A D V A N C E D C O U R S E S OF MATHEMATICAL A N A L Y S I S
I
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EDITORS
A , AIZPURU-TOMAS Fa L E O N - S A A V E D R A Universidadde Cadiz, Spain
P R O C E E D I N G S OF T H E F I R S T I N T E R N A T I O N A L S G H O O L
A D V A N C E D C O U R S E S OF MATHEMATICAL ANALYSIS I Cadiz, Spain
N E W JERSEY
*
22-27 September 2002
LONDON
World Scientific 1 : SINGAPORE
BElJlNG
*
SHANGHAI
HONG KONG
*
TAIPEI * CHENNAI
Published by
World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA oftice: 27 Warren Street, Suite 401402, Hackensack, NJ 07601 UK oftice: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
ADVANCED COURSES O F MATHEMATICAL ANALYSIS I Proceedings of the First International School Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or p u t s therevJ may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-060-2
Printed in Singapore by World Scientific Printers (S)Pte Ltd
V
Preface An idea, a dream and then a reality. This is the manner we can describe the gestation of the First International Course of Mathematical Analysis in Andalucia. Our small research group, working in the University of Cadiz, considers that, nowadays, Andalucia has a remarkable scientific and cultural importance as well as an indisputable international presence and this idea should take shape in some type of periodic event. Our main aims were: (i) That the different research groups in Andalucia working on mathematical analysis should meet each other and collaborate. (ii) That the young researchers working in these groups could have access to the most advanced research lines. Besides, we consider it of great importance to unite efforts in order to guarantee a solid education in our young researchers. The Course was held in 2002, from September 23 to 27, in the historical part of Cadiz, a beautiful city surrounded by the Atlantic Ocean. This book is the first volume in a series of advanced courses of Mathematical Analysis. The authors of this collection are recognized experts with an extensive research and educational experience. The authors of the first volume are: Yoav Benyamini, Manuel Gonzblez Ortiz, Vladimir Miiller, Simedn Reich (co-authored with E. Matouskova and A. J. Zaslavski) and Angel Rodriguez Palacios. The article by Benyamini is the only updated survey of the exciting and active area of the classification of Banach spaces under uniformly continuous maps. The article by Gonzblez is a pioneer introduction to the theory of local duality for Banach Spaces. The paper, Genericity in nonexpansive mapping theory, by Eva Matoubkov6, Simedn Reich and Alexander J. Zaslavski, provides an up-to-date detailed overview of the applications of the generic method to nonexpansive mapping theory. The article by Vladimir Muller provides a survey of results and ideas concerning orbits of operators and related notions of weak and polynomial orbits. The Scott Brown technique to obtain invariant subspaces is carefully exposed. The survey paper by Rodriguez Palacios collects the results on absolutevalued algebras since the pioneering works of Ostrwski, Mazur, Albert,
vi
and Wright to the more recent developments. The celebrated UrbanikWright paper on the topic is fully reviewed. The survey contains in addition some new results, and several new proofs of known results. As a matter of fact, it will be the authoritative reference for the optimal version of results scattered in the literature through many separate papers. We want to express our gratitude to all of them. We also want to thank Professors Tomas Dominguez Benavides and Angel Rodriguez Palacios for being the first in supporting the.idea of celebrating the Course. We thank from the heart every participant for their presence: the mature researchers as well as the young ones still in their educational period. Of all of them we keep pleasant memories. Besides, we want to thank Professor Maria Victoria Velasco for assuming the responsibility of organizing the second course in September 2004 in the gorgeous city of Granada. Finally, we want to thank the publishing house, World Scientific, for making possible the interesting contents of our seminars to be enjoyed by all the mathematical community.
The Editors
vii
Contents Introduction to the Uniform Classification of Banach Spaces Y. Benyamini An Introduction to Local Duality for Banach Spaces
1
31
M. Gonzdez Orbits of Operators V. Muller
53
Genericity in Nonexpansive Mapping Theory E. Matouikovci, S. Reach and A . J . Zaslavski
81
Absolute-Valued Algebras, and Absolute-Valuable Banach Spaces A . R. Palacios
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1
INTRODUCTION TO THE UNIFORM CLASSIFICATION OF BANACH SPACES
YOAV BENYAMINI* Department of Mathematics, Technion
-
Israel Institute of Technology
Haifa 32000, Israel e-mail:
[email protected]. ac.il
This is an introductory survey of the classification of Banach spaces as metric spaces, where the maps are (nonlinear) uniformly continuous maps or, more specifically, Lipschitz maps. We describe basic results which show that the uniform theory and the linear theory are different but that, nevertheless, some linear features of a Banach space are preserved under uniform homeomorphisms.
1. Introduction
The norm on a Banach space induces on it a topology and a metric. In the linear theory of Banach spaces the natural maps are continuous linear maps, and these maps “respect” not only the topology but also the metric structure: they are bounded, i.e., Lipschitz maps. Of special interest are the linear maps which preserve the metric, i.e., isometries. One can, however, disregard the linear structure of the Banach spaces and consider them as a special class of topological spaces or as a special class of metric spaces. In this setup the maps are no longer required to be linear and we consider continuous maps in the first case and uniformly continuous, Lipschitz or isometric maps in the second. The following two theorems define the “natural boundaries” for a meaningful theory. The first is due to Mazur and Ulam 32 and the second to Kadec 26 (for separable Banach spaces) and to Toruliczyk 39 (for general density characters).
Theorem 1.1. Let E and F be Banach spaces. Iff : E isometry satisfying f(0) = 0, then f is linear.
-+F
is a surjective
‘This work was supported by the Technion Fund for the Promotion of Research.
2
Theorem 1.2. Two infinite-dimensional Banach spaces are homeomorphic to each other iff they have the same density character. These two theorems say that the topological and isometric classifications of Banach spaces are “trivial” in two opposite ways: Theorem 1.1 implies that the linear structure can be completely recovered from the metric structure and thus reduces the (seemingly) nonlinear isometric classification problem to the linear one. Theorem 1.2 says that the topology gives no information whatsoever on the linear structure. The uniform classification of Banach spaces is lLinbetween”these two extreme theories. As we shall see in these notes this is an interesting theory which, on the one hand, gives some information on the linear structure of the spaces but, on the other hand, it is rich enough to be genuinely different from the linear theory. A word of warning before we go on. While the topological or nonlinear isometric theories are “trivial” from the point of view of classification, they are certainly far from being trivial. Infinite-dimensional topology is a rich theory, see the books by Bessaga and Pekzy6ski lo and van Mill 40. Similarly, there are, of course, many problems concerning isometric (or, more generally, 1-Lipschitz) maps, such as into-isometries, extensions, nonexpansive retractions, fixed points etc., which we do not consider here a t all. The purpose of this survey is to describe the main results, ideas and techniques in the area. The idea is to draw the attention of the reader to this exciting area and t o stimulate further reading and interest. We did not try to give “best” results or to analyze variations of the concepts and techniques. For these and for many more topics that are not discussed here a t all we refer the reader to the recent book by the author and J. Lindenstrauss. We also refer to for unexplained notation and terminology. The article is divided to three sections. The first is a general description of the area in the spirit of an expanded colloquium talk. In the other two sections we describe in more detail and with proofs, or sketches of proofs, some of the main results on the Lipschitz classification (in Section 3) and on the uniform classification (in Section 4) of Banach spaces. These notes are an expanded version of lectures delivered a t the “First International Course of Mathematical Analysis in Andalucia” , which was held in Cadiz, Spain, in September 2002. It is a pleasure to thank the organizers and the participants of the course for the pleasant and inspiring atmosphere. I also thank the referee for his useful comments.
3 2. A General Overview Historically, the first result in infinite-dimensional topology is the following theorem of Mazur 31. Denote the closed unit ball of L p ( p ) by B p ( p ) .
Theorem 2.1. Let p and u be two measures and let 1 5 p , T < 00. If L p ( p ) and L T ( u )have the same density character, then their unit balls, B p ( p )and BT(u), are uniformly homeomorphic t o each other. The homeomorphism of B p ( p ) onto BT(p)(for the same measure p ) is given explicitly by the map f -+ IflP/'sign(f). For different measures we first use these maps pass to B2(p) and B2(u) and then note that these balls are even isometric to each other. Note that the L, spaces are not linearly isomorphic to each other for different values of p . Hence the uniform classification (at least of the unit balls of Banach spaces) is different from the linear classification of the spaces. We shall see later that L, and L, (for p # q), as well as L, and 1, (for p # 2) are also not uniformly homeomorphic to each other. Thus the uniform classification of subsets of Banach spaces is also different from the uniform classification of the whole spaces. The next result, which generalizes Mazur's theorem, was proved for spaces with an unconditional basis by Ode11 and Schlumprecht 35 and for general lattices by Chaatit 12. For proofs and more details see also Sections 9.1-9.3 in '.
Theorem 2.2. Let E be a Banach lattice which does not contain 1,'s uniformly. Then its unit ball is uniformly homeomorphic t o the unit ball of a Halbert space. The condition that E does not contain loo's uniformly is, in fact, also necessary. This was proved much earlier by Enflo 15. Aharoni and Lindenstrauss gave the first example of two Lipschitz equivalent Banach spaces which are not isomorphic to each other. It follows that the Lipschitz classification of Banach spaces is different from their linear classification. We now describe the structure of the example. Let q : E -i F be a linear quotient map from E onto F with kernel 2.A linear operator T : F -+ E is called a lifting of q if qT = id^. Such a lifting is an isomorphism of F into E whose image, T F , is complemented in E (by the projection Tq). The kernel 2 is complemented in E (by the projection i d E - Tq) and E is isomorphic to F @ 2.If T is a (nonlinear) Lipschitz lifting, then the same
4
formulas yield similar Lipschitz consequences: F is Lipschitz equivalent to a Lipschitz retract of E , the kernel 2 is a Lipschitz retract of E and E is Lipschitz equivalent to F @ 2. The example of Aharoni and Lindenstrauss is a Banach space E which does not contain a subspace isomorphic to @(I?) (for some uncountable set I?), but such that there is a surjective linear quotient map q : E -+ @(I?) with kernel co and such that q admits a Lipschitz lifting. It follows that E is Lipschitz equivalent to %(I?) @ co = @(r), but they are certainly not isomorphic to each other: E does not even contain an isomorphic copy of CO(F).
Godefroy and Kalton l7 have recently introduced a “categorical” construction, which, among other things, gave a systematic method of constructing more examples of the same nature. (See also Kalton 27 where the approach is extended to spaces of Holder and more general uniformly continuous functions.) We now describe the construction in 17. Let E be a Banach space and denote by Lipo(E) the Banach space of Lipschitz functions f : E -+ R satisfying f ( 0 ) = 0. The norm of f is its Lipschitz constant. Let F ( E ) denote the closed linear subspace of the dual Lipo(E)*,spanned by the point evaluations &(f) = f(~).The space F ( E ) is called the Lipschitz free space over E. Lipschitz maps between Banach spaces induce, in a natural way, linear maps between their free spaces and F ( E ) enjoys some useful properties of free objects in a category. The map 6~ : E + F ( E )is a (nonlinear!) isometry of E into F ( E ) . One checks easily that the linear map PE : F ( E ) + E , given by P ~ ( z a , 6 , ~ )= Canz,, is a surjective quotient map with llP~ll= 1. Clearly 6~ is an isometric lifting of PE, hence F ( E ) is Lipschitz equivalent to @ ker(PE). The next theorem yields the required examples. Recall that a Banach space E is weakly compactly generated (WCG) if there is a weakly compact subset K c E whose linear span is dense in E. Since the unit ball of a reflexive space is weakly compact, reflexive spaces are certainly WCG.
Theorem 2.3. Let E be a nonseparable WCG space. Then F ( E ) does not contain a subspace isomorphic to E . I n particular F ( E ) ,which is Lipschitz equivalent to E @ ker(PE), is not isomorphic to it. All the examples known to date of Lipschitz equivalent Banach spaces which are not isomorphic to each other are nonseparable and nonreflexive. The main open problem in this area is
5
Problem 2.1. If two separable Banach spaces are Lipschitz equivalent to each other, are they necessarily isomorphic? What i f they are also reflexive? Uniformly convex? It turns out that the method used to construct nonseparable examples cannot work in the separable case. This is one consequence of the following remarkable theorem of Godefroy and Kalton 17.
Theorem 2.4. Let F be a separable Banach space and assume that it is the image of some Banach space E under a surjective quotient map q : E --+ F . If q admits a Lipschitz lifting, then it also admits a linear lifting. In the linear theory the space Q is considered to be a “small” space. For example, it does not contain reflexive subspaces or any of the other classical Banach spaces. This is no longer true in the Lipschitz category. Aharoni proved that it is actually “universal”. The proof of the following theorem is presented in Section 3.
Theorem 2.5. E v e y separable metric space is Lipschitz equivalent t o a subset of C O . While we do not know whether the Lipschitz and linear classifications of separable Banach spaces coincide, the uniform classification is certainly different from both. The following result is due to Aharoni and Lindenstrauss 3 , improving on a previous result of R b e 38. For a proof, more details and extensions see Section 10.4 in 8 .
Theorem 2.6. Let 1 I p , q , p n < 00 with p , --f p and let E = (C@lpn)q. Then E is uniformly homeomorphic to E @ 1,. If p # q and p, # p f o r all n, then E and E @ 1, are not isomorphic. If p = 1 and q , p , > 1 for all n, then E is reflexive while E @ 11 is not, hence (see Corollary Z.l(i) below) they are not even Lipschitz equivalent. The results up to this point showed the difference between the uniform and the linear classification. We now discuss some results in the opposite direction, namely, instances in which at least some of the linear structure is preserved. The most natural way to “linearize” a mapping is by differentiation. Recall that a mapping f : E --t F is said to be Giiteaux differentiable at a point 2 if the limit Du = limt+o (f( x tu) - f ( x ) ) / t exists for every u E E and is a bounded linear operator as a function of u. The operator D is called the Gdteaux derivative of f at x and is denoted by Of(.).
+
6
If f is a Lipschitz mapping with constant K , which is G6teaux differentiable a t some point x , then the derivative D = D f ( x )is bounded by the same constant K . Moreover, iff is a Lipschitz equivalence (Le., a Lipschitz map which also satisfies the lower estimate 11 f ( x )- f (y)II 2 (12- y”/K for every x , y E E ) , then D is also bounded from below by the same constant, and is thus an into-isomorphism. (Note that D f ( x )is only bounded from below. Its image may very well be a proper subspace of F even when f was assumed to be a surjective Lipschitz equivalence.) It follows from this discussion and from the next theorem that in many cases the existence of a Lipschitz embedding of E into F implies that there is also a linear isomorphism of E into F . (The theorem, due independently to Aronszajn 5 , Christensen l 3 and Mankiewicz 30, will be discussed in detail in Section 3.) Theorem 2.7. Let E be a separable Banach space and assume F has the Radon-Nikodgm property (RNP). I f f : E t F is a Lipschitzfunction, then there is a point x E E where f is G6teaux differentiable. It is obvious that some assumption is needed in Theorem 2.7 on the space F . For example, one cannot take F = co because by Theorem 2.5 every separable Banach space is Lipschitz embedable in CO, but “most” Banach spaces are not linearly embedable in it. The assumption that F has RNP is actually essential. Indeed, one of the characterizations of RNP is that F has RNP iff every Lipschitz map from R to F is differentiable at some point of R (or, equivalently, differentiable almost everywhere). Thus for the theorem to hold we must assume that F has RNP. Since reflexive spaces have RNP we obtain Corollary 2.1. (i) If a separable Banach space E is Lipschitz equivalent to a subset of a reflexive Banach space F , then E is isomorphic to a subspace of F .
(ii) If E is Lipschitz equivalent to a subset of a Halbert space, then it is isomorphic to a Hilbert space. (iii) If p > 1 and r 2 1, then L, is Lipschitz equivalent to a subset of L, iff it is isomorphic to a subspace of L,, i.e., iff r = 2 or 2 2 r 2 p . Part (ii) was originally proved by Enflo
16. To deduce it from Theorem 2.7 we need the fact that E is isomorphic to a Hilbert space iff all its separable subspaces are. Part (iii) also holds for p = 1, but this requires an additional argument.
7
Differentiation results are, of course, not available for general uniformly continuous maps. We now describe some of the techniques used in their study. More details, including the definition and some basic facts on ultraproducts, will be given in Section 4. A basic useful property of uniformly continuous mappings is that they satisfy a Lipschitz condition for large distances. More precisely Proposition 2.1. Let f : E -+ F be uniformly continuous. T h e n f o r every a > 0 there is a constant K = K ( a ) > 0 such that [ l f ( z ) - f ( y ) ( [5 K[[z-y[( whenever (111: - yl[ 2 a.
One way to apply the proposition is to use it to create Lipschitz maps from uniformly continuous ones: Assume that f : E -+ F is uniformly continuous and assume, as in the proposition, that I \ f ( z ) - f ( y ) I I 5 K J [ z - y l l whenever 1 1 : - yJJ2 1. Put fn(z)= f(nz)/n.Then the map fn satisfies Ilfn(z)- fn(y)II I K \ J z- y)) whenever JIz- y\I 2 l/n. It follows that if U is a free ultrafilter on N,then g = (fn)uis a K-Lipschitz map from the ultrapower ( E ) u into ( F ) u . Moreover, if f is a uniform homeomorphism, then one can apply the same procedure to f-l. One checks directly that ( f n ) - l = ( f - l ) n , thus the procedure gives a Lipschitz inverse to g . We have thus proved the following theorem of Heinrich and Mankiewicz 21
Theorem 2.8. If E and F are uniformly homeomorphic, then they have Lipschitz equivalent ultrapowers.
As we shall see in Section 4, this result, together with the results of Section 3, combine to give a simple proof of the following theorem of Ftibe 37. Roughly speaking, the theorem says that uniformly homeomorphic Banach spaces have “the same” finite-dimensional subspaces or, in the language of Banach spaces theory, that they have the same local linear structure.
Theorem 2.9. Let E and F be two uniformly homeomorphic Banach spaces. Then they are crudely finitely-representable in each other, i.e., there
8
is a constant C > 0 so that for every finite-dimensional subspace El of E there is a finite-dimensional subspace Fl c F with d(E1,F l ) 5 C and vice versa. In particular, since L, and L, do not have the same local structure when p # r (as follows, for example, by computing their type and cotype), it follows that they are not uniformly homeomorphic to each other. Proposition 2.1 can also be applied through the study of approximate midpoints.
Definition 2.1. Fix x , y E E and 6 midpoints between x and y is
> 0. Then the set of &approximate
Mid(z,y,b)= { z : ) ) x - z l I , J ) ~ - z l l5
(~+~)IIxTYII/~}
When 6 = 0 we say that z is a midpoint (or, when we want to emphasize, exact midpoint) between x and y. It is clear that exact midpoints are mapped by isometries to exact midpoints. More generally, if f is K-Lipschitz and two points x and y happen to satisfy l)f(x)- f (y))I = K J J x-yll, then f maps exact midpoints between x and y t o exact midpoints between their images. The following proposition generalizes this fact. Proposition 2.2. Let f : E -+ F be a uniform homeomorphism and let 0 < 6 < 1/2. Then there are points z, y with 112 - yII arbitrarily large, so that
f(Mid(GY76)) c Mid(f(x), f ( d , 5 6 ) . Proof. We only sketch the proof and leave the exact computations, that give the estimate 56, to the reader. Let K ( a ) be the Lipschitz constant of f for distances larger than a. Then K ( a ) is a nonincreasing function of a. Pu t K = lima-,mK(a). Since f - l is also uniformly continuous, it follows that K > 0. Fix tu so large that K ( a / 2 ) K . Choose x , y with llx - yII > a such that (1 f (z) - f (y)(I K(lx- yII and let z E Mid(x, y, 6). Then N
N
llf(4- f (.>I1
I K ( Q / 2 ) l I X - zll Kllx - 41 5 (1+ W l l x - Yll/2 (1 + 6)ll.f
(x)- f (Y)11/2.
9
We now explain the idea of how the approximate midpoint sets can be used to show that some Banach spaces E and F cannot be uniformly homeomorphic to each other: Compute the approximate midpoint sets for points in E and in F . By the proposition a uniform homeomorphism f : E -+ F will have to take some approximate midpoint set in E into an approximate fked point set in F . If it so happens that approximate midpoints sets in E are “large” sets and approximate midpoint sets in F are %mall”, then this would contradict the uniform continuity of f-’. To demonstrate how this idea is implemented we shall compute in Section 4 the approximate midpoint sets in 1, and L,, and then use the right notions of “large” and “small”, appropriate for the different situations, to prove Theorem 2.10. For every 1
5 p < 03, p # 2, the spaces L, and 1, are not
uniformly homeomorphic to each other. This method was introduced by Enflo (unpublished), who used it to prove the case p = 1 of the theorem. The case 1 < p < 2 is due to Bourgain l1 and the case p > 2, which required a new notion of “large” and “small” is due to Gorelik 19. The problem of characterizing Banach spaces which are uniformly homeomorphic to a subset of a Hilbert space was completely solved by Aharoni, Maurey and Mityagin 4. The result is, in fact, more general and holds for linear metric spaces and not only for Banach spaces. For example, it follows from the next theorem and from known results in the linear theory that L p ( p ) is uniformly homeomorphic to a subset of a Hilbert space iff 05p52. Theorem 2.11. A real linear metric space is uniformly homeomorphic to
a subset of a Hilbert space iff it is linearly isomorphic to a subspace of L o ( p ) for some measure p. We shall not discuss this theorem in these notes. The interested reader is referred to Chapter 8 in The last problem that we discuss in this section is the uniform and Lipschitz classification of balls and spheres in Banach spaces. Benyamini and Sternfeld proved that for every infinite-dimensional Banach space E there is a Lipschitz retraction from the unit ball B ( E ) onto the unit sphere S ( E ) . Equivalently, S ( E ) is Lipschitz contractible and there is a Lipschitz map on B ( E ) with no approximate fixed point. (The results in followed Nowak 34, who proved them for some special
10
spaces. See also Azagra and Cepedello-Boiso for a smooth version of these results and for generalizations to starlike sets.) This result leads naturally t o the following open problem.
Problem 2.2. Let E be an infinite-dimensional Banach space, are its unit ball B ( E ) and unit sphere S ( E ) Lipschitz equivalent? Are they uniformly equivalent? The problem is open for any Banach space (including Hilbert space!) except for one “pathological” counter-example: Gowers and Maurey 2o constructed a separable reflexive Banach space, G M , which is not isomorphic to any of its subspaces. For this space the unit ball cannot be Lipschitz equivalent t o the unit sphere. Indeed, i f f : B -+ S were a Lipschitz equivalence, then since the space is separable and reflexive f would be Giiteaux differentiable at some point by Theorem 2.7. The derivative, D, would be an isomorphism from G M into a proper subspace of itself (as is easy to deduce from the fact that the image of f is in the sphere, which “looks” locally like a subspace of codimension one). But there is no isomorphism of G M onto a proper subspace of itself! It is easy to see that for any Banach space E the unit sphere S ( E ) is Lipschitz homogeneous, i.e., for any two points x,y in the sphere there is a Lipschitz homeomorphism of the sphere onto itself taking x to y . It follows that if the ball and the sphere of E are Lipschitz (or uniformly) equivalent, then the ball should also be Lipschitz (or uniformly) homogeneous. The only nontrivial advance on Problem 2.2 is the following result of Nahum 33. See Section 9.4 in for a proof and further discussion.
’
Theorem 2.12. Let E be a Banach space which is isomorphic to E @ R, then B ( E ) is Lipschitz (respectively, uniformly) equivalent to S ( E ) iff it is Lipschitz (respectively, uniformly) homogeneous. Here is a simple application of the theorem.
Corollary 2.2. Let E and F be two Banach spaces which are isomorphic to E @ R and F @ R respectively. Then the balls B ( E ) and B ( F ) are Lipschitz (respectively, uniformly) equivalent to each other iff the spheres S ( E ) and S ( F ) are. Proof. It is clear that if f is an equivalence between the spheres, then its homogeneous extension gives an equivalence between the balls. Conversely, assume that f : B ( E ) -+ B ( F ) is a Lipschitz (or uniform) equivalence. I f f
11
takes S ( E ) onto S ( F ) ,then we are done. So assume that there is a point z 6 S ( E )such that Ilf(z)II < 1, and choose a point y E E with llyll < 1 and llf(y)ll < 1. (Any point y with llyll < 1 which is close enough to z certainly satisfies this condition.) Let g be a Lipschitz homeomorphism of B ( F ) onto itself which takes f(z)onto f ( y ) . Then f-' o g o f maps the point z in S ( E )to the point y E E with JJyJJ < 1. Since pairs of points in S ( E ) can be mapped to each other by a Lipschitz equivalence of B ( E ) and the same is true for pairs in B ( E )\ S ( E ) ,it follows that B ( E ) is homogeneous and the same is true for B ( F ) which is Lipschitz (or uniformly) homeomorphic to it. By the theorem B ( E ) and B ( F ) are Lipschitz (or uniformly) homeomorphic to S ( E )and S ( F ) repectively. Thus all the four sets involved are equivalent to each other and, in particular, so are S ( E ) and S ( F ) . 0 3. Lipschitz Maps
The main result in this section will be the proof of Theorem 2.7 as well as some further comments and variations. But we start with a proof, essentially due to Assouad 6 , of Theorem 2.5. Proof of Theorem 2.5. A map f : X --t Q is given by a sequence of real-valued functions f(z)= (fl(z), fi(s), ...), where the fn's satisfy (i) f,(z)-+ 0 for every z E X. When X is a metric space, then f is a Lipschitz map iff (ii) The fn's are Lipschitz with a common Lipschitz constant. And f is a Lipschitz embedding if also (iii) There is a constant C > 0 so that for every z # y in X there is a n such that Ifn(2) - fn(Y)I 2 Cllz - YII. + In the construction we shall use functions of the form (a-d(z, M ) ) for suitably chosen sets M C X and constants a. Such functions are Lipschitz with constant 1. To present the idea of the proof in a somewhat simpler setup, let us first assume that X is compact and that its diameter is 1. For each n 2 0 let {zy : i 5 m,} be a finite 2-,-net in X and put f,,i(z)= (2-n+1 - d ( z , z y ) ) + . Then (i) and (ii) hold because Ifn,i(z)I5 2-n+1 + 0 and Ilfn,i((~ip = 1. To check (iii) fix z # y. Let n satisfy 2-(n+1) 5 d(z,y) 5 2-n and choose i such that d ( z , ~ 1 + < ~ )2-(n+3). Then fn+3,i(Z) = 2 -(n+3)+1 - d(z, ..+3) - 2-(n+3)+l >
- 2-(,+3) == 2-(n-k3) > - d(z,y)/8.
12
On the other hand
2 d(z, y) - d(z, xy+3) > d(z, y) - 2-(n+3) 2 2-(n+l) - 2-("+3) = 3.2-(n+3)
d(y,
hence fn+3,i(y)
=0
and (iii) holds with C = 1/8.
For a general separable metric space X we need to replace the points xy by sets. The main difficulty is that we cannot do this with finitely many sets for each fixed n and this makes it more difficult to fulfill both conditions (i) and (iii) simultaneously. We shall assume that X is a separable Banach space. This is possible because every separable metric space is isometric to a subset of the Banach space C(0,l). Denote the unit ball in X by B. We construct a function F : X 4 co with IIFl[m, l l F l l ~ i5~ 1 so that FIB = 0 and so that 8 5 112 - yII 5 16 implies that llF(x) - F(y)II 2 112 - yll/l6. Once this is done put FO = F and Fn(x)= 2nF(x/2n)for n # 0 in Z.We then write co as (Cz @Q)O and the required embedding is f(x) = (..., F-l(x), Fo(x),Fl(x),...). To see that the Fn's satisfy condition (i) notice that for each fixed x we have F n(x) = 0 and also Fn(x)= 0 whenever n satisfies 2n > IIxlI. It is clear that all the Fn's, hence also f, are 1-Lipschitz. To check condition (iii) fix x # y and choose n such that 8 5 11(x - ~ ) / 2 ~ 5l l16. Then
IIFnk) - Fn(Y)II = 2"11F(x/2") - F(Y/2n)ll 2 2n[1(x- ~)/2~11/16 = llx - ~11/16. To construct F let {Ai}i>obe a sequence of 1-balls that cover X . Denote the concentric balls with radius 2 by 2Ai (and the ball of radius 2 centered at the origin by 2B). Put MO= 2Ao \ 2B and
Mi
= 2Ai
\
( U M j ) \2B = 2Ai \ ( U 2 A . j )\2B j
Then X \ 2B
(*)
=
for i 2 1.
j
u Mi and the disjoint sets Mi have the crucial property
For every x
E
X the set {i : d(x,M i ) < 1) is finite.
Indeed, fix k such that x E Ak. If d ( z , M i ) < 1, choose z E Mi with I(x- zll < 1. Then z E 2Ak, hence, by the construction of the M's, cannot belong to Mj for any j > k. Thus i 5 k. + Define now F = ( f o , f l , ...) : X -+ co by fi(x) = (1 - d ( x , M i ) ) . Then Ilfillm, I I f i I l ~ iI ~ 1 and F is well defined because for every x E X the condition (*) gives that f i ( x ) # 0 for finitely many i's only. Clearly FIB EE0.
13 Finally, fix two points z , y with 8 5 1111: - yII 5 16. We may assume that IIzII 2 2. If i is such that x E Mi, then fi(x) = 1. On the other hand fi(y) = 0 because d ( y , M i ) 2 11z,yll - diam(Mi) 2 8 - 4 > 1. It follows 0 that fi(z)- fi(y) = 1 2 ( ( z- y11/16. We now move to the discussion of Theorem 2.7. The proof of the theorem actually gives more than just one point of Giiteaux differentiability: the function is Giiteaux differentiable “almost everywhere”. This requires an explanation. There is no “good” measure on infinite-dimensional spaces (as the next lemma shows), hence no natural notion of “almost everywhere”. The heart of the proof of the theorem is the introduction of a family of “negligible” sets in a general separable Banach space. These sets play the role that sets of measure zero play in Rn and the theorem then says that the function is Giiteaux differentiable on the complement of such a negligible set.
Lemma 3.1. Let E be a n infinite-dimensional Banach space and let K be a compact subset of E . T h e n there i s a point y E E so that all the translates { K t y : t E R} are pairwise disjoint. I n particular there is n o translation invariant a-finite regular Borel measure p o n E .
+
Proof. The linear subspace V spanned by K is a proper subset of E. Indeed, it is contained in the a-compact, hence proper (by Baire’s theorem) subset UnL, where L is the closed convex symmetric hull of K . Choose y E E \ V . If ( K t y ) n ( K sy) # 0, then there are XI,5 2 E K such that z1 t y = 2 2 s y , i.e., y = (z1 - z2)/(s - t ) E V , a contradiction. If p is a regular Borel measure, then there is a compact set K such that p ( K ) # 0. Chose y as above for this set K . Being a-finite p cannot have nonzero mass on all the uncountably many disjoint translates of K in the direction of y. Thus p is not translation invariant. 0
+
+
+
+
Several different notions of negligible sets have been introduced. Aronszajn, Christensen and Mankiewicz introduced three such notions and another useful notion, the Gauss null sets, was introduced later by Phelps and is, perhaps, the easiest to define. A Borel set A is Gauss null if p ( A ) = 0 for every non-degenerate Gaussian measure p on E . (A measure p is a non-degenerate Gaussian measure if every functional 0 # x* E E x has a non-degenerate Gaussian distribution with respect to p . ) The analysis of the structure of negligible sets belongs to the area of infinite-dimensional geometric measure theory and leads to many interesting results. (See Chapter 6 of for a systematic presentation of this topic.) It is a deep recent
14
result of Csornyei l4 that the two notions introduced by Aronszajn and by Mankiewicz as well as the notion of Gauss null sets are actually equivalent to each other. We shall follow Christensen and will now present what he called “Haar null” sets and their basic properties. This notion was introduced again much later (under the different name of “shy sets”) in 22 (see also 23). We work in separable Banach spaces, but the definition and the basic properties of Haar null sets hold in any abelian Polish group. Definition 3.1. A Borel subset A of a separable Banach space E is called Haar null if there is a regular Borel probability measure p on E such that p ( A z) = 0 for all z E E , equivalently, p * X A = 0. We denote the family of Haar null sets by N.
+
Note that if E is finite-dimensional, then a set is Haar null iff its Lebesgue measure is zero. Indeed, for sets of Lebesgue measure zero just take p to be any probability measure equivalent to the Lebesgue measure. Conversely if p is a test measure for A as in the definition, then JJXA(Z z ) d z d p ( z ) = J p ( A z ) d z = 0 , hence the inner integral J x ~ ( z z ) d z is zero p-a.e. But this inner integral is identically equal to the Lebesgue measure of A. The family N is closed under finite unions. Indeed, if p i are test measures for {Ai}i5n,then p1 * ... * p n is a test measure for UAi. It is also closed under countable unions. To prove this one needs to choose the test measures pi so that the infinite convolution p = IIy * p i converges, and then p is a test measure for UAi. See Proposition 6.3 in for details. It follows that Haar null must have empty interior. Indeed, if A is open, then E can be covered by a countable number of translates of A . Thus A E N would imply that also E E N,which is false.
+
+
+
Proposition 3.1. If A is a Borel set and A $ N,then A - A contains a neighborhood of 0. Proof. Assume not and choose xj $ A - A with IIzjll < 2-j. Let C = (0, l}N be the Cantor group (i.e., the group operation is addition modulo 2 in each coordinate). Put qn = (0, ...0, 1 , O ...), where the 1 is in the nth position and denote the Haar measure on C by A. Define cp : C E by cp(d1,62, ..,) = C b j z j . The condition IIzjll < 2 - j implies that the series converges and that cp is continuous. Since A $ N,it follows that there is a point y E E such that X(cp-l(A y)) # 0. Indeed, otherwise the image of --f
+
15
X under cp, namely, the measure defined by p ( B ) = X(cp-l(B)))could serve as a test measure to show that A is Haar null. A classical theorem in measure theory yields that the difference set U = cp-l(A y) - cp-l(A y) contains a neighborhood of 0 in C, and thus vn E U for large enough n. For such n the set cp-'(A+y) contains two points that differ only in the nth coordinate, hence xn E (A+y) - (A+y) = A-A, a contradiction. 0
+
+
Corollary 3.1. Let E be a n infinite-dimensional Banach space. Then (i) If A is a Borel proper subspace of E , then A E N . (ii) If A c E is compact, then A E N . Part (i) is obvious, and (ii) follows from the fact that the compact set A - A cannot contain an open set. Since it is easy to construct, in any separable Banach space, a compact set K and a Gaussian measure p such that p ( K ) # 0, it follows from (ii) that the notions of Haar null and Gauss null are not equivalent. An important property of Haar null sets is that they satisfy a weak version of Fubini's theorem. Lemma 3.2. Let V be a finite-dimensional subspace of E and denote the Lebesgue measure on V by A. W e also use the same notation X for its translates to translates of V . I f X(A n {V y}) = 0 for all y E E , then
+
AEN. Proof. As a test measure just take any probability measure on V which is equivalent to A. 0
To prove Theorem 2.7 we shall show that there is a Haar null set A such that f is GSteaux differentiable on E \ A. As a preparation we first present a simple technical lemma and then a slight generalization of the classical theorem of Rademacher that a Lipschitz map between finite-dimensional spaces is differentiable almost everywhere. Lemma 3.3. Let G be a dense additive subgroup of E and let x E E . If f : E -+ R is a Lipschitz function such that limt+o (f( x tu) - f ( x ) ) / t exists for all u E G and is a n additive function of u E G, then f is Giteaux differentiable at x . I n particular the set of Gdteaux differentiability points o f f is a Borel set.
+
16
+
Proof. The functions ht(u)= (f(z tu) - f ( z ) ) / tare Lipschitz with the same constant as f. Hence their convergence on the dense set G implies that the limit exists everywhere. The additivity of the limit is clear, and the homogeneity follows from a change of variable: ht(au)= ah,t(u). To see that the set of differentiability points is Borel we choose the dense group to be countable and fix any u E G. P u t ht(z,u)= (f( z + t u ) - f ( z ) )/ t and denote the set { ( s , t ) E Q x Q : Isl,Itl < l/m} by Dm. Then the derivative a t z o E E in the direction u exists iff
zo E
nun
{z :
-
N } . If y E G1/PB(E1),then llxfyll = (IIxllP+ IlyllP)l/p 5 (1+6)l/P 5 1+6, i.e., y E Mid(%,-x,6) . For the other inclusion, assume that y satisfies 112 f yI( 5 1 + 6. Write y = zo z1, where zi E Ei,and chose a sign 0 = f l so that llx 0zoII 2 1. Then
+
+
+ 6IP L llx + @/(IP Hence llzlll 5 ( ( 1 + 6 ) P - 1)"' (1
+ BZOI(P + ( ( 2 1( ( PL 1 + ( I Z l ( l P .
= ((z
5 (2pb)'lP for 6 < 6(p). If the support of x is not finite, we use the argument above for a finitely supported vector which approximates x (where the degree of approximation depends only on 6). (ii) Inductively divide the interval [0,1] to 2j disjoint subsets {Ai,j : i 5 2 j } such that Ai,j = Azi-l,j+l U A2i,j+1 and such that JAi,j1x1 = 2-j. Then define z j ( t )= (--l)zx(t) for t E Ai,j and i = 1 , . . . , 2 j .
24 (iii) We only prove the case z = 1. The general case is similar but requires some more technical details. Let F 2 be the span of the Rademacher functions. By Khintchine’s inequality there is a constant A, such that ApllyllPI ((yI(2for every y € F2. Take F1 t o be the closed subspace of L, orthogonal to span{ 1,F2}, i.e., F1
=
{z E L, : J z ( t ) ( l + y ( t ) ) d t = 0
Fix z E Mid(%,-x, 6) and write it as z Assume, as we may, that a 2 0. Then
+ 6 2 111+
1
hence a
Zll,
+
2 111 zllz
=
5 6. By the same inequality 1+6 2
=
for every y E
a
F2
}
+ x i + x2 where xi E Fi.
+ + + 2 1+a I 1 + 6 < 2, hence
l((1 a )
21
52112
11x2112
II(1+ a ) + 2 1 + 22112 2 111+ 22112
= (1
+ 11x211~)1/22 1 + Ilx2ll;/8 2 1 + A ~ l l ~ 2 l l ~ / 8
and 11x211p 5 (86/Ag)1/2. Thus dist(z, F1) I ]la x2ll
+
I 6 + (86/A,) 2 1/2 .
0
We are now ready to sketch the proof of Theorem 2.10. We refer to Theorem 10.13 in for details and for the proof for 1 < p < 2 which we do not discuss here. Proof of Theorem 2.10 for p = 1. Assume that f : L1 -+ I1 is a surjective uniform homeomorphism. Then f-’ satisfies a Lipschitz condition for large distances and we choose a constant K so that JIy- zll 2 1 implies K - l I l y - 41 I Ilf(y) - f(z)ll. Fix 6 > 0. By Lemma 4.1 the set Mid(%,-x, 0) is a “large” subset of L1 for every x E L1: it contains an infinite IIxlI-separated set. We now apply Proposition 2.2. After translating the points to x and -x and assuming, as we may, that llxll 2 1, we obtain for this x that this large set is supposed to be mapped by f into Mid(f(z),f(-z),G), which is a “small” set in 11: it is a finite-dimensional perturbation of a set of small diameter. But this is impossible. Indeed, compactness of bounded sets in finite-dimensional spaces would then imply that the images of two of the IIxlI-separated points y, z E Mid(x, -x,O) would satisfy llf(y) - f ( z ) l l 6llxll - and for small 6 0 this would contradict llf(y) - f ( z ) l l 2 K-llly - zll = K-lIIx((.
-
Proof of Theorem 2.10 for p > 2. This case requires a new way to measure “large” and “small” sets. We first give a heuristic formulation
25 of the so called "Gorelik principle", which uses a mixture of topological and metric conditions t o compare the "size", and use it to give a heuristic argument for the case p > 2. We then formulate a precise quantitative version of the principle and apply it to give a precise proof.
The Gorelik principle: A uniform homeomorphism between two Banach spaces cannot take a large ball in a subspace of finite codimension in one space into a small neighborhood of a subspace of infinite codimension in the other. Assume now that f : 1, -+ L, is a surjective uniform homeomorphism. By Lemma 4.1 and Proposition 2.2 f should map a Sl/p((zl(-ballin a subspace El of finite codimension in 1, into a 61/211z11-neighborhoodof an infinite-codimensional subspace F1 of L,. But this would contradict the Gorelik principle because p > 2 implies that for small S the radius of the ball, bl/PIIzII, is much larger then the size, S 1 ~ 2 ~of~the z ~ neighborhood. ~, We now give the precise formulation of Gorelik's theorem
19.
Theorem 4.1. Let E and F be Banach spaces, and assume that 'p : E -+ F is a homeomorphism with a uniformly continuous inverse 'p-l. Assume that El c E is a subspace of finite codimension and Fl c F a subspace of infinite codimension. If a ,P > 0 satisfy
+
q(aB(E1))c F1 P B ( F ) = { z E F : dist(z, F1) I P } (*) then the modulus of continuity of 'p-l satisfies w,-i(2P) 2 a/4. To deduce the case p > 2 of Theorem 2.10 we use the fact that f-' satisfies a Lipschitz condition for large distances to choose K > 0 so that t 2 1 implies w f - l ( t ) 5 K t . Fix S > 0 and use Proposition 2.2 and Lemma 4.1 to choose z with 261/211z11 2 1 so that f maps a 61/pIlzll-ball in a subspace El of finite codimension in I , into a 61/211zll-neighborhoodof an infinite-codimensional subspace Fl of L,. Theorem 4.1 with a = S1/p((z(( and P = 61/211z11then gives w f - l ( 2 6 1 ~ 2 ~ ~2z 6~ 1~ )~ q x ~ = ~ f51/p-1/2 /4
. N 2llxll/4
But wf-l (261/211z11) I K.261/211z11by the choice of K , and this is impossible for small enough 6 (because p > 2).
Proof of Theorem 4.1. The proof will follow from two facts: (i) There is a compact set A c ; B ( E ) such that whenever f : A -+ E is a continuous function satisfying 11 f (z) - 211 1. a/4 f o r every z E A , then f(A) n Ei # 0.
26 Indeed, by a theorem of Bartle and Graves (see Proposition 1.19(ii) in ) there is a (nonlinear) continuous right-inverse : E/E1 -+ E to the quotient map IT : E 4 E/E1 so that maps ? B ( E / E l ) into $ B ( E ) . The set A = + ( F B ( E / E 1 ) )is compact because E/E1 is finite-dimensional, If f : A 4 E satisfies Ilf(z)- 211 I a/4 for every z E A, then the map g(y) = y - ~ f ( + ( y ) ) maps %-B(E/E1) into itself because ~ $ J ( y v = ) y, 1 1 1 ~ 1 1 = 1 and $(y) E A yield
+
+
By Brouwer's fixed point theorem g has a fixed point yo, and it follows that Tf(+(YO)) = 0, i.e.1 f(G(Y0)) E f ( A )n El. (ii) Let B be a compact subset of F . Then there is a point y E F with IIyyII < 2P and dist(B
+ y, F I )> P.
Indeed, let B1 be a finite PIZdense set in B . The subspace G spanned by F1 and B1 is a proper subspace of F (because F1 has infinite codimension), hence there is a point y E F with llyll < 2P and dist(y,G) > 3 p / 2 . Given z E B , choose z1 E B1 with llz - zlll < P / 2 . If z E F1, then z1 - z E GI hence JIz y - zll 2 dist(y, G) - llz - z111 > p.
+
To prove the theorem, find a compact set A C ; B ( E ) as in (i). Then use (ii) for the compact set B = cp(A), and find y E F with llyll < 2,B for which dist(cp(A) y, F1) > P. The map f ( z ) = cp-'((p(s) y) maps A into E , and if we had w,-1(2p) < a / 4 , then f would satisfy
+
+
Ilf(z) - zll
=
Ilv-l(cp(4+ Y)
-
cp-'(cp(4)II
I W,-1(IIYII> 5 W,-1(2P) < 4 4 for every z E A. By (i) there is an zo E A with f ( z 0 ) E El. In fact ~ ( x o E) aB(E1) because I l f ( ~ o ) l l I Ilf(zo)- z o l l + llzoll < 3 a / 4 < a. Condition (*) then implies that cp(zo) y = cp(f(z0)) E FI PB(F), contradicting dist(cp(A) y, F I ) > ,B.
+
+
+
We finish by quoting three results on spaces whose linear structure is uniquely determined, or is determined up to finitely many possibilities, by their metric structure. The proofs use the techniques of this section as well as quite deep results in the linear theory of Banach spaces. Parts (i) and (iii) are due to Johnson, Lindenstrauss and Schechtman 2 5 , part (ii) is due to Godefroy, Kalton and Lancien 18.
27
< 00. If E i s uniformly homeomorphic t o I,, t h e n it i s isomorphic t o it. (ii) If E i s Lipschitz equivalent t o Q, then it is isomorphic t o it. (iii) For every n there i s a Banach space whose u n i f o r m class contains exactly 2n different linear structures. More precisely, there are 2n mutually nonisomorphic Banach spaces E l , ...,En, which are uniformly homeomorphic t o each other, and so that a n y Banach space which i s uniformly homeomorphic t o t h e m i s isomorphic t o one of the Ei 's.
Theorem 4.2. (i) Let 1 < p
It is unknown whether the analog of (i) also holds for L, (when p # 2). In other words, it is unknown whether a Banach space which is uniformly homeomorphic to L, is necessarily isomorphic to it. It is also unknown whether (i) holds for 11. In fact, it is not even known whether a Banach space which is Lipschitz equivalent to 11 is necessarily isomorphic to it. (The point is that it is unknown whether a space which is Lipschitz equivalent to a dual space is necessarily isomorphic to a dual space. If E is a dual space which is Lipschitz equivalent to 11, then it is isomorphic to it by Theorem 3.2 and the decomposition method.) It is unknown whether part (ii) holds for spaces which are uniformly homeomorphic t o CO. In part (iii) one can take El = T ( p l ) @ ... @ T ( P n ) , where T ( p ) is the pconvexified Tsirelson space, and 1 < p l , ...,pn < 00 are different from each other. The 2" mutually nonisomorphic but uniformly homeomorphic spaces are El @ ( CjEJ @ l p j ) , where J is an any subset of (1, ...,n } . It is not known how t o construct a space with k different linear structures when k is not a power of 2. For proofs see also Theorem 10.15 (for (i)), Theorem 10.17 (for (ii)) and Proposition 10.40 (for (iii)).
*
References 1. I. Aharoni, Eve? separable metric space is Lipschitz equivalent to a subset ofco, Israel J. Math. 19 (1974), 284-291. 2. I. Aharoni and J. Lindenstrauss, Uniform equivalence between Banach spaces, Bull. Amer. Math. SOC. 84 (1978), 281-283. 3. I. Aharoni and J. Lindenstrauss, An extension of a result of Ribe, Israel J. Math. 52 (1985), 59-64. 4. I. Aharoni, B. Maurey, and B. S. Mityagin, U n i f o m embeddings of metric spaces and of Banach spaces into Halbert spaces, Israel J. Math. 52 (1985), 251-265. 5. N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147-190.
28 6. P. Assouad, Remarques sur un article de Israel Aharoni sur les prolongements Lipschitziens duns cg, Israel J. Math. 31 (1978), 97-100. 7. D. Azagra and M. CepedelleBoiso, Smooth Lipschitz retractions of starlike bodies onto their boundaries in infinite-dimensional Banach spaces, Bull. London Math. SOC.33 (2001), 443-453. 8. Y . Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. 9. Y . Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. SOC.88 (1983), 439-445. 10. C. Bessaga and A. Pelczyriski, Selected Topics in Infinite Dimensional Topology, PWN, Warsaw, 1975. 11. J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, Israel GAFA Seminar 1985-86 (J. Lindenstrauss and V. D. Milman, eds.), Lect. Notes in Math., vol. 1267, Springer, 1987, pp. 157-167. 12. F. Chaatit, O n the uniform homeomorphisms of the unit spheres of certain Banach lattices, Pacific J. Math. 168 (1995), 11-31. 13. J. P. R. Christensen, O n sets of Haar measure zero i n abelian Polish groups, Israel J. Math. 13 (1972), 255-260. 14. M. Csornyei, Aronszajn null and Gaussian null sets coincide, Israel J. Math. 111 (1999), 191-202. 15. P. Enflo, O n a problem of Smirnov, Ark. Mat. 8 (1969), 107-109. 16. P. Enflo, Uniform structures and square roots in topological groups I, Israel J. Math. 8 (1970), 230-252. 17. G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math., to appear. 18. G. Godefroy, N. J. Kalton and G. Lancien, Subspaces of co(N) and Lipschitz isomorphisms, GAFA 10 (2000), 798-820. 19. E. Gorelik, The uniform nonequivalence of L , and l,, Israel J. Math. 87 (1994), 1-8. 20. W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. SOC. 6 (1993), 851-874. 21. S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), 225251. 22. B. R. Hunt, T. Sauer, and J. A. Yorke, Prevalence: A translation-invariant “almost every” on infinite dimensional spaces, Bull. Amer. Math. SOC. 27 (1992), 217-238. 23. B. R. Hunt, T. Sauer, and J. A. Yorke, Prevalence: A n addendum, Bull. Amer. Math. SOC.28 (1993), 306-307. 24. D. J. Ives and D. Preiss, Not too well differentiable Lipschitz isomorphisms, Israel J. Math. 115 (ZOOO), 343-353. 25. W. B. Johnson, J. Lindenstrauss and G. Schechtman, Banach spaces determined by their uniform structure, Geom. Funct. Anal. 6 (1996), 430-470. 26. M. I. Kadec, A proof of topological equivalence of all separable infinite-
dimensional Banach spaces (Russian), Funk. Anal. i. Prilozen 1 (1967), 6170. 27. N. J. Kalton, Spaces of Lipschitz and Holder functions and their applications, t o appear. 28. J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263-287. 29. J. Lindenstrauss, E. Matouskovb, and D. Preiss, Lipschitz image of a measure-null set can have a null complement, Israel J. Math. 118 (2000), 207-219. 30. P. Mankiewicz, On the differentiability of Lipschitz mappings in Fre'chet spaces, Studia Math. 45 (1973), 15-29. 31. S. Mazur, Une remarque sur l'home'omorphie des champs fonctionnels, Studia Math. 1 (1929), 83-85. 32. S. Mazur and S. Ulam, Sur les transformations isome'triques d'espaces vectoriels norm&, C.R. Acad. Sci. Paris 194 (1932), 946-948. 33. R. Nahum, On the Lipschitz equivalence of the unit ball and the sphere of a normed space, preprint. 34. B. Nowak, O n the Lipschitzian retraction of the unit ball in infinite dimensional Banach spaces onto its boundary, Bull. Acad. Polon. Sci. SBr. Sci. Math. Astrono. Phys. 27 (1979), 861-864. 35. E. Ode11 and Th. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259-281. 36. A. Pelczyliski, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Mathematicae, vol. 58, PWN, 1968. 37. M. Ribe, On uniformly homeomorphic normed spaces, Ark. Mat. 14 (1976), 237-244. 38. M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), 139-147. 39. H. Toruriczyk, Characterizing Hikbert space topology, Fund. Math. 110 (1981), 247-262. 40. J. van Mill, Infinite Dimensional Topology, North-Holland, 1989.
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31
AN INTRODUCTION TO LOCAL DUALITY FOR BANACH SPACES *
MANUEL GONZALEZ Departamento de Matemdticas Facultad de Ciencias Universidad de Cantabria 39071 Santander (Spain) E-mail:
[email protected] We present an introduction to the study of the local dual spaces of a Banach space. We describe with detail the main properties of this concept and give several characterizations. These characterizations allow us to show examples of local dual spaces for many classical spaces of sequences or functions.
Introduction.
A (closed) subspace 2 of the dual of a Banach space X is a local dual of X if for every couple of finite dimensional subspaces F of X* and G of X, and every 0 < E < 1, there is an operator L : F Z satisfying:
-
(a)(1 - E)Ilfll 5 (b)
IlLfll I (1 + E ) I l f l l , Yf E F ;
Wf,4 = (f,4, Yx E G, Yf E F ;
(c) L ( f ) = f, b'f E F
n 2.
A trivial example is given by 2 = X*. The PRINCIPLE OF LOCAL REFLEXIVITY (P.L.R., FOR SHORT) gives a nontrivial example. This principle can be stated by saying that X ,as a subspace of the second dual X * * , is a local dual of X*. One reason to study local dual spaces of X is that in many cases X* is a big space and for we do not have a good representation of X*. In this paper we present an introduction to the study of local dual spaces of Banach spaces. Our aim is to help to understand this concept. *Work partially supported by the DGI (Spain). Proyecto BFM2001-1147.
32
So from the beginning we have explained with some detail the meaning of the conditions that appear in the definition of local dual. We have included many examples of local dual spaces, some of them elementary, and some other that could be interesting because they are far more simple that the whole dual space. We have also given several characterizations of the subspaces of X* which are local duals of X and some other results which provide additional properties or techniques to find examples of local duals. We have tried to write the paper in such a way that could be read by people that have attended a course of functional analysis. We have included some proofs that are not too complicated and help to understand the concept of local dual. Since many of these proofs need the concept of convergence over an ultrafilter, we have included at the end of the paper a short appendix describing this concept. This should be enough to follow the proofs. Our material is organized into eight short sections plus an appendix. In the first section we analyze the properties that define the concept of local dual space. The second section contain a characterization in terms of finite dimensional subspaces, which is a weakening of the definition of local dual, and some characterizations of “global” type, in which only the whole spaces appear. These global characterizations are easier to deal with than the definition. However there is a disadvantage: the second dual X** appears in the statements. In the third section we show that being a local dual is a symmetric relation: 2 is a local dual of X if and only if 2 has a local d u d isometric to X . Later in the paper we will apply many times this property and the characterizations given in Section 2 to find examples of local dual spaces. In Section 4 we give some natural examples: C[0,1]is a local dual of L1[0,1] and the Bore1 measures on [0,1] which are absolutely continuous with respect to the Lebesgue measure form a local dual of C[O,11 isometric to L1[0,1]. We also give some examples of local duals for the spaces e , ( X ) and &(X) of vector sequences and for the spaces C ( K , X ) ,L1(pL,X) and L,(p,X) of vector-valued functions. In Section 5 we show local duals for spaces with a basis. Section 6 contains some additional properties of local dual spaces. Every separable space admits a separable local dual and some spaces admit a smallest local dual. We have included these properties so late in the paper in order to have at our disposal examples to illustrate them. Section 7 contains local dual spaces for projective and injective tensor
33
products of Banach spaces. Note that the spaces C ( K , X ) and L1(p,X) are tensor products 5 . In the eighth section we introduce the concept of ultrapower of Banach spaces. The P.L.R. for ultrapowers, due to Heinrich 15, give us an interesting example of local dual. Moreover, we give characterizations of the subspaces of X * which are local duals of X in terms of ultrapowers. Finally we give an appendix on ultrafilters. It contain the definition of convergence over an ultrafilter and the fact that the ultrafilters in a compact set converge. This result is an essential tool in the previous sections. Throughout the paper we use standard notations: X and Y are Banach spaces, Bx the closed unit ball of X , and X * the dual space of X . Subspaces are always closed. We identify X with a subspace of X**. For A c X we consider the set A I : = { f ~ X *. . (,z) f = 0 for every z E A}. Analogously, for C c X * we define the set C_Lof X . We denote by B ( X ,Y ) the space of all (bounded linear) operators from X into Y , and by K ( X ,Y ) the subspace of all compact operators. Given T E B(X,Y ) ,N ( T )and R(T) are the kernel and the range of T , and T* is the conjugate operator of T . We denote by W the set of all positive integers.
WARNING: Sometimes we say that some concrete space Z is a local dual of X without specifying the position of Z inside X . In these cases it should be clear to what copy of Z we are referring to. A simple example is C[O,11 is a local dual of L1[0,1]. A not so simple example: L1[0,1] is a local dual of C[O,11. For the latter case see Example 3. The concept of local dual space has been introduced recently 11, but there are some related concepts that have been previously studied by many authors. It would have been difficult to acknowledge all the relations with previous results. So we have mentioned some of them throughout the paper and we mention here some of the authors that have studied these properties: Jerry Johnson 16, Stefan Heinrich 15, Nigel Kalton 18, Gilles Godefroy 7 , Asvald Lima l9 and Santiago Diaz '. Many of the results we present here are joint work of Antonio Martinezl2 13. Abej6n and the author in I am grateful to Antonio Aizpuru and Fernando Le6n-Saavedra for their invitation and for their effort to create a pleasant atmosphere for all the people which attended the first international course of mathematical analysis in Andalucia.
'' ''
34 1. Basic properties
Here we study the properties of the local dual spaces of a Banach space which are a direct consequence of the definition.
Definition 1. A subspace 2 of the dual of a Banach space X is a local dual of X if for every couple of finite dimensional subspaces F of X* and G of X and every 0 < E < 1, there is an operator L : F 2 satisfying
-
( a ) (1 - &)llfll
5 IlLfll I (1+ &)llfll, Yf E F ;
( b ) ( L f , z )= (f,z),Yx E G, Vf E F ; ( c ) L ( f )= f, Vf E F n 2. Clearly the dual space X * is a local dual of
X.
Example 1. PRINCIPLE OF LOCAL REFLEXIVITY of X**, is a local dual of X * . For a short enlightening proof of the P.L.R. see
(I7):
x as a subspace
2o
Remark 2. Let 2 be a local dual of X. (1) Property ( a ) says that X* is finitely represented in 2: the isometric properties of each finite dimensional subspace F of X* are arbitrarily close to the isometric properties of some finite dimensional subspaces L ( F ) of X . ( 2 ) Property ( b ) says that L preserves the duality when we pass from the pair (G, F ) to the pair (G, L ( F ) ) . (3) The meaning of property (c) is more difficult to grasp. It says that the operators L fix points of 2,and it is related with the fact that ' 2 is complemented in X**. See Theorem 5.
Exercise 1. Let 2 be a local dual of
X.
(1) The subspace 2 is norming. This means that for every z E X ,
II4I = SUP{I(Z, .)I (2) Let z** E 2 ' .
: z E 2 ; llzll = 1).
Then llz** - 211 2 llzll for all z E X .
Clearly, X* is the only local dual of X when X is reflexive. Moreover, this can also be true for some non-reflexive spaces.
35 Exercise 2. Apply part (2) of Exercise 1 to show that dual of q,.
C1
is the only local
NOTES AND REMARKS (1) Being a local dual of X is an isometric property and depends on the position of 2 inside X * . It is not preserved under renormings of X. (2) Two local dual spaces for a Banach space can be very different from an isomorphic point of view. Indeed, for the space el the copy of Q inside loo is a local dual of X , and also each predual of el is isometric to a subspace of loo which is a local dual of C1. However, there are preduals of C1 which do not contain subspaces isomorphic to CO. See Bourgain’s spaces in ’.
2. Some characterizations Here we give some characterizations of the subspaces of X* which are local dual spaces of X . These characterizations are useful to understand the concept of local dual. The first characterization should be compared with the definition. Here we replace the “exact” conditions (b) and (c) by the “approximate” conditions (b’) and (c’). As we shall see, this is very useful to simplify some arguments.
Theorem 3. (Local Characterization) A subspace Z of X* is a local dual of X if and only if f o r every couple of finite dimensional subspaces F of X* and G of X , and e v e y 0 < E < 1, there is a n operator L : F 2 satisfying
-
PROOF. A direct proof can be done using an argument similar to the one in part (b) of Proposition 2 in proof of Theorem 2.5 in ll.
13.
For an indirect argument we refer to the 0
Example 2. We can apply Theorem 3 to show directly that co is a local dual of .!?I
36
-
Let Pn : em Q be the projections defined by Pnek = ek for k I n and Pnek = 0 for k > n. It is immediate to check that these projections satisfy the following properties. (1) llPnll = 1 and IIPnzIIm 4 llzlloo for all z E em. ( 2 ) llPnZ - ~ 1 + O1 for~ all z E Q. (3) (PnZ,y) 4 ( z , y ) for all z E loo and all y E e l .
Let F c C, G E el and E > 0. CLAIM:There exists an integer n such that Pn satisfies the properties of the map L associated to F , G and E in Theorem 3. Indeed, taking into account that the unit ball of a finite dimensional space is compact, (a) follows from (l),(b’) follows from (2) and (c’) follows from (3). Although the definition of local dual space is “local”, we shall see in Theorem 5 that it is possible to give “global” characterizations. The following result will be useful. Proposition 4. Let 2 be a local dual of X . Then every compact operator T : 2 --+ Y admits a compact extension T : X* Y with = IlTll.
I]?]
--+
PROOF.Let A be the set of all the pairs CY = ( F ,G) of finite dimensional subspaces F c X * and G c X . For each CY = (F,G) E A, we denote by L, the map from F into 2 associated to F , G and 6 = (dim E, dim Fa)-’ in the definition of local dual. Note that for every f in the unit ball of F and every CY E A, TLaf is contained in the compact set 2T(Bz). Thus taking an ultrafilter U on A refining the filter associated to the inclusion, we can define T : X * 4 Y as in Remark 29
+
Tf
:=
lim TL,f.
Ol-U
It is enough to check that this operator satisfies the required conditions. 0 Exercise 3. Complete the proof of Proposition 4.
-
Let Y be a subspace of X. An operator x : Y * X* is an extension operator if ( x f ) I Y = f for every f E Y*.Observe that 11 f 11 5 Ilxfll. The operator x is an isometric extension operator if IIx f I( = llfll for every f E Y*.
37 Exercise 4. Let Y be a subspace of X and let J : Y + X denote the embedding of Y into X . Show that an operator x : Y* X * is an extension operator if and only if J * x is the identity on Y * . In this case, x J * is a projection from X * onto x ( Y * ) .
-
In the following result we give “global” characterizations of the property of being a dual local. These characterizations are easier to deal with than the definition. However there is a disadvantage: the second dual X** appears in the statements. Theorem 5. (Global characterizations) For a subspace Z of X * , the fol-
lowing statements are equivalent:
-
(1) Z is a local dual of X . (2) there is an isometric extension operator x : Z* R(X) 3 (3) there exists a norm-one projection P : X** N ( P ) = Z* and R ( P ) 2 X ; (4) there exists a norm-one projection Q : X*** R(Q) = 2’’ and N ( Q ) c X I .
x;
PROOF.
Let
L
: 2 -+
X** so that
X** such that
X*** such that
X * denote the inclusion.
(1) =+ (2) For every finite dimensional subspace H of Z* we consider the quotient map q H : 2 -+ Z/H*. By Proposition 4 there exists an extension QH : X * -+ Z/HL of QH with 1 1 Q ~ l l = 114Hll. Since ( q H ) * is the embedding of H into Z * , X H := ( Q H ) *: H 4 X** is an isometric extension operator. Indeed, llx~ll= ( ( q h ( = ( 1. Moreover, let h E H and z E 2.
(XHh1 2) = ( h1 Q H Z ) = ( h 1 q H z ) = ( (qH)*h> 2) = ( h z ) . Now we take a nontrivial ultrafilter V on the set of all finite dimensional subspaces of H of Z* refining the filter associated to the inclusion. Since Bx.. is weak*-compactl we can define a map x : Z* -+ X** by
x := weak*-H-V lim
XH.
Clearly x is an isometric extension operator. It remains to check that X ( Z * )3 Let x E X . Then L * X E Z* and for each f E X * ,
x.
38 Thus x satisfies the required conditions.
(3) Let x be the operator given in (2). The kernel of the conjugate (2) operator L* : X** + Z*is ZL, and L*X is the identity on Z*.Therefore XL* is a norm-one projection on X * * , N ( x L * )= N ( L * )= ZL and R ( x L * = )
R(X) 3 (3)
x.
+ (4) Take Q = P*.
(4) ivity to details.
(1) It is essentially an application of the principle of local reflexfor
Z c Z**= 2". We refer to the proof of Theorem 2.5 in
''
0
Remark 6. Recall that X is a local dual of X * by the P.L.R. In this case the decomposition given by the projection in the third part of Theorem 5 is
x*** =X*@XL, and the projection P on X*** is just the operator restriction to X .
Remark 7. The range of the projection P which appears in part (3) of Theorem 5 is weak*-dense because R ( P ) 3 X . Therefore, in the nontrivial case Z # X * there is no subspace M of X * such that R ( P ) = M I .
NOTES AND REMARKS If in the definition of local dual we delete property (c), then we obtain the concept of finite dual representability (f.d.r.) studied in lo. There are examples of Banach spaces X and subspaces Z of X * such that X * is f.d.r. in 2,but 2 is not a local dual of X . Note that clearly 21 c 2 2 and X * f.d.r. in 21 implies X*f.d.r. in 22.However,' 2 is complemented when Z is a local dual, and 2 : complemented does not imply 2; complemented. See Example 2.11 in
3. Symmetry Here we shall show that being a local dual space is a symmetric property in some sense. This fact is a useful tool to find examples of local dual spaces for some concrete Banach spaces, as we shall see throughout the paper. Let Z be a local dual of X. Let L : Z -+ X* and J : X embedding maps. We introduce the map
T = L * J :X -+ Z*.
-+ X**
be the
39 Since
Z is norming (Remark 1)]T is an isometry.
Theorem 8. Let Z be a local dual of X. Then T(X) is a local dual of Z isometric to X.
-
PROOF. Let x : Z* X** be an isometric extension operator such that R ( x ) 3 X . Then L*X is the identity on Z*and XL* is a projection on X** with R(xL*)= R(x) 3 J(X). Therefore, for every x E X I xTz = XL*JX = J x. CLAIM:The map $J : T(X)* --+ Z** defined by
($Jf,d:= (XSI f
O
f E T(X)*I 9 E Z*l
T);
is an isometric extension operator and R(+) II 2. Clearly
ll$Jll 5 1. Moreover] for every f
E T(X)*and every Tx E T(X),
($f,Tz) = (XTZ,f 0 T) = ( J x f 0 T) = (f 0 T, x) = (f]Tx). ]
Thus the claim is proved. Let y E 2. We can write y
=f o
(+fl9) = (xg1 f
for every g E Z * ; hence
O
TI for some f 6 T(X)*.Then
T) = (xg1 Y) = (91Y),
R(+)II 2.
0
4. Some natural examples
The theorem of representation of Riesz (Theorem 6.19 in 2') allows us to identify the dual space of the space C[O,11 of the continuous functions on the unit interval [O,11 with the space M[O,11 of all the Bore1 measures on [0,1]. More precisely, for each F E C[O,1]*there exists a meamre p~ E M[O,11 so that
for every g E C[O,11. Moreover, the map F 4 p~ is a bijective isometry. By the Radon-Nikodym theorem (Theorem 6.10 in 21)1 every p E M[O,11 can be decomposed into two parts p = pa p s 1where pa and p, are absolutely continuous and singular, respectively, with respect to the Lebesgue measure on [0,1]. Moreover] JJpJJ = IIpaJJIlpsJ) and there exists a function f E L1[0,1] such that dpa = f dt and \\pall = IJflll.
+ +
40
In this way we obtain a decomposition
M[O,11 3 -Wzc[O, 11 el Msing[O, 11 Example 3. M,,[O, 11 is a local dual of C[O,11 isometric to L1[0,1].
PROOF. Along this proof we identify M,,[O, 11 and L1[0,1]. For every integer n we consider the partition of [0,1] given by the intervals [(i - 1)/2", i/Y) if i = 1,. . .2" - 1, I? = [(2, - 1)/2,, 11 if i = 2,.
{ xF
Let us denote by the characteristic function of I?. We define an operator G, : M[O,11 -+ L1[0,1] by
i=l
Then we check that the maps G, satisfy (1) Each G, is a norm-one projection. (2) The sequence (Gnf) converges in norm to f, for every f E L1[0,1]. (3) (G,X) converges to X in the weak*-topology and ))G,XII 4 IlXll, for every X E M[O,11. Finally we proceed as in the proof of Example 2.
0
Remark 9. It is possible to give an alternative proof of the fact that M,,[O, 11is a local dual of C[O,11taking limits with respect to an ultrafilter. We define the map
x: M,,[O, by
x h := weak*-lim G;
1]*= L,[O, 11
-
M[O,1]*
h.
Clearly llxll 5 1. So it is enough to show that x is an extension operator satisfying R(x) 3 C[O,11. Let h E L1[0,1]* and f E L1[0,1].
(xh, f ) = lim(G: Thus x h is an extension of h.
h, f ) = lim(h, G, f ) = (h, f).
41
Let g E C[O,11 and p E M[O,11.
0
Hence x g = g and R(x) 3 C[O,11.
Example 4. C[O,11 c L,[O, 11 is a local dual of Li[O, 11.
PROOF. We have just seen that M,,[O, I] e L1[0,1] is a local dual of C[O,11. Thus applying Theorem 8, it is enough to show that T(C[O,11) is the copy of C[O,11 in L,[O, 11. Let L : L1[0,11 = M,,[O, 11 -+ M[O,11 and J : C[O,11 -+ C[O,1]**be the inclusion maps. Recall that T = L*J. Let g E C[O,11. For every f E L1[0,1] (L*JS, f) =
Jd
1
(L*Jg)(t)f(t)dt
and ( L * J S , f ) = ( J g , L f ) = (Lf79) =
Thus ( L * J g ) ( t ) = g ( t ) a.e. Hence
L*
J g = g.
I'
g(t)f(t)dt. 0
Remark 10. Both spaces L1[0,1] and C[O,11 admit a separable local dual, although their dual spaces are nonseparable. We will see later that this is a consequence of a general result: Each separable Banach space admits a separable local dual. Example 5. (1) l , ( X * ) is a local dual of l,(X). (2) & ( X ) is a local dual of el@*). First case: For every couple a := (E,F ) of finite dimensional subspaces of ll(X*) and l,(X**), we select a pair of sequences of finite dimensional subspaces (En) in X * and (F,) in X** so that E c ll(En)and F c l,(Fn). We denote la1 := dim(E) dim(F). The principIe of local reflexivity allows us to find, for every n, an [a[-'isometry SE : Fn X so that ( S z f , e ) = ( e , f ) for every e E En and f E F,, and S;(f) = f for every f E F, n X . We consider the map S" : F -+ l,(X) given by S"(zn) := ( S z ( Z n ) ) .
-
+
42
Let U be an ultrafilter in the set of all couples a = ( E , F ) of finite dimensional subspaces E of l l ( X * ) and F of l,(X**) refining the order filter. We define an operator A : l,(X**) l,(X)** by
-
A(a,) :=weak*- lim S 0 ( t n ) . a-U
Note that A is an isometry and A(y,) = (y,) for every (y,) E l,(X**). Therefore, A is an isometric extension operator. Moreover, A((zn)) = (z,)~ if (z,) E C,(X). In particular A(l,(X**)) 2 l,(X). Second case: Observe that the operator T : l,(X) duced in Section 3, is the natural inclusion.
-
l , ( X * ) * intro-
The first part of the previous example is particularly useful because the dual space of & ( X ) is a "big" space and does not admit a good representation. This is also the case for the spaces considered in the next example.
Example 6. (See
and
12)
Let p be a finite measure.
(1) L l ( p , X * ) is a local dual of L m ( p , X ) . (2) L,(p, X * ) is a local dual of L1(p,X ) .
NOTESAND REMARKS From
it follows
M[O,11" = Mac[O,111
@ ,
MsingI0, 111
However, this fact does not allow us to apply Theorem 5 to derive that Ma,[O,11 is a local dual of C[O,11. Indeed, Msing[O,11' does not contain C[O111. 5. Spaces with a basis
Recall that a sequence ( e n ) in a Banach space X is a (Schauder) basis of X if for every z E X there is a unique sequence (a,) of scalars so that x = C z l aiei. Let (e,) be a basis of X . The projections P, : X -+ X defined by P,(CeO az. ea. ) := Cy=laiei are continuous. The basis (en) is monotone if llPnll = 1 for every n.
43 The linear functionals e: defined by e : ( x & aiei) = a, are continuous. We denote by [e;] the closed subspace of X * generated by {e; n E W}.
Theorem 11. Let X be a Banach space with a monotone Schauder basis (e,). T h e n [e:] is a local dual of X . PROOF.Let U be an ultrafilter on W. Since the unit ball of X** is weak*compact, we can define a map P on X** by
Pz :=weak*- lim P;*z, n+U
z E X**.
Note that PnPk = PkP, = P, for n 5 k. Thus P,**P = PP;* = P, for every n. Since each P, is a norm one projection, P is a norm one projection on X**.Also it is clear that R ( P ) I X . Now N ( P ) c N(P,) for every n. Thus N ( P ) c nr="=,N(P;*). But the sequence of subspaces (R(P,*))is increasing and U&R(P,*) is dense in 0 [e:]. Therefore [e:]' = n&N(P,**) c N ( P ) . Thus [e:]l = N ( P ) . The following example is similar to those provided by Theorem 11.
Example 7. The subspace 2 of Lm[O,11 generated by the characteristic functions xn,i of the dyadic intervals
is a local dual of L1[0,1]. To prove it we consider the sequence (P,) of projections defined by 2n
Pnf := C(2nXn,i, f)Xn,i, i=l
and repeat the argument in the proof of Theorem 11. We refer to l 1 for more details. It is not difficult to show that this subspace 2 is isometric to the space C(A) of the continuous functions on the Cantor set A in [0,1]. NOTESAND REMARKS It was proved by Casazza and Kalton that a separable Banach space X has the M.A.P. if and only if we can find a commuting approximating sequence in X ; i.e., a sequence of finite rank operators T, acting on X so that lim IIT,a: - a:((= 0 for all 5 E X ,
n+m
44
(b)
lim llTnll = 1 and
n-co
( c ) TnTk = TkTn = Tmin{k,n); Using this fact we can prove the following result. 11) Let X be a separable Banach space with the M.A.P., and let (T,) be a commuting 1-approximating sequence o n X . Then U,"==, R(T,*)is a local dual of X and has the M.A.P.
Theorem 12. (Theorem 2.15 in
Remark 13. Let X be a space with a monotone basis. Then the projections Pn : X + X form a commuting 1-approximating sequence on X . 6. Further properties
The first result in this section extends Theorem 11. It is essentially a special case of Lemma 111.4.3 in 1 4 . Its proof in l4 is based in some ideas contained in ". Theorem 14. Every separable Banach space admits a separable local dual
space. The space L1[0,1] does not admit a smallest local dual; i.e., there exists no local dual z d contained in every local dual of L1 [o, 11. Indeed, we have seen in the previous section that L1[0,1] admits two local dual spaces C[O,11 and C(A). These subspaces of L,[O, 11 have empty intersections. Here we shall see that this cannot happen to spaces that contain no copies of C,. Theorem 15. If
X contains no copies of el, then it admits a smallest local
dual. For dual spaces we have a similar result, which is an application of Proposition v.1 in 7. Proposition 16. (Proposition 2.22 in 11) Suppose that X i s isometric t o a
dual space. Then X admits a smallest local dual z d i f and only i f it admits a smallest normang subspace 2,. I n this case z d = Zn, and this space is the unique isometric predual of X .
NOTESAND REMARKS Let dens(X) stand for the density characterof X, defined as the smallest cardinal K for which X has a dense subset of cardinality K .
45 The following result gives an extension of Theorem 14. It is essentially a special case of Lemma 111.4.4 in 14.
Proposition 17. Every subspace L of X* is contained in a local dual ZL of X with dens(2L) = max{dens(l), dens(X)}. 7. Tensor products Here we describe some local dual spaces of injective or projective tensor products of Banach spaces. For information on tensor products of Banach spaces we refer to 5 . Let X and Y be Banach spaces. Let B(X; Y) denote the vector space of all bilinear maps on X x Y and let B(X; Y)* denote the space of all linear functionals on B ( X ;Y). For each pair z E X and y E Y we define z @ y E B(X;Y)* by (z @ y
, A ) := A ( x ,y).
Definition 18. The tensor product X @ Y of the spaces X and Y is the subspace of B(X; Y)* generated by {z @ y : z E X, y E Y}. From the norms on X and Y we can derive many norms on X @ Y. The most popular ones are the projective norm 11 . llT and the injective norm
II . I l e .
Let a E X@Y. Observe that the representation a = zl@yl+. . *+zn@yn is not unique in general, e.g., (x y) @ z = z z y @ z . We define the projective norm 1) . \IT on X @ Y by
+
+
Definition 19. The projective tensor product X&Y of X and Y is the completion of (X Y , 11 . llT). We can identify (XG,,Y)* with the space B(X, Y*) of all the operators from X into Y* by defining
( T , z @ d= (Tz,y). In the previous section we considered the M.A.P. for separable Banach spaces. In general, we say that a Banach space X has the M.A.P if for every E > 0 and every compact set K in X ,there is a finite rank operator T on X such that llTll 5 1 and IITz - 211 5 E for every x E K .
46
Most of the classical Banach spaces have the M.A.P.; for example, the space L 1 ( p ) of integrable functions with respect to a finite measure p, the space C ( K )of continuous functions on a compact space K and its respective dual spaces have the M.A.P. The following result is proved using some ideas of 16. Theorem 20. Suppose that Y * has the M.A.P.T h e n the subspace K ( X , Y * ) of the compact operators in B ( X , Y * ) is a local dual of X G j , Y .
PROOF.Since Y * has the M.A.P. there exists a net (A,) of finite rank operators on Y* with IIA,II 5 1, so that lim, llA,g - 911 = 0 for every g E Y * . Note that (by compactness) we can assume that (A,) is weak*convergent in K(Y * ) **. Now, given T E B ( X , Y * ) and @ E I c ( X , Y * ) * ,the expression ( @ T ,A ) :=
(a, AT)
-
defines @T E K ( Y * ) * .Thus we can define A : K ( X ,Y*)*
(A@,T ) := lim(@,A,T) a
B ( X ,Y * ) *by
= lim(A,, @ T ) . Q
Note that for every f @ g E X * @ Y* we have
(A@,f G3 9 ) = l ai d @ ,Aa(9) . f) = (@,
f @ 9).
So A is an isometric extension operator. Analogously, we can check that for every z 8 y E X&,Y c B ( X , Y * ) * ,we have
A(. @ Y I K ( X , Y * ) ) = 5 @ Y. Thus XG,Y c A ( K ( X ,Y * ) * )and , it is enough to apply Theorem 5.
0
NOTES AND REMARKS We can define injective tensor product X & Y of X and Y as the completion of ( X @ Y , 11 . / I E ) , and obtain some results similar to those given for projective tensor products. We observe that the description of ( X & Y ) * is more complicated. It can be identified with the space Z ( X ,Y * )of all the integral operators from X into Y * with its natural norm. Theorem 21. dual of X & Y .
Suppose that Y*has the M.A.P.T h e n X * & Y * i s a local
Moreover, we can identify the space L l ( p , X ) of vector-valued integrable functions with X&L1 (p)and the space C ( K ,X ) of continuous vector Valued functions with X & C ( K ) 5 . From these facts, taking into account that
47
L 1 ( p ) *and L , ( p ) are C ( K )spaces and that C ( K ) *is a & ( p ) space, we obtain additional examples. 8. Ultrapowers of Banach spaces
The ultrapowers of Banach spaces allow us to deal with some properties of a Banach space X defined in terms of finite dimensional subspaces, i.e., the local properties. Many times a local property of X corresponds to a global property of an ultrapower XU of X with respect to some ultrafilter U .We refer to l5 for a good survey on ultrapowers of Banach spaces. See also
'.
Here we consider the ultrapowers of Banach spaces to give new examples of local dual spaces and to present some characterizations of the subspaces of X* which are local dual spaces of X . We saw in part (a) of Remark 2 that X* is finitely represented in each local dual 2 of X . This is equivalent to say that there exists an ultrapower ZU of Z such that X* is isometric to a subspace of ZU. In order to define the ultrapowers of Banach space we fix an infinite set I and a nontrivial ultrafilter U on I . We refer to the Appendix for some basic information on ultrafilters. Let X be a Banach space. We denote by l , ( I , X ) the set of all the bounded families ( x i ) i E ~ in X . The space l , ( I , X ) endowed with the supremum norm
is a Banach space. For each ( x i ) i EE~ l , ( I , X ) , (Ilxill)iEl is a bounded family of real numbers. Thus Theorem 28 implies that the limit 1imi-u IIxill exists. It is not difficult to see that
N u ( X ) := {(xi) E &,(I, X )
:
lim llzill = 0)
Z-U
is a closed subspace of & ( I , X).
Definition 22. The ultrapower XU of X (with respect to the ultrafilter
U)is defined as the quotient space
48 We will denote by [xi] the element of It is not difficult to check that =
II[.illl
Xu associated to (xi) E &,(I, X).
p%Ibill.
Remark 23. (1) The space X can be identified with the subspace of all the constant classes in X u . Equivalently, the map
z
E
x
-
[ x , x , .. .] E xu
is an isometric embedding. (2) The ultrapower ( X * ) u of X * can be identified with a subspace of the dual ( X u ) *of X u by means of the map defined by ( [ f i l ,[xi]) := p+%fi(Zi),
where [ f i ] E ( X * ) u and [xi]E Xu. In general ( X * ) uis strictly contained in (Xu)*. See the Remarks and Notes at the end of this section.
Example 8. (P.L.R. for ultrapowers)15 The ultrapower (X*),is a local dual of Xu. The following result is an extension of the P.L.R. for ultrapowers.
Theorem 24. l3 A subspace Z of X * is a local dual of X i f and only if Zu is a local dual of Xu. The are other characterizations of the subspaces of X* which are local dual spaces of X in terms of ultrapowers. Let Q : Zu -+ X* be the map defined by
Q[zi] :=weak*- lim zi E X*. 2 - 4
-
Theorem 25. A subspace Z c X* is a local dual of X if and only if there Zu such that exist a n ultrafilter U and a n isometric operator T : X *
QT = 1x8 and T ( z ) = [ z ,z , z , . ..], Vz E Z. NOTES AND REMARKS
(1) Superreflexivity (see for its definition) is the local property that corresponds to reflexivity. More precisely,
X superreflexive H Xu reflexive; .3 ( X * ) u =
(Xu)*.
49
(2) There is a P.L.R. for ultrapowers of operators. Every operator T : X + Y induces an operator TU : XU + YU defined by Tu[zi]:= [Tzi]. It is easy to see that (Tu)*is an extension of (T*)u.Thus
N ( ( T * ) u )c N ( (Tu)*) = (Yu/R(Tu))*. Example 9. Let T E B ( X ,Y ) . Then N ( (T*)u) is a local dual of (YU/R(Tu))*. lo that N ( (Tu)*)is f.d.r. in N ( (T*)u). See the section of Notes and Remarks in Section 2 for the definition 0 of f.d.r. The proof of the present result is similar.
PROOF.It is proved in
9. Appendix: Ultrafilters Here we will see a short introduction to the theory of ultrafilters which could be useful to read the previous sections. For additional information we refer to or l5. The ultrafilters are a tool that allows us to get a limit for some families of elements of a compact set. Each of these families have adherent points, and the ultrafilter selects one adherent point for each family in such a way that the linearity and some other properties of the limit are preserved. Let I be a nonempty set. A filter 3 o n I is a non-empty family of subsets of I satisfying (1) A , B E 7 + A n B E 3. (2) A E 3 , Ac B c I +B E
F.
Definition 26. An ultrafilter U o n I is a filter on I which is maximal with respect to ordering by containment. That is,
3 filter on I and U c 3 implies 3 = U . An ultrafilter on U on I is said to be nontrivial if it contains no finite subsets of I . We can define the convergence with respect to an ultrafilter as follows.
Definition 27. Let U be an ultrafilter on a set I and let K be a topological space. We say that a family (ICi)iE1c K converges over U t o k E K if for every neighborhood W of k , { Z E I : kiEW} EU.
50 In this case we write Ic = 1imi-U
Ici.
The property of the ultrafilters which is the most useful for us is given in the following result.
Theorem 28. Let K be a compact Hausdorff space and let U be an ultrafilter on I . For each family ( k i ) i E l c K , the limit 1imi-u ki exists in K . Finally let us give a detailed description of an application of Theorem
28 which appears several times in the paper.
Remark 29. Let U be a nontrivial ultrafilter on the set Fi(X) of all finite dimensional subspaces of X refining the filter associated t o the inclusion. For each F E Fi(X), let LF be an operator from X into a dual Banach space Y * such that
{IILFII : F
E
Fi(X)I
is bounded. By Alaoglu’s Theorem, the closed unit ball of Y * is compact for the weak*-topology. Therefore, for every x E X ,{LFx : F E Fi(X)} is contained in a weak*-compact set of Y* and we can define
Lx := weak*-
lim
E-iU,xE E
LFx.
Note that each 5 E X belongs eventually t o some subspace F . In this way we define a map L : X --+ Y * ,and we can write
L =weak*- lim L F . E-iU
References 1. A.G. Aksoy and M.A. Khamsi. Nonstandard methods in fixed point theory. Universitext. Springer-Verlag,New York, 1990. 2. J. Bourgain. New classes of &,-spaces. Springer Lecture Notes in Math. 889, 1981. 3. P.G. Casazza and N.J. Kalton. Notes on approximation properties in separable Banach spaces, London Math. SOC.Lecture Notes 158, Cambridge Univ. Press (1990), 49-63. 4. S. Diaz. A local approach to functionals on L m ( p , X ) , Proc. Amer. Math. SOC. 128 (2000), 101-109. 5 . J. Diestel and J.J. Uhl, Jr. Vector measures. Math. Surveys 15. Amer. Math. SOC.,Providence, 1977.
51
6. J. van Dulst. Reflexive and superreflexive Banach spaces. C.W.I. Math. Tracts 59. Amsterdam 1989. 7. G. Godefroy. Existence and uniqueness of isometric preduals: a survey, Amer. Math. SOC.Contemporary Math. 85 (1989), 131-193. 8. G. Godefroy and N.J. Kalton. Approximating sequences and bidual projections, Quarterly J. Math. Oxford 48 (1997), 179-202. 9. G. Godefroy, N.J. Kalton and P.D. Saphar. Unconditional ideals in Banach spaces, Studia Math. 104 (1993), 13-59. 10. M. Gonzalez and A. Martinez-Abej6n. Local reflexivity of dual Banach spaces, Pacific J. Math. 56 (1999), 263-278. 11. M. Gonzalez and A. Martinez-Abejbn. Local dual spaces of a Banach space, Studia Math. 142 (2001), 155-168. 12. M. Gonzalez and A. Martinez-Abej6n. Local dual spaces of Banach spaces of vector-valued functions, Proc. Amer. Math. SOC.130 (2002), 3255-3258. 13. M. Gonzalez and A. Martinez-Abej6n. Ultrapowers and subspaces of the dual of a Banach space, Glasgow Math. J., to appear. 14. P. Harmand, D. Werner and W. Werner. M-ideals in Banach spaces and Banach algebras. Lecture Notes in Math. 1547.Springer-Verlag, Berlin, 1993. 15. S. Heinrich. Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104. 16. J. Johnson. Remarks o n Banach spaces of compact operators, J. Funct. Anal. 32 (1979), 304-311. dir 17. W.B. Johnson, H.P. Rosenthal and M. Zippin. O n bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. 18. N.J. Kalton. Locally complemented subspaces and &-spaces f o r 0 < p < 1, Math. Nachr. 115 (1984), 71-97. 19. A. Lima. T h e metric approximation property, norm-one projections and intersection properties of balls, Israel J. Math. 84 (1993), 451-475. 20. A. Martinez-Abej6n. An elementary proof of the principle of local reflexivity. Proc. Amer. Math. SOC. 127 (1999), 1397-1398. 21. W. Rudin. Real and complex analysis, 3rd ed. McGraw-Hill, 1986. 22. B. Sims and D. Yost. Banach spaces with m a n y projections, Proc. Cent. Math. Anal. Austral. Nat. Univ. 14 (1986), 335-342.
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53
ORBITS OF OPERATORS
VLADIMiR MULLER* Institute of Mathematics, Czech Academy of Sciences, iitnli 25, 115 67 Praha 1, Czech Republic E-mail:
[email protected] The aim of this paper is to give a survey of results and ideas concerning orbits of operators and related notions of weak and polynomial orbits. These concepts are closely related to the invariant subspace/subset problem. Most of the proofs are not given in full details, we rather try to indicate the basic ideas. The central problems in the field are also formulated.
Mathematics Subject Classification: primary 47A05, 47A15, 47A16, secondary 47A11. Keywords: orbits, invariant subspace problem, hypercyclic vectors, weak orbits, capacity, Scott Brown technique. 1. Introduction
Denote by B ( X ) the algebra of all bounded linear operators acting on a complex Banach space X . Let T E B ( X ) . By an orbit of T we mean a sequence {Tnx : n = 0,1, , ..} where x E X is a fixed vector. The concept of orbits comes from the theory of dynamical systems. In the context of operator theory the notion was first used by Rolewicz [30]. Orbits of operators are closely connected with the local spectral theory, the theory of semigroups of operators [26], and especially, with the invariant subspace problem, see e.g. [4]. The invariant subspace problem is the most important open problem of operator theory. Recall that a subset M c X is invariant with respect to an operator T E B ( X ) if T M c M . The set M is nontrivial if (0) # M # X.
*Supported by grant no. 201/03/0041 of GA CR.
54
Problem 1.1. (invariant subspace problem) Let T be an operator on a Hilbert space H of dimension 2 2. Does there exist a nontrivial closed subspace invariant with respect to T? It is easy to see that the problem has sense only for separable infinitedimensional spaces. Indeed, if H is nonseparable and x E H any nonzero vector, then the vectors x , T x ,T 2 x , .. . span a nontrivial closed subspace invariant with respect to T . If dim H < 00, then T has at least one eigenvalue and the corresponding eigenvector generates an invariant subspace of dimension 1. Note that the existence of eigenvalues is equivalent to the fundamental theorem of algebra that each nonconstant complex polynomial has a root. Thus the invariant subspace problem is nontrivial even for finite-dimensional spaces. Examples of Banach space operators without nontrivial closed invariant subspaces were given by Enflo [8],Beauzamy [3] and Read [28]. Read [29] also gave an example of an operator T with a stronger property that T has no nontrivial closed invariant subset. It is not known whether such an operator exists on a Hilbert space. This “invariant subset problem” may be easier than Problem 1.l.
Problem 1.2. (invariant subset problem) Let T be an operator on a Hilbert space H . Does there exist a nontrivial closed subset invariant with respect to T? Both Problems 1.1 and 1.2 are also open for operators on reflexive Banach spaces. More generally, the following problem is open:
Problem 1.3. Let T be an operator on a Banach space X . Does T* have a nontrivial closed invariant subset/subspace? It is easy to see that an operator T E B ( X ) has no nontrivial closed invariant subspace if and only if all orbits corresponding to nonzero vectors span all the space X (i.e., each nonzero vector is cyclic). Similarly, T E B(X)has no nontrivial closed invariant subset if and only if all orbits corresponding to nonzero vectors are dense, i.e., all nonzero vectors are hypercyclic. Thus orbits provide the basic information about the structure of an operator.
55 Typically, the behaviour of an orbit {Tnz : n = 0,1,. . . } depends much on the initial vector x E X. An operator can have some orbits very regular and other orbits extremely irregular.
Example 1.4. Let H be a separable Hilbert space with an orthonormal basis {eo, e l , . . . }. Let S be the backward shift, i.e., S is defined by Seo = 0 and Sei = ei-l (i 2 1). Consider the operator T = 2 s . Then:
c H such that IIT"z1I -+ 0 (z E M I ) ; (ii) there is a dense subset M2 c H such that llTnzll 00 (x E M z ) ; (iii) there is a residual subset M3 c H such that the set { P z : n = 0 , 1 , . . . } (i) there is a dense subset M I
4
is dense in H for all z E M3.
As the set MI it is possible to take the set of all finite linear combinations of the basis vectors ei. Properties (ii) and (iii) follow from general results
that will be discussed in the subsequent sections. The paper is organized as follows. In the following section we study regular orbits. Of particular interest are the orbits satisfying I(Tnzll 4 00. It is easy to see that if an operator T has such an orbit, then it has a nontrivial closed invariant subset {Tnz : n = 0 , 1 , . . . }-. In the third section we study the other extreme - hypercyclic vectors, i.e., the vectors with very irregular orbits. In the subsequent sections we study weak and polynomial orbits. A weak orbit of T is a sequence {(Tnrc,rc*) : n = 0,1,. . . } and a polynomial orbit of T is a set of the form {p(T)a: : p polynomial}, where z E X and
x* E
x*.
Polynomial orbits are closely related with the notions of capacity and local capacity of an operator. These concepts are studied in Section 6. In the last section we discuss the Scott Brown technique which is the most efficient method of constructing invariant subspaces of operators. On an illustrative example we show the basic ideas of the method, which are closely connected with orbits. For simplicity we consider only complex Banach spaces. However, some results concerning orbits remain true also for real Banach spaces. In particular, all results based on the Baire category theorem remain unchanged for real Banach spaces. Although the invariant subspace problem is usually formulated for complex Hilbert spaces, the corresponding question for real spaces (of dimen-
56
sion 2 3) is also open; it is very easy to find an operator on a 2-dimensional real Hilbert space without nontrivial invariant subspaces. 2. Regular orbits
Let X be a complex Banach space and let T E B ( X ) . If T is power bounded (i.e., sup, llTnll < 00) then all orbits are bounded. The converse follows from the Banach-Steinhaus theorem. Theorem 2.1. Let T E B ( X ) . Then T is power bounded if and only if sup, llTnxll < 00 for all x E X .
A more precise statement is given by the following theorem. Recall that a subset M c X is called residual if its complement X \ M is of the first category. Equivalently, M is residual if and only if it contains a dense G6 subset. Theorem 2.2. Let T E B ( X ) and let (a,) be a sequence of positive numa, = 0 . Then the set of all points x E X with the bers such that property that llTnxll 2 a,llTnll f o r infinitely many n is residual. Proof. The statement is trivial if T is nilpotent. In the following we assume that T n # 0 for all n. For k E N set hfk =
x : there exists n 2 k such that IIT"x[I > a,llTn(l}. is an open set. We prove that Mk is dense. Let x E x and
{x
E
Clearly Mk > 0. Choose n 2 k such that < 1. There exists z E X of norm one such that llTnzll > u , E - ~ I I T ~ Then ~~.
E
+
and so either llTn(z EZ)II > a,lITnll or llT"(x - E Z ) ~ )> u,llTnll. Thus either z EZ E Mk or z - E Z E Mk, and therefore dist {z, Mk} 5 E . Since x and E were arbitrary, the set Mk is dense. By the Baire category theorem, the intersection k f k is a dense G6 set, hence it is residual. Clearly each z E Mk satisfies llT"x11 2 u,llTnll for infinitely many n. 0 Denote by r ( T ) = max{IXI : X E a(T)} the spectral radius of an operator T E B ( X ) . By the spectral radius formula we have T ( T ) = limn+co (ITn II l / n = inf, IITn(ll/n. Recall that r(Tn) = ( r ( T ) ) " for all n.
+
57 For z E X let r,(T) denote the local spectral radius defined by r,(T) = limsup, jo3 IIT"II1/" (the limit limn+o3 IIT"zlI1lnin general does not exist). The local spectral radius plays an important role in the local spectral theory. Note that the resolvent z H ( z - T)-l = $& is analytic on the set { z : IzI > r ( T ) } .Similarly, the local resolvent z H (z-T)-lz = C:=, can be analytically extended to the set { z : IzI > r,(T)}. It is easy to see that r,(T) 5 r ( T )for all z E X. Corollary 2.3. cf. [31] Let T E B ( X ) . Then the set {z E X : r,(T) = r ( T ) } is residual.
c,"==,
Proof. Let a, = n-l. By Theorem 2.2, there is a residual subset M c X such that for each z E M we have llTnzll 2 n-lIITnll for infinitely many n. Thus
for all z E M .
0
As we have seen, it is relatively easy to construct vectors x such that infinitely many powers Tnx are large. It is much more difficult t o construct orbits such that all powers Tnz are large in the norm. The result (and many other results concerning orbits) is based on the spectral theory and therefore it is valid only for complex spaces. For real Banach spaces see Remark 2.14. Denote by ae(T)the essential spectrum of T E B ( X ) ,i.e., the spectrum , K ( X ) is the ideal of all of p(T) in the Calkin algebra B ( X ) / K ( X ) where compact operators on X and p : B ( X ) B ( X ) / K ( X )is the canonical projection. Equivalently, a e ( T )= {A E C : T - X is not Fredholm}. Let r e ( T )denote the essential spectral radius, r e ( T )= max{IXI : X E a e ( T ) } . If X is an infinite dimensional Banach space and T E B ( X ) then a e ( T ) is a nonempty compact subset of a ( T ) . Moreover, the difference a(T) \ a,(T) is equal to the union of some bounded components of C \ ae(T) and of at most countably many isolated points. In particular, if X E a ( T ) belongs to the unbounded component of C \ ae(T)then X is an isolated point of the spectrum a ( T ) ,it is an eigenvalue of finite multiplicity and the corresponding spectral subspace is finite dimensional. Denote further by ane(T)the essential approximate point spectrum of T , i.e., a,,(T) is the set of all complex numbers X such that
-
inf{ l
l ( -~ ~ ) z l:lz E M , IIzlI = I }
=
o
58
for every subspace M c X with codimM < 00. It is easy to see that X $ a T e ( T )if and only if dimKer (T - A) < 00 and T - X has closed range, i.e., if T - A is upper semi-Fredholm. It is known [14] that aTe(T)contains the topological boundary of the essential spectrum a,(T). In particular, one(T)is a nonempty compact subset of the complex plane for each operator T on an infinite dimensional Banach space X. The elements of the essential approximate point spectrum a,,(T) are ) are "approxvery useful for the study of orbits. For each X E a T e ( Tthere imate eigenvectors" - vectors x E X of norm 1 such that II(T - X)xll is arbitrarily small. Moreover, the approximate eigenvectors can be chosen in an arbitrary subspace of finite codimension. This property is particularly useful in various inductive constructions. The following result was proved in [20]; for Hilbert space operators see 141.
Theorem 2.4. Let T be an operator on a Banach space X , let E > 0 and let (a,) be a sequence of positive numbers such that limn+m a, = 0. Then:
(i) there exists a vector z E X such that llxll a, . r ( F ) for all n;
< sup, a,
+E
and llTnxll 2
(ii) there exists a dense subset of points x E X such that llTnxll 2 anr(Tn) for all but a finite number of n. Outline of the proof. Let X E a ( T ) satisfy two cases:
1x1 = r ( T ) . We distinguish
(a) Suppose that r ( T ) > r e ( T ) .Then X is an eigenvalue of T . The corresponding eigenvector 2 of norm one satisfies ((T"lcll= IIX"z(( = r(T") for all n. Moreover, X is an isolated point of the spectrum of T and the spectral subspace Xo corresponding to X is finite dimensional. Let u E X \ X O . It is easy to verify that there is a positive constant c = c(u) such that llTjull 2 c . r ( T j ) for all j. Thus in this case the set of all points satisfying (ii) is even residual. for all E > 0, n E N and A4 c X of (b) Let r ( T ) = r e ( T ) .Since X E uTe(T), finite codimension there exists x E M such that IIxlI = 1 and II(Tj -Xj)xlI < E ( j 5 n). These "approximate eigenvectors" are basic building stones used in the construction of the vector with the required properties. We indicate the proof for Hilbert space operators.
59 Without loss of generality it is possible to assume that 1 > a1 > a2 > r ( T ) = 1 and X = 1. For k 2 0 set r k = min{j : aj < 2-'}. w e construct inductively vectors X k E of norm 1 such that Tjxk = xk ( j 5 rk) and e
,
x
< k , j 5 Tk) (note that the subspace { u : Tju I Tjxi (i < k , j 5 r k ) } is of finite T'Xk 1@ X i
(2
codimension) . 2-2+lzi. Set x = Let rk-1 < j 5 7-k. Since Tjxi 1Tjxk
xFl
(i # k), we have
Thus x satisfies llTjxll 2 aj for all j. The full statement of Theorem 2.4 can be obtained by a modification of this argument; we omit the details. 0 For Banach spaces the proof is a little bit more complicated. The basic idea is t o use instead of the orthogonal complement of a finite dimensional subspace (which was in fact used here) the following lemma.
Lemma 2.5. Let X be a Banach space, let F c X be a finite dimensional subspace and let E > 0. Then there exists a subspace M c X of finite codimension such that
I b + f II 2 (1 - E)max{llflli
lle11P)
for all f E F and m E M .
An immediate consequence of Theorem 2.4 is the following corollary.
Corollary 2.6.Let T E B ( X ) . Then:
(i) the set { x E X : liminf,,, JJT"xJJ1/" = r ( T ) }is dense; llTnxlllln = r ( T ) } is residual; (ii) the set { x E X : limsup,,, (iii) the set of all x E X such that the limit limn+, I(Tnxl('lnexists (and is equal to r ( T ) )is dense. As another corollary we get that the infimum and the supremum in the spectral radius formula
60 can be exchanged.
Corollary 2.7. Let T E B(X).Then
Example 2.8. Let H be a separable Hilbert space with an orthonormal basis ( e j : j = 0,1,. . . } and let S be the backward shift, Seo = 0, Sej = ej-1 llS"zlll/" = 0} is ( j 2 1). Then the set {z E H : liminf,,, residual. Since r ( S ) = 1 and the set {z E H : limsup,,, IIS"zlll/" = l} is also residual, we see that the set {z E H : the limit limn+, IISnzII1/"exists} is of the first category (but it is always dense by Corollary 2.6). Proof. For k E N let
Mk
=
{ x E X : there exists n L k such that IlS"x1I < k-"}.
Clearly Mk is an open subset of X . Further, Mk is dense in X. To see this, let u E X and c > 0. Let u = CEoajej and choose n 2 k such that
Cj"= lql2 , < c2. Set y
= CyIi a j e j . Then IIy - uII < E and S"y = 0. Thus y E Mk and Mk is a dense open subset of X. By the Baire category theorem, the set A4 = nE"=,k is a dense G6 subset of X, hence it is residual. Let x E M . For each k E N there is an n k 2 k such that IJSnkzll< and so liminf,,, IIS"zJJ'/n= 0. 0
It is also possible to combine conditions of Theorems 2.2 and 2.4 and to obtain points z E X with llTnxl\ 2 a,. IITnII for all n; in this case, however, there is a restriction on the sequence (a,).
Theorem 2.9. Let T E B ( X ) , let (a,) be a sequence of positive numbers such that C,U;'~< 00. Then there exists x E X such that llTnxll 2 anlITnll for all n. There is a dense subset L c X such that for each x E L there is a k E N with the property that
llTnxll 2 anllTnll
(n2 k).
Outline of the proof: Fix k E N. We indicate the construction of a vector x satisfying llTjxll _> ajIITjll ( j 5 k). The vector satisfying this relations for all n can be then obtained by a limit procedure.
61
Let k E N be fixed. For j = 1 , 2 , . . . ,k fix a vector z j E X of norm one which almost attains the norm of T j , i.e., llTjxll IlTjII (we omit the exact calculations). L e t A = { X = ( X l , ...,X k ) E ~ ' : l X j 1 5 a ~ ' 3 f o r a l l j } . F o r X E A l e t k U A = Cj=l X j z j . Consider the Lebesgue measure p on A. For j = 1,.. . , k let Aj = {A E A : llTju~II< ajIITjll}. The basic idea of the proof is to show that p ( A \ Us=l,Aj) > 0, which means that there exists X E A such that llTju~II2 aj IITJ11 for all j = 1 , . . . ,k. For details see [24].
A better estimate can be obtained if we replace the norm llSJJof an operator 5' E B(X) by the quantity IISIIp = inf{ llSlMll : M c X,codimM
0, a, -+ 0, then there exists a dense subset L c X such that for each constant c > 0 with llTnzl(> ca,r(T)" for all n.
2
E L there is a
Theorem 2.9 can be modified in the following way: let T be an operator on a real Banach space X, let (a,) be a sequence of positive numbers such that C,a~" < 00. Then there exists z E X such that IITnz((2 a,(lT"(( for all n. There is a dense subset L c X such that for each z E L there is a k E N with the property that IIT"zll 2 anllTnll
( n 2 k).
3. Hypercyclic vectors Vectors with extremely irregular orbits are called hypercyclic. More precisely, a vector z E X is called hypercyclic for an operator T E B ( X ) if the set {Tnz : n = 0, 1, . . . } is dense in X. An operator T is called hypercyclic if there is at least one vector hypercyclic for T . Recall also that a vector z E X is called cyclic for T E B ( X ) if the set { p ( T ) z: p polynomial} is dense in X, and supercyclic for T if {AT"%: X E c,n = 0 , l . . . }- = These notions make sense only for separable Banach spaces. It is easy to see that an operator in a non-separable Banach space can not have cyclic (supercyclic, hypercyclic) vectors. Moreover, it is not difficult to show that there are no hypercyclic operators on finite-dimensional Banach spaces (this follows from the fact that T* has eigenvalues, cf. the proof of Theorem 3.2 below). In the rest of this section we assume that all Banach spaces are infinite dimensional and separable. It is easy to find an operator that is not hypercyclic. For example, any contraction (or more generally, a power bounded operator) is not hypercyclic. On the other hand, if T has at least one hypercyclic vector then almost all vectors are hypercyclic.
x.
63
Theorem 3.1. Let T E B ( X ) be a hypercyclic operator. Then there is a residual set of vectors hypercyclic for T . Proof. Let x E X be hypercyclic for T . For each Ic E N the vector T k x is hypercyclic for T , and so the set of all hypercyclic vectors is dense. Let ( U j ) be a countable base of open subsets in X . It is easy to see that the set of all vectors hypercyclic for T is equal to U, T - n U j , which is a Gs subset. 0
nj
Theorem 3.2. Let T E B ( X ) be a hypercyclic operator. Then there exists a dense linear manifold L c X such that each nonzero vector in L is hypercyclic for T . Proof. We show first that T* has no eigenvalues. Suppose on the contrary that there are X E C and a nonzero vector x* E X * such that T*x*= Ax*. Let x E X be a hypercyclic vector for T . Then C = {(T,x,x*) : n = 0 , 1 ,
. . . } - = ( x , x * ) * ( X ~ : ~ = O , ,... ~ }-.
It is easy to see that the last set can not be dense in C. Thus X is not an eigenvalue of T * ,and so (T - X)X is dense in X for each X E C. Let x be a hypercyclic vector for T . We show that p(T)x is also hypercyclic for each nonzero polynomial p . Write p ( z ) = a ( z - XI) . . . (2 - A), where a # 0, XI,. . . ,A, E C. Then
(T"p(T)x: n = 0,1,. . . } = a(T - X i ) . . . (T - X,){Tnx : 72 = 0 , 1 , . . . }. The last set is dense in X since x is hypercyclic for T , and the operators T - X j have dense ranges for each j. Thus p(T)x is hypercyclic for T . 0 The following criterion provides a simple way of constructing hypercyclic vectors, see [16], [ l l ] . It also implies property (iii) in Example 1.4.
Theorem 3.3. Let T E B ( X ) . Suppose that there is an increasing sequence of positive integers (nk)such that: (i) there is a dense subset X O c X such that lim~--tmTnkx 4 0 for all xE (ii) T"kBx is dense in X , where Bx denotes the closed unit ball in X . Then T is hypercyclic. By Theorem 3.1, this means that the set of all hypercyclic vectors is residual.
x,; Uk
Conversely, suppose that T is hypercyclic. Then it is not difficult to show that T satisfies both conditions (i) and (ii), but not necessarily for
64
the same subsequence ( n k ) . Thus the conditions in Theorem 3.3 are close to the notion of hypercyclicity (cf. Problem 3.12).
A similar criterion may be used to construct closed infinite dimensional subspaces consisting of hypercyclic vectors, see [19], [12] and [17].
Theorem 3.4. Let T E B ( X ) . Suppose that T satisfies the conditions of Theorem 3.3 and that the essential spectrum a,(T) intersects the closed unit ball. Then there is a closed infinite dimensional subspace M c X such that each nonzero vector in M is hypercyclic for T . Theorems 3.1 - 3.4 indicate that hypercyclic vectors and operators are quite common, that in some sense it is a typical behavior of an orbit. Similarly as in Theorem 3.1 it is possible to show that the set of all hypercyclic operators on a Banach space X is a Gg set. It is not dense since the operators with IlTll < 1 can not be hypercyclic. Thus the set of all hypercyclic operators is a residual subset of its closure. By [15],it is possible to characterize the closure of hypercyclic operators on a separable Hilbert space. For Banach spaces such a characterization is not known.
Theorem 3.5. Let H be a separable Hilbert space, let T E B ( H ) . Then T belongs to the closure of hypercyclic operators if and only if the following conditions are satisfied: (i) the set a w ( T )U { z E C : IzI = 1) is connected; (ii) ao(T)= 0; (iii) ind (A - 5") 2 0 for all X E C such that X - T is semi-Fredholm.
nKEK(H)
a(T+ Here a w ( T )denotes the Weyl spectrum of T , a w ( T )= K ) . Equivalently, X $ aw ( T )if and only if T - X is Fredholm and ind (T A) = 0. Recall that an operator S is called semi-Fredholm if it has closed range and either dim ker S < co or codim SX < 00. The index of a semi-Fredholm operator S is defined by ind S = dim ker S - codim SX. Furthermore, ao(T) denotes the set of all isolated points of a ( T ) such that the corresponding spectral subspace is finite dimensional.
Theorem 3.6. Let T E for T . Then:
B(X)be an operator and let z E X be hypercyclic
65 (i) z is hypercyclic for T" for each n E N;
(ii) z is hypercyclic for AT for each X E C, 1x1 = 1; (iii) if T is invertible then T-' is hypercyclic. The first statement of Theorem 3.6 was proved in [2]. For (ii) see [MI. The third statement follows from the observation that T is hypercyclic if and only if for all nonempty open subsets U, V c X there exists n E N such that T"U n V # 0. Although it is relatively easy to construct an operator with a residual set of hypercyclic vectors (see Example 1.4), it is extremely difficult to construct an operator with all nonzero vectors hypercyclic. The first example of this type was constructed by Read [29] on the space e l . Equivalently, such an operator has no nontrivial closed invariant subset. It is an open problem whether this can happen in Hilbert spaces, cf. Problems 1.2 and 1.3. The next result shows that such an operator must satisfy certain rather narrow conditions on the norms IITn(l.
Theorem 3.7. Let T be an operator on a Banach space X which has no nontrivial closed invariant subsets. Then r ( T ) = re(T)= 1, supn llT"ll = 00, C , ((Tnll-2/3 < 00 and C, llTn(lil< 00. If X is a Hilbert space then
C,
JJTnJJ-' < co and
C,
JJTn]1,1'2< 00.
Indeed, if T does not satisfy the conditions above, then either T is power bounded or there exists a vector z E X with llTnzll + 00, see Theorems 2.4, 2.9, 2.10 and 2.11. Hence {Tnz : n = 0 , l . . , }- is a nontrivial closed invariant subset with respect to T . Thus it is a very interesting question for which operators there are orbits satisfying llTnzll + co.
Problem 3.8. What are the best exponents in Theorem 3.7? Example 3.9. There is an operator T on a Hilbert space H such that llTn(l+ co and there is no z E H with \lTnxll + 00, see [4]. As an example it is possible to take a unilateral weighted shift with suitable weights; the operator satisfies ((T"1)= (lnn)1/2. It is also possible to construct an operator T E B ( H ) such that inf, IIT"zII = 0 and SUP, llTnzll = 00 for all nonzero vectors z E H .
66 We finish this section with some other open problems. Problem 3.10. Let T be a Hilbert space operator such that limn+w llTnll = 00 and the norms llTnll form a nondecreasing sequence. Does there exist a vector z E H such that IIT"zll --t oo? Problem 3.11. Is the characterization of the closure of hypercyclic operators (Theorem 3.5) true also for Banach spaces? Problem 3.12. Does there exist a hypercyclic operator T E B(X)that does not satisfy conditions of Theorem 3.3? There are other equivalent formulations of this problem. The most interesting reformulation is: does there exist a hypercyclic operator T such that T @ T is not hypercyclic, see [5]? Problem 3.13. An operator T E B ( X ) is called weakly hypercyclic if there exists a vector z E X such that the orbit {Tnz : n = 0,1,. . . } is weakly dense in X, see [9]. Does there exists a weakly hypercyclic operator that is not hypercyclic? Must a weakly dense orbit be norm dense? Note that the corresponding notion of weakly cyclic vectors makes no sense since a weakly closed linear manifold is automatically closed by the Hahn-Banach theorem. 4. Weak orbits
Weak orbits were introduced and first studied by van Neerven [26]. Many results for orbits of operators can be modified also for weak orbits. For a survey of results see e.g. [26], [24]. The following three results are analogous t o the corresponding statements for orbits. Theorem 4.1. Let T E B(X)and let (a,) be a sequence of positive numbers such that limn--rwan = 0. Then the set of all pairs (x,z*)E X x X* with the property that I(Tnz,z*)l > a,IITnll for infinitely many n is a residual subset of X x X*. Theorem 4.2. Let T be an operator on a Banach space X, let (a,) be a sequence of positive numbers such that C , < 00. Then there exist z E X and x* E X* such that I(Tnz,z*)l 2 anIITnll for all n. Theorem 4.3. Let T E B(X).Then:
67 (i) the set {(z,z*) E X x
X* : liminf I(Tnz,z*)(l/n = r ( T ) }is dense; n-im
(ii) the set { (x,z*) E X x
X* : limsup I(Tnx,x * ) ( l l n= T ( T ) }is residual; n-+w
(iii) the set of all pairs
(5, z*)
E X x X*such that the limit lim
I(Tnx)I1/n
n-+m
exists (and is equal t o r ( T ) )is dense. The statement analogous to Theorem 2.12 for weak orbits is not true:
Example 4.4. There exists an operator T on a Hilbert space H such that < co for all x , y E H.
c n
As an example it is possible to take the operator T = @ElSk, where
Sk is the shift operator on a (k + 1)-dimensional Hilbert space, see [24].
The statement analogous t o Theorem 2.4 for weak orbits is an open problem:
Problem 4.5. Let T E B ( X ) , let (a,) be a sequence of positive numbers satisfying lim,-,man = 0. Do there exist x E X and z* E X* such that I(Tnx,z*)) 2 a n r ( T n ) for all n? The statement is false for real Banach spaces. A partial positive answer is given in the following case which is important from the point of view of the invariant subspace problem. Some other partial results were given in P61.
Theorem 4.6. Let T be an operator on a Hilbert space H such that 1 E a ( T ) and Tnz -+ 0 for all z E H . Let (a,) be a sequence of positive numbers satisfying limn+man = 0. Then there exists z E H such that Re (T"x,5) > a, for all n. Using Theorem 4.6 and techniques of [18] it is possible to obtain the following result.
Theorem 4.7. Let T be a power bounded operator o n a Hilbert space H satisfying r ( T ) = 1. Then there is a nonzero vector x E H such that x is not supercyclic. Moreover, T has a nontrivial closed invariant positive cone, i.e., there is a nontrivial closed subset M c H such that T M c M , M + M c M and t M c M (t 2 0 ) .
68
It is a natural question whether the previous result can be improved in order to obtain an invariant real subspace.
Problem 4.8. Let T be a power bounded operator on a Hilbert space such that r ( T ) = 1. Does T have a nontrivial closed invariant real subspace, i.e., does there exists a nontrivial closed subset M c H such that T M C M , M M c M and t M c M (t E R) ?
+
Problem 4.9. Is Theorem 4.7 true f o r operators on reflexive Banach spaces ? 5. Polynomial orbits If 2 is an eigenvector of T , T x = Xz for some complex A, then p(T)a: = p(X)x for every polynomial p , and so we have a complete information about the polynomial orbit {p(T)z: p polynomial}. Unfortunately, operators on infinite dimensional Banach spaces have usually no eigenvalues. The proper tool appears to be the notion of the essential approximate point spectrum a?re(T).
The following result is an analogue of Theorem 2.4.
Theorem 5.1. [22] Let T be a n operator o n a Banach space X , let X E axe(T).Let (a,) be a sequence of positive numbers with a, = 0. Then: (i) there exists x E X such that llP(T)X11 2 a d e g p IP(X>l
f o r every polynomial p ; (ii) let u E X , E > 0. Then there exists y C = C(E)such that IIy - uII 5 E and IlP(T)Yll 2 C ' a d e g p '
E
X and a positive constant
b(X)l
f o r every polynomial p . In the previous theorem we expressed the estimate of llp(T)x11 by means of Ip(X)l where X was a fixed element of axe(T).Next we are looking for Since da,(T) 3 aTe(T), an estimate in terms of max{lp(X)( : X E aTe(T)}. by the spectral mapping theorem for the essential spectrum ae we have
69 An important tool for the results in this section is the following classical lemma of Fekete [lo]. It enables to estimate the maximum of a polynomial on a (in general very complicated) compact set u&?) by means of its values at finitely many points.
Lemma 5.2. Let K be a non-empty compact subset of the complex plane and let k 2 1. Then there exist points U O ,ul,. . . ,U k E K such that m={Ip(z)I : E K } I (k
+ 1). Om= IP(Ui)l liSk
for every polynomial p with degp 5 k. Moreover, we have
k. for all polynomials p with degp I By using the previous lemma we can get [21], [23] Theorem 5.3. Let T be an operator on a Banach space X, let k 2 1. Then: (i) if carda,,(T) 2 k
E
2 0 and
+ 1 then there exists x E X with ((Ic((= 1 and
for every polynomial p with degp I k. (ii) let x E X and E > 0. Then there exists C = C(E)such that IIy - 5 E and
XI(
IMT)YII2 c
y E
X and a positive constant
(1 + degp)-(l+E) re(p(T))
for every polynomial p . The proof of Theorem 5.3 is much simpler for operators on Hilbert spaces. The same result for Banach space operators can be obtained by a Dvoretzky’s theorem type argument. A simpler proof based on Lemma 2.5 is available for weaker estimates Ilp(T)xII 2 z(k;t)2re(p(T)) and llp(T)z/(_> C . (1 degp)-(2+E)r,(p(T)),respectively. The estimates in Theorem 5.3 (i) are the best possible.
+
Example 5.4.[23] Let k E N. There exists a Banach space X and an operator T E B ( X ) such that for each x E X of norm one there is a polynomial p of degree 5 k with llp(T)xll I (k l)-lr,(p(T)).
+
70 6. Capacity
The notion of capacity of an operator (or more generally, of a Banach algebra element) was introduced and studied by Halmos [13]. If T E B ( X ) then capT = lim (capkT)llk= inf(capkT)l/k, k
k-+m
where capkT = inf { Ilp(T))): p E
PL}
and P; is the set of all monic (i.e., with leading coefficient equal to 1) polynomials of degree k. This is a generalization of the classical notion of capacity (sometimes also called Tshebyshev constant) of a nonempty compact subset K of the complex plane: cap K = lim (capkK)llk= inf(capkK)l/k k-cc
k
where capkK = inf { l l p l l ~: p E P l }
and
llpl)= ~ sup{lp(z)l : z E K}.
The classical capacity cap K is equal to the capacity of the identical function f ( z ) = z considered as an element of the Banach algebra of all continuous functions on K with the sup-norm. Another connection between these two notions is given by the following main result of [13]. Theorem 6.1. capT = capa(T) for each operator T E B ( X ) . Let x E X. The local capacity of T at x can be defined analogously. We define capk(T,x) = inf { IIp(T)xll p E P i } and cap (T,x) = lim sup capk(T,x)l/k k+cc
(in general the limit does not exist). It is easy to see that cap(T, x) 5 cap T for every x E X . Note that there is an analogy between the spectral radius and the capacity of an operator:
71
r,(T) = limsup IIT~x))'~', k-+m
capT = lim (cap kT)llk= inf(cap kT)'lk, k+m
cap (T,x) = lim sup (cap k(T,z)) I l k . k-+m
Furthermore, capT 5 r(T) and cap(T,x) Theorem 6.2. Let T E B ( X ) . Then:
[a)
the set (x E
x : liminfk,,
I r,(T)
for all z E X.
capk(T,z)l/k= capT} is dense an
x;
(ii) the set {x E X : cap(T, x) = cap T} is residual in X;
(iii) the set {x
E
x : limk,,
cap k(T,x)llk = cap T} as dense in
x.
Outline of the proof. By Theorem 5.3, there is a dense subset of vectors X with the property that IIp(T)zll 2 C * (1 degp)-2re(p(T)) for all polynomials p . Thus we have
+
II: E
caPk(T,z)
inf{llP(T)zll : P E
pi}
2 C . (1+ k)-2inf{re(p(T)) : p E P i } = C(1+ k)-2capkae(T).
Hence liminf(cap k ~ ) l / k 2 liminf(cap k a , ( ~ ) ) l /=~cap ae(T). k-cm
k-m
Using the general relations between o(T) and a,(T), it is possible to see that capo,(T) = capa(T). Hence, by Theorem 6.1,liminfk,,(cap (T,z))'lk= capT for all x in a dense subset of X. The second statement requires a more refined arguments, see [24]. 0 An operator T E B ( X ) is called quasialgebraic if and only if cap T = 0. Similarly, T is called locally quasialgebraic if cap ( T ,II:) = 0 for every z E X. It follows from Theorem 6.2 that these two notions are equivalent. Corollary 6.3. An operator is quasialgebraic if and only if at is locally quasialgebraic. Corollary 6.3 is an analogy to the well-known result of Kaplansky: an operator is algebraic (i.e. p ( T ) = 0 for some non-zero polynomial p ) if and only if it is locally algebraic (i-e.,for every II: E X there exists a polynomial p , # 0 such that p,(T)a: = 0).
7. Scott Brown technique The Scott Brown technique is an efficient way of constructing invariant subspaces. It was first used for subnormal operators but later it was adapted
72 to contractions on Hilbert spaces and, more generally, to polynomially bounded operators on Banach spaces. Some results are also known for n-tuples of commuting operators. The basic idea of the Scott Brown technique is to construct a weak orbit {(T"x, x*): n = 0, 1,. . . } which behaves in a precise way. Typically, vectors x E X and x* E X* are constructed such that
(T"x,x*)=
{
0 1
nZ1; n=0.
Equivalently, (1)
(P(T)X,x*) = P ( 0 )
for all polynomials p . Then T has a nontrivial closed invariant subspace. Indeed, either T x = 0 (and x generates a 1-dimensional invariant subspace) or the vectors {Tnz : n 2 1) generate a nontrivial closed invariant subspace. The vectors x and x* satisfying the above described conditions are constructed as limits of sequences that satisfy (1) approximately. Let D = { z E CC : JzI < 1) denote the open unit disc in the complex plane and T = { z E CC : IzI = 1) the unit circle. Denote by P the normed space of all polynomials with the norm IJpJJ = sup{)p(z)): z E D}. Let P* be its dual with the usual dual norm. Let $ E P*. By the Hahn-Banach theorem, $ can be extended without changing the norm to a functional on the space of all continuous function on T with the sup-norm. By the Resz theorem, there exists a Bore1 measure p on T such that ((pL(( = 11$11 and $ ( p ) = J p d p for all polynomials p . Clearly, the measure is not unique. Let L1 be the Banach space of all complex integrable functions on T with the norm llflll = (27r)-'J:", If(eit))dt. Of particular interest are the following functionals on P: (i) Let X E D. Denote by Ex the evaluation at the point A, i.e., Ex is defined
by E x b ) = P ( 4
( P E PI.
(ii) Let f E L1. Denote by M f E
~ f ( p=) (27r)-1/"
P*the functional defined by p(e"">(eit)dt
(pE
P).
--x
Then IlMfll F Ilflll. The evaluation functionals Ex are also of this type. Indeed, for X E D we have Ex = M p x , where Px(eit) = 5- 1-JX12 is the Poisson kernel. In
73
particular, if g = 1 then Mg(p)= p ( 0 ) for all p , and so Mg is the evaluation at the origin. (iii) Let k > 0 and let T : X -+ X a polynomially bounded operator with constant k, i.e., T satisfies the condition Ilp(T)II 5 IclJpJJ for all polynomials p. Fix x E X and x* E X * . Let x @ x* E P* be the functional defined by
.(
@ X*>(P) =
(p(T)x,x*)
(P E P).
Since T is polynomially bounded, x @ x* is a bounded functional and we have llx @x*II5 kllzll . llz*II. Of course the definition of x@x* depends on the operator T but since we are going to consider only one operator T , this can not lead to a confusion. By the von Neumann inequality, any contraction on a Hilbert space is polynomially bounded with constant 1. More generally, every operator on a Hilbert space which is similar to a contraction is polynomially bounded. Recall that there are polynomially bounded Hilbert space operators that are not similar to a contraction. This was shown recently by Pisier [27] who gave thus a negative answer to a well-known longstanding open problem given by Halmos. Denote by L" the space of all bounded measurable functions on with the sup-norm. Since P c Loo = (L1)*,the space P inherits the w*-topology from L". Of particular importance for the Scott Brown technique are those functionals on P that are w*-continuous, i.e., that are continuous functions w*) to @. Equivalently, these functionals can be represented by from (P, absolutely continuous measures. The next result summarizes the basic facts about w*-continuous functionals on P . Theorem 7.1.
(i) Let ( p n ) C P be a sequence of polynomials. Then p , s O if and only i f (pn) is a Montel sequence, i.e., supn JJp,)I< 00 and p n ( z ) + 0 ( z E ID); (ii) The w* closure of P in L" is the Hardy space H" of all bounded functions analytic o n JD. (iii) 1c, E P* is w*-continuous if and only i f it can be represented by a n absolutely continuous measure. By the F. and M. Riesz theorem, in this case each measure representing 11, is absolutely continuous. By the RadonNilcodym theorem, there exists f E L1 such that llflll = II11,II and 11, = M f .
74
(iv) Let @ E P* be w*-continuous. Let A c D be a dominant subset, i.e., supxe,, If ( X ) l = 11 f l l for all f E H". Let E > 0. Then there are numbers XI,. . . ,A, E A and 0 1 , . . . , a , E C such that CZ1jail 5 ll@ll and
[I@ - c:=, 11 < E . &xi
Let T E B ( X ) be a polynomially bounded operator such that IIT"uII -+ 0 for all u E X . Then all the functionals x @ x* can be represented by absolutely continuous measures. Equivalently, these functionals are w*continuous, i.e., they are continuous on the space (P,w * ) . These results can be shown using classical results from measure theory. We summarize the results in the following theorem. Conditions (iii) and (iv) are not necessary for our purpose, we include them only for the sake of completeness.
Theorem 7.2. Let T be a polynomially bounded operator o n a Banach space X . Suppose that IIT"uII .+ 0 f o r all u E X . Then: (a) x @ x* can be represented by an absolutely continuous measure for all x E X and x* E X * . Equivalently, x @ x* is w*-continuous;
(ii) the set {p(T)x: p E P , llpll 5 1) is precompact f o r all x E X ; (iii) the functional p H (p(T)x,x*) extends to the w*-closure of P in Loo, i.e., to the Hardy space H" of all bounded functions analytic o n D; (iv) it i s possible to define the HOO-functional calculus, i.e., an algebraic homomorphism @ : H" --+ B ( X ) such that Q(1) = I and @ ( z ) = T . Moreover, @ i s ( w * , S O T ) continuous, i.e., the mapping h -+ @ ( h ) xis a continuous function from ( H m ,w * ) to X for each x E X . Now we are able to give an illustrative example how the Scott Brown technique works.
Theorem 7.3. Let T be a contraction o n a Hilbert space H such that a(T)n D i s dominant in D and IIT"xII --+ 0 f o r all x E H . Then T has a nontrivial closed invariant subspace. Outline of the proof. Without loss of generality we may assume that neither T nor T* has eigenvalues. In particular, a,,(T) = a ( T ) . The first step in the proof is that we can approximate (with an arbitrary precision) the evaluation functionals Ex for X E ane(T) by the functionals of the type x @ x with x E H .
75
(a) Let X E ore(T),E
> 0, let z E H ,
IIzlI = 1 and II(T - X)zll
< E . Then
Indeed, we have
2ke
I ( p ( T ) z4, - P(X)I = I (q(T)(T- X)z, 4 1 I Il4J(T)II II (T - X > 4 l I q. *
For the approximation procedure we need a stronger version of (a).
. . ,u, E H be given. Let X E a&!') and E (b) Let u1,. exists z E H of norm 1 such that z I( 2 1 1 , . . . , u,} and 1 1 5 €3
> 0. Then there
z - 8x11 < E ,
112€3Uill < &
llut €3 511
0. We say that a subset E c Y is porous in (Y,d ) if there exist Q E ( 0 , l ) and rg > 0 such that for each r E (0, rg] and each y E Y , there exists z E Y for which
& ( z , Qr) c &(Y, r ) \ E. A subset of the space Y is called 0-porous in (Y,d ) if it is a countable union of porous subsets in (Y,d). Other notions of porosity have been used in the literature [2, 341. We
83 use the rather strong notion which appears, for instance, in [7]-[9], [27], [28],and which turns out to be appropriate for our purposes. Since porous sets are nowhere dense, all a-porous sets are of the first Baire category. If Y is a finite-dimensional Euclidean space R", then the Lebesgue density theorem implies that a-porous sets are of Lebesgue measure 0. The existence of a non-a-porous set P c R" which is of the first Baire category and of Lebesgue measure 0 was established in [33,341. It is easy to see that for any a-porous set A c R", the set A IJ P c R" also belongs t o the family E of all the non-a-porous subsets of R" which are of the first Baire category and have Lebesgue measure 0. Moreover, if Q E E is a countable union of sets Qi c Rn, i = 1,2,.. . , then there is a natural number j for which the set Q j is non-a-porous. Evidently, this set Qj also belongs to 1. Therefore one sees that the family E is quite large. Also, every complete metric space without isolated points contains a closed nowhere dense set which is not a-porous [35]. To point out the difference between porous and nowhere dense sets, note that if E c Y is nowhere dense, y E Y and r > 0, then there are a point a E Y and a number s > 0 such that B d ( a , s) c Bd(y, r ) \ E. If, however, E is also porous, then for small enough r we can choose s = ar, where a E ( 0 , l ) is a constant which depends only on E. In general, when we look for either genericity or porosity results we are faced with the problem of choosing an appropriate metric with the goal of obtaining a large set of "good" elements. Usually, a given (function) space can be equipped with several natural metrics and an optimal choice is not possible. For example, in a genericity result we want the set of "good" elements to be dense with respect t o a strong topology and to be a Gg set with respect t o a weak topology. To overcome this difficulty, we used a two-topology approach in [25]. In the setting of Banach space geometry such an approach was first used in [ll]. In the context of porosity we need to use a two-metric approach. The concept of porosity with respect t o a pair of metrics was introduced in [36]and used in [29,301. To define this concept, assume that Y is a nonempty set and that d l , d2 : Y x Y + [ O , o o ) are two metrics which satisfy d l ( z , y ) 5 dz(z,y) for all 5,y E Y . We say that a subset E c Y is porous in Y with respect to the pair (d1,dz) (or just porous in Y if the pair of metrics is understood) if there exist (Y E (0,l)and T O > 0 such that for each r E (0, TO] and each y E Y , there exists z E Y for which & ( z , y) 5 r and B d l ( a , ar) n E = 8. A subset of the space Y is called a-porous in Y with respect to ( d l , dz)
84
(or just a-porous in Y if the pair of metrics is understood) if it is a countable union of porous (with respect to ( d l ,d 2 ) ) subsets of Y . Note that if d l = d2, then by Proposition 1.1 of [36] our definitions reduce to those in [7]-[9],[27], [28]. Notice also that the porosity of a set with respect to one of the metrics dl or d2 does not imply its porosity with respect to the other metric. However, it is shown in [36], Proposition 1.2, that if a subset E c Y is porous with respect to ( d l , d 2 ) , then E is porous with respect t o any metric d which is between d l and d2, that is, c l d l 5 d 5 c2d2 for some positive c1 and c2. In the next section we discuss the concept of null sets in infinitedimensional Banach spaces. In such spaces the relationships between aporous sets and null sets are not yet fully understood. For instance, in [20] the authors construct a subset M of the separable Hilbert space l 2 which is porous in a very strong sense (arbitrarily close to each z E M , there is a “hole”, that is, a ball of radius one, which is completely outside M ) , but at the same time very large in the sense of measure. This construction was generalized in [21] to separable superreflexive Banach spaces. Such phenomena are impossible in a finite-dimensional setting.
3. Null sets In the previous two sections we reviewed two related useful methods for finding points which are plentiful in either a topological or a stronger, metric sense. This approach to proving that a certain subset is dense is, however, not always suitable. For instance, let f : R” -+ R be a Lipschitz function. Then, by a theorem of Rademacher, f is differentiable almost everywhere. Nevertheless, the set of points where f is differentiable can be of the first category. This raises the question of how we are going to prove a generalization of Rademacher’s theorem to infinitely many dimensions, as, obviously, Baire’s theorem is not going to be of help. A natural idea, which indeed turns out to work, is to introduce a concept analogous to Lebesgue null sets, and hence also a notion of a property holding almost everywhere, in separable infinite-dimensional Banach spaces. Here are some natural requirements such a family of null sets should satisfy: (i) Every translate of a null set is null. This is necessary for the application we have in mind. Let a Lipschitz function f be defined on a set A and let B be a translate of A. Define a new Lipschitz function g on B by composing f with an appropriate translation. Then the
85 set of points where g is not differentiable is just a translate of the set where f was not differentiable. (ii) Nonempty open sets are not null. This is natural, as we want to use null sets to prove that certain subsets of their complements are dense. (iii) A countable union of null sets is null. This will be a useful tool. Often we can describe the desired property by excluding countably many “bad” cases. If each of them happens only on a null set, then their union is also a null set, and hence the desired property takes place almost everywhere. All of the above properties are, of course, satisfied by Lebesgue null sets in Rn. Hence it seems natural in infinite-dimensional Banach spaces to also look for a translation-invariant Borel measure p which would play the role of the Lebesgue measure and then to consider its null sets. This idea, however, does not pan out. To see this, suppose for a contradiction that p is such a measure in a separable infinite-dimensional Banach space X . As p is outer regular with respect to open sets, there is an open U c X such that 0 < p ( U ) < 00. Since X is infinite-dimensional, there are a number r > 0 and pairwise disjoint balls B1,Bz, . . . of radius T inside U . As X is separable, there exist balls El, &, . . . of radius r so that U c &. Hence 0 < p ( U ) 5 p ( g i ) , and since p is translation-invariant, we have
u -
0 < p(B1) = p(B1) = p(B2) = p(B2) = .. ..
p(Bi) = m, which is indeed a contradiction. Consequently, p ( U ) 2 This means that we need to define null sets differently, not using a single measure. Here is a simple definition introduced by Christensen [4]. A Borel subset A of a separable Banach space X is called Huur nu12 if there exists a Borel probability measure p on X so that p ( A x) = 0 for all z E X . A translate of a Haar null set is, clearly, Haar null. The probability measure witnessing that a given set is null can be taken with its support contained in an arbitrarily small ball. Hence open sets are not null. Suppose that { A , } is a sequence of null sets, and that for each n the measure p, witnessing that A, is null is a probability measure which in the weak topology is “close enough” to the probability measure supported at the origin. Then it is not difficult to see that the convolution p1 * p2 * p3 * . . . witnesses that UA , is Haar null.
+
86
Haar null Borel sets coincide with Lebesgue null Borel sets in finitedimensional Banach spaces. Christensen introduced this family to show that a Lipschitz function f defined on a separable Banach space X is Giiteaux differentiable almost everywhere. This means that it is differentiable outside a Haar null set, and so, in particular, on a dense set. Let us briefly sketch a proof of this, in order to understand how one uses null sets and finite-dimensional results to prove density results in infinitedimensional spaces. Let {u,} be a sequence of unit vectors dense in the unit sphere of X , and let A, be the normalized Lebesgue measure supported on the unit ball of the span L, of (211,. . . , un}. Let A, c X be the set of all points z E X such that f restricted to z + L, is not differentiable at z. By Rademacher’s theorem, X,(A, y) = 0 for all y E X . Hence each A, is Haar null. It is not difficult to see that f is Giiteaux differentiable at every point outside the Haar null set A = UA,. There are many other notions of null sets in infinite-dimensional Banach spaces. Aronszajn [ I ] and cube null sets [19] were introduced similarly to Haar null sets in order to prove Giiteaux differentiablity of Lipschitz functions, and later on turned out to be different definitions of the same family of sets [5]. A generalization of Aronszajn null sets and the so-called r-null sets [18] were introduced to examine the FrBchet differentiability of Lipschitz functions. Rather than going into the sometimes quite technical definitions of these famillies of null sets, we finish this section with a few simple observations about Haar null sets, in order to give the reader some more feeling for them. Any closed proper (affine) subspace L and, in particular, any hyperplane, of a separable Banach space X is Haar null. To see this, it is enough to take u E X \ L and the normalized Lebesgue measure X on [0,u]. Suppose K is a compact subset of an infinite-dimensional separable Banach space X . As above, we will find a point u so that any line in the direction of u intersects K in at most one point; the Lebesgue measure X on [O,u]will show that K is Haar null. As K is compact, so is its closed convex symmetric hull k.Since X is infinite-dimensional, the interior of k is empty. Hence, as we have seen in the first section, by Baire’s theorem, Unk # X.Now choose any u E X \ Ung. It turns out [22] that in reflexive spaces all closed convex sets with empty interior are Haar null; actually this is a characterization of separable reflexive spaces. Even in Hilbert space, however, there is a convex set
+
87 which is Haar null, but this fact is not witnessed by any measure with a finite dimensional support. To see this, let Q = { C a i e i E & : ai 2 0) be the positive cone of &. Then Q is closed, convex and it has empty interior; hence it is Haar null by [22]. At the same time, Q contains a translate of the unit ball of every finite-dimensional subspace of l 2 . Finally, let us mention that by far not all “good” properties of Lebesgue null sets are shared by Haar null sets. If F : R” -+ R” is a Lipschitz mapping, then the image of a Lebesgue null set is a null set again. There exists, however, a mapping F of f$ onto itself such that both F and its inverse are Lipschitz, and at the same time the complement of F ( A ) is Haar null for some Haar null Bore1 set A c & [17].
4. Nonexpansive and contractive mappings
Recall that a Lipschiz mapping with Lipschitz constant equal to one is said to be nonexpansive. Nonexpansive mapping theory has flourished during the last forty years or so with many results and applications. See, for example, [12], [13], [15], and the references mentioned therein. In contrast with the iterates of nonexpansive mappings which in general do not converge, it is known that the iterates of contractive mappings (see the definition below) converge in all complete metric spaces [23]. However, it is also known [6] that in Banach spaces the iterates of most nonexpansive mappings (in the sense of Baire’s categories) do converge to their unique fixed points. In the next section we explain this result by showing that most nonexpansive mappings are, in fact, contractive [26]. As a matter of fact, it turns out that our result holds for all complete hyperbolic spaces, a notion which is of independent interest and which we now recall. Let (X, p ) be a metric space and let R denote the real line. We say that a mapping c : R -+ X is a metric embedding of R into X if p ( c ( s ) ,c ( t ) )= 1s - t ( for all real s and t. The image of R under a metric embedding will be called a metric line. The image of a real interval [a,b] = {t E R : a 5 t 5 b} under such a mapping will be called a metric segment. Assume that (X,p ) contains a family M of metric lines such that for each pair of distinct points x and y in X, there is a unique metric line in M which passes through x and y. This metric line determines a unique metric segment joining x and y. We denote this segment by [x,y]. For each 0 5 t 5 1, there is a unique point z in [x,y] such that p(x,z ) = tp(x,y) and p(z, y) = (1 - t)p(z,y). This point will be denoted by (1- t ) x @ ty. We will say that X, or more
aa precisely ( X ,p, M ) , is a hyperbolic space if
1 2
1 2
p(-x @ -y,
1
1 1 63 )-. 2 2
I ,P(Y,
--2
.>
for all x , y and z in X . A set K c X is called pconvex if [z,y] c K for all x and y in K . It is clear that all normed linear spaces are hyperbolic. A discussion of more examples of hyperbolic spaces and, in particular, of the Hilbert ball can be found, for example, in [24]. In the sequel we will repeatedly use the following fact (cf. [13], pp. 77 and 104, and [24]): If ( X ,p, M ) is a hyperbolic space, then p((1 - t ) x @ t z , (1 - t)?4@ t w ) I (1 - t M x , Y)
+ M.,
w)
(1)
for all x , y , z and w in X and 0 5 t I 1. Assume that ( X ,p) is a hyperbolic complete metric space and let K be a bounded closed pconvex subset of X . Denote by A the set of all operators A : K + K such that p(Ax, Ay)
5 p(x, y) for all x,y
E
K.
(2)
In other words, the set A consists of all the nonexpansive self-mappings of K . Set
d ( K ) = SUP{P(Z,Y)
:
x,y E K).
We equip the set A with the metric h(., .) defined by h(A, B ) = sup{p(A~,Bz): z E K } , A, B E A. Clearly, the metric space (A,h ) is complete. We say that a mapping A E A is contractive if there exists a decreasing function 4 A : [0,d ( K ) ]+ [0,1] such that
$A(t)< 1 for all t E (O,d(K)]
(3)
and d A x , AY) 5 $A(P(GY))P(x, Y) for all 2 , y E K .
(4)
The notion of a contractive mapping, as well as its modifications and applications, were studied by many authors. See, for example, [16]. We now quote a convergence result which is valid in all complete metric spaces [23].
Theorem 4.1. Assume that A E A is contractive. Then there exists X A E K such that Anx -+ X A as n -+ 00, uniformly on K .
89 For each A, B E A and each a E (0, l ) , define the operator a A @ ( l - a ) B by
( a A @ (1 - CX)B)Z = AX @ (I - a ) B z , z E K.
(5)
Note that a A @ (1 - a ) B E A by (1). Next, we note the following simple fact.
Proposition 4.1. If A E A is contractive, B E A and a E (0, l ) , then the composition operators AB, B A and the p-convex combination a A @ (l - a ) B are also contractive. 5. The set of contractive mappings contains a residual subset Now we show that most of the mappings in A (in the sense of Baire's categories) are, in fact, contractive.
Theorem 5.1. There exists a set 3 which is a countable intersection of open, dense sets in A such that each A E 3 is contractive. Proof. Fix B E K . For each A E A and each y E (0, l ) , define A, E A by
A,z
=
(1 - ~ ) A @ z 70, 2
E
K.
(6)
Clearly, the set {A, : A E A, y E ( 0 , l ) ) is dense in A. Let A E A and y E (0,l). Inequality (1) implies that
P(A-97 A,Y) I (1 - Y ) P ( G Y)
(7)
for all x,y E K . For each integer i 2 1, define
U(A, 7, i) = { B E A : h(A,, B ) < 4-2-lyd(K)}.
(8)
We will show that for each A E A, y E (0, l ) , and each integer i 2 1, the following property holds: ( P l ) For each B E U ( A , y , i ) and each x , y E K satisfying p(x,y) 2 4-Zd(K), the inequality p ( B x ,By) 5 (1 - 2-'y)p(x, y) is valid. Indeed, let A E A, y E (0,l ) , and let i 2 1 be an integer. Assume that
B
E
U ( A ,y,i),x,y E K , and p(x,y) 2 4-24K).
BY (8), (9), and (7), p ( B x , BY) I p(A,x, A,Y)
+ 2-'
. 4 - i y d ( K ) I (1 - Y)P(Z,Y)
(9)
90
Thus property (Pl) holds. Now define
3 = n g l U { U ( A ,y,q ) : A E A, y E (0,1)}. Clearly, 3 is a countable intersection of open, dense sets in A. We claim that any B E 3 is contractive. To prove this, assume that q is a natural number. There exist A E A and y E ( 0 , l ) such that B E U(A, y,q). By property (Pl), for each z,y E K satisfying p ( z , y ) 2 4 - 4 d ( K ) , we have p(Bz,By) 5 (1 - 2-ly)p(z, g). Since q is an arbitrary natural number, we can conclude that B is contractive. Theorem 5.1 is proved. 0
A variant of Theorem 5.1 has recently been established by De Blasi and Pianigiani [lo] who used the Baire category method to solve the prescribed singular values problem with Dirichlet boundary data. We now remark in passing that an extension theorem of theirs, namely, Theorem 3.1 in [lo], can be improved. Theorem 5.2. Let H and L be two Halbert spaces and let D be a subset of H . T h e n each contractive mapping A : D t L with a bounded range has a contractive extension B : H -+ L to all of H .
Proof. Let A satsify (4) for all x,y E D,where the decreasing function + A : R+ 3 [0,1] satisfies (3) for all t > 0, and let v : R+ -+ R+ be its modulus of continuity; that is, let
v ( t ):= S U ~ { ~ A-XAgl : Z, Y E D,)Z- Y)_< t } . Clearly, v has a bounded range and v(t) 5 t for all t E R+.We claim that v ( t ) < t for all positive t. Indeed, suppose v ( s ) = s for some s > 0. Then there would be sequences {xn} and {yn} in D such that 12, - l,y 5 s for all n E N and limnhco (Az, - Ay,I = s. Since
IA% - AYnI 5 $JA(I.n - Ynl)lzn - Ynl
I 1% - Ynl 5 s,
we conclude that (z, - yn( -+ s and that +A(Izn - g,)) + 1 as n -+ m. But this means that ( 2 , - yn) t 0, so that s = 0, which is a contradiction. Thus v(t) < t for all t > 0, as claimed. Now we can follow the argument in [lo] and obtain a concave function w : R+ --f R+ such that w ( 0 ) = 0, v ( t ) 5 w ( t ) for all t 2 0, and w ( t ) < t for t > 0. By the Grunbaum-Zarantonello extension theorem [14] (see also [2], p. 18), A has a uniformly continuous extension B : H -+ L with modulus of continuity o(t) 5 w ( t ) , t 2 0. Define $B : [0, m) [0,1] by @(O) := 1, $ B ( t ):= w ( t ) / t , t > 0. Then $B(t)< 1 for all t > 0 and $B is decreasing --f
91
+
because w ( s ) 2 (1 - s/t)w(O) ( s / t ) w ( t ) = ( s / t ) w ( t ) , 0 5 s concavityofw. Also, IBz-By1 I P(lz-yl) 5 w ( l ~ - y l ) = $ B ( l for all z, y E H . Thus B is indeed contractive, as asserted.
< t , by the
~-~l)l~-~I 0
Note that if A : D 4 L satisfies / A x - Ayl < Iz - yI for z # y and D is compact (as assumed in [ l o ] ) then , A is necessarily contractive. This, however, is no longer true if D is not compact. 6. The complement of the set of contractive mappings is
a-porous In the previous section we have explained the result in [6] by showing that most nonexpansive mappings are, in fact, contractive. However, it is shown in [7] that the complement of the set of power convergent nonexpansive mappings is not only of the first category, but also a-porous. In this section we strengthen all of these results by showing that the set of all noncontractive mappings is not only of the first category, but also a-porous [28]. For simplicity, we will consider only self-mappings of a subset of a Banach space. However, just as in the previous section, our result also holds for the wider class of hyperbolic complete metric spaces. Assume that (X, 1) . ) I ) is a Banach space and let K be a bounded closed convex subset of X. We continue to use the notation of Section 4 with the understanding that p(z,y) = 1111: - yI I.
Theorem 6.1. There exists a set 3 c A such that A \ 3 i s a-porous in (A,h ) and each A E 3 is contractive. Proof. For each natural number n, denote by A, the set of all A E A which have the following property: ( P 2 ) There exists K E ( 0 , l ) such that IlAz - Ayll 5 1c11z- yII for all z,y E K satisfying ) ) z- y J J_> d ( K ) ( 2 n ) - ' . Let n 2 1 be an integer. We will show that the set A \ A, is porous in (A,h). To this end, set (u
= 8-' min{d(K), 1 } ( 2 n ) - l ( d ( K )
+ l)-l.
(10)
Fix 0 E K . Let A E A and r E ( 0 , 1 ] . Set
y = 2 - ' r ( d ( ~ )+ I)-'
(11)
and define
A,z
=
(1-y)Az
+ 70, z E K.
(12)
92 Clearly, A, E A,
w,,A) I 7 d ( K ) , and for all z, y E
(13)
K,
IIA+ - ArYll I (1 - 7)llAz -AYll
I (1 - 7)llz - YII.
(14)
Assume that B E A and that
h(B,A,) 5 ar. We will show that B E Indeed, let
(15)
A.
z , y E K and I(z-
YII
2 (2n)-'d(K).
(16)
It follows from (14) and (16) that 1111: - YII
- IIA,z
-
(17)
ArYll 2 Y l l Z - YII L 7 4 K ) ( W - l .
BY (15), IIBz - BY11 I IlBz - A+Il
+ I I 4 z - A,Yll+ I P , Y
I llA+ - A,yll+
2ar.
When combined with (17), (11) and 113: - YII -
llBz - B Y l l 2
- BY11
113: - YII
(lo), this implies that - IlA+ - A,Yll - 2 a r
2 yd(K)(2n)-'
- 2ar
= 2-'r[(2n)-'d(K)(d(K)
2 2T1rd(K)(4n)-'(d(K)
+ 1)-' - 4a] + 1)-l.
Thus
JJBx- By11 I
1 1 2-yJJ -
+
r d ( K ) ( d ( K ) 1)-'(8n)-'
I llz - Yll(1- .(8n)-'(d(K)
+ I)-').
Since this holds for all z,y E K satisfying (16), we conclude that B E A,. Thus each B E A satisfying (15) belongs to A,. In other words,
{ B E A : h(B,A,) 5 o r } C A,.
(18)
If B E A satisfies (15), then by (13), ( l o ) , and (11) we have
h(A, B ) 5 h ( B ,A,)
+ h(A,,
A ) 5 ar + y d ( K ) 5 8-'r
+ 2T'r
Thus
{ B E A : h ( B ,A,)
I a r } C { B E A : h(B,A) I r } .
5 r.
93 When combined with (18), this inclusion implies that A \ A, is porous in (A,h). Set .F = n,",ld,. Clearly, A \ .F is a-porous in (A,h). In view of property (P2), each A E F is contractive. 0
Remark. If X is a Hilbert space, then the set of all strict contractions (that is, mappings with Lipschitz constant strictly less than one) is a-porous in (A,h ) [7]. It would be of interest to determine if this continues to hold in all Banach spaces, as well as for rotative mappings [15], Chapter 11. Analogous results to Theorem 6.1 and to the above remark for setcontractions with respect to Kuratowski's measure of noncompactness were established in [3].
7. Well-posedness In this section we show that the fixed point problem for any contractive mapping is well posed [31]. Therefore Theorem 6.1 also yields Theorem 8 in [7]. Let ( K ,p) be a bounded complete metric space. We say that the fixed point problem for a mapping A : K K is well posed if there exists a unique X A E K such that A X A= X A and the following property holds: if {x,},",~ c K and p(xn,Ax,) -+ 0 as n + 03, then p(x,, X A ) 0 as ---f
---f
n
4
03.
The notion of well-posedness is of central importance in many areas of mathematics and its applications. In our context this notion was studied in [7], where generic well-posedness of the fixed point problem is established for the space of nonexpansive self-mappings of K . We continue to use the notation introduced in Section 4.
Theorem 7.1. Assume that a mapping A : K the fixed point problem for A is well posed.
-+
K is contractive. Then
Proof. Since the mapping A is contractive, there exists a decreasing function qbA = qb : [O,d(K)]-+ [0,1] such that (3) and (4) hold. By Theorem 4.1, there exists a unique X A E K such that
A X A= X A . Let {x,},"=~
c K satisfy lim p(x,,Axn) = 0.
n-oo
(19)
94
We will show that x, -+ X A as n + 00. Suppose this were false. By extracting a subsequence, if necessary, we might then assume without loss of generality that there exists E > 0 such that p(x,,xA) 1 E for all integers n 2 1.
(21)
It follows then from (19), (4), (21) and the monotonicity of the function 4 that for all integers n 2 1, p(xA,Zn) _< P(ZA,Azn) +p(A%,%)
I p(Axn, zn) + 4(p(z,,
I p(Azn, 2 , )
ZA))P(G,
%A)
(22)
+ +(E)P(ZA,xn).
Inequalities (22) and (21) imply that for all integers n 2 1, E(1
- 4(E))
5 (1- 4(E))P(xA,xn) 5 p(Axn,xn),
a contradiction (see (20)). The contradiction we have reached proves Theorem 7.1. n 8. Strict contractions
Let K be a nonempty, bounded, closed and convex subset of a Banach space ( X ,II . 11). Set rad(K) = sup(llx1l : x E K } . For each A : K
-+ X
(23)
, let
LiP(A) = sup{llAz
-
AYIlAb - YII :
Z,Y E
K , x # Y}
(24)
be the Lipschitz constant of A. We continue to denote by d the set of all nonexpansive mappings A : K -+ K , that is, all self-mappings of K with Lip(A) 5 1, or equivalently, all self-mappings of K which satisfy llAx - Ayll 5 llx - yII for all x , y E K .
(25)
We say that a self-mapping A : K -+ K is a strict contraction if Lip(A) < 1. We have already seen that although the set of all strict contractions in is small, the set of power convergent nonexpansive mapthe space (d,h) pings is large. In Sections 5, 6 and 7 we have singled out a property (that of being contractive), which on the one hand turns out to be possessed by most nonexpansive mappings and on the other ensures power convergence to unique fixed points as well as well-posedness of the fixed point problem.
95 It may seem strange that the set of contractive mappings is large while the set of strict contractions is small. Needless to say, the answer to the question whether a certain set is small or large depends on the metric (topology) under consideration. Indeed, in this section we show that if the space A is endowed with another natural complete metric, then the set of strict contractions is, in fact, large [32]. Our new metric is defined by d(A,B) = sup{llAz - BzlJ : z E K } +Lip(A - B ) ,
(26)
where A, B E A. It is not difficult to see that the metric space (A,d) is complete. Theorem 8.1. Denote by 3 the set of all strict contractions A E A. Then A \ 3 is porous.
Proof. Fix a number
Q
> 0 such that Q < (1 + 2rad(K))-l32-l
(27)
and fix 0 E K . Let A E A and let r E (0,1]. Set
y = (1+ 2rad(K))-lr/S
(28)
and put A,z=(l-y)Az+yB,
XEK.
(29)
I (1 - 7)Ilz - YII.
(30)
Clearly, A, E A and for each z, y E K ,
IlA+
- 4 V l l = (1 - r)llAz - AYll
By (29), (23), (24) and (26), for each z E K ,
I J A ,~ A ~ J=I11(i- r ) ~ z+ y e I2yrad(K),
- ~ z l= l rile -~ z l l
96
We now assume that B E A and t h a t
d(B,A,) 5 ar.
(33)
In view of (33), (26), (30) and (28), we see that Lip(B) ILip(A,)
ILip(A,)
+ Lip(B - AY) 5 Lip(A,) + d ( B , A,)
+ ar 5 (1 - y) + ar I 1 - (r/8)(1+2rad(K))-'
+ r(32(1+ 2rad(K))-l and so B E
I 1 - (r/16)(1+ 2rad(K))-l < 1,
F. Clearly, by (33), (32) and
(27),
+
d(B, A ) 5 d ( B , AY) d(A,, A) I ar Thus for each B E A satisfying (33), B E completes the proof of Theorem 8.1.
F
+ r/8 I r.
and d ( B , A ) 5 r. This 0
References 1. N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57, 147-190, 1976. 2. Y . Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Amer. Math. SOC.,Providence, RI, 2000. 3. J. Carmona AlvArez and T. Dominguez Benavides, Porosity and k-set contractions, Boll. Un. Mat. Ital. 6, 227-232, 1992. 4. J.P.R. Christensen, Topology and Bore1 Structure, North-Holland, Amsterdam, 1974. 5. M. Csornyei, Aronszajn null and Gaussian null sets coincide, Israel J. Math. 111, 191-201, 1999. 6. F.S. De Blasi and J. Myjak, Sur la convergence des approximations successives pour les contractions non lindaires dans un espace de Banach, C . R. Acad. Sci. Paris, 283, 185-187, 1976. 7. F.S. De Blasi and J. Myjak, Sur la porositt! de l'ensemble des contractions sans point fixe, C. R. Acad. Sci. Paris 308, 51-54, 1989. 8. F.S. De Blasi and J. Myjak, O n a generalized best approximation problem, J. Approximation Theory 94, 54-72, 1998. 9. F.S. De Blasi, J . Myjak and P.L. Papini, Porous sets in best approximation theory, J. London Math. SOC.44, 135-142, 1991. 10. F. S. De Blasi and G. Pianigiani, O n the Baire category method for the prescribed singular values problem, Preprint, 2003. 11. V.P. Fonf, Sets of the super-first category in Banach spaces, Functional Analysis and its Applications 25, 66-69, 1991. 12. K. Goebel and W.A. Kirk, Topics in Metric Faxed Point Theory, Cambridge University Press, Cambridge, 1990. 13. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
97 14. F. Griinbaum and E. H. Zarantonello, On the extension of uniformly continuous mappings, Michigan Math. J. 15, 65-74, 1968. 15. W.A. Kirk and B. Sims (Editors), Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001. 16. M.A. Krasnosel'skii and P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. 17. J. Lindenstrauss, E. MatouSkovB and D. Preiss, Lipschitz image of a measurenull set can have a null complement, Israel J. Math. 118, 207-219, 2000. 18. J. Lindenstrauss and D. Preiss, On Fre'chet differentiability of Lipschitz maps between Banach spaces, Annals of Math. 157, 257-288, 2003. 19. P. Mankiewicz, On the differentiability of Lipschitz mappings in Fre'chet spaces, Studia Math. 45, 15-29, 1973. 20. J. MatouSek and E. MatouSkovB, A highly non-smooth n o r m on Halbert space, Israel J . Math. 112, 1-27, 1999. 21. E. MatouSkovB, An almost nowhere Fre'chet smooth n o r m o n superreflexive spaces, Studia Math. 133, 93-99, 1999. 22. E. MatouSkovB, Translating finite sets into conuex sets, Bull. London Math. SOC.33, 711-714, 2001. 23. E. Rakotch, A note o n contractive mappings, Proc. Amer. Math. SOC.13, 459-465, 1962. 24. S. Reich and I. Shafrir, Nonexpansiue iterations in hyperbolic spaces, Nonlinear Anal. 15, 537-558, 1990. 25. S. Reich and A.J. Zaslavski, Convergence of generic infinite products of nonexpansive and uniformly continuous operators, Nonlinear Anal. 36, 10491065, 1999. 26. S. Reich and A.J. Zaslavski, Almost all nonexpansiue mappings are contractive, C. R. Math. Rep. Acad. Sci. Canada 22, 118-124, 2000. 27. S. Reich and A.J. Zaslavski, T h e set of divergent descent methods in a Banach space is a-porous, SIAM J. Optim. 11, 1003-1018, 2001. 28. S. Reich and A.J. Zaslavski, T h e set of noncontractiue mappings is a-porous in the space of all nonexpansiue mappings, C . R. Acad. Sci. Paris 333, 539544, 2001. 29. S. Reich and A.J. Zaslavski, Well-posedness and porosity in best approximation problems, Topological Methods in Nonlinear Analysis 18, 395-408, 2001. 30. S. Reich and A.J. Zaslavski, Porosity of the set of divergent descent methods, Nonlinear Anal. 47, 3247-3258, 2001. 31. S. Reich and A.J. Zaslavski, Well-posedness of fixed point problems, Far East J. Math. Sci., Special Volume (Functional Analysis and its Applications), Part 111, 393-401, 2001. 32. S. Reich and A.J. Zaslavski, Many nonexpansiue mappings are strict contractions, Preprint, 2002. 33. L. Zajieek, Sets of a-porosity and sets of u-porosity ( q ) , Casopis Pest. Mat. 101, 350-359. 1976. 34. L. ZajiEek, Porosity and u-porosity, Real Analysis Exchange 13, 314-350, 1987. 35. L. ZajiEek, Small non-u-porous sets in topologically complete metric spaces,
98 Colloq. Math. 77, 293-304, 1998. 36. A.J. Zaslavski, Well-posedness and porosity in optimal control without convexity assumptions, Calculus of Variations and Partial Differential Equations 13, 265-293, 2001.
99
ABSOLUTE-VALUED ALGEBRAS, AND ABSOLUTE-VALUABLE BANACH SPACES *
ANGEL RODR~GUEZPALACIOS Universidad de Granada, Facultad de Caencias, Departamento de Andlisis Matema'tico, 180'71-Granada (Spain) E-mail:
[email protected] Absolute-valued algebras are fully surveyed. Some attention is also payed to Bai nach spaces underlying complete absolute-valued algebras.
Introduction Absolute-valued algebras are defined as those real or complex algebras A satisfying 11zy/11 = IIzlIIIy/I] for a given norm 1) . 11 on A , and all z,y E A . Despite the nice simplicity of the above definition, absolute-valued algebras have not attracted the attention of too many people. A reason could be that, in the presence of associativity, the axiom (1xyll = llxllIlyll is extremely obstructive. Indeed, according to an old theorem of S. Mazur 66, there are only three absolute-valued associative real algebras. Nevertheless, when associativity is removed, absolute-valued algebras do exist in abundance. Some facts corroborating the above assertion are that every complete normed algebra is isometrically algebra-isomorphic to a quotient of a complete absolute-valued algebra (Corollary 3.2), and that every Banach space is linearly isometric to a subspace of a complete absolute-valued algebra (Theorem 5.1). Anyway, the quantity and quality of works on absolute-valued algebras seemed to us enough to deserve a detailed survey paper like the one we are just beginning. Our paper collects the results on absolute-valued algebras since the pioneering works of Ostrowski 7 5 , Mazur 66, Albert 3 , and Wright log (see Subsection 1.3) to the more recent developments. The inflexion point in the
'
'This work is partially supported by Junta de Andalucia grant FQM 0199 and Projects I+D MCYT BFM2001-2335 and BFM2002-01810
100
theory, namely the Urbanik-Wright paper lo6,is fully reviewed (see Subsections 2.1, 2.2, 2.3, and 3.1). Among the recent developments, we emphasize the solution 57 t o Albert's old problem if every absolute-valued algebraic algebra is finite-dimensional (see Subsection 2.7), and the study of Banach spaces underlying complete absolute-valued algebras, done in and 69 (see Section 5). A special attention is also payed to the intermediate works of K. Urbanik (lol to lo4) and M. L. El-Mallah (35 to 46). This is done in Subsections 2.4, 2.5, 2.7, 3.1, 3.2, and 3.4. Contributions of other authors (including the one of this paper) are also reviewed (see mainly Subsections 1.4, 2.6, 3.5, and 3.6, and the whole Section 4). The clarifications of the theory at some precise points, done by Gleichgewicht 49 and ElduqueP6rez 331 are inserted in the appropriate places (see Subsection 3.4, and Subsections 1.3, 2.1, and 3.5, respectively). Our paper contains also some new results, and several new proofs of known results. Known proofs have been included only when they seemed to us specially illuminating. As far as we know, absolute-valued algebras have been surveyed in exclusive several times (see 86, 'l, and 'lo), but in references not easily available, and never in English. Moreover, references 91 and 'lo are relatively short, and references and 'lo become today rather obsolete. On the other hand, there are also survey papers on more general topics, devoting to absolutevalued algebras some attention (see 87 and 8 8 ) . Finally, let us note that the Ph. Theses 35, 63, and 78 are devoted to absolute-valued algebras, and contain both reviews of other people's results and proofs of results of their respective authors. 1. Finite-dimensional absolute-valued algebras 1.1. Some basic definitions and facts
By an algebra over a field F we mean a vector space A over F endowed with a bilinear mapping (x, y) -+ xy from A x A to A called the product of the algebra A . Algebras in this paper are assumed t o be nonzero, but are not assumed to be associative, nor to have a unit element. We suppose that the reader is familiarized with the basic terminology in the theory of algebras. Thus, terms as subalgebra, ideal, or algebra homomorphism are not defined here. For an element x in an algebra A, we denote by L , (respectively, R,) the operator of left (respectively, right) multiplication by x on A. The algebra A is said to be a division algebra if, for every nonzero element 2 of A, the operators L, and R, are bijective. An algebra is said t o be alternative if it satisfies the identities xfx2 = x1(x1x2) and
101 ( 2 1 2 2 ) 2 2 = x ~ x g . We note that alternative algebras are “very nearly” associative. Indeed, by Artin’s theorem (see Theorem 2.3.2 of ‘13), the subalgebra generated by two arbitrary elements of an alternative algebra is associative. It is also worth mentioning that every alternative division algebra has a unit (see page 226 of 31). By an algebra involution on an algebra A we mean an involutive linear operator x -+ x* on A satisfying (zy)* = g*x* for all x , y E A . Now, let K denote the field of real or complex numbers. An algebra norm (respectively, absolute value) on an algebra A over K is a norm 11 . I( on the vector space of A satisfying llxyll 5 IIxlI ((yI((respectively, llzy[l = Ilxl[[[y[() for all x , y E A . By a normed (respectively, absolutevalued) algebra we mean an algebra over IK endowed with an algebra norm (respectively, absolute value). We note that absolute-valued j n i t e dimensional algebras are division algebras. We also note that, if there exists an absolute value o n a finite-dimensional algebra A over K,then we can speak about “the” absolute value of A , understanding that such an absolute value is the unique possible one o n A . This is a straightforward consequence of the easy and well-known result immediately below. The proof we are giving here is taken from 26.
Proposition 1.1. Let A be a normed algebra over K,let B be an absolutevalued algebra over K,and let $ : A + B be a continuous algebra homomorphism. T h e n $ is in fact contractive. Proof. Assume to the contrary that q5 is not contractive. Then we can choose a norm-one element z in A such that Il$(x)l[> 1. Defining inductively 2 1 := x and x,+1 := xi, we have 11$(x,)1) = 11q5(x)112n-14 00 . Since Ilxnll 5 1, this contradicts the assumed continuity of $. 0 Looking at the above proof, we realize that Proposition 1.1remains true if B is only assumed to be a normed algebra over K satisfying ((y2(( =( ( ~ 1 1 for every y E B , and 4 : A -+ B is only assumed to be a continuous linear mapping preserving squares. As a consequence of Proposition 1,1, every continuous algebra involution o n a n absolute-valued algebra is isometric. Let A be a normed algebra. An element z of A is said to be a left (respectively, right) topological divisor of zero in A if there exists a sequence { x n } of norm-one elements of A such that {xz,} 4 0 (respectively, {z,x} + 0). Elements of A which are left or right (respectively, both left and right) topological divisors of zero in A are called one-sided
~
102
(respectively, two-sided) topological divisors of zero in A. The element IC E A is said to be a joint topological divisor of zero in A if there exists a sequence {zn}of norm-one elements of A such that { x z n } -+ 0 and {xnx} -+ 0. We note that both absolute-valued algebras and normed division alternative algebras have no one-sided topological divisor of zero other than zero. (This is clear in the case of absolute-valued algebras, and is easily verified in the case of normed division alternative algebras, by keeping in mind the fact already pointed out that division alternative algebras have a unit element, and applying the properties of “invertible” elements of unital alternative algebras given in page 38 of 94.) We will review in Theorem 1.1 a much deeper fact implying that, conversely, normed alternative algebras without nonzero joint topological divisors of zero are division algebras. 1.2. Quaternions and Octonions
Surveying absolute-valued algebras, we should write something about the algebra W of Hamilton’s Quaternions, and the algebra 0 of Octonions (also called “Cayley numbers”). These algebras, together with the fields of real and complex numbers (denoted by R and C, respectively), become the basic examples of absolute-valued algebras. The algebras C, W,and 0 can be built from R by iterating the so-called “Cayley-Dickson doubling process” (see for example pages 256-257 of 31). Thus, if A stands for either R, @, or MI, and if * denotes the standard algebra involution of A (which, for A = R, is nothing other than the identity mapping), then we can consider the real vector space A x A with the product given by (21,52)(2.3,24)
:= (51x3 - - 4 x a , x T 2 4
+
23x2)
7
obtaining in this way a new real algebra which is a copy of either 6,H, or 0, respectively. In this doubling process, the standard involution * and the absolute value 11 . I( of the new algebra are related to the corresponding ones of the starting algebra by the formulae
respectively. It follows from the last formula that the absolute values of R, C, W , and 0 come from inner products. It is also of straightforward verification that the algebra W is associative but not commutative, whereas the algebra 0 is alternative but not associative. The joint introduction of IHI and 0done above is surely the quickest possible one. However, concerning W, there is another more natural approach.
103 Indeed, in the same way as C can be rediscovered as the subalgebra of the algebra M2(R) (of all 2 x 2 matrices over R) given by
W can be rediscovered as the real subalgebra of Mz(C)given by
(see for example page 195 of 3 1 ) . Regarded W in this new way, the standard involution of W corresponds with the transposition of matrices, and the absolute value of an element of W is nothing other than the nonnegative square root of its (automatically nonnegative) determinant. The algebras MI and 0 are very far from being only exotic objects in Mathematics. By the contrary, they solve many natural problems in the field of the Algebra, the Geometry, and the Mathematical Analysis. Thus, as a consequence of the Robenius-Zorn theorem (see for example pages 229 and 262 of 3 1 ) , R, C, MI, and 0 are the unique finite-dimensional alternative division real algebras. On the other hand, we have the following. Theorem 1.1. Every normed alternative real algebra without nonzero joint topological divisors of zero is algebra-isomorphic to either R, C, W,or 0. Theorem 1.1has been proved by M. L. El-Mallah and A. Micali 45 by applying the forerunner of I. Kaplansky 6o (see also 17) for associative algebras. Keeping in mind the uniqueness of the absolute value on a finite-dimensional algebra, pointed out in Subsection 1.1, it follows that R, C, W,and 0 are the unique absolute-valued alternative real algebras. A refinement of the fact just formulated (see Theorem 2.4) and other interesting characterizations of R, C, W,and 0 (see Theorems 2.1, 2.5, 2.6, and 3.4) will be reviewed later. The reader interested in increasing his knowledge on Quaternions and Octonions is referred to the books 22 and 31, and the survey papers and 98. These works and references therein will provide him with a complete panoramic view of the topic. Nevertheless, let us emphasize the abundance of historical notes and mathematical remarks collected in 31, and take some samples of them. Thus, in a note written with the occasion of the fifteenth birthday of the Quaternions, W. R. Hamilton says: “They [the Quaternions] started into life, or light, full grown, on the 16th of October, 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge.” (see
104
page 191 of 3 1 ) . It turns out also curious to know that Hamilton tried for many years to built a three dimensional division associative real algebra. In fact, shortly before his death in 1865 he wrote to his son: “Every morning, on my coming down to breakfast, you used to ask me: ‘Well, Papa, can you multiply triplets?’ Whereto I was always obliged to reply, with a sad shake of the head: ‘No, I can only add and subtract them’.’’ (see page 189 of 31). It is not less curious how, in a very elemental way, one can realize that the attempt of Hamilton just quoted could not be successful. Indeed, refining slightly the content of the footnote in page 190 of 3 1 , we have the following.
Proposition 1.2. Let A be a (possibly nonassociative) division real algebra of odd dimension. Then A has dimension 1, and hence it is isomorphic to R.
A\ {0}, and let x be in A. Then the characteristic polynomial of the operator LY1 o L, must have a real root (say A), which becomes
Proof. Fix y
E
an eigenvalue of such an operator. Taking a corresponding eigenvector z # 0, we have (z - Xy)z = 0, which implies z = Xy. Since 2 is arbitrary in A, we have A = R y . 0 We note that the above proof is nothing other than a natural variant of the usual one for the fact that finite-dimensional division algebras over an algebraically closed field IF are isomorphic to IF. According to the information collected in page 249 of 31, the Octonions were discovered by J. T. Graves in December 1843, only two months after the birth of the Quaternions. Graves communicated his discovery to Hamilton in a letter dated 4th January 1844, but did not publish it until 1848. In the meantime, just in 1845, the Octonions were rediscovered by A. Cayley, who published his result immediately. For a more detailed history of the discovery of Octonions the reader is referred to pages 146-147 of 6.
1.3. The pioneering work of Ostrowski, Mazur, Albert, and Wright We already know that R, @, W,and 0 are the unique absolute-valued alternative real algebras. As a consequence, B,@, and W are the unique absolute-valued associative real algebras (a fact first proved by S. Mazur 66). More particularly, we have the following.
Proposition 1.3. Let A be a n absolute-valued, associative, and commutative algebra over R. Then A is isometrically isomorphic to either R or @.
105
Proof. Since A is an integral domain, we can consider the field of fractions of A (say IF), and extend (in the unique possible way) the absolute value of A to an absolute value on IF. Now IF is an absolute-valued field extension of R,and hence it is isometrically isomorphic to R or C (see Lemma 1.1 below). Since A is a subalgebra of IF, the result follows. 0 According t o the information collected in pages 243 and 245 of 31, the above proposition and proof are due t o A. Ostrowski 75, who seems to have been the first mathematician considering absolute-valued algebras as abstract objects which are worth being studied. The following lemma (today a consequence of the famous Gelfand-Mazur theorem) is also due t o him.
Lemma 1.1. Every absolute-valued field extension of isomorphic to either R or @.
R is isometrically
The first paper dealing with absolute-valued algebras in a general nonassociative setting is the one of A. A. Albert 2 , who proves as main result the following.
Proposition 1.4.R,C, W,and 0 are the unique absolute-valued finitedimensional real algebras with a unit.
A surprisingly short proof of Proposition 1.4, based on early works of H. Auerbach and A. Hurwitz 5 3 , can be given. However, since such a proof was not noticed by Albert, nor by anybody at his time, we prefer to postpone it until the conclusion of Subsection 2.2, and continue here with the chronological narration of facts. As we will see in Proposition 1.6 below, Proposition 1.4 was refined shorty later by Albert himself. Thus, the actual interest of Albert’s paper relies on both the introduction of the notion of “isotopy” between absolute-valued algebras, and the proof of the following proposition.
Proposition 1.5. Let A be a n absolute-valued finite-dimensional real algebra. Then A is isotopic to either R,C, W,or 0. Therefore A has dimension 1 , 2 , 4 , or 8, and the absolute-value of A comes from a n inner product. According to Albert’s definition, two absolute-valued algebras A and B over K are said t o be isotopic if there exist linear isometries 4 1 , 4 2 , 4 3 from A onto B satisfying & ( z y ) = # Q ( x ) ~ ~ for ( Y )all z , y in A . Albert derives Proposition 1.5 from Proposition 1.4 in a clever but quite simple way. Indeed, choosing a norm-one element a E A , and defining a new product 0 on the normed space of A by s a y := R;’(z)L;’(y), we obtain
106
a finite-dimensional absolute-valued algebra, which is isotopic to A and has a unit (namely, a 2 ) . The argument just reviewed has been recently refined in the paper of A. Elduque and J. M. PQrez 33, yielding Lemma 1.2 immediately below. As we will see later, such a lemma has turned out to be very useful in the theory.
Lemma 1.2. Let A be a n absolute-valued algebra over K such that there exist a , b E A satisfying a A = A b = A . Then A is isotopic to an absolutevalued algebra over K having a unit element. Proof. We may assume that llall = llbll = 1. Then, defining a new product 0 on the normed space of A by z 0 y := R b 1 ( z ) L ; ' ( y ) , we obtain an absolute-valued algebra over K, which is isotopic to A, and has a unit 0 (namely, ab).
Concerning the assertion in Proposition 1.5 about the dimension of absolute-valued finite-dimensional real algebras, it is worth mentioning that, some years after Albert's paper (just in 1958), it was proved the following. Theorem 1.2. Every finite-dimensional division real algebra has dimension 1 , 2 , 4 , or 8.
The paternity of Theorem 1.2 seems to be rather questioned. Indeed, according t o 31, 48, and 6 , such a theorem was first proved by Kervaire and Milnor 6 8 , Adams l, and Kervaire 62 and Bott-Milnor 13, respectively. Anyway, in contrast with the case of Proposition 1.2, all known proofs of Theorem 1.2 are extremely deep. A second paper of Albert contains as main result the following refinement of Proposition 1.4. Proposition 1.6. Let A be a n absolute-valued algebraic real algebra with a unit. Then A is equal to either R, C , W,or 0.
We recall that an algebra A is called algebraic if all single-generated subalgebras of A are finite-dimensional. As we will see later, Proposition 1.6 has been also refined, in two different directions, and at two very distant dates (see Theorems 2.1 and 2.11). Therefore, Proposition 1.6 has today the unique interest of having been, some years later, one of the key tools in the original proofs of more relevant results in the theory of absolute-valued algebras. Among these results, we limit ourselves for the moment to review the one of F. B. Wright log which follows.
107
Theorem 1.3. An absolute-valued algebra over i K i s finite-dimensional i f (and only i f ) it i s a division algebra. Albert’s paper also contains the particular case of Theorem 1.3 that absolute-valued algebraic division algebras are finite-dimensional. However, the proof given in for this result seems to us not to be correct. To conclude the present section, let us note that Propositions 1.3 and 1.6, and Theorem 1.3 above become “ap6ritifs” for Section 2 below.
1.4. Classification
A,
For A equal to either C, W, or 0, let us denote by *A,and A* the absolute-valued real algebras obtained by endowing the normed space of A with the products x @ y := x*y*, x 0 y := x * y , and x @ y := x y * , respectively, where * means the standard involution. It follows easily from * Proposition 1.5 that C, C,*C,and C* are the unique absolute-valued twodimensional real algebras. Therefore, to be provided with a classification (up to algebra isomorphisms) of all finite-dimensional absolute-valued real algebras, it would be enough to obtain such a classification in dimension 4 and 8. Whereas for dimension 8 the problem seems to remain open, the case of dimension 4 has been solved in the paper of M. I. Ramirez 77, by applying Proposition 1.5 and the description of all linear isometries on W (see page 215 of 31). To this end, the so-called principal isotopes of W are considered. These are the absolute-valued real algebras W l ( a ,b), W z ( a ,b), W3(a, b), and &(a, b) obtained from fixed norm-one elements a , b in W by endowing the normed space of W with the products x 0y := axyb, z 0y := ax*y*b, x @ y := x*ayb, and x 0y := axby*, respectively. Then it is proved the following.
Proposition 1.7. E v e y four-dimensional absolute-valued real algebra is isomorphic to a principal isotope of W. Moreover two principal isotopes W i ( a ,b) and Wj(a’, b’) of H are isomorphic if and only if i = j and the equalities a‘p = &pa and b‘p = Spb hold for some norm-one element p E W and some E , S E (1, -1). Proposition 1.7 can be also derived from ”. A refinement of it can be found in 19. The paper 77 also contains a precise description of all four-dimensional absolute-valued real algebras with a left unit, as well as many examples of four-dimensional absolute-valued algebras containing no two-dimensional subalgebra.
108
Eight-dimensional absolute-valued real algebras with a left unit have been systematically studied in the recent paper of A. Rochdi 79. As a first basic result, Rochdi proves the following.
Proposition 1.8. The finite-dimensional absolute-valued real algebras with a left unit are precisely those of the form A,, where A stands for either R, @, IHI, or 0, p : A -+ A is a linear isometry fixing 1, and A, denotes the absolute-valued real algebra obtained by endowing the normed space of A with the product x 0 y := cp(x)y. Moreover, given linear isometries cp, 4 : A 4 A fixing 1, the algebras A, and A4 are isomorphic i f and only if there exists an algebra automorphism $ of A satisfying 4 = $ 0 cp 0 $-’. It is proved also in 79 that, for A and cp as in Proposition 1.8, subalgebras of A, and cp-invariant subalgebras of A coincide. Moreover, a linear isometry p : 0 -+ 0 fixing 1 can be built in such a way that 0 has no four-dimensional pinvariant subalgebra. It follows that there exist eight-dimensional absolute-valued real algebras with a left unit, containing no four-dimensional subalgebra. Such algebras are characterized, among all eight-dimensional absolute-valued real algebras with a left unit, by the triviality of their groups of automorphisms. Such algebras seem to become the first examples of eight-dimensional division real algebras containing no four-dimensional subalgebra. In Subsection 3.4 we will review in detail the results concerning those absolute-valued real algebras A endowed with an isometric algebra involution * which is different from the identity operator and satisfies xx* = x*x for every x E A. In the finite-dimensional case, such algebras have been classified in The classification theorem has a flavour similar to that of Proposition 1.8. Right Moufang algebras are defined as those algebras satisfying the identity x ~ ( ( x I x ~ = ) x(~( )x ~ x I ) x ~ Absolute-valued )x~. right Moufang algebras are considered by J. A. Cuenca, M. I. Ramirez, and E. SBnchez 2 4 , who show that such algebras are finite-dimensional. More precisely, they prove Theorem 1.4 immediately below. The formulation of such a theorem involves the notation introduced in Proposition 1.8 above, as well as the result of N. Jacobson 55 that both IHI and 0 have an “essentially” unique involutive automorphism different from the identity operator.
Theorem 1.4. The absolute-valued right Moufang real algebras are R, @,
IHI, 0,*@, and the algebras A,, where A stands for either JHI or 0, and cp denotes the essentially unique involutive automorphism of A different from
109
the identity operator. 2. Conditions on absolute-valued algebras leading to the finite dimension
2.1. The noncommutative Urbanik- Wright theorem Despite the constant scarcity of works on absolute-valued algebras along the history, a relatively short paper of K. Urbanik and F. B. Wright lo6, appeared in 1960 and announced the same year in lo5, attracted the attention of many people because of the nice simplicity of its powerful results. In fact, Urbanik-Wright theorems have become the key tools in the later development of the theory of absolute-valued algebras. The first surprising result in the Urbanik-Wright paper is the following.
Theorem 2.1. For an absolute-valued real algebra A , the following conditions are equivalent: (1) There exists a E A \ (0) satisfying a x = xu, a ( a x ) = a2x, and (xu). = xu2 for evenJ x E A. (2) A has a unit element. (3) A is equal to either R,@, W,or 0.
We shall call the crucial implication (2) 3 (3) in Theorem 2.1 above the noncommutative Urbanik-Wright theorem. Such a theorem immediately “works havoc” in the theory. For instance, it follows from it, and Albert’s ideas about isotopes, that an absolute-valued algebra A over R is finite-dimensional if (and only i f ) there exists a E A satisfying a A = Aa = A. This refinement of Wright’s Theorem 1.3 attains a better form whenever Lemma 1.2 replaces Albert’s ideas. Thus we have the following.
Theorem 2.2. An absolute-valued algebra A over K is jinite-dimensional if (and only i f ) there exist a, b E A satisfying a A = Ab = A . Even, applying an easy argument of completion (see 26 for details), we derive from Theorem 2.2 a still better form of Theorem 1.3. Indeed, an absolute-valued algebra A is finite-dimensional i f (and only i f ) there exist a, b E A such that a A and Ab are dense in A. Theorem 2.2 was first proved by the author 85 with other techniques. The proof given here is taken from the Elduque-P6rez paper 33.
110
2.2, Kaplansky 's prophetic proof of the noncommutative
Urbanik- Wright theorem Concerning the proof of the noncommutative Urbanik-Wright theorem, the interested reader could go into the original paper lo6to see how Urbanik and Wright apply, to commutative subspaces, Schoenberg 's characterization 95 of pre-Hilbert spaces as those normed spaces X satisfying 1 1 5
+ Y1I2 +
1 1 2-
Yll 2 2 4
for all norm-one elements z , y E X (see Remark 2.1 later), and how then, after some technical arguments, they show that the algebra satisfies the requirements in Albert's Proposition 1.6. However, it seems to us more instructive to sketch how a proof of the noncommutative Urbanik-Wright theorem can be tackled by the light of the present knowledge. Actually, the proof of the noncommutative Urbanik-Wright theorem can be divided into two parts. The first one, of a purely analytic type, consists in realizing that absolute-values on unital algebras come from inner products. This question was completely clarified twenty years ago. Indeed, it is easy to show that unital absolute-valued algebras become particular cases of the so-called smooth-normed algebras (see the proof of ( b ) + ( a ) in Corollary 29 of 82), and it follows from Theorem 27 of 82 that the norm of every smoothnormed algebra derives from an inner product (see also Section 2 of 84 for a considerable simplification of the arguments in 82). We recall that a normed space X over K is said to be smooth at a norm-one element z E X if the closed unit ball of X has a unique tangent real hyperplane at z, and that smooth-normed algebras are defined as those normed algebras A over K having a norm-one unit 1 such that the normed space of A is smooth at 1. Incidentally, we note that C is the unique smooth-normed complex algebra, and that R, C, W,and 0 are the unique smooth-normed alternative real algebras (see 82 and 84, and references therein). We also remark that other arguments of more autonomous nature, showing as well that unital absolute-valued algebras are pre-Hilbert spaces, have been found later by El-Mallah 40 and the author 85 (see Theorems 3.2 and 3.5, respectively). Now that we know that absolute values on unital algebras over K derive from inner products, the second (and last) part of the suggested proof of the noncommutative Urbanik-Wright theorem (now of a purely algebraic type) begins with an easy observation. Indeed, if an absolute value o n a (possibly nonunital) real algebra A comes from an inner product, then we are provided with a nondegenerate quadratic form q o n A (namely, the mapping z -+ llzl12)satisfying q ( s y ) = q ( z ) q ( y )for all z, y E A. In this way, we nat-
111
urally meet the so-called composition algebras, and the problem of classifying them. This problem was already considered and solved by Hurwitz 53 under the additional requirements of finite dimension and existence of a unit. Later Kaplansky “ proved that the assumption of finite dimension in Hurwitz’s theorem is superfluous (see also Chapter 2 of ‘13). Applying the Hurwitz-Kaplansky theorem, we obtain that the unique unital composition real algebras are R, C, R2 (with coordinate-wise multiplication), MI, M2(R), 0, and a certain eight-dimensional alternative nonassociative algebra 0’ which (as for the case of R2 and M2(R)) has nonzero divisors of zero. Since this last pathology is prevented in the case of absolute-valued algebras, the proof of the noncommutative Urbanik-Wright theorem is then concluded. In the paper just quoted, which was published seven years before the one of Urbanik and Wright, Kaplansky prophesies both the noncommutative Urbanik-Wright theorem and a proof similar to that we have sketched above. Even, it seems that he thinks that the noncommutative Urbanik-Wright theorem was already proved at that time. Thus, he says that “Wright log succeeded in removing the assumption [in Albert’s Proposition 1.61 that the algebra is algebraic”. Since we know that the above assertion is not right, we continue reproducing Kaplansky’s words with the appropriate corrections and explanations: “Wright proceeds by proving that the norm [of a unital absolute-valued DIVISION algebra] springs from an inner product [see Lemma 3.2 later], and then that the algebra is algebraic. ... Thus Albert’s finite-dimensional theorem [i.e., Proposition 1.41 can be proved by combining Wright’s result with Hurwitz’s classical theorem on quadratic forms admitting composition [see also Proof of Proposition 1.4 below]”. Immediately, Kaplansky motivates his work by saying that “The main purpose of this paper is to make a similar method possible in the infinite-dimensional case by providing a suitable generalization of Hurwitz’s theorem.” Concerning the proof of the noncommutative Urbanik-Wright theorem just sketched, let us also comment that, really, the two parts in which we have divided it overlap somewhat. This is so because the proofs of the results in 82, 40, and 85, implying that unital absolute-valued algebras are pre-Hilbert spaces, give simultaneously a rich algebraic information, which is also provided by a part of the proof of the Hurwitz-Kaplansky theorem. In fact, with such an additional information in mind, the proof of the noncommutative Urbanik-Wright theorem can be concluded by applying the Frobenius-Zorn theorem instead of the one of Hurwitz-Kaplansky (see
112
Remark 31 of s2 and Remark 4 of s5 for details). To conclude the present subsection, let us show how actually Albert could have derived his Proposition 1.4 from Hurwitz’s theorem, if he were aware of a result of Auerbach (see also Theorem 9.5.1 of ”) implying that finite-dimensional transitive normed spaces are Hilbert spaces. We recall that a normed space X is called transitive if, given arbitrary norm-one elements z, y E X , there exists a surjective linear isometry T : X + X such that T ( z )= y. The notion of transitivity just introduced will be revisited more quietly in Subsection 5.1.
Proof of Proposition 1.4. Let A be an absolute-valued algebra over K. If A is a division algebra, then the normed space of A is transitive, since for all norm-one elements x , y E A we have T ( z )= y, where T := L R G ~ ( is vl a surjective linear isometry on A . Therefore, when A is finite-dimensional, Auerbach’s result applies, giving that the norm of A comes from an inner product. Finally, if R = R,if A is finite-dimensional, and if A has a unit, then A is equal t o either R,@, W,or 0 (by Hurwitz’s theorem). 0 The argument in the above proof is taken from page 156 of 6 , where no reference to the works of Albert and Auerbach is done. In fact, Proposition 1.4 appears as Theorem 1 of 6 , and is directly attributed there to Hurwitz 53, including shorty later the above argument as a part of the complete proof of such Hurwitz’s theorem. We do not agree with this attribution. Indeed, as far as we know, the observation that absolute-valued division algebras have transitive normed spaces appears first in the proof of Lemma 4 of Wright’s paper log (fifty five years after Hurwitz’s paper). On the other hand, Aurbach’s result, published thirty six years after Hurwitz’s paper, seems to us non obvious. 2.3. The commutative Urbanik- Wright theorem
The second surprising result in the Urbanik-Wright paper ing.
Theorem 2.3. R, C, and real algebras.
lo6is
the follow-
are the unique absolute-valued commutative
We shall call Theorem 2.3 above the commutative Urbanik-Wright theorem. We know no proof of Theorem 2.3 other than the original one in lo6. Starting with a new application of Schoenberg’s theorem 95, such a proof is really clever and easy. Therefore we do not resist the temptation of reproducing it here. Some unnecessary complications are of course avoided.
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Proof. Let A be an absolute-valued commutative real algebra. Since for all norm-one elements z, y E A we have 4 = 411zYll = I(.
+ d2- .(
- Y)"I
i 112 + Y1I2
+ 11%
- YII
2 7
Schoenberg's theorem applies giving that A is a pre-Hilbert space. On the * other hand, since R, @, and @ are the unique absolute-valued commutative real algebras of dimension 5 2 (see Subsection 1.4), it is enough to show that the dimension of A is 5 2. Assume to the contrary that we can find pair-wise orthogonal norm-one elements u,w,w in A. Then we have 11u2- v211 = IIu +v1111u - 1111 = 2. Since llu211 = 11v211 = 1, the parallelogram law implies that u2+v2 = 0. Analogously, we obtain u2+w2 = v2+w2 = 0. It follows u2 = 0, and hence also u = 0, a contradiction. With the help of Lemma 2.4 below, the commutative Urbanik-Wright theorem can be refined as follows. There is a universal constant K > 0 such that every absolute-valued real algebra A satisfying llzy - yzll 5 Kllzllllyll
f o r all z, y
EA
is in fact equal to either R, @, or
(see Corollary 1.4 of
59).
Remark 2.1. For a normed space X over K, consider the property P which follows:
( P ) There exists a normed space Y over R, together with a bilinear mapping ( a , b ) + ab from X x X to Y satisfying ab = ba and llabll = llall llbll for all a, b E X. Arguing as in the beginning of the proof of Theorem 2.3, we see that, if the normed space X satisfies Property P , then X is a pre-Hilbert space. The converse is also true (see Theorem 4.4 of 8). 2.4. Power-associativity
Let A be an algebra over a field IF. We say that A is of bounded degree if there exists a natural number n such that all single-generated subalgebras of A have dimension 5 n, and power-associative if all single-generated subalgebras of A are associative. In the case that the characteristic of IF is different from 2, we will consider the algebra A" whose vector space is the same as that of A , and whose product is defined by z @ y := $ ( ~ y yz). We remark that both the bounded degree and the power-associativity pass from A to A".
+
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Lemma 2.1. Let A be a normed algebra over K satisfging llx211 = llx112 for every x E A, and such that A" is power-associative and of bounded degree. Then A has a norm-one unit. Proof. Since A" is a commutative power-associative algebra of bounded degree, and has no nonzero element x such that x 2 = 0, it follows from Proposition 2 of 21 that A" has a unit element (say 1). Moreover, since 11111 = 111211 = 111112,we have 11111 = 1. Then, since A is a normed algebra, both L1 and R1 lie in the closed unit ball of the normed algebra C ( A ) of all continuous linear operators on A. Since i ( L 1 R1) = I A (the identity operator on A ) , and I A is an extreme point of the closed unit ball of C ( A ) (by Proposition 1.6.6 of 9 3 ) , it follows that L1 = R1 = I A , i.e., 1 is a unit element for A. 0
+
Now we can prove the main result in this subsection. It is due to ElMallah and Micali 45, and reads as follows. Theorem 2.4. R, C,W, and 0 are the unique absolute-valued powerassociative real algebras. Proof. Let A be an absolute-valued power-associative real algebra. By Proposition 1.3, A is of bounded degree. Then, by Lemma 2.1, A has a unit. Finally, by the noncommutative Urbanik-Wright theorem, A is equal to either R, C,W,or 0. 0
The original proof of El-Mallah and Micali differs not too much of the above one. Of course, they did not know Lemma 2.1, which has been proved here by the first time. Thus, in the El-Mallah-Micali proof, Lemma 2.1 was replaced with a simpler purely algebraic result (see Lemma 1.1 of 45). Anyway, both Lemma 1.1 of 45 and Proposition 2 of l7 (which has been one of the tools in the proof of Lemma 2.1, and is also of a purely algebraic nature) have a common root, namely the proof of Lemma 5.3 of 94. Before the appearance of the Urbanik-Wright paper, Wright knew that R, C,W, and 0 are the unique unital absolute-valued power-associative real algebras (see the introduction of log). This (today doubly unsubstantial) result was rediscovered by L. Ingelstam 54 (four years after the appearance of the Urbanik-Wright paper!) with a proof essentially identical to the one suggested by Wright in log. Anyway, the Wright-Ingelstam argument has some methodological interest. Indeed, it shows that, in an autonomous
115
proof of Theorem 2.4, the noncommutative Urbanik-Wright theorem can be replaced with Albert’s forerunner given by Proposition 1.6. As we commented in Subsection 2.2, smooth normed algebras are preHilbert spaces. A converse to this fact is proved in Proposition 2.1 immedi(see ately below. The key tools are Lemma 2.1 and the result of B. Zalar also Theorem 3 of ‘12) that IR and C are the unique pre-Hilbert associative commutative real algebras A satisfying llz211= )1z112f o r every z E A.
Proposition 2.1. Let A be a normed real algebra. Then the following conditions are equivalent: (1) A is a smooth-normed algebra.
(2) A is power-associative, the norm of A derives from a n inner product, and the equality 1)z211= )1z112holds f o r every z E A . (5’) A” is power-associative, the norm of A derives from an inner product, and the equality llz211= ))z)12holds f o r every z E A . Proof. The implication (1) + (2) is a consequence of Theorem 27 of 8 2 , whereas the one (2) + (3) is clear. Assume that Condition (3) is fulfilled. Then, by Zalar’s result quoted above, the algebra As is of bounded degree. Therefore, by Lemma 2.1, A has a norm-one unit. Since pre-Hilbert spaces are smooth at all their norm-one elements, it follows that A is a smoothnormed algebra. 0 The following result of Zalar ‘11 follows straightforwardly from Proposition 2.1 above and Hurwitz’s theorem (see Subsection 2.2). Theorem 2.5. Let A be a n absolute-valued real algebra whose norm springs from a n inner product, and such that AS is power associative. Then A is equal to either W,C, W,or 0.
In relation to Proposition 2.1, it is worth mentioning that smooth normed algebras are precisely those unital normed algebras A satisfying 111 - z211 = 111 zlllll - zll f o r every z E A , as well as those unital normed algebras A satisfying IIVz(y)II = 1 1 ~ ) ) ~ )f o) yr Jall I z,y E A , where U%(y):= z(yz) (yz)z - yz2 (see Corollary 29 of ”). Another characterization of smooth normed algebras is given in the next proposition.
+ +
Proposition 2.2. Let A be a normed real algebra. Then the following conditions are equivalent: (1) A is a smooth-normed algebra.
116
(2) A is power-associative, and the equality IIUz(y)II = llx11211~llholds for all x, y E A.
Proof. In view of Proposition 2.1 and the comments immediately above, it is enough t o show that (2) implies that A has a unit. Assume that (2) is fulfilled. Then for x, y in any single-generated subalgebra of A, we have
Therefore, all single-generated subalgebras of A are absolute-valued algebras. By Proposition 1.3, A is of bounded degree. Finally, by Lemma 2.1, A has a unit. 0 Proposition 2.2 was first proved by M. Benslimane and N. Merrachi lo with slightly different techniques. More information about smooth normed algebras can be found in Subsection 3.5. To conclude the present subsection, let us comment that Theorem 2.4 is “almost” contained in the early paper of Urbanik lo2. Indeed, it could have been very easy for him to establish such a theorem by selecting, among the many auxiliary results in that paper, the appropriate ones for the goal. However, Urbanik does not do this, since he completely devotes his paper lo2 to characterize R, @, IHI, and 0 in terms conceptually far from the power-associativity. An element x of an algebra A is said to be reversible if there exists y E A satisfying x y - xy = x y - yx = 0. The algebra A is said to fulfill the reversibility condition if all its elements, except those in some countable set, are reversible. Now the main result in lo2 reads as follows.
+
+
Theorem 2.6. R, @, W,and 0 are the unique absolute-valued real algebras satisfying the reversibility condition. Note that for A equal to either R, @, IHI, or the unit of A, are reversible.
0, all elements of A, except
2.5. Flexibility
An algebra is said to be flexible whenever it satisfies the identity ( 2 1 2 2 ) q = x 1 ( z 2 q ) . Since single-generated subalgebras of flexible algebras are commutative, the commutative Urbanik-Wright theorem applies successfully t o single-generated subalgebras of absolute-valued flexible algebras. After a lot of work, involving the information obtained from the procedure just pointed out, El-Mallah and MiCali 46 prove the following.
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Lemma 2.2. Absolute-valued flexible algebras are finite-dimensional. Later, El-Mallah, in a series of papers (see 36, 37, 38, 39, and 40), refines deeply the result just reviewed, by considering absolute-valued algebras satisfying the identity xx2 = x2x (which is of course implied by the flexibility), and proving the following.
Theorem 2.7. For a n absolute-valued real algebra A, the following assertions are equivalent: (1) A is flexible. (2) A i s a pre-Hilbert space and satisfies the identity x2x = x x 2 . (3) A i s finite-dimensional and satisfies the identity x 2 x = xx2. * * * (4) A i s equal to either R, C, C, Ell, H, 0, 0, or the algebra pseudo-octonions.
P of
According to Theorem 2.7 just formulated, the algebra P of pseudooctonions is the unique absolute-valued flexible real algebra which has been not still introduced in our development. Such an algebra was discovered by S. Okubo 72 (see also pages 65-71 of 70). The vector space of P is the eight-dimensional real subspace of M3 (C) consisting of those trace-zero elements which remain fixed after taking conjugates of their entries and passing t o the transpose matrix. The product 0 of P is defined by choosing a complex number p satisfying 3p(1 - p ) = 1, and then by putting 1 x 0y := p ~ y (1 - / L ) ~-xz T ( x y ) l .
+
Here T denotes the trace function on M3(C), 1 stands for the unit of the associative algebra I@(@), and, for x , y in P,x y means the product of x and y as elements of such an algebra. If for x , y E P we define ( x l y ) := i T ( x y ) , then (.I.) becomes an inner product on P whose associated norm is an absolute value. In relation to Theorem 2.7, it seems t o be an open problem (see the abstract of 41) if every absolute-valued real algebra satisfying the identity x2x = xx2 is finite-dimensional. According to Theorem 2.7 itself, the answer is affirmative if A is a pre-Hilbert space. The answer is also affirmative if A is algebraic 41, but, as we will see in Subsection 2.7, this result is today unsubstantial. As a more ambitious problem, we can wonder whether every absolute-valued algebra satisfying some identity is finite-dimensional. The classification of absolute-valued flexible real algebras contained in Theorem 2.7 was tried in 63, with a partial success. Actually, such a clas-
118
sification can be derived from Lemma 5.3, Proposition 1.5, and 73. Theorem 2.7 has inspired the result in 34 that finite-dimensional composition algebras satisfying the identity x 2 x = xx2 are an fact flexible. 2.6. H*-theory
The following theorem has been proved by J. A. Cuenca and the author
26.
Theorem 2.8. Let A be an absolute-valued algebra over K. Assume that there exists a complete inner product (.I.) o n A, together with an involutive conjugate-linear operator * o n A, satisfying (zylz) = ( x l z y * ) = (y(x*z)f o r all x,y , z E A . Then we have: (1) A is finite-dimensional.
(2) The Hilbertian norm. x + is a positive multiple of the absolute-value of A. (3) The operator * is an algebra involution o n A . (4) The equality x*(zy) = (yx)x* = holds f o r all x,y, z E A. With the terminology of 2 5 , the assumptions on (.I.) and * in Theorem 2.8 mean that, forgetting the absolute value of A, (A,(.[.),*) is a semi-H*-algebra over K, The conclusion, that * is in fact an algebra involution, then reads as that ( A ,(.I.), *) is an H*-algebra over K. Besides a little H*-theory 2 5 , the proof of Theorem 2.8 involves some results on absolute-valued algebras previously reviewed (as Wright’s Theorem 1.3), and others to be reviewed later (as for example Theorem 3.8). Such a proof, as well as that of Theorem 2.9 below, also includes some easy facts first pointed out in 86. Among these, we emphasize the following one for later reference. L e m m a 2.3. Let A be an absolute-valued algebra over K. Assume that the absolute-value of A comes from an inner product (.I.), and that, for every x E A, there exists x* E A satisfying (xylz) = (ylz*z) for all x,y, z E A. f o r all x,y , z E A. Then we have x * ( x y ) = Proof. For x,y E A, we have (zylxy) = Ilxl12(yly). Linearizing in the variable y, we obtain that the equality ( z z l x y ) = IIx112(zly) holds for all x,y,z E A . Since ( x z I x y ) = (zlx*(xy)), we deduce ( z I z * ( z y ) )= 11x112(z1y), which, in view of the arbitrariness of z , yields z * ( x y ) = l l ~ 1 1 ~ y . 0 The Cuenca-Rodriguez paper 26 also contains a precise determination of the algebras A in Theorem 2.8. Since the case that K = C is unsub-
119
stantial (see Subsection 2.8 later), only the case that R = R merits to be considered. Thus, in view of Theorem 2.8, we are dealing in fact with an absolute-valued finite-dimensional real algebra A endowed with an algebra involution *, and whose norm derives from an inner product (.I.) satisfying x*(xy) = (yx)x* and (xylz) = (xlzy*) = (ylx*z) for all z,y,z E A. Since * is isometric, we can consider the isotope of A (say B ) consisting of the normed space of A and the product x @ y := x*y*. Now, we trivially realize that the absolute-valued real algebra B is flexible and satisfies (x0ylz) = (zly 0z ) for all z, y, z E B . Then, we deduce from El-Mallah’s * * * Theorem 2.7 that B is equal to either R, @, W,0, or P. Moreover, * becomes an algebra involution on B , and the correspondence (A,*) 4 ( B ,*) is categorical and bijective. After the laborious classification of algebra in-
* * *
volutions on @, W,0, and P made in 26, the determination of the algebras in Theorem 2.8 concludes. In this way, three new distinguished examples of absolute-valued finite-dimensional real algebras appear. These are the natural isotopes of MI,0, and P (denoted respectively by W,0, and @ built as follows. For every absolute-valued algebra A, and every linear isometry $ on A, the $-twist of A is defined as the absolute-valued algebra consisting of the normed space of A and the product x @ y := $(x)$(y). For A equal to either JH[ or 0, we define as the $-twist of A, where $ stands for the essentially unique involutive automorphism of A different from the identity operator (see Subsection 1.4). On the other hand, there exists an “essentially” unique algebra involution 0 on P, which allows us to define @ as the 0-twist of P. Now we have the following. A
h
Theorem 2.9. Let A be an absolute-valued real algebra fulfilling the requirements in Theorem 2.8. Then A is equal to either R,@, ,; JHI, fi, 0, 6,o r @ . A slight variant of the proof of Theorem 2.9 sketched above, involving Corollary 7 of 74 instead of Theorem 2.7, can be seen in Remark 2.9 of 26.
2.7. Alge braicit y Albert’s Proposition 1.6, although obsolete after the noncommutative Urbanik-Wright theorem, has had the merit of encouraging the work on the question if every absolute-valued algebraic algebra is finite-dimensional. Since for complex algebras such a question has an almost trivial affirmative answer (see the concluding paragraph of Subsection 2.8 below), the interest centers in the case of real algebras. Some partial affirmative answers have
120
been provided by El-Mallah. Thus, an absolute-valued algebraic real algebra is finite-dimensional whenever there exists a nonzero idempotent in A commuting with every element of A 36, or there exists a continuous algebra involution * on A satisfying xx* = x*x f o r every x E A 3 9 J or A satisfies the identity xx2 = x2x 41. We note that the result in 36 would become later a consequence of the one in 39 (see El-Mallah's Theorem 3.2), and that the result in 41 was already commented a t the conclusion of Subsection 2.5. To specify that an algebra A is of bounded degree, let us say that A is of degree n E N if n is the minimum natural number such that all single-generated subalgebras of A have dimension 5 n. It follows from Proposition 1.4, that absolute-valued algebraic real algebras are of bounded degree, and, more precisely, of degree 1,2,4, or 8. Then, since IR is the unique absolute-valued algebraic algebra of degree 1 (see again the concluding paragraph of Subsection 2.8), the strategy of studying separately the cases of degree 2,4, and 8 could seem tempting in order t o answer affirmatively the question we are considering. Unfortunately, such an strategy has turned out t o be unsuccessful for the moment, unless for the case of degree 2, for which we have the following result of the author 89. * Theorem 2.10. The absolute-valued real algebras of degree two are @, @,
h, *HIJ MI*, 0,6,*0,O*, and IF'.
*@, @*, W,
Via the commutative Urbanik-Wright theorem, Theorem 2.10 above contains both Theorem 2.4 and the classification of absolute-valued flexible real algebras included in Theorem 2.7. However, this is quite deceptive because, in fact, the proof of Theorem 2.10 involves Theorem 2.4 and the whole Theorem 2.7. In any case, by keeping in mind again the commutative Urbanik-Wright theorem, Theorem 2.10 shows by the first time that, for absolute-valued algebras, power-commutativity and flexibility are equivalent notions. We recall that an algebra is said to be power-commutative if all its single-generated subalgebras are commutative, and that flexible algebras are power-commutative 7 6 . Theorem 2.10 has inspired the classification of composition algebras of degree two, done in 34. Returning to the general problem if absolute-valued algebraic algebras are finite-dimensional, we must say that, six years ago, A. Kaidi, M. I. Ramirez, and the author 57 succeeded in solving it. Thus we have the following. Theorem 2.11. A n absolute-valued real algebra is finite-dimensional if
(and only i f ) it is algebraic.
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We know no proofs of Theorem 2.11 above others than the original one in 57, and the slight variant of it given in 58. We do not enter here the details of such proofs, nor even give a sketch of them. Referring the reader to 58 for such a sketch, we limit ourselves here to say that both arguments are long and complicated, and involve in an essential way the techniques of normed ultraproducts 52. Thus, by the first time in the theory of absolute-valued algebras, the following folklore result shows useful.
Lemma 2.4. The normed ultraproduct of every ultrafiltered family of absolute-valued algebras over R becomes naturally an absolute-valued algebra over R. Concerning the proof of Theorem 2.11, let us also revisit a minor auxiliary result (namely, Lemma 4.2 of 5 7 ) . Such a result can be refined as follows.
Lemma 2.5. Let X be a normed space over R, let F : X -+ X be a linear contraction, and let M be a finite-dimensional subspace of X . Assume that F is the identity o n M , and that X is smooth at every norm-one element of M . Then there exists a continuous linear projection IT from X onto M such that ker(7r) is invariant under F . Proof. Let M * denote the dual space of M . By a theorem of Auerbach (see also Lemmas 7.1.6 and 7.1.7 of 92), there are bases {ml,...,m k } and (91, ...,gk} of M and M * , respectively, consisting of norm-one elements and satisfying gi ( m j )= S i j . Extending each gi to a norm-one linear functional & on X (via the Hahn-Banach theorem), and considering the mapping 2 + ~ ~ = , q 5 i ( z )from m i X to M , it is easily seen that such a mapping satisfies the properties asserted for 7r in the statement of the lemma (see the proof of Lemma 4.2 of 57 for details). 0
Lemma 2.5 above was proved in 57 under the additional assumption that the restriction to M of the norm of X springs from an inner product. The refinement we have just made does not matter there because, when the lemma applies, X is an absolute-valued algebra, and M is a subspace of a finite-dimensional subalgebra of X , so that the superfluous requirement in the original formulation of the lemma is automatically fulfilled (by Proposition 1.5).
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2.8. A remark on complex algebras
All conditions we have considered above, leading absolute-valued real algebras to the finite-dimension, in the case of absolute-valued complex algebras yield that the algebra is C. Indeed, if an absolute-valued complex algebra fulfills some of those conditions, then, by restriction of scalars, we obtain an absolute-valued real algebra satisfying the same condition, and hence the corresponding result applies. But we know that absolute-valued finite-dimensional algebras are division algebras, and that C is the unique finite-dimensional division complex algebra. In some cases, the result obtained in this way can be refined still more. For example, the complex version of Theorem 2.2 is that, i f A is an absolutevalued complex algebra, and i f there exists a E A such that a A is dense in A, then A = C (see Lemma 1.1of 2 6 ) . On the other hand, the joint complex version of Theorems 2.8 and 2.9 is that, if A is an absolute-valued complex algebra, and i f there exists a complete inner product (.I.) on A making the product continuous, and an involutive conjugate-linear operator * o n A satisfying (zylz) = (zlzy*) for all x , y , z E A, then A = C (see Theorem 1.2 of ")). None of the two results just quoted remains true (with the finite-dimensionality of A instead of A = C in the conclusion) whenever real algebras replace complex ones. Concerning the second result, in the real case nor even can be expected the Hilbertian norm z + to be equivalent to the absolute value of A (see Example 1.7 of 2 6 ) . These pathologies give rise to an interesting development of the theory of absolutevalued algebras, which will be reviewed in Subsection 3.5. As a consequence of Theorem 2.11 and the comments at the beginning of the present subsection, C is the unique absolute-valued algebraic complex algebra. However, this can be proved elementarily. Indeed, notice that, by the same comments, absolute-valued algebraic complex algebras are of degree one, and that, if F is a field containing more than two elements, if A is an algebra over F of degree one, and if there is no nonzero element x E A with x2 = 0, then A = F (see for example page 297 of 57). 3. Infinite-dimensional absolute-valued algebras 3.1. The basic examples
The first example of an absolute-valued infinite-dimensional algebra appears in the celebrated paper of Urbanik and Wright lo6. Indeed, they show that the classical real Hilbert space l 2 becomes an absolute-valued algebra under a suitable product. Looking at their argument, many other
123
similar examples can be built. To get them, let us start by fixing an arbitrary nonempty set U , and a mapping 6 : U x U --f X , where X = X ( U , K ) stands for the free vector space over K on U . We denote by A = A(U,6,K) the algebra over K whose vector space is X , and whose product is defined as the unique bilinear mapping from X x X to X which extends 6. From now on, we assume that U is infinite, and, accordingly to such an assumption, we choose 6 among the injective mappings from U x U to U or, more generally, of the form f g , where g : U x U ---t U is injective and f : U x U + K satisfies If(u,v)l = 1 for every ( u , v ) E U x U . With these restrictions in mind, we are going to realize that there are “many” absolute values on A. To this end, let us involve a new ingredient, namely an extended real number p with 1 5 p 5 00. Then, for x in A, we can think about the family {x,},~u of coordinates of x relative to U ,and define llzllp := (CuEu Iz,IP)t if p
< 00 and IIxllm := max{1z,I
:uE
U}.
Invoking the properties of 6, we straightforwardly verify that )I . 1, is an absolute value on A. We denote by dp= dp(U,19, K)the absolute-valued algebra over K obtained by endowing A with the norm 11. .,1 By considering the completion of A,, we obtain a complete absolutevalued algebra over R, denoted by C, = C,(U, 6, K), whose Banach space is nothing other than the familiar space l,(U, K) if p # 00,or Q(U,K)otherwise. Now, the UrbanikWright example is just the algebra Cz(N,6, R), with 6 : N x N --f N equal to any bijection. Returning to our general setting, let us remark that, since C, is a Hilbert space if and only if p = 2, it follows from the above construction that composition algebras need not be finite-dimensional, and that, contrarily to what is conjectured in 40, absolute-values need not come from inner products. Another consequence of our construction is that there exist complete absolute-valued algebras without uniqueness of the (noncomplete) absolute value. Indeed, for 1 5 p < q 5 00, the complete absolute-valued algebra C, can be algebraically regarded as a subalgebra of C,, but the topology of the restriction of the absolute value of C, t o C, does not coincide with the natural one of C,. The straightforward fact, that 11 1, 2 11 . 1, on C,, is not anecdotic. Indeed, as a consequence of Theorem 3.8 below, every complete algebra norm o n an absolute-valued algebra is greater than the absolute value. In particular, two complete absolute values on the same algebra must coincide. The refinement of the Urbanik-Wright example, done above, is implicitly known in some works on Banach spaces (see for instance the proof of
-
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Theorem 3.a.10 of 6 5 ) . The interest of such a refinement in the theory of absolute-valued algebras seems to have been first pointed out in 85. A real algebra A is said to be ordered if it is provided with a subset A+ of positive elements, which is closed with respect t o multiplication by positive real numbers and with respect to addition and multiplication in A , and satisfies A+ n ( - A + ) = 0 and A+ U ( -Af) = A \ (0). In lo4,Urbanik shows that IR is the unique absolute-valued finite-dimensional ordered real algebra. Nevertheless, he also proves the following.
Theorem 3.1. There exists a com.plete absolute-valued infinite-dimensional ordered real algebra.
A simplification of Urbanik’s argument is the following. Proof. Let 6 : N x N + N be defined by 6(n,m):= 2n3m, and let us fix 1 5 p 5 00. Since 6 is injective, we can consider the complete absolutevalued infinite-dimensional real algebra C, = C,(N, 6,R). The natural inclusion N -+ C, converts N into a Schauder basis of C,. For z E C,, let { z n } , E ~ stand for the family of coordinates of z relative t o such a basis, and, when z # 0, define n ( x ) := min{n E N : IC, # 0). Finally, put C t := {x E C, \ (0) : zrn(%) > 0). Keeping in mind that 19 is increasing in each one of its variables, it is easily seen that C$ fulfils the properties required above for the sets of positive elements of ordered real algebras. 0 3.2. h e normed nonassociative algebras
Let us fix a nonempty set V . Nonassociative words with characters in V are defined inductively (according t o their “degree”) as follows. The nonassociative words of degree 1 are just the elements of V . If n 2 2, and if we know all nonassociative words of degree < n , then the nonassociative words of degree n are defined as those of the form ( W I ) ( W ~ )where , w1 and w2 are nonassociative words with deg(wl)+deg(w2) = n. Although the use of brackets is essential in the above definition, some natural simplifications in the writing are usually accepted. For example, brackets covering a word of degree 1 are omitted, and words of the form (w)(w), for some other word w ,are written as ( w ) ~ Two . nonassociative words are taken to be equal only if they have exactly the same writing. Thus for example, for v E V , the nonassociative words vv2 and v2v are different. Now, denoting by U the set of all nonassociative words with characters in V , and by 6 the mapping ( ~ 1 , 2 0 2-+ ) (w1)(w2) from U x U to U , we can think about the
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algebra A(U, 6, K)constructed in the preceding subsection. Since such an algebra depends only on V and K,we denote it by F(V,K). The algebra .F(YiK)] called the free nonassociative algebra on V over K,contains
V in a natural manner, and is characterized up to algebra isomorphisms by the following “universal property”: If A is any algebra over K,and if cp : V + A is any mapping, then ‘p extends uniquely to an algebra homomorphism from F(V,K) to A (see Theorem 1.1.1 of ‘13). Now, since the mapping 29 above is injective (by Proposition 1.1.2 of ‘13), we invoke again the preceding subsection to realize that there are “many” absolutevalues o n F(V,IK). In the original proof Io4 of Theorem 3.1, Urbanik already knows that, when V reduces to a singleton, F(V,R) becomes an absolutevalued algebra under the norm 1). 112. The general case of such an observation is due to M. Cabrera and the author (who announced it in 16), and appears formulated with the appropriate precisions first in 8 5 . For 1 5 p 5 co, we denote by Fp(V,IK)the absolute-valued algebra over IK obtained by endowing F(V, K) (= d(U,6,K)for U and 29 as above) with the absolute value 11 . I l p . As we are seeing in the proof of Proposition 3.1 immediately below, the absolute-valued algebra F1 (V, K)has a special relevance in the general theory of normed algebras. Proposition 3.1. Let V be a nonempty set. Then, up to isometric algebra isomorphisms, there exists a unique normed algebra N = N ( V ,K) over K satisfying the following properties: ( 1 ) V is a subset of the closed unit ball of N. (2) If A is any normed algebra over IK, and if cp is any mapping from
V into the closed unit ball of A , then cp extends uniquely to a contractive algebra homomorphism from N to A . Moreover, we have: (3) The normed algebra N is in fact an absolute-valued algebra. (4) The set V consists only of norm-one elements of N .
Proof. Take N = Fl(V,K). Clearly N satisfies Properties ( I ) , (3), and (4) in the statement. Let A be a normed algebra over IK, and let cp be a mapping from V into the closed unit ball of A. Since, forgetting the norm, N is nothing other than F(V,IK), the universal property of this last algebra provided us with a unique algebra homomorphism 11, : 3 1(V,K)--f A which extends cp. Let x be in N . We have x = CwEU x w w , where U denotes the set of all nonassociative words with characters in V , and { I C , } , ~ ~stands
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for the family of coordinates of x relative to
U.Therefore
(Starting from the fact that $ ( V ) is contained in the closed unit ball of A , the inequality ll$(w)II 5 1 just applied is proved by induction on the degree of w.) Now that we know that N also satisfies Property (2), let us conclude the proof by showing that Af is the “unique” normed algebra over K satisfying (1) and (2). Let N’be a normed algebra over K satisfying (1) and (2) with N ’instead of N , Then we are provided with contractive algebra homomorphisms 4 : N -+ N ’and 4’ : N ’+ N fixing the elements of V . Therefore $04 and 40# are contractive algebra endomorphisms of N and N ‘, respectively, extending the corresponding inclusions V 4 N and V -+ N I.By the uniqueness of such extensions, we must have 4’ 0 4 = IN and c$ o 4’ = I,v It follows that @ is an isometric algebra isomorphism from N onto N ’respecting the corresponding inclusions of V in each of 0 the algebras. I .
Now, if A is a normed algebra over K, if V denotes the closed unit ball of
A , and if CP : N(V,K) -+ A is the unique contractive algebra homomorphism which is the identity on V, then we easily realize that the induced algebra homomorphism N(V,K)/ ker(@) A is a surjective isometry. Therefore, we have the following. -+
Corollary 3.1. Every normed algebra over K is isometrically algebraisomorphic to a quotient of an absolute-valued algebra over K.
The absolute-valued algebra N(V,K) in Proposition 3.1 has its own right to be called the free normed nonassociative algebra on the set V over K. The variant of Proposition 3.1, with Lccompletenormed” instead of “normed” everywhere, is also true, giving rise to the free complete normed nonassociative algebra on the set V over K. This algebra is implicitly involved in the proof of the following result. Corollary 3.2. Every complete normed algebra over K is isometrically algebra-isomorphic to a quotient of a complete absolute-valued algebra over K. Proof. Let A be a complete normed algebra over K. Choose any subset V of A whose closed absolutely convex hull is the closed unit ball of A . By Proposition 3.1, N(V,K) is an absolute-valued algebra over K whose
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closed unit ball contains V , and there exists a contractive algebra homomorphism from N(V,IK)to A fixing the elements of V . By passing to the completion of N(V,IK), and invoking the completeness of A, we are in fact provided with a complete absolute-valued algebra B over IK whose closed unit ball contains V , and a contractive algebra homomorphism @ : B A fixing the elements of V . Let A1 and B1 denote the closed unit balls of A and B , respectively. Since a ( & ) is an absolutely convex subset of A containing V, and A1 is the closed absolutely convex hull of V, the closure of @(B1)in A contains A l . Now, from the main tool in the proof of Banach’s open mapping theorem (see for example Lemma 48.3 of 11) we deduce that @(B1)contains the open unit ball of A. Since : B -+ A is a contractive algebra homomorphism, it follows from the above that the induced algebra homomorphism B / ker(@) 4 A is a surjective isometry. 0 -+
Of course, the most confortable choice of V in the above proof is the one V = A1 . However, finer selections of V allow us to realize that the absolute-valued algebra B can be chosen with the same density character as that of A. We recall that the density character of a topological space E is the smallest cardinal among those of dense subsets of E. Gelfand-Naimark algebras are defined as those complete normed complex algebras A endowed with a conjugate-linear algebra involution * satisfying 11x*x11 = 1 1 ~ 1 for 1 ~ every x E A. Their name is due to the celebrated Gelfand-Naimark theorem 30 that there are no Gelfand-Naimark associative algebras others than the closed *-invariant subalgebras of the Banach algebra C ( H ) of all continuous linear operators o n some complex Hilbert space H , when this last algebra is endowed with the involution * determined by (x(q)I 0 are called nearly absolute-valued algebras. Let A be a normed finite-dimensional algebra over K. Then, by the compactness of spheres, A is nearly absolute-valued if (and only i f ) it is a division algebra. Moreover, if this is the case, then A is isomorphic to C when K = C, and the dimension of A is equal to 1 , 2 , 4 , or 8 when K = R (by Theorem 1.2). On the other hand, by Hopf’s theorem (see page 235 of 31), nearly absolute-valued finite-dimensional commutative real algebras have dimension 5 2. We note that, since every finite-dimensional algebra over K can be endowed with an algebra norm, nearly absolute-valued finite-dimensional real algebras and finite-dimensional division real algebras essentially coincide. By Theorem 1.1, every nearly absolute-valued alternative real algebra is isomorphic to either R, @, W, or 0,so that no much more can be said about such an algebra. The consequence that nearly absolute-valued alternative algebras are algebra-isomorphic to absolute-valued algebras is no longer true if alternativeness is removed (even for finite-dimensional normed algebras A wit p ( A ) near one). For instance, for 0 < E < consider the normed real algebra A consisting of the normed space of W and the product IC 0y := (1 - E)ZY + E ~ Z .Then we have p ( A ) 2 1 - 2 ~ , but A cannot be algebra-isomorphic to an absolute-valued algebra. Indeed,
4,
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A is not associative and has a unit, whereas every absolute-valued fourdimensional real algebra with a unit is isomorphic t o W (by Theorem 2.1). The above example shows in addition how a theory of nearly absolutevalued algebras parallel to that of absolute-valued algebras cannot be expected. Another notice in the same line is that, in contrast with Theorem 2.3, there exist nearly absolute-valued infinite-dimensional commutative algebras over K. Indeed, as proved in Example 1.1 of 59, for every infinite set U and every injective mapping 6 : U x U + U ,the complete normed real algebra A obtained by replacing the product of &(U, 6, K) (see Subsection 3.1) with x 0y := ; ( x y y z ) satisfies p ( A ) 2 2-a. Despite the above limitations, in the paper of Kaidi, Ramirez, and the author 59 just quoted we wondered whether there could be a theory of nearly absolute-valued algebras “nearly” parallel to that of absolute-valued algebras. More precisely, we raised the following.
+
Question 4.1. Let P be anyone of the purely algebraic properties leading absolute-valued real algebras to the finite dimension. Is there a universal constant 0 5 K p < 1 such that every normed real algebra A satisfying P and p ( A ) > K p is finite-dimensional? We were able to answer Question 4.1 for the most relevant choices of Property P. Thus we have the following.
Theorem 4.1. 59 Question 4.1 has an afirmative answer whenever Property P is equal to the existence of a unit, the commutativity, or the algebraicity. Moreover, for such choices of P , the universal constant K p can be (uniquely) chosen in such a way that there exists a normed infinitedimensional real algebra satisfying P and p ( A ) = K p . Now, fundamental Theorems 2.1, 2.3, and 2.11 are “nearly” true when nearly absolute-valued algebras replace absolute-valued algebras. As a consequence, a normed power-commutative real algebra A is finite-dimensional whenever p ( A ) is near one. Many other results of the same flavour can be obtained (see for example the variants of Theorems 2.2, 2.8, and 3.8 proved in Corollaries 3.2 and 3.4, and Theorem 3.3 of 59, respectively). We already know that K p 2 2 - i when P means commutativity. When P means algebraicity or existence of a unit, we do not know whether or not the equality K p = 0 holds. In any case, Nearly absolute-valued complex algebras are isomorphic to C whenever they have a left unit or are algebraic (see Remark 2.8 of 59 and our Subsection 2.8, respectively). If they are commutative, then a result similar to the one given by Theorem 4.1 (with
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P equal to the commutativity) holds. A normed space X is said to be uniformly non-square if there exists 0 < u < 1 such that the inequality min{l)z y J JJ]z , - yII} < 2 0 holds for all x,y in the closed unit ball of X . We note that pre-Hilbert spaces are uniformly non-square, and that the completion of every uniformly nonsquare normed space is a superreflexive Banach space (see Theorem VII.4.4 of 27). We also remark that neither absolute-valued algebras nor nearly absolute-valued finite-dimensional algebras with a unit need be uniformly non-square (by our Subsection 3.1 and Example 2.1 of 59, respectively). These facts give its own interest to the following variant of Theorem 3.5.
+
Theorem 4.2. 59 Let A be a normed real algebra with p ( A ) > 2-a. If there exists a E A such that a A is dense in A , then A is uniformly non-square. 4.2. Other deviations
By a trigonometric algebra we mean a pre-Hilbert real algebra A satisfying llzy1I2 + (4Y)2 = 11~11211Y112(or, equivalently, IlzYIl = ll~llllYllsin% where Q is the angle between z and y ) for all z, y E A \ (0). By cleverly applying Hurwiz’s theorem (see Subsection 2.2), P. A. Terekhin looshows that the dimensions of finite-dimensional trigonometric algebras are precisely 1, 2, 3, 4, 7, and 8. The existence of complete trigonometric algebras of arbitrary infinite Hilbertian dimension is implicitly known in 59. Indeed, if U is an infinite set, and if 6 is an injective mapping from U x U to U , then the real Hilbert algebra obtained from the absolute-valued algebra C2(U,6, IR) (see Subsection 3.1) by replacing its product with the one x @ y := Iy--y2 4 becomes a trigonometric algebra (see Remark 1.6 of 59 for details). Actually, all infinite-dimensional -trigonometric algebras can be constructed in a transparent way from the absolute-valued real algebras with involution considered in Subsection 3.4 (see 9). By a triple system over a field IF we mean a nonzero vector space T over IF endowed with a trilinear mapping (. . .) : T x T x T 4 T . Absolute-valued triple systems over K are defined as those triple systems T over K endowed with a norm )I . 11 satisfying [I(zyz)ll= llzllIlyllllzll for all z, y , z E T . Each absolute-valued triple system gives rise to “many” absolute-valued algebras. Indeed, if T is an absolute-valued triple system, and if u is a norm-one element in T, then the normed space of T becomes an absolute-valued algebra under the product x 0y := ( m y ) . As pointed out in 18, this implies (by Proposition 1.5) that the n o r m of every absolutevalued finite-dimensional real triple system springs f r o m a n inner product.
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It follows that, if T is an absolute-valued finite-dimensional real triple system, then the mapping q : x --f ) ) x ) )is2 a quadratic form on T satisfying q ( ( x y z ) ) = q ( x ) q ( y ) q ( z )for all x,y , z E T . Now we can mimic Albert’s definition of isotopy (see Subsection 1.3), and apply the main result in K. McCrimmon’s paper 67, to get that, u p to a n isotopy, the absolutevalued finite-dimensional real triple systems are R,C (with ( x y z ) = x y z in both cases), W (with (xcyz) equal to either x y z , x z y , or y x z ) , and 0 (with ( x y z ) equal to either ( x y ) z , ( x z ) y , ( y x ) z , ~ ( g z )~, ( z y ) or , y ( x z ) ) . This result is a sample of how the study of absolute-valued triple systems can be promising (see l8 and ‘O). An algebra A is said to be two-graded if it can be written as A = A0 @ A l l where A0 and A1 are nonzero subspaces of A satisfying AiAj Ai+j for all i , j E Zz. Two-graded absolute-valued algebras are defined as those normed two-graded algebras A = A0 @ A1 over K satisfying IlxixjII = Ilxillllxjll for all i , j E Zz and every (xi,xj)E Ai x Aj. The work of A. J. Calder6n and C. Martin l9 deals with these objects, and starts with the observation that, if A is a two-graded absolute-valued algebra, then, in a natural way, A0 i s a n absolute-valued algebra, and A1 i s a n absolute-valued triple system. As a consequence, two-graded absolute-valued finite-dimensional real algebras have dimension 2, 4, 8, or 16.
5. Absolute-valuable Banach spaces In this concluding section we are going to deal with those Banach spaces which underlie complete absolute-valued algebras. Such Banach spaces will be called absolute-valuable. The finite-dimensional side of this topic is definitively solved by Albert’s Proposition 1.5. Indeed, the absolutevaluable finite-dimensional real Banach spaces are precisely the real Hilbert spaces of dimension 1, 2, 4, and 8. On the other hand, it is clear that C is the unique absolute-valuable finite-dimensional complex Banach space. Therefore, the interest of absolute-valuable Banach spaces centers into the infinite-dimensional case.
5.1. The isometric point of view We already know that, for every infinite set U , the classical Banach spaces lp(U,K) (1 5 p < co) and Q(V,JK) are absolute-valuable (see Subsection 3.1). In fact, the rol played there by JK can be also played by any absolute-valued algebra, and hence we have that, given a n infinite set U and a Banach space X , the Banach spaces l p ( U , X ) (1 5 p < m) and
145
co(U,X ) are absolute-valuable whenever so is X . Even, the value p = 00 is allowed above (by Proposition 5.2 below). Other stability properties of the class of absolute-valuable Banach spaces can be also derived from previously reviewed results. For example, it follows from Lemma 2.4 that the normed ultraproduct of every ultrafiltered family of absolute-valuable Banach spaces is a n absolute-valuable Banach space. More examples of absolute-valuable Banach spaces are given in Proposition 5.1 immediately below. As usual, given Banach spaces X and Y over K,we denote by C ( X ,Y ) the Banach space over R of all bounded linear operators from X to Y, and by K ( X , Y) the closed subspace of C ( X lY) consisting of all compact operators from X to Y. Moreover, we write X * , L ( X ) , and K ( X ) instead of L ( X , R ) , C ( X ,X ) , and K ( X , X ) , respectively. Proposition 5.1. Let 1 5 p 5 m, let U1 be a n infinite set, and let X 1 stand f o r lP(U~,IK). Then X,* is absolute-valuable. Moreover, if U2 is another infinite set, and if X2 stands f o r e P ( U 2 , R ) , then C ( X l , X , ) and K ( X 1 ,X 2 ) are absolute-valuable.
As a consequence, infinite-dimensional Hilbert spaces over R are absolute-valuable, and moreover, i f H1 and H2 are infinite-dimensional Hilbert spaces over IK, then C ( H 1 ,H z ) and K ( H 1 , H2) are absolute-valuable. The paper of J. Becerra, A. Moreno, and the author 7 , from which we have taken Proposition 5.1, also contains Theorem 5.1 immediately below. Given a topological space El we denote by dens(E) the density character of E (see Subsection 3.2). Theorem 5.1. Every Banach space X over R is linearly isometric to a subspace of a Banach space Y over R with dens(Y) = dens(X) and such that Y, Y*, C(Y), and K(Y) are absolute-valuable. The ideas in the proof of Theorem 5.1 are not far from those in Proposition 5.2 immediately below. I n what follows, R will denote a compact Hausdorff topological space.
Proposition 5.2. Assume that there exists a continuous surjection from R to R x R, and let X be an absolute-valuable Banach space. Then C(R, X ) is absolute-valuable. Proof. Let us choose a product (x,y) 4 zy on X converting X into an absolute-valued algebra, let q5 : R + R x R stand for the continuous surjection whose existence is assumed, and, for i = 1,2, let 7ri : R x R 4 R
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denote the i-th coordinate projection. Then the product o on C(R,X) defined by (f o g ) ( w ) := f(7rI(q5(w)))g(7rz($(w)))(for every w E R and all f,g E C(R, X ) ) converts C(R, X)into an absolute-valued algebra. We note that the choice R = [0,1] is allowed in Proposition 5.2 (see and references therein). According to that proposition, the existence of a continuous surjection from R to R x R is a suficient condition for C(R, K) to be absolute-valuable. In 69, a partial converse is shown. Indeed, we have the following Theorem 5.2. If C(fl,K) is absolute-valuable, then there exists a closed subset F of R, and a continuous surjection from F to R x 52.
As a consequence, C(R,K) is not absolute-valuable whenever R is the one-point compactification of a discrete infinite space (a fact first proved in 7). In particular, the classical space c of all real or complex convergent sequences is not absolute-valuable. Theorem 5.2 is also applied in 69 to prove that, in the case that R is metrizable, C(R,K) is absolute-valuable i f and only if R is uncountable. The arguments for Theorem 5.2 mimic those in the proof of a theorem of W. Holsztynski on nonsurjective isometries between C(R)-spaces (see Section 22 of 96). Given a Banach space X , we denote by Q the group of all surjective linear isometries on X . We recall that a Banach space X is said to be transitive (respectively, almost transitive) if, for every (equivalently, some) norm-one element u in X , B(u) is equal to (respectively, dense in) the unit sphere of X . The reader is referred to the book of S. Rolewicz 92 and the survey papers of F. Cabello l5 and Becerra-Rodriguez * for a comprehensive view of known results and fundamental questions in relation to the notions just introduced. Hilbert spaces become the natural motivating examples of transitive Banach spaces, but there are also examples of non-Hilbert almost transitive separable Banach spaces, as well as of non-Hilbert transitive nonseparable Banach spaces. However, the Banach-Mazur rotation problem, if every transitive separable Banach space is a Hilbert space, remains unsolved to date. Since almost transitive finite-dimensional Banach spaces are indeed Hilbert spaces, the rotation problem is actually interesting only in the infinite-dimensional setting. Then, since infinite-dimensional Hilbert spaces are absolute-valuable, we feel authorized to raise the following strong form of the Banach-Mazur rotation problem. Problem 5.1. Let X be an absolute-valuable transitive separable Banach space. Is X a Hilbert space?
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We hope Problem 5.1 to have an affirmative answer in the next future. In the meantime, we must limit ourselves to review the following.
Proposition 5.3. There exists a non-Hilbert absolute-valuable almost transitive separable Banach space X such that C(X, Y) and K(X,Y) are absolute-valuable for every absolute-valuable Banach space Y . Actually, the space X in Proposition 5.3 can be taken equal to L1([0,1])). Proposition 5.3 implies (applying Lemma 2.4 among other tools) that there exists a non-Hilbert absolute-valuable transitive non-separable Banach space. One of the tools in the proof of Proposition 5.3 is the following.
Lemma 5.1. Let X and Y be Banach spaces over K. Assume that the complete projective tensor product XGTX is linearly isometric to a quoY) tient of X, and that Y is absolute-valuable. Then C(X, Y) and F(X, are absolute-valuable. Here 3 ( X ,Y) stands f o r the space of all finite-rank operators f r o m X to Y. We conclude the present subsection by applying Lemma 5.1 to prove the following
Theorem 5.3. Every Banach space X over JK is linearly isometric to a quotient of an absolute-valuable Banach space Y over JK satisfying dens(Y) = dens(X), and such that L(Y,Z), and Ic(Y,Z) are absolutevaluable for every absolute-valuable Banach space Z over K. Proof, Let U be a set whose cardinal number equals dens(X), and let Y stand for the absolute-valuable Banach space l l ( U ,K). Clearly, we have dens(Y) = dens(X). On the other hand, it is well-known that X is linearly isometric to a quotient of Y. (In fact, noticing that X becomes a complete normed algebra under the zero product, we had to show a little more when we proved Corollary 3.2.) Finally, noticing that YG,,Y = lI(U,JK)&ll(U,JK) = l l ( U x U,K) = l,(U,K) = Y (see Ex 3.27 of ") and that Y* has the approximation property (see 5.2 of 28, the proof is concluded by applying Lemma 5.1 (see 5.3 of 28). 0 5 . 2 . The isomorphic point of view
Most isomorphic properties on Banach spaces considered in the literature are inherited by quotients and/or subspaces. Therefore, by Theorem 5.1
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and Corollary 5.3, none of such properties can be implied by the absolute valuableness. Now, recall that a Banach space X is called weakly E~ countably determined if there exists a countable collection { K n } nof w*-compact subsets of X** in such a way that, for every x in X and every u in X**\ X, there exists no such that x E K,, and u @ Kn0. If X is either reflexive, separable, or of the form c o ( r ) for any set r, then X is weakly countably determined. In fact, the class of weakly countably determined Banach spaces is hereditary, and contains the non hereditary class of weakly compactly generated Banach spaces (see Example VI.2.2 of 29 for details). Among the results proved in concerning the isomorphic aspects of absolute-valuable Banach spaces, the main one is the following.
Theorem 5.4. Every weakly countably determined real Banach space, different from R, is isomorphic to a real Banach space X such that both X and X* are not absolute-valuable. We do not know if the requirement of countable determination can be removed in Theorem 5.4. A Banach space X is said to be hereditarily indecomposable if, for every closed subspace Y of X , the unique complemented subspaces of Y are the finite-dimensional ones and the closed finite-codimensional ones. According to the paper of W. T. Gowers and B. Maurey 51, the existence of infinite-dimensional hereditarily indecomposable separable reflexive Banach spaces over K as not in doubt. On the other hand, we have proved in that infinite-dimensional hereditarily indecomposable Banach spaces over R fail to be absolute-valuable. Thus, since the hereditary indecomposability is preserved under isomorphisms, we are provided with an infinite-dimensional Banach space over R which is not isomorphic to any absolute-valuable Banach space. In other words, the property of absolute valuableness is not isomorphically innocuous. In the case K = C, more can be said. Indeed, we have the following.
Proposition 5.4. Let X be an infinite-dimensional hereditarily indecomposable complex Banach space. Then X cannot underlie any complete normed algebra without nonzero two-sided topological divisors of zero. Proof. By Corollary 19 of 51, X is not isomorphic to any of its proper subspaces. Assume that, for some product, X is a complete normed algebra without nonzero two-sided topological divisors of zero. Then, for every x E X \ {0}, L, or R, is an isomorphism onto its range, and hence it is
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bijective. Now, X is a quasi-division algebra, and hence finite-dimensional (by Proposition 3.6). Let us say that a Banach space X over K satisfies the Shelah-Steprans property whenever X is not separable and, for every F E L ( X ) ,there exist X = X(F) E IK and S = S ( F ) E L ( X ) such that S has separable range and the equality F = S Xlx holds. In our present discussion, Banach spaces enjoying the Shelah-Steprans property play a rol similar to that of infinitedimensional hereditarily indecomposable Banach spaces. Indeed, reflexive Banach spaces satisfying the Shelah-Steprans property do exist (see 97 and lo8),the Shelah-Steprans property is preserved under isomorphisms, and Banach spaces over W fulfilling the Shelah-Stepmns property fail to be absolute-valuable '. By the way, the proof of the result of just reviewed can be slightly refined to get the following.
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Proposition 5.5. Let X be a Banach space over K satisfying the ShelahSteprans property. Then X cannot underlie any complete normed algebra without nonzero two-sided topological divisors of zero. Proof. First, we note that, for F in L ( X ) , the couple ( X ( F ) , S ( F )given ) by the Shelah-Steprans property is uniquely determined, and that the mappings X : F -+ X ( F ) and S : F -+ S ( F ) from L ( X ) to W and L ( X ) , respectively, are linear. Now, since X is not separable, and ker(X) consists of those elements of L ( X ) which have separable range, we have X ( F ) # 0 whenever the operator F on X is an isomorphism onto its range. Assume that, for some product, X is a complete normed algebra without nonzero two-sided topological divisors of zero. Then, since L, or R, is an isomorphism onto its range whenever 5 is in X \ {0}, it follows that the linear mapping x -+ (X(L,),X(R,)) from X to K2 is injective. Therefore X is finite0 dimensional, a contradiction. Concerning the topic of the present section, let us say that the study of absolute-valuable Banach spaces is just started, so that there are more problems than results on the field. Since we have mainly emphasized the results, let us conclude the paper with one of the problems non previously collected. Let us say that a Banach space is nearly absolute-valuable if it underlies some complete nearly absolute-valued algebra (see Subsection 4.1). It is easy to see that the near absolute valuableness is preserved under isomorphisms. Consequently, isomorphic copies of absolute-valuable Banach spaces are nearly absolute-valuable. However, we do not know whether
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or not every nearly absolute-valuable Banach space is isomorphic t o an absolute-valuable Banach space.
Acknowledgements The author thanks the organizers of the ‘$First International Course on Mathematical Analysis in Andalucia”, Professors A. Aizpuru and F. Leon, for inviting him t o deliver lectures in that Course and write the paper which concludes now. The discussions began during the Course with Professor J. Benyamini enriched very much the papers and 69. Consequently, since the review of such papers done in Section 5 above has benefited from such an enrichment, special thanks are due to him. The author is also grateful t o J. Becerra, M. Cabrera, J. A. Cuenca, A. Fernfindez, A. Kaidi, A. Kaplan, J. Martinez, C. Martin, M. Martin, J. F. Mena, A. Moreno, R. Payfi, A. Peralta, M. I. Ramirez, A. Rochdi, A. Slin’ko, M. V. Velasco, and the anonymous referee. In one or other way, all they have helped him very much in the writing of this paper.
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