, 558 [9] Montgomery-Smith, S.J. 74, 81, 84 [31]; 249, 259, 265 [6,7]; 523, 527, 529 [48]; 531 [109]; 1118, 1129 [54]; 1142, 1153–1155, 1173 [41,46]; 1174 [61]; 1175 [84–86] Moors, W.B. 1521, 1534, 1537, 1538, 1544 [10,11]; 1545 [26,37]; 1804, 1811 [180,181] Moreno, J.P. 664, 665 [16]; 799, 833 [101]; 1758, 1784, 1804, 1811 [158] Morris, P. 633, 667 [89,90] Morton, A. 374, 390 [8] Morzocchi, M. 425, 434 [17] Moschovakis, Y.N. 1019, 1068 [98] Muckenhoupt, B. 579, 596 [52] Muhly, P.S. 237, 249, 250, 265 [12,13]; 1445, 1455 [12]; 1494, 1496, 1511 [16]; 1514 [127] Mujica, J. 677, 705 [35]; 1800, 1812 [215] Müller, C. 926, 938 [102] Müller, P.F.X. 130, 131, 159 [78–80]; 1693, 1694, 1696, 1698, 1699, 1702 [33–37] Murray, F.J. 129, 159 [81]; 484, 494 [115]; 1770, 1812 [216] Muscalu, C. 1484, 1514 [129] Musiela, M. 374, 391 [41] Musielak, J. 515, 531 [110] Nachbin, L. 88, 120 [99]; 1712, 1741 [56] Nagasawa, M. 702, 705 [36] Naimark, M.A. 607, 669 [132] Nakano, H. 87, 89, 120 [100] Namioka, I. 87, 88, 120 [101]; 798, 834 [140]; 1745, 1751, 1752, 1789, 1791, 1792, 1794, 1802–1804, 1807 [33]; 1810 [147,151–155]; 1812 [217–220]; 1814 [257] Naor, A. 845, 869 [48] Narayan, S.K. 320, 360 [8] Nash-Williams, C.St.J.A. 1076, 1097 [17] Natanson, I.P. 507, 531 [111]; 1749, 1801, 1812 [221] Nathanson, M. 207, 231 [33] Nawrocki, M. 1666, 1669 [45] Nazarov, F. 184, 192, 194 [34] Negrepontis, S. 139, 157 [12]; 659, 669 [133]; 1019, 1046, 1068 [77,96]; 1600, 1602 [48]; 1745, 1755, 1756, 1758, 1763, 1767, 1774, 1775, 1784, 1801, 1802, 1806 [14]; 1812 [206,222,223] Neidinger, R.D. 1280, 1296 [44,45] Nelson, E. 1464, 1481, 1514 [130,131] Nemirovski, A.M. 812, 813, 834 [141]

1839

Neufang, M. 1813 [224] Neuwirth, S. 1505, 1513 [84] Newman, C.M. 147, 157 [13]; 230 [4]; 342, 361 [23]; 463, 491 [10]; 748, 773 [23]; 866, 867, 868 [2] Newman, D.J. 1700, 1702 [38] Neyman, A. 906, 938 [103] Ng, K.F. 87, 122 [143] Ng, P.W. 1457 [71] Nguyen Nhu 1112, 1127 [5] Nica, A. 345, 357, 358, 366 [178]; 1445, 1458 [103]; 1466, 1487, 1516 [196] Niculescu, C. 480, 494 [117] Nielsen, N.J. 146, 159 [82]; 268 [98]; 277, 279, 286, 304, 315 [71,77–79]; 518, 531 [112]; 1240, 1245 [28]; 1447, 1457 [54]; 1510, 1513 [99]; 1600, 1602 [49]; 1739, 1739 [14] Nikishin, E.M. 516, 531 [113]; 1302, 1316, 1331 [50,51] Nikol’ski˘ı, S.M. 575, 597 [53]; 1363, 1398, 1399, 1409, 1420 [7] Nilsson, P. 1153, 1173 [31] Nirenberg, L. 1398, 1421 [66] Nissenzweig, A. 1749, 1813 [225] Nordgren, E.A. 101, 119 [48] Nördlander, G. 793, 834 [142] Norin, N.V. 1528, 1545 [38] Novikov, I. 523, 531 [114]; 595, 597 [54,55] Novikov, S.Ya. 527, 531 [115,116] Nussbaum, R.D. 99, 112, 120 [102,103]; 121 [104] Oberlin, D. 685, 705 [37] Odell, E. 59, 83 [22]; 133, 139, 141, 143–145, 147, 149, 150, 154, 156 [2]; 158 [51,52]; 159 [83–85]; 255, 256, 267 [58]; 276, 279, 313 [4,17]; 565, 581, 595 [1]; 596 [23]; 612, 645, 669 [134]; 813, 820, 821, 833 [117]; 834 [143,144]; 839, 862, 868, 868 [1]; 1016, 1019, 1022, 1025, 1030, 1034, 1036, 1044–1046, 1048, 1050, 1052, 1056, 1060, 1061, 1063–1065, 1065 [2,3]; 1066 [30]; 1068 [66]; 1069 [99–105]; 1081, 1082, 1090, 1097 [18,19]; 1110, 1129 [66]; 1133, 1162, 1175 [87]; 1251, 1260, 1296 [46]; 1336, 1339, 1343, 1347–1349, 1351, 1352, 1354, 1355, 1357, 1358, 1359 [2,17,18,22]; 1360 [36–46]; 1748, 1759, 1775, 1813 [226–230] Odlyzko, A.M. 345, 362 [77]; 364 [127] Ogrodzka, Z. 1414, 1422 [67] Oikhberg, T. 275, 285, 314 [48]; 1454, 1457 [50,72–75]; 1551, 1602 [50] Oja, E. 1748, 1813 [231] Oleszkiewicz, K. 460, 493 [96]; 1505, 1513 [84] Olevskiˇı, A.M. 213, 231 [34]; 251, 269 [114,115] Olin, R.F. 341, 364 [128] Olsen, G.H. 618, 668 [122]

1840

Author Index

Oncina, L. 1791, 1792, 1813 [232] O’Neil, R. 505, 511, 531 [117,118] Oprea, A.G. 1493, 1512 [44] Ordower, M. 340, 362 [63] Orihuela, J. 792, 834 [139]; 1756, 1762, 1764, 1766, 1767, 1778, 1789–1792, 1801, 1803, 1812 [209–213]; 1813 [233–235] Orlicz, W. 458, 473, 494 [118] Ørno, P. 92, 121 [105]; 1749, 1813 [236] Ornstein, D. 1409, 1422 [68] Ortynski, A. 1666, 1669 [45] Ostrovskii, M.M. 1143, 1144, 1174 [63] Otto, F. 350, 364 [129] Ovchinnikov, V.I. 1151, 1154, 1174 [78,80]; 1175 [88]; 1464–1467, 1514 [132–134] Oxtoby, J.C. 1813 [237] Ozawa, N. 1447, 1452, 1454, 1455, 1456 [28]; 1457 [55,71]; 1458 [76,77] Pajor, A. 175, 179, 193 [14]; 194 [33]; 454, 492 [54]; 718, 722–724, 728, 750, 756, 759, 762, 767, 769, 773 [19]; 776 [105,117]; 777 [130,131,146–148,155,156]; 854, 868 [15]; 924, 938 [97]; 946, 968, 974 [10]; 1225, 1226, 1246 [40] Paley, R.E.A.C. 250, 269 [116]; 1664, 1669 [46]; 1688, 1702 [39] Pallaschke, D. 1113, 1129 [67] Palmon, O. 767, 778 [157] Panzone, R. 107, 118 [34] Paouris, G. 765, 778 [158] Papadimitrakis, M. 164, 193 [22]; 725, 774 [69]; 918, 938 [104] Papadopoulou, S. 660, 669 [135] Papini, L. 650, 655, 666 [45] Parisi, G. 354, 364 [123] Parrott, S.K. 339, 364 [130] Partington, J.R. 1775, 1813 [238] Pastur, L.A. 343, 344, 349, 353, 357, 364 [118,131–133] Paulsen, V.I. 320, 362 [64]; 364 [134]; 1427–1432, 1435, 1438, 1440, 1442, 1445, 1455 [12]; 1456 [13,14]; 1458 [78–81] Pavlović, M. 1677, 1682, 1701 [20,32] Payá, R. 642, 666 [35]; 1046, 1068 [86] Pearcy, C.M. 322, 364 [135]; 536, 539, 558 [27]; 559 [28]; 1170, 1175 [89] Pechanec, J. 1795, 1813 [239] Peck, N.T. 278, 301, 302, 314 [54]; 403, 435 [57]; 1101, 1104, 1107, 1111, 1113–1115, 1125, 1128 [30]; 1129 [55,56,68]; 1157, 1158, 1174 [62]; 1666, 1668 [31] Pedersen, G.K. 1461, 1510 [1]

Peetre, J. 481, 494 [119]; 1136, 1140, 1145, 1173 [32]; 1174 [76]; 1175 [90,91]; 1363, 1382, 1400, 1407, 1422 [69–71]; 1466, 1468, 1471, 1472, 1514 [116,135] Peirats, V. 518, 529 [60] Pelant, J. 832 [62]; 1745, 1747–1758, 1761–1766, 1768, 1769, 1771, 1772, 1774, 1775, 1777–1780, 1782, 1785, 1787–1789, 1792, 1795–1797, 1802, 1804, 1805, 1808 [62,80]; 1809 [118]; 1813 [240] Pełczyński, A. 96, 119 [50]; 125, 129, 132, 135, 142, 146, 147, 156, 158 [58,66,67]; 159 [86]; 251, 255, 256, 268 [105]; 269 [117–119]; 274, 277, 290, 300, 305, 315 [80–82]; 440, 443, 445, 458, 459, 462–466, 473, 479, 491 [31]; 492 [40]; 494 [104,114,120–125]; 510, 530 [73]; 577, 580–582, 596 [35,41,51]; 597 [56,57]; 603, 648, 665 [9,10]; 675, 688, 703, 704, 705 [39]; 792, 830 [17]; 858, 869 [23]; 875, 876, 878–880, 883, 886, 887, 889, 894, 896 [3]; 897 [18,19,23,27,28]; 901, 905, 911–915, 935 [18]; 951, 963, 974 [21,27]; 1018, 1028, 1029, 1045, 1066 [16]; 1069 [106,107]; 1133, 1136, 1137, 1173 [35]; 1219, 1222, 1223, 1240, 1244 [7]; 1246 [41,42]; 1249, 1253, 1254, 1280, 1295 [7,10]; 1296 [47]; 1301–1304, 1330 [8]; 1331 [32,39]; 1366, 1367, 1372, 1377, 1380–1383, 1387, 1393, 1396, 1400–1403, 1410–1413, 1417–1419, 1420 [6]; 1421 [47,58]; 1422 [72–74,76–81]; 1478, 1495, 1498, 1509, 1514 [103,111]; 1525, 1545 [39]; 1552, 1560, 1569, 1571, 1580, 1590, 1593, 1594, 1601 [15]; 1602 [42,51–54]; 1639, 1642, 1652, 1668 [35]; 1669 [47–49]; 1674, 1675, 1697, 1701 [13,27,29]; 1709, 1711, 1717, 1723, 1727, 1728, 1732–1735, 1740 [35,44,45]; 1741 [57–59]; 1745, 1761, 1765–1767, 1796, 1800, 1803, 1806 [24,25]; 1807 [46]; 1813 [241,242] Peller, V.V. 1500, 1501, 1505, 1506, 1510 [2]; 1514 [136,137]; 1515 [138,139] Pena, A. 769, 773 [21]; 1127, 1127 [10] Peressini, A.L. 87, 121 [107] Perissinaki, I. 722, 774 [70] Perron, O. 98, 121 [108] Persson, A. 462, 494 [126] Petrushev, P. 579, 596 [39]; 597 [58] Pettis, B.J. 444, 492 [41]; 792, 834 [145]; 1579, 1601 [25] Petty, C.M. 164, 177, 193 [13]; 194 [35]; 725, 778 [159]; 918, 935 [16] Petunin, Yu.I. 74, 78, 81, 83 [13]; 87, 120 [83]; 1133, 1174 [69] Pezzotta, A. 662, 667 [70] Pfaffenberger, W.E. 558 [12] Pfitzner, H. 1477, 1515 [140] Phelps, R.R. 34, 35, 83 [26]; 435 [61]; 610, 611, 614, 626, 627, 634, 640, 641, 645, 646, 659, 664,

Author Index 665 [13]; 667 [72]; 668 [121]; 669 [136–140]; 793, 798, 832 [69]; 834 [140]; 1019, 1025, 1067 [52]; 1069 [108]; 1532, 1535, 1537, 1538, 1545 [40,41]; 1733, 1739 [7]; 1745, 1747, 1749, 1752, 1783, 1789, 1795, 1798, 1809 [99]; 1812 [188]; 1813 [243]; 1814 [257]; 1823, 1823 [3a] Phillips, N.C. 1431, 1457 [62] Phillips, R. 338, 359, 364 [116]; 1451, 1457 [69] Piasecki, M. 264, 269 [120,121] Picardello, M. 1466, 1512 [67] Pichorides, S.K. 482, 494 [127] Pietsch, A. 250, 269 [122]; 439, 440, 444, 448, 451–455, 458, 459, 462–464, 466, 469–471, 475, 476, 480, 484, 493 [102]; 494 [126,128–136]; 495 [137–139]; 590, 597 [59]; 929, 935 [19]; 943–945, 947, 950, 951, 953, 954, 957, 961, 966, 969, 973, 974 [7,28–32]; 1222, 1246 [43]; 1301, 1331 [52]; 1466, 1468, 1515 [141,142]; 1563, 1601 [22] Pigno, L. 576, 577, 596 [49] Pinsker, A.G. 87, 106, 107, 120 [75] Pintz, J. 207, 230 [5] Pisier, G. 47, 51, 53, 83 [17]; 206, 231 [35,36]; 235, 237, 269 [123,124]; 276, 277, 284–286, 305, 314 [49]; 315 [76,83–85]; 320, 338, 346, 365 [136–141]; 466, 468, 471, 473–475, 477, 479, 483, 489, 494 [112]; 495 [140–150]; 518, 523, 531 [106,119]; 695, 705 [38]; 710, 726, 752, 754, 760, 767, 770–772, 776 [124,125]; 777 [149]; 778 [160–164]; 793, 804, 805, 812, 831 [54]; 834 [136,146–149]; 843, 845, 849, 870 [49–51]; 879, 882, 886, 888, 891, 893, 894, 897 [29–32]; 964, 974 [33]; 1129 [69]; 1133, 1144, 1157, 1164, 1173 [39]; 1175 [92–98]; 1181, 1188, 1190–1192, 1194, 1199 [15,34,37]; 1200 [38–42]; 1222, 1224–1226, 1230–1232, 1234, 1241, 1242, 1245 [16]; 1246 [38,44]; 1303–1306, 1315, 1320, 1322, 1331 [44,45,48,53–55]; 1332 [56–61]; 1406, 1422 [82]; 1429, 1431, 1432, 1438, 1441, 1442, 1444, 1446–1451, 1457 [56,73]; 1458 [82–92]; 1471, 1478, 1483–1493, 1495, 1499, 1500, 1502, 1503, 1508, 1513 [80,81]; 1514 [121]; 1515 [143–157]; 1599, 1601 [21]; 1607, 1610, 1625, 1628, 1634 [47–49]; 1726, 1739 [12]; 1741 [60]; 1818 [2a] Pitt, L.D. 94, 121 [109]; 905, 936 [42] Pittenger, A.O. 257, 269 [125] Pitts, D.R. 341, 362 [65,66]; 365 [142] Plemmons, R.J. 98, 113, 118 [35] Plichko, A. 304, 313 [2]; 1759, 1764, 1765, 1767, 1769, 1789, 1800, 1803, 1804, 1806 [3]; 1807 [35]; 1813 [244–248]; 1817, 1818 [1a] Pliska, S.R. 374, 377, 391 [30]

1841

Plotkin, A.I. 901–903, 905, 910, 937 [86,87]; 938 [105–109] Plymen, R.J. 1466, 1515 [158] Pol, R. 1019, 1069 [109]; 1745, 1750, 1751, 1754, 1757, 1758, 1762, 1792, 1794, 1802, 1806 [5]; 1807 [38]; 1812 [220]; 1813 [249–254] Polya, G. 907, 938 [110,111]; 956, 974 [12]; 1173 [44] Polyrakis, I.A. 116, 118 [14] Pommerenke, Ch. 701, 705 [40] Pompe, W. 573, 597 [60] Poornima, S. 1403, 1422 [83] Popescu, G. 341, 365 [143] Popovici, I.M. 97, 98, 121 [110,111] Pospíšil, B. 1753, 1807 [36] Poulsen, E.T. 617, 669 [141] Power, S.C. 339, 341, 362 [67]; 365 [144]; 1496, 1515 [159] Preiss, D. 37, 42, 84 [39]; 396, 399, 413, 418, 425, 433 [8]; 434 [37]; 435 [50,62]; 793, 798, 820, 821, 829, 830 [19]; 833 [107]; 834 [140,150,151]; 1529–1534, 1538, 1539, 1541, 1543, 1544 [4,22]; 1545 [25,28–31,36,42–47]; 1747, 1752, 1777, 1789, 1813 [255,256]; 1814 [257,258]; 1818 [2a] Prékopa, A. 1607, 1634 [50] Privalov, A.A. 578, 597 [61] Privalov, I.I. 694, 705 [41] Protter, P. 376, 391 [42] Prüss, J. 250, 268 [113]; 269 [126,127] Przelawski, K. 793, 830 [13] Przeworska-Rolewicz, D. 445, 495 [151] Ptak, V. 1052, 1069 [110] Pukhlikov, A.V. 732, 776 [102] Pustylnik, E.I. 87, 120 [79] Quenez, M.C. 382, 390 [24] Quinzii, M. 115, 120 [95] Rabinovich, L. 917, 938 [112] Radjavi, H. 99, 101, 119 [48]; 121 [112]; 542, 549, 559 [29,30] Raghavan, T.E.S. 98, 113, 118 [33] Rainwater, J. 609, 669 [142]; 1783, 1814 [259] Raja, M. 805, 828, 834 [152]; 835 [153]; 1787, 1790–1794, 1798, 1814 [260–263] Randrianantoanina, B. 515, 531 [120]; 901, 905, 936 [49]; 938 [113,114] Randrianantoanina, N. 1490, 1492, 1493, 1498, 1509, 1515 [160–165] Range, R.M. 675, 683, 705 [42] Ransford, T.J. 1126, 1128 [20] Rao, M. 612, 669 [143] Rassias, T.M. 1108, 1128 [31]

1842

Author Index

Raynaud, Y. 137, 139, 140, 158 [43]; 159 [87–89]; 519, 523, 527, 528 [8,9]; 531 [121–126]; 532 [127,128]; 1479, 1509, 1515 [166–169] Read, C.J. 98, 102, 121 [115–117]; 274, 295, 297, 299, 301, 315 [86]; 549, 550, 555, 556, 559 [31–36]; 1216, 1240, 1246 [45]; 1254, 1296 [48]; 1644, 1669 [50] Reese, M.L. 1103, 1129 [70] Reif, J. 1814 [264] Reinov, O. 462, 484, 495 [152,153]; 1817, 1818, 1818 [4a] Reisner, S. 527, 532 [129]; 759, 775 [79]; 778 [165]; 896, 897 [12] Retherford, J.R. 953, 957, 961, 964, 965, 974 [15] Revalski, J. 434 [31] Revuz, D. 372, 373, 378, 379, 391 [43] Rezniˇcenko, E.A. 1747, 1814 [265] Reznick, B. 916, 917, 938 [115] Rhandi, A. 112, 121 [113,114]; 433 [11] Ribarska, N.K. 1789, 1803, 1814 [266,267] Ribe, M. 829, 835 [154,155]; 1104, 1107, 1108, 1110, 1129 [71,72]; 1157, 1175 [99]; 1347, 1360 [47] Ricard, E. 1457 [74] Rieffel, M.A. 484, 495 [154]; 634, 669 [144] Riesz, F. 442, 495 [155,156] Riesz, M. 244, 269 [128] Ringrose, J.R. 339, 365 [145]; 439, 495 [157]; 1463, 1514 [104] Riss, E.A. 1530, 1545 [48] Rivière, N.M. 1411, 1420 [28]; 1422 [84] Roberts, J.W. 403, 435 [57]; 603, 669 [145]; 1101, 1107–1109, 1111–1114, 1117, 1118, 1125, 1129 [56–58,73–76]; 1255, 1296 [33]; 1666, 1668 [31] Robinson, A. 535, 558 [10] Robinson, D.W. 620, 665 [23]; 1466, 1511 [25] Robinson, P.L. 1466, 1515 [158] Rochberg, R. 703, 705 [43]; 706 [44,45]; 1120, 1128 [15]; 1133, 1158, 1159, 1161, 1172 [20]; 1173 [28,49]; 1175 [100,101] Rodé, G. 641, 669 [146,147]; 1814 [268] Rodin, V.A. 519, 532 [130]; 1155, 1175 [102] Rodriguez-Salinas, B. 518, 527, 529 [61]; 530 [62–64] Rogalski, M. 611, 669 [148]; 1048, 1069 [111] Rogers, C.A. 177, 193 [29]; 458, 492 [44]; 631, 668 [94]; 720, 735, 774 [54]; 918, 937 [73]; 1303, 1330 [11]; 1626, 1633 [12]; 1745, 1761, 1788, 1789, 1791, 1794, 1803, 1804, 1810 [146,147,151–155]; 1811 [156,157]; 1814 [269] Rogers, L.C.G. 376, 383, 391 [44,45] Rohlin, V.A. 616, 669 [149] Rolewicz, S. 445, 495 [151]; 769, 778 [166]; 1101, 1102, 1129 [77,78]

Romberg, B.W. 1104, 1113, 1120, 1128 [23] Ropela, S. 575, 597 [62] Rørdam, M. 325, 362 [78]; 363 [79] Rosenoer, S. 340, 365 [146] Rosenthal, H.P. 20, 59, 84 [40]; 125, 128, 129, 131, 134, 140, 145–148, 150, 151, 153, 154, 157 [11,21,37]; 158 [54,68]; 159 [90–93]; 204, 231 [37]; 255, 265 [22]; 268 [99]; 269 [119]; 274, 277, 279, 290, 291, 293, 294, 296, 300, 305, 314 [50,51]; 315 [81,87]; 445, 466, 473, 480, 492 [40]; 494 [105]; 495 [158,159]; 521, 532 [131]; 576, 581, 595 [11]; 596 [33]; 612, 614, 636, 644, 645, 665 [19]; 669 [134,150,151]; 813, 821, 833 [105]; 835 [156,157]; 868, 870 [52]; 875, 876, 897 [33]; 1015, 1016, 1018–1020, 1022–1025, 1028, 1030, 1031, 1034, 1036, 1044–1048, 1051, 1055, 1056, 1058, 1060, 1064, 1065, 1066 [26,33]; 1067 [34]; 1068 [66,93]; 1069 [102,112–120]; 1076, 1081, 1096 [1]; 1097 [20,21]; 1242, 1245 [17]; 1260, 1261, 1296 [41]; 1301–1303, 1330 [8]; 1332 [62]; 1339, 1340, 1349, 1352, 1359 [18,31]; 1360 [38,48,49]; 1454, 1455 [3]; 1457 [66,75]; 1458 [93]; 1506–1509, 1513 [82]; 1515 [170]; 1551, 1560, 1561, 1580, 1581, 1587, 1589, 1592, 1594, 1595, 1598–1600, 1601 [10,34,40,41]; 1602 [44,50,55–62]; 1652, 1654, 1669 [48,51]; 1697, 1702 [40]; 1708, 1714, 1717, 1722, 1723, 1725, 1727–1729, 1732, 1740 [28,46,47]; 1741 [61–64]; 1748, 1755, 1758, 1762, 1767, 1771, 1772, 1800, 1802, 1804, 1805, 1808 [68]; 1811 [170]; 1813 [228]; 1814 [270–274] Rosenthal, P. 101, 119 [48]; 542, 549, 559 [29,30] Rosinski, J. 905, 938 [116] Ross, K.A. 886, 897 [24] Rosset, S. 730, 778 [167] Roth, K. 200, 231 [25] Rothschild, B.L. 1074, 1097 [13] Royden, H.L. 6, 13, 15, 20, 62, 83 [18]; 1751, 1814 [275] Ruan, Z.J. 1427, 1432–1436, 1438–1440, 1442–1447, 1452–1455, 1456 [15,26,28–41]; 1457 [54,55,57,65]; 1458 [94–97]; 1510, 1512 [60]; 1513 [99] Rubio de Francia, J.L. 250, 269 [129–131] Rudelson, M. 737, 767, 778 [168–170] Rudin, M.E. 1751, 1752, 1754–1756, 1780, 1806 [21]; 1812 [207] Rudin, W. 19, 37, 83 [19]; 147, 159 [94]; 197, 205, 206, 231 [38]; 359, 365 [147]; 603, 669 [152]; 883, 897 [34]; 901, 902, 938 [117]; 1419, 1422 [85]; 1502, 1516 [171]; 1674, 1675, 1677, 1679, 1694, 1700, 1701 [28]; 1702 [41,42] Rudnick, Z. 345, 365 [148] Ruelle, D. 354, 360 [1]; 619, 620, 669 [153]

Author Index Ruiz, C. 527, 530 [65,66] Russo, B. 1600, 1601 [27] Russu, G.I. 1466, 1516 [172] Ruston, A.F. 449, 450, 484, 495 [160–162] Rutkowski, M. 374, 391 [41] Rutman, M.A. 99, 120 [82] Ruzsa, I. 205, 231 [39] Ryan, R. 263, 269 [132] Rychtáˇr, J. 1782, 1814 [276,277] Ryff, J.V. 1151, 1175 [103] Ryll-Nardzewski, Cz. 906, 938 [101] Saab, P. 95, 120 [73] Saakyan, A.A. 575, 596 [36] Saccone, S.F. 682, 685, 686, 706 [46,47] Sagher, Y. 505, 529 [39]; 1141, 1158, 1172 [20]; 1173 [30,33] Sahakian, A. 578, 596 [44] Saint Raymond, J. 633, 669 [154]; 759, 778 [171]; 793, 831 [38] Saito, K.S. 1465, 1496, 1497, 1513 [93]; 1514 [127]; 1516 [173–177] Sakai, S. 1463, 1516 [178] Saks, S. 444, 490 [4] Salem, R. 873, 897 [17] Salinas, N. 324, 332, 333, 360 [11]; 362 [62]; 365 [149]; 536, 539, 558 [27] Saloff-Coste, L. 1363, 1423 [95] Samet, D. 1730, 1741 [65] Samuel, C. 286, 305, 315 [88]; 316 [89]; 1560, 1569, 1602 [63] Samuelson, P.A. 370, 391 [46] Saphar, P.D. 285, 310, 314 [31]; 461, 495 [163,164]; 792, 829, 832 [81] Sarason, D. 1495, 1497, 1516 [179]; 1700, 1702 [43] Sarnak, P. 338, 345, 359, 364 [116]; 365 [148]; 1451, 1457 [69] Savage, L.J. 616, 667 [87] Scalora, F.S. 260, 269 [133] Schachermayer, W. 263, 267 [82]; 374, 377–386, 389, 390 [10–16]; 391 [47,48]; 404, 435 [49]; 633, 636, 669 [155,156]; 798, 799, 832 [73]; 835 [158]; 1767, 1800, 1803, 1809 [97]; 1813 [234]; 1814 [278] Schaefer, H.H. 87, 88, 92, 99, 110, 111, 121 [120–124]; 821, 835 [159] Schäffer, J.J. 805, 835 [160] Schatten, R. 439, 484, 486, 495 [165–169] Schauder, J. 273, 316 [90]; 442, 495 [170] Schechter, M. 1159, 1175 [104] Schechtman, G. 47, 48, 51, 53, 83 [16]; 129–132, 134, 140, 148, 150, 153, 154, 156, 157 [21]; 158 [50,53,55]; 159 [78,79,95–97]; 176, 194 [36]; 208, 231 [31]; 257, 268 [96]; 278, 314 [46]; 336,

1843

338, 346, 347, 363 [98]; 364 [124]; 365 [150]; 471, 475, 494 [113]; 511, 521–523, 530 [70–72]; 531 [108]; 576, 581, 595 [11]; 710, 713, 718, 735, 737, 739–742, 745, 746, 748, 749, 755, 769, 770, 775 [96,97]; 776 [118]; 777 [150–152]; 778 [172–175]; 792, 793, 821, 829, 833 [106,107]; 840, 844, 845, 847, 848, 851–853, 855, 857–859, 862–864, 866, 867, 869 [21,22,29–34,46,47]; 870 [53–56]; 884, 897 [26]; 906, 925, 936 [45,46]; 1018, 1019, 1066 [26]; 1157, 1173 [50]; 1186, 1192, 1199 [19]; 1200 [43]; 1234, 1242, 1245 [15]; 1246 [39]; 1310, 1315, 1317, 1325, 1326, 1331 [49]; 1338, 1360 [33]; 1403, 1422 [86]; 1539, 1543, 1544 [4,22]; 1606, 1608, 1610, 1612, 1614, 1624–1626, 1628–1630, 1633 [13,26–29]; 1634 [45,46,51–55]; 1642, 1652, 1654–1656, 1660–1664, 1668 [27]; 1669 [52]; 1698, 1699, 1702 [37]; 1716, 1740 [29]; 1796, 1805, 1811 [169] Schep, A.R. 108, 121 [125] Scherer, K. 1404, 1420 [20] Schipp, F. 576, 597 [63] Schlüchtermann, G. 1478, 1512 [58] Schlumprecht, Th. 139, 143, 144, 159 [84,85]; 176, 177, 193 [18]; 194 [36]; 813, 820, 821, 833 [117]; 834 [143,144]; 919, 935 [29]; 1061–1063, 1069 [103,104,121]; 1081, 1082, 1090, 1097 [18,19]; 1110, 1129 [66]; 1133, 1162, 1175 [87]; 1251, 1256, 1260, 1296 [46,49]; 1335, 1339, 1343–1349, 1352, 1354, 1355, 1357, 1359 [3,4,18,22]; 1360 [38–44,50,51]; 1775, 1813 [229,230] Schmidt, E. 439, 441, 496 [171]; 716, 778 [176] Schmuckenschläger, M. 1625, 1634 [53] Schnaubelt, R. 112, 121 [114] Schneider, R. 710, 720, 726, 734, 778 [177,178]; 911, 921, 938 [118–120]; 1607, 1634 [56] Schoenberg, I.J. 906, 938 [121] Scholes, M. 369, 371, 390 [4] Schonbek, T. 1471, 1472, 1512 [42] Schreier, J. 1051, 1064, 1069 [122]; 1581, 1602 [64] Schur, I. 439, 442, 496 [173,174] Schütt, C. 140, 158 [63,64]; 159 [88]; 471, 496 [172]; 525–527, 530 [84,85]; 532 [128,132,133]; 855, 856, 869 [39]; 933, 934, 937 [67,71]; 1234, 1245 [19]; 1478, 1516 [180]; 1655, 1669 [53] Schwartz, J.T. 106, 107, 119 [54]; 439, 492 [42]; 1366, 1388, 1395, 1420 [25]; 1639, 1669 [42] Schwartz, L. 471, 496 [175] Schwarz, H.U. 87, 121 [126]; 488, 496 [176] Schweizer, M. 387–389, 390 [10,26] Sciffer, S. 1537, 1545 [26]; 1804, 1811 [180,181] Sedaev, A.A. 527, 532 [134]; 1477, 1511 [37] Segal, I. 1461, 1516 [181] Seidel, J.J. 916, 936 [34]

1844

Author Index

Seifert, C.J. 479, 491 [36] Semadeni, Z. 625, 669 [157]; 1714, 1740 [25]; 1745, 1803, 1814 [279,280] Semenov, E.M. 74, 78, 81, 83 [13]; 87, 120 [83]; 519, 523, 527, 531 [114,116]; 532 [130]; 597 [55]; 812, 813, 834 [141]; 1133, 1155, 1174 [69]; 1175 [102] Sersouri, A. 798, 835 [158]; 1019, 1069 [123] Shapiro, J.H. 470, 471, 496 [177–180]; 1104, 1113, 1120, 1129 [59,60,79,80]; 1683, 1684, 1702 [44–46] Sharpley, C. 1404, 1420 [21] Sharpley, R. 505, 515, 528 [10]; 1133, 1136, 1147, 1172 [4,5]; 1404, 1407, 1420 [4] Shashkin, Yu.A. 640, 669 [158] Shatalova, O.A. 917, 937 [92] Shcherbina, M. 349, 361 [45]; 364 [133] Shelah, S. 1255, 1297 [50,51]; 1759, 1814 [281] Shevchyk, V.V. 1817, 1818 [1a] Shields, A.L. 322, 364 [135]; 559 [28]; 1104, 1113, 1120, 1128 [23]; 1700, 1701 [3] Shiga, K. 440, 490 [2] Shilov, G.E. 902, 908, 920, 936 [31] Shimogaki, T. 1175 [105] Shiryaev, A.N. 374, 391 [49] Shlyakhtenko, D. 360, 365 [151]; 1447, 1448, 1458 [92] Shreve, S. 374, 391 [33] Shteinberg, A. 1172 [13] Shultz, F. 627, 665 [3] Shura, T.J. 136, 157 [28]; 276, 313, 313 [18]; 812, 830 [27]; 1065, 1066 [31]; 1094, 1096 [7]; 1255, 1295 [9] Shvartsman, P. 1373, 1383, 1420 [15,16] Sidorenko, N.G. 1412, 1415, 1416, 1418, 1421 [51]; 1422 [87] Sierpinski, W. 1019, 1068 [88]; 1560, 1602 [45]; 1722, 1741 [54] Silver, J. 1049, 1069 [124] Silverstein, J.W. 349, 353, 365 [152] Silverstein, M.L. 266 [44] Simon, B. 439, 496 [181]; 1465, 1516 [182] Simon, P. 576, 597 [63]; 1747, 1755, 1814 [258,282] Simoniˇc, A. 105, 121 [127,128]; 535, 559 [37] Simonovits, M. 1625, 1633 [30]; 1634 [39] Simons, S. 644, 669 [159] Sina˘ı, Ya.G. 349, 365 [153] Sinclair, A. 1431, 1440–1442, 1444, 1445, 1456 [15,20,21] Singer, I. 1020, 1069 [125]; 1639, 1669 [49]; 1745, 1777, 1814 [283,284] Singer, I.M. 333, 334, 363 [99]; 859, 869 [35] Sisson, P. 1114, 1129 [81] Sjölin, P. 595, 597 [64] Skorochod, A.V. 1526, 1545 [52]

Skorohod, A.S. 384, 391 [37] Sledd, W.T. 1681, 1702 [47] Smickih, S.V. 99, 122 [146] Šmídek, M. 798, 833 [97]; 1752, 1758, 1810 [149] Smith, B. 576, 577, 596 [49] Smith, K.T. 535, 558 [4] Smith, M. 1777, 1814 [285] Smith, R. 1431, 1440, 1442, 1456 [16]; 1458 [81] Smoluchowski, M. 369, 391 [50] Smolyanov, O.G. 1527, 1528, 1545 [38,49–51] Šmulyan, V.L. 793, 835 [161] Snobar, M.G. 463, 493 [81]; 929, 936 [48] Sobczyk, A. 1454, 1458 [98]; 1553, 1602 [65]; 1705, 1707, 1716, 1739 [8]; 1741 [66] Sobecki, D. 1142, 1173 [36] Sobolev, A.V. 87, 120 [80] Sobolev, S.L. 1365, 1373, 1398, 1422 [88] Sobolevsky, P.E. 87, 120 [79] Sohr, H. 250, 267 [83,84]; 269 [127] Sokolov, G.A. 1767, 1814 [286] Solecki, S. 1532, 1545 [53] Sölin, P. 223, 231 [15] Solonnikov, V.A. 1415, 1422 [89] Somasundaram, S. 1534, 1537, 1545 [37] Sondermann, D. 387, 390 [27] Soria, J. 1161, 1172 [16] Soshnikov, A.B. 346, 349, 365 [153,154] Souslin, M.M. 1019, 1069 [126] Spalsbury, A. 545, 559 [38] Sparr, G. 1151, 1175 [106,107]; 1466, 1468, 1514 [135] Speicher, R. 1491, 1511 [19] Srinivasan, T.P. 702, 706 [48] Srivatsa, V.V. 1792, 1814 [287] Starbird, T. 129, 131, 147, 157 [31,36]; 822, 830 [15]; 1115, 1128 [24]; 1652, 1668 [14]; 1754, 1806 [22] Stegall, C. 129, 158 [65]; 483, 496 [182]; 798, 835 [162–164]; 1020, 1069 [127]; 1252, 1296 [38]; 1532, 1545 [35]; 1591, 1600, 1601 [39]; 1602 [66]; 1717, 1727, 1740 [40]; 1752, 1758, 1766, 1803, 1812 [196]; 1814 [288–290] Stegenga, D.A. 1681, 1702 [47] Stein, E.M. 215, 231 [40]; 244, 267 [72]; 269 [134–136]; 505, 532 [135]; 894, 897 [35]; 1363, 1371, 1373, 1398, 1399, 1403, 1405, 1410, 1422 [90–92]; 1531, 1546 [54]; 1688, 1696, 1701 [11]; 1702 [48] Stephani, I. 452, 491 [25]; 958, 959, 973, 973 [3] Stephenson, K. 905, 938 [122] Stepr¯ans, J. 1255, 1297 [51]; 1759, 1814 [281] Stern, R. 412, 434 [15,16] Sternfeld, Y. 618, 625, 668 [122]; 669 [160] Stezenko, V.Ya. 88, 118 [32] Stiles, W.J. 1116, 1129 [82]

Author Index Stinespring, W. 1461, 1516 [183] Størmer, E. 620, 669 [161,162] Stout, E.L. 675, 706 [49] Stout, W.F. 1610, 1634 [57] Strassen, V. 1188, 1190, 1198 [6] Straszewicz, S. 628, 669 [163] Stratila, S. 1463, 1516 [184,185] Strichartz, R. 217, 231 [41] Stricker, C. 374, 385, 389, 390 [1,6,7,10]; 391 [51] Strömberg, J.O. 575, 597 [65]; 1531, 1546 [54] Study, E. 712, 774 [45] Sucheston, L. 260, 267 [69]; 1036, 1066 [27,28]; 1074, 1096 [6]; 1304, 1306, 1307, 1310, 1330 [4,5] Sudakov, V.N. 756, 778 [179] Sukochev, F. 515, 531 [107]; 1477, 1478, 1484, 1492, 1506–1509, 1511 [37–40]; 1512 [41,50,55–58,65,66]; 1513 [82]; 1516 [186–188] Sullivan, F.E. 1752, 1810 [130] Sunder, V.S. 106, 119 [67]; 327, 365 [155] Swart, J. 484, 491 [34] Synnatzschke, J. 92, 106, 121 [129] Szankowski, A. 275, 283–285, 305, 316 [91–94]; 489, 496 [183]; 735, 766, 776 [115]; 778 [180]; 1440, 1458 [99]; 1478, 1516 [189] Szarek, S.J. 146, 159 [98]; 273, 295, 299, 301, 302, 316 [95]; 322, 331, 345, 358, 359, 361 [24]; 363 [95]; 365 [156–164]; 460, 496 [184]; 580, 597 [66]; 736, 737, 749, 765–767, 770, 773 [19,35]; 774 [55]; 778 [181–186]; 779 [187]; 859, 868 [16]; 870 [57]; 969, 973 [6]; 1158, 1175 [108]; 1212, 1213, 1218, 1220–1222, 1224, 1225, 1235, 1236, 1239, 1243, 1244, 1244 [4]; 1245 [29]; 1246 [46–55]; 1254, 1297 [52,53]; 1305, 1332 [64,65]; 1525, 1545 [39]; 1681, 1688, 1702 [49] Szego, G. 907, 938 [111] Szlenk, W. 805, 835 [165]; 1020, 1069 [128]; 1560, 1561, 1602 [67]; 1719, 1741 [67]; 1765, 1813 [242] Szulga, J. 1617, 1633 [31] Tacon, D.G. 1789, 1815 [291] Takesaki, M. 1463, 1477, 1516 [190] Talagrand, M. 96, 121 [130]; 181, 194 [37]; 201, 231 [42]; 338, 344, 346, 350, 354, 359, 364 [112]; 365 [165,166]; 366 [167–169]; 472, 473, 493 [97]; 496 [185]; 523, 530 [87]; 633, 665 [20]; 737, 740, 755, 756, 766, 769, 776 [109]; 778 [186]; 779 [188,189]; 828, 835 [166]; 844, 845, 848, 849, 869 [41]; 870 [58,59]; 885, 898 [36]; 925, 939 [123]; 1019, 1020, 1047, 1066 [25]; 1067 [61]; 1069 [129]; 1117, 1129 [83]; 1179, 1180, 1183, 1185, 1186, 1188, 1190–1196, 1199 [30–32];

1845

1200 [44–47]; 1221, 1246 [53]; 1614–1617, 1619, 1623–1625, 1630, 1632, 1634 [35,58–63]; 1745, 1748, 1758, 1763, 1767, 1775–1777, 1790, 1791, 1794, 1795, 1802, 1806 [30]; 1809 [119]; 1815 [292–297] Tam, S.C. 1106, 1129 [84] Tamarkin, J. 968, 974 [13] Tang, W.K. 793, 835 [167]; 1019, 1068 [83,84]; 1782, 1801, 1815 [298,299] Tao, T. 222, 223, 231 [29,30,43,44] Taylor, A.E. 265 [17] Taylor, B.A. 1608, 1632 [6] Taylor, P.D. 470, 496 [180]; 1684, 1702 [46] Temme, D. 1700, 1702 [50] Terenzi, P. 1784, 1815 [300–302] Terevcak, I. 1107, 1128 [30] Terp, M. 1464, 1466, 1470, 1473–1476, 1480, 1516 [191,192] Thompson, A.C. 664, 666 [46] Thomson, J.E. 341, 364 [128] Thorbjørnsen, S. 358, 363 [88,89]; 1449, 1457 [47]; 1819, 1820 [4a] Tišer, J. 1529, 1531, 1533, 1541, 1543, 1545 [36,45,46]; 1546 [55] Todorˇcević, S. 1019, 1069 [130]; 1745, 1748, 1802, 1815 [303–305] Tokarev, E.V. 527, 531 [116] Tolias, A. 1065, 1066 [12]; 1255, 1295 [3] Tomas, P. 215, 231 [45] Tomczak-Jaegermann, N. 43, 45–47, 51, 53, 60, 64, 66, 83 [20]; 84 [36]; 266 [55]; 277, 279, 287, 315 [57,72,73,78]; 316 [96]; 358, 364 [117]; 365 [164]; 459, 464, 471, 473, 496 [186–188]; 710, 722, 726, 736, 743, 749–751, 756, 757, 765–767, 770, 774 [49,60]; 776 [116,121]; 777 [155,156]; 779 [187,190–195]; 793, 831 [33]; 835 [168]; 841, 850, 853, 854, 858, 864, 868 [17]; 869 [25]; 870 [60]; 930–934, 937 [67–69]; 1060, 1063, 1065, 1068 [97]; 1069 [105,131]; 1103, 1128 [17]; 1216, 1222–1226, 1230–1234, 1236, 1240, 1242–1244, 1245 [11,19,30–35]; 1246 [40,54–56]; 1252, 1296 [34,35]; 1305, 1306, 1330 [14]; 1332 [65,66]; 1350–1358, 1359 [30]; 1360 [34,45,46,52,53]; 1392, 1423 [93]; 1477, 1480, 1481, 1483, 1484, 1512 [45,70]; 1516 [193–195]; 1735–1737, 1741 [68]; 1745, 1815 [306] Tonge, A. 47, 55, 60, 65, 66, 72, 83 [9]; 455, 458, 459, 464, 466, 471, 474, 491 [35]; 518, 529 [40]; 675, 681, 690, 705 [15]; 867, 868 [18]; 879, 881, 882, 887, 897 [10]; 1222, 1244 [5]; 1378, 1387–1389, 1420 [22]; 1452, 1456 [24]; 1710, 1711, 1724, 1736, 1739 [16] Toninelli, F.L. 1819, 1820 [3] Töplitz, O. 1667, 1668 [32]

1846

Author Index

Topping, D. 1170, 1175 [89] Torrea, J.L. 250, 269 [131]; 505, 529 [53] Torunczyk, H. 799, 835 [169]; 1797, 1799, 1811 [159]; 1815 [307] Tracy, C.A. 349, 358, 366 [170,171] Tran Van An 1112, 1127 [5] Trautman, D.A. 519, 526, 527, 528 [28,29]; 529 [49] Triana, M.A. 1116, 1117, 1124, 1128 [16] Triebel, H. 463, 495 [138]; 958, 973 [4]; 1363, 1400, 1404, 1423 [94]; 1515 [142] Troitsky, V.G. 102, 121 [131,132]; 556, 558, 559 [39,40] Troyanski, S.L. 635, 669 [164]; 792, 805, 814, 821, 832 [83,84]; 834 [122,132,139]; 835 [170]; 1759, 1767, 1778, 1781–1785, 1787, 1788, 1790–1792, 1798, 1805, 1808 [86]; 1809 [120–122]; 1810 [124]; 1811 [184]; 1812 [209–214]; 1815 [308–314] Tsarpalias, A. 1019, 1050, 1058, 1064, 1065, 1066 [11]; 1755, 1812 [223] Tsay, J. 348, 360 [17] Tsirelson, B.S. 136, 140, 159 [99]; 276, 316 [97]; 799, 812, 835 [171]; 1060, 1069 [132]; 1250, 1255, 1297 [54]; 1335, 1342, 1343, 1360 [54]; 1649, 1669 [54] Tsolomitis, A. 722, 774 [70] Turett, B. 505, 528 [31] Turpin, P. 1101, 1113, 1119, 1120, 1122–1124, 1129 [85,86]; 1130 [87] Tzafriri, L. 7, 10–14, 18, 21–27, 30, 33, 50, 51, 78–80, 83 [14,15]; 84 [41]; 87, 92, 120 [86,87]; 121 [133]; 125, 133–135, 140, 143, 146, 147, 158 [50,69,70]; 159 [71,72,100]; 257, 258, 268 [106,107]; 276, 279, 281, 282, 284, 289, 300, 301, 305–307, 309, 313 [12,15]; 315 [64–67]; 336, 337, 361 [42–44]; 505, 511, 514, 515, 518, 523, 529 [51]; 530 [70,94]; 531 [95–99]; 574, 577, 580, 596 [42,43]; 681, 705 [33]; 784, 834 [130]; 853–855, 859, 860, 862–864, 866, 867, 868 [7,9–13]; 869 [25,29]; 870 [61]; 906, 928, 936 [46]; 937 [77]; 965, 974 [23]; 1116, 1119, 1129 [64]; 1130 [88]; 1151, 1155, 1167, 1174 [72]; 1245 [21]; 1252, 1256, 1260, 1263, 1264, 1285, 1296 [39]; 1304, 1305, 1331 [40]; 1336, 1359 [25]; 1380, 1383, 1387, 1420 [32]; 1421 [60]; 1480, 1481, 1514 [115]; 1639, 1641–1643, 1646, 1647, 1649–1652, 1654–1658, 1660–1664, 1667 [6]; 1668 [11,27,36–38]; 1708, 1711, 1714, 1715, 1724, 1727, 1740 [48–50]; 1745, 1748, 1757, 1759, 1764–1768, 1771, 1775, 1800, 1801, 1804, 1805, 1812 [197,198] Uglanov, A.V. 1528, 1546 [56]

Uhl, J.J. 35, 36, 38–40, 83 [8]; 106, 107, 119 [51]; 259, 260, 263, 266 [56]; 479, 480, 483, 484, 491 [37]; 558 [14]; 793, 831 [50]; 1396, 1420 [23] Uriz, Z. 1103, 1127 [9]

Vaaler, J.D. 175, 194 [38] Valdivia, M. 1755, 1763, 1764, 1766, 1767, 1771, 1784, 1788, 1790, 1792, 1800, 1801, 1803, 1812 [211–213]; 1813 [234,235]; 1815 [315–321] Valette, A. 1450, 1458 [100] van Dulst, D. 515, 529 [50] van Mill, J. 603, 669 [165] van Rooij, A.C.M. 91, 121 [118,119] Vanderwerff, J. 434 [36]; 828, 835 [172]; 1749, 1766, 1767, 1771, 1797, 1800, 1806 [27]; 1815 [322–324] Varadhan, S.R.S. 1188, 1200 [48] Varga, R. 555, 559 [41] Vargas, A. 222, 231 [44] Varopoulos, N.Th. 1363, 1423 [95] Vašák, L. 1764, 1770, 1815 [325] Vaserstein, L. 916, 917, 938 [93] Vasin, A.V. 905, 939 [124] Veech, W.A. 1454, 1458 [101]; 1716, 1741 [69] Vega, L. 222, 231 [44] Veksler, A.I. 87, 89, 119 [39,40] Venakides, S. 349, 362 [70] Venni, A. 250, 267 [60–62] Vera, G. 1807 [33] Vershynin, R. 1784, 1815 [326] Vesely, L. 650, 655–657, 667 [71]; 669 [166] Vilenkin, N.Ya. 919, 936 [32] Villa, R. 716, 772 [11]; 1607, 1632 [4] Villadsen, J. 627, 669 [167] Villani, C. 350, 364 [129] Vincent-Smith, G.F. 612, 669 [168] Virasoro, M.A. 354, 364 [123] Vlasov, L.P. 663, 670 [169]; 1823, 1823 [1a] Vodop’yanov, S.K. 1372, 1420 [35] Voiculescu, D. 323, 326, 330–332, 345, 347, 356–360, 360 [9,10]; 366 [172–178]; 1445, 1451, 1458 [102,103]; 1466, 1487, 1516 [196] von Koch, H. 451, 493 [87] von Neumann, J. 439, 447, 484, 494 [115,116]; 495 [169]; 670 [170]; 906, 936 [47] von Weizsäcker, H. 1527, 1545 [50,51] Vu, V.H. 346, 363 [105]; 1819, 1819 [1a] Vukotić, D. 1682, 1702 [51] Vulikh, B.Z. 87, 88, 91, 92, 106, 107, 120 [75,76]; 121 [134–136] Vuza, D.T. 97, 98, 121 [110,111]

Author Index Wachter, K.W. 343, 353, 366 [179] Wade, W.R. 576, 597 [63] Waelbroeck, L. 1113, 1123, 1130 [89,90] Wage, M. 1751, 1752, 1754–1756, 1780, 1806 [21] Wagner, G. 769, 779 [196]; 925, 939 [125] Wagner, R. 1060, 1065, 1069 [105]; 1352, 1356, 1360 [35,46,55] Walsh, B. 95, 121 [137] Wang, G. 215, 230 [1,2]; 259, 265 [9,10]; 269 [137] Wang, J.-K. 702, 706 [48] Wang, X. 1521, 1544 [11] Wassermann, S. 332, 359, 366 [180,181]; 1431, 1449, 1452, 1458 [104,105] Wattbled, F. 1471, 1516 [197] Webster, C. 1438, 1456 [42] Wegmann, R. 358, 366 [182] Weil, W. 906, 911, 936 [35]; 938 [120]; 939 [126] Weinberger, W.F. 339, 362 [68] Weis, L. 445, 446, 496 [189]; 860, 870 [62]; 1589, 1602 [68] Weiss, B. 619, 620, 667 [76] Weiss, G. 336, 361 [30]; 505, 532 [135]; 859, 868 [3]; 1133, 1158, 1159, 1161, 1170, 1171, 1172 [20]; 1173 [37,49]; 1175 [101,109,110]; 1363, 1371, 1399, 1403, 1406, 1420 [18]; 1422 [92]; 1682, 1691–1693, 1701 [12] Wells, J.H. 906, 939 [127] Wenzel, J. 250, 269 [122,138,139]; 471, 476, 495 [139]; 590, 597 [59] Wermer, J. 673–675, 704 [1]; 706 [50] Werner, D. 143, 158 [60]; 821, 829, 833 [93,114]; 1046, 1067 [65]; 1357, 1359 [21]; 1745, 1810 [142] Werner, E. 798, 835 [158] Werner, J. 116, 118 [24] Werner, W. 829, 833 [93]; 1046, 1067 [65]; 1745, 1810 [142] West, G. 1496, 1497, 1499, 1514 [124–126] Weston, A. 1127, 1130 [91] Weyl, H. 326, 366 [183]; 439, 496 [190]; 948, 974 [34] Wheeler, R.F. 636, 666 [49]; 1745, 1789, 1802, 1808 [67] White, M.C. 968, 974 [35] Whitfield, J.H.M. 413, 434 [37]; 792, 798, 812, 832 [63,83,84]; 834 [125]; 835 [173]; 1766, 1767, 1784, 1785, 1791, 1795, 1797, 1798, 1800, 1803, 1808 [87,88]; 1809 [118,120–122]; 1812 [189]; 1813 [239]; 1815 [324] Whitney, H. 484, 496 [191]; 1373, 1423 [96] Wickstead, A.W. 91, 93, 95, 96, 105, 118 [13,18–21]; 121 [138]; 122 [139–142] Widder, D.V. 605, 670 [171] Widom, H. 349, 358, 366 [170,171] Wiegerinck, J. 1700, 1702 [50]

1847

Wielandt, H.W. 328, 363 [96] Wiener, N. 1495, 1497, 1516 [198] Wigner, E.P. 342, 358, 366 [184,185]; 1221, 1246 [57] Williams, D. 376, 391 [45] Williams, L.R. 906, 939 [127] Williamson, J.H. 1113, 1130 [92] Willinger, W. 374, 390 [8] Willis, G.A. 286, 309, 314 [34]; 316 [98] Wils, I.M. 334, 366 [186] Winkler, S. 1438, 1456 [43] Wittstock, G. 1428, 1458 [106] Wo-Sang Young 576, 597 [67] Wodzicki, M. 1171, 1173 [37] Wojciechowski, M. 1366, 1367, 1377, 1380, 1381, 1383, 1387, 1392, 1400–1403, 1409, 1411–1413, 1417–1419, 1420 [6]; 1422 [76–81]; 1423 [97–100] Wojtaszczyk, P. 7, 13, 16, 58, 66, 83 [21]; 84 [42]; 146, 149, 157 [34]; 159 [82]; 274, 279, 311, 313 [16]; 315 [79,82]; 459, 471, 477, 496 [192]; 518, 532 [136]; 566, 567, 569, 570, 573, 575, 577, 578, 580, 596 [34]; 597 [68–74]; 674, 675, 686, 687, 689, 690, 702, 703, 706 [51]; 841, 870 [63]; 905, 936 [30]; 1119, 1129 [53]; 1130 [93]; 1389, 1396, 1423 [101]; 1600, 1602 [49]; 1644, 1645, 1664–1667, 1667 [1]; 1668 [19,21,30]; 1669 [55,56]; 1677, 1682, 1691, 1692, 1696, 1697, 1701 [15,23]; 1702 [52–54]; 1745, 1800, 1816 [327] Wolenski, P. 412, 434 [16] Wolfe, J.E. 1599, 1602 [69]; 1714, 1741 [70–72] Wolff, T. 221, 223, 231 [46]; 232 [47]; 1407, 1423 [102] Wolfson, H. 767, 771, 772, 773 [40]; 777 [153,154] Wolniewicz, T. 704, 706 [52]; 1681, 1688, 1694, 1702 [49,55] Wolnik, B. 573, 596 [28] Wong, Y.C. 87, 122 [143] Wood, G.V. 278, 314 [55] Woodroofe, M. 1192, 1199 [33]; 1629, 1634 [37] Wo´zniakowski, K. 577, 578, 597 [69]; 1371, 1423 [103] Wulbert, D. 626, 668 [123]; 1600, 1602 [43] Xu, Q. 477, 495 [150]; 700, 705 [32]; 879, 898 [37]; 1133, 1174 [67,68]; 1175 [98]; 1405, 1406, 1421 [52]; 1438, 1447, 1457 [54]; 1458 [107]; 1465, 1468, 1472, 1483–1485, 1487, 1491–1493, 1497, 1503, 1507–1510, 1513 [99–102]; 1515 [156,157,169]; 1516 [188,199–203]; 1517 [204–206] Yahdi, M. 828, 835 [174] Yan, J.A. 380, 391 [52]

1848

Author Index

Yeadon, F.J. 1517 [207] Yin, Y.Q. 344, 353, 358, 360 [18,19]; 361 [20]; 366 [187] Yood, B. 558 [12]; 559 [42] Yor, M. 372, 373, 378, 379, 391 [43] Yosida, K. 443, 496 [193] Yost, D. 793, 830 [13]; 835 [175]; 1745, 1757, 1759, 1765, 1789, 1799, 1803, 1807 [35]; 1813 [248]; 1816 [328,329] Yurinskii, V.V. 1182, 1200 [49,50]; 1612, 1634 [64] Zaanen, A.C. 87, 89, 99, 106, 110, 120 [93,94]; 122 [144,145] Zabre˘ıko, P.P. 87, 99, 120 [79]; 122 [146] Zachariades, Th. 139, 157 [12] Zahariuta, V.P. 1667, 1669 [57] Zahorski, Z. 1539, 1546 [57] Zaidenberg, M.G. 901, 939 [130] Zajíˇcek, L. 435 [63]; 798, 828, 832 [64]; 833 [97]; 834 [151]; 1532, 1533, 1536, 1545 [47]; 1546 [58,59]; 1752, 1758, 1786, 1808 [89]; 1810 [149] Zalgaller, V.A. 710, 712, 726, 774 [43] Zanco, C. 662, 667 [70] Zastavnyi, V.P. 909, 939 [128,129] Zeitouni, O. 346, 363 [87] Zelazko, W. 1125, 1130 [94,95] Zelený, M. 1530, 1546 [60] Zemanek, J. 959, 974 [25] Zhang, G. 177, 194 [39]; 918, 919, 939 [131–133] Zhao, D. 1019, 1068 [83] Zhong, Y. 101, 119 [48] Zhou, X. 349, 362 [70] Zhu, K. 471, 494 [107]; 1677, 1682, 1683, 1701 [31]; 1702 [56]

Zhu, Q. 418, 433 [9] Ziemer, W.P. 1367, 1423 [104]; 1526, 1546 [61] Zimmermann, F. 250, 269 [140] Zinger, A.A. 905, 939 [134] Zinn, J. 52, 84 [34]; 129, 158 [55]; 176, 194 [36]; 1181, 1182, 1185, 1186, 1190–1192, 1194, 1195, 1198 [3,8]; 1199 [9–11,23,33]; 1200 [42,51]; 1610, 1624, 1629, 1633 [29]; 1634 [37,54,55] Zinnmeister, M. 1019, 1069 [133] Zippin, M. 19, 84 [43]; 131, 134, 140, 141, 146, 149, 158 [54,56,57]; 159 [101]; 274, 277, 279, 290, 293, 294, 296, 300, 314 [50]; 316 [99]; 581, 596 [33]; 829, 833 [108]; 1242, 1245 [17]; 1454, 1458 [108]; 1596, 1598–1600, 1601 [34–36]; 1602 [70,71]; 1640, 1641, 1650, 1668 [39]; 1669 [58]; 1707, 1714, 1717, 1721, 1726, 1731–1733, 1740 [28,30–33]; 1741 [73–77] Zizler, V. 6, 7, 13, 20, 33, 34, 36, 41, 42, 57, 63, 83 [6,11,12]; 84 [44]; 406, 408, 409, 413, 415, 434 [24,25,37]; 476, 491 [33]; 644, 666 [41]; 783, 789, 792, 793, 798, 799, 805, 812–814, 820–822, 828, 831 [45,46]; 832 [60–65,83,84]; 833 [90,92,102]; 835 [173,176]; 1537, 1544 [18]; 1600, 1602 [72]; 1745, 1747–1769, 1771–1792, 1795–1802, 1804, 1805, 1807 [55–58]; 1808 [75–85,88–91]; 1809 [92,118,120–122]; 1810 [129,140]; 1811 [159–164]; 1813 [239]; 1815 [324]; 1816 [330–332] Zlatov, P. 1107, 1128 [30] Zobin, N. 1373, 1423 [105,106] Zolotarev, V.M. 907, 939 [135] Zsidó, L. 1463, 1498, 1516 [185]; 1517 [208] Zvavitch, A. 840, 845, 869 [48]; 870 [56,64] Zygmund, A. 214, 232 [48]; 245, 266 [45,46]; 269 [141]; 522, 531 [105]; 678, 706 [53]; 875, 898 [38]; 1175 [111]; 1370, 1412, 1423 [107]

Subject Index approximation property (AP), 12, 97, 275, 488, 1478 – bounded (BAP), 12, 274 – bounded compact, 308 – commuting bounded (CBAP), 12, 291 – commuting compact, 310 – commuting metric, 291 – commuting unconditional metric, 295 – compact, 308 – metric (MAP), 12, 287, 488 – metric uniform, 307 – positive, 286 – stochastic, 1811 – unconditional, 291, 295 – uniform (UAP), 60, 305 – uniform projection (UPAP), 305 arbitrarily distortable, 1063 arithmetic diameter, 891, 893 Asplund space, 410, 411, 795, 1140, 1537, 1754 associate space, 512 asymptotic p space, 1060, 1063 asymptotic c0 space, 1060, 1063 asymptotic freeness of matrices, 356, 359 asymptotic models, 1358 asymptotic order, 320 asymptotic set, 1250 asymptotic structure, 1352 atomic, 1106 atomic space, 1103 Auerbach lemma, 45 automorphism, 3 Azuma inequality, 1317, 1610

α-th oscillation of f , 1031 A-convex, 404 A-martingale, 402 Aδ -set, 404 w∗

A , 1746 Ap -condition, 1638 absolutely continuous vector measure, 39 abstract Lp space, 22 abstract M space, 22 adjoint – Banach ideal, 457 – ideal norm, 457 admissible class of perturbations, 395, 398, 399 admissible cones of perturbations, 401 Aldous theorem, 515 Alexandrov theorem, 420 Alexandrov–Fenchel inequalities, 727 algebra – C ∗ , 627 – Jordan–Banach, 627 allocation, 113 almost isometric embedding, 925 almost sure convergence, 343, 352, 353, 357, 360 Amir–Cambern theorem, 702 analytic continuation, 908, 921 analytic decomposition of unity, 699 analytic distribution, 921 analytic family, 1158 analytic function on Banach space, 675, 806 – m-homogeneous polynomials, 676 analytic map, 806 analytic operator valued functions, 1496 analytic Radon–Nikodým property (ARNP), 236, 262, 638, 1484 analytic subset, 1010 analytic UMD, 1495 Androulakis–Odell lemma, 1052 angelic compact, 1748 anisotropic Sobolev space, 1408 approximate identity, 882 approximate ultrafilters, 1086, 1087 approximating sequence, 288, 292, 296, 297 approximation numbers, 452, 945

β-differentiable, 406 þ Bp,q (Rn ), 1400 (B(X, Y ), τ ), 281, 282 BV(k) (Ω), 1366 B2 -sequences, 917 Baire space, 1786 Baire’s space Σ := NN , 1760 balayage, 401, 610 Banach couple, 74, 1133, 1141, 1142 1849

1850 Banach function spaces, 1141 Banach ideal, 448 – adjoint, 457 Banach lattice, 21, 60, 89, 125, 1696 – λ-lattice injective ((λ-L)-injective), 1715 – p-concave, 27, 504, 855, 964 – p-concavity constant, 27 – p-convex, 27, 504, 841, 855, 964 – p-convexity constant, 27 – absolute value, 21 – discrete, 1715 – functional calculus, 26 – lattice injective (L-injective), 1715 – order complete, 23 – order continuous, 23, 1716 – order continuous, functional representation, 24 – symmetric, 21, 81 Banach space, 1062, 1063 – B-convex, 52, 474, 894 – C(K), 19, 1547 – Cσ (K), 1600 – c1 (Σ × Γ ), 1770 – D, 1756 – JL2 of Johnson and Lindenstrauss, 1757 – K convexity constant, 53 – K-convex, 53, 1320 – L∞ (μ), 14 – Lp (μ), 1 p < ∞, 13 – Lp (μ, X), 1 p < ∞, 37 – cotype q, 48, 1307, 1315 – genus, 1650 – Lorentz space, 21 – Orlicz space, 21 – predual of L1 (μ), 1599 – predual of C(K) and L1 (μ), 20, 625, 657 – quasi, 1665 – reflexive, 10, 31, 443 – Schlumprecht’s space S, 1335, 1344 – separable conjugate, 10, 38, 635, 639 – smooth, 30 – Sobolev space, 585 – stable, 515, 519 – strictly convex, 30 – superreflexive, 33, 56, 235, 237, 479 – Tsirelson’s space, 1335 – Tsirelson’s space T , 1342 – type p, 48, 845, 1306, 1315 – uniformly convex, 31 – uniformly smooth, 31 – weakly sequentially complete, 4 – with enough symmetries, 74 Banach–Dieudonné theorem, 381, 1751 Banach–Mazur compactum, 765

Subject Index Banach–Mazur distance, 3, 43, 858, 862, 924, 1626, 1627 Banach–Saks property, 444, 1581 Bang lemma, 185, 188, 192 barrier, 638 – PSH, 638 – strong, 638 barycenter, 604 barycentric calculus, 1048 basic sequence, 7, 1104 basis, 7, 274, 287, 585 – K-equivalence, 8 – asymptotically non-equivalent, 1656 – bimonotone, 7 – boundedly complete, 10, 275, 285 – conditional, 1639 – constant, 7 – equivalence, 8 – Faber–Schauder, 9, 564 – Haar, 9, 125, 250 – Markuschevich, 13 – monotone, 7, 255, 256, 277 – perfectly homogeneous, 1650 – problem, 273 – quasi-equivalent, 1667 – shrinking, 10, 278, 294, 299 – subsymmetric, 11, 1650 – symmetric, 11, 837, 854–858, 1637 – unconditional, 9, 126, 251, 274, 301, 855, 858, 1637 – unconditionally monotone, 9 – universal, 11 basis constant, 1675 basis constant of n-dimensional space, 1212 – asymptotically sharp estimate, 1212 Beck’s convexity, 1181 Berg technique, 331 Bernoulli selectors, 338 Bernstein theorem, 606 Berry–Esséen theorem, 934 Besicovitch set, 198, 220 Besov space, 575, 585, 971, 1400 Bessaga–Pełczyński theorem, 648 Bessel inequality, 950, 951 Bessel process, 378, 384 Beurling–Ahlfors transform, 236, 259 biorthogonal functionals, 7 biorthogonal system, 1765 biquasitriangular operators, 332 Bishop–Phelps theorem, 34, 385, 395, 396, 641, 1750 Black–Scholes model, 371 Blaschke–Santaló inequality, 728 block basis, 7, 131, 134

Subject Index BMO, 389, 591, 695 Bochner integrable functions, 36, 969 Bochner integral, 36, 402 Bochner theorem, 906, 911 Bochner–Riesz multiplier, 222 Borell lemma, 717 Borwein–Preiss theorem, 396 boundary, 639 – Choquet, 640 – minimal, 640, 656 – of the spectrum, 1308 boundedly complete finite dimensional decompositions, 299 Bourgain algebra, 687 Bourgain projections, 699 Bourgain theorem, 687, 689 Bourgain–Stegall minimization principle, 414 Boyd index, 257, 258, 514, 527, 1151, 1155, 1663 Brascamp–Lieb inequality, 164 Brenier map, 713 Bröndsted and Rockafellar theorem, 395 Brown–Douglas–Fillmore theorem, 320, 323, 331 Brownian motion, 369 Brunn–Minkowski inequality, 178, 711, 1606, 1608 Brunn–Minkowski theorem, 921 bump function, 400, 794, 1746 Busemann–Petty problem, 901, 918, 919 C (k) (Ω), 1364 (k) C0 (Ω), 1364 C0S (Ω), 1408 C1 , 285 C2 , 295, 301 Cp , 279, 280 conv K, 1746 C[0, ω1 ], 1756 C S (Ω), 1408 C 1 function, 37 C n function, 37 C ∞ function, 37 C(Tn )-module, 1393 (c)-sequence, 1023 c0 -index theorem, 1039 c0 -theorem, 1028 c.b. multilinear maps, 1442 C.C.C. (countable chain condition), 1774 Calderón couple, 80, 1133, 1144, 1150–1153 Calderón–Mitjagin theorem, 1150 Calderón–Zygmund singular integral operator, 245, 894, 1405 Calkin algebra, 323, 334 Calkin theorem, 447

1851

Cameron–Martin–Girsanov theorem, 373 canonical embedding, 4, 21, 1367, 1409 Cantor–Bendixson index, 1062 Carathéodory theorem, 602, 913–915, 927 CBAP, 1440 ˇ Cech complete ball, 1802 central limit theorem, 850, 1188, 1195 change of density, 1630 characteristic function, 6 Chebyshev’s inequality, 1611 Choquet, Bishop–de Leeuw theorem, 610 Choquet, integral representation, 607 Christensen–Sinclair factorization, 1442 Ciesielski–Pol CP space, 1758 Clarkson inequality, 479, 1479 closed – hull of an ideal, 441 – ideal, 441 coanalytic set, 1011 codomain, 440 coefficient converging, 1030 coefficient problem, 163, 183, 189 Cohen idempotent theorem, 874 column vectors, 1431 commodity space, 113 commutant, 539, 540 commutator, 557 commutator estimate, 1159, 1167 commuting projections, 1323 compact family of finite subsets of N, 1049 compact operator, see operator, compact compactification – Bohr, 645, 876 – Stone–Czech, 645 compatible couple, 1404 complementary function, 512 complemented subspaces, 4, 129, 143, 155, 837, 839, 863–868 complemented subspaces of H1 (D), 1698 complemented subspaces theorem, 928, 965, 967 complete contraction, 1429 complete isometry, 1429 completely bounded, 1427, 1428 completely isometric, 1429 completely isomorphic, 1429 complex convexity, 637 complex interpolation, 1158, 1160, 1407 complex interpolation method, 1138 complex interpolation spaces, 964, 1142 component of a vector in a lattice, 23 component of an operator ideal, 441 composition inequality, for p-summing operators, 65 composition operator, 470, 1683

1852

Subject Index

concentration, 744, 1177, 1603, 1605, 1606, 1621, 1622, 1629, 1630, 1632 concentration inequality, 737, 1186, 1606, 1614, 1617, 1623–1625, 1627, 1628, 1630 conditional expectation, 1610 cone generating, 87 constructivity, 330, 332, 358–360 continuum hypothesis, 1746 contraction principle, 848, 850 convergence in distribution, 343, 355 convergence in probability, 343, 352 convex block basis, 1024 convex body, 164, 169, 174, 646, 710 convex function, 1535, 1614, 1615 convex unconditionality, 1064 convolution inequalities, 163 corona problem, 674 Corson compact, 1754 coset ring, 874, 875 Cotlar’s trick, 1499 cotype, 48, 126, 286, 505, 526, 882, 892, 1177, 1188, 1307, 1315 – Gaussian, 472, 1710, 1738 – Haar, 477 countable tightness, 1748 coupon collector’s problem, 847 covering number, 756 critical point theory, 423 crossnorm, 485 – general, 486 – reasonable, 485 – uniform, 486 CSL algebra, 340 cubature formulas, 916 current, 1524 cylindrical measure, 475 Δ2 condition, 513 Δ02 condition, 513 D(Ω), 1364 Davie theorem, 673 Davis interpolation method, 1643 Day’s norm, 1783 decomposition method, 14, 125, 129, 135, 149, 151, 866, 1565, 1641 decomposition, p , 12 decomposition, monotone, 12 decoupling inequalities, 338 decreasing rearrangement, 1611, 1659 DENS space, 1760 density, 840 – change of, 837 – Lewis change of, 840, 849

– Maurey change of, 842 – Pisier change of, 843 density character or density (dens T ), 1746 density hypothesis, 225 dentable, 35, 397, 634 derivation, 340, 1012 descriptive, 1793 deviation inequality, 1618 diagonal argument, 1645 differentiability – almost Fréchet, 1543 – Fréchet (F), 37, 405, 788, 1537, 1541 – Gâteaux (G), 37, 408, 788, 1536, 1539 – metric, 1543 – of a convex function, 41, 409, 1535 – of a vector measure, 39 – of Lipschitz functions, 42, 1534 – weak∗ , 1543 differential games, 426 differential subordination, 253, 258 dilation, 610 dimension conjecture, 703 direct sum, 4 direct sum, infinite, 5 Dirichlet problem, 622 Dirichlet series, 198 discrepancy theory, 927 disk algebra, 673, 874, 879, 1667 distance maximal, 837, 859 distortion – λ-bounded, 1349 – λ-distortion, 1337 – arbitrary, 1337 – biorthogonal, 1344, 1348 – of p , 1335, 1348 – of Hilbert space, 1335 – problem, 1335 distribution function, 5 distributional partial derivative, 1364 domain – (ε, δ), 1372 – Lipschitz, 1373 – quasi-Euclidean, 1372 – with the segment property, 1365 domination problem, 94 Doob–Meyer theorem, 383 Doubling strategy, 376, 377 DP (= the Dunford–Pettis property), 1396 dual ideal, 441 duality of operator spaces, 1435 Duhamel integral formula, 218 Dunford theorem, 107 Dunford–Pettis property (DP), 61, 444, 687, 1056, 1579

Subject Index Dvoretzky theorem, 47, 475, 735, 740, 844, 915, 1338, 1606, 1625 Dvoretzky–Rogers factorization, 737, 1221 Dvoretzky–Rogers lemma, 47, 720 dyadic – martingale, 476 – representation, 453 dyadic tree, 0-separated, 56 ε-(c.c.) sequence, 1032 ε-entropy, 891 Eberlein compact, 1753 Effros–Borel structure, 1018 eigenfunctions, 213 eigenvalues, 213, 943 Ekeland theorem, 395, 396 ellipsoid of maximal volume, 46, 164, 719, 1626 energy function, 1619 entropy, 1194, 1620 entropy function, 1110, 1161, 1162 entropy numbers, 202, 946, 958 epigraph of f , 1794 equiangular lines, 917, 931 equiangular vectors, 931 equimeasurability theorem, 902, 903, 912 evolution case, 426 exactness, 1447 exchange economy, 113 expander graphs, 1632 expectation, 5 expectation, conditional, 6 exposed point, 35, 628, 790 extension, 1155, 1161 extension of isometries, 902 extension property (EP), 1705 – C(K) (C(K) EP), 1709, 1730, 1732 – λ (λ-EP), 1705, 1707 – λ-C(K) (λ-C(K) EP), 1709, 1729–1734 – λ-separable (λ-SEP), 1722, 1723 extension theorem, 902, 904 extrapolation principle, 15 extrapolation theorem, 1143 extremal vectors, 545, 546, 602 F (X) = F (X, X), 280 F (X, Y ), 280 ϕ-function, 514, 527 F. and M. Riesz theorem, 878 Fabes–Rivière criterion, 1412 face, 602 – closed, 614 factorization of operators, 96 factorization property, 443

1853

far-out convex combinations, 1025 Fatou norm, 89 filter, 55 filtration, 236, 253, 254 fine embeddings, 1627 finite decomposition, 837, 865–868 finite dimensional decomposition (FDD), 11, 140, 296, 1353, 1354 finite dimensional expansion of the identity, 300 finite geometries, 359 finite nuclear norm, 966 finite rank operator, 106, 441, 880 finite variation, measures of, 39 finitely representable, 136, 138, 1306 first Baire class, 1013 first order Hamilton–Jacobi equations, 426 first order smooth minimization principle, 406, 408 first order sub- and super-differentials, 415 fixed point property, 526 Fock space, 356 Föllmer–Schweizer decomposition, 389 Fourier transform, 481, 901 – restriction to surfaces, 198, 216 Fourier type, 1407 fractional Brownian motion, 383 fractional derivative, 920 fragmented, 1803 Fréchet-differentiable norm, 408, 418 Fredholm operator, 1264 Fredholm resolvent, 441 free central limit theorem, 355 free Poisson distribution, 353 free probability, 354–359 freeness, 355 function – affine, 607 – affine continuous, 612 – almost periodic, 644 – completely monotonic, 605 – concave, 251, 608 – convex, 608 – first Baire class, 611 – Haar, 476 – infinitely divisible, 606 – Lipschitz, 1338 – plurisubharmonic, 397, 637 – positive definite, 606 – Rademacher, 460 – stabilizes, 1337, 1338 – upper envelope of, 608 – upper semicontinuous, 608 – upper semicontinuous envelope of, 612 function spaces on compact smooth manifolds, 583–588

1854

Subject Index

– decomposition of function spaces, 587 – decomposition of the manifold, 586 – spaces on subsets, 585 – spaces with boundary conditions, 586 function, biconcave, 236, 251 function, biconvex, 235, 237 function, maximal, 237 functional – w ∗ -support, 628 – support, 628, 641 functions – stabilization principles, 1338 fundamental biorthogonal system, 1804 fundamental lemma, 1145 fundamental polynomial, 1409 fundamental theorem of asset pricing, 374, 379 γ1 , 858 γ2 , 841 G-viscosity superdifferential, 419 Gagliardo complete, 1134, 1149, 1159 games in Banach spaces, 1090–1093 Garsia conjecture, 209 Gâteaux differentiability space, 1537 Gâteaux differentiable norm, 408, 788 Gauss measure, 475, 905, 1623, 1624 Gaussian correlation problem, 179 Gaussian isoperimetric inequality, 1185 Gaussian processes, 338, 351, 848, 1624, 1625, 1630, 1631 Gaussian variables, 5, 16, 68, 839, 850, 853 Gelfand numbers, 454, 945, 954 general equilibrium, 113 general perturbed minimization principle, 397, 398 generalized Hankel operator, 681 generating cone, 87 GL (= Gordon–Lewis property), 1222, 1388 gl constant, 837, 858 Glicksberg problem, 680 Gordon–Lewis local unconditional structure (G-L l.u.st.), 59, 278–280, 302, 466, 680, 834, 872 Gorelik principle, 826 Gromov–Hausdorff distance, 1144 Grothendieck constant KG , 67, 467, 860, 1639 Grothendieck inequality, 67, 190, 842, 860, 1639 Grothendieck space, 505 Grothendieck theorem, 338, 467, 688, 872, 1748 Grothendieck–Maurey theorem, 1389 Grothendieck-type factorization theorem, 1488 H p -convex, 1483 h-elliptic polynomial, 1411

Haagerup tensor product, 1440 Haagerup’s approximation theorem, 1462, 1476 Haar – function, 476 – polynomial, 476 Haar measure, 1605, 1608, 1609, 1624–1626 Haar null set, 42, 1531 Haar system, 1664, 1666 Hadamard lacunary sequence, 875, 883, 886 Hahn–Banach extension property, 1104 Hahn–Banach theorem, 1616 Hamilton–Jacobi equation, 419 Hamming cube, 1612, 1625 Hamming distance, 1616 Hamming metric, 1613, 1617, 1618 Hankel operators, 1500 Hardy inequality, 500, 887, 894, 956 Hardy operators, 1154 Hardy space, 874, 1666 – dyadic, 1666 harmonic measures, 1625 Hausdorff dimension, 220 Hausdorff metric, 924, 1609 Hausdorff–Young inequality, 77, 504, 970 Hedging problem, 116 Henkin measure, 1394 hereditarily indecomposable Banach spaces, 1062, 1089, 1094, 1263, 1759 hereditary family, 1049 Hilbert space, 1189 – characterization of, 965 Hilbert transform, 235, 244, 481, 1169, 1638 Hilbert–Schmidt norm, 328, 1608 Hille–Tamarkin kernel, 469 Hindman’s theorem, 1082 Hoeffding inequality, 519 Hölder continuous, 425 holomorphic semi-group, 1324 homogeneous function, 910 homogeneous polynomials, 915 Hörmander–Mikhlin criterion, 1370 hyper-reflexive, 339 hypercontractivity, 50, 350 hyperfinite II1 factor, 1465, 1476, 1481, 1506, 1510 hyperplane C0 [0, ω1 ], 1756 hyperplane problem, 722, 924 ideal, 21, 90, 440 – p-Banach, 448 – Banach, 448 – closed, 441 – dual, 441

Subject Index – idempotent, 444 – injective, 446 – quasi-Banach, 448 – regular, 446 – Schatten–von Neumann, 447 – sequence, 446 – surjective, 446 – symmetric, 441 – ultrapower-stable, 455 ideal p-norm, 448 ideal norm, 448 – adjoint, 457 – non-normalized, 448 ideal property, 61 ideal quasi-norm, 448 ideals of operators on Hilbert space, 66 idempotent ideal, 444 idempotent measure, 875 indecomposable space, 1263 independent, 5 indicator function, 5 induction, 1614, 1615, 1617 inevitable set, 1250, 1260 information theory, 350 injection, 444 injective, 18, 58, 285, 1452 – hull of an ideal, 445 – ideal, 446 – tensor norm, 486 injective tensor product, 882 injective, separably, 18 insurance, 115 integral operator, see operator, integral intermediate space, 75 interpolation method, 1134 interpolation of non-commutative Lp -spaces, 1466, 1471 interpolation pair, 75 interpolation space, 502, 511, 526, 527, 1280 interpolation, K-method, 78 interpolation, complex method, 76 intersection body, 918, 919 invariant mean, 872, 876, 877 invariant subspace, 98, 533 invariant under spreading, 1311 inverse Blaschke–Santaló inequality, 759 inverse limit, 624 isometric embedding, 906 – into Lp , 524, 901 – into lp , 911 isometry, 3, 515, 526 isomorphic classification of non-commutative Lp -spaces, 1506–1510 isomorphic symmetrization, 759

1855

isomorphism, 3 isoperimetric inequality, 163, 173, 346, 715, 1605, 1607–1609 – approximate, 1605–1608, 1625 – Levy’s, 1608 isoperimetric problem, 1613, 1614 isotropic – constant, 723 – measure, 722 – position, 722 – vectors, 911 J -functional, 1135 James theorem, 34, 385, 643 James tree space JT , 1758 James’ weak compactness theorem, 1747 Jayne–Rogers selector J , 1788 Jensen theorems, 421 JET = Jones Extension Theorem, 1372 John position, 718 John representation of the identity, 721 John theorem, 46, 169, 170, 718 Johnson–Lindenstrauss subspace JL0 , 1757 joint densities of eigenvalues, 344, 349 Josefson–Nissenzweig theorem, 1749 K-closed couple, 1404 K-convexity, 53, 483, 845, 894, 1320 k-cube, 1657 K-divisibility theorem, 80 K-functional, 78, 502, 523, 1135, 1404 k-intersection body, 923 Kσ δ subset, 1778 K-divisibility, 1149 K-divisibility principle, 1145 K-monotone interpolation space, 1145, 1149 Kadets distance, 1143 Kadets–Klee norm, 1782 Kadets–Klee property, 515, 527, 1782 Kadison–Singer problem, 333, 859 Kahane–Khintchine inequality, 50, 472, 1307 Kakeya maximal function, 221 Kakutani representation theorem, 22 Kashin decomposition, 359, 360 Kato theorem, 1324 KB-space, 89 Khintchine inequality, 16, 26, 460, 472, 519, 717, 850, 934, 1486, 1654 Kislyakov theorem, 679, 691 Knöthe map, 712 Kolmogorov number, 454 Kolmogorov rearrangement problem, 208 Krein–Milman property (KMP), 633

1856

Subject Index

Krein–Milman theorem, 602, 928, 1111, 1713 Krein–Rutman theorem, 99 Krein–Šmulian theorem, 87, 381 Krivine theorem, 48, 1310, 1339, 1353 Kunen’s C(K) space, 1758 Kwapień–Schütt inequality, 855 Ky Fan norms, 328 Λϕ,w (I ), 527 λ-equivalent, 1037 Λp -set, 197, 854, 872 Λ(p)cb -set, 1504 L2 (p ), 523 Lϕ , 511, 527 Lϕ (0, 1), 518 p L(k) (Ω), 1364 Lp (Lq ), 523, 527 Lp,∞ , 500, 518 Lp,q , 500, 505, 523 Lp,q (Rn ), 1399 p LS (Ω), 1408 Lw,q , 524, 526, 527 p,∞ , 500, 505, 519 p,q , 500 ϕ , 518 w,q , 525 L-functions, 198 -position, 751 ξ ξ p , (resp. c0 ) spreading model, 1057 p -spaces – finite direct sums of, 1644, 1646 – infinite direct sums of, 1646 L∞ -space, 1598 L1 -space, 302 Lp,λ -space, 57, 129 Lp -space, 57, 126, 146, 279, 287 Laplacian on the torus, 213 large deviations for random matrices, 344, 346, 348 lattice norm, 89, 820 lattice of measurable functions, 694 – BMO-regular lattice, 696 lattice order, 613 lattice-convexity, 1118 left approximate identity, 286 Legendre polynomials, 926 length of a finite metric space, 1612 Leontief model, 113 Levi norm, 89 Lévy families, 744 Lévy processes, 370 Lewis lemma, 45 Liapunov theorem, 602 Lidski˘ı trace formula, 451, 463

lifting property, 17, 1454, 1708, 1726 limit order of an ideal, 469 Lindenstrauss Lifting Principle = LLP, 1382 linear extension operator, 1372 linear perturbation principle, 403 Liouville theorem, 905 Lipschitz constant, 1605, 1608, 1618, 1621–1623, 1626 Lipschitz function, 1539, 1605, 1614, 1620–1622, 1624 Lipschitz isomorphic Banach spaces, 826, 1539 Littlewood–Paley decomposition, 879 local basis property, 302, 303 local martingale, 378, 380 local reflexivity in operator spaces, 1451 local reflexivity principle, 53 local theory, 0, 43, 321, 455, 710 local unconditional structure, 1387, 1478 localization lemma, 1625 locally bounded, 375 locally depends on finitely many coordinates, 1794 locally uniformly convex norm, 784 locally uniformly lower semicontinuous, 418 locally uniformly rotund (LUR), 1782 logarithmic Sobolev inequality, 350, 1620, 1622, 1623 long James space, 411 Lorentz function space Lw,q (I ), 524 Lorentz sequence space, 519, 955, 957, 971, 1642 – non-locally convex, 1666 Lorentz space, 1399 Lorentz spaces, isometries of, 526 low M ∗ -estimate, 749 lower p-estimate, 504 lower q-estimate, 504, 514 lower semi-continuous, 1021 Lozanovskii factorization, 1162, 1163 Luxemburg norm, 511 M-admissible, 1061 M-admissible families, 1061 M-allowable, 1061 M-basis, 1765 M-ideal, 310, 1046 M-ellipsoid, 759 Mackey–Arens–Katˇetov theorem, 1749 majorizing measures, 181, 338, 1625 Marchenko–Pastur distribution, 353 Marcinkiewicz interpolation theorem, 502, 503, 505, 510, 1134 Marcinkiewicz set, 879, 880, 887 marketed space, 115 Markushevich basis, 1104, 1765

Subject Index martingale, 6, 235, 253, 380, 401, 476, 630, 1610, 1612, 1616, 1628 – analytic, 263 – difference sequence, 6, 236, 476 – dyadic, 242 – inequality, 852 – simple, 239 – square function, 256, 257 – tangent, 264 – transform, 235, 237, 262, 880 martingale difference, 1610 matrix splitting, 859 Matuszewska–Orlicz indices, 514 Maurey extension theorem, 1710, 1735–1738 Maurey–Khintchine inequalities, 510 Maurey–Nikishin–Rosenthal factorization theorem, 872, 883, 884 Maurey–Pisier theorem, 51 maximal quasi-Banach ideal, 456 maximum principle, 426 Mazur map, 1347 Mazur’s intersection property, 1804 mean value estimates, 1542 mean value theorems, 422 measurable function, 36 measure – Banach space valued, 39 – concentration of, 321, 338, 346, 349, 735 – determination by balls, 1529 – differentiable, 1526 – ergodic, 609, 615 – Gauss, 475, 905 – Haar, 617, 873, 933 – Hausdorff, 1522 – Jensen, 637 – maximal, 609, 613 – quasi-invariant, 1521 – space automorphism, 1658 – stable, 901 – T -invariant, 615 – total variation of, 39 – unique maximal probability, 613 – Wiener, 476 measure preserving involution, 1609 Mergelyan theorem, 673 metric (see also approximation property) – entropy, 331, 338 – injection, 444 – π -property, 295, 296, 300 – surjection, 445 metric probability spaces, 1603 metric space – finite, 1612 Michael selection theorem, 1731

1857

Milne theorem, 676 Milutin lemma, 1552 Milutin theorem, 702, 1551 minimal and maximal structures, 1434 minimal extension, 1107, 1156, 1171 minimal mean width position, 725 minimal surface position, 724 minimal tensor product, 1432 Minkowski box theorem, 174 Minkowski compactum Mn , 1208 – diameter of, 1208 Minkowski functional, 918 Minkowski sum, 711, 844, 924 Mityagin–Pełczyński theorem, 688 mixed discriminant, 731 mixed homogeneity, 1411 mixed volumes, 726 mixing invariant, 1216 modified Schlumprecht space, 1063 modular space, 858 modulus of – continuity, 971 – convexity, 31, 413, 1480, 1607, 1608 – convexity of power type p, 413 – operator, 90 – smoothness, 789, 1400 – uniform convexity, 785 moment method, 344 monotone, 1618 Montgomery conjectures, 223 mountain pass theorem, 424 Muckenhaupt condition, 389 multi-index, 1364 multiplicity, 943 multiplier, 872, 1369 multiplier transform, 1369 N-function, 511 Nagasawa theorem, 702 Nagata–Smirnov theorem, 1793 Namioka property, 1792 Nash-Williams’s theorem, 1077–1080 near unconditionality, 1064 needlepoint, 1111 nest algebra, 339, 1496 Nikishin factorization theorem, 516, 1316 Nikolski˘ı inequality, 464 No Arbitrage (NA), 378, 384 No Free Lunch with Vanishing Risk (NFLVR), 379 non-commutative Λ(p)-sets, 1501, 1502 non-commutative Burkholder–Gundy inequalities, 1491 non-commutative Doob’s inequality, 1493

1858

Subject Index

non-commutative Grothendieck theorem, 1488 non-commutative Hilbert transform, 1498 non-commutative Kadec–Pełczyński dichotomy, 1509 non-commutative Khintchine inequalities, 1486 non-commutative martingale, 1490 non-normalized ideal norm, 448 non-smooth calculus, 414 non-trivial weak-Cauchy, 1022 nonlinear Schrödinger equation (NLS), 197 normal semifinite faithful trace, 1463 normal structure, 515 normed vector lattice, 89 nuclear, 0, 1449, 1455 – representation, 449, 461 ω (f ; ·), 1400 .p Lp (Ω), 1367 .|α|k C(Ω), 1367 .|α|k |α|k M(Ω), 1367 oscα f , 1031 OAP, 1440 operator, 3 – 1-integral, 457 – 1-nuclear, 45, 449 – α-nuclear, 488 – γ -summing, 475 – k-normal, 331 – (k, β)-mixing, 1213 – Lp -factorable, 465 – lp -singular, 445 – p-concave, 27 – p-concavity constant, 27 – p-convex, 26 – p-convexity constant, 27 – p-integral, 71, 462, 488 – p-integral norm, 72 – p-nuclear, 461 – p-summing, 63, 220, 459, 475, 677, 929, 950, 957, 959–961, 1639, 1736 – p-summing norm (πp (T )), 63 – (p, 2)-summing, 959 – (p, q)-summing, 459, 677, 950 – (q, p, X)-summing, 693 – ρ-summing, 475 – A-universal, 443 – absolute integral, 106 – absolutely summing, 63, 677 – almost commuting, 320 – almost integral, 106 – approximable, 441 – Banach–Saks, 1581

– band irreducible, 110 – biquasitriangular, 332 – block diagonal, 324, 330 – compact, 4, 19, 94, 281, 316, 442, 535, 538, 542, 658, 943, 957 – compact friendly, 103 – completely continuous, 442, 686 – composition, 470 – conditional expectation, 1648, 1662 – convolution, 973 – creation/annihilation, 356 – diagonal, 469 – dilation, 1663 – dominated, 93 – essentially normal, 323 – factoring through, 14 – factorization of, 14, 96 – finite nuclear, 966 – finite rank, 441 – fixing a space Z, 1580 – Fourier type p, 481 – Fredholm, 1264, 1645 – Fredholm operator, index of, 63 – Gaussian cotype q, 472 – Gaussian type p, 472 – Haar cotype q, 477 – Haar type p, 477 – Hilbert–Schmidt, 439, 470, 949, 950 – Hilbertian, 465 – Hille–Tamarkin, 969 – integral, 106, 457, 462, 475, 969 – lattice homomorphism, 21 – lattice isomorphism, 21 – lattice-factorable, 466 – lifting of, 17 – monotone, 1535 – nearly commuting, 320 – nearly dominated, 687 – nuclear, 45, 286, 449, 461, 881, 929, 959 – order bounded, 90 – Paley operator, 678 – positive, 21, 88 – power-compact, 943, 953, 959, 969 – quasi-p-nuclear, 878 – quasinilpotent, 90 – quasitriangular, 332 – Rademacher cotype q, 472 – Rademacher type p, 471 – regular, 90 – regular averaging, 1554 – related, 944 – Riesz, 943, 958, 959 – singular integral, 235, 244, 249 – strictly Lp -factorable, 465

Subject Index – strictly p-integral, 462, 677 – strictly cosingular, 445 – strictly singular, 62, 445, 1263, 1645 – super weakly compact, 479 – transitive, 549 – translation invariant, 875 – UMD, 476 – uniformly p-smooth, 478 – uniformly q-convex, 478 – uniformly convex, 478 – uniformly convexifiable, 479 – uniformly smooth, 478 – universal, 443, 483 – weakly compact, 4, 95, 442 – weakly singular, 970 operator algebras, 543 operator Hilbert space OH, 1445 operator ideal, 440 – p-Banach, 448 – Banach, 448 – quasi-Banach, 448 operator of type 2, 1191 operator space structures, 1429 operator spaces, 354, 357, 1427 operators commuting with translations, 873, 879, 880, 883 optimal control or differential games, 419 optimal control theory, 426 optimal portfolio, 115 optimal sequence associated to f , 1041 option, 369 order continuity, 89 order continuous norm, 89 order of the derivative, 1364 ordered vector space, 87 ordinal index, 139, 154 Orlicz class, 511 Orlicz function, 855–857, 1153, 1155 Orlicz norm, 512 Orlicz property, 465 Orlicz sequence space, 140, 512, 518, 522, 658, 855, 1168, 1642 – non-locally convex, 1666 Orlicz space, 511, 523, 527, 855, 910, 1153, 1664 Orlicz spaces, isometries of, 515 Orlicz–Lorentz space, 527 orthogonal matrix, 1656 orthonormal system, 201 oscillation index, 1035 p-atom, 1685 p-Banach space, 403 p-concavification, 30

1859

p-convex lattice, 504 p-convexification, 30 ϕ-function, 514, 527 p-integral operator, see operator, p-integral p-nuclear, see operator, p-nuclear p-stable random variable, 861, 1627–1629 p-stable random vector, 906 p-subadditive, 402 p-summing operator, see operator, p-summing (p, 2)-bounded, 126, 147, 149, 150 PSH p -martingales, 403 p (X) , 284 π1 , 858 πp , 840, 861 π -property, 295–301, 307 πλ -property, 295, 307, 310, 312 Palais–Smale around F , at altitude c, 423 Paley inequality, 678, 679, 894 Paley’s projection, 1682 parabolic Hamilton–Jacobi equations, 428 Pareto optimal allocation, 114 Parseval equality, 460, 921 path of complemented subspaces, 866 paving problem, 334 payoff operator, 116 Peetre’s theorem, 1382, 1383 Pełczyński property, 685 perfectly homogeneous, 134 periodic boundary conditions, 218 permutation group, 1612, 1619 Perron–Frobenius theorem, 98 perturbed minimization principle, 395, 397 Pettis integral, 518 Pietsch factorization theorem, 64, 459, 840, 877, 950 Pisier’s lemma, 1406 PL-convexity, 1483 plank problem, 182, 183 plurisubharmonic functions, 397, 637 plurisubharmonic perturbed minimization principle, 403 Poincaré inequality, 350, 1620–1623 point – w ∗ -exposed, 628, 640 – w ∗ -support, 628 – denting, 634 – exposed, 601, 628, 656 – extreme, 601, 602, 605, 640 – farthest, 663 – nearest, 662 – PSH-denting, 638 – smooth, 30, 640 – strongly exposed, 628 – support, 601, 628

1860 point of continuity property (PCP), 636 polar decomposition, 66, 945 Polish ball, 1802 Polish space, 1009, 1801 polydisk algebra, 874 polynomial map on a Banach space, 806 polytope, 650, 924 – α, 659 – β, 659 – ε-approximating tangent, 659 portfolio, 116 positive cone, 87 positive curvature, 923 positive definite distribution, 919, 923 positive definite function, 901, 906, 909, 911 predictable σ -algebra, 375 preference relation, 113 price space, 113 primary, 133, 865, 1594 prime space, 1116 principal ideal, 90 probabilistic method, 358 product of ideals, 444 product space, 1614, 1615, 1623 projection, 4 – contractive, 255, 256, 1323 – partial sum, 7 – partial sum for a decomposition, 11 – Rademacher, 52, 1321 – Riesz, 9 projection constant, 71, 902, 931, 933 – absolute, 928 – relative, 928, 965 projectional resolution of identity (PRI), 1758 projective, 1452 projective tensor norm, 485 projective tensor product, 285, 882 Prokhorov’s inequality, 1610 property (u), 1028, 1048 property T of Kazhdan, 332, 359, 619 property C, 1750 property P(norm topology, weak topology) (P( · , w)), 1791 proximal subgradients, 412 pseudocompact, 1747 Ptak’s theorem, 1052 pure state, 333 put option, 116 q (X) , 284 quadratic perturbations, 396 quantum limit, 214 quartercircle law, 343

Subject Index quasi-p-nuclear operator, 878 quasi-Banach ideal – maximal, 456 – ultraproduct-stable, 456 quasi-Banach space, 402, 1099 quasi-Cohen set, 875, 889, 890 quasi-convex bodies, 769 quasi-interior point, 104 quasi-linear map, 1155 quasi-Marcinkiewicz set, 879 quasi-minimal Banach spaces, 1094, 1095 quasi-norm, 402 quasicomplemented, 1770 quasidiagonality, 320, 324, 330 quasiidempotent measure, 875 quasireflexive space, 646 Quermassintegral, 727 quotient of ideals, 444 quotient of subspace theorem, 752 quotient space, 3 ρX (τ ), 31 r-summing operator, 1389 RA-hierarchy, 1053 Rademacher – cotype q, 472, 1307 – functions, 16, 125, 126, 460, 848, 850, 853, 934, 1641 – projection, 482, 845, 1321 – theorem, 42 – type p, 471, 1306 Rademacher projection, 1327 Rademacher series, 1155 Radon–Nikodým derivative, 1610 Radon–Nikodým property (RNP), 35, 38, 154, 236, 259, 260, 402, 405, 414, 483, 601, 629, 1140, 1141, 1535 Rainwater–Simons theorem, 1749 Ramanujan graphs, 338, 359 Ramsey’s theorem, 1073 random n-dimensional space, 1206 – random space Xn,m , 1206 – random space Yn,m , 1207 random matrices, 319, 321, 341, 969 – edge of the spectrum, 345, 348 – global regime, 345 – local regime, 345 – spectral gaps, 345, 358 random matrix ensembles, 343, 347 random orthogonal factorizations, 766 random variable – r-stable, 6, 17 – Gaussian, 5, 16, 68 – Gaussian, standard, 5

Subject Index – symmetric, 6 random walk on the free group, 358 rank-one operator, 106 rapidly increasing sequence of n1 s, 1259, 1270 re-iteration theorem, 1140 real interpolation, 502, 1404 real interpolation method, 1135 real variable Hardy spaces, 879 rearrangement invariant space, 21, 257, 1637 reasonable crossnorm, 485 reflexive algebra, 338 regular – hull of an ideal, 445 – ideal, 446 – norm, 91 relative Dixmier property, 335 repeated averages hierarchy, 1053 replicate, 372 representable Banach space, 1016, 1017 representable, λ-, 53 representable, finitely, 53 representation – 1-nuclear, 449 – p-nuclear, 461 – dyadic, 453 – finite, 441 – Schmidt, 446 representing matrix, 625 reproducible, 132 restricted invertibility, 837–839, 854, 859–862 restricted unconditionality, 1055 retraction continuous affine, 623 reverse – Brascamp–Lieb inequality, 171 – Hölder condition, 389 – isoperimetric inequality, 163, 169 – metric approximation property, 293 Ribe space, 1108 Ricci curvature, 1608 rich subspace of C(Q, E), 1396 Riemann ζ function, 345 Riesz projection, 1169, 1498, 1638 Riesz–Kantorovich formulas, 90 Riesz–Thorin interpolation theorem, 75, 1134 right inverse, 1382 risk-neutral, 373 Rochberg theorem, 703 Rodin–Semenov theorem, 519 Rosenthal 1 theorem, 18, 1079, 1080 Rosenthal compact, 1801 Rosenthal inequality, 128, 149, 521 Rosenthal property, 445 rough norm, 796, 1538 row vectors, 1431

1861

Ruan’s theorem, 1433 Runge theorem, 673 S # , 1408 S(Rn ), 1369 SS(X, Y ), 63 span K, 1746 span · K, 1746 ∗ span w K, 1746 σ -fragmented, 1803 ∼ - is isomorphic to, 1373 s-number ideals, 948 (s)-sequence, 1022 Saccone theorem, 686 scattered space, 1754 Schatten–von Neumann classes, 447, 895, 1446, 1465 Schauder bases, 7 – bases with vector coefficients, 588–590, 594, 595 – – X-basis constant, 589 – – equivalent X-bases, 594 – – unconditional X-basis constant, 589 – negative results, 580 – non-explicit existence results, 581–583 – unconditionality in Lp , 569–579 Schauder decomposition, 11, 304 Schauder–Tichonoff theorem, 603 Schlumprecht space, 1062, 1256 Schmidt representation, 446 Schoenberg problem, 901, 906 Schreier classes Sα , 1351 Schreier families, 1051 Schreier unconditional, 1055 Schrödinger group, 197 Schur multipliers, 1500 Schur multipliers on S p , 1503 Schur property, 9, 443 Schwartz class, 1369 second order Hamilton–Jacobi equation, 430, 433 second order smooth minimization principle, 406, 424 second order subdifferential, 419 second order superdifferential, 420 security, 115 selection continuous affine, 622 self-extension, 1157 semi-concave function, 420 semi-continuous functions, 1021 semi-martingale, 375, 376, 382 semicircular distribution, 343 separable complementation property, 311 separable perturbation of M, 1393 separably injective, 1454, 1709

1862 separating polynomial, 412 separation theorem, 601, 1011 sequence, 1022, 1028, 1030, 1046, 1054 – Sξ unconditional, 1055 – basic, 7 – boundedly convexly complete, 1059 – convexly unconditional, 1055 – nearly unconditional, 1055 – normalized, 7 – Schreier, 1585 – semi-boundedly complete, 1056 – seminormalized, 7 – series-bounded, 1056 – unconditionally basic, 10 – weakly Cauchy, 4 sequence ideal, 446 sequentially separable, 1758 set – 1-norming, 655 – Gδ , 608 – σ -directionally porous, 1533 – σ -porous, 1533 – Γ -null, 1534 – w-compact convex, 635 – antiproximinal, 664 – Aronszajn null, 1532 – Chebyshev, 662 – closed convex bounded (CCB), 601 – compact convex, 602 – compact convex metrizable, 612 – convex stable, 601, 660 – covered by δ-convex hypersurfaces, 1534 – dentable, 634 – Gauss null, 1532 – Haar null, 1531 – independent, 206 – interpolation, 644 – Korovkin, 640 – locally compact, 629 – norming, 646 – porous, 1533 – proximinal, 601, 662 – strongly antiproximinal, 664 – thin, 646, 647 – universally measurable, 631 – valued map, 622 – weak* compact convex, 611 – with dense extreme points, 661 short exact sequence, 1155 shrinking, 1765 Sidak Lemma, 176, 179 Sidelnikov inequality, 932 Sidon constant, 885 Sidon problems, 200

Subject Index Sidon sets, 205, 644, 872, 885–888, 891–893 sigma-martingale, 379, 380 simple function, 36 simple growth process, 112 simple predictable, 375 simplex, 613 – Bauer, 615, 616 – compact, 601, 613 – compact prime, 620 – finite-dimensional, 613 – Poulsen, 618, 619 simplexoid, 626 singular numbers, 446, 945 skipped block sequence, 141, 144 SLD map, 1791 Slepian lemma, 229, 849 Slepian–Gordon lemma, 350 slice, 35, 397, 634 slicing problem, see hyperplane problem small isomorphism, 156 small perturbations, principle, 8 Smirnov domain, 701 smooth minimization principle, 406, 417 smooth partitions of unity, 1797 smooth point, 30, 640 smooth variational principle, 1752 smoothness, 515 – n-dimensional, 1408 – anisotropic, 1408 – isotropic, 1408 – non-degenerate, 1412 – of Ascoli type, 1417 – of order k, 1412 – reducible, 1412 Šmulyan’s lemma, 1749 Sobolev embedding, 469, 1377, 1380, 1397 Sobolev projection, 1369, 1409 Sobolev space, 970, 1364 Sobolev spaces of measures, 1366 Sobolev–Besov imbedding theorems, 972 social endowment, 113 space – C(0, 1), 1642 – Hp (Tm ), 1666, 1667 – K-convex, 1320 – U1 , 1642 – Xp , 1652 – λ-injective, 1705 – λ-separably injective, 1707, 1717 – ζ -convex, 235, 237 – P1 , 1706, 1712, 1713 – Pλ , 1705, 1706, 1714 – ARNP, 236, 262, 263

Subject Index – AUMD, 236, 264 – Fréchet, 1667 – Gurarii, 626 – hereditarily indecomposable, 1263 – Hilbert transform (HT), 235, 245 – indecomposable, 1263 – injective, 1705 – James tree, 636 – locally convex, 603 – modular, 1650 – normed incomplete, 641 – nuclear, 1667 – of securities, 115 – polyhedral, 650 – quotient, 651 – rearrangement invariant, 21, 257, 500, 574 – RNP, 236, 259, 260, 263, 483, 601, 629 – separably injective, 1707, 1716, 1717 – UMD, 244, 250, 253, 264, 590, 894 space c∞ (Γ ), 1756 spaces of vector-valued functions, 588–595 – with values in a UMD space, 590 spaces with mixed norm, 107 spectral distance, 320, 327 spectral methods, 1619 spectral radius formula, 959 spectral theorem, 945 spherical design tight, 917 spherical harmonics, 926 – addition formula, 926 spherical isoperimetric inequality, 715 spherical Radon transform, 921 spin glass theory, 354 splitting of atoms, 846, 847, 850 spreading family, 1049 spreading models, 125, 136, 1035, 1036, 1076, 1310, 1339, 1340, 1353, 1650 – conditional, 1036 – unconditional, 1036 square function, 27, 863, 932 squares sets of, 206 stable, 137, 1479 – embedding, 524 – uniform algebras, 703 stable type p, 1628 star body, 918–920, 923 state space, 626 stationary case, 426 Stein restriction conjecture, 219 Stein theorem, 883 Steiner symmetrization, 712 Steinhaus variables, 934 Stieltjes transform method, 344, 357 stochastic exponential, 383

1863

stochastic interval, 375 stopping time, 375 Strichartz inequality, 197, 217 strict Aδ -functions, 404 strict Aδ -set, 404 strictly – convex norm, 784, 1773 – positive functional, 22 strictly singular operator, 1263 striking price, 115 strong Eberlein compacts, 1754 strong minimum, 398, 406 strong minimum at x0 , 420 strong type (r, s), 1369 strongly exposed point, 35, 628, 790 strongly summing, 1028 subadditive, 1618 subdiagonal algebra, 1495 subdifferentiable function, 795 subdifferential calculus, 414 submartingale, 401 submeasure, 1117 subspace – rich, 687 – smooth, 655 – tight, 682 subspaces of Lp – finite dimensional, 837 – Fourier transform characterization, 906 – infinite dimensional, 140 successive approximation, 1627, 1629 Sudakov inequality, 756 sufficiently Euclidean, 306 summation operator – finite, 479 – infinite, 443 summing basis, 1022 super – ideal, 455 – property, 56 – weakly compact operator, 479 super-property, 1306 support of a measure, 1751 support of a vector in a lattice, 23 surjection, 445 surjective – hull of an ideal, 445 – ideal, 446 symbol of an operator, 874 symmetric basis, 11, 298, 854, 1631, 1632, 1805 symmetric ideal, 441 symmetrically exchangeable random variables, 1653

1864

Subject Index

symmetrization, 1608–1610 – two point, 1609 systems of functions, 563 – Daubechies wavelets, 567 – Faber–Schauder system, 9, 564 – Franklin system, 564, 574, 575, 594 – Haar system, 9, 564 – Meyer wavelets, 566, 576, 577 – polynomial bases, 577–579 – Rademacher system, 16, 125, 460, 563, 848 – rational bases, 579 – spline wavelets, 567 – systems of analytic functions, 569 – tensoring, 568 – trigonometric polynomials, 577, 881, 889, 896 – trigonometric system, 13, 564 – Walsh system, 564, 575, 576, 1321 Szegö type factorization, 1496 Szlenk index, 802, 1018, 1559, 1719

transport of measures, 350 transportation cost, 1625 tree, 139, 143, 144, 154, 1009, 1583, 1754 – well-founded, 1583 triangular matrices, 1496, 1498 triangular truncation, 319, 329, 360 Tsirelson space, 276, 709, 1060, 1255, 1649 – 2-convexification, 1649 – p-convexification, 1649 twisted Hilbert space, 1157 twisted sum, 1155 two arrows space, 1756 two point symmetrization, 1609 type, 49, 125, 126, 137, 139, 472, 505, 526, 1177, 1181, 1188, 1194, 1306, 1315, 1339 – doubly generated, 1339 – Fourier, 481 – Gaussian, 472, 1710, 1738 – Haar, 477

T 2 , 279, 305 T p -smooth, 412, 413 tail distribution, 6 Taylor expansion of order p, 412 tensor – norm, 485 – – injective, 1563 – product, 153, 484, 916, 1123 – stability, 959 tensor products of C ∗ -algebras, 1449 the Brunel–Sucheston theorem, 1074, 1075 the Graham–Rothschild theorem, 1074 three space property, 1751 tight embeddings, 1628 tiling, 601, 661 – bounded, 661 – convex, 661 total variation of a measure, 39 totally disconnected compact, 1800 totally incomparable, 63 trace, 450, 1381, 1463 – duality, 44, 351, 456 – formula, 451, 463, 968 – matrix, 968 – spectral, 968 trace theorems, 1381 trace-class, 1170, 1171 transference theorem, 1371 transfinite oscillations, 1032 transitive algebra, 536–538 translation invariant – space, 873 – subspace, 872

Up , 279 ultrafilter, 55 ultrapower, 55, 455, 1307 ultrapower-stability, 455 ultraproduct, 55, 455, 1650, 1736 ultraproduct of operators, 55 ultraproduct, of Banach lattices, 55 ultraproduct-stability, 456 UMD Banach space, 253, 590, 894, 1494 unconditional – 1-unconditional over X, 1340 – basic sequence, 125, 128, 131 – basis, 9, 126, 251, 274, 277–279, 301, 302, 304, 855, 858, 1631, 1632, 1691, 1759 – constant, 250, 251, 256 – constant, complex, 250, 256 – convergence, 9 – finite dimensional decomposition, 295, 298, 301, 304 – finite dimensional expansion of the identity, 274, 278 – structure, local (l.u.st.), 59 unconditional basis constant, 1675 uniform – algebra, 673 – convexity, 413, 515, 527, 785 – Kadets–Klee property, 526 – retract, 876 – structure of a Banach space, 876 uniform convexity, 1607 uniform Eberlein compact, 1753 uniform homeomorphisms, 1805 uniform PL-convexity, 1483 uniformly

Subject Index – A-dentable, 397 – F -smooth norm, 789 – convergent Fourier series, 895 – Gâteaux-smooth norm, 789 – integrable, 17, 143 uniformly convex, 1479 uniformly Gâteaux differentiable (UG), 1777 uniformly rotund in every direction (URED), 1781 uniformly smooth, 1479 unimodal, 178 unique – non-atomic lattices, 1665 – rearrangement invariant structure, 1658, 1662– 1664 – symmetric basis, 1637, 1641, 1642 – unconditional basis, 1637, 1639, 1640 – unconditional basis, up to a permutation, 1637, 1644 uniqueness in – finite-dimensional spaces, 1651 – general spaces, 1637 – non-Banach spaces, 1665 uniqueness of complements, 837, 865–868 uniqueness theorem for measures, 902 unit ball of np , 1624 unit in a Banach lattice, 90 unitarily invariant norm, 328 unitary ideal property, 66 unitary orbit, 326, 327 universality conjecture, 345, 346 upper p-estimate, 504, 514 upper semi-continuous, 1021 upper semicontinuous function, 608 Urysohn inequality, 728 usco map, 1761 v(μ), 1366 Vaaler theorem, 175, 176, 179 vacuum state, 356 vacuum vector, 356 Valdivia compact, 1801 valuation, 732 Vapnik–Cervonenkis class, 1193 viscosity solutions for Hamilton–Jacobi equations, 427 viscosity subsolutions, 427, 429, 430 viscosity supersolutions, 427, 429, 430 volume ratio, 169, 171, 172, 174, 748, 1224 volumetric invariant vk, , 1225 w∗ -Hδ -sets, 404 Walsh system, 1321

1865

Walsh–Paley martingale, 476 wavelet, 1691 wavelet basis, 565 wavelet set on Rd , 566 weak Lp , 500, 505, 523 weak Asplund space, 1537, 1752 weak cotype 2, 1230 weak Fatou norm, 89 weak Hilbert space, 277, 305, 313, 968 weak star uniformly rotund (W∗ UR), 1778 weak type (1,1), 879–881 weak type (r, s), 1369 weak∗ usco, 1535 weak∗ -locally uniformly convex, 408 weakly compact M-basis, 1765 weakly compactly generated (WCG), 1760 weakly complete dual, 1045 weakly continuous harmonic functions, 404 weakly countably determined or a Vašák space (WCD), 1760 weakly Lindelöf (WL), 1760 weakly Lindelöf determined (WLD), 1760 weakly Lindelöf M-basis, 1765 weakly locally uniformly rotund, 1792 weakly realcompact, 1803 weakly sequentially compact, 1747 weakly unconditionally Cauchy, 1023 weakly unconditionally convergent (wuc) series, 686 weakly unconditionally summing, 1022 weakly uniformly rotund, 1781 weight function, 524 weighted norm inequalities, 387 weighted shift, 322, 323, 329 Welfare theorems, 114 well founded, 1009 well founded closed tree, 1037 well-founded tree, 1038 well-posed, 395 Wermer theorem, 673 WET = Whitney Extension Theorem, 1372 Weyl inequality, 943, 948, 953, 957, 963 Weyl numbers, 945, 954, 956, 957, 970, 972 Weyl–von Neumann–Berg theorem, 324, 328, 330 Wiener criterion, 1410 Wiener measure, 476 Wigner semicircle law, 342 Wolf’s theorem, 1407 ξ -generate a spreading model, 1038 ξ -generating the 1 -basis, 1046 ξ -generating the c0 -basis, 1046

1866 ξ -th variation, 1040 (ξ, M)-convergent, 1054 Yan theorem, 380 Young inequality, 512

Subject Index ζ -function, 198 zonoids, 768, 844, 902, 911, 924, 1631 – approximating by zonotopes, 925 zonotopes, 768, 902, 924

Handbook of the Geometry of Banach Spaces : Volume 2

Handbook of the Geometry of Banach Spaces, Volume Volume 2

Handbook of the geometry of Banach spaces