Amenability (Mathematical Surveys and Monographs)

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Amenability Alan L. T. Paterson

American Mathematical Society

MATHEMATICAL SURVEYS AND MONOGRAPHS SERIES LIST Volume

I The problem of moments, J. A. Shohat and J. D. Tamarkin

16 Symplectic groups, 0. Timothy O'Meara

2 The theory of rings, N. Jacobson 3 Geometry of polynomials, M. Marden 4 The theory of valuations, O. F. G. Schilling 5 The kernel function and

17 Approximation by polynomials with integral coefficients, Le Baron O. Ferguson

conformal mapping, S. Bergman

6 Introduction to the theory of algebraic functions of one variable, C. C. Chevalley

7.1 The algebraic theory of semigroups, Volume 1, A. H.

Clifford and G. B. Preston 7.2 The algebraic theory of semigroups, Volume II, A. H. Clifford and G. B. Preston

8 Discontinuous groups and antomorphic functions, J. Lehner

9 Linear approximation, Arthur Sard 10 An introduction to the analytic theory of numbers, R. Ayoub 1i Fixed points and topological degree in nonlinear analysis, J. Cronin 12 Uniform spaces, J. R. Isbell 13 Topics in operator theory, A. Brown, R. G. Douglas, C. Pearcy, D. Samson, A. L. Shields; C. Pearcy, Editor 14 Geometric asymptotics, V. Guillemin and S. Sternberg

15 Vector measures, J. Diestel and J. J. Uhl, Jr.

18 Essentials of Brownian motion and diffusion, Frank B. Knight

19 Contributions to the theory of transcendental numbers, Gregory V. Chudnovsky 20 Partially ordered abelian groups with interpolation, Kenneth R. Goodearl 21 The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors 22 Noncommutative harmonic analysis, Michael E. Taylor

23 Introduction to various aspects of degree theory in Banach spaces, E. H. Rothe

24 Noetherian rings and their applications, Lance W. Small, Editor 25 Asymptotic behavior of dissipative systems, Jack K. Hale

26 Operator theory and arithmetic in H°O, Hari Bercovici 27 Basic hypergeometric series and applications, Nathan J. Fine 28 Direct and inverse scattering on the lines, Richard Beals, Percy Deift, and Carlos Tomei

Amenability

Mathematical Surveys and Monographs

Volume 29

Amenability

Alan L.T. Paterson

American Mathematical Society Providence, Rhode island

2000 Mathematics Subject Classification. Primary 43-02, 43A07, 47H10, 22D25, 22D40, 22E15, 22E25, 03E10; Secondary 03E25, 20F05, 20F16, 20F18, 20F19, 20F24, 22D10, 22D15, 22E27, 28C15, 28D05, 43A60, 43A65, 43A80, 46H05, 46H25, 46J10, 46L05, 46L10, 47A35, 53C42, 54D35, 60B05, 60G50, 62F03.

Library of Congress Cataloging-In-Publication Data Paterson, Alan L. T., 1944Amenability.

(Mathematical surveys and monographs; no. 29) Includes bibliographies and Index. 1. Harmonic analysis. 2. Locally compact groups. I. Title. It. Series. 1988 QA403.P37 515'.2433

88-14485

ISBN 0-8218-1629-6 (als. paper)

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society.

Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to repzint-permissioneams. org.

Copyright © 1988 by the American Mathematical Society Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. ISO The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at URL: http://vw.ams.org/

1098765432

0403020100

Dedicated to

MARLON M. DAY

Nicht die Neugierde, nicht die Eitelkeit, nicht die Betrachtung der Nutzlichkeit, nicht die Pflieht and Gewiesenhaftigkeit, sondern ein unaueloschlicher, ungliicklicher Durst, der sick auf keinen Vergleich enldsst, fuhrt uns zur Wahrheit. G. W. F. Hegel

Contents Preface

xi

List of Further Results

xv

Plan

xvii

CHAPTER 0. Introduction: Basic Concepts and Problems of Amenability Theory

1

CHAPTER 1. Amenable Locally Compact Groups and Amenable Semigroups

25

CHAPTER 2. The Algebra of Invariant Means

51

CHAPTER 3. Free Groups and the Amenability of Lie Groups

95

CHAPTER 4. Folner Conditions

125

CHAPTER 5. Ergodic Theorems for Amenable Locally Compact Groups

195

CHAPTER 6. Locally Compact Groups of Polynomial Growth

217

CHAPTER 7. Sizes of Sets of Invariant Means

269

APPENDICES:

A. Nilpotent, Solvable and Semidirect Product Groups B. Lie Groups

301

C. Existence of Borel Cross-Sections

329

D. Mycielski's Theorem E. On the Density of the Exponential Map

331

305

333

Some Abbreviations

337

Bibliography

339

Supplementary Bibliography ((S])

397

Sketched Solutions to Problems

413

Index of Terms

443

Index of Symbols

451

ix

Preface The subject of amenability essentially begins with Lebesgue (1904). One of the properties of his integral is a version of the Monotone Convergence Theorem, and Lebesgue asked if this property was really fundamental, that is, if the property follows from the more familiar integral axioms. Now the Monotone Convergence Theorem is equivalent to countable additivity, and so the question is concerned with the existence of a positive, finitely (but not countably) additive, translation invariant measure p on R with p((0,11) = 1.

The classical period (1904-1938) is therefore concerned with the study of finitely additive, invariant measure theory. The study of isometry-invariant measures led to the Banach-Tarski Theorem (1924) and the theory of paradoxical decompositions. The class of amenable groups was introduced and studied by von Neumann (1929) and used to explain why the Banach-Tarski Paradox occurs only for dimension greater than or equal to three.

The modern period begins in the 1940s and continues with increasing energy

to the present. The main shift is from finitely additive measures to means: integrating a positive, finitely additive measure p on a set X with p(X) = 1 gives rise to mean m on X, that is, a continuous linear functional on lo, (X) such that m(1) = 1 = jlmfl. The connection between p and m is given by p(E) = m(XE), and the correspondence p --* m is bijective. This shift is of fundamental importance, for it makes available the substantial resources of functional analysis (and eventually of abstract harmonic analysis) to the study of amenability. Von Neumann's definition translates into the language of means and gives the familiar definition of an amenable group: a group G is amenable if and only if there exists a left invariant mean m on G. Here, a mean m is left invariant if m(ox) = m(4,), where 4,x(y) = 4,(xy) (y E G) for 4' E l,,(G), x E G. The above definition also applies to semigroups, a semigroup admitting a left invariant mean being called left amenable. In the 1940s and 1950s, the subject of amenable groups and amenable semigroups was studied by M. M. Day, and his 1957 paper on amenable semigroups is a major landmark.

Since that time the subject has developed at a fast pace, as the bibliography will testify. The plan at the beginning of this book will give some idea of the range of mathematics in which the amenability phenomenon has been observed xi

xii

PREFACE

and proved relevant. The amenability phenomenon is protean and is not readily "pigeon-holed". However, the roots of the subject lie in functional analysis and abstract harmonic analysis, and it draws its coherence from these.

The ubiquity of amenability ideas and the depth of the mathematics with which the subject is involved seems evidence to the author that here we have a topic of fundamental importance in modern mathematics, one that deserves to be more widely known than it is at present. Until 1984, the only book available on the subject was the influential short account by F. P. Greenleaf 12] (1969). We should also mention Chapter 8 of Reiter's book (R] (1968) and Day's indispensable survey papers 14],19] of 1957 and 1968 respectively. In addition, §17 of the work on abstract harmonic analysis IHR.1] by Hewitt and Ross discusses amenable semigroups and groups. More recently (1984), a book by Sean-Paul Pier on amenable locally compact groups has appeared. As one might expect, there is overlap between Professor Pier's book and the present one. However, the points of view adopted in the two books are very different, as also are much of the contents. Another work that has recently appeared (1985), relevant to amenability, is the elegant book by Stan Wagon dealing with the Banach-Tarski Theorem. The main objectives of the present work are to provide an introduction to the subject as a whole and to go into many of its topics in some depth. While the main area of amenability lies in analysis on locally compact groups, the

subject applies to a much wider range of mathematics than that. We have tried to bear this in mind. In particular, applications in the areas of statistics, differential geometry, and operator algebras will be found in Chapters 4, 7, and 1, 2 respectively. Chapter 4 also contains discussions of the two most outstanding

theorems in amenability established since 1980-these are in the areas of von Neumann's Conjecture and the Banach-Ruziewicz Problem. We have attempted to describe the main lines of development of the subject, showing what progress or lack of progress has been made in solving the main problems, and raising a number of open problems. It is hoped that the present work will stimulate further research in the subject. The introduction gives an informal, nontechnical discussion of amenability as

a whole. It is recommended that the reader read this chapter carefully before going on with the rest of the book. Chapters 1 and 2 establish the basic theory of amenability. Chapter 3 investigates amenability for Lie groups. (Lie theory is very important in Chapters 3 and 6. The Lie theory we need is briefly sketched in the Appendix B.) The last three chapters deal with amenability and ergodic theorems, polynomial growth, and invariant mean cardinalities respectively.

The main prerequisites for reading the present work are a knowledge of abstract harmonic analysis (such as is contained in [HR.1)) and of functional analysis. Of course, we also presuppose a sound understanding of undergraduate mathematics. At the same time, we hope that the book is accessible to workers in other areas of mathematics.

PREFACE

xiii

Chapters 1, 2 and 4 could be used for a graduate course in amenability, assuming a background in basic functional and harmonic analysis, but not in Lie theory. Some of the results in Chapters 5 and 7 could also be included in such a course. A course on groups of polynomial growth could be given based on Chapters 6 and 3, and presupposing acquaintance with the Lie theory sketched in Appendix B. Throughout the book we use results from certain standard textbooks. These

textbooks are listed, together with their abbreviated forms of reference, at the beginning of the Bibliography. Each chapter is divided into three parts. The first and main part is intended to give a coherent account of the area dealt with by the chapter. The second part ("Further Results") deals with important developments of the ideas in the first part. The third part is a list of problems; these vary from the routine to the quite difficult, and substantially increase the scope of the book. The author hopes that readers will try to solve some of the problems. References and sketched solutions (where appropriate) appear in the "Sketched Solutions to Problems" at the end

of the book. Problem b of Chapter a is referred to in the text as: Problem a-b. In addition to the main bibliography, there is a supplementary bibliography [S] which contains many of the most recent papers in the subject. In view of the tangled and wide-ranging nature of the subject, it seems likely that inaccuracies will have occurred. Of course, absence of a specific reference for a result does not imply any claim to originality on the part of the author. The author will be grateful to readers who take the trouble to point out to him errors in the text. I am very grateful to many mathematicians for mathematical discussions and advice. These include Rob Archbold, James Bondar, Ching Chou, Mahlon Day, G. Dales, John Duncan, David Edwards, William Emerson, Edmond Granirer, John Hubbuck, Barry Johnson, Anthony Lau, Michael Leinert, Mick MacCrudden, Paul Milnes, Isaac Namioka, A. Yu. Ol'shanskii, John Pym, Michael Rains, Guyan Robertson, A. van Rooij, Joseph Rosenblatt, Stan Wagon, David Wallace, Alan White, John Williamson, and James Wong. Thanks are also due to Louise Thompson, who coped heroically with masses of handwritten pages and typed most of the book. I am especially indebted to my friends John Duncan, Ed Granirer, Anthony Lau, Paul Milnes, and John Pym, whose kindness and interest in this work encouraged me to keep writing when the going was difficult. My thanks are also due to the editorial staff of the American Mathematical Society for their helpfulness and skill. Finally, I am grateful to my wife Christine, whose encouragement and support have meant more than I can say. University of Mississippi August 1988

Alan L. T. Paterson

List of Further Results (1.29)

The Sorenson Conjecture

(1.30)

Amenable Banach algebras

(1.31)

Amenable C'-algebras

(2.31)

On the L-space structure of 31(E)

(2.32)

Invariant measures and the Translate Property

(2.33)

The nonintroversion of C(G)

(2.34)

Left invariant means and non-Archimedean analysis

(2.35)

Amenable von Nenrnann algebras

(2.36)

Invariant means and almost periodicity

(2.37)

On the range of a left invariant mean

(3.11) A group in NF but not in EG (3.12)

Three problems considered by Banach

(3.13)

Two results on equidecomposability

(3.14)

Konig's Theorem and division

(3.15)

Tarski's Theorem on amenability and paradoxical decompositions

(3.16)

The Banach-Tarski Theorem

(4.22)

Semigroups and Folner conditions

(4.23)

The groups G for which . (G) = Z(G)

(4.24)

Amenability and statistics

(4.25)

The Hunt-Stein Theorem

(4.26)

Amenability and translation experiments

(4.27)

Uniqueness of invariant means

xv

xvi

LIST OF FURTHER RESULTS

(4.28)

The Banach-Ruziewicz Problem: the S"-version

(4.29)

The Banach-Ruziewicz Problem and the Kazhdan Property (T)

(4.30)

Von Neumann's Conjecture

(4.31)

On the norms of operators in Ct (G)

(4.32)

An amenability criterion for finitely generated groups

(4.33)

Ol`shanskii's counterexample to von Neumann's Conjecture

(4.34)

Amenability and the Fourier algebra of a locally compact group

(5.21)

Other ergodic theorems

(6.40)

Discrete groups and polynomial growth

(6.41)

Polynomial growth and fundamental groups

(6.42)

Exponentially bounded groups and the Translate Property

(6.43) A Folner condition for exponentially bounded groups (6.44)

Polynomial growth and the nonzero fixed-point property

(6.45)

The representation theory of solvable Lie groups and the Type R condition

(6.46)

Poisson spaces, amenability, and the Type R condition

(6.47)

Random walks and amenability

(6.48)

Symmetry, the Wiener Property, and polynomial growth

(6.49) A symmetric algebra associated with an exponentially bounded group (6.50)

Polynomial growth and the symmetry of LI (G)

(6.51) A discrete, nonsymmetric group of polynomial growth (7.28)

Exposed points and left invariant means

(7.29)

Nonmeasurability of some L, (G)-convolutions

(7.30)

Locally compact groups that are amenable as discrete

(7.31)

Amenability and the radicals of certain second dual Banach algebras

Invariant means realised as

Amenability and the fundamental group of a Riemannian manifold

statistics

Applications of amenability to probability and

Cardinalitiea of ads of invariant means

Poisson spaces

Groups of polynomial growth

Almost pmiodkity

Symmetry of LI (0)

Ergodic theory for locally compact amenable groups

Amenable Bausch algebras

The weak containment property

Almost convergence

Growth and Feiner conditions

Amenable Lie groups

compact groups

Amenable local ly

11

t

Fixed-point theory

PLAN

Amenability and operator aigebrna

I

Amenability and noncommntative harmonic analysis

Amenability and subgroups of locally compact groups

This plan is an attempt to give a reasonable indication of the range of contexts to which amenability applies. It is not comprehensive. A strongrelationship between two topics is indicated by a line joining them. The topics vary greatly in scope.

locally compact semiaronas

Amiable

on /3S and VG)

semigroups - probability measures

Amenable

/

Functional analytic context for amenability: invariant means

I

Classical amenability: finitely additive invariant measures, Banach's three problems, Banach-Tarski Theorem, amenable groups and von Neumann's Conjecture

AMENABILITY

CHAPTER 0

Introduction: Basic Concepts and Problems of Amenability Theory Amenability essentially began in 1904 when Lebesgue gave a list of properties that uniquely specified his integral on R. All of these, except one, are elementary

properties of the R.iemann integral. The exceptional property is a version of the Monotone Convergence Theorem (MGT). Lebesgue naturally asked if the integral was still uniquely specified if MCT is dropped. A moment's thought will convince the reader that MCT is actually equivalent to countable additivity, and Lebesgue's question can be rephrased: Is the Lebesgue integral still unique if the MCT condition is replaced by mere finite additivity?

Banach later answered this question negatively by producing a finitely additive, invariant integral on R different from the Lebesgue integral. (As we shall see in (4.29), for certain compact groups the answer to the corresponding question is positive.) In his proof, Banach used the existence of a finitely additive, invariant, positive integral of total mass one on the circle group T.

It turned out that there exists such an invariant measure p on the real line R as well. Two remarkable facts about µ are, first, that it is defined on all subsets of R (not just on the Lebesgue measurable ones) and, second, that R has finite p-measure (µ(R) = 1), contrasting with the fact that the Lebesgue measure of R is oo! In modern language, R is "amenable as a discrete group," and the measure ,u is an invariant mean. In the 1920s and 1930s, the question of the existence of an invariant mean for a group G acting on a set X was investigated by Banach and Tarski. Tarski (1938) showed that such a mean exists if and only if X does not admit a "G-paradoxical decomposition." This deep theorem will be discussed in (3.15).

In his study of the Banach-Tarski Theorem, von Neumann introduced and studied the class of amenable groups (amenable = mittelbar (German) = moyennable (French) = aMeHa6enbaaa (Russian)). The term amenable was introduced by M. M. Day (as a pun) in 1950. In more recent times, invariant means have proved relevant for answering an amazing variety of questions (especially in analysis on locally compact groups).

We therefore start by investigating some of the more basic properties of invariant means. These means relate to most of the results discussed in the rest 1

INTRODUCTION

2

of this work, and this enables the present chapter to serve as an introduction to the book as a whole. Thus we will take the opportunity to introduce at a basic level ideas and results which will later be developed in greater depth. It is hoped that the present chapter will also serve as a (somewhat informal) motivation for the rest of the work. The classical period of amenability (1904-1939) defined (left) invariant means as we have done above, that is, as finitely additive measures on a group G. This is still useful, sometimes, in the modern theory, but in general, suffers from serious drawbacks. For a start, the powerful theorems of countably additive measure theory are no longer available, and the finitely additive versions of these theorems are pale reflections of the originals. (We note (Problem 1-10) that there are no countably additive invariant means on noncompact groups.) M. M. Day revolutionised the subject by showing that an invariant mean can be regarded as a continuous linear functional on L0,(G). Before discussing how this is done, let us recall some basic facts from abstract harmonic analysis. (Our standard reference for this area of mathematics is IHR1).) Let G be a locally compact group. There exists a left Haar measure A on G: so A is a nonzero, positive, regular Borel measure on G, and A(xE) = A(E) for all x e G and all E E R (G), the o-algebra of Borel subsets of G. The space LI (G) = LI (G, A) of complex-valued, integrable functions on G is a Banach space with norm given by IJfIII =1G IfIdA. A very important fact for our purposes is that the dual of L, (G) can be identified with the space of bounded, A-measurable, complex-valued functions l on G with the "ess sup"-norm: 11011. =

xEG

(Here, if 0: G --+ R is bounded, rthen l

e x4EG ss

pO(x)

mf {

up

N is locally null } ,

a subset N of G being locally null if it is A-measurable and A(N ft C) = 0 for all C E W(G), the family of compact subsets of G. Ess info (x) is defined similarly.) mc-G and the dual L1(G)' of L1(G) is given The above correspondence between

by associating 0 E L (G) with the functional f f Of dA (f E L, (G)). The family of A-measurable subsets of G is denoted by £(G). Let p be a positive, finitely additive measure on £(G) vanishing on locally null sets with A(G) = 1. We can regard p as a member m of La,(G)' as follows. Obviously we must define m(XE) = µ(E) for E E 4'(G). More generally (Problem 0-2) we can define m on the span A of {XE : E E 4'(G)} by defining n

m

(iX)E, ==7

n

_ E aiµ(Ei)

(ai E C, E; E O (G)).

i=1

ITwo functions in L. (G) are identified if they differ only on a locally null set.

INTRODUCTION

3

Now m is continuous on A and since A is norm dense in we can extend m to all of L. (G). The functional m is a mean (or state) on L,,. (G), that is, an such that p(l) = I = 11pfl. Obviously m is determined by element p E µ, and, conversely, 11 is determined by m: µ(E) = m(XE). The advantages of looking at means from the point of view of m rather than ,u are obvious. The machinery of dual Banach spaces (e.g., weak"-compactness) becomes available, and means fit profitably into functional and harmonic analysis. Like most fundamental ideas, the change in perspective from measures to

functionals, due to Day, is natural in retrospect, but without it, amenability could not have developed as it has done.

When we do wish to regard m from the measure point of view-this will normally be when G is a discrete group (or semigroup)-we write m(E) in place of m(XE) (E E.O(G)). will be denoted by 911(G). A serious problem The set of means on is how can we get hold of elements of 91(G)? The reason why this problem is serious is because means are elements of Lo, (G)', and the space L (G)' is a "very large" space that is badly understood. To get a grip on 91(G) we need to pick out a subset that is accessible and yet, in a suitable sense, is big enough for us to go from it to any mean. This subset is P(G)A, which we now introduce. Recall that L. (G) = LI (G)'. Of course, LI (G) is isometrically embedded

in its second dual LI (G)" = L (G)' by the map f -+ j, where j(46) = 0(f) (0 E L,,. (G)). We are therefore led to ask which functions f E LI(G) are such that f E 9J1(G)? The set of such functions is denoted by P(G). Here are some fundamental facts about 91(G) and P(G).

is a mean if and only if further, if n E 911(G) and

(0.1) PROPOSITION. (i) An element n E n(1) = 1 and n(¢) > 0 whenever 46 > 0 in

E L,o(G) is real-valued, then ess sup ¢(x) < n(O) < ess sup ¢(x) wEG

zEG

(ii) 9R(G) is a weak* compact, convex spanning subset of

(iii) P(G) = (f E Li (G) : f >- 0, f f d)t = 1) and P(G)^ is weak* dense in 91(G) .

PROOF. The space L (G) is a commutative C"-algebra under the pointwise product, and in C'-terminology, mean = state. The results (ii) and the first part of (i) are well known to be true for states on any (unital) C'-algebra [D2, 2.1.9, 2.6.4). The last assertion of (i) follows from the first since both E0-

(esssuP(x))1) , ((esssuP(x)) 1 -

pr\\\ove

We now

vEG

(iiiii). Using (1),

P(G)={fELI(G): f fdA=land

ffd)0if0inLoo(G)}.

INTRODUCTION

4

Now if f f o d x > 0 for all 0 >_ 0 in L (G), it follows that f > 0 a.e., that is, that f >- 0 in LI (G). The converse is obviously true, and the first assertion of (iii) follows.

For the second assertion, suppose that TI(G) does not equal the weak* closure K of P(G)^. Let mo E 0R(G) - K. Separating mo from K [DS, V.2.10, V.3.9], we can find Qio E L (G) such that Ref(¢o) 0 for all r and E00 a, = 1. What we want to do is to construct a sequence { fn} in P(G) with at least one of its weak* cluster points an invariant

mean. Clearly, we need the fn's to become "more and more" invariant under Z-translates as n gets large. A reasonable choice for fn would be the "Cesbro sum" :

fn=2n+1

r=-n

br

Then if ¢ E 1 (Z) and s > 0 in Z, we have

(

n

E

s) - m(r)))

i!n(4,s) - Awl = 2n 1+ 1 r

n++aa-1

1

2n + 1

< 2sMl 2n + 1

1 (-n

-a 0

n+a

F,

n+I

as n - oo.

A similar result holds if a < 0, and we see that every weak* cluster point of {in) in WI(Z) is a left invariant mean! In particular, Z is amenable.

INTRODUCTION

6

(0.4) EXAMPLE: G = R. Modifying (0.3), we take fn = X(_n,nj/2n. Then

forx>_0and 5EL,,(R),wehave, asinf(0.3), V-(Ox) -! (-0)I = Zn (2)

1

(4(x+t) - 0(t)) dtj

1

1. We will see ((6.17)) that every nilpotent G has polynomial growth. The theory of groups of polynomial growth is impressive, and Chapter 6 is devoted to this topic. These groups can be regarded as "super-amenable" and have rather special amenability properties. We will be content, for the present, to prove that polynomial growth implies amenability.

(0.13) PROPOSITION. Every locally compact group of polynomial growth is amenable.

PROOF. Let G have polynomial growth. Let xi,... , x,. E G, and let C E (G) be such that each xi, xj-i E C. Let an = \(C"). There exists a polynomial

p such that 0 < an < p(n). We can suppose that p(n) < Kn' for some K > 0 and some fixed r E P, the set of positive integers. Note that an < an+i since C1+1 contains a left translate of C" and A is left invariant. Now 0 < h(C)1/"
0. [This and its converse is proved in (2.32).] 6. Let be a sequence of nonnull measurable sets in a locally compact group G. Prove that for x E G, ) (xA 0 if and only if A(zA fl 1.

7. Let G be a locally compact group. A mean m on G is called symmetrically

invariant if 1(xm + x-lm) = m for all z E G. Prove that every symmetrically invariant mean on G is actually a left invariant mean.

INTRODUCTION

8. Let A,, be as in (0.5). Show that A(xA,.

21

0 for all x E S2.

9. Let G be an amenable locally compact group. Prove that there exists a net {g6} in P(G) such that {g6 : ¢} converges pointwise to a constant function on G for each 0 E L. (G).

10. Prove that if G is noncompact and m E £(G), then m(Co(G)) = {0}, where Co(G) is the space of continuous complex-valued functions on G vanishing

at infinity. Deduce that m(C) = 0 for all C E i'(G). Prove that £(G)f1P(G)^ 0 if and only if G is compact (c.f. (1.20)).

11. Let G be abelian and {f6} satisfy the condition of (0.8). Let r be the dual group of G. Prove that {f6 } converges pointwise on r to the function that is 1 at the identity and 0 everywhere else.

12. Prove that if G is compact then £(C(G)) is the singleton set {Al.

13. A sequence ¢: P - C is called almost convergent (with limit 1) if The set of such sequences is 1 cp(r) = t uniformly in n.

limp...,o p 1 Er

denoted by AC. Show that AC is a closed subspace of l,,,, containing 1, and that if ci E AC has limit 1, then {l} = {m(ci): m E £(P)}. (See Problem 4-15 for the converse.) Give examples of sequences in AC, and show that AC is not norm separable.

14. Prove that .C (R), £(P), and £ (S2) are infinite.

15. Let G be an infinite discrete group, m E £(G), and E C G be such that JET < JGj. Prove that m(E) = 0. 16. Let G be a connected locally compact group and C E W'(G). Prove that G has polynomial growth if and only if there exists a polynomial p such that A(CI) < p(n) for all n > 1.

17. Let G be a (discrete) finitely generated group. Let F be a finite set of generators for G. Prove that G has polynomial growth if and only if there exists a polynomial p such that jF^l < p(n) for all n > 1.

18. Prove that every (discrete) abelian group has polynomial growth. Let G be the free abelian group on two generators x, y and E = {x, y, x-1, V-1). Calculate 1E81.

19. Prove that S2 does not have polynomial growth. 20. A locally compact group G is said to be exponentially bounded if .\(Cn)1js

- 1 for all C E W, (G). Prove that if C has polynomial growth, then G is exponentially bounded. Show that exponential boundedness implies amenability.

21. A discrete group G is said to be locally finite if every finite subset of G generates a finite subgroup. Prove that every locally finite group is amenable.

22

INTRODUCTION

22. Prove that the group of those permutations of a set which leave all but a finite number of elements fixed is amenable. 23. Let G be a discrete group for which there is a normal series

G"={e} such that G;/G%+I is amenable for 1 < i < n - 1. Show that G is amenable. Deduce that a semidirect product of amenable groups is amenable and that every solvable group is amenable.

24. Let {G;: i E I} be a family of finite groups such that supiE1 JG,J = K < oo. Prove that the Cartesian product jj;EX Gi is amenable. 25. Prove that the isometry group G,, of R", regarded as discrete, is amenable

if n = 1, 2. (G" is not amenable as discrete if n > 2-cf. Problem 3-5.) 26.

Show that every amenable subgroup of a group G is contained in a

maximal amenable subgroup of G and that there is a normal amenable subgroup N of G which contains every normal amenable subgroup of G and is contained in every maximal amenable subgroup of G. 27. Prove that if every nontrivial, homomorphic image of a group G contains a nontrivial, amenable, normal subgroup, then G is amenable. (Supersolvable groups (Baer [1), M. Hall 111) satisfy this property.) 28. Let FS2 be the free semigroup on two generators x, y. (So a typical element of FS2 is uniquely of the form ulu2 ... u,,, where u, E {x, y}.) Prove that FS2 is not left amenable.

29. Is every subsemigroup of an amenable group necessarily left amenable?

30. Prove semigroup versions of (2), (4), and (5) of (0.16). 31.

Give an example of a semigroup that is left amenable but not right

amenable.

32. Let S be a semigroup. Prove that the map m - µ,,, (cf. Problem 1 above) is a bijection from .C(S) onto the set of finitely additive, positive measures µ on

S of mass I such that

p(s-'E) = µ(E) for all E C S, s E S. [Note: s-IE = {t E S: st E E}j. 33. Show that if S is a semigroup and in E .C(S), then m(I) = 1 for every right ideal I in S. 34. Let G be a locally compact group and it a strongly continuous representation of G on a Hilbert space J5 such that SUPZEG JJir(x)JJ < oo and each r(x) is

an invertible element of B(fj). Show that there exists a unitary representation 7r' of G on J5 such that 7r and ir' are equivalent, that is, there exists an invertible element A E B(Sj) such that vr(x) = Avr'(x)A-1 for all x E G (c£. Problem 1-40).

INTRODUCTION

23

35. Suppose that the locally compact group G is the union of an upwards directed family of open, amenable subgroups. Show that G is amenable.

CHAPTER 1

Amenable Locally Compact Groups and Amenable Semigroups (1.0) Introduction. In (0.16) we showed that amenability is preserved for discrete groups under the processes of taking subgroups and forming quotient groups, group extensions, and direct limits. We saw in Problem 0-35 that a version of the direct limit result holds in the nondiscrete case. The other three processes, however, pose problems, and one of the main purposes of this chapter is to show that topological versions of these three processes preserve amenability in the nondiscrete case also. The proofs depend on two crucial facts. The first is that the amenability of G is equivalent to the existence of a left invariant mean on any one of a wide range of subspaces of The second is that the amenability of G is equivalent to the existence of a topologically left invariant mean ((0.9)) on We finally investigate some of the properties of left amenable semigroups.

Let us start by looking at the difficulties that arise when we try to extend (1) and (2) of (0.16) to the locally compact group case. Let G be an amenable locally compact group and H be a closed subgroup of G. If G is discrete, then we obtain n E £(H) by writing n(O) = m(;"), where m E £(G), -0 E and ,b'(hb) = ,b(h) (b E B), where B is a transversal for the set of right H-cosets. In the nondiscrete case we run into serious measurability problems for 0': it is not clear that B can be taken to be even Borel in G. The problems arising with (2) of (0.16) are not so serious, but nevertheless non-trivial results are needed to push through the discrete argument. In both cases, life would be made easier by replacing by C(G), the space of bounded, continuous, complex-valued (in the natural way), and every functions on G. Of course, C(G) C m E £(G) restricts to an element of £(C(G)). The converse is not immediately clear.

It turns out that to solve the above problems, we have to consider not only C(G) but also the space U,(G) of right uniformly continuous, complex-valued functions on G. In addition, we will show that G is amenable if and only if there exists a topologically left invariant mean on G. It is convenient, before dealing with these matters, to reformulate the actions of L, (G) on L. (G) (see (0.8)) in terms of measures. This not only enables us 25

AMENABLE LOCALLY COMPACT GROUPS

26

to extend the action to one of M(G) on L (G) but also clarifies and smooths out many of the arguments involving such an action. Of course, the function viewpoint is also useful.

(1.1) The action of M(G) on L..(G) (cf. [HR1, §201). Let X be a locally compact Hausdorff space and M(X) be the space of complex-valued, regular Bore] measures on X. Then M(X) is a Banach space under the total variation norm. By the Riesz Representation Theorem, M(X) is canonically the Banach space dual of Co(X), the space of functions 0 E C(X) which vanish at infinity, each p E M(X) corresponding to the functional 0 - f 0 du on Co (X). When X = G, the space M(G) is a convolution Banach*-algebra, the product being given by

jj

f Od(u*v)=u *v(O)= JJ O(xy)dµ(x)dv(y)

(¢ECo(G))

GxG

The involution u -+ p- on M(G) is given by u~(E) = p(E-1) (E E ..V(G)) and is isometric. Closely related to u^' is u*, where u*(E) = p(E-1). By the Radon-Nikodym Theorem, L1 (G) is identifiable with the space of measures

in M(G) that are absolutely continuous with respect to A. Thus a function f e L1 (G) corresponds to the measure A f E M(G) determined by pf (40) =

f(x)f(x)dA(x)

(E Co (G)).

We will normally identify f E Ll (G) with pf, the context making clear if we are dealing with a function or a measure. Note that x * f, f * x (defined in (0.7)), as measures, are just 6 * u f, of * &, where Sx is the point mass at x. We will often write x * u, u * x or even xu, px in place of 5 *,a, p * b

An important fact is that L1 (G) is an ideal in M(G), so that M(G) acts on L1 (G) by convolution. The product f * g of functions, f, g E L1 (G) is given by:

f * g(x) = fG f (t)g(t-lx) dA(t) (x E G). The dual of this M(G)-action leads to an action of M(G) on LA(G), and the formulae for this action are as follows: if p E M(G) and 40 E L.. (G), then 0p, pfi as members of L1(G)' are given by

¢u(v)=0(p*u)= f 0d(p*v) (1)

f bd(v*,)

(v E Ll (G)),

while Op, u, as functions in L, .. (G) are given by OA(X) = x-0(u) = f ¢(yx) dp(y), (2 )

A0(X) = u(lx) = f(xY)d/4(ls). We note that this action of M(G) dualizes to give an action on LA(G)'. The algebra Ll (G) is a *-ideal of M(G), the involution on Ll (G) in function

terms being given by the map f - f", where f- (x) =

t) and


27

is the modular function on G. The "adjoint" of this map on L.(G) is the map ¢ --+ ¢ , where ¢*(z) _ ¢(a 1). So

f

r (x) f (z) da(x) _ ¢(x) f'" (x) da(x). 1 Here are two illustrations of why it is often convenient to regard elements of Ll (G) as measures. In the first place, it makes clear the essential difference (3)

between the actions of G on Ll (G) and L. (G): the one is the dual of the other, and the fact that x * f is defined similarly to ox-1 is accidental. In the second place, it often obviates the need for calculations involving the modular function. For example, if f E L, (G), p E M(G), then, as a function,

f * JA(y) = f f (yz-') A (z ') dp(z), whereas the measure version is just the simple convolution of * p.

Our next result gives an important property that characterizes Ll (G) as a subset of M(G).

(1.2) PROPOSITION [HRI, (19.27)]. Let p E M(G). Then p E LI(G) if and only if the map x - x * p is norm continuous from G into M(G). We now introduce the spaces Ur(G), U1(G), and U(G) of uniformly continuous functions. These spaces link up well with topologically invariant means. For example, as we shall see, every left invariant mean on Ur(G) is actually topologically left invariant! (1.3) DEFINITIONS. Recall that C(G) is the space of bounded, continuous, complex-valued functions on G. A function ¢ E C(G) is said to be left [right] x0 [z -+ Ox] is norm continuous from uniformly continuous if the map x G into C(G). The set of left [right] uniformly continuous functions in C(G) is denoted by U,(G) [Ur(G)J, and the elements of U(G) = U1(G) fl Ur(G) are said to be uniformly continuous. Our discussion will proceed in terms of U*(G); of course, analogous results hold for US(G). These spaces are often discussed in terms of the uniformities on G (cf. [HR1,

§4]). However, the two approaches are equivalent, and the approach that we adopt is usually easier to work with and applies directly to the semigroup case.

Let ¢ E Ur(G) and xo E G. If x6 -+ x in G, then I](zoO)xs - (xoo)x]] < 0, and II(0xo)x6 - (-0xo)xJJ = II zoxs - Oxoxll -+ 0 so that

I106 -'xII

xoO, 4ixo E Ur(G). So Ur(G) is an invariant subspace of C(G). (Of course, C(G) is an invariant subspace of L,o(C).)

(1.4) PROPOSITION. If 0 E L,,(G), 't' E Ur(G), and p,v E L1(G), then 0v E Ur(G), go E Ur(G), and p0v E U(G). PROOF. If x, y E G, then for z E G, (1)

I($v)x(z) - (0v)y(z)I 0, nets {x6) in G, {hb) in.H,

andxEGsuch that x6-+x and (1)

IQ(xbh6) - /3(xh6)I ? e.

We can suppose that there exists C E F(G) such that x6 E C for all b. Let L be the compact set (z E CH: 6(z) > 0}- ((1)). If a subnet of {h6} is eventually inside a compact subset of G, then a contradiction of (1) follows easily.

Otherwise, we eventually have Ch6 fl L = 0, and again (1) is contradicted. This establishes the norm continuity of the map x - (,6x) In, and using (i) again, it follows that O(C(H)) C C(G). From the left invariance of .\x, we have 4>(¢ho) = 4>((k)ho for all 95 E C(H), ho E H. Thus m o E £(C(H)) whenever m E Z(C(G)), so that H is amenable ((1.10)).

(1.13) PROPOSITION. If H is a closed normal subgroup of G, then G is amenable if and only if both H and G/H are amenable PROOF. If C is amenable, then so is H by (1.12). The amenability of G/H follows as in (0.16): £(C(G)) 96 0 implies that £(C(G/H)) # 0. Conversely, suppose that both H and G/H are amenable. The argument follows that of its discrete version (0.16.(3)). Note that this works only because we can use continuous and uniformly continuous functions.

Let p E £(C(H)), and for 0 E Ur(G) define 4/}p: G/H -r C by setting Op(W) = p((I,t)IH) (t E G). Since the quotient map from G onto G/H is open and the map t (0t)If e C(H) is norm continuous, it follows that Op belongs to C(G/H). Then for q E £(C(G/H)), the map 0 q(,kp) belongs to £(Ur(G)), and G is amenable ((1.10)). We now turn to the elegant theory of left amenable semigroups. This area of amenability is largely the creation of M. M. Day, E. E. Granirer,1. Namioka, J. Sorenson, A. H. Frey, T. Mitchell, C. Chou, A. T. Lau, J. C. S. Wong, and M. Klawe.

To make progress with this subject, it is essential to have at our disposal the amenability fixed-point theorem known as Day's Fixed-Point Theorem (Day 15)).

Amenability fixed-point theorems will be studied in detail in the next chapter. However, we feel justified in proving Day's theorem here, partly because we need it now and also because it exhibits the basic ideas of these theorems very clearly.

32


Let E, X be locally convex spaces and K a compact convex subset of E. A continuous map T: K X is called affine if T(akl + (1 - a)k2) = aTk1 + (1- a)Tk2 for all kl, k2 E K and a E 10, 1]. The set of affine maps from K to C is denoted by Af(K). Note that {FIK +C1: F E E'} C Af(K). The set K is called an affine left S-set, where S is a semigroup, if K is a left S-set with each map k sk affine. An obvious (and important) affine left S-set is the set of means Ell(S) under the natural action. (In that case, E = I., ,(S)' with the weak* topology.)

(1.14) THEOREM (DAY'S FIXED-POINT THEOREM). Let S be a left amenable semigroup and K an affine left S-set. Then there exists an S-fixedpoint in K.

PROOF. Let m E £(S) and let (f6} be a net in P(S) such that fb - m weak* in Let ko E K and k6 = Saes fa(s)(sko). Since K is convex and S-invariant, it follows that k6 E K. We will show that {k6} converges to a

fixed-point for S. To this end, let 3 be the "weak topology" induced on K by

Af(K): so ab --a a in (K, 3) if and only if F(ab) -+ F(a) for all F E Af(K). Let 31, 32 be respectively the relative topologies on K induced by the given and weak topologies of E. Then 31 >- 3 > 32 and since (K, 31) is compact and 32

is Hausdorff, we have 31 = 3. For F E Af(K), let 'OF E l,o(S) be given by OF (8) = F(sko). Then, by the affineness of F,

F(kb) = E fo(e).bF(8) = f6 (OF) --' m(q5F), 8ES

and since 31 = 3, we see that {k6} converges to some kl E K, and F(k1) _ m(¢F). If so E S, then Fso, where Fso(k) = F(sok) (k E K), is in Af(K), and so F(sok1) = m(95F,,) = m(OF8o) = m(-OF) = F(k1) So sold = kl, and k1 is an S-fixed-point.

O

(1.15) Fundamental problems for amenable semigroups. We start our discussion of the basic theory of amenable semigroups by listing a number of natural questions which a satisfactory theory should be able to answer. (1) When is a finite semigroup left amenable? (The answer fdr groups is always.)

(2) When does a semigroup S admit a left invariant mean in P(S)^? (An infinite group never admits such a mean.) (3) When does a subset of a left amenable semigroup "support" a left invariant mean?

(4) How "close" is a left amenable semigroup to being a subsemigroup of an amenable group? (5) Which subsemigroups of an amenable group are left amenable?

(6) When does a semigroup admit a multiplicative left invariant mean?. (A nontrivial group never admits such a mean.)


33

We will give reasonable answers to questions (1)-(5) in (1.19), (1.20), (1.21),

(1.27), and (1.28). The sixth question will be considered in the next chapter ((2.29)).

We start with our first question. It turns out that with little extra effort, we can answer the topological version of this question, where finite is replaced by compact. Before proving this version, we recall some facts about the structure of compact semigroups.

(1.16) Compact semigroups. Let T be a jointly continuous, compact, Hausdorff semigroup. (By "jointly continuous" we mean that the map (s, t) -+ at from T x T into T is continuous.) It is obvious that T contains a unique minimal, closed ideal K, called the kernel of T. A good account of the structure theory of the kernel is given in M. Rosenblatt [1, Chapter 5]. The kernel K is the disjoint union of the family of left [right] minimal ideals of T. Every minimal right ideal of T is closed and of the form eT for some idempotent e E K and conversely. For such an e, eTe = Te n eT is a compact group. and so it makes sense to The space C(T) is an invariant subspace of talk of left invariant means on C(T). The following result is due to Rosen [1, 2].

(1.17) PROPOSITION. There exists a left invariant mean on C(T) if and only if T has exactly one minimal right ideal. PROOF. Let m E .C(C(T)) and suppose that R1 and R2 are (closed) minimal right ideals with Rl # R2. Then R1 f1R2 = 0, and since T is compact Hausdorff, we can find, using Urysohn's Lemma, functions 0i E C(T) (i = 1, 2) with 0
a880 = &({aso)) = &(8{88)) > &({so}) = k. It follows that L is a left ideal in S. Further, since 1L4k < E8ES a8 = 1, we have that L is finite. So (i) implies (ii).

Now suppose that (ii) holds. Let L be a finite left ideal in S. Applying the later result (1.22), we see that L is also left amenable. Hence, by (1.19), it contains exactly one minimal right ideal R, and R is the kernel of L. Let e be an idempotent in R. Then eL = R and Le = Len eL is a finite left ideal group. in L, so that (ii) implies (iii). Finally, suppose that (iii) holds. Let G be a finite left ideal group in S and 6 = 1GI-3 F_xEG x E P(S). Now check that 4 E .C(S) to give that (iii) implies (i).

We now characterize those subsets of a left amenable semigroup that support

left invariant means. A subset E of a semigroup S is said to be left thick if whenever F E -9-(S), the family of finite subsets of S, then there exists s E S such that Fs C E. Similarly, E is said to be right thick if, for each F E -9-(S), there exists s E S such that sF c E. Left thick sets were introduced and studied by Mitchell [2), and the following proposition is due to him. Left thick subsets of an amenable group G are normally big. For example, if G is infinite and E is left thick in G, then [El = [Cl (Problem 1-18). The family of right thick subsets of a semigroup rarely forms a filter (Problem 7-19).

(1.21) PROPOSITION. A subset E of a left amenable semigroup S is left thick if and only if there exists m r= .C(S) with m(E) = 1.

PROOF. Suppose that E is left thick in S. If F E .r(S) and 8F E S is such that FsF C E, then sp(XES) = i for all s E F. It follows that if mo is a weak* cluster point of {§F} in !I(S), we have mo(XES) = 1 for all s E S, or equivalently, smo(XE) = 1 (s E S). Let K = {n E 931(5): sn(XE) = I for all s E S}. Clearly, K is a nonvoid weak* compact, convex subset of I.. (S)' that is left invariant for S. By Day's Fixed-Point Theorem, there exists m E K n C(S). So m(XE) = m(XES) = 1.

Conversely, suppose that m r= .C(S) is such that m(E) = 1 and that E is not left thick in S. Then we can find sl, ... , s,, in S such that {sl, ... , s E S. So I XES; (s) < (n - 1) for all s, and hence nm(E) _


35

m(E i XESi) : J1 E i XEsi4l : (n - 1), contradicting the equality m(E) _ 1. 0 (1.22) COROLLARY. If L is a left ideal in S, then L is left amenable if S is.

PROOF. The set L is left thick in S, so that there exists m E C(S) with m(L) = 1. Now show that mIL is in £(L). a We note that the converse to (1.22) is also true, as is the (easy) result that left amenability is inherited by right ideals (Problems 0-33, 1-19). We now consider our fourth problem: How close is a left amenable semigroup S to being a subsemigroup of an amenable group? To tackle this, it is reasonable

to try to quotient out by some congruence on S in order to bring some "cancellativeness" to the situation. In the theory of semigroups, the condition that we need to achieve such a quotient is that of left reversibility. The semigroup S is called left reversible [CP2, p. 194] if every pair of right ideals in S has nonempty intersection (or equivalently, if the family of right ideals of S has the finite intersection property). Similarly, S is right reversible if every pair of left ideals in S has nonempty intersection.

(1.23) PROPOSITION. Every left amenable semigroup is left reversible. PROOF. If m E .C(S) and R1, R2 are right ideals in S, then, by Problem 0-33,

m(Ri)=1=m(R2). So m(RinR2)=1 andRinR2g£0. 0 Recall that a congruence on a semigroup S is an equivalence relation - on

S such that if r - t, then sr - st, rs -- is for all s re S. Note that S/ is a semigroup in the natural way and that the quotient map is a surjective homomorphism. The following result on left reversible semigroups is given in [CP1, p. 35]. The reader may wish to regard it as an exercise. (1.24) PROPOSITION. Let S be a left reversible semigroup, and let .

be the

relation on S defined by setting x x y if there exists s E S for which as = ys. is a congruence on S, and the semigroup S/N is right cancellative. Then The next result, in a sense, reduces the study of left amenable semigroups to the study of right cancellative, left amenable semigroups. (We shall see ((1.29)) that a reduction to cancellative semigroups is not possible.) (1.25) PROPOSITION. A left reversible semigroup S is left amenable if and only if S//. is left amenable.

PROOF. The semigroup S/ is a homomorphic image of S, so that S is left amenable. Conversely, suppose that S/%zr is left amenable. Let K = 97i(S). If Si, ... , sn E S, then, since S is left reversible, we can find t E n i (siS), so that n 1(siK)

D f;`_1(siSK) D tK. So the family {sK: s E S} has the finite intersection property, and hence Ko = n{sK: s E S) is nonempty. Further, if 81, 82 E S and ri,r2 E S are such that sirs = s2r2, then s1Ko C sir1K = s2r2K C s2K.


m

So s1Ko C Ko, and Ko is an affine left S-set. Now if s1u = 82U in S, then since Ko c uK, it follows that SIm = 82m for all m E KO. Hence, in a natural way,

Ko is an affine left (S/n)-set, and any fixed-point in Ko for S/-- ((1.14)) is in .C(S).

0

The following well-known theorem (Ore's Theorem) from semigroup theory will be needed in (1.27). References for Ore's Theorem are [CP1, p. 351 and Ljapin 11, p. 392).

Again the reader may wish to regard the result as an exercise. (1.26) PROPOSITION. Every cancellative, left reversible semigroup S can be embedded as a subsemigroup of a group G such that G = {st'1: s, t E S}.

The next result is due to Wilde and Witz [1j. (1.27) PROPOSITION. Every left amenable, cancellative semigroup S is a subsemigroup of an amenable group G such that S is a generating left thick subset of G.

PROOF. A left amenable, cancellative semigroup is left reversible and so by (1.26) can be embedded as a subsemigroup of a group G, with G generated by S. We now show that G is amenable. In fact, we will show that there exists in E .C(G) such that m(S) = 1, and the desired result will follow by (1.21). Let m1 E .C(S), and define mo E 211(G) by setting mo(¢) = ml(01S). Then smo = mo = s-Imo for all s E S, and since S generates G, we see that mo E .C(G). So G is amenable, and since mo(S) = 1, S is left thick in G. O

Problem 2-10 deals with a topological version of the preceding result (and (1.28)).

With the above results in mind, we see that the fourth problem can be settled

most satisfactorily if we can bridge the gap between right cancellative + left amenable and full cancellativity. Indeed, from (1.25), S is left amenable if and only if the right cancellative semigroup is left amenable. If we know that S/.. is actually cancellative, then (1.27) identifies it with an amenable subsemigroup of an amenable group. The conjecture that every right cancellative, left amenable semigroup is cancellative is called Sorenson's Conjecture. We will see in (1.29) that the conjecture is not true in general. Finally, we turn to our fifth problem: which subsemigroups of an amenable

group are left amenable? The answer is elegant and algebraic-the subsemigroup has to be left reversible! There are familiar amenable groups containing subsemigroups that are not left amenable (Problem 0-29). The following result is due to Frey [1j. (1.28) PROPOSITION. Let G be amenable and let S be a subsemigroup of G. Then S is left amenable if and only if S is left reversible.

PROOF. By (0.16), we can suppose that S generates G. Suppose that S is left amenable. By (1.23), S is left reversible.


37

Conversely, suppose that S is left reversible. By the Hahn-Banach Theorem, the linear functional aXs -+ a (a E C) on the subspace CXS of I,o(G) extends to give a mean p E 9Jt(G) with p(S) = 1. Let

K = {m E M(G): m(S) = 1}. Then K is a nonvoid, compact convex subset of ER(G). One readily checks that

SK C K. Let K0= n{sK: s E S}. Then Ko is a nonvoid compact, convex subset of Tt(G). As in the proof of (1.25), the left reversibility of S yields that Ko 0 0 and that SKo C Ko. We now show that GKo = Ko. As G is a group, each of the maps k - sk (k E K) is one-to-one. Hence

KDsKo=s( ntK)=n(stK)D n(tK)=K0. tes

tES

tES

So sKo = Ko = s-1Ko for all s E S. Since S generates G, we have GKo = Ko. Now apply Day's Fixed-Point Theorem to obtain mo E Ko fl £(G). Noting that

mo E Ko C K, we have mo(S) = 1, and amenable. 0

E £(S), so that S is left

References Chou [4], Day [1), [2), [4], [5), [7], [9], Dixmier [1], Folner [2], Frey [1], Granirer [9), [10], Greenleaf [2], Hewitt [2), Hulanicki [4], Jenkins [3), Klawe [1], [2], Lau [3), [6], Mitchell [2], Namioka [2], Reiter [14], [R], Rickert [2], Rosen [1], [2], Specht [1].

Further Results (1.29) The Sorenson Conjecture. Recall ((1.27)) that Sorenson's Conjecture asserts every left amenable, right cancellative semigroup is cancellative. The results (i) and (ii) below give some support to the conjecture. However, (iv) below demonstrates that the conjecture is false. The conjecture still seems to be open for finitely generated semigroups. We note that the conjecture is equivalent to a conjecture involving the "strong Folner condition" ((4.22)). (i) A finite, left amenable, right cancellative semigroup is a group.

[Since S is right cancellative and finite, Su = S for all U E S. So S is the kernel of S. Using (1.17), S is a left and right minimal ideal in itself and so is a group.)

(ii) (Sorenson [2), Klawe [2]). Let S be left amenable, right cancellative, and such that there exists m E WE(S) for which m(sA) = m(A) for all s E S, A C S. Then S is cancellative. [Suppose that S is not left cancellative, and find x, y, sl E S such that x ¢ y and s I x = sly. Using Zorn's Lemma and the fact that S is right cancellative, we

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can find a maximal, nonvoid subset A of S such that xAf1 yA = 0. If z re S - A, then, since S is right cancellative and A is maximal, either xz E yA or yz E xA. So S = A U Ai U A2, where Al = x-1(yA) and A2 = y-1 (zA). Since xA1 C yA and yA2 C xA, we have m(Ai) < m(A) (i = 1, 2). So 1 = m(S) < 3m(A), and

m(A) > 3. But since zA fl yA = 0 and s1x = sly, we have 2m(A) = m(xA U yA) = m(sixA U s1yA) = m(A) so that m(A) = 0 and a contradiction follows.] A semigroup S is called left measurable if there exists m E f73l(S) such that m(sA) = m(A) for all s E S, A C S. The properties of left measurable semigroups are explored by Sorenson and Klawe. (iii) Let S and T be semigroups and let p be a homomorphism from T into the semigroup of endomorphisms of S. The semidirect product semigroup S x, T is

defined in the obvious way: as a set, S xo T is just S x T, and multiplication is given by (s,t)(s',t') = (sp(t)s',tt'). A semidirect product of locally compact groups is always amenable. A semigroup version of this result is the following, which is due to Klawe 12].

Let S and T be left amenable and let p(t) be surjective for all t E T. ThenS xoT is left amenable. {For t¢ E l (S), t E T, and m E !Ui(S), define Op(t) E l,, (S) and p(t)m E 971(S) by setting Op(t)(s) = 0(p(t)(s)) (a E S) and (p(t)m)(¢) = m(op(t)). Then

ll(S) becomes an affine left T-set with respect to the map (t, m) -+ p(t)m.

Let s E S and t e T, and find so E S such that p(t)(so) = s. As p(t) is a homomorphism, we have, for x E S, (Os)p(t)(x) = ¢(p(t)(sox)) = op(t)so(x), and so (sp(t)m)(0) = m(Op(t)so) = p(t)(som)(0). It follows that C(S) is also an affine left T-set, and by Day's Fixed-Point Theorem, there exists mo E C(S) for which p(t)mo = mo for all t re T. For 0 E im(S xvT) and s E S, define Vi, e l.(T) by O3(t) = sy(s,t). Let ml E £(T), and define m1 E lo,(S) by If x = (sl,t1) E S xoT, then (bx),(t) = midi(s) = It follows that if ' = Ox E l,o(S xDT), then m10' = ((m10)s1)p(t1) and the functional i,b- mo(m1>j') belongs to Z(S xoT).] Maria Klawe also shows that the above result is false in general if the requirement that each p(t) be surjective is dropped (Problem 1-29). (iv) (Klawe (2]). There exists a right cancellative, left amenable semigroup that is not cancellative. (Let F be the free commutative semigroup on an infinite, countable set which is enumerated {u,,: n E P}. Define a surjective homomorphism a of F by setting

ui'+"2u23...un,1, and let p(n) = a" (n E P). Let S = F xv P. The semigroup S is left amenable by (iii). We now show that S is right cancellative but not cancellative. Suppose that (wl,n1)(w,n) = (w2,n2)(w,n)

in S. Then (w1p(ni)(w),n1 + n) = (w2p(n2)(w),n2+n) so that n1 = n2 and wip(nl)(w) = w2P(n2)(w). Since F is cancellative, wl = w2 and S is right cancellative. However (u1,1)(u1,1) = (u1, 2) = (u1,1)(u2i 1) so that S is not cancellative.)


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(v) Klawe [3] gives an example of a cancellative right amenable semigroup that is not left amenable, thus answering a question of Granirer. The example is also a semidirect product of semigroups.

(1.30) Amenable Banach algebras. Amenable Banach algebras were introduced and studied by B. E. Johnson in his definitive monograph [2]. This class of Banach algebras arises naturally out of the cohomology theory for Banach algebras, the algebraic version of which was developed by Hochschild [1, 2,

3). Other relevant papers are Guichardet [SI, S2], Johnson [3, 5, 82, 83, S4], Kamowitz [1), Khelemskii [S1, S2], Khelemskii and Sheinberg [1], Lau [11), and Racher [1]. The author is grateful to G. Dales for a helpful communication. Amenability has proved particularly fruitful in the category of operator algebras-amenable operator algebras are briefly discussed in (1.31) and (2.35). Let A be a Banach algebra and X a Banach space that is an A-module. Then X is called a Banach A-module if there exists K > 0 such that IIaCII < KI]aj[ IICII, IICaII < KIICII IIa1I for all a E A, C E X. (A good example of a Banach A-module

is afforded by X = Li (G) and A = M(G), with convolution operation as in (1.1).) The dual space X' of a Banach A-module X is itself a Banach A-module in the natural way: a f (C) = f (Ca), f a(C) = f (aC) (f E X', C E X, a E A). We say that X' is a dual Banach A-module. A derivation from A to a Banach A-module X is a norm continuous linear X such that D(ab) = aDb + (Da)b for all a,b E A.2 If mapping D: A

C E X, then it is easily checked that the map Df, where De(a) = aC - Ca, is a derivation on A. Such derivations DE are called inner. The set of [inner] X is a linear subspace of the Banach space B(A,X) of derivations D: A bounded linear operators from A to X. We are interested in those Banach algebras A for which "many" of the derivations are inner. It is unrealistic to expect there to be many interesting algebras A with every derivation D: A -+ X inner for every Banach A-module X. However, dual Banach A-modules have pleasant weak*-compactness properties, and modifying the above suggestion leads to the definition of amenable Banach algebras. The Banach algebra A is called amenable if every derivation from A into any dual Banach A-module is inner. The use of the term amenable in this context may seem strange since, a priori, there seems to be no obvious connection between invariant means on a locally compact group G and derivations on a Banach algebra. However, Johnson proved the remarkable result ((iv) below) that G is amenable if and only if Li (G) is an amenable Banach algebra. In Johnson's Banach-algebra version of Hochschild's cohomology theory for associative algebras, the algebra A is amenable if and only if every first cohomology group Hi (A, Y) = {0} for every dual Banach A-module Y. The classes of Banach algebras A for which H" (A, Y) = {0} for every dual Banach A-module Y (n > 1) does not seem to have been studied in any detail. 2See Willis )1) for automatic continuity results on derivations on amenable group algebras.

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We now discuss some of the properties of amenable Banach algebras. The discussion is based on Johnson [2]. (i) Every amenable Banach algebra contains a bounded (two-sided) approximate identity. [Let A be an amenable Banach algebra. Let X be the Banach A-module A'

with the usual right action and zero-left action. Then Y = X' = A" is a dual Banach A-module with zero-right action, and by amenability, the derivation a a must be inner. So a = aE for some E E A". Approximating E weak* by a bounded net in A and then following (with slight modification) the proof of (0.8), we produce a bounded right approximate identity {f} for A. By taking X with the usual left action and zero-right action, we produce a bounded left approximate identity {ea} for A. Then (as observed by P. G. Dixon) the net {e.,0}, where e.,,6 = e. + fL - fae,,,, is a bounded (two-sided) approximate identity for A.] The above result (i) means that for developing the theory of amenable Banach algebras, we can restrict attention to those algebras A with a bounded approximate identity. The extension of Cohen's Factorisation Theorem (HR2, (32.22)] is thus available, and a consequence of this is that for proving the amenability of such an algebra A, we need only consider dual modules X' where X is neo-unital

(or essential). The Banach A-module X is called neo-unital if for all C E X, there exist a, a' E A, 77, 77' E X such that arl = = ,'a'. The advantage of dealing with a neo-unital module X is that X can be made into a A(A)-module, where A(A) is the multiplier (or centraliser) algebra of A (discussed below). For example, this gives that a neo-unital Lt (G)-module is also a G-module, and the L1(G)-action is obtained by "integrating up" the G-action (cf. the relationship between the nondegenerate *-representations of Ll (G) and the continuous, unitary representations of G). (ii) Let A be a Banach algebra with a bounded approximate identity. Then A X' is inner whenever X is is amenable if and only if every derivation D: A a neo-unital Banach A-module. (Let X be a Banach A-module. Since A has a bounded approximate identity

{eb}, the extension of Cohen's Theorem applies to give that the set {ae: a E A, 6 E X} is a closed subspace Z of X. Clearly, Z is also a Banach A-module, and applying the "right-action" version of the above extension gives that Y = {aeb: a, b E A, 6 E X} is a closed submodule of X. By Cohen's original theorem [HR2, (32.26)] every z E A is a product ab in A, and it follows that Y is neo-

unital. Let D: A - X' be a derivation and Q1: X'

Z' be the restriction map. Since Ql preserves the A-module action, it follows that D3 = QI o D is a derivation into Z'. We claim that D is inner if D1 is inner. For suppose

that al E Z' is such that Dj(a) = aal - ala(a E A) and let a E X' be such that Ql (a) = al. Let D. be the inner derivation associated with a. Then D' = (D - D.) is a derivation into the weak * closed submodule Zi =

{f EX':f(Z)={0}} of V. Since l3(a{)=0for follows that ZIA = {0}. So for all a E A, D'(aea) = aD'eb, and if fo is a weak


41

cluster point of {D'e6}, then fo E Z -L and

D'(a) = afo = afo - foa and D' is inner. Hence D = D. + D' is inner, and so D is inner if D1 is inner. Now follow the right-hand version of the above argument with D, X, Z replaced by D1i Z, Y to obtain that D1 is inner if Q2 o D1: A

Q2: Z'

Y' is inner, where Y' is the restriction map. Since Y is neo-unital, the non-trivial part

of the proof is complete.] The theory of multipliers (or centralisers) of a Banach algebra is developed

by Johnson [Si]; see also Larsen IS]. We sketch briefly the relevant facts that we require. Let A be a Banach algebra with a bounded approximate identity {e6} with sup6 IIe6II = M (< oo). A pair (T1iT2) E B(A) x B(A) is called a multiplier if for all x, y E A, we have (T1 x) y = T1(xy), xT2y = T2 (xy), and xT1 y = (T2x)y. The set A (A) of multipliers on A is a unital Banach algebra with multiplication and norm given by (T1,T2)(S1,S2) = (T1S1,S2T2), II(T1,T2)II = max{IITIII, IIT2II}. Another natural and useful topology on A(A) is the multiplier

topology a net T6 .- T in the multiplier topology if T6a -* To in norm for all a E A. Here a E A is identified with the pair (La, Ra), which we now introduce. There is a canonical algebra homomorphism a --* (L0, Ra) from A

into &(A), where Lax = ax, Rx = xa (x E A). Trivially, this map is onto if A has an identity element. In general, the map is faithful and bicontinuous: indeed IIaI I = lim I Iae6II = lim IILae6I I

0 such that [[xgyll < KIIell for all g E X. Also, using (1.2) and the continuity of the M(G)-actions for the multiplier topology, we see that the maps (x, y) - xl:y are norm-continuous for each e E X. A Banach space which is a G-set with the above properties is called a Banach G-module. So a neo-unital Banach LI (G)-module is, in the natural way, a Banach G-module. Conversely, a Banach G-module becomes a neo-unital Banach L1(G)-module by integrating the G-action. If the G-action

arises from a Banach L1(G)-module X as above, then integration brings us back

to where we started. (The proof is formally the same as that establishing the equivalence of nondegenerate *-representations of L1 (G) with the continuous, unitary representations of G (D2, 13.3].) Similarly, a derivation D : L1(G) -, X', where X is a neo-unital Banach L1(G)-module, determines, and is determined by, a G-derivation, that is, a weak* continuous, norm bounded map 6: G X' satisfying 6(xy) = b(x)y+ x6(y) for all x, y E G. Indeed, if {u45} is a bounded approximate identity for L1(G), then, for each x E G, {D(x *,u6)} converges weak*, and we define b(x) to be the weak*

limit. Very explicitly, we have b(x)(aCb) = [bD(x * a) b E L1(G)). Noting that for µ E L1(G), f D(x * a) dp(x) = D(,ua), we see that integrating 6 brings us back to D again. More generally, if A is a Banach algebra

with a bounded approximate identity, then a derivation D: A -t X', where X is a neo-unital Banach A-module, extends in a natural way to a derivation D': A(A) X'. We now come to the remarkable result of Johnson that justifies the use of the term "amenable" as applied to Banach algebras. Khelemskii and Sheinberg [1] interpret this result in terms of "flat" modules. (iv) Let G be a locally compact group. Then G is amenable if and only if L1(G) is an amenable Banach algebra.


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[Suppose that G is amenable and let X be a Banach G-module and b: G X' a G-derivation. By (ii) and the preceding comments, the amenability of L1 (G) will follow once we have shown that 6 is inner in the sense that, for some ao E X', 6(x) = xao - aox (x E G). Define a new Banach G-module structure on X with

operations: C.z = z-' x, x.g = £ (x E G, C E X). The dual actions on X are given by z.a = xax-1, a.z = a (a E X'). Further, 6': G - X', where 6'(x) = b(x)z-1, is a G-derivation for this new module structure, and 6 is inner

if and only if 6' is inner since 6(x) = xa - ax e* 6'(z) = x.a - a.x. Define /3: X C(G) by /3(£)(x) = 6'(z)(l;) (x E G). Let m E £(G) and a E X' be given by a(e) = m(f(g)). Then for xo E G, (xo.a a(t.zo - zo.t;) _ m(fi(e.zo - g)). Since 0 =

Q(c.xo - e)(z) =

(6'(xoz) - 6'(xo).x - 6'(x))(f)

= S(e)xo(x) - 6'(zo)(e) - tgR)(x),

we have (xo.a - a.xo)(f) = m(fl(f)zo) - 6'(xo)(e) - m(Q(l;))

so that for all x0 E G, 6'(xo) =

ao.xo where ao = -a. So 6' is inner.

Hence L1(G) is amenable. Conversely, suppose that Ll (G) is amenable. The following argument is due

to J. It. Ringrose. The space X = U(G) is a Banach G-module with trivial left action and the usual right action: (¢, x) Ox. Since Cl is G-invariant,

Y = X/Cl is also a Banach G-module, and Y' = {a E X': a(l) = 0}. Let v = 6e E X' and let 6: G Y' be the derivation given by 6(x) = xv - v. Since L1(G) is amenable, 6 is inner, and so there exists A E Y' such that xv - v = 6(z) = xµ -,u. Then e = (v - µ) ¢ 0 and is left invariant in X'. So there exists a left invariant mean on U(G) (see (2.2)) and G is amenable by (1.10).] We note that Johnson [5] shows that amenability of a Banach algebra A is "stable" under small perturbations of the multiplication. Khelemskii [S) has given characterisations of amenable Banach algebras. The author understands that related work in this area has been done by Curtis and Loy [Si) and by Dales and Esterle (in a Monograph under preparation). Khelemskii's paper also contains the following result due to Steinberg: a uniform algebra A is amenable if and only if A is a C(X) (X compact Hausdorjf). The important notion of weak amenability for a Banach algebra A was introduced in the commutative case by Bade, Curtis and Dales [S]. The notion readily extends to the non-commutative case (Johnson [S5]) and is defined as follows: a Banach algebra A is called weakly amenable if every derivation D: A - A' is inner. Of course, A' is a dual Banach A-module in the obvious way, and so every amenable Banach algebra is weakly amenable. The converse is false: as Johnson [S5] points out, if I < p < oo, then In with pointwise multiplication is weakly


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amenable (using the fact that the span of its idempotents is dense) yet 1" is not amenable since it does not contain a bounded approximate identity. Bunce and Haagerup have shown that every C'-algebra is weakly amenable (Haagerup [31). An interesting open question is: is L1 (G) weakly amenable for every locally compact group G? Johnson [S5] has proved that L1 (G) is weakly amenable if G is either discrete or [SIN] or one of the groups GL(n, C). Weak amenability for the Fourier algebra A(G) of G is considered by Forrest [S], [S3]. Characterisations of weak amenability and applications to semigroup algebras are given in Groenbaek [SJ. In the latter context, Duncan and Namioka [1) have shown that 11 (S) is never amenable if S is an infinite semilattice.

(1.31) Amenable C'-algebras. The class of amenable C*-algebras has a particularly rich theory. We shall be content to discuss briefly some of the results of the theory. (Amenable von Neumann algebras are discussed in (2.35).) Let G be a locally compact group and C* (G) be the enveloping C*-algebra of G. There is a canonical, continuous *-homomorphism from L1 (G) onto a dense

subalgebra of C*(G). If G is amenable, then ((1.30(iv))) L1(G) is amenable, and it easily follows (Problem 1-38) that C*(G) is an amenable C*-algebra.. More generally, if it is a continuous, unitary representation of G, then the norm closure of ir(Li(G)) (which is also 7r(C*(G))) is amenable. In particular, Ct (G) = 7r2(C*(G)) is amenable, where ire: G U(L2(G)) is the left regular representation: ir2 (x) f (y) = f (x-1 y), and U(L2 (G)) is the unitary group of B(L2(G)). The above comments and Problem 1-38 then give the following (Johnson [2], Bunce [3]). If G is amenable, then C*(G) is amenable. If G is discrete, then G is amenable if and only if Ct (G) is amenable. It follows that if G is discrete and nonamenable (for example, if G = F2), then Ct (G) is not amenable. Johnson shows in [2] that every Type 1 C* -algebra is amenable. Other classes of amenable C*-algebras are the UHF-algebras (Sakai [1, p. 73]) and the Cuntz algebras 9n (Cuntz [Si]). Further, the class of amenable C*-algebras is closed under inductive limits and spatial tensor products. Rosenberg [2] has also shown that amenability is preserved under crossed products by amenable discrete groups. (A more general result, following from a theorem of Green, is briefly discussed below.)

Johnson [2] introduced the class of strongly amenable C*-algebras. For mo-

tivation, we note that if A is an amenable C*-algebra, then every derivation D: A -+ X', where X is a Banach A-module, is inner. However, a priori, we do not know if there is any control over KD = inf{)[ao((: D(a) = acao - aoa for all a E A}. Now if G is an amenable locally compact group and A = C* (G) with X neo-unital then D can be regarded as a G-derivation S, and it easily follows from the proof of the first part of (1.30(iv)) and (0.1) that there exists such an ao with

ao belonging to the weak* closure of co{-S(x)x-1: x E G}. This suggests the following definition: a unital C*-algebra A is called strongly amenable if, whenever X is a Banach A-module with i = £ = e1 for all e E X and D: A X' is a


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derivation, then there exists ao in the weak* closure of co{-D(u)u* : u E U(A)},

such that D(a) = aao - aoa (a E A). (Here, U(A) = {u E A: uu* = 1 = u*u} is the unitary group of A.) There is a corresponding definition of strong amenability in the nonunital case. Much of the theory of strongly amenable C*-algebras parallels the amenable theory. However Rosenberg [2) showed that the Cuntz

algebras ®n (n > 2) are amenable but not strongly amenable. Haagerup [3] shows that amenable C*-algebras are stably isomorphic to strongly amenable C*-algebras-in fact, if A is amenable, then the C*-tensor product A OK, where K is the algebra of compact operators on an infinite-dimensional Hilbert space, is strongly amenable. Bunce [2] gives the following elegant "Fixed-Point Theorem" characterisation of strong amenability (cf. Problem 1-39): A unital C* -algebra A is strongly amenable if and only if, whenever X is a Banach A-module and C is a nonempty, weak* compact, convex subset of X' such that uCu* C C for all u c- U(A), then there exists c E C such that ucu* = c for all u E U(A). It also follows from the work of Haagerup [3] that if A is an amenable C*-

algebra, then Ka of the preceding paragraph is < IIDII; indeed, we can find an appropriate ao in the weak* closure of {a*D(a) : a E A, Nail < 1). The same paper of Haagerup, together with a result of Connes [41, establishes the remarkable result that the classes of amenable and nuclear C* -algebras coincide.

(An earlier partial result was obtained by Bunce and Paschke [1].) The proof uses the deep result (Connes [1], Choi and Effros [3], Elliott 12]) that a nuclear C*-algebra has approximately finite-dimensional second dual, as well as Pisier's generalisation of Grothendieck's inequality. It seems appropriate, therefore, to discuss briefly the topic of nuclear C*-algebras. A good reference for this topic is Lance 11], to which the reader is referred for more details and references. Let A, B be C*-algebras, and let A ® B be the algebraic tensor product of A and B. Clearly A ® B is a *-algebra in the natural way. A C* -norm on A ® B is a norm y on A0 B such that -t(x*x) = 7(x)2 for all x E A ®B. Any such 7 is a cross-norm, that is, 7(a ® b) = Nail IIbI1 for all a E A, b E B. It is easy to see that the projective tensor product A®B is a Banach *-algebra and has a bounded approximate identity. It therefore has an enveloping C*-algebra C [132, (2.7)] in which A®B is faithfully embedded, and the restriction II 11maX of the norm of C to A ® B is a C*-norm. This is the maximum C*-norm on A®B. The completion of A®B with respect to II.11max is denoted by A®maxB. There is also a minimum C*-norm I I I Imia on A0 B, and this can be obtained as follows. Realise A, B as C*-algebras of operators on Hilbert spaces S5, .&. Then A ® B is realised as a *-algebra of operators on S5 ® A. The operator norm of A ®B C B(f3 (9 S) is 11 I Imm (which is independent of the particular spaces 55, . on which A, B are realised). The completion of A® B with respect to II Ilmi" is -

denoted by AOmj,, B. The C*-algebra A is called nuclear if 11.11max = 11 llm;n for

every C*-algebra B, or equivalently, if there is exactly one C*-norm on A ® B for every C*-algebra B. As commented above, a C*-algebra is amenable if and only if it is nuclear.

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Phillip Green (S, Proposition 14] has shown, using Rieffel's "tensor product"

theory of induced representations, that if H is a closed subgroup of a locally compact group G with G/H amenable (in the sense of Eymard [2) that there exists a G-invariant mean on C(G/H)) and (G, A, 3') is a "twisted covariant system", then C ' (G, A, 1) is nuclear if C ' (H, A, 3) is nuclear. Since nuclear = amenable, two immediate corollaries are (a) every cross-product A xa G is amenable if A is an amenable C'-algebra and G is amenable, and (b) C' (G) is amenable if there is a closed subgroup H of G with C` (H) amenable and G/H amenable. Lau and Paterson [S3) give a more general result than Green's above using direct amenability arguments (cf. Rosenberg [2)). In particular, these techniques give a very straightforward proof of (b) above (Problem 1-41). The result (b) above suggests a question which is of interest from a locally compact group point of view. Let d be the class of locally compact groups G for which C* (G) is amenable. The class say is large. Indeed, there seem to be three main subclasses of d. Let G be a locally compact group. Then

(i) G E d if G is almost connected. [The definition of "almost connected" is given in (3.7). The result when G is separable and connected follows from a deep result of Connes [1], and also uses

Choi and Effros (S) and the fact that nuclear = amenable. The extension to the almost connected case is given in Lau and Paterson [S3), and uses results of Lipsman [S), Milidid 181 and Batty [S).) (ii) G E sag if G is Type 1. (iii) G E sad if G is amenable.

[This, and (ii), were discussed above.) In view of (1.13), a natural question is is it true that if H is a closed, normal subgroup of G, then G E sad if and only if both H, G/H E sad? This seems to be an open question. If it has a positive answer, then we can construct many more groups G E d by using group extensions involving (i)(iii). (Note that (Effros and Lance [1)) the Type 1 group SL(2, R) contains the nonamenable group F2 as a discrete subgroup ((3.2)) so that closed subgroups do not always inherit the property of being in ,std.) Here are three partial results proved in Lau and Paterson [S3]. Note that it is easy to show that G/H E d if G E St?. As mentioned earlier, Green proved the nuclear version of (a) below. Let H be a closed normal subgroup of G.

(a) If H E .V and G/H is amenable, then G E d.

(b) If GEd,then HEd. (c) If G is separable, H is of Type 1, and G/H is almost connected, then GEsa?. The results (a) and (the separable case of) (b) are set as Problems 1-41, 1-42. All three results (a), (b), (c) are special cases of more general results.


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Problems 1 I. Establish (1) and (2) of (1.1). 2. Show that Co(G) C Ur(G).

3. Show that C(G) = Ur(G) if and only if G is either compact or discrete. Prove also that Ur(G) is finite-dimensional if and only if G is finite. 4. Show that £t (G) _ {A} if G is compact. 5.

Prove that a semidirect product G x,, H of locally compact groups is

amenable if and only if both G and H are amenable. 6. Prove that every solvable locally compact group is amenable.

7. Prove that the following statements are equivalent for a locally compact group G: (i) G is amenable; (ii) the closure of every compactly generated subgroup of G is amenable; (iii) the closure of every finitely generated subgroup of G is amenable.

8. Let G be amenable, and let H be an open subgroup of G. Let T be a transversal for the right H-cosets in G and mo E £(C(G)). For ¢ E C(H), V) E C(G) and for p r= £(C(H)) let tb1, E C(G) be given by Op(x) = p((Ox)IH),

and ¢T e C(G) be given by ¢T(ht) = 4,(h) (h E H, t E T). Show that we can define maps a: £(C(G)) -{ £(C(H)) and 3: £(C(H)) -+ .C(C(G)) by a(m)(4') = m(4T), f3(p)(0) = mo(t/lp). Show that fi is one-to-one and that if G has equivalent left and right uniform structures (that is, U1(G) = Ur(G)) then there exists a one-to-one map -i: £(U(H)) -+ £(U(G)). Obtain relationships between the cardinals I£(C(G))I, I£(C(H))I and the cardinals I£(U(H))I and I£(U(G))I 9. Let G be a locally compact amenable group, H a closed subgroup of G, and AGIH a quasi-invariant measure on G/H as in (1.11). Show that G/H is amenable (in the sense that on L,o(G/H, AG1H) there exists a G-invariant mean). 10. Let G be a locally compact group and µ a positive, finitely additive, left invariant measure on the family R(G) of Borel sets with y(G) = 1. Prove that p is countably additive if and only if G is compact. (This result shows that

invariant means are never countably additive in the noncompact case. Hint: deal with the a-compact case first.) 11. Let 88 be a left amenable semigroup with a continuous left action on a compact Hausdorff space X. Show that there exists v E PM(X) such that v(Os) = v(¢) for all s E S, 10 E C(X). [Here, PM(X) is the (weak* compact,

convex) set of probability measures on X.)


48

12. Let G be a locally compact group that is amenable as discrete. Prove in two ways that G is amenable: (i) using Day's Fixed-Point Theorem, and (ii) using (1.10). 13. Let G be a locally compact group that is amenable as discrete. Let B be a right invariant subspace of LA(G) containing 1. Prove that the map m mis is a surjection from £(G) onto .C(B).

14. Let S be an amenable semigroup and K a compact convex subset of a locally convex space that is a (two-sided) affine S-set. Show that there exists ko E K such that sko = ko = kos for all s E S. 15. Let X be a Banach space and S a left amenable semigroup with a right action on X such that for each s E S, the map x -+ xs is linear (x E X), and, for some M > 0 we have llxsll < Ml lxll (x E X, s E S) (that is, X is a right Banach S-module). Let Y be a right invariant subspace of X, and let f E Y' be left invariant for S. Prove that there exists left invariant F E X' such that Fly = f and l JFll < Mll f ll. Establish a converse. 16. Give examples of jointly continuous, compact Hausdorff semigroups T with .C(C(T)) # 0 and of left amenable, finite semigroups. 17. Determine the left thick subsets of N.

18. Let G be a discrete group. Prove that every left thick subset of G has cardinality JGl.

19. Prove that a semigroup S is left amenable if it contains a left amenable left ideal L. 20.

Show that if a semidirect product S = U x. T of semigroups is left

amenable, then so also are U and T.

21. Suppose that T is a subsemigroup of a semigroup S and that m(T) > 0 for some m E .C(S). Prove that T is left amenable and that mlT/m(T) E 5(T). 22. (Semigroup version of (1.13)). Let T be a subsemigroup of S. Let us say that s tit a' in T if there exists sa , ... , s,, E S and t1, ... , t,,-,, ti, ... , tn_ 1 in T such that s = 31, s' = s,,, and s1t1 = S2t1, 82t2 = s3t2,..., Sn_ltn_1 = San-1. Show that -1 is a left congruence on S. Let -', be the right version of -1. Prove that if -1 = ^-, and both T and S/ ^-1 are left amenable, then S is left amenable. 23. Let S be a left amenable, cancellative semigroup. Prove that every subsemigroup of S is left amenable if and only if FS2 (Problem 0-28) is not a

subsemigroup of S. 24. Let S be a subsemigroup of an amenable discrete group G such that s-'(Ss) c S for all s E S. Show that S is left amenable.


49

25. A semigroup S is called an inverse semigroup if for each s E S, there exists a unique element S* E S such that ss's = s, s'ss = 8*. (a) (i) Show that the set of partial 1-1 maps on a set is an inverse semigroup in the natural way. (ii) Prove that a *-semigroup of partial isometries on a Hilbert space is an inverse semigroup. (This is why inverse semigroups are of interest to functional analysts.) (b) Let S be an inverse semigroup and E be the set of idempotents in S. (i) Show that s8* E E for all s E S. (ii) Show that E is a semilattice (commutative, idempotent semigroup) in S.

(iii) Show that if I, J are right ideals in S, then I n J 96 0. (iv) Show that x -- y (in the sense of (1.24)) if and only if ex = ey for some e E E.

(v) Deduce that S/a is a group, the maximal group homomorphic image of S.

(c) Prove that an inverse semigroup is left amenable if and only if its maximal group homomorphic image is amenable.

26. By using an "Arens product" type argument (c.f. Problem 2-6), show that an amenable locally compact group always admits a (two-sided) invariant mean.

27. Let G be an amenable locally compact group and E a Borel subset of G.

Prove that m(E) = 1 for some m E Zt(G) if and only if, given C E '(G) and e > 0, there exists t E G such that 1(C - Et-1) < e. (Assume the Fixed-Point Theorem (2.24).)

28. Let E be an open subset of R. Write E = U{(an, bn): n E Z, an < bn < ant,}. Prove that m(E) = 1 for some m E £t(R) if and only if given M, e > 0, there exist n, < n2 in Z such that n2

n2-

n=nl

n=nl

7' (bn - an) > M,

E (an+1 - bn) < e.

29. Give an example of a semidirect product S x p T of left amenable semigroups which is not left amenable. (Of course, by (1.29(iii)), p(t) cannot be surjective for all t.)

30. Let G be a locally compact group, and note that M(G)' = Co(G)" is a unital C'-algebra. Prove that there exists a left G-invariant state on M(G)' if and only if G is amenable. (Here M(G)' has its natural dual G-action.) 31. Let A, B be Banach algebras and t: A -+ B a continuous homomorphism with 4i(A) dense in B. Prove that B is amenable if A is amenable. 32. Complete the proof of (1.30(iii)).

33. Prove that the projective tensor product A®B of two amenable Banach algebras A, B is also amenable.


50

34. Let A be a Banach algebra and A the Banach algebra obtained by adjunction of an identity 1 to A. Prove that A is amenable if and only if A is amenable.

35. Let A be an amenable Banach algebra, X a Banach A-module, and Y a closed submodule of X. Suppose that h E Y' is such that ah = ha for all a e A. Prove that there exists an extension k E X' of h such that ak = ka for all a E A.

36. Let A be a Banach algebra and x: A®A A the bounded, linear map specified by ar(a 0 b) = ab. Let A®A be the Banach A-module with module action given by a(b 0 c) = ab ® c, (b (D c)a = b ® ca, and note that it is an A-module map. A virtual diagonal for A is an element M E (A(bA)" such that

aM=Ma,zr"(M)a=8=aar"(M)for all aEA. (i) Show that A has a bounded approximate identity if A has a virtual diagonal.

(ii) Show that if A has a virtual diagonal, then there exists a virtual diagonal M for A such that M(a ®b - D(ab) (a)) = 0 whenever X is a neo-unital Banach A-module, l; E X and D: A X' is a derivation. (iii) Prove that A is amenable if and only if A has a virtual diagonal. In the following problems, H is a closed subgroup of a locally compact group G and sag is the class defined in (1.31).

37. Prove that G/H is amenable (in the sense that there exists a G-invariant mean on C(G/H)) if and only if, whenever X is a Banach G-module, then a G-derivation D: G -+ X' is inner if DIH is inner. 38. Show that C* (G) is amenable if G is amenable, and that if G is discrete, then G is amenable if and only if Ci (G) is amenable. (Hint: use Problem 35 above.)

39. Show that if A is a unital, strongly amenable C'-algebra, then there exists a tracial state on A.

40. Let A be a unital, strongly amenable C'-algebra, .ft a Hilbert space, and

rr: A - B(.) a continuous homomorphism with 7r(1) = I. Prove that 4r is similar to a *-representation of A on . (cf. Problem 0-34).

41. Prove that if H E sa7 and G/H is amenable, then G E S1. (Hint: use Problem 37 above.) Deduce that an extension of an amenable locally compact group by a group in sag is also in s/.

42. Let G be separable and let H be normal in G. Prove that H E St if G E sad. (Hint: use the theory of induced representations and the fact that nuclear = amenable.)

CHAPTER 2

The Algebra of Invariant Means (2.0) Introduction. We have already seen how useful Day's Fixed-Point Theorem is. When we try to obtain topological versions of this theorem, we find ourselves involved in consideration of Arens-type products involving means.

Such products arise in other amenability matters and are the central theme of this chapter. The idea of applying such products to amenability is due to Day. We start by looking at the fixed-point space 3, (G) for a locally compact group

The space 3,(G) is actually an abstract L-space ((2.31)) G acting on and is spanned by .C(G) ((2.2)). We then introduce the algebraic manipulations of the Arens type in the context of left invariant means. This leads to the theory of left introverted spaces, examples of which are Ur(G) ((1.3)) and the space AP(G) of almost periodic functions on G. Left introverted spaces are particularly well suited for Arens-type arguments. After pausing briefly to prove the basic facts concerning amenability and the space of (weakly) almost periodic functions, we prove topological amenability fixed-point theorems. We then discuss how means can be regarded as (countably additive) probability measures on a compact Hausdorff space X. When S is a discrete semigroup, X is just the Stone-Cech compactification /3S of S. We conclude by solving the sixth problem of (1.15): when does S admit a multiplicative left invariant mean? The elegant solution is given in (2.29): the semigroups in question are those for which every finite subset has a right zero. The class can also be characterised in terms of right zeros in,6S. We first discuss the relationship between .C(G) and the space 31(C) of left invariant elements of Of course, 3,(G) is a vector space containing .C(G) as convex subset. In some contexts 3,(G) is more appropriate than .C(G); for example, in cardinality questions it is sometimes more useful to look at the dimension of 31 (G) than at the cardinal IC(G)1. As we shall see, however, £(G) actually spans 3, (G), so that the study of 3, (G) usually "boils down" to that of .C(G).

Our first result is a simple application of the Hahn-Banach Theorem. References are Dixmier [1] and Folner [2). 51

THE ALGEBRA OF INVARIANT MEANS

52

(2.1) PROPOSITION. The space 31(G) ,-4 {0} if and only if

D(G)=Span{¢-Os:¢EL,,(G),sEG} is not norm dense in

PROOF. Let F E E and only if F(O - ¢s) = 0, it follows that

and s E G. Since sF(0) = F(4) if

31(G) = {F E L. (G): F(ti) = 0 for all tP E D(G)}. It follows from the Hahn-Banach Theorem that

D(G)- = {ti E L. (G): F( O) = 0 for all F E 31(G)}.

(1)

The result now follows.

0

Easy modification of the above proof gives that £(G) 0 0 if and only if III -'I{ > 1 for all 0 E D(G). The next result is due to Day [2) and Namioka [3). The same proof gives a corresponding result for other of (such as U(G)). (2.2) PROPOSITION. The space 31(G) is spanned by .C(G).

is a commutative C`-algebra, it can be identified with PROOF. Since We also identify C(4(C)), where 4(G) is the carrier space of with M((I(G)) by the Riesz Representation Theorem. Let v be a nonzero element

of 31(G) (c M(S(G))). Write

where vI and v2 are respectively the real and imaginary parts of v, and for i = 1, 2, vt and vi are respectively the positive and negative variations of V j. It is sufficient to show that each nonzero vt, vq is a multiple of an element in .C(G). (Note that since v 0 0, at least one of the vT, vt is nonzero.) By taking suitable scalar multiples of v, we need only deal with the case vi 0 0. Suppose,

then, that vi '40. It is sufficient to show that vi E 11(G) since then vi /Ilvi II will belong to real-valued, and s E C, then

.C(G). Now if ¢ E

v1(¢-Os) =Rev(O-Os) =0,

so that vi E 31(G). Further, vI = sv1 =svi - svi , and svi , sv > 0. By the minimum property of the Jordan decomposition, it follows that suit > vi ,

svi > vi. But then Ilsvi - vi II = (svi - vi)(1) = 0 since IIFII = F(1) for a positive functional F on vi E31(G) as required. 0

Thus suit = vi , and

The next result continues with the "Hahn-Banach" theme. It is essentially due to Dixmier [1]. We formulate the proof so that it applies in the semigroup case.


53

(2.3) PROPOSITION. Let G be a locally compact group and the space of real-valued functions in L. (G). Then G is amenable if and only if the following condition (*) holds:

(*) whenever n > 1, 01, ... , On E L,p(G, R) and 811... , sn, t1, .... to E G, then

ess sup

(1)

aEG

(i=1

- ¢:ti)(x) J > 0. isi

J

PROOF. Suppose that .C(G) 34 0 and let E L (G, R) and 81, ... , sn, tl, ... , to E G. Then form E ,C(G), m(l', (4isi - 4,iti)) = 0 and

(1) follows from (0.1(i)).

Conversely, suppose that (*) holds. Let A be the span in L (G, R) of the set of functions {(¢s - 0t): 4' E L (G, R), s, t E G}. If 0 E A, then -0 E A, and so is of the form Z= 1(0isi - 4iti). From (1), 111 - VII > 1. By the Hahn-Banach Theorem, we can find m E with m(A) = (0). Let so E G. Then som E 9Jt(L (G, R)) and for ¢ E L (G, R), s E G, som(4,s) = m(4,sso) = m(4,so) = som(¢)

so that som E R)). Now extend som to L., (G) in the obvious way to obtain an element of £(G). o We now introduce the Arens-type products mentioned earlier. The natural context for discussing this is that of a semigroup action on a Banach space.

(2.4) Notations. Let S be a semigroup and X be a right Banach S-space.1 This means that X is a right S-set, that each map f --y Cs (s E S) is in B(X) and there exists M > 0 such that (1)

116,911:5 MIIeII

for all sES,CEX.For each CEXand FEX',define FC(s) = F(Cs)

(s E S).

Of course, using (1), (2)

IIFFII Os from S into C(S) is weakly continuous is denoted by WUr(S). Thus if 0 E C(S), then 0 E WU,(S) if and only if F4, E C(S) for all F E C(S)'. Once we know that WUr(S) is left introverted, then it is evident that WU,(S) is the maximal, left introverted subspace of C(S). (When S is a locally compact group, then WUr(S) = Ur(S) ((2.33)).) A function 0 E C(S) for which S¢ = {80: 8 E S} is relatively compact in the norm [weak) topology of C(S) is said to be almost periodic [weakly almost periodic]. The set of almost periodic [weakly almost periodic] functions on S is denoted by AP(S) [WP(S)].

We now show that the four spaces U,(S), WUr(S), AP(S) and WP(S) are left introverted subspaces of C(S) (C l.(S)). Note that by the Hahn-Banach Theorem, a right invariant subspace A of C(S) is left introverted if and only if F0 E A whenever F E C(S)', ¢ E A. References for the following result are Eberlein [1], Glicksberg and de Leeuw [S], Burckel [S), and Berglund, Junghenn, and Milnes [1].

(2.11) PROPOSITION. Each of the sets Ur(S), WU,(S), AP(S), and WP(S) is an invariant, left introverted, closed subspace of C(S) containing 1.

PROOF. We omit the proof for Ur(S) as this is an easier version of the WU,(S) proof. It is obvious that WU,(S) is a subspace of C(S) containing 1. Let {4n) be a sequence in WU,(S) with On 0 in norm in C(S). Let F E C(S)' and s6 s in S. Then, for each n, [F(066 - 0,9)1:5 JF(4,s6 - 4'.s6)[ + [F(4,n36 - 4,ns)[ + [F(4'ns - 05)[ : [F(4,nsb - q5ns)I + 2[[4,n - 4'[[ [[F[[,

and it readily follows that F(¢s6 - 4,s) 0. So 0 E WUr(S), which is therefore closed in C(S). Also, if -0 E WU,(S) and so E S, then F((rbso)s6) =

F(s'sos6) -+ F((ayso)s), and F((soO)s6) = (Fso)(,0s6) - F((so,)s) so that WUr(S) is invariant. Further, if P E C(S)', then the functional Q, where Q(i,b) = F(P4/i), belongs to WUr(S)'. Thus

F((PVb)s6) = Q(bs6) - Q(tbs) = F((P1b)s) and Pay E WU,(S). So WUr(S) is left introverted.

We now turn to WP(S). If al, a2 E C and 01, 02 E WP(S), then S(a14,1 +a202) is contained in the sum of two weakly compact subsets of C(S),

and it follows that WP(S) is a subspace of C(S). Clearly, 1 E WP(S). Also,

TAE ALGEBRA OF INVARIANT MEANS

57

if 0 E WP(S) and s E S, then S(s4) C So, so that s¢ E WP(S). Since the map -+ abs is weakly continuous on C(S), it follows that ¢s E WP(S). So WP(S) is invariant. Now let 0,a - 0 in norm in C(S) with On E WP(S) for all n. To show that 0 E WP(S), it is sufficient, by the Eberlein-S`mulian Theorem [DS, V.6.1J to show that if {sk} is a sequence in S, then {44'} has a weakly convergent subsequence in C(S). The same theorem, coupled with the "Cantor Diagonal Process", gives a subsequence (salt)} of {sk} and, for each n, a function 0)n E C(S) with sa(t)4' -, Olin weakly as I -+ oo. If F E C(S)', then JF(On - bm)I = limt- JFs0(t)(4n - om)I 5 IIFIJ Jl0n - 4'm11, so that {0,,} is a (norm) Cauchy sequence in C(S) and so converges in norm to some b E C(S). A routine triangular inequality argument gives that s.(1)0 --> weakly. Thus ¢ E WP(S) and WP(S) is closed in C(S). We now prove that WP(S) is left introverted. Let 0 E WP(S). By the Krein$mulian Theorem [DS, V.6.4], the weak closure C,, of co(SiO) in C(S) is weakly compact. Since the norm and weak closures of co(SO) coincide and as WP(S) is norm closed in C(S), we have Co C WP(S). Let m E M(S), and let {ag) be a net in coS C II(S) such that as - m weak* ((0.1(i)). Since each &a' E C,1,, we can suppose that for some 0' E WP(S), {&60} converges weakly, and hence pointwise, to 0. Since &60 m pointwise, we have mO E WP(S). It follows ((0.1(ii))) that WP(S) is left introverted. 0

We will discuss the amenability properties of WP(S) and AP(S) in (2.36). For the moment, we turn to a result on left introverted spaces essentially due to Mitchell [21 and Granirer and Lau [1J. The result in question is (2.13).

(2.12) Discussion. Suppose that S is a (discrete) semigroup, and let B be a left introverted subspace of I.,, (S) containing 1. If m E .C(B) and 0 E B, then m4' = m(4')1. Approximating m weak* by elements in co S, we can approximate the constant function m(4')1 by convex combinations of functions of the form so (s E S). This is made precise in the semigroup version of the theorem below, which establishes the (more difficult) converse characterising the "amenability" of B in terms of the existence of a constant function in the closure of co SO in the topology of pointwise convergence on S. In the case where B is a subspace of L. (G), with G a locally compact group, there is an obvious problem with this result since we cannot define the pointwise topology on (as are equivalence classes of "genuine" functions on G). This is overcome by replacing co{b, : s E G} by P(G). (2.13) THEOREM. LetB be a left introverted subspace of L,,,, (G) with 1 E B. Then (i) .C(B)

0 if and only if, for each ¢ E B, there exists a constant function in

the closure C4, of the set {MO: p E P(G)} in the topology of pointwise convergence on G;

(ii) if .C(B) 96 0, then, for each 0 E B, (1)

{o E C: al E CO} = {m(¢): m E C(B)).

58


PROOF. (i) Suppose that £(B) 96 0. Let m E £(B) and ¢ E B. Then from the first equality of (2.5(v)), m(¢)1 = m4) E B. Now from (0.1), we can find a net {µ6} in P(G) with As -+ m weak' in Y. Using (2.5(vii)), p4) = A60 - mo pointwise on G. Hence the constant function m(o)1 belongs to CO. Conversely, suppose that there exists a constant function in Co for all ¢ E B. For each 0 E B, let D(4)) = {m E 1)1(B): m¢ E Cl) and

D= fl{D(4)): 0 E B}. Obviously, D C C(B), and so it is sufficient to show that D 0 0. Again, using (2.5(vii)), we see that each D(4)) is weak* compact in 91i(B). So we need only show that the family {D(4)): 0 E B} has the finite intersection property. Let 1, ... , 0, E B. Now if {p6} is a net in P(G) with A601 converging pointwise to a constant function, then every weak' cluster point of {µs} belongs to D(¢1). So we can find m1 E D(4)1). Now since B is left introverted, m102 E B, and so, by the above, we can find m2 E D(ml¢2).Then if m = m2m1 EJJ1(B), mq51 = m1q51 E C1,

m02 = m2(m) 4)2) E Cl

so that m E D(¢1) f1D(¢2). Applying this procedure as many times as required, we can find ml,..., m E 9)2(B) such that if m' = Mn . m2m1, then m'4)= is

constant (1 _< i C n). Thus m' E fl 1 D(4)j), and this concludes the proof of (i).

(ii) Suppose that £(B) 54 0. Let ¢o E B, and suppose that a E C is such that al E Coo. Now adapt the last part in the argument for (i) to show that the family {E(4)): ¢ E B} has the finite intersection property, where E(4)) = {n E D(¢): nqSo = a1}.

This produces m E £(B) with m(¢o) = a. The rest of the proof is obvious. Our next application of the "Arens product" idea relates invariant means to certain projection operators. Let S be a semigroup. (2.14) Discussion. Invariant means are useful for producing projection operators on a Banach space. Indeed, if X is a right Banach S-space, then the map F -> PF on X', given in (2.4), is, in a suitable sense, an invariant projection if P is a left invariant mean on X (S). This, and its converse, are established in our next theorem. An interesting application of this result to operator algebras is given in (2.35). Another application, dealing with the complementation of certain subalgebras of Loo (G), is discussed in Problem 2-20. Finally, we note that the result is related to the Mean Ergodic Theorem for amenable groups and semigroups in Chapter 5. Let X be a right Banach S-space with 11f sll < 11C11 for all e E X, s E S. Let

T8F = sF for F E X (a E S). Of course, T. E B(X') and lIT,ll < 1.


59

(2.15) THEOREM. Suppose that 1 E X(S). Then (i) and (ii) are equivalent.

(i) £(X(S)) 0 0; (ii) there exists P E B(X') such that (a) P(X') = 31(X), and (b) there exists a net {A6} in co{T,: s E S} such that A6F - P(F) weak' for all F E X'. If P satisfies (a) and (b) of (ii), then P is a projection (that is, P2 = P) and "P11 < 1.

PROOF. Let B = X(S) C 1,,(S). Suppose that (i) holds. Let m E £(B), and define

P(F)=mF

(1)

(FEX').

Clearly, P E B(X'). Now for s E S, E E X, we have

T,F(le) = (sF)(e) = F(es) = s(Fg). Now use (0.1) to approximate m by convex combinations of elements s and then

use (2.5(viu)) to establish (ii)(b). Clearly, P(X') is contained in 31(X) and PF = F for F E 31(X). This gives (i)(a) so that (i) implies (ii). Conversely, suppose that (ii) holds. Let P and As be as in (ii). Suppose that for 1 < i:5 n, we have as E C, Gi E X', and rli e X such that n

EaiGirli = 0

(in

i=1

n ai(T,Gi)(r/i) _ E 1 ai(Girli)(s) = 0, we have, for each ai(A6Gi)(r) = 0, and from (ii)(b), F,1 aiP(Gi)(rli) = 0. Hence we

Observing that

6, Zt

1

1

can define a linear functional m on B = Span{Fe: F E X', f E X} by setting

m(Fe) = P(F)(f)

(2)

For t<j 0 giving a contradiction. It follows that D is E-invariant.

(iii) Let K = PM(X), the set of probability measures in M(X). Then K is convex. Now X is compact, and so K is weak" compact in M(X). Also, in the obvious way, M(X) can be regarded as a subspace of the Banach E-space M(4?(E)).

Let p E K. We show that sit E K. Indeed, if U is open in 4i(E) with Uf1X=0, and u.EU,then we can find 0ELa,(E)with 0 2 and subsets A1,. .. , AN- 1 of S such that (i), (ii), and (iii) of (2.28) hold. Let m E .C(S) be multiplicative. Then for 1:5 i < N - 1, we have, using (2.28(iii)), m(Ai+2) = m(x-2Ai+2) >- m(Ai), and it follows that m(Ai) = 1/(N - 1) for all i. But m(Ai)2 = m(XA,)2 = m(XA) = m(Ai) so that m(Ai) is 1 or 0. This is a contradiction. So S is a singleton, and trivially every finite subset of S has a right zero. Cl (2.30) COROLLARY. The only right cancellative, ELA semigroup is the trivial singleton semigroup.

A more general version of the ELA condition is the n-ELA condition: a semigroup S is n-ELA if there exists a finite, left ideal group G in fiS with )G) = n. Thus ELA = 1-ELA. Problem 2-35 gives an algebraic characterization of n-ELA semigroups.

References Berglund, Junghenn, and Milnes [1), Day [2), [4], [5), Dixmier [1), Folner [1), [3),

Furstenberg [1), Granirer [5)-[7), Granirer and Lau (1), Lau [5), [8), de Leeuw and Glieksberg [1), [S], Mitchell [2), [6), Narnioka [3), von Neumann [1), Rickert [2), Wong [1), [4).


67

Further Results (2.31) On the L-space structure of 3,(E). Let E be a semigroup or a locally compact group. Instead of considering the complex vector space 31(E), we consider 31(E,R), the real subspace of 11(E) spanned by C(E). (In fact, use of (2.2) (or an easy direct argument) shows that 31 (E) is the complexification of (E, R).)

We will show that 31(E, R) is an abstract L-space. We briefly recall the definition of such spaces. We state the axioms in the form most suitable for our purposes. A good account of this topic is given in (Sch). A real Banach space X, with an ordering 0, we have

(a) if x p(A) = 1.

Let pB be the measure obtained by restricting y to.9(B) and vB = pBfp(B). Then we can identify vB with a mean on X and n

n

0:5 LB E aiXs,A = L.i aivB(SiA) _ (E i.1

i=1

s-1

1

ail

Ip(B)

Thus (a) implies (b). Conversely, suppose that (b) holds. Let 1 (X, R) be the space of real-valued functions in and C the linear span in l,o(X,R) of the set {XBA: s E G}. E G are such Suppose that al,... , E R and

that

n

m

E iXs;A = EYjXt,A j=1

iq=1

Then (E 1 c Xs;A-Zj 1 RjXt5A) = 0 2! 0, and s0 by (b), 1 ai-gym 1 Yj > 0. Similarly, Ly'--1 Qj - E 1 ai >_ 0, and so Z", ai = Z9 1 Ij. It follows that we can define a positive linear functional p on C by defining n

p

aiXs,A icl

= E ai. i=1

It is readily checked that C is a right invariant subspace of 1 (X, R) (under the usual G-action) and that p is right invariant. Let us say that a subset E of X is bounded if it is contained in the union of a finite number of translates sA, and that f e 1 (X, R) has bounded support if it vanishes outside a bounded set. Let D be the subspace of such functions R). f, and note that C C D and that D is a right invariant subspace of

Give the algebraic dual Da of D its "weak" topology-this topology is, of course, locally convex, and G acts affinely on D'1 with the dual left action. Let K = {# E Da : /3 is positive, ,Dic = p}. Clearly, K is convex and G-invariant. Krein's Extension Theorem ((R!, p. 2271, Peressini (S, (2.8)1) gives that K 0.

Finally, K is weak' compact-indeed, the usual proof of Alaoglu's Theorem


69

[DS, V.4.2] applies directly, noting that if 6 E K and if f E D vanishes outside a set of the form E = U 1 siA, then [$(f)[ >-1

for all 0 E B.

Define j E l (P, K) by j(n) = nl

(n E P).

Then II j 11 = 1. It is interesting that j plays such a central role in what follows.

(ii) .C(lo,(P,K)) = 0. (Suppose that m E Then m(TO) = m(¢) for all 0 E Thus as T j (n) = (n + 1)1 = j (n) + 1,

0=m(Tj-j)=m(1)=1 giving a contradiction.]

If Y is a linear subspace and g an element of a Banach space X over K, then

we say that e 1 Y if

Of -'7II- Hill for all rl E Y. The notion of orthogonality is, of course, familiar in the Hilbert space context. Our next result characterises those subspaces B of is not empty.

for which .C(B)

(iii) Let K be spherically complete, and let B be an invariant subspace of l (P, K) which contains 1. Then .C(B) ¢ 0 if and only if j 1 B. {Suppose that Z (B) 96 0 and let m E e(B). Suppose that it is not true that j 1 B. Then there exists ¢ E B such that jlj - 011 < 11j 11 = 1. So there exists e > 0 such that In1-¢(n)I E' such that for C E X, the map x p(x) (g) 1 >_ 11011 = ess supZEX IIp(x)II, and 4 ()(x) = p(x)(g) a.e. x. In is in particular, p(x) E El a.e. We claim that p is the required -y-invariant section so that X is G-amenable. We have to show (f) for each a E G, -y` (x, a) p(x, a) = p(x) a.e. and (g) p(x) E A. a.e. We will prove (f) first. For any essentially bounded, measurable function -i: X E, define (',S) E L,O(X x G) by {-i,S)(x,a) = S(x,a)(O(x)). If E X, E L,. (X), and O(x) = ¢(x)£, then using (c),

a((b,S))(x)=of{ p(x)(i'(x)) a.e. By approximating general I, by linear combinations of functions of the form we obtain that x(3)

o((4,S))(x) = p(x)(O(x)) a.e.

Applying (3) with 1,(x) = -y(xa-1, a)-1 and using (d),

p(xa)(-y(x,a)-1e) = p(xa)(Il'(xa))

(4)

= u(( , S))(xa) = o(a(IJi, S))(x). Now for x E X, b E G, we have, using the cocycle identity for -y,

a(,S)(x,b) _ (b,S)(xa,ba) = S(xa,ba)(1b(xa)) ry'(xa,a-Ib-I)c(xb-1)(-p(x, = c(xb-I)((-y(x,a)-r(xa,a lb-1))-1t) = c(xb-I)(- (x,b-1)-lf) _ {e,S)(x,b). So using (4), we obtain (f): ry'(x,a)p(xa)(e) = o((,S))(x) = p(x)(e)

a.e.

We now turn to (g). If K is a weak* compact, convex subset of E' and a E E' K, then a and K are separated in the sense that there exists e E E such that Re Vi(a) < inf {Re (Q) : 0 E K}. From Problem 2-30, there exists a sequence

84


{bra} of Borel functions from X into EI such that for a.e. x, {bn(x): n > 1} is weak' dense in A. Let Y be a countable dense subset of E, l; E Y, q E Q, and

Xo= {,EX: infn>IRe &(bn(x))>q}. If, for some x' E X, p(x') 0 Ax,, then we can separate p(x) from Ax, using some e E Y, and noting that infra>I ReE(bn(x)) = inf{£(p): I E Ax} for a.e. x E X, we will have p(x) E A. a.e. provided we can show that {x E Xo : e(p(x)) >_ q} has null complement in Xo. To this end, we can obviously suppose that jt(Xo) > 0.

Since S(x,a) E A. a.e., we have for almost every x E X0, Re(C, S) (x, a) = Re e(S(xa)) >_ q. Using (c) and the fact that a is positive and unit preserving, we have Rea((E,S)xxa 0 1) ? a(gxx0 0 1) = gXX0, so that for a.e. X E Xo, Ret;(p(x)) =Rea((e,S))(x) > q. This gives (g).) In the case where X is a singleton, the above result applies even when G is not countable, and becomes a well-known result of Schwartz [1): A discrete group G is amenable if and only if VN(G) has Property P. (Schwartz extended this result in Schwartz [S].) It is natural to ask if the above result in Schwartz [1J is valid for general locally compact groups G. It follows from (C) that G amenable . VN(G) has Property P. Since there exist nonamenable groups of Type 1 (for example, SL(2, R)), the converse is certainly not true. The quest for the missing ingredient that, together with Property P, will ensure that G is amenable leads to the study of a weaker kind of invariant mean on G that is particularly relevant for operator algebras. The means in question are inner invariant means.

(H) Inner invariant means. A mean m on L,o((j) is called inner invariant if m(xox-I) = m(o) for all 0 E Lw(G), x E G. Inner amenability is definitely an Lam-phenomenon-the mean 6e is always inner invariant on C(G), and since there exist familiar groups (such as SL(2, R)) that are not inner amenable, there is no hope of an inner invariant version of (1.10) holding! Every invariant mean is inner invariant, so that all amenable locally compact groups are inner amenable. In general, inner amenability is much weaker than amenability.

With our definition, every discrete group G is inner invariant with be an inner invariant mean. E. Effros, whose paper [1] originated the study of inner invariant means, defines a discrete group to be inner amenable if there exists an inner invariant mean m on l,o(G) with m 0 be. This fits in well with Property r of von Neumann algebras (Problem 2-33). However for locally compact groups in general, our definition of inner amenability seems preferable. We shall say that

a discrete group G is trivially inner amenable if 6e is the only inner invariant mean on lo,(G). There is now substantial literature on inner amenability-see, for example, Akemann and Walter (3), M. Choda [Si], [S2], H. and M. Choda [S), Bedos and de la Harpe [S), Lau and Paterson [S2], Losert and Rindler [S2), [S3], Paschke [1), Pier [S), and Yuan [S]. Some of these results are covered in Problems 232-2-34. Losert and Rindler [S2) (and Grosser, Losert, and Rindler [S]) show


85

that a connected locally compact group G is inner amenable if and only if G is amenable. (The class of such groups is determined in (3.8).) More generally, amenability and inner amenability are equivalent for G in the class sad of (1.31). Indeed, if G E sad, then the amenability of C*(G)" implies that VN(G) is also amenable, and the result below applies. This result shows that inner amenability is the "missing ingredient" alluded to above. (Lau and Paterson). The locally compact group G is amenable if and only if VN(G) has Property P and G is inner amenable. [If G is amenable, then G is inner amenable, and by (C), VN(G) has Property P. Conversely, suppose that VN(G) has Property P and that G is inner amenable. Let m E 271(G) be inner invariant. Then mm = mx for all z E G, and the usual (0.8) theme gives a net {f6) in P(G) such that f Ix * f6 - f6 * x111 0 for each x E G. Let g6 = f6112 and note that 1196112 = 1. Let 7r, be the right regular representation of G on L2 (G): so 7r, (x)g(t) = A(x)1/2g(tx) (x, t E G, g E L2 (G)). Then (x * f6)1/2 = 7r2(x)g6, (f6 * x)'/2 = 7rr(x-1)96, and (cf. (4.3(1))) -'rr(x_1)96112 117r2(x)96

0, and let 60 be such that 1172(x 1)g6 - 7r,(x}96112 < 'whenever 6 > 6o. For 0 E LA(G), U E H, we have Lot-, = 7r2(x)L07r2(x-1), and so, since U and 7r2(x) commute,

I6(0x_')(U) = (UL0z-iU*96,96) = (U7r2(x)Lm r2(x-')U*gs,g6) = (UL#U*7r2(x-')96,1r2(z-1)96) Also WY6(¢)7rr(x-1)(U) = (ULdU*7rr(x)g6,7rr(x)g6} and so by asimple triangular inequality argument, for 6 > 6o,

< 21101111.2(x-')96 - 7r,- (X)96112 0 in A, there exists a G-invariant element a E A. such that a(a) > 0. Lau (9] investigates this topic using almost periodicity ideas. See also Stormer [1], Doplicher, Kastler, and Stormer [1], and Komlbsi [1].

(2.37) On the range of a left invariant mean. Let S be a left amenable semigroup and m E .C(S). The range R,,, of m is the set {m(A): A C S}. Obviously, R,,, C 10, 11. Granirer (8] essentially made the following conjecture:

R,,, = [0, 1] for all m E £(S) if and only if S is not n-ELA for any n. Granirer established the conjecture for all cases except when S/x is an infinite, periodic group with the property that each infinite subgroup is not locally finite (where r is as in (1.24)). Chou [3] established the conjecture when S is right cancellative. The truth of the conjecture in general follows using Chou's method.

(i) Let X be a set, m E 9Y (X) and suppose that the measure th on ;6X (defined as in (2.24)) is continuous, that is, m({p}) = 0 for all p E fX. Then {m(A) : A C X} _ 10, 11.

[By (HR.1, (11.44)), if E E R (,8X), then (1)

[0, 7%(E)] = {yn.(F): F E R (,OX), F C E}.

88


Let B C X. Now j3X is zero-dimensional. Further, every open and closed subset

of pX is of the form A-, where A C S and A- is the closure of A in fX. It follows, using (1) and the regularity of m, that (2)

[0,m(B)] = {m(A): A C B}

Let a E (0,1) and let {an} be a sequence in (0,1) such that a

a, and for all

n, (3)

a2n-l > a2n+l > a > a2,+2 > a2n

We construct recursively a sequence {A,,) of subsets of X such that for all n > 1, (4)

a2n+1 < m(A2n_l) < a2.-1, a2n < m(A2n) < a2n+2

and

A2n-3 D A2n-1 D A2n A2n-2 Suppose then that A1,. .. , A2k (k > 1) have been constructed so that (4) and (5) are valid whenever they make sense for 1 < n:5 k. The following argument easily adapts to give the sets A1i A2 to start the recursion. From (4) and (3), (5)

we have

m(A2k-1) > a2k+1 > a2k+3 > a > a2k+4 > a2k+2 > m(A2k) Applying (2) with B replaced by A2k-1 -r A2k, and noting that, from (4) and A2k), we can find (6), 0 < a2k+3 - m(A2k) < a2k+1 - m(A2k) < m(A2k-1 C C A2k-1 - A2k such that (6)

a2k+3 - m(A2k) < m(C) < a2k+l - m(A2k) Setting A2k+1 = A2k U C, we obtain

a2k+3 < m(A2k+1) < a2k+l

Similarly, we can find D C A2k+1 ^- A2k such that a2k+2 - m(A2k) < m(D) < a2k+4 - m(A2k), and setting A2k+2 = A2k U D, we obtain a2k+2 < m(A2k+2) < a2k+4.

So (4) and (5) are valid whenever they make sense for 1 < n < k + 1, and the construction of the sets {An} is completed. Now let A= f,°=1 A2n_1. From (5), A2n_1 D A A2n for all n, so that, using (4), a2.-1 > m(A) > a2n, and since an -t a, m(A) = a. This completes the proof.]

(ii) The range Rm of every left invariant mean on a left amenable semigroup S is [0,1) if and only if S is not n-ELA for any n. (Suppose that R,n = (0,1] for all m E C(S). If S is n-ELA for some n, then we can find m E C(S) such that fn = n =1 by; for some pi E /S, and Rm = {r/n: 0:5 r < n}. A contradiction results. Conversely suppose that S is not n-ELA for any n, and let m E C(S). By Problem 2-35, 7h is continuous. It follows from (i) that R,n = [0, 1].)


89

(iii) Granirer [6] showed that for "most" infinite, right cancellative, left amenable semigroups S, there exists a family {A(t) : t E [0, 1]) of subsets of S such that (a) A(s) C A(t) if s < t, (b) m(XA(t)) = t for all m E £(S). (In particular, XA(t) is left almost convergent.) Further results in this direction are obtained by Snell [S1], [S2].

Problems 2 Throughout G is a locally compact group and S is a discrete semigroup.

1. Let D(G) be as in (2.1). Show that G is amenable if and only if D(G) is not norm dense in L. (G), and that if G is amenable, then

D(G)+C1 = {0 E L. (G): m(¢) = n(¢) for all m, n E £(G)}. 2. Let B be a left introverted subspace of L,o(G) which is right invariant for P(G), is closed under complex conjugation and contains 1. Prove that there exists a topologically left invariant mean on B if and only if whenever 0 E B is real-valued and p, v E P(G), then nf(OA - ¢v)(x) < 0. mEG

3. Let 31t (G) _ {F E L.. (G)': 1tF = F for all p E P(G)}. Prove that 3,t (G) is an L-space and that £t(G) spans 31t(G) (cf. (2.31)).

4. Show that C is amenable if and only if whenever n > 1, E',.. . , En are Borel subsets of G, and x1, ... , xn, Y], .... Yn E G, then n

esEGf {=1 5.

-Xy-.E)(x) < I.

Show that G is not amenable if and only if there exists a real-valued

function 0 E UT(G) and, for some n, elements ri, si (1 >1. i=1

6. Let A be a Banach algebra. Show that the second dual A" of A is a Banach algebra under the following Arens product: if F1, F2 E All, f E A', and a, b E A, we define F1 F2 (f) = F1(F2 f ), where F2f E A' is given by F2 f (a) = F2 (f a) with fa(b) = f (ab). (There is another Arens product on A" but this will not concern us.) Show that the Arens product on 11(S)" coincides with the Arens product on 1.(S)' obtained by regarding 1 (S) as a left introverted subspace of itself.

Interpret the Arens product on the second dual of a Banach algebra A as a special case of the canonical product ((2.8)) on the dual of a left introverted subspace B of some 1o,(S).


90

7. A mean m on is said to be inversion invariant if m = m*, where m* (0) = m(¢*) and 0* (x) = O(x-I). Show that if G is amenable then G admits an inversion invariant mean in 0(G).

8. Let p E M(G), F1, F2 E L,, (G)', and let L,, (G)' = LI(G)" be given its Arens product. Recall that L,(G)' is an M(G)-set ((1.1)). Prove that (i) p(FIF2) = (zFi)F9, (Fjp)F2 = Fi (iF2), (FIF2)p = FI(F2R); (ii) V) (G) is a subsemigroup of (iii) if Q E 31t(G) (see Problem 2-3), then FIQ = FI(1)Q.

9. Prove that AP(S) C U(S), where S is a separately continuous semigroup. 10. Let S be an open, generating subsemigroup of G. Show that if .C(U(S)) j4

0, then there exists m E £t(G) with m(S) = 1. Deduce that if S is an open subsemigroup of G and G is amenable, then £(U(S)) 96 0 if and only if S is left reversible. (These are topological versions of (1.27) and (1.28).) 11. Let Y be a left Banach S-space with l{syll 5 Ilyll for all y E Y, let s E S,

and let X = Y. Suppose that I E X(S) and C(X(S)) # 0. Suppose further that yo E Y and that the weak closure Cv, of co Syo is weakly compact in Y. Show that there exists an S-fixed-point in Cyo. 12. Let Y, X, yo and C90 be as in Problem 11. Show that if m E M(S), then rnya E (C90)".

13. Let B be a left invariant, closed subspace of t (S) containing 1. Let E B be such that the weak closure Co of co S¢ is weakly compact in B. Deduce that m¢ E Co (C B) for all m E OJt(S).

14. Let C be a-compact. Prove that G is amenable if and only if there exists a G-invariant probability measure on every (jointly continuous), separable, compact metric left G-space. Show also that C is amenable if and only if, whenever E is a separable right Banach G-space with jointly continuous, isometric action, then every G-invariant, weak* compact, convex subset of E' contains a fixed-point for G. 15. Let S be a separately continuous, locally compact Hausdorff semigroup and B a left introverted, closed subspace of C(S) containing 1. Prove that B is right invariant for PM(S) (Problem 1-11) and that every left invariant mean on B is also left invariant for PM(S). (A version of this result for the case B = Ur(G) was proved in (1.8).)

16. Let S be separately continuous, and let 9e be the family of sets K E ((2.21)) for which the set of maps k -. sk (a E S) is equicontinuous with respect to the (unique) uniformity for K which gives the topology of K (Kelley [2]). (i) Prove that OJt((AP(S)) E Xe1 (ii) Prove that there exists an S-fixed-point in every member of 3e if and only if .C(AP(S)) 3& 0.


91

17. Let (Z, 9, v) be a o,-finite measure space such that Z is a left G-set with G a-compact and (i) ez = z for all z E Z; (ii) the map (x, z) -+ xz from G x Z into Z is a measurable transformation

from (G x Z,4f(G) x0) into (Z,3); (iii) v(zE) = v(E) for all x e G, E ER. (a) Show that the map f -+ f . x (x e G), where f . z(z) = f (xz), is an isometric right Banach space action of G on LI (Z), and that the dual left action of G on is given by x . 4>(z) = O(x-1z).

(b) Let K be a nonempty, weak" compact, convex, 0-invariant subset of Prove that if G is amenable, then K contains a G-fixed-point.

18. Let K be a compact, convex subset of a Banach space X and S a left amenable semigroup of nonexpansive maps from K into K (so that jjsz - syjj < jjz - yjj for all s e S, x, y E K). Let M be a compact subset of X. Show (i) if jMj > 1, then there exists n E co M such that

sup jju - mjj < diam M; mEM (ii) there is an S-fixed point in K. 19. If S is a separately continuous, left reversible semigroup, show that there exists a left invariant mean on AP(S). 20. Let B be a norm closed, right invariant subspace of The space B is called invariantly complemented if there exists a right invariant closed subspace

C of L (G) such that L (G) = B 9 C (vector space direct sum). (i) Prove that B is invariantly complemented if and only if there exists a continuous, linear projection P from L,0(G) onto B such that P(4>x) = P(4>)x

(¢ E L,, (G), x E G).

(ii) Now suppose that B is also a weak" closed. C*-subalgebra of

(a) Show that if p E P(G) and {µa} is a bounded approximate identity for Ll (G) in P(G), then for each 0 E B, ¢µ6 -+ 0, µs0 - 0 weak', and Op E B.

(b) Show that ifB0{O}, then IEB. (c) Show that if B 0 {0}, then N = {x E G: z4> = 0 for all 0 E B} is a closed

subgroup of G, and B = {4,EL.(G):z¢=0forallxEN}. (iii) Prove that G is amenable if and only if every right invariant, weak" closed, C'-subalgebra of L,, (G) is invariantly complemented.

21. Show that Z = {p E Jag: the map q -+ pq is continuous on J3Z}.

22. Show that .C(G) is weakly compact in

if and only if 31(G) is

finite-dimensional.

23. Let G be amenable as discrete. Show that there exists a finitely additive positive measure p on PE(G) such that p(xE) = µ(E) for all z E G, E C G and the restriction of p to ,if (G) coincides with A.


92

24. Let H be a group which is also a locally compact Hausdorff space with separately continuous multiplication. Show that H is a locally compact group. 25. Let K be a complete, non-Archimedean field which is spherically complete.

Show that (a) there exists a maximal invariant subspace B,,, of t (P, K) containing 1

and such that e(B..) i4 0; (b) if

is maximal in the sense of (a), then £(Bm) is a singleton.

In Problems 26-28, A, fj, X, and H are as in (2.35(C)). 26. Let H be given the strong operator topology. (i) Show that H is a topological group. (ii) Show that X(H) C U1(H).

27. Let A be AFD. (So A contains an increasing sequence {Mn} of finitedimensional C'-algebras with U 1 M,, strongly dense in A.) Show that Z(Ur(H)) 36 0. Deduce that AFD implies Property P. Show that if A = B(fj), then A is AFD yet .C(C(H)) = 0. 28. Prove that if A is abelian then A has Property P. More generally, show that if A is of Type 1, then A has Property P. 29. Prove that a Type 1, discrete group is amenable. 30. Let x -+ A. be a Bore] field of weak' compact, convex subsets of r, as in of Bore] functions from X to (2.35(G)). Show that there exists a sequence E;, such that {bn(x): n > 1}- = AZ a.e. (Hint: use the von Neumann Selection Theorem.)

31. Let A be a unital, nuclear C'-algebra of operators on a Hilbert space i5. Show that A` is injective. Deduce that A" is injective. 32. (i) Show that F2 is trivially inner amenable. (ii) Let it be the "inner regular" unitary representation of a discrete group G on 12(G): so 7r(x) f (t) = f (x-ltx). Let C, (G) be the C'-algebra generated by ir(G) and Pe the orthogonal projection from 12(G) into Ce. Show that G is not trivially inner amenable es P, 0 C, (G). 33. Let G be a discrete group. Prove that G is not trivially inner amenable if VN(G) has Murray and von Neumann's Property r, that is, given T1, ... , T, E

VN(G) and e > 0, there exists U E U(VN(G)) such that r(U) = 0 and IjUT,U' - TjJ 2 < e (1 < j < n), where r(T) = (Te, e) and JITI12 = r(T`T)1/2 (T E B(12(G))).

34. Let G be a locally compact group, and let vr,,: G x¢x-1. Let AC', be the commutant of in

be given Prove that G is inner amenable if and only if AC', contains a nonzero compact operator. by


93

35. Let S be a semigroup. (i) Show that there exists m E £(S) with th not continuous if and only if pS contains a finite left ideal [group]; (ii) Prove that if S is left amenable, then S is n-ELA ((2.30)) if and only if n is the smallest positive integer with the property that whenever F E -47-(S), then there exists E C S with JEJ < n and FE C E; (iii) Show that S is n-ELA if and only if S is left reversible and S/-- ((1.24)) is a group of order n; (iv) Give examples of n-ELA semigroups.

36. Prove that S is ELA if (i) S is a semilattice; (ii) S is the set of transformations s on an infinite set X for which the set {z E X : sx 0 x} is finite; (iii) S is the semigroup of all nonempty, countable subsets of a group G with multiplication given by (A, B) --* AB; (iv) S is the Cartesian product of a family of ELA semigroups;

(v) S = P x P with multiplication given by

(ml,nl)(m2,n2) = (m2,n2) = (m2,n2)(ml,n1) (ml,nl)(m1,n2) = (m1,n1 +n2) 37. Prove that if S is ELA, then

(ml < m2),

ExtL(S) _ {p E $S: Sp = {p}}. 38. Prove that a subset E of a left amenable semigroup S is left thick if and only if the closure t of E in J3S contains a left ideal of fS. 39. (An amenability fixed subspace theorem.) Let X be a Banach space and H a closed linear subspace of X of codimension n < oo. Suppose that Y C X

is such that Y fl (g + H) is a compact convex set for all £ E X and that Y contains an n-dimensional linear subspace of X. Show that if G C B(X) is a group of invertible transformations on X that is. amenable (as a discrete group) and that leaves Y invariant, then there exists an n-dimensional subspace L of X

with LCY andGL=L.

CHAPTER 3

Free Groups and the Amenability of Lie Groups (3.0) Introduction. The simplest example of a nonamenable group is the free group F2 on two generators. In this chapter, we explore the extent to which nonamenability is related to the presence of F2. We concentrate our discussion on groups that are either connected or "nearly" connected, leaving the discrete case (von Neumann's Conjecture) to (4.30)-(4.33). The main results of the chapter are the theorems (3.8) and (3.9). The result (3.8) characterises amenable, almost connected groups. (A locally compact group G is almost connected if GIGe is compact.) The result (3.9) determines which connected Lie groups are amenable

as discrete groups. These results are proved by a reduction process, the last steps of which involve considering the Lie groups PSL(2, R) and SU(2). The reduction process requires the introduction of the radical of a locally compact group ((3.7)), and uses an important theorem ((3.4)) on the identity component of the automorphism group of a compact group. The cases of PSL(2,R) and SU(2) are dealt with using (3.2): PSL(2, R) contains F2 as a discrete subgroup, and SU(2) contains F2 as a subgroup. The following simple lemma is useful when discussing the existence of free subgroups in a group.

(3.1) LEMMA. Let G and G' be locally compact groups and ': G G' a continuous homomorphism. If 1P is surjective and G' contains F2 as a [discrete) subgroup, then so also does G. If G contains F2 as a subgroup and ker t C Z(G), the centre of G, then G' contains F2 as a subgroup.

PROOF. Suppose that D is surjective and that G' contains a subgroup H' that is a free group on two generators u', V. Let u, v E G be such that $(u) = u',

4a(v) = V. Then the subgroup of G generated by u and v is free and, by the continuity of ', is discrete in G if H' is discrete in G'. Now suppose that kerb C Z(G) and that G contains a subgroup H which is free on two generators. Then

H fl Z(G) C Z(H) _ {e)

so that F2. o

E H is one-to-one. Hence the subgroup 4 (H) of G' is isomorphic to

95

FREE GROUPS AND THE AMENABILITY OF LIE GROUPS

96

B references in the following are references to Appendix B. Recall (B19) that

if n > 1 and F is either R or C, then the Lie groups SL(n, F) and SU(n) are defined as follows:

SL(n,F) = {A E GL(n,P): detA =1}, SU(n) = (A E SL(n, C) : A*A = I = AA'), where det A and A' are the determinant and adjoint of A respectively. Of course, SU(n) is a compact Lie group. We shall also be concerned with the group PSL(2, R) of linear fractional transformations z -f

az+b cz+d

defined on C°O such that a, b, c, d E R and (ad - bc) = 1. It is elementary that PSL(2, R) is the group of linear fractional transformations preserving the upper half plane.

Now, in a natural way, PSL(2,R) can be identified with SL(2,R)/{-I,I}. Indeed, the map Q, where

Q{

c

b

dj)(z)=cz+d'

is a homomorphism from SL(2, R) onto PSL(2, R). Of course PSL(2, R) is given the quotient topology. Under this topology PSL(2, R) becomes a Lie group. It is readily checked that for each z E Coo, the map T -+ T (z) from PSL(2, R) into COO is continuous.

We shall use the elementary result (Rudin 13, Chapter 14]) that if C is a circle in the complex plane and T E PSL(2,R), then T(C) is either a line with 0o included or is, again, a circle, and that in the latter case, T maps the interior of C either onto the interior or onto the exterior of T(C) in COO. As we shall see later, the question of which connected Lie groups are amenable (as discrete) reduces to the consideration of PSL(2, R) (SU(2)). The next result shows that F2 is a subgroup (discrete in case (i)) of PSL(2, R) (SU(2)), and we will be able to deduce the same result for every connected Lie group which is nonamenable [as discrete]. The proof of (i) was pointed out to me by Peter Waterman and probably goes back to Poincare. A reference for (iii) is Dekker (1); it can also be deduced from the result of Tits stated in (3.10). (3.2) PROPOSITION. (i) PSL(2,R) contains F2 as a discrete subgroup. (ii) The centre of PSL(2, R) is trivial. (iii) SU(2) contains F2 as a subgroup.

PROOF. (i) Let C1, Cl , C2, CC be four circles in the complex plane such that each is exterior to all of the others, and, for i = 1, 2, let Ti be a linear fractional transformation in PSL(2, R) carrying Ci onto Cj and the interior (exterior) of C= onto the exterior [interior] of C. For example, we could take C1, Ci , C2, C2' to


97

be the circles [z + 21 = 1, [z - 21 = 1, [z + 51 = 1, and Iz - 51 = 1, respectively, and 2z+3 T2 ( Z ) 5z+24

T

i(

z) =

= Z+5

z+2 '

Obviously, if z E C is exterior to both the circles C,, Cj', then T,(z) is in the interior of C' and T,-1 (z) is in the interior of Q. Further, T; (z) and Ti -I (z) are exterior to both circles Cs, C? where j # i. Let zo be exterior to all of the four circles, and let H be the subgroup of PSL(2,R) generated by TI and T2. An easy induction argument on the length of words now shows that if To E H corresponds to a nontrivial reduced word in {TI, T;-I, T2i TZ I } starting with T, [Ti-'], then To(zo) belongs to the interior of Ci' [C;}, and so

[To(zo) - zol > d > 0, where d is the distance of zo from CI U Cl' U C2 U C. Since the function T -+ T(zo) is continuous on PSL(2, R), it follows that H is a discrete subgroup of G. Clearly, H is isomorphic to F2.

(ii) This fact is elementary. Suppose that T E Z(PSL(2, R)). Let Q: SL(2, R) -r PSL(2, R) be the quotient map, and let A E SL(2, R) be such that Q(A) = T. For all B E SL(2, R), ABA- IB-I E {I, -I} so that

ABA-' E RB

(1)

(B E SL(2, R)).

By scaling, (1) is true for all B E E, where

E={CEM2(R): detC>0}. Now E is an open subset of M2(R). By considering eigenvalues for the linear transformation X - AXA-I (X E M2(R)), we see that there exists \ E R and an open, nonvoid subset U of M2(R) such that for all X E U,

AXA'I = \X.

(2)

Since U spans M2 (R), (2) is true for all X E M2(R), and by putting X = A in (2), we obtain k = 1, and A E Z(M2(R)) n SL(2, R) = RI n SL(2, R) = {I, -I}. Thus T is trivial. (iii) We first choose a certain angle 0 and then an element z E T. We then define, in terms of 9, z, two matrices A, B E SU(2) that generate a copy of F2 in SU(2).

Choose 0 E R - {0} so that 7r/9 is irrational. Let E be the set of all finite products in R of elements from the set

{cosr9:rEZ}U(sin s9:SE Z^-{0}}. Since 7r/9 is irrational, it follows that 0 ¢ E. Let S be the (countable) additive subsemigroup of R generated by E. Let FE be the set of functions f : T - C given by a formula of the form

(wET), n=p


98

where p < q in Z and every an E S. The function f is said to be nontrivial if an 00 for some n96 0. For each nontrivial f E FE, the set {z E T: f (z) E {0,1}} is finite, and since S is countable and T uncountable, we can find z E T such that for all nontrivial f E FE, f (z) 0 10, 11. Now let A, B E SU(2) be given by sin 8

cos 8

A= -sin8 cos8j'

_z

B- [0

0

z-1 Let L be the subgroup of SU(2) generated by A and B. It will be shown that L is free on the generators A, B. We start by obtaining information about an element T E L of the form An' Bm' ... Ank Bmk,

(3)

where k > 1, and ni, m, E Z. First, an elementary calculation shows that zm3 cosnl8 z-m, sin n18 An' 'B"1 = (4)

-z'ni sinnl8 z mi cosn1O

To find an expression for (3) for general k, we can proceed as follows. Let R be the set of k-tuples r = (ri, ... , rk ), where, for each i, ri is either mi or -mi. Define

Y={rER:rk=-mk}. X = {r E R: rk = mk}, 1 ri. Note that different is can give the same For each r E R, let a(r) = a(r). By either using (4) and induction on k, or the formula for the (i, j)th component of a matrix product, we can write (5)

T=

aTza(r)

1 E,Ex L EEX brza(r)

[..rEY C,z*(r)1

ErEY

_ [p(z) q(z)

s/(z) 1

t(z)

where a b c d, E E U {0} and the functions p, q, s and t belong to FE. a product of terms from the set {cosni8: 1 < i < Each of k} U {sin(fni8) : 1 < i < k} and so belongs to E if ni i6 0 for all i. We now prove that L is free on A, B. Let w be a nontrivial reduced word

in {A, B, A-1, B-') and T the element of L corresponding to w. It has to be shown that T 96 I. We can suppose that T is not of the form AnI B'"3 since (4) and the choice of z, 8 deal with the latter case. This leaves four cases to be considered: (a) T = A" B" ... An-, B'-', (b) T = An' Bm' ... Bmk_ I Ank (c) T = Bm' Ana Bma ... Ank Bmk (d) T = Bm' Ana ... Bmk-' Ank

where k > 2 and every ni, mi is nonzero.

Suppose that T is as in (a). Then ro = (1mjI,...,fmkl) belongs either to X or Y. Suppose that ro E X. Then, referring to (5), the highest power of z occurring in the expansion of q(z) (and p(z)) is za(,0), and since every mi 0 0, ro is the only r E X with a(r) = a(ro). Hence the coefficient of z°(r-) in the


99

expansion of q(z) is o,.. Since n, # 0 for all i, a,., 0 0. Thus q is nontrivial in FE, and by the choice of z, q(z) i4 0. Hence T # I. A similar argument applies with re E Y, q being replaced by a. (With case (c) in mind, we have chosen q, s rather than p, t in this paragraph.)

Now suppose that T is as in (b) and that T = I. Then An, Bm1 ... Ank-I Bmk-I = A-nk -

cos(-nkO) L - sin(-nkO)

sin(-nkO) cos(-nkO) j

By the argument of the preceding paragraph, we can find f E FE, where f is the appropriate q or s, such that (a) the highest power of z occurring in the expansion of f (z) is zd, where (Q) the coefficient of zd in the above expansion belongs to E; and

(y) f(z)=aEE. Then z -y (f (z) - a) is nontrivial in FE, and so f (z) - a # 0. This is a contradiction. So again T j4 I.

Now suppose that T is as in (c) and that T = I. Then as in the preceding case,

An2B"' ... AT' Bmk = B-" _

Z mI 0

0' zm

In this case we obtain an equation f (z) = 0 and a contradiction results as before.

Finally, case (d) reduces to case (a) or the An' B,, -case by considering

T-I. D Note that the proof of (iii) actually provides uncountably many copies of F2 in SU(2)-simply vary 0 and z appropriately. We can also choose the generators A, B for F2 arbitrarily close to I. The Lie algebras of PSL(2, R) and SL(2, R) are the same, since PSL(2, R) is the quotient of SL(2, R) by the discrete subgroup {I,-I}. Recall (B19) that the Lie algebra of SL(2, R) is the Lie subalgebra sl(2, R) of M2(R), where

sl(2,R) = {A E M2(R): trA = 0). Further, the Lie algebra of SU(2) can be identified with the Lie algebra so(3, R) of SO(3, R), where

so(3, R) = {A E M3(R): A' = -A). Each of these Lie algebras has a natural basis such that the Lie products of the basis elements have particularly simple form. These Lie products, of course, determine all Lie products on the algebra in question. In the case of sl(2, R), we have the basis }}{H, E, F) where

H=10 -0j,

E=

r

10

0]

and F= l10

Then (6)

[H, E) = 2H,

[H, F) = -2F,

[E, F) = H.

01


109

Of course, every 3-dimensional, real Lie algebra with a basis satisfying the preceding equalities is isomorphic to s1(2, R). A typical element of so(3, R) has the form 0

-a

a

b

0

c

(a,b,cER).

-b -c 0

Let {A, B, C} be the basis of so(3, R) given by 0

0 0 0

1

A= -1 0 00

B=

0

0

1

fl

fl

fl

-1 0

and C=

0

0

00 0 0

-1

1

.

0

Elementary matrix multiplication shows that [B, C) = A,

[A, B] = C,

(7)

[C, A) = B.

The latter equalities, of course, specify so(3, R) as a 3-dimensional, real Lie algebra. (3.3) PROPOSITION. (i) If t is a real, semisimple, compact Lie algebra, then t contains so(3, R) as a subalgebra.

(ii) If go is a semisimple, real, noncompact Lie algebra, then go contains sl(2, R) as a subalgebra.

PROOF. (i) Let g be the complexification of a real, semisimple compact Lie algebra t. Then g is a complex, semisimple Lie algebra. Let A, Ha, and X. be as in B53-B55. Then a compact real form gk of g is obtained (B57) by setting

9k= ER(iHa)+1: R(X0-X_a)+1: R(i(xa+X_a)). aE0

QED

aEA

Let a E A. Then (B55), a(H0) > 0 and (1)

1Ha, Xaf --a(H0)X0,

[Ha, X-a] = -a(H0)X_a,

1X0, X-a] = Ha.

Let

B' = (Xa - X-.)l (2a(Ha))1/2,

A' = (1Ha)Ia(Ha), and

C' = i(X0

+X_a)I(2a(Ha))112.

Using (1), we have (2)

[A', B') =C,

[B', C) = A',

fC' A`J = B',

and comparing (7) of (3.2) with (2) above, there is an obvious isomorphism between so(3, R) and the subalgebra of gk spanned by {A', B', C'). Finally, since t is trivially also a compact, real form of g, it follows from B58 that I is isomorphic to gk. (ii) Let to + po be a Cartan decomposition of go, and to + ao + no the corresponding Iwasawa decomposition of go (B58, B59). Since go is noncompact,

it follows from B60 that no 0 (0). Let X E no - {0}. By B60, adX is a


101

nonzero, nilpotent element of L(go). From B63, sl(2, R) is a subalgebra of go as required. O The following theorem ((3.4)) due to Iwasawa [1) is of great intrinsic interest, and will be needed in establishing one of the main results of this chapter ((3.8)). Recall that the group Aut G of continuous automorphisms of C has a natural topology under which it becomes a topological group [HR1, §26). Indeed, a base of neighbourhoods of the identity i of Aut G is provided by sets of the form (3)

for E AutG: a(c) E Uc and a'1(c) E Uc for all c E Cl,

where C is a compact subset of G and U is a neighbourhood of e. If K is a closed, normal subgroup of G, and x E G, then we can define a1 E Aut K by setting az (k) = xkx 1. We write ax for aG. If A C G, then IA(K) is defined to be the set {ay : x E Al. Thus the group of inner automorphisms I (G)

of G is just IG(G). The map ': G

I(G), where '(x) = ax, is a continuous

homomorphism, I(G) being given the relative topology. If H is a topological group with identity e, then He is the identity component of H.

(3.4) THEOREM. Let G be a compact group. Then the identity component (AutG); of AutG is equal to 1G1(G). PROOF. The proof proceeds by considering three cases.

Case 1: G is a connected Lie group. The radical R of C is compact, connected, and solvable, and so is abelian (B51). It is therefore of the form T^ (B21). So AutR is discrete (B21) and therefore has trivial identity component. Now as the property of being a maximal, connected solvable, normal subgroup of G is preserved under the action of an automorphism of C, we have a(R) = R for all a E Aut C (that is R is characteristic in G). It is easily shown that the map a -» a f R from Aut G into Aut R is continuous and so maps (Aut G), into the (trivial) identity component of Aut R. Hence a(r) = r for all a e (Aut G);, r E R. Since G is connected and the map fi: G -» 1(G) is continuous, we have

ax E (Aut G); for all x E C, and so ax (r) = r for all x E G, r c- R. Hence R C Z(G). For 6 E Aut G, define 0' E Aut (G/R) by setting

/r(xR) =,6(x)R

(x E G).

It is easily shown that the map,8 -+ Y from Aut G into Aut(G/R) is a continuous homomorphism. Let a E (AutG)1. Then a' belongs to the identity component of Aut G/R, and since G/R is a connected, semisimple Lie group, it follows from

B42 that a' is an inner automorphism. So we can find xo E G, and, for each x E G, an element ax E R such that a(x) = (xoxxo 1)ax. Since R C Z(G) and a(xy) = a(x)a(y) (x, y E G), we see that axy = axay and a. = e if r E R. Hence we can define a map Q:G/R - R by Q(xR) = ax.

102


Now Q is clearly a continuous homomorphism from the semisimple Lie group G/R into the solvable Lie group R, and so, by B42, the induced homomorphism between the Lie algebras of G/R and R is the zero map. Since G/R is connected,

we can apply B11 and B18 to obtain Q(xR) = e for all x E G. Hence ax = e for all x E G, and a = a... Thus (AutG)1 C I(G). The reverse inclusion is obvious, so that (Aut G); = I (G) (= IG, (G)) as required. Case 2: G is a Lie group. Define subsets A, N of G by

A= {a E AutG: a(x) E xGe for all x E G},

N={aEA: a(y)=y forallyEGe}. It is easily checked that A is a subgroup of Aut G. Now A is open and therefore closed in Aut G. Indeed, since G is a Lie group, Ge is open in G. Further, xGe = Gex for all x E G and since Ge is a characteristic subgroup of G, a(x) E xGe

if and only if a-1(x) E xGe. By taking t1 = Ge and C = G in (3) of (3.3), we see that A is open as required. Obviously, N is a closed subgroup of Aut G and,

by definition, N C A. If y E Gef then yxy-1 = x[x-1yx]y-1 E xGe, and so IG, (G) C A.

We claim that it is sufficient to prove (1)

IN/(NnIG,(G))I ,W) = ah for some h E H,(e), and if x E Ge

is such that 7r(x) = h, then x E B,,. So B is a nonempty, compact subset of G. If Tr1, ... , irn E R(G), then consideration of it = 7r1 9 . ® ir shows that nn 1 B,,; ¢ 0. Hence we can find z E n{ B,,: it E R(G)}. Since R(G) separates the points of G, we have f3 = az. This concludes the proof.

D

(3.5) COROLLARY. Let G be a connected, locally compact group, K a com-

pact, normal subgroup of G, and H = {x E G: xk = kx for all k E K). Then G = KeH. PROOF. Let 4 3 unanswered. The problem for n > 3 was only solved in the 1980s and is much the deepest of the three problems considered by Banach. The remarkable solution to the Banach-Ruziewicz Problem (for the n-sphere) will be considered in Chapter 4.

(3.13) Two results on equidecomposability. We shall state two straightforward results on equidecomposability that are needed to prove Tarski's Theorem on paradoxical decompositions ((3.15)). A third equidecomposability result is also needed for Tarski's Theorem. This result is deeper than the other two, and depends on Konig's Theorem which will be stated and proved in (3.14).

Let G be a group and X a left G-set with ex = x for all x E X. Following Wagon [3) we say that subsets A and B of X are G-equidecomposable (or simply equidecomposable) if there exist partitions {A1,. .. , A,,, } and (B1, ..., B,n) of

of G such that siAi = B, for A and B respectively and elements I < i < m. If A and B are equidecomposable, then we write A = B. We say A S B if A C for some subset C of B. The two following results appear in Banach and Tarski [1). See also Stromberg [1). The proofs of these two results are set as Problem 3-13.

(i) = is an equivalence relation on 9(X).


(ii) Let A, B E S (X) be such that A B and B

113

A. Then A - B.

(3.14) Konig's Theorem and division. Let G and X be as in (3.13). The following result will be called (Tarski [21) the Division Theorem (for equidecomposable sets), since it is the equidecomposable version of the trivial arithmetical

fact: if a, b E R, r E P, and ra = rb, then a = b (which, of course, is obtained by dividing by r)!

(i) (The Division Theorem). Let A1,... , A B1, ... , B,. E _9(X) and be such that

(a) Aif1Aj =0=Bif1B1 whenever i 0 j; (b) Ai - Al and Bi B1 for all i;

Bi)

(c) (Ur ,1 Ai) B1. Then Al

This result was proved by Banach and Tarski in the case where n is a power of 2. The general case depends on a Theorem due to Kong [1] and is proved in (iii) below. Kong's proof of his Theorem is rather involved; the modem approach is substantially simpler and uses the combinatorial result known as the "Marriage Theorem." Our account is based on the treatment of the Marriage Theorem by Bondy and Murty 151 and uses ideas in Konig [1). We have also been greatly influenced by Wagon's treatment of Konig's Theorem (Wagon [S2, Theorem 8.111). To ease the exposition, it is convenient to express Konig's Theorem in graphtheoretic terms.

Let X, Y, and E be disjoint sets. We think of X and Y as sets of vertices (points) and E as a set of edges joining points of X to points of Y. Let F: E X x Y. If F(e) = (x, y), then we say that a joins z to y (or y to x), and z, y are the ends of e. Note that z, y can be the ends of more than one edge. A path in (X, Y, E) is a finite sequence of edges (e1,. .. , en) together with a finite sequence

(ao, ... , an) of vertices with ai an end-point of ei and ei}1 (1 < i < n - 1) and ao, an end-points of e1, en. We say that the path joins ao to an. The diagram below illustrates a path joining ao to as.

ao = a4

a2

ag

Let k E P. We say that (X, Y, E) is a k-regular bipartite graph if each x E X is an end of exactly k edges and each y E Y is an end of exactly k edges. A matching of subsets A of X and B of Y is a subset F of E such that every a r= A [b E BI is an end of exactly one f E F and the ends of every f E F are in A U B. When X and Y are finite, Konig's Theorem below implies the colourful classical "Marriage Theorem": if every girl in a village knows exactly k boys and

114


every boy knows exactly k girls, then the boys and girls can be married off in such a way that each boy marries a girl he knows. In the following proof, the reader will probably find it helpful to have a sketch of a bipartite graph in front of himself/herself. (ii) (Kong's Theorem) Let (X, Y, E) be a k-regular bipartite graph. Then there is a matching of X and Y. Let G' = X U Y. For S C G', let N(S) be the set of vertices in G' joined by an edge to a point of S. There are two cases to consider. Case 1: G is finite. By k-regularity, kJXJ = JEJ = kJYJ so that JXJ = JYJ = n for some n E P. (a) JN(S)J >_ JSJ for all S C X. The number of edges joining vertices in S to vertices in N(S) is kJSJ. So EtEN(S) nt = kJSJ, where nt is the number of edges joining t E N(S) to members of S. If JN(S)J < JSJ, then nt > k for some t, which is impossible.

(b) There exists a matching of X and Y. Let A, B be subsets of X, Y admitting a matching F with JFJ maximum possible. Suppose that A 96 X and let x E X -r A. Consider the set Z of points z E G' for which there is a path (e1 i ... , en) with the properties: (1) the path joins z to z, (2) the ei are alternately in F and E - F. If x, a1, ... , z are the vertices associated with such a path, then all of the X-vertices x, 62, a4.... lie in A (except for x) and all of the Y-vertices a1, as, ... lie in B except possibly for z. Suppose that z E Y. We show that by the maximality of F, the vertex z lies, in fact, in B. Indeed n = 2k+ 1 is odd, and if z ¢ B and we take F = (F - {e2, e4, ... , elk}) U {e1, e3, ... , en} then F' is a matching of A U {x} with B U {z} and JF'J = JFJ + 1 > JFJ, contradicting

the maximality of F. So T = Z ft Y C B. Let S = Z fl X. It is easy to check that T = N(S) and that JTJ = JSJ - 1. Then JN(S)J < JSJ, contradicting (a). So A = X, B = Y as required. Case 2: G' is infinite. Clearly, being joined by a path gives an equivalence relation on G', and by k-regularity, each equivalence class is a countable, kregular bipartite graph. If we can find a matching for each such class, then a matching for G' follows. So we can assume that G' is denumerable and every pair of vertices is path-connected. Let (x, y) E X X Y and define subsets Z, of G' by: Z1 = {x, y} U N({ x, y}),

Zi+1 = N(Zi). Clearly, by connectedness, G' = U°-1 Z,. Let Xi = X fl Z,, Y, = Y fl Z,. Note that Z, C Z,+1, so that X, C Xit1, Y, C Y,+1. Let i > 1. By adding new vertices to X1 , Yj and new edges, we obtain a finite k-regular bipartite graph (X,, Y, Ej). [[We illustrate the case j = 1, k = 2 below. 1/

111

1/2

bi

b?


115

Here, N(x) = {y1, y2}, N(y) = {xi i x2}, and points x'1, x2, y', y2 are added on

to X and Y. The new edges introduced are dotted.) By Case 1, there exists a matching of X? and YJ . For i < j , this matching induces a matching of Xi with a subset of Y.

An i-matching is a matching of Xi with a subset of Y. The above shows that there is an i-matching for all i. If i < j and M,, is a j-matching, then M,, induces an i-matching MJ I X; . We now use "Kong's Infinity Lemma" to obtain a matching of X and Y. Since the set of 1-matchings is finite, there exist

r-matchings M, (r > 1) such that M*) x, = M11. Let M1 = MI. Similarly, there exist r-matchings M, (r > 2) such that M2 ) x, = M1 and M, ) x, = M22. Let M2 = M2. An obvious induction argument produces a sequence {Mi} of i-matchings with Mi+1 I x, = Mi. Then M = U7 1 Mi is a matching for X and Y.)

(iii) (Proof of the Division Theorem.) Let G, X, r, Ai, Bi be as in (i). Let Oi : Ai

A1,iki:Bi

B1, and 0. U 1 Ai

(, f

1 Bi be the equidecompos-

ability bijections associated with (b) and (c) of (i). We take 01,61 to be the identity maps. We define an r-regular bipartite graph (X1,Y1,E1) as follows. Let X1 = Al x (0), Y1 = B1 x {1}. (The reason for the "x{O}" and "x{1}" is that of ensuring disjointness for X1 and Y1.) Assign an edge joining (a, 0) E X1

to (b, 1) E Y1 for each pair i, j such that ¢0= 1(a) E B, and ,(ij'qi 1(a) = b. (The diagram below will help clarify the situation.

03

Here, r = 3, a1 = a, a2 = c21(a), a3 = 031(a), and bi = 0(ai).

Also

b'I = 02(b1), b2 = b3(b2) and b3 = '02(b3). Note that 0 need not map A, into Bi, and that the bL need not be distinct. We assign edges joining (a,, 0) to (b1,1), (b2,1) and (b3,1).) By Konig's Theorem, there exists a matching M for X1 and Y1. For a E A1, let a(a) E B1 be such that (a(a),1) is the other end of the edge in M containing (a, 0). Then a(a) = jt4' (a) for some i, j. Since ?, ¢ and ¢; are given by G-translates, a(a) is of the form y? n,8kxs ,1a. There are only a finite number of possibilities for the y,,,,,, sk and xi,t in G, and we obtain a partition K1, ... , KQ of Al and elements zz, each of the form y?,m8kxt i1, such that {z} K1, ... , zQK4} is a partition of B1. So Al - B1.)


116

(3.15) Tarski's Theorem on amenability and paradoxical decompositions. (Tarski [2), [4]). Let (G, X, A) be as in (2.32). Tarski's Theorem is concerned with the equivalence between the existence of an invariant measure for (G, X, A) and the nonexistence of a paradoxical decomposition of A. Tarski's beautiful proof depends on showing that such a measure a exists if and only if there exists a certain homomorphism on a semigroup E associated with (G, X, A) and then using the Division Theorem of (3.14) to complete the proof of his theorem.

(Is there a short proof of Tarski's Theorem? This question seems open.) Let

_{BCX: there exist nEPand s1i...,snEGsuch that BCU", siA}. The elements of 9 were called bounded in (2.32). From the last sentence of (2.32), we see that there exists an invariant measure on (G, X, A) if and only with if there exists a positive, G-invariant, finitely additive measure v on v(A) = 1. Reference to the definition of a paradoxical decomposition of A (given in (3.12)) shows that 3 partitions of A are involved. To cope with this, it is convenient to have disjoint copies of A available. For this reason, we introduce

Y = X x P, where X is identified with X x {1}. Let R(P) be the group of permutations of P (that is, the group of bijective maps from P onto P). Let H = G x II(P). Then H acts on Y by means of the formula: (s,p)(x,n) = (sx,p(n)) ((s,p) E H, (x,n) E Y). Let I be the identity permutation on P. Then (e, I) is the identity of H and

(e,I)y=y forallyEY. Let

Y={CE.'{(Y):CCBxFforsomeBE 1,FEJr(P)}. Clearly, rY is H-invariant and is a ring of subsets of Y. (i) There exists an invariant measure for (G, X, A) if and only if there exists an H-invariant, positive, finitely additive measure v on Y with v(A) = 1. [Every N E ' can be written uniquely in the form

`J C, x {ji} i=1

where 1 < ji < j2 < .. < jr, Ci E 0 and each Ci # 0. Note that if (s, p) E H, then

r

(SIP) (C) _ [J sCi x {ki} i=1

where the ki are distinct. Every invariant measure v for 2 gives rise to an appropriate invariant measure £ on Y. where r

C(N) _ E v(Ci). i=1

Conversely, given an invariant measure f on A', we obtain an invariant measure v on . C ./Y by restriction and by identifying each s E G with (s, I) E H.]


117

Let us write N1 N2 in Y whenever N1 and N2 are H-equidecomposable. By (3.13(i)), - is an equivalence relation Let E = Y/-, and r: ./Y E be the canonical surjection. Let a = r(A). We make E into an abelian semigroup as follows. Let N1, N2 E X. There exists h E H such that h(N1) n N2 = 0. Define N1 +h N2 = h(NI) U N2.

One readily checks that if N1 - N{, N2 = Nz, and k E H is such that k(N{) n NZ = 0, then N1 +h N2 - Nl +k N2'. Hence it makes sense to define an element r(N1) + r(N2) by setting it equal to r(N1 +h N2). It is readily checked that E is an abelian semigroup under the map (7, r) - (ry + r). The next result reduces the problem of measure for (G, X, A) to that of the existence of a certain homomorphism from E into the (additive) semigroup [0, oo). (ii) There exists an invariant measure for (G, X, A) if and only if there exists a homomorphism f : E -» [0, oo) such that f (a) = 1. [Suppose that there exists an invariant measure v for (H, Y, A). Since v(N1) _ v(N2) if N1 = N2 in .4', it follows that we can define a map f : E --+ [0, oo) by

f (r(N)) = v(N)

(N E Y).

If Ni, N2 E Y with N1 n N2 = 0, then v(N1 UN2) = v(N1) +v(N2). Referring

to the definition of addition in E, we see that f is a homomorphism. Since v(A) = 1, it follows that f (a) = 1. Conversely, suppose that g: E - [0, oo) is a homomorphism with g(a) = 1. For N E ./Y, define g(N) = g(r(N)). Then F(A) = 1 and since, for h E H and N E Y, N - hN, it follows that g(hN) = £(N). If Ni, N2 E Y with N1 nN2 = 0, then r(N1 U N2) = r(NI) + r(N2), so that e(N1 U N2) = e(N1) + C(N2). Obviously e(0) = 0, so that g is an H-invariant, positive, finitely additive with e(A) = 1. measure Now use (i) to complete the proof.] The following extension result for homomorphisms on an abelian semigroup

S will be required. We shall say s < t in S for s, t E S if either s = t or there exists w E S such that s+ w= t. For related extension results, see Ross [1], [2] and Comfort and Hill [1].

(iii) Let T be a subsemigroup of the abelian semigroup S and F : T - [0, oo)

an (additive) homomorphism. Let e E T be such that F(e) = 1 and, for each s E S, there exists n E P such that s < ne. Then there exists a homomorphism g: S - [0, oo) with g 17. = F if and only if F(s) jg(mw + kwl) - g(kw2 )]/m

= g(w) + k[(g(wl) - g(w2))/m). Taking the supremum (over all choices of w1i w2, m satisfying (3)) of the righthand side of (4) and using (1) with a in place of x, we have (5)

g'(x) > g(w) + kg'(a).

Conversely, suppose that v1, v2 E W and n E P are such that (6)

v1 < v2 + nx.

Then v1 < (V2 + nw) + nka, and so (7)

g'(a) > l9(vl) - g(v2 + nw)]/(nk)

= k[(9(vl) - g(v2))/n] - kg(w).


119

Taking the supremum of the right-hand side of (7) over all choices of V1, V2, n satisfying (6), we obtain

g'(a) ?'9'(x) - "9(w) So

(8)

g'(x) < g(w) + kg'(a).

The first equality of (2) now follows from (5) and (8), and the second equality of (2) from an easier version of the above proof. The resulting contradiction gives W = S. This completes the proof.]

We now continue with the proof of Tarski's Theorem.

(iv) There exists an invariant measure for (G, X, A) if and only if ka ¢ Ia whenever k, t E P with k # 1. [Suppose that there exists an invariant measure for (G, X, A). By (ii), there

exists a homomorphism f : E -+ 10, oo) such that f (a) = 1. If k,1 E P with

k01,then f(ka)=k, f (Ia) = I so that ka 96 La. Conversely, suppose that ka i4 Ia whenever k, I E P with k # 1. Let T = {na: n E P}. Then T is a subsemigroup of E. If /3 E T, then there exists exactly

one n E P with /9 = na. Define F: T - [0,oo) by F(/3) = n. Obviously F is a homomorphism, and F(a) = 1. We now show that the hypotheses of (iii) are satisfied with S, e replaced by E, a.

Let -y E E. We require to show that there exists 6 E E and r E P such that ' + 6 = ra. To this end, let C E Y be such that r(C) = 7. We can write C as a disjoint, finite union of sets B,,, x {m}, where B,,, E 2, and since r(C) = Er(B,,,), we can suppose that C E .9. So C C U'_1 siA for some si E G. Write C as a disjoint union Us_, C;, with Ci' C s1A and find hi E H

such that hi(siA) l h?(s,A) = 0 if i i4 j. Let Bi' = hi(siA) - hi(C,) and B'=(}B;. Then -y + 7 (B') = ra and we can take 6=r(B'). We now show that if s, t E T with s < t in E, then F(s) < F(t). Suppose, then, that s = na and t = ma with s < t in E. If n < m, there is nothing to prove. Suppose that n > m. Let Cn, C,n E with r(C,,) = na, r(C,) = ma. Since no = ma + (n - m) a, it follows that Cn, 5 Cn (where we use the notation of (3.13) with respect to the action of H on Y). Since s < t, we can find (3 E E with na+/3 = ma, and it follows that C. C,,,. By (3.13(ii)), we have C,, = C,n and no = ma. But by hypothesis, no 0 ma since n m, and a contradiction

results. So n < m, and F(s) < F(t). Thus (iii) applies to give a homomorphism g: E

10, oo) with g(a) = 1. Hence, by (ii), there exists an invariant measure for (G,X,A).] (v) There exists an invariant measure for (G, X, A) if and only if a 0 2a. [By (iv) we need only establish that if a 96 2a, then ka 0 la whenever k 0 1 in P. Suppose, then, that a # 2a and that there exist k, t E P with k i4 I such that ka = Ia. We will derive a contradiction. Without loss of generality, we can

120


suppose that k < 1. Let d = k + r with r > 1. Then by adding on ra at each stage, ka = (k + r)a,

(k + r)a = (k + 2r)a,...

and we obtain ka = (k + pr)a for all p > 1. Putting p = k, we obtain ka = we see that the k[(1 + r)a]. Translating this equality into the context of Division Theorem of (3.14) applies to give a = (1 + r)a. If r = 1, then a = 2a and the desired contradiction follows. If r > 1, then by adding on (r -1)a to both sides of the equality a = (1 + r)a, it follows that r(a) = r(2a), and again using the Division Theorem, we obtain a = 2a and the desired contradiction.)

(vi) (Tarski's Theorem). There exists an invariant measure for (G, X, A) if and only if A does not admit a paradoxical decomposition ((3.12)) (with respect to G). [If there exists an invariant measure for (G, X, A), then, by (3.12), A does not admit a paradoxical decomposition. Conversely, suppose that G does not admit a paradoxical decomposition. By (v) it is sufficient to show that a 3& 2a. Suppose, on the contrary, that a = 2a. Then A = AU (A x {2}) in./' with respect to H. (Recall that A is identified with A x { 1}.) So there exists a partition A1,. .. , Am, Bi,... , Bn of A and elements

of H such that pi(1) = 1, q j(1) = 2, and {(sl,p1)AI, , (sm,pm)Am, (t1,g1)B1, , (tn,gn)B,,} is a partition of A U (A x {2}). It follows that {siAi,...,smAm} and {t1B1,...,t,,Bn} are partitions of A so that A admits a paradoxical decomposition. This contradiction establishes the Theorem.) Tarski's Theorem, applied with (G, G, G) in place of (G, X, A) yields the following remarkable characterisation of amenable groups.

(vii) The group G is not amenable if and only if there exists a partition

{A1,...,Am,B1,...,B,} of G and elements xi,...,xm,yl,.... y, of G such that {x1A,,...,x..A,,,) and {y1BI,...,ynBn} are partitions of G. Is there a similar characterisation of amenability for discrete semigroups? See Problem 3-15 for the locally compact group version. Rosenblatt [11) discusses paradoxical decompositions in the context of Boolean algebras. A version of Tarski's Theorem involving countably additive measures is given by Chuaqui [1].

(3.16) The Banach-Tarski Theorem. The Banach-Tarski Paradox (Theorem) was established in the paper of Banach and Tarski [1). We will be content to base our account on the short proof in Wagon [S2). A good account of the

classical proof, which uses the fact that Z2 * Z3 is a subgroup of SO(3,R), is given by Stromberg [1].

(The Banach-Tarski Theorem). Let A and B be bounded subsets of R each having non-empty interior. Then A - B (with respect to the action of G3 on R3).


121

The "Paradox" associated with this Theorem can be expressed in the following dramatic form: a billiard ball can be chopped into a finite number of pieces and the pieces fitted together to form a life-size statue of Banach! The flaw in the "paradox" is that the "real" world is being confused with its geometrical model. (Observe, for example, that at least one of the "pieces" has to be nonmeasurable (since Lebesgue measure preserves volumes)!)

The key to the proof of the Theorem is that SO(3, R) contains a subgroup H isomorphic to F2. This is an easy consequence of (3.2(iii))-see Problem 3-5. Each element of H - {I} is a rotation A about an axis IA through 0, and IA intersects the sphere S2 in two points aA,19A. Let D be the (countable) set of such points aA, QA. Then H acts freely on S2 -r D-the free condition means that A = I whenever Ax = x for some x E S2 - D. By Tarski's Theorem, H admits a paradoxical decomposition (p.d.)-in fact, it is easy to write down an explicit p.d. for F2 acting on itself (Problem 3-12). Let T C S2 - D be such that IT f1 HxI = 1 for every x E S2 - D. (The existence of T uses the Axiom of Choice.) Then S2 - D admits a p.d. for H-indeed, if A=, BB implement a p.d. for H, then AiT, BST implement an S2 - D-p.d. We would like to have S2 itself having an SO(3, R)-p.d.-the problem is that of dealing with D. To this end, since D is countable, there exists a line t through 0 intersecting S2 at points of S2 --- D. Further, there exists an angle 8 such that

the rotation p through 8 with axis I is such that the sets (p' (D) : n > 0} are pairwise-disjoint. Let D = U°O_o p"(D). Then p(D) U D = D, and it readily follows that S2 = S2 - D (for SO(3,R)), and that S2 has an SO(3,R)-p.d. If Ai, B1 implement a p.d. for S2 (with respect to 80(3, R)), then X, Bj' implement

a p.d. for B - {0}, where B is the unit ball of R3, and for L C S2, we define L` = {ax: a E (0, 1], x E L}. Arguing as above with p replaced by a suitable rotation whose axis does not pass through 0, we can readily show that B has a p.d. for G3. It follows that B is equidecomposable with the union of a finite number n > 1 of disjoint balls of radius 1, and an application of (3.13(ii)) yields the Banach-Tarski Theorem (cf. Problem 3-14).

R. M. Robinson IS) has shown that there exists a p.d. of B into five sets and that this is best possible. Robinson's work was extended by Dekker and de Groot )1). A full discussion of this work is given in Chapter 4 of Wagon 1S21.

Problems 3 1. Prove that F2 is isomorphic to a subgroup of a Cartesian product l0EA G,,

of finite groups. Deduce that the Cartesian product of a family of amenable groups need not be amenable.

2. Show that there exists a compact group that is not amenable as discrete and yet contains a dense subgroup that is amenable as discrete.


122

3. Show that if G is the free product *QEAZ, (n, > 1), then G F2 if and only if either JAI = 2 and na = 2 for all a E A or JAI = 1. (Hint: use the Kurosh Subgroup Theorem.) 4. Let G be a group presented by a set {xi : i E I) of generators and defining

relations x'!j = e (j E J) where nj > 2 and J C I. Show that G is amenable

if and only if either III=2=jJjandn,=2(jEJ)orIII=1. Show that von Neumann's Conjecture is true for the class of such groups G. (Hint: use Problem 3-3.) 5.

Assuming (3.2(iii)), give an easy proof that SO(3,R) contains F2 as a

subgroup. Give an example of elements A, B which generate F2 in SO(3, R).

6. Check Iwasawa's Theorem (3.4) directly when G is T" (n > 1). 7. Let us say that a finite-dimensional Lie algebra g over R is amenable if its Levi-Malcev Decomposition s ® t is such that s is a compact Lie algebra (B48). Show that a connected Lie group G is amenable if and only if its Lie algebra is amenable. 8. Let G be a connected Lie group with g as Lie algebra. The group G is said to be of Type R if for every X E g, the eigenvalues of ad X are purely imaginary

(that is, Sp(adX) C iR). Give examples of Type R groups. Is the "ax + b" group of Type R? Show that G is amenable if G is of Type R. (For more on the Type R theme, see Chapter 6.)

9. Show that every connected Lie group that, as a discrete group, has polynomial growth ((0.12)), is solvable.

10. Let G be a discrete group and H,* G. Show that G is locally finite if and only if both H and G/H are locally finite. (This was used in (3.11).) 11. Consider the Lebesgue problem as stated in (3.12). Why is there no point

in replacing 4 (R) by .,'(R)? 12. Give a paradoxical decomposition for F2 acting on itself.

13. Prove ((3.13(i),(ii)) that = is an equivalence relation on a(X) and that the "Cantor-Bernstein Theorem" holds for 95.

14. Let (G, X, A) be as in (3.12), and suppose that B C A is such that A is contained in the union of a finite number of G-translates of B. Show that if A admits a paradoxical decomposition (p.d.) then so does B and that A = B. Deduce that every subset of S" with nonempty interior admits an SO(n+ 1, R)p.d. and that all such subsets are SO(n + 1, R)-equidecomposable. (Use 3-18 below.)

15. (i) Let G be a locally compact group. Show that if an open subgroup H of G admits a Borel p.d. then so also does G. Show that G admits a Borel p.d. if a quotient group of G does.


123

(ii) Prove that a locally compact group G is not amenable if and only if it admits a Borel paradoxical decomposition. 16. Let G be a nonamenable locally compact group. Show that if a: L,o(G) C is a translation-invariant linear functional, then a(1) = 0.

17. Fill in the details of the Banach-Tarski Theorem ((3.16)).

18. Show that if n > 3, S' has a paradoxical decomposition under the action of SO(n+ 1, R). Deduce that the unit ball in R1 has a paradoxical decomposition

for G..

CHAPTER. 4

FOIner Conditions (4.0) Introduction. In this chapter, we examine in detail the growth conditions on an amenable locally compact group G known as the Folner conditions. These were briefly discussed in the introductory chapter. The starting place for these conditions is the Reiter-type condition which was proved in (0.8): there

0 for all x E G. Our exists a net J f6} in P(G) such that (Ix * f6 - fells first objective ((4.4)) is to show that the f6 can be taken to be such that the convergence to 0 is uniform on compacta. Our next objective is to replace the f6's by scaled characteristic functions XK/X(K). This leads to the remarkable condition of (4.10), which roughly asserts that

G is amenable if and only if whenever C E '(G), then we can find nonnull K E '(G) that, relative to its size, is "almost invariant" under left translation by elements of C. The deepest of the conditions considered here is (4.13). In (4.16), we show that when G is amenable, we can form useful averages of functions in L ,,o (G); these averages are analogues of the averages ( _n 0(r))/(2n + 1) in the case where G = L. We complete the main section by discussing three important applications of

the Reiter-type conditions that characterise the amenability of G. The first ((4.18)) determines the precise distance of coif *x: x E G} from 0 for f E LI (G):

the distance is just If f dtl. The second and third are related to the relevance of amenability for the representation theory of G. Our second application ((4.19)) shows that II f 111 = 117r2 (f) fI for all f > 0 in L1(G), where 1r2 is the left regular representation of G. The result (4.20(ii)) gives a converse: G is amenable if there exists a self-adjoint f > 0 whose support generates a dense subgroup of G and for which If 11, = IIir2(f)II. (This will be important in our discussion of von Neumann's conjecture in (4.32).) The third application ((4.21)) is the "weak containment" property, which can be neatly expressed as asserting that Ct (G) = C'(G), that is, the representation theories of G and the (accessible) C'-algebra Ct (G) coincide. (See Chapter 0.) Let G be a locally compact group. Our first result is proved in exactly the same way as (0.8), taking in to be in .it(G) rather than C(G). (Recall (1.10).)

(4.1) PROPOSITION. The group G is amenable if and only if there exists a net {f6} in P(G) such that Ily * I6 - fells --+ 0 for all u E P(G). 125

FOLNER CONDITIONS

126

(4.2) Notes. Examination of the proof of (0.8) shows that if m E Zt (G), we can actually ensure that 16 -+ m weak*. Obvious modifications of the proof, using m E :It (G), yield that G is amenable if and only if there is a net { f6} in P(G) such that for all p E P(G), we have his * f6 - f6111 -a 0 and ll f6 * Fs - f6111 --; 0. The natural right-handed versions of (4.1) (for example, involving llf6 * is - f611, rather than lip * f6 - f6l11) hold. For 1 - 0, Ilf[IP =1}.

Note that P1(G) = P(G). We recall [HR1, §201 that LP(G) becomes a left M(G)-space with action given by

p* f(x) = f f(y-1 x)dp(y) and that his * f lip 0, and then using (2), with b = f (x)1/P, a = g(x)1/P, and Holder's inequality. References for the result below are Reiter [l1, [3), 1141, [R), Dieudonne [21, Stegeman [1), Hulanicki [4], and Day [8]. The characterisation of amenability given in (4.4(ii)) is often called Reiter's condition or Property Pp. (2)

FOLNER CONDITIONS

127

(4.4) THEOREM. Let G be a locally compact group and 1 0, then there exists f E PP(G) such that

IIx*f - flip<e

(1}

(ZEC);

(iii) there exists a net {g6} in PP(G) such that IIIA * g6 - g61IP -y 0 for all E PM(G). PROOF. We prove that (i) implies (ii). Suppose that (i) holds. Let C E F(G) and e > 0. Let g E P(G) and 17 > 0. Using the continuity of the map x - x * g from G to L1(G), we can find an open, relatively compact neighbourhood U of e such that llu * 9 - 9111 < 77

(2)

(u E U).

Let xj,... , xn. in G be such that x1 = e and C C UL 1 xjU. Since xi * g E P(G) for each i and as G is amenable, we can find, by (4.1), an element go E P(G) such that

(1- A(cK) = X(K) for any c E C. So

A(CKL K) =A(CK - K) + A(K - CK) _ (A(CK - K) + A(K) - A(K n CK))

0. By (4.4(ii)), we can find f E P(G) such that llx* f -f III < q for all x E C. Then we can find a sequence {gn} of simple, positive functions

in LI (G) such that l{gn - f III -+ 0. Now if g E L1(G) is a simple function Z:iN

_17;Xs;,whereryL>Ofor all i,EtnEJ=QSif ij,and 0 1 - s/2.

(3)

Our next objective is to prove that amenable locally compact groups satisfy the deep F9lner condition of (4.13). We require two rather technical lemmas. References for the first are [R, Chapter 8) and Milnes and Bondar [1]. Let G be a locally compact group.

(4.11) LEMMA. Let V be a relatively compact subset of G with nonempty interior. Then there exists a subset T of G and an integer R > 0 such that (i) G= U{tV : t E T}, and (ii) for each x E G, we have i{t E T: x E tV}[ R. PROOF. Suppose first that V is a compact, symmetric neighbourhood of e. If G is compact, then trivially we can find a finite subset T of G such that (i) and (ii) are satisfied. Suppose, then, that G is not compact.

Suppose, next, that G = UZ, V". Let Z E 9,(G) be such that Z is symmetric and Z2 C V. We construct recursively sequences {ti} in G and {Ni} in P such that for all n (a) N.:5 )a(V"Z)/A(Z);

(b) t, 0 Ul-I tiV (j > 1);

(c) tj EVkif 1<j 0 such that (i) and (ii) are satisfied with G, T replaced by H, S. Let A be a left transversal of G with respect to the subgroup H. Then the theorem is established by taking T = {as: a E A, s E S}. Finally, we now deal with the case where V is no longer assumed to be a symmetric compact neighbourhood of e. Let W E W.(G) be symmetric. Find a subset T' of G and an integer R' > 0 such that (i) and (ii) are satisfied with T, R, V replaced by T', R', W. Since V- is compact with VI 34 0 and as W E Te (G), we can find n, m E P and elements xi (1 < i < n) and yi (1 < i < m) in G such

that n

m

WCUxiVCUWY'i=1

i=1

One readily checks that (i) and (ii) are satisfied with T = {t'xi : 1 < i < n,

t'ET'}andR=nmR'. 0

Now let C E F,(G) and C' = CC'1C. Let T and R be as in (4.11) with V = C. Let M = max{©(x): x E C-'}, where A is the modular function of G, and b = "A (C). The next lemma is analogous to (4.7). This result and (4.13) are due to Emerson and Greenleaf [1]. Leptin [1], [2], [5] studied the quantity I(G) of (4.14).

(4.12) LEMMA. Let 17 E (0, (2A(C-1))/(MA(C'))) and suppose that K' E W (G) is nonnuil and is such that A(xK' L1 K')/A(K') < ri

(1)

(z E C').

Let b E (nb, A(C'1)/M) and define

K'(b) = {x E K': A(C' - K'x-1) < b}. Then we have

(i) K'(S) is compact and A(K'(5))/.1(K') >- 1 - (i?b)/b; (ii) A(CK'(b))/a(K') < 1 + [5MR/(A(C-1) - SM)].

PROOF. Since the map x .1(C' -r K'x-1) = A(C') - .1(C' ft K'x-1) is continuous (Problem 4-5), the set K'(b) is obviously compact. Let E(5) =

K' - K'(b). (So E(b) = {x E K': A(C' ^- K'x 1) > b}.)

Using Fubini's

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134

Theorem,

6A(E(6)) < JE(6) A (C' - K't-1) da(t) (2)

=

f

E(6)

daft) f01Xc,,,,x,t-' (x) dA(x)

=f A(E(6) - x'1K') dA(x). Now for x E C',

A(E(6) - a-1K') < A(xK'

K') = X(xK') - a(xK' n K') _'§A(xK'AK') < 1rla(K') and so

using (4.6(i)) and (1). Using (2), we have 6.\(E(6))
A(K')(1- (2r/.1(C')/6)) (r7b)/6).

The inequality of (i) follows. We now turn to (ii).

Using (4.11(1)), G = U{tC: t E T} = U{C-1t-1 : t E T}. Letting I(6) _ 0 0 if and only ((K'(6))-1C-1) nT and noting that for t E T, K'(6) if t E 1(6), we have K'(6) c U{C`1t'1 : t E I(6)}, so that nC'1t-1

CK'(6) C U{CC-1t-1: t E I(6)}.

(3)

For each t E 1(6), choose yt E C'1t-1 n K'(6). Recalling that C' = CC-1C, noting that t'1 = cyt for some c E C, and using (3) and the definition of K'(6), we have?

A(CK'(6)) 0, and H, K, be as above. Then: (i) G is amenable if Hµ = G and JJir2(p)II =#(G); (ii) G is amenable if K; = G, p- = p, and IIir2(p)II = A(G)PROOF. (i) Suppose that H; = G and 11ir2(p)I1 = µ(G). We can suppose that µ(G) = 1. Let v =,u- * A. Then .7r2(v) > 0, and we have 1=117r2(v)11= sup{(7r2(v)f,f): f r= L2(G), 11f 112 =1}.

Since v > 0, there exists a sequence {fn} in CC(G) with fn > 0, Ilfn112 = 1 such that (1r2(u) fn, fn) - 1. Let 97, (x) = (2r2(x) fn, fn). (In the notation below, gn = fn * fn.} Then 0 1}. Then L- D S(v). Since for x E G, (1)

112 * fn - fnJ12 = (x * fn - fn,x * fn - fn) = t 1211fn112 - 2gn(x)),

it follows that L = {x E G: Ijx * fn - fnl12

0). Since 11x*fn-f2112=11x-'*fn-fnll

and

11xy*fn-fnll2 :5 IIy*fn-fnJJ2+1Ix*fn-fn112, we see that L is a subgroup of G with L' S(v). Since S(v) generates the dense subgroup H, of G, it follows that L is dense in G. Now if we knew that L was all of G, then a "p = 2" Reiter-type condition would immediately give G amenable. Unfortunately, we only know that L- = G. To overcome this difficulty, let hn = fn E P(G) (n > 1). Then ((4.3(1)))

Ilx * hn - hn111 -+ 0 for each x E L. Any weak' cluster point m of {hn} in 01t(Ur(G)) then satisfies: xm = m for all x E L. Since the left action of G an OR(U,(G)) is jointly continuous for the weak' topology ((2.22)), we obtain xm = m for x E L' = G. Hence m E .L`(Ur(G)) and G is amenable ((1.10)). (ii) Suppose that K; = G, p"' = p, and Il7r2(p}Il = µ(G). We can also suppose

that p(G) = 1. Let A = S(µ) (= S(µ)'1) and N = H;, the closure of the

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143

subgroup generated by A2. Since A generates Kµ, Ku = G, and aA2a-1 C N

for all a E A, it follows that N is normal in G. Let ao E A. We claim that G = N U aoN. Indeed, if x E G N, then since K; = G, there exists a net x in G where each xa is the product of an odd number of elements of A. Then xs = ao(aolx,,) E aoN, and so x E aoN. Thus GIN is of order < 2 and N is open in G. Clearly, the amenability of G will follow from that of N. We xa

will now show, using (1), that N is amenable. Consider the measure aop. Clearly, S(aop) C A2 C N. Further, if a, b E S(p),

then a-1b = (aoa)-1(aob) E S(aop)-1S(aop), so that the latter set generates a dense subgroup of N. By (i), with G, p replaced by N, aop, we will have N amenable if we can show that JIir2 (aop)IJ = 1, where ire is the left regular representation of N on L2 (N). If N = G, then 7r2 (aop) = 1r2 (aop) = 1r2 (ao)1r2 (p), and we obviously have JINN (aop) II = 1 as required. Suppose then that N 36 G, so that G is the disjoint union N U Nao. Then AGI N is a left Haar measure on N, and it is easily checked that L2(G) can be identified with L2 (N) 03 L2 (N), and under this identification, ir2(aop) = 7r2N (aop) ® r2 (aop). Then 1 = II7r2(aou)II = Bit (aop)Il as required. We conclude this section by proving the "weak containment" characterisation of amenability. (For "weak containment" see (4.29).) Recall that the reduced C-algebra of G is Cl* (G), the C--algebra generated by ire (LI (G)). The universal C--algebra C' (G) can be described as the completion of the Banach *-algebra L3 (G) under its largest C'-norm 11- 11 (5 11-111) (D2, (13.9)). Since II f JJ > 111r2(f)ll

(f E Li (G)), we have a canonical homomorphism Q2 from C* (G) onto Ct (G).

We shall say that C' (G) = Ct (G) if Q2 is an isomorphism. We discussed in Chapter 0 the significance for representation theory of the equality C-(G) _ Ct (G). Before proving the theorem, we briefly discuss some fundamental facts about representations of and states on L3 (G) and C` (G). References for these facts are

JHR2J and (D2). A continuous function ¢: G - C is called positive definite if, whenever x1, ... , x a E G and a1, ... , an E C, then Fs j ai&9O(xi x,-1) > 0. Such functions are precisely the matrix coefficients arising from continuous unitary representations of G. Thus, if it is such a representation on a Hilbert space 55 and £ E S7, then the function x ---* (a(x){, ) is positive definite. Conversely, every positive definite function 0 can be so realised. Every such 0 satisfies 11011oo = ¢(e), and the set of states on L1(G) (or C` (G)) can be identified with those ¢ for which 4'(e) = 1. If 0, 0 are positive definite on G, then so also is oft. The 4's arising from 1r2 are rather special. Indeed, for such a ¢, there exists f E L2 (G) such that for all x E G, 4'(x) = (1r2(x)f, f) =

1G

i(')7() = f * f-(x) = g * gt(x)

where g = I E L2(G) and 91 (x) = g(x-1) (= f* (x)).

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144

We require an important result due to Godement [D2, (13.8.6)]: if ¢ E CM(G) is positive definite, then there exists g E CM(G) with 0 = g * gt. References for the theorem below are Godement [1], [D2), Greenleaf [2), Hulanicki [2), and [R]. It is sometimes called the "weak containment" theorem for amenable groups.

(4.21) THEOREM. The locally compact group G is amenable if and only if C* (G) = Ct (G).

PROOF. Suppose that G is amenable. Let F be a state on C* (G). Then F[ L, (G) is a state on L1(G), and there exists a (continuous) positive definite function 0 on G with 4'(e) = 1 and

F(f) =

(f L1(G)).

Let f E L1(G). By (4.19) and the proof of (4.20)(i) with p = [fI +beF we can find a sequence {fn} in CM(G) with fn > 0, []fn]P2 = 1 such that f,, * ft, -+ 1 pointwise u"' *,u a.e. and hence A-a.e. on the support off (c S(lz- * is)). Now (f,, * f,t,)4' is positive definite and of compact support, and so by Godement's Theorem, there exists k,, E C,(G) with 1Ik,,112 = 1 such that (fn* f,t,)4' = kn*k,t,.

So kn * kt - ¢ a.e. on the support of f, and by the Dominated Convergence Theorem,

r

]F(f) I = lim (G f (kn * k,t) d ,1 = lim 1(7r2 (f) kn, k, )1 5 Nit (f) [[ d

So for g E L1(G),

u[g[12 = ][g * g-11 = sup{F(g * g'): F is a state on C*(G)} 5 117r2(g * 9-)]] = ]]7r2(9)]]2 < ][9[[2,

giving C* (G) = Ct (G).

Conversely, suppose that C* (G) = Cl* (G). Let 1(f) = f f dl (f E L1(G)). Then 1 is a continuous positive functional on Ct (G). A simple separation argument ([D2, 3.4.4)) shows that we can approximate 1 weak* by convex combinations of functionals of the form f -+ (1r2(f)g,9) (g E CM(G), 119112 = 1). Godement's result gives that any such combination is also of the form f - (ir2 (f )g, g). The upshot is that we can find a net {96} in C,,(G), 1196112 = 1, 96 > 0 such that 96 * gt5 1 pointwise on G. (Indeed, the convergence is uniform on compacta by [D2, (13.5.2)).) Now use (1) of (4.20) to obtain a Reiter condition for G and hence the amenability of G. D

It easily follows from the above proof that G is amenable if and only if 1 is weakly contained in are. (See Problem 4-28.) M. Bekka [S] has recently proved that G is amenable if and only if 1 is weakly contained in 7r 0 fr for every continuous, unitary representation ar of G. (Here, fr is the conjugate representation of r ([D2, 13.1.5]).) While on the "tensor product" theme, we note that if C is discrete, then r2 0 r is a multiple of are for every unitary representation ar of G. The latter result is useful in K-theory (Cuntz [S21).

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145

The weak containment property can be formulated for inverse semigroups. An interesting, open question is that of characterising those inverse semigroups with this property. The papers Paterson [1), Duncan and Paterson [1), [S] investigate this question for Clifford semigroups, the class of inverse semigroups which are unions of groups.

References Berg and Christensen [2], Bondar and Milnes [1], Chou [4], Day [2], [3], [4], [6], Dieudonne [2], del Junco and Rosenblatt [1], Emerson [2], Emerson and Greenleaf [1], Folner [2], Frey [1], Glicksberg [2], Godement [1], Greenleaf [4], Greenleaf and Emerson [11, Hulanicki [4], Jerison [2], Kesten [1), Leptin [1), [2), [5), Milnes and Bondar [1), Namioka [1], Raimi [3], and Reiter [1), [13), [14).

Further Results (4.22) Semigroups and Folner conditions. The results of this paragraph are due to Argabright and Wilde [11. Let us say that a semigroup S satisfies (FC) ("Folner condition") [(SFC) ("strong Folner condition" A if for every C E _0-(S)

and e > 0, there exists nonvoid K E -9-(S) such that [xK - KJ/JKJ < e [[K - xK[/[K[ < e] for all z E C. Note that we obtain a condition equivalent to (SFC) if K - xK is replaced by xK AK in the statement of the (SFC) condition. [Indeed, [K -r xK[ = [K[ - [K n xKJ and [xK - K[ = [xKJ - [K n xKJ, so that

[K - xK[ - JxK -- K[_[K[-[xKJ>0. Thus 21K- xK[>JxKAKJ> [K - xK[, and this establishes the desired equivalence. It also follows that (SFC) implies (FC) (thereby justifying the "S" in (SFC)).] Now according to (4.9), S satisfies (FC) if S is left amenable. The converse to the latter statement is easily seen to be false, since, with K = S, every finite S, left amenable or not ((1.19)), satisfies (FC)! However, S is left amenable if S satisfies (SFC). Argabright and

Wilde also show that every left amenable semi group that is either left cancellative or finite satisfies (SFC). (See Problem 4-32.) They also show that every abelian semigroup satisfies (SFC).

More information about (SFC) is given in Rajagopalan and Ramakrishnan [1). However, Klawe [2] shows that left amenability is equivalent to (SFC) if and only if Sorenson's Conjecture is true. Since the latter conjecture is false ((1.29)),

it follows that there exists a left amenable semigroup which does not satisfy (SFC). Indeed, every right cancellative, left amenable semigroup which is not left cancellative does not satisfy (SFC). [An example of such a semigroup is the semigroup Fx pP of (1.29(iv)). Let S be right cancellative, left amenable, and not left cancellative. We can find r, s, t E S such that rs = rt and s # t. Suppose that

S satisfies (SFC). Let e = 1/8. Then we can find nonvoid K E f(S) such that [K - xKI < elKJ for x e {r, s, t}. Thus for x E {r, s, t}, JxK n K[ > (1- e)[K[, and so J{k E K: xk E K}[ > (1-e)[KJ. Hence if U = {k E K: sk 6 K, tk E K},

146

FOLNER CONDITIONS

then JUJ > (1- 2e)JKJ. Let W = sU U W. (So W C K.) If k E U, then sk i4 tk, yet r(sk) = r(tk). Thus if r- is the equivalence relation on W defined by w - w' whenever rw = rw', then JrWJ = JW/ - J, and as each equivalence class contains at least two elements, we have J W J > JrW J - 2. It follows that JrW J
- JsUJ > JsKJ - Js(K - U)J > (1 - e)IKI - JK > (i - e)JKJ - 2eJKJ = (1- 3e)JKJ

UJ

J W J. Now

z

So

(1- e)IKI < JrKJ < JrWJ + Jr(K -- W)J < 1 JKJ + 3e1KJ. Hence (1- e) < 1 + 3e and we have contradicted the choice of e.j Recently, long-standing problems posed by Namioka [1) on Folner conditions for semigroups have been solved by Z. Yang [S2]. Rosenblatt ([11, [2]) discusses a Folner type condition in connection with the general problem of measure and the translate property ((2.32)).

(4.23) The groups G for which £t(G) = 9tt(G). (Paterson [31, Milnes [4], Kotzmann and Rindler [1].) The following conjecture is made by Paterson: if G is an amenable locally compact group, then £t (G) = t(G) if and only if G E [FCJ-. We say (Palmer [1]) that a locally compact G is an [FC]- group (or G E [FC]-) if the closure of every conjugate class of G is compact. (Such groups are amenable and unimodular (Problem 6-7, (6.9) ).

The condition that £t(G) _ 'Rt(G) is easily seen to be equivalent to the condition every topologically left invariant mean on G is actually topologically invariant. The conjecture is obviously true when G is either abelian or compact. (Abelian and compact groups belong to [FC]-.)

The conjecture will be shown to be true when G is o-compact or discrete ((iii)). For general G, the conjecture seems to be open. (G) (i) Let G E [FCJ', and be either a-compact or discrete. Then £t (G) [Since m E £t (G) if and only if m* E tt(G), it is sufficient to show that £t (G) C 9tt(G). (For m*, see Problem 2-7.) Let {K6} (b E A) be a summing net for G with K61 = Kb for all b (Problem

If G is a-compact, take ((4.15)) A = P (so that {K6) is a summing sequence). Let {x6} be a net in G and u,5, £({xb}) be as in the paragraph 4-10).

preceding (4.17). By (4.17), it suffices to show that £({xb }) C 94(G). Since G is unimodular, A(K6xb) = A(Kb) and u,5 * x6 = XXsxa/A(Ko). If t E G, then [[pb * x6 * t - tts * xs[[ = A(K6x6t 6 K6xs)/,1(K6) = A(K6x6txa 1 0 K6)/X(K6) = A([K6z6tx616 K6]-1)/a(K5)

_.(xst-1x6 1K6 AK6)/A(K6), where we have used the symmetry of the K6 and the fact that G is unimodular.

Since G E [FCJ', we can find C E r'(G) such that x6t-1x6 1 E C for all b. Hence

A(x6t-1x61K6 A Ks)/A(K6)

0

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147

by (4.15(3)). It follows from the above that A(K6xot A K6x6)/a(K6) --' 0.

(1)

Using Problem 4-8 and the equality

A(K6x6t © K6x6)/.(K6) = X(t-1(xs 'KT') A xa 1Ka 1)/A(x61K61) to cope with the a-compact case, we see that the convergence in (1) is uniform (in t) on compacta. Referring to the earlier equalities, it follows that

IIA6*x6*t-P6 *x611-+0 uniformly (in t) on compacts, and thus the right-handed version of (4.3), with p = 1, gives

IIp6*x6*v-96 *x611-*0 for each v E P(G). Hence if m E £({x6}) (that is, m is a weak` cluster point of {(p6 * x6)^}), then my = m, and m E'Rt(G) as required.] (ii) Let G be an amenable, locally compact, a-compact group. Let {Kn} be a sequence in '(G). Suppose that x E G is such that (Cl(x))- is not compact, where Cl(z) is the conjugate class of x. Then there exists a sequence {xn} in G such that if A = U°n°__1 Knxn, then x ¢ A'1A.

(The sequence {xn} is constructed recursively. Let n E P and suppose that the elements x, (s < n) have been constructed so that if A8 = U;=1 Kx then x ¢ A,-'A.. If V E G, then (An-1

UKy)-1(An_1 UKny)

2

=A- 11An_1 U An

UKny U y-1Kn 1An_1 U y-1K; 1 Ky.

Let D = {y E G: yxy-1 ¢ Kn 1Kn}. If D is relatively compact, then, as C1(x) C DxD'1 U K; 'K,, we would have Cl(x) relatively compact, and so a contradiction. Hence D is not relatively compact, and we can find y E D such that x ¢ 1An_1 Setting xn = y and referring to (2), we have x 0 An1An This completes the construction of {xn}.] (iii) Let G be an amenable, locally compact group which is either a-compact

or discrete. Then £1(G) _ P(G) if and only if G E [FC]-. [By (i), if G E [PC] -, then Zt (G) = N(G). Conversely, suppose that £1(G) = 9(G). We deal with the a-compact and discrete cases separately.

Suppose that G is a-compact and not in [PC)-. Let {Kn} be a summing sequence for G and An = as usual. Since G V [FC]-, we can find x E G with (Cl(x))- not compact. Let {xn} and A be as in (ii), and m E Z({xn}). For each n, Knxn C A, and noting that XA E L..(G), we have 1 > (µn * Zn )^ (XA) = An (XAx ') >_ A.

1. Hence m(XA) = 1. Further,

(Fin * xn)^(x-1XA) = ({in * xn)^(XAz) = A(Knxn)-1.1(Ax n Knxn) = 0

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148

since Ax fl Knxn C Ax n A = 0 by (ii). So m(x-1XA) = 0 and m E .t(G) - 9t(G) c £c(G) - !(G). The resulting contradiction establishes the desired result in the a-compact case.

Suppose, then, that G is discrete and not in [FCJ- (= [FCJ). Then we can find x E G and a countable subset B of G with the set {bxb-1: b E B} infinite. Let H be the countable (and so a-compact!) subgroup of G generated by B U {x}, and note that H 15 [FC]. We shall derive a contradiction by showing

that £(H) ='3t(H). We will use the map 6: Z (H) - .C(G) of Problem 1-8. So l(p)(ti) = mo(,ip), where mo E 1r(G) is fixed, a n d b (x) = p((Ox))I H) for p E e(H), b E 1"(G), and x E G. Let T be a transversal for the left cosecs of H in G. For qi E 1... (H), define TO E 1.(G) by Tgs(th) _ ¢(h)

(t E T, h E H).

Let m E .C(H). We will eventually show that m E 9c(H). For to E T, ho E H, we have

(T0)m(toho) = m(((T.0)toho)[ H) = m(4ho) = m(4),

and so /3(m)(Tg5) = m(o). Since 8(m) EE(G), and, by hypothesis, £(G) _ 91(G), it follows that for hl E H, (3)

Q(m)(hl(T46)) = fj(m)(Tc) = m(q )-

But if z = toho with to E T, ho e H, then hi (T4')(x) = Tlp(xhi) = T4(to(hohl)) _ 0(hohi) = h10(ho) so that h, (TO) = T(hiO) Hence from (3), m(h10) =,6(m) (h, (T95)) = m(qS)

and m E 91(H). Thus E(H) = 91(H) giving a contradiction.]

(4.24) Amenability and statistics. The paper by Bondar and Milnes {1] is a useful source of information about the applications of amenability to mathematical statistics. Two typical such applications are discussed in detail in (4.25)(4.26). See also Problem 4-19. The first, and most well-known, of these applications is the Hunt-Stein Theorem ((4.25)). This theorem was proved during the Second World War by G.

A. Hunt and C. Stein and appeared in the unpublished paper Hunt and Stein [1). The usual proof of the Hunt-Stein Theorem involves the existence of an asymptotically invariant sequence of probability measures on a a-compact locally compact group G. A sequence {p,,} of probability measures in M(G) is called asymptotically right invariant (cf. Wesler [1]) if (Jun (Bx)-pn(B)) -+ 0 for all B E R (G) and x E G. If {pn} is such a sequence, then, since each qi E C(G) is the u n i f o r m limit of a sequence in Span {XB : B

.

( G ) } , A. (X`0 g5 - 0) --* 0 for

all ¢ E C(G), and any weak' cluster of {µn}, where µn is regarded as an element

of C(G)', is in 1(C(G)). Also, if G is amenable, then Property PI ((4.4(ii))) yields a sequence {vn} in P(G) with l[x * vn - vn[li 0 for all x E G, and it

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149

is asymptotically right invariant. Consequently, there exists follows that an asymptotically right invariant sequence for G if and only if G is amenable. However, as with Bondar and Milnes, we have preferred to use a fixed-point theorem in place of such a sequence when proving the Hunt-Stein Theorem. (See Bondar and Milnes [1, p. 111] for a converse to the Hunt-Stein Theorem.)

Applications of amenability to the theory of symmetric random walks on groups are given in Kesten [1], [2] and C. Stone [1].

An application of summing sequences to informational futures is given in Pickel [1]. Emerson (5], [61 makes interesting use of summing sequences in his develop-

ment of ideas in abstract probability due to J. C. Kieffer.

(4.25) The Hunt-Stein Theorem. (Hunt and Stein [1], Peisakoff [1], Wesler [1], Lehman [1].) Following Wesler [1], we shall discuss the theorem in the language of testing problems. Schmetterer's book [S] provides a useful source for the theory of hypothesis testing. Let (Z,.3, v) be a measure space with v a-finite, Cl a set, G a or-compact,

locally compact group, and P a mapping from Cl into the set of probability measures on (Z,R). We set P,,, = P(w) (w E fl). We require that both Z and fl be left G-sets, with e acting as the identity transformation on both Z and Cl. We also require

(1) the map (x, z) - xz from (G x Z,4 (G) xR) into (Z,3) is measurable; (2) v(xE) = v(E) for all x E G, E E .;

(3)P,,avfor allw,and PP,,,(xE)=P,,(E)for all xEG,wEfl,and E E R. (In particular, P,, E L1(Z) (= Li (Z, v)) so that P,,, E La, (Z)'.) A test is a measurable function 0: Z 10, 1). (So every test is in L.,(Z).) A test q5 is called invariant if, for all x E G, ¢(xz) = O(z) v-a.e. Let flo, Cl E .2 be G-invariant and be such that Cl is the disjoint union Co U f11. For each -1 E [0, 11, let

T.r={0:0is atest and E,,(¢) aF(¢) + (1 - a)F(V)) for all x E G, 0, E T., and a E 10, I], and so A.r is convex and G-invariant. Now apply (a) to obtain an invariant, minimax test for T.I.)

(4.26) Amenability and translation experiments. The definitions below are based on Le Cam [1], [2]. In the discussion below, we could have taken A and B to be abstract L-spaces. However, every abstract L-space can be realised as some LI (X, is), so that there is no loss of generality in the concrete approach adopted below. Let X be a locally compact, Hausdorff space and Ato a positive, regular, Borel

measure on X. Let A = LI(X) (= LI(X,µo)), and P(A) = P(X,1o). Thus

P(A)={zELI(X):v>0, v(X)=1). An experiment for A (over 8) is a function P : 8

P(A), where A is some

nonempty set.

Classically we can think of 8 as a set of possible theories about a certain physical system on which we perform an experiment. We take X to be the set of possible results of the experiment, and the probability that, on the basis of a theory B, the result belongs to a measurable subset E of X (an "event") is just

P(8)(E). Now suppose that Y is a locally compact, Hausdorff space, vo is a positive, regular, Bore) measure on Y, and B = LI (Y) (= LI (Y, vo)). Suppose that we have an experiment Q for B over the same set (of "theories") 8. Both experiments P, Q will give us information about 8. In order to compare the amounts of information provided by experiments over

e, Le Cam introduced a "distance" function 6(P, Q). For motivation, observe that to effect the comparison, we need to "lift" the information from one space into the other. This "lifting" is achieved by means of "transitions." A transition from A to B is a linear mapping T: A - B such that

(a)T/.i>0if u2:0in A; (b) [[T,c[[ = [[j.c[U if u > 0 in A.

The set of transitions from A into B is denoted by T(A, B). (Such operators are called stochastic in [Sch, p. 191}.)

FOLNER CONDITIONS

151

Let T E T (A, B). We claim that T E B(A, B) with 11T1I = 1. [This is well known (cf. [Sch, Chapter II, (5.3))) and can be proved as follows. If it E A = L1(X), then, using the Jordan decomposition, we can write y = µl -Ill+i(A3

-

144) with µi >- 0 in A and Dill 5 IIufl Applying (b), we see that [[Tµ[I O in L.(X) if ¢(y) > 0 for all u 2: 0 in LI(X), it follows that T*0 2: 0 whenever' 0 in L,, (Y). Further, T"1(u) = µ(X) (by (b)) for all It E L1(Y) so that T'1 = 1. Evaluating (T` b)^ at points of the maximal ideal space of LA(X) gives states on L. (Y), and it follows that T*, and therefore T, is norm decreasing.) Clearly, T(A, B) is a convex subset of B(A, B). A measure of the amount of information yielded by P relative to Q is provided by the real number b(P, Q), where we define 5(P, Q) by

b(P, Q) =inf 10Ee sup [[Q(8) - T(P(8))[I : T E T(A, B) I . In general, the computation of 6(P, Q) is difficult. However, the situation is

better when we consider translation experiments. (Such an experiment is just a variant of a "translation family" introduced in Problem 4-19.) Let G be a locally compact group. A translation experiment (for G) is an

experiment P for Ll (G) over G of the form for some p E P(G), P(x) = xp for all x E G. We will write p in place of P. Writing T(Li(G)) in place of T(L1(G), L3 (G)) we have, for µ, v E P(G),

b(µ, v) = inf sp[lxv - T (xp)I[ : T E T(L1(G))

= inf

sup [[ v

IzEG

- x-1T (xp)[I : T E T(L1(G))

l 11

The next result shows that when G is amenable, then, for µ, v E P(G), b(p,v) is much more manageable. For this result, see Boll [1), Torgersen [1], and Paterson [7). What can be said when G is not amenable? What can be said, even, for G = F2? Paterson [7) gives a result corresponding to the one below for left amenable semigroups.

Let G be amenable and s, v E P(G). Then (2)

b(p, v) = inf{[Iv - yCll : g E P(G)}.

[The essential ideas of the proof of (2) are as follows. The space B(L1(G)) is a left Banach G-space, where we define (3)

(xT)(e) =

xT(x_1£)

(la E L1(G)),

and T (L1(G)) is a convex subset of B(L1(G)). If, somehow, we could "find" a weak' topology on B(L1(G)), we might hope to have T(L,.(G)) weak' compact and to use an amenability fixed-point theorem to obtain a fixed-point To in T(L1(G)). Hopefully, with equation (1) in mind, we can choose To so that supxEG 11 v - x-1To(xp)[I is close to b(p, v). Since To is a fixed-point for G, we

FOLNER CONDITIONS

152

see from (3) that To(xf) = xTo(e) (g E L1(G)), that is, To is a multiplier of L1(G). Thus by Wendel's Theorem (HR2, (35.5)J, we can find 17 E M(G) such that To (p) = prl. Hopefully, we can take i E P(G) and we would have sup llv - x-1To(xA)ll = iiv - 91711

MEG

and so the right-hand side of (2) is less than or equal to its left-hand side. The reverse inclusion is easy.

This crude argument falls down in two crucial aspects. First, if we want a weak* topology, we ought to replace

B(L1(G)) = B(L1(G), L1(G)) by B(L1(G), M(G))

since the latter space has the natural weak* topology, which it inherits when identified with the dual space (L1 (G)®Co(G))'. Even with this replacement and T(L1(G)) replaced by T(L1(G),M(G)), there is a second problem, since the lack of an identity in Co(G)' in general, jeopardises the weak" compactness of PM(G) and hence of T (L1(G), M(G)). We will suppose that G is not compact, since the latter case is easier. To overcome the above difficulties, we replace Co(G) by C(G,o), where G. is the one-point compactification of G. Then C(G(,,) can be identified with the space Co(G) a C1 of functions in C(G) that tend to a limit as x - oo. Clearly, can also C(G.,) is a G-invariant subspace of C(G). The G-action on be obtained in the canonical way from the natural action of G on

zoo=oo=oox (xEG). We will identify M(G) in the obvious way with the following subspace of

M(G ): {µ E

µ({oo}) = 0}.

Clearly (4)

M(G)

CS,,

where S is the point-mass at oo, and the right-hand side of (4) is a Banach space direct sum. We now proceed to the proof of (2). Let e > 0, and define K, as follows:

K, = {T E T(L1(G), M(G )) : Jlxv - T (zp) ll < 6(jj, v) + e for all x E G}. Since

T(L1(G)) C T(L1(G), KK is not empty. Clearly K, is convex in X = B(L1(G), Now X is canonically identified with (L1 an operator T E X corresponding to the linear functional f 0 0 -. (TC)" (0). We now show that K, is weak' compact in X. Let T6T weak' in X with T6 E Ke for all 6. Then for µ > 0 in L1(G) and ¢ >- 0 in we have (Tp)"(O) = lim6(T6p)"(.0) > 0.

F$LNER OONDITIONS

153

So Tp>0,and IIT,I1= (Tu)^(1) =1i n(T6p)^(1) = [[lull.

So T E

and Ke is weak* closed in X. Since KE is also bounded in X, it is weak' compact. In order to apply an amenability fixed-point theorem, we need an action of G on KE. In fact, we can make X into a left Banach G-space (with KE G-invariant) is a right Banach G-space under the action: as follows. The space

(zEG), commented earlier. where we regard C(G) as Dualising this action, we obtain that X is a left Banach G-space. One readily checks that this action is given by (5)

(xT)(e) = x[T(x 1C)J

(x E G).

Note that IIvT[I = IITII for all y E G, T E X. We claim that C U(G). Indeed, C(G,o) = Co(G) ® Cl, and since CM(G) is norm dense in Co(G) and is contained in U(G), it follows that Co(G), and so C(G,o), is contained in U(G) as claimed. x in G and T6 T weak' in a bounded subset of X then, since, for If x6

each 0 E C(G ), E E Li(G), the maps ¢ --> ¢t, t - t-1g (t E G) are norm continuous, it follows using (5) that

(z6T6)(e)(cb) =T6(x61)(4'xs) - (zT)(V¢) so that the action of G on every bounded subset of X is jointly continuous for the weak' topology. If T E KE and x, y E G, then lxv - (yT)(xp) fl =1I(y-1x)g - T((y-1x)u)ll 2 so that µ 0 m. Hence .0(X) 0 {Ii).

FOLNER CONDITIONS

156

For the converse, it is sufficient to show that if X does not admit arbitrarily small, almost invariant sets, then .C(X) = {µ}. Suppose, then, that X does not admit arbitrarily small, almost invariant sets. Then using the countability of G, there exist e, k > 0 and a finite subset F of G such that if A E a is nonnull and p(sA A A)/p(A) < £ for all s E F, then u(A) > k. Let us say that a sequence {Bn} of nonnull, measurable subsets of X is a Folner sequence if p(sBn L Bn)/p(Bn) -* 0 for all s E G. If {Bn} is a Folner sequence, then, from the argument of the first paragraph of the present proof, every weak' cluster point of the sequence {(XBn Jp(Bn))A} belongs to Z(X). Let m E .E(X). We need to prove that m = A. The proof of this falls into two parts

(a) and (b). In the first, we suppose that m is such a weak' cluster point, and in the second, we deal with the general case.

(a) Suppose that m E £(X) is a weak' cluster point of {pn}, where pn = and {Bn} is a Folner sequence. We can suppose that

p(sBn A Bn)/p(B,) < £

for all s E F and all n, so that we have p(B,,) > k for all n. For C E R. we have

P.- (C) = U XB du1) /(p(Bn)) = p(C fl Bn)/p(Bn) k-iJ///p,

and so pn 0. Let B = A - s-'A = s'i (sA - A). Then p(B) > 0 and f (b) < 1, f (sb) > 1 for all b E B. Since m(B) = m(sB), we have

fB f dp =

f

B

fdp

So

p(B) > f fdp = f fdp ? p(sB) = p(B)

s

eB

giving a contradiction. So p(sA - A) = 0 for all s E G, and p(A - sA) _ p(s-1A - A) = 0 also. Thus p(sA A A) = 0 for all s E G, and since G is ergodic, p(A) = 0 or 1. If p(A) = 1, then

m(X)= 1 fdp 1}) = 0

so that f = 1 p-almost everywhere, and m = it as required.

FOLNER CONDITIONS

157

We note that, from the above, µ is the only weak* cluster point of {p,,}, so µ weak*. that pn (b) Now suppose that m is a general point in .>r(X). Suppose also that m 34 µ. We will eventually derive a contradiction.

Let a = sup{;8 > 0: m >:,6p on .3}. Then (m - aµ) > 0. Since Jim - aAJJ _

(m - aµ)(1) = 1 - a and m

µ, it follows that a < 1. By subtracting aµ

from m and dividing by (1- a) we can suppose that if Q > 0 then it is not true that ,8µ < in. For each p E P, it follows, by putting 8 = 1/p, that there exists nonnull CP e.9 such that µ(C,) > pm(CP).

(2)

We now adapt ideas in (0.8) and (4.2). We can find a net {gs} in P(X) such that ga --> m weak*. (Here, of course, P(X) is the set of probability measures in L1(X).) For each p, pg6(CP) --* pm(CP) < .(CP) by (2). Using the countability of G, we can construct a sequence {f,,} in P(X) with

Iis*fn-fnit1-'0

(3)

(sEG)

and

(4)

p(CP) > pfn(CP)

(Of course, s * f (x) = f

(s_1x).)

(n > p)

Our next task is to modify the proof of (4.7). We will assume that G is denumerable, the case where G is finite being easier.

Enumerate G = {x,: r > 1}. Let n > 1. By choosing f, for suitably large r and then approximating f, by simple functions, we can find a simple function

N

N gn=E,$n(xB,/u(Bn))

E$

=1)

f=1

f=1

///

such that BnCB; if 15 r<s5 Nn, y(Bn)>0for 1_ p).

Arguing as in (4.7), with summation over {x1, ... , xn} replacing integration over C,

N.

n JJ}}

f=1

E({t(xiB, A B)/u(BB)) < n- n-' = n-1. i=1

Let U. = {r: Ej 1(,A(xiBn A Bn)/u(Bn ))
, An E(,a(xiBn A Bn)/u(B, )) 2! (rEVn E 6n) 2n 1, rEV.

i=1

JJlll

so that rEV

>4r < 2

In particular, since Z;_"1$ = 1, it follows that Un 9' 0.

FOLNER CONDITIONS

158

Summarising, we thus have, for each n, a nonvoid subset U of [1; Nn] and for each r E U,,, a nonnull set B* E 3 and a number P > 0 such that (6)

µ(xiB; Ll B,)/µ(Br) < 2n'1

(1 < i < n, r E Un), 31

rEU 0:5 1` 6 n(XB; /p(B, )) < g,.

(8)

rEU,

By choosing a suitable subsequence, we can suppose that ErEU Q; - t where, from (7), z < t < 1. We now claim that for each C E ,3, (9)

max E(XB; (C)/Fi(Br )) - p(C) M -+ 0

rEU

as n -- oo. For if not, we could, using (6), construct a Folner sequence {An), where each A,n is some BT, with a weak* cluster point n for which n(C) µ(C), and the conclusion of (a) is contradicted. This establishes (9) and it follows from

(7) and (9) that

a; (XB,^/A(Bn))^(C) - (E

rEU

p:)

(C)l

0.

12i

Recalling that F,,EU, f; -- t, we obtain rEU

Q:`{XB /u(Br ))^ - til

weak*. But this is impossible, since a consequence of (8) and (5) is that for every p E P, IA(C,) lim

n-00

1 'r (XB, /p(B"))A(Cp) 5 P

rEU,

and so t < p-' for all p. But t > .1. This contradiction establishes (i).] A related question is the following. Let G be a compact group. A function E L,,,(G) is said to have a unique left invariant mean if m(o) = A(qS) for all m E £(G). Since ,C(C(G)) it is trivial that every 0 E C(G) has a unique left invariant mean. A larger set of functions in with unique left invariant mean is the set of Riemann measurable functions on G (Problem 4-34). See Talagrand [S2], [S4).

A sufficient condition for a function 0 E L. (G) to have a unique left invariant mean is the following: given e > 0, there exist n > 1 and elements x1,. .. , xn E G

such that (10)

n-1 (tz) - (1.0d,\) 111_ <e. 11

FOLNER CONDITIONS

159

(By applying m E .C(G) to the L.-function on the left-hand side of (10), the condition is obviously sufficient.) Talagrand (5] asserts that this condition can be used to construct for many compact groups G, functions 0 which are not Riemann-measurable and yet have a unique left invariant mean. Rubel and Shields (1J raised the following question: suppose that 0 E Lo, (G) and that 00 has a unique left invariant mean for all ' E C(G). Does it follow that 0 is Riemann-measurable? Talagrand shows that this is not the case for many compact groups, using (10) in his proof. See Rosenblatt and Yang (SJ for recent work on functions with unique left invariant mean.

(4.28) The Banach-Ruziewicz Problem: the S"-version. In the sequel we shall often abbreviate "the Banach-Ruziewicz Problem" to "the B-R Problem". Recall ((3.12), (4.27)) that this problem (for R") is: does there exist a finitely additive, Gn-invariant, positive measure µ on .4b(R") such that An Q0,

1]n) = 1 and µ 34 An?

With the uniqueness result (i) of (4.27) in mind, it is desirable to change the problem to one in which we have a probability measure in place of the infinite measure An. This is achieved by studying the corresponding problem on

the (compact) set Sn-1. Recall that Sn-1 is invariant under the action of the orthogonal group 0(n) C Gn. The required probability measure µn_1 on Sn-1 is obtained from An as follows. For each subset E of S11'1, define E' C R" by

E'={ax:xEE, 0 2 being similar. (Note that S" has an Hn-paradoxical decomposition for n > 2 by Problem 3-18.) As in the proof of Tarski's Theorem ((3.15)), let y = S2 x P and H = H2 x II(P). Identify S2 With S2 x {1). Then H acts on Y. Let m E P. Since S2 has a paradoxical decomposition, m S2 = U (S2 x {r}). r=1

(1)

For such an m, "slice" the sphere into parts A0,.. . , Anz_ 1 by the planes

y cos(2r7r/m) - x sin(2r7r/m) = 0

(0 < r < (m -1)).

(The process is rather like separating a skinned orange into its divisions.) More precisely, let A,.

((cos ¢)

1 --z2, (sin ¢)

1 - z2, z)

- 1 < z < 1, 2r7r/m < 0:5 2(r + 1)1r/m}

.

Then S2 - {(0, 0, 1), (0, 0, -1)} is the disjoint union of the A,.'s. Further, A,. A0 with respect to H2 since rotation about the z-axis through an angle of 2r7r/m carries A0 onto A,_ Since singletons are µ2-null and 142 is H2-invariant, it follows

that µ2(A,.) = (92(S2))/m = 1/m. The measure µ also vanishes on singletons (since any point can be rotated to infinitely many other positions and µ(S2) = 1.)

Hence µ(A,.) = 1/m as well. Finally, as in the proof of the Banach-Tarski Theorem ((3.16)), S2 - ({(0, 0, 1), (0, 0, -1)}) = S2 with respect to H2. Hence uf_01 A, = S2 with respect to H2, and using (1) and the Division Theorem ((3.14)) (for the action of H on Y), we obtain (2)

Ao = S2.

FOLNER CONDITIONS

161

Now let E E 3 (S2) be p2-null. From (2), we can find a partition E1,... , Ek

of E and elements x1, ... , xk E H,, such that xiEi n x; E1 = 0 if i # j and Uk 1 xiEi C AO. Normally, one cannot relate p to the sets Ej since these sets will not, in general, belong to 4'(S2). However, in the present case, Ei is a subset of a p2-null set, and so itself is p2-null (and, in particular, belongs to

£(S2)). Thus

p(E)=p(IJE1) =p(uxiEi1 0, C E'(G), we can find t;E E J5,r with 1le.11 =1 and 11ir(x)ee - CE11 < e for all x E C. But then 10r(x)CE, SE) -11= I((ir(x) - I)eE, CE)I -< llir(x)CE - EE11 < e,

and since 1 is the only function associated with iro, it follows from the definition of the G#-topology that irn -+ iro.]

A much stronger notion than that of "weak containment" is that of "containment." If Tr,ir1 E RepG, then we say that ir1 is contained in it if there exists a ir(G)-invariant, closed subspace 55' of 55, such that the representation x , ir(z)l f t is unitarily equivalent to 7r1. It is almost trivial that if Ir1 is contained in ir, then ir1 is weakly contained in it.

(b) Definition of Property (T). The group G is said to have Property (T) if, whenever it,, -+ rro in G#, then it,, contains iro eventually. We now discuss some facts about groups with Property (T). These facts are due to Kazhdan [1], and detailed proofs are given in Delaroche and Kirillov [1]. We shall not reproduce the full proofs here. However, for the convenience of the reader acquainted with the representation theory of locally compact groups, the easier proofs will be sketched. If G has Property (T), then {iro} is an open subset of G. [To see this, suppose that G has Property (T) and that {Tro} is not an open subset of G. Then using the first countability of G, there exists a sequence it,, -+ 7ro in G with it,, # iro. Since it,, -+ iro in G#, Property (T) gives that Trn contains 7ro eventually, and so _ 7ro eventually by irreducibility. This is a contradiction.]

The converse to the preceding result is also true (using disintegration theory) so that G has Property (T) if and only if {aro} is an open subset of G. Obviously, if G is compact, then G has Property (T), since in that case, G is discrete. Also, if G is abelian, then G has Property (T) if and only if G is compact, since the dual of an abelian, locally compact group is discrete if and only if the group is compact. Now if H is a closed, normal subgroup of G, then the canonical map from (G/H)^ into G is continuous, so that G/H has Property (T) if G has Property (T). Combining the last two assertions, we obtain that if C has Property (T) and H is the closure of the commutator subgroup of G, then G/H is compact. This easily implies that groups with Property (T) are unimodular. We now come to the two results involving Property (T) that are of particular importance for solving the B-R Problem.

Recall from (1.11) that if H is a closed, unimodular subgroup of G, then there is a canonical measure aG1H on G/H associated with the function q(z) =

164

FOLNER CONDITIONS

AG(x)-1. The subgroup H is said to have cofinite volume if )1GIX(G/H) < oo. If H has cofinite volume, then \GI U is G-invariant.

(c) Let H be a closed, unimodular subgroup of G with cofinite volume. if G has Property (T), then H also has Property (T). (From the Mackey theory, each ir' E H# induces a representation ¢(ir') E G. Further, ¢ is continuous. Let ira be the trivial representation of H. Then the finiteness of )\G/H (G/H) yields that ¢(1ra) contains iro. (Indeed, the space of #(ar'o) is just L2(G/H), and every constant function in L2(G/H) is fixed under the action of ¢(irfl)(G).) So if ,r;, ira in H#, then, using the definition of the G#-topology, ¢(ir;,) Tro in G#. Since G has Property (T), ¢(ar,,) contains Tro eventually. By looking at the formula relating ir,, to b(ir;,), we see that 1r;, contains Tra eventually. Hence H has Property (T).] The second result on Property (T) that we shall need in the solution of the BR Problem lies considerably deeper and involves the theory of algebraic groups. We shall not attempt to sketch this theory. References for the theory are Borel

[1], Borel and Tits [1], and Bruhat and Tits [1). An excellent, introductory account is given in the book by Humphreys [1]. We will be content to state the theorem and then comment briefly on it. (d) Let G be an algebraic, semisimple, Lie group over a nondiscrete, locally compact field F, and be such that its simple components are of rank > 2. Then G has property (T). The rank of a simple algebraic group is defined to be the dimension of a maximal torus of G. The basic examples of a group G with rank > 2 are SL(3, F) and Sp(4, F), and the proof of (d) in general reduces to the consideration of these two cases. The next two results enable us to use the uniqueness result of (4.27(i)) to obtain uniqueness results for uncountable groups through the mediation of countable subgroups with Property (T). These results are stated in Margulis [2]. See also Rosenblatt [14] and Sullivan [1]. (iii) Let (X, R, z) be a probability measure space and G a group of invertible, measure-preserving transformations of X (as in (4.27)). Let M be a countable subgroup of G that is ergodic on X and (as a discrete group) has Property (T). Then f is the only G-invariant mean on [Let 7r: G - B(L2(X)) be given by

(zr(s)f)(x)= f(s'1x)

(sEG, f EL2(X), rEX).

It is routine to check that it is a unitary representation of the discrete group G on L2(X). Since every G-invariant mean on L,,(X) is also M-invariant, it is sufficient to show that

{P}=.CM(X) (={mEW(X):sm=mforallsEM}). Applying (4.27(i)) (with M in place of G), it is sufficient to show that there do not exist arbitrarily small, almost invariant sets in X for the action of M.

FOLNER CONDITIONS

165

Suppose that there do exist such sets. Then we can find a sequence {An} of nonnull, measurable subsets of X such that /.c(An) -+ 0 and

#(xA. & An)/u(A,) - 0

(3)

(x E M).

Let

LO(X)= {9EL2(X): f gdli=0}. (Note that since µ(X) < oo, L2 (X) C LI(X) and the preceding integral makes sense.) Clearly, LO (X) is a closed subspace of L2 (X) that is invariant under

r(G). Let fn = (XA - p(An)1) E L02(X)./ Then for s e M, (4)

IIr(3)fn - fnII2 = f IXA.13_1x) - XAn (x)12 dg(x)

=

f

dfb(x) = Fi(sAn A An).

Also (5)

11f- 1122 =

ftxA

- 2,u(An)XA + [ i(An))21) d i

= p(An) - 2[,u(An)]2 + [t4An)]2 = p(An)I1 - p(An)J.

Since tz(An) -+ 0, we can suppose that p(An) < 1. Let g,, = fn/!Ifn0I2. From (3), (4), and (5), it follows that for each s E M, (6)

117r(s)g. - gn112 = p(sAn L An)/[p(An)(1- u(An))) -' 0.

Let 15 be the closed, ar(M)-invariant subspace of LO (X) generated by fib,:

n E P}. Let r(s) = r(s) I j for s E M. Then r' E M#. It follows from (6) and (ii), (a) that w,, ro in G#, where ir,, = r' for all n > 1. Since M has Property (T), 7ro is actually contained in r'. Hence there exists f E LO(X) with

Iifli2=land ir(s)f = f

(s E M).

Now f e LI (X), and the positive and negative parts of Re f and hn f are also r(M)-invariant. Also M is ergodic on X and arguing as in the proof of (4.27(i)), we see that each of these parts is constant almost everywhere. Hence f is constant

almost everywhere, and since f f du = 0, we obtain f = 0 almost everywhere. This contradicts the fact that HfII2 = 1, and the desired result follows.)

(iv) Let G be a compact group, M a subgroup of G, X a locally compact, Hausdorff space that is a left G-set, and let µ E PM(X) be such that sx is measurable and measure preserving (a) for each s e G, the map x with respect to lc;

(b) for each z E X, the map $ - sx is continuous from G into X; (c) G acts transitively on X (that is, if x, y E X, then there exists s E G such that sx = y);

166

FOLNER CONDITIONS

(d) M has Property (T) (as a discrete group), and is a countable, dense subgroup of G.

Then µ is the only G-invariant mean on p). [The idea of the proof is to identify X with a quotient GIL, where L is a closed subgroup of G. This enables us to identify p with AGIL, and the result becomes a domestic matter for G. Let as E X and L be the stabiliser of xo; thus

L = {sEG: sxo=xo}. Using (b), L is a closed subgroup of G, and using (c), the map sL -+ sxo is a bijection from GIL onto X. The latter map is continuous, since the quotient map Q: G -+ GIL is open and (b) holds. It follows that X is compact (since G is), and since X is also Hausdorff, X and GIL are homeomorphic. Thus we can (and shall) identify X and GIL as left G-sets. We now claim that p = AG/L, where AG, ay, and AG1y are connected by Weil's formula ((1.11(1))). For let p' be the regular, Borel measure on G given by

j

IGIL dp(xL) JL cb(xl)dAL(l) for ¢ E C(G). Clearly, by (a), p' is a left Haar measure on G with total mass 1, and so coincides with AG. Now (7) is valid with ' replaced by XU, where U is open in G, and the Monotone Class Lemma yields the validity of (7) for ' = XA with A E. (G). Let E E.9(GIL). With ' = in (7) and using Weil's formula, we obtain p = AG/L as required. (7)

We now show that the action of G on X is ergodic and that for each A E .9 (X) , the map s - p(sA A A) is continuous on G. Indeed, with A = AG, p(sA A A) = .1(Q-' (sA IJ A)) = A(sQ-' (A) A Q-' (A)), and the desired results follow by (1.2) and the ergodicity of the action of G on G.

(For the latter fact, note that if E E R (G) with A(E) > 0 and A(sE A E) = 0 for all a E G, then the measure A -+ A(E ft A) is a Haar measure on G.) Finally it will be shown that M is ergodic on X. The fact that µ is the only p) will then follow from (iii). G-invariant mean on Suppose that A E . ' (X) is such that p(tA A A) = 0 for all t E M. Since the

map a -. p(sA A A) is continuous on G and M is dense in G, it follows that p(sA A A) = 0 for all s E G. Thus p(A) = 0 or 1, and so M is ergodic on X as required.]

The preceding results give a criterion for determining when the answer to the B-R Problem is affirmative. Recall that Hn = O(n+ 1).

(v) Let n E P and suppose that there exists an algebraic, simple Lie group G' over R with rank > 2 and a countable group M such that (a) M is a subgroup of both G' and Hn; (b) M is a discrete subgroup of G' and of cofinite volume in G';

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167

(c) M is dense in H,,.

Then £(S") = [From (ii) (d), G' has Property (T). Using (ii) (c), M also has Property (T). It is now sufficient to check that the conditions of (iv) are satisfied with G = H",

X = S", and p = µn. Conditions (a) and (b) of (iv) are obviously satisfied. From the above, M has Property (T). The latter fact, together with (v) (c), gives (iv)(d).

It remains to show G acts transitively on S". Let p,q be distinct points of S", and 0 be the origin of R"+1. Then the orthogonal transformation x

xI2(p-q).x

l Ilp - q[12

(P

q)

reflects points of Rn}1 in the hyperplane through 0 perpendicular to PQ and carries p into q. This establishes (iv)(c).J

(vi) Sketch of Sullivan's solution to the B-R Problem for n > 3. This solution is given in Sullivan [1]. Margulis [2] gives another solution. (Drinfeld [S) has settled the remaining cases n = 2,3 so that for all n > 2, £(S") = {µ.,i}.) The solution is obtained by applying (v). The group G' is taken to be O(n - 1, 2), where O(n - 1, 2) is the group of (n + 1) x (n + 1) invertible, real matrices preserving the quadratic form

/2

r

xp2 +X1 +...+xn_2 - V Lxn_1 - v Z2. Then O(n - 1, 2) is an algebraic, simple, Lie group over R with rank >- 2. Now let M be the set of elements of O(n - 1, 2) with matrix coefficients of

the form (n + ms) (n, m E Z). Results from the theories of arithmetic and algebraic groups give that M is a discrete subgroup of cofinite volume in G. It remains to identify M with a dense subgroup of O(n+1). For A E M, let A

be the matrix obtained by replacing each matrix entry (n + mf) by (n - mf ) and let k = {A: A E M}. Then k is isomorphic to M in the obvious way,

and since the map (r + sv) - (r - sf) is an automorphism of Q(f), the elements of k preserve the quadratic form

r 220+x1+ +xn_2+v1 xn_1+fxn.

It is elementary that we can find an invertible (n + 1) x (n + 1) matrix C such that, for all B E M, the matrix CBC-1 preserves the quadratic form

G+x2i

n.

Thus the map A CAC-1 enables us to identify M with a subgroup of Hn. Again using the theory of algebraic groups, it can be shown that M is a dense subgroup of Hn. Thus (v) applies.

xT (vii) Other related results. For our first such result, let T" = T x be the n-dimensional torus. Then T" is a compact, abelian group. Let H = Aut T". Since T" is compact, it follows that each element of H preserves the

168

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Haar measure u on T". By B21, H contains SL(n, Z) as a subgroup. Let n > 3.

We can apply (ii)(d) to obtain that SL(n,R) has Property (T). The discrete subgroup SL(n, Z) of SL(n, R) has cofinite volume in SL(n, R) (cf. Humphreys [2, Chapter 3]) and so, by (ii)(c), also has Property (T). Finally, the action of SL(n, Z) on T' is ergodic, and we obtain from (iii) the following result (Losert and R.indler [1], Rosenblatt [131, Schmidt (11, and Margulis [2]):µ is the unique H-invariant mean on La, (T", p). Another application, pointed out by Rosenblatt, concerns the uniqueness of left invariant means on a compact group G. Of course, . (G) = {A }, but what can one say about Z(G)? If G is finite, then, obviously, £(G) = {AG}. However, if G is infinite and amenable as a discrete group, then .C(G) is infinite ((7.19)). The natural question is can it ever happen that .E(G) = {Ac} with G infinite? The answer to this question is affirmative. Indeed, let G = O(n + 1), where n > 3. From (vi), G contains a countable, dense subgroup with Property (T). Now apply (iv) with X = G, p = AG to obtain that .C(G) = {.1G}.

(4.30) Von Neumann's Conjecture. The most celebrated conjecture in the history of amenability is von Neumann's (or Day's) Conjecture ((0.16)): a (discrete) group is not amenable if and only if it contains F2 as a subgroup. Evidence in favour of the conjecture is provided by (3.9), Problem 3-4, and the

groups G. However, A. Yu. Ol'shanskii ([2], [4]) has constructed a group Go that is not amenable and does not contain F2 as a subgroup, so that the conjecture is false. Our discussion.of this remarkable piece of mathematics falls into two parts, in (4.31) and (4.32) respectively. The actual construction of the group Go giving the counterexample involves a long and intricate argument in combinatorial group theory, and we shall not attempt to describe the details. A brief discussion of the group is given in (4.33).

From the construction of Go, it is clear that Go does not contain F2 as a subgroup. How are we going to show that Go is not amenable? Now the group Go is finitely generated-indeed, it is generated by two elements-and so what is needed is a criterion for deciding when a group G on k generators ul, ... , uk is amenable. We can obviously take k > 1. Given such a group G, there is a canonical homomorphism Q: Fk G, where Fk is the free group on k generators and Q carries the canonical generators of Fk onto the G-generators {u,. .. , uk }. Let N = kerQ, so that G = Fk JN. One would expect that the "size" of N (in a suitable sense) is critical in determining whether or not G is amenable. If, for example, N is too "large" (e.g., if N contains all commutators, so that G is abelian), we will have an amenable group G that is too special. If, however, N is too "small," then G will be too "close" to Fk to allow it to be amenable. To make these intuitions precise, define, for each n E P, an integer y" > 0,

where ry" is the number of elements in N of length n (in the generators of

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169

Fk). Suppose that N is nontrivial. We will show that 72n > 0 eventually, and limn....,oo(72n)1/2n ='y exists. It is easy to show that 'y> (2k The following deep theorem, due to Grigorchuk (and Cohen), is proved in 1)1/2,

(4.32): (*)

G is amenable if and only if lim (72n )1/2n = (2k - 1).

n-co

(*} now provides a criterion for establishing the nonamenability of Go. For the detailed construction of Go enables one to estimate (72n)1/2, for Go and to show that limn.oo(72n)1/2n = 7, is not 2.2 - 1 (= 3)! The proof of (*) proceeds by a surprising route. Recall that 1r2 : G B (12 (G)) is the left regular representation of G. It follows from (4.20 (ii)) that if g E 11(G) is > 0, self-adjoint, and is such that its support generates G, then G is amenable if and only if 11vr2(9)11 = 119111-

In order to use this result, we have to pick a suitable g E 11 (G) and calculate 111r2 (9) 11 in terms of 7. We choose 9=uk1+uk11+...+11i1}u1 +...+uk_1+uk.

{1)

To prove (*) it will be shown that

11r2(9)II = 7+ (2k - 1)/7.

(2)

This easily gives (*), since for 7 >

2k - 1 > 1, 7 + (2k - 1)/7 = 2k (= u ghh1)

if and onlyif -y = (2k-1). The proof of (2) is somewhat involved. An expression for 1H1r2(f)jJ, where

f = f"' in G, will be obtained in (4.31). This expression involves the natural trace on the C'-algebra Ct (G) generated by 1r2 (G), and facilitates the calculation of II1r2(f)HH. We will also require the value of the norm of 1r2(h) E B(12(Fk)), where h E 11(Fk) is defined as in (1). Finally, a power series argument, described in (4.32), gives the desired equality (2).

(4.31) On the norms of operators in Ct (G). Recall that 7r2: G B(12(G)) is the left regular representation of a (discrete) group G and that CI(G) is the C'-subalgebra generated by 1r2(G). The map 1r2 extends to a norm-decreasing, '-representation of the Banach '-algebra 11(G). This extension will also be denoted by 1r2. For T E Ct (G), define

trT = (T(e),e). Here, of course, e, and in general, any element of 11(G), is identified with an element of 12 (G) in the obvious way.

In the language of operator algebras )D2, (6.1.1)), the restriction of tr to the positive part of CI (G) is a "finite trace". It is easy to show that tr is a state on Ct (G), and tr(TS) = tr(ST) for all T, S E Ct (G). Clearly, trT is easier to calculate, in general, than 11T11. The next result shows

that we can use tr to calculate the norms of operators V2 (A where f = f"' E 11(G). For this result, see Kesten (2).

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170

(i) Let f E 11 (G) with f"' = f 96 0. Then (1)

1172(f)N =

lin00[tr(7r2(f)2n)]1112n).

(Let A be the commutative C'-algebra generated by T = 7r2 (f) and I, and let X be the carrier space of A. Now it is routine that SPA(T) = Sp(T), the spectrum of T in B(12 (G)), and since T generates A, the map f -+ f (T) from X onto Sp(T) is a homeomorphism. We thus can, and shall, identify X with Sp(T) C: R. So t(t) = t for all t E X: The linear functional S -+ tr S is a state on C(X), and so, by the Riesz Representation Theorem, there exists i E PM(X) such that

trS=f Sdp x

for all S E A. We now show that X is the support S(p) of 1A. For suppose otherwise. Then X S(µ) is a nonempty, open subset of X, and so there exists non-zero W E A with W > 0 and W (t) = 0 for all t E S(µ). Let V = W11'. Then 0 = tr W = (W (e), e) = 11 V (e)112

so that V(e) = 0. Recall ((2.35(H))) that it,. is the right regular representation of G and that ir,(ll (G)) is in the commutant of VN(G), and so of Ct (G). Hence, for all z E G, (2)

V(x) =V7r,. (x) (e) = 7r, (x) (V (e)) =0.

It follows that V(12(G)) _ {0}, so that V and W are 0. This contradiction establishes that $(A) = X. We now prove (1). For all n > 1,

tr(T2n) = f ten d,c(t)

(3)

x

Let M = 11T211 (= 11-2 (f * fi)11- 11v2(f)112). Note that M > 0 since f 7x2 is one-to-one on 11(G). Let e E (0, M) and define

0 and

UE={tEX:t2>M-e}. Then UE is an open, nonempty subset of X, and since S(µ) = X we have Fi(UE) _ kE > 0. So, using (3), 1172(f)II = urn lIT2n111/2n > limsQ p(tr(T2n))1/2n

n-oo lim n (tr(T2n))1/2n 1f \ n-oo

= lim n f fu. ten dµ(t) + lL

//

f

ten dA(t)

1/2n

x-_Ua

1/2n

liminf ` t2n dA(t) lim onf((M - )nk£)1/2n n-oo fu, / (M-e)1/2 = 11n2(f)11(1 -elM)1/2.

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171

The equality (1) now follows by letting e -+ 0.1

In (4.32), we will require the value of an operator norm associated with the left regular representation of a free group. (Indeed this value will be used to obtain the norm of the corresponding operator for any finitely generated group.) Let Fk be the free group on k generators x1, ... , xk. To any element x E Fk, we can assign a "length" 1(x), where I(x) is the length of the reduced word in {xk1, ... , xi 1,x1, ... , xk} corresponding to x. Let En be the set of elements in Fk of length n. For example, Eo = {e} and E1 = {xk 1, ... , xi xli... , xk}. For each n > 0, define hn E I1(Fk) by

hn=

x.

zEE

Soho=eandhl=(xkl+...+xil+xi+...+xk). Seth=hl. Our next result shows that each power hn of h can be expressed as a linear combination

n

E ai,nhi i=0

with each ai,n an integer, and obtains a number of simple, useful facts about the ai,n. These integers ai,n will also be used in (4.32). (ii) (Cohen 11[). There exist integers a,,n > 0 (0 0) such that

for each n, hn

(4)

= Eai,nhi. i=0

The integers ai,,

are uniquely determined by (4), and the following properties

hold:

(a) for all n, an,n = 1;

(b) for all n, ai,n = 0 if (n - i) is odd; (c) for all n > 1, ao,n+1 = 2kal,n; (d) for i > 1 and n > (i + 1), ai,n+1 = ai-1,n + (2k - I)ai+1,n; (e) if ry E (0,1), then, for all i < n,

a,,n < -i1('7-1 + (2k - 1)-,)n. [We start by establishing the following equalities: (5)

h*hn=hn+1+(2k-1)hn_1

(6)

h2 = h2 + 2kho.

(n>2),

The proof of (6) is very easy. To prove (5), let n > 2. For y E E1 i let Ay [By[ be

the set of elements in Fk of length n in - 1) not beginning with l I [y[. Then (7)

yhn = E yx + E x`. zEA

z'EB

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172

Summing the equalities (7) with y ranging over El gives (5). The equality (4) is trivially true for n = 0. If n > 1 and we have n

hn=r ai,nhi, i=0

where ai,n E N, then, using (5) and (6),

hn+l = E ai,nh * hi i=0 (8)

ai,n (hi+1 + (2k - 1) hi_1)

= ao,nh + a1,n (h2 + 2kho) + i=2

and we can write

n+1 hn+1 = E ai,n+1 hi, i=o

where ai,n+l E N. Thus, by induction, (4) is established for all n. By considering

the length of the group elements involved in each hn, it is evident that (4) determines uniquely the integers ai,n.

Now (8) gives an+l,,+1 = an,n, ao,n+l = 2ka1,n, an,n+l = an=1,n, and ai,n+l = ai-l,n + (2k - I)ai+l,n (i > 1, n > (i + 1)). The results of (a), (c), and (d) now follow. As for (b), note that if (n + 1) - i is odd, then so also are n - (i - 1) and n - (i + 1). Now use (d) and induction to yield (b). We now prove (e). Let -y c- (0,1) and ,8 = (^y 1 + (2k -1)-y}. We have to show

that for each n > 0,

ai,n 5 'f/3'

(9)

(0 < i < n).

The proof is, again, by induction. If n = 0, then 1 = ao,o = yo,60

and if n = 1, then ao,1=0570,81R,

1=a1,1=7. I 1. Then since 0 < ry < 1, we have, using (c),

ao,n+1 = 2ka1,n < 2k-jlfln < (7-i + (2k - 1)'7),3n ='Yop"+l.

IfI< (n-1), then ai,n+l = ai-1,n + (2k - I)ai+l,n

Y'-1Q'a+(2k-1)7i+ln 1+(2k-1)1)t=y=jjn+l =ry`

{7

Finally, since an,n+l = 0 and an+l,n+l = 1, we see that (9) is true with n replaced by (n + 1). This completes the proof.)

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173

Cohen comments that there does not seem to be a conveniently computable expression for the ai,n. Our next result entails that HH1r2(h)II = 2(2k -1)1f2,

where 1r2 is the left regular representation of Fk. This result is due to Kesten (2], and also follows from Akemann and Ostrand [1]. The elegant proof given here is due to Cohen (1]. (iii) Sp(1r2(h)) = (-2(2k - 1)1/2 2(2k - 1)1/2).

(10)

(Let T = 1r2(h) and A be the C'-subalgebra of Cl (Fk) generated by T and 1. Using (4), we see that A is the norm closure of Span{7r2 (hn) : n > 0). Define 4P:A

12 (Fk) by 40(W) = We. Ifao,...,aNEC,then N

N

L. an1r2(hn) n=0

=

Eanhnn=0

Since' is continuous and {hn) is obviously an orthogonal sequence in 12(Fk), it

follows that for each W E A, t(W) is of the form E. ;6nhn, where 00

I0ni211hn112 < oo.

n=0

We now turn to the proof of the equality (10).

Let a E R be such that jai < 2(2k - 1)1/2. We shall show that a E Sp(T). Suppose, on the contrary, that a 0 Sp(T). Since Sp(T) = SPA(T), there exists

SEA such that (T - aI)S = I. Let

0 "D(S) = J Anhn n=0

as above. Now if W E A is a polynomial in T with real coefficients, then one checks directly that the coefficients of hn in the expression for 4(W) are all real. Since S is the norm limit of a sequence of such polynomials W, it follows that f3 E R for all n. Now

e = le = (T - a)(Se) = (7'- a) (E /3nhn J n-0

.

JJJ

Hence using (5) and (6), 00

00

n=o

n=0

e=E8nh*hn-aE5nhn 00

cc

AA,.

= Qoh +;61(h2 + 2ke) + E fin(hn+1 + (2k - 1)hn-1) - a n=2

n=0

174

FOLNER CONDITIONS

Since {hn} is an orthogonal sequence, we can equate the coefficients of each hn on both sides of (11), obtaining = (r + 1)01 - alto,

(12)

(n > 1),

0 = n-1 + rfln+1 - a#n

(13)

where r=2k-1. Thus (14)

th

1

Qn+1

+I/3o+ r+1:

a

q L--' r-1 (n?1). rNn-

Clearly {#n} is determined by 60i and direct checking shows that the solution to (14) is of the form: On = pr-(n/2) sin(nO + q)

(15)

where p, q E R satisfy (17) below, and the angle 9 satisfies the equalities (4r 2)1/2 (16} cos B = , sin 9 =

2

2

f

Note that since Iai < 2r1/2, it follows that 4r - 2 > 0. It also follows that we can take 0 < 9 < r. The numbers p, q have to satisfy the equalities (17)

go=psinq,

a X30+r +L'I pr-1/2sin(9+q)= r+1

Now from the definition of hn in (i), IIhnUI2 = IEnI

Elementary combinatorics shows that IEnI = 2k(2k - 1)n-1, so that

II7'nhnll2 = per-" sin2(n9+q)(r+ = per1(r + 1) sine (n9 + q).

1)rn-1

Since 0 < 9 < ir, {sin 2(n9 + q)} does not converge. Hence E°n°--o Il)nhnIl2 diverges. This is impossible since F ,'o IIf`nhnll2 = IIj>(S)II2 < oo.

We thus obtain that (-2(2k -1)1f2, 2(2k - 1)1/2) C Sp(T), and since Sp(T) is closed, it follows that (18)

[-2(2k -1)1/2, 2(2k - 1)1/2) C Sp(T).

Let b be the spectral radius of T. The equality (10) will follow from (18) once we have shown that b:5 2(2k - 1)1/2. This is proved using (i). For each n, we see from (4) that 2n

tr(T2n) = tr Z at,2n7r2(ht)) = ao,2n t=o

since 7r2(h0) = I and (7r2(hi)e,e) = (hi, e) = 0 if i 96 0. Using (i) and recalling that T is hermitian, (19)

b = IITII = nliln00(4,2n)1/2n

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175

Let 7 E (0,1). From (ii) (e), (ao,2n)hI2n < -y-' + (2k - 1)7.

Put 7 = (2k - 1)-1J2 and use (19) to obtain that b:5 2(2k - 1)1/2 as required.) The preceding proof also established the following result. (iv) 2(2k -1)1f2 = lim,i...,oo(ao,2n)1/2n

(4.32) An amenability criterion for finitely-generated groups. This remarkable criterion is due to Grigorchuk [2). We shall follow the proof of Cohen [2). The author is grateful to Joel Cohen for very helpful correspondence on this result.

Let G be a group generated by the finite set {u1,...,uk), where k > 1. (Clearly, there is no point in looking at the case k = 1, since every abelian group is amenable!) Let Fk be the free group on k generators {x1, ... , xk}. Let Q: Fk -+ G be the canonical homomorphism from Fk onto G determined by the equalities Q(xi) = u, (1 < i 0, let

En= {x E Fk: l(x) = n}. Let N = kerQ, and Nn = N fl En. Let 7n = [Nn). Obviously, N = Un°=1 N. Now, of course, some of the N might be empty. Indeed, if N = {e} (so that Fk = G), then Nn = 0 for all n > 1. In our next result, we shall show that if N {e}, then 72n 0 eventually, and that 7 = limn-m('Y2n)1/2n exists. The number ' can be regarded as a measure of "how far away" G is from Fk. (i) Suppose that N 0 {e}. Then there exists no E P such that 72n > 0 for all n > no, and 2n)1/2n

'Y = n n (

exists, and belongs to the interval [(2k - 1)1/2, (2k - 1)). [The following two inequalities will be proved first for all m, n > 0: (1) (2)

(2k - 2) (2k -

< '1m+2n,

7n'Ym :5 '1m+n+2

Let Ak = {xkl, ... , xi 1,x1, ... , xk) and x E Nn. Write x in reduced form wn, with wi E Ak. Let Bn be the set of elements y E Fk which can be written in reduced form v1 v, with vn i4 wi 1, w,n 0 vn. Then yxy-1 = v1 ... yawl ... wmv; 1 ...vi (3) w1

is in reduced form, so that 1(yxy-1) = (2n + m). We now count the number of elements in B,. First, [Ak[ = 2k. In the expression v1 vn for y, there are (2k - 2) choices for vn and (2k - 1) choices for every other v,. Hence [Bn[ = (2k - 2)(2k - I)n-1. Since N is normal in Fk, it follows that yxy-1 E N2n+m.

176

FOLNER CONDITIONS

It is clear from (3) that a different choice of (x, y) (x E N,,,, y E Bn) gives rise to a different element yxy-1. Thus (4)

'72n+m >- (2k - 2) (2k -

1)n-IINmI

giving (1).

We now turn to (2). Let x' E N,,, and y' E Nn have reduced forms w1 and vl v;, respectively. If u E Ak - {(w,n)-1, (vl)-1,v;,}, then r -1 r r r ...W,nUV1...ynu, 1 )=9171 2(uy9t

is in reduced form, and so belongs to Nm+n+2. By considering all possible choices of x', y' we obtain (2). We now show that for some no, 72n > 0 for all n > no. To prove this, note

first that 7m > 0 for some m E P since N 34 {e}. It follows from (2) that '72(m+i) > 0, so that we can take m to be even. Let m = 2p (p E P). Using (1), 7'2r > 0 for all r > p, so that we can take no = p. It will now be shown that {('y2n)i/2n} converges. In this argument, it will always be assumed that n, m > no. Let an = log12n ? 0. Since log[('y2n)1/2n) = an/2n, it is sufficient to show that {an/n} converges. From (2) (5)

an + am 5 an+m+1

Let bn = an-1- Clearly, {an/n} converges if {bn/n} converges. We will prove the latter. From (5), (6)

Thus 2bm (7)

bn + bm < bn+m

b2m, and, by induction, rb,n < brn,

(r > 1).

From (6) we also have that {bn} is increasing. Now

'7n = INnI 5 JEnj = 2k(2k - 1)n-1,

so that

bn = an-1 = log'y2n-2 -< log[2k(2k - 1)2n-3]. Since 2k < (2k - 1)3, (8)

bn < log(2k) + (2n - 3) log(2k - 1) < 2nlog(2k - 1).

Thus the sequence {bn/n} is bounded above. Let a = lim sup,,, a, (bn/n) and 6 = lim So a >- 0. Let rl > 0 and p E P. Then we can find m such that b,n/m > (a -?I), and then find n > pm with bn/n < (9 + ?7). Write n = rm + q where r E P and 0 < q < m. Note that r > p. Using (7) and the fact that {bn} is increasing, we have:

a - 17 < bm/m = rb,n/rm < b,.m/rm < bn/rm.

= (bn/n)(n/m)(1/r) = (bn/n)(r + q/m)(1/r) (e+97)(1+r-1) < (/3+,7)(1+p 1). < (bn/n)((r+1)/r)
- (2k - 2)(2k -

1)n-1,no >- (2k - 2) (2k -

1)n-1

so that

= Finally, using (8),

(2k -1)1/2

n

7 = lim exp(an/(2n)) = 1im exp(bn/(2n)) < (2k -1).) n-+oo n- oo

It is easy to show that, for any group G with k generators u1, ... , uk, one of (a), (b), (c) below holds:

(a)-yn=0 for alln>0 (that is,N={e}); (b) 12n-1 = 0 for all n > 1, and, for some no, 12n > 0 for all n > no; (c) for some n1 i -yn > 0 for all n > n1. An easier version of the above argument shows that in case (c),

l

('Y.)1'n

exists. Our objective is to prove the following criterion for the amenability of G (with N j4 {e}):

G is amenable if and only if 'y = (2k - 1). This is proved in (vii) below. Let g=uk1+...+U 1+U,+...+uk.

Obviously, g > 0 in 11(G), g^' = g, and S(g) generates G. Further, JJgJJ1 = 2k. We will calculate 11ir2(g)11 and then apply (4.20) to establish the criterion.

Let ai,n be the integers introduced in (4.31(ii)). (ii)117r2(g)11= limn_oo(E o ai,2n

Yi)1J2n.

[Let T = 1r2(g). By (4.31(4)),

2n (9)

tr(T2n) = E ai,2n tr(ir2(Q(hi))), i=O

where Q is extended, in the obvious way, to a linear map, also denoted Q, from 11(Fk) onto 11(G). Of course, Q(h) = g. Now, from the definition of hi,

tr(x2(Q(h,'))) = E (7r2(Q(z))e, e) XEE,

(10)

=J{xEEE:Q(x)=e}J =JN=J=-yi

Now use (9), (10) and (4.31(1)).) For large even i, (-yi)I/i is close to -y. This suggests replacing E?='o ai,2n'yi in (ii) by El no ai,2. Yi. Motivated by the formula for the radius of convergence of

FOLNER CONDITIONS

178

a power series, we are led to consider the power series in z E C for each fixed

s>0: 00

L

(11)

a.(s)zn

n-0

rL

where an (s) = En 0 a2,ns1. We shall also have to consider the power series: 00

(12)

ao,nzn.

n=0

Let p, and p be the radii of convergence of the power series in (11) and (12) respectively. Since ao,n 1, (13)

an+1(s) = (2k)-1ao,n+1 - ((2k - 1)/8)ao,n + (s + ((2k - 1)/s))an(s).

(b) If jzj 1 and r = (2k - 1). Then using (d), (a), (b) and (c) of (4.31(ii)): an+1 (s) = ao,n+l + ai,n+18 +- + an+l,n+18n+l = a0,n+1 + (ao,n + ra2,n)s + (al,n + ra3,n)s2 + (a2,n + ra4,n)s3

+ ... + (a, _4,n + ran_2,n)sn-3 + (an-3,n +

+ (an-2,n +

ran,n)sn-1

+ an,n+l sn +

ran-l,n)8n-2

an.+l,n+l,n+l

= a0,n+1 + a0,ns + al,ns2 + (r/s + s)a2,nS2 + (r/8 + S)a3,ns3

+ ... + (r/s + s)an_2,nsn-2 + r5n-1f+ Sn+l = ao,n+1 + ao,ns + al,ns2 + (s + r/s) 1 Z aims' Y-2

ao,n+1 + ao,ns + al,ns2

- (5 + r/8) (ao,n + al,ns) + (s + r/s)an(s) = ao,n+l [1 + s2(2k)-1 - (s + r/s)s(2k)-1j + a0,n(s - S - r/s) + (s + r/s)an(s) = ao,n+1 (1 - (2k - 1)/(2k)) - (r/s)ao,n + (s + r/s)an (s) = (2k)-1ao,n+i - ((2k - 1)/s)ao,, + (s + ((2k (b) Using (13), 00

f,(z) = ao(s) + al(s)z + E[(2k)-1ao,n+1 - ((2k - 1)/s)ao,n n=1 + (a + (2k - 1)/s)an(s)jzn+l = 1 + 8z + (2k)-1[g(z) - 11 - ((2k - 1)/s)z[g(z) - 11 + z(s + (2k - 1)/s)[f,(z) - 1j.

FOLNER CONDITIONS So

179

2k[1- (s+ (2k -1)/s)z]fs(z) = 2k{1 + sz + g(z)1(2k)-1 - ((2k - 1)/s)z] - (2k)-1 + ((2k - 1)/s)z - z(s + (2k - 1}/s)}

= 2k{1 -

(2k)-1

+ (2k)-1g(z)[i - (2k(2k -1)/s)z]}

= (2k - 1) + (1 - (2k(2k - 1)z/s)]g(z).]

We now prove an elementary result on real sequences.

(iv) Let {an} and {bn} be sequences with an > 0 for all n, and L,K E R with 0 < L < K, jbnj < Ln eventually, limsupn_ a;, /n = K, and (15)

an+1 = bn + Kan.

Then an/n

K. [Dividing both sides of (15) by Kn+1 gives: an+l/Kn+1 = bn/Kn+1

+ an/K"Replacing an, bn by an/Kn and bn/Kn+1 respectively, we can suppose that

L 0

be such that L < (1- e). Let N1 E P be such that both: jbnj < Ln

(17)

[L/11

(n > NI),

L

)]n

< 1 (n > N1).

Now find N > NI such that a17 >_ 1- e. Let m E P. Adding up the equalities (16) for N < n < (m +N), we have m+N

am+N+1 - aN = L..: bn. n=N

So, using the first inequality of (17), m+N

am+N+1 >- aN -E Ln >_ (1 - e)N - LN /[1 - L]. r=N Using the second inequality of (17), line inf(a.+N+l )1/(m+N+1)

> ,,,ia1(1 - e)N - LNl [I So lim inf n-

¢n/n = 1-- lim SUpn_. an

1.

and an/n

1]

(v) Lets > 0. Then p, _ (s + (2k - 1)/s)-1.

[Recall that ps [p] is the radius of convergence of the power series in (11) [(12))

giving f,(z) [g(z)), and that ps < p. Now ao,n = 0 for odd n by (4.31(ii)(b)), and from (4.31(iv)), j P = I limsup(ao,2n)1/2n, 11 n-00

1

=

1

o (ao,2n)1/2n

In-00

= (2k - 1)-1/2. 2

I80

FOLNER CONDITIONS

For X E (0, oo), let rx = (x + (2k -1)/x)-I. By elementary differentiation, the function x -+ rx has a unique maxiumum on (0, oo) at x = (2k - 1)1/2, so that

rx < r(2k-1)2/z = p for all x. If )zi < r then, from (14), the power series obtained by multiplying out

(2k)-1 E[(s+(2k-1)/s)z]n

I

(2k -1) + [i - (2k(2k - i)z/s)]

(

ao,n,z"

n-o has to coincide with the power series in (11) for f,(z). This means that the n=o

JJJ

JJJ

radius of convergence p8 of the power series in (11) is > r8. So r8

p8 < p.

-1)-1/2). We now show that either p, = r8 or p, = p (= (2k Suppose that p8 > r,. Substituting z = r, in z(14), we see that

(2k - 1) + [1 - (2k(2k - 1)r,/s))g(r,) = 0. Then the function

M)

I

(r8

-

z)-1[(2k

- 1) + [1 - (2k(2k - 1)z/s)]g(z)]

extends to an analytic function on the disc {z E C: IzI < p} using the theorem on removable singularities and, by (14), is given by the power series expansion of (11), thus coinciding with f,. So if p8 > re, then p, = p. So either p8 = r8 or p, = p. We shall now show that p, = re. To this end, let

A = {x E (0, oo) - {(2k - 1)'I'): (2k -1) + (1 - (2k(2k - 1)rx/x)]g(rz) = 0}. Since for x E A, rx < r(2k-1)v2 = p, we see that g(rx) makes sense. Let x E A. We claim that x is isolated in A. To prove this we suppose that x > (2k - 1)1/2. (A more complicated version of the argument applies to the case where x < (2k - 1)1/2 letting x -* 0 rather than oo in (18).) We can find an open, connected set U in C containing ((2k - 1)1J2, oo) such that the following function h: U C is analytic: h(z) _ (2k - 1)+ 1-

2k(2k - 1)

z

z2+(2k-1)j g(Z2+(2k-1)

(The function h(z) is obtained by replacing z, s in the right-hand side of (14)

by (z + (2k - 1)/z)-1, z.) Suppose that x is not isolated in A. Then x is a nonisolated zero of h in U, so that h is identically 0 on U. So (18}

0=zimoh(x)= (2k-1)+g(0)=2k

giving a contradiction. So the elements of A are isolated.

If s ¢ A, then p8 cannot be > r8i for then substituting z = re in (14) gives h(s) = 0. Since p, is either r, or p, we have ps = r,.

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181

1)1/2)-1 = If s = (2k - 1)1/2, then r8 = (2(2k p, so that P8 = re = P Suppose, finally, that s E A - {(2k - 1)1J2}. Then we can find z A with

x # (2k - 1)1J2, x < s. Since

0

6i,rzxi < F,' o ai,nsi and z

A, we have

Ps- 0 and choose 17 E (0, e) such that both y - ?7, y + rl E (0, oo) {(2k - 1)1f2) and 1 < (y - ri). Since ,y2n2n -+ y, we can find N E P such that t1)

(j ? N).

(_y

Recalling that ai,2n = 0 whenever i is odd, we have for n > N, 2n

2n

2n

A+ E ai,2n(y-r,); <Eai,2nyi N, 1/2n

2n

(B+a2n(y- 7))112n
(2k- 1)1/2 > 1, it follows that G is amenable if and only if -y = (2k- 1).]

(4.33) OV`shanskii's counterexample to von Neumann's Conjecture. The result of (4.32) suggests an approach for resolving von Neumann's Conjecture negatively. Suppose that we have a finitely generated group Go that does not contain F2 as a subgroup, and that, we suspect, is nonamenable. If we have detailed information about the relations satisfied by the generators

ul,... , uk of Go, then we might hope to be able to estimate 12n and hence 'y = If y j4 (2k - 1), then Co is, indeed, a counterexample to the Conjecture. OI'shanskii 12) constructed a group Go for which the above procedure can be limn-oo(-Y2n)1/2n.

carried out. Here are some of the properties possessed by Go. (a) Go is generated by two elements {a, b} and there exist equality and conjugacy algorithms for Go; (b) Go is a simple, infinite group; (c) every proper subgroup of Go is infinite cyclic; (d) Go is Noetherian, that is, there are no infinite, ascending chains of subgroups;

(e) root extraction is unique, that is, if x, y e Go and x" = yn for some n E P, then x = y. The construction of this group requires a formidable induction argument involving 80 lemmas and a large number of parameters. It relies on the well-known work by Adian and Novikov on the Burnside problem (concerning the existence of a finitely generated, infinite group H such that for some n E P, x" = e for all x E H). It also involves the use of geometrical methods, originating in a result of

FOLNER CONDITIONS

183

van Kampen, and embodied in the so-called "small cancellation theory" (Lyndon and Schupp [1]). We shall not attempt to describe the argument. The details of the construction enable Ol'shanskii 141 to estimate the 7, for

Go with respect to the generators {a, b} of (a). He shows that the parameters defining Go can be chosen so that ,yn
(Loo(G))? To answer this, simply note that 4)(:B) = rr2(x). Since {i: x E G} generates Loo(t), it follows that 4,(Loo(G)) = VN(G). This and the observations that C'(G) - C0(G) and that Co(G)' = M(G) give a way of defining A(G) and B(G) for general G.

184

FOLNER CONDITIONS

So let G be an arbitrary locally compact group. We define A(G) = VN(G)., the predual of VN(G), and B(G) = C'(G)'. Now the positive functionals on C'(G) correspond to continuous positive definite functions on G [D2, (13.4)], and since every element of C` (G)' is a linear combination of positive functionals, it follows that B(G) can be identified with Span{ 0: 0 is continuous and positive

definite on G} C C(G). Turning to A(G) = VN(G)., we observe that Ci (G) is strongly dense in VN(G), and so by the Kaplansky Density Theorem [Dl, Part 1, Chapter 3, Theorem 3), A(G) can be regarded as a closed subspace of C, (G)'. Since CI (G) is a homomorphic image of C' (G), A(G) can be regarded as a closed subspace of C' (G)' and hence as a closed subspace of B(G). The context will make clear if B(G) (or A(G)) is being regarded as a space of linear functionals or as a subspace of C(G). The natural question now is can we identify those functions in B(G) C C(G)

that belong to A(G)? To answer this, we recall ((2.35(B))) that the predual of VN(G) consists of all those functionals a of the form F°_, Wh;,k,, where (TN., k,) h;,ki e L2(G), Z=_1 Ilhtll2 < oo E 1 I1kdI2 < oo, and (T E VN(G)). When a is identified with a function 0 E C(G), we have ¢(x) = a(7r2(x)) (since wh,,k,(1r2(f)) = f f(x)(ir2(x)ht,kt)dA(x) for f E L1(G)), and so it follows that O(x)

(ir2(x)h4-, k4). The following useful formulae are

easily checked:

(f,g E L2(G),x E G), where, as in (4.20), g'(y) = g(y-1) (y E G), gt = g'. So (ir2(x)h.i,ki) _

f * gt(x) _ (f,ir2(x)g),

1* gr(x) _ (f,7r2(x)9)

ki * hL (x), and A(G) is the set of all functions of the form F,°_1 fi * gi , where < co. (In fact Eymard [Nil fi,gi E L2(G) and E 111h1122 < 00, 2 1 Il9ill2 shows that A(G) _ {f * gt : f, g E L2(G)}.) We will use 11 11 to denote the norm of B(G) or A(G); the C(C)-sup-norm will be denoted by E1 11.. Now if f E L1(G) and E1 11. is the C* (G)-norm on LI(G), then for ' E B(G), I0(f)I 0 with compact

support in M(G). To this end, we first show that given K E F(G) and e > 0, there exists 0 E A(G) such that 10(x) -11 < E,

(2)

(x E K),

11 Oil 0, then there exists h E A(G) such that [h(x) -1[ < e (x E C) and [[h[[ S 1.

Then ifxECand6,cEA, [Tu6(x) - Tu,(x)[ 0,

then there exists a nonnull symmetric set K E '(G) such that L C K and A(xK A K)/A(K) < e for all x E C. (Hint: distinguish between the unimodular and nonunimodular cases.) Deduce that every amenable G admits a symmetric summing net. 11. Prove that G is not amenable if and only if I(G) = oo ((4.14)).

12. Suppose that G is amenable. Show that if nonnull sets CI, C2 E F (G) and e > 0, then there exist nonnull sets K1, K2 E 9'(G) such that A(C1K1 A C2K2)/(.(K1)+ A(K2)) < e. 13. Prove that G is amenable if and only if there exists a net {E6} of closed

subsets of G with 0 < A(E6) < oo for all b such that for all C E T(G), a(n.EC(cE6))/A(E6) - 1. 14. Let G be amenable. A function r¢ E U(G) is called left almost convergent if {m(¢): m E .L'(Ur(G))} is a singleton. The set of such functions is denoted by AC1(G).

(i) Show that AC1(G) is a closed, right invariant subspace of Ur (G) containing W (G).

(ii) Show that if {K,5) is a summing net for G and ts6 = XK, /.1(K6 ), then 4 E AC1(G) if and only if 11¢p6 - a1IJ. - 0 for some a E C. (iii) Show that AC!(G) admits a unique left invariant mean m, and

=16m.1(K6)-1 f ¢(t) dl (t), x,

where {K6} is as in (ii).

15. We can define the notion of almost convergence for a semigroup S as in Problem 4-14. We also defined almost convergence for sequences in Problem 0-13. Show that the two notions for S = P are equivalent. 16. Let G be amenable. A function q5 E Ur(G) is called Bohr almost periodic

(¢ E BAP(G)) if given e > 0, there exists C E W(G) such that for all x E G, there exists c= E C such that (1)

f1x0 - cx0IIoo < E.

FOLNER CONDITIONS

190

Show that

AP(G) = U(G) n BAP(G) c BAP(G) c ACI (G). 17. (Ly-conjecture for amenable groups). Let us say that for p > 1, LP(G) is closed under convolution if for f, g E Lp(G), the function f * g, where

f * g(x) = f f(y)g(y-lx) dA(y) is well defined and belongs to L, (G). Show that if G is amenable and p E (1, oo), then Lp(G) is closed under convolution if and only if G is compact.

18. For 0 E L,,. (G x G) and v E L1(G) define functions v o1 0, v 02 0 on G by

vol 0(s) = J 0(xs, s) dv(x),

v o2 q5(s) = f 5(s, xs) dv(x).

Show that v o1 0, v o2 40 E L (G). Let m E 9R(G). Sh o w that m E Cc(G) if and

only if m(v 02 0) = m(v- of ¢) for all v E P(G), 0 E L (G x G). 19. Let p E P(G) and P: G -. P(G) be the "translation family" given by P(s) = ps. Let m E '1) (G). A posterior for m (relative to P) is a map

Q: G -# P(G) such that for every bounded measurable function 0: G x G

C,

we have

m(8

PA(8)(00)) = ml'(x - Qn (x)(0x)),

where ml E JJl(G) is given by m1(E) = m(8 - P(8)(E)), e0(y) = 0(8,y), 0. (y) = 0(y,x) and it is assumed that the functions x -+ Qn(x)(Ox) are in L,o(G). (Show that 8 -+ P" (e.)(eO) is in for each 0.) Show that if G is amenable and m E .t(G), then the translation family Q associated with p- is a posterior for m relative to P.

20. Let H be a closed normal, amenable subgroup of G and QH : Ll (G) L1(GJH) be the canonical *-homomorphism of (1.11). Show that IIQHfIII = d(O, Cf(H)) for all f E L1(G), where Cf(H) -;- co{f *x: x E H} and d(0, Cf(H)) = inf{IIgII1: g E Cf(H)}. Prove also that G is amenable if and only if for all f e L1(G), I f f d.I equals the distance of co{x * f : x E G) from the zero function.

21. Let H be a closed, normal subgroup of G. Suppose that G is amenable and that {K5}j is a summing net for H. Show that if f E L1(G), then

Jc/H

(1)

dAG/R(xH)I fH f(xh)d.H(h)I

= libm\H(Ks)-1 f dA(x) J

c

xa

f*

Deduce the classical summation formula: if f E L1(R), then ,N

00

E f(x + n) J0 1

n=-oo

dx=N-oJ_

00

AI-- 1

E f(x+n) n=-N

dx.

FOLNER CONDITIONS

191

22. The group G is said to be uniformly distributed (u.d.) if there exists a sequence {x,} in G such that for all f E L, (G), lim n-oo

1n _F,f*xr rc1

=I

ffdal.

Such a sequence {x,} is said to be u.d. in G. (i) Show that if G is u.d., then G is amenable. (ii) Show that G is amenable and separable if and only if G is u.d.

23. Let G be a compact group. Let {x,,} be a sequence in G and suppose that for all ¢ E C(G),

n'1

(1)

r.a

0(xr) - J

bdA.

J

Show that {x,} is u.d. for G. (Problem 4-22). Suppose that G = R/Z (= 10,1) with mod 1 addition). Prove Weyl's Criterion: {Zr} satisfies (1) if and only if n-1 Zr 1 exp(2,rikx,) -+ 0 for all k E l - {0}. Give an example of such a sequence {x,}. 24. Prove that G is amenable if and only if i E Spar2(µ) for all µ E P(G).

25. Prove that if 1 < p < oo, then (4.20(i)) is true with ir2(p) replaced by iry(µ).

26. Check directly that 1 E Spir2(µ) for all i E P(G) when G is abelian (Problem 4-24).

27, Give a direct proof of the weak containment equality C*(G) = C! (G) when G is abelian and when G is compact. 28. Show that G is amenable if and only if there exists a net {06 } in CC (G) such that 06 * 06 -+ 1 pointwise on G. Show also that G is amenable if and only if the map f f f dA is continuous for the Cl (G)-norm on LI(G).

29, Let G be separable and of Type 1. Prove that G is amenable if and only if the support of the Plancherel measure on G is the whole of G. Illustrate this result when G is (a) abelian, (b) the Heisenberg group of real 3 x 3 matrices of the form 1

x3

xa

0 0

1

x2

0

1

(Xi E R).

30. Let G be a locally compact group containing a nonamenable open, proper

subgroup H. Let S1(G) and S(G) be the sets of states on Cl (G) and C* (G) respectively. We can regard S1(G) C S(G). Let 1 be regarded as a state on C*(G): 1(f) = f f dA (f E L1(G)). Show that St (G) C co(St(G) U 1) * S(G).

192

FOLNER CONDITIONS

31. Show that a separable, amenable locally compact group G has Property (T) ((4.29)) if and only if G is compact.

32. Show that a semigroup S is left amenable if S satisfies (SFC) ((4.22)). Show also that if S is left amenable and is either left cancellative or finite, then S satisfies (SFC). 33. Let S be a left amenable semigroup and To be a countable subset of S. Show that there exists a countable left amenable subsemigroup T of S with To c T.

34. Let G be compact. A function , E LA(G) is called Riemann measurable if there exists a null set E C G such that 0 is continuous at each point of G - E. Show that every Riemann measurable function on G has a unique left invariant mean.

35. We proved in (1.12) that a closed subgroup H of an amenable locally compact group G is itself amenable. Give a quick proof of this result using Problem 4-28 and Godement's Theorem ((4.20)).

36. Let G be an amenable locally compact group. When does there exist a right summing net for G, that is, a net {Ka} satisfying (4.15) with Ksa in place of xKb?

37. Prove that if G is amenable, then C, (G) is not a simple C'-algebra.

38. Let L°(G) = {f E L1(G): f f dA = 0} be the "augmentation ideal" of L1(G). Prove that L°(G) has a bounded approximate identity if and only if G is amenable.

39. Show that A(G) = B(G) if and only if G is compact.

40. For a Banach *-algebra A, let Prim. A be the set of ideals of A that are kernels of nonzero, topologically irreducible *-representations of A on a Hilbert space. (So if A is C'-algebra then Prim A = Prim. A). (i) Show that the hull kernel topology for Prim. (A) makes sense. (For the rest of the question, Prim. A is given this topology for A = L1(G), C' (G).)

(ii) Show that the map TG: Prim C' (G) -+ R(L1(G)), where We (I) _ I n L1(G), is a continuous bijection onto Prim. L1(G).

(iii) Let ('Il be the class of locally compact groups G for which q'a is a homeomorphism. Prove that G is amenable if G E [T). [Note: Problem 6-25 asserts that G E IT] if G has polynomial growth.)

41. Assume that G is separable. Show that G is amenable if and only if it possesses the following "weak Frobenius" property: whenever N is a closed

normal subgroup of G, a E G, o = TIN, and U° is the representation of G obtained by inducing o to G, then U° weakly contains ir.

FOLNER CONDITIONS

193

42. Let P(A(G)) be the semigroup of positive definite functions 0 E A(G) with q5(e) = 1. (i) Let f, g E Ce (G) with g vanishing outside C E r'(G) and 111112 = 1 = 118112.

Let 4' be as in (4.34(i)(b)). Show that

4' (ft.rc ® (tf)gda(t)) _ (f * ft)(g * gt). Deduce that

11(f*ft)(g*gt)-g*gt)II 0, and C E 91,,(G). Let K1 E '(H) be nonnull and (such that / AH(CHK1 AKi)/AH(Ki) 0.

Since kerir is a discrete subgroup of Z(N) = R"', we can find r > 0 and vectors e1,.... e,. in R"', independent over Z, such that

kerr

It nie,:niEZ

It follows that Z(N) Z) Z(N)/kerr = Rm ' x T', and since, by (5.15), K(N) C K(G) = {e}, we have r = 0. Hence r is an isomorphism and N = N is simply connected.

Let R be the radical of G, and r and n be the Lie algebras of R and N respectively. By B37, fr,t] C n, so that r/n is abelian. Hence R/N is abelian. Thus R/N = R' x TQ for certain integers p, q > 0. Let Q : R R/N be the canonical quotient map and S = Q-'(RP). Then S is obviously closed in G and is simply connected since both N and S/N are simply connected (B45). We claim that S is a normal subgroup of G. To this end, let (G, r') be the simply connected covering group of G. Since S is simply connected, S C G is isomorphic under r' to S. Let DG be the commutator subgroup of G. Then, by B24, Op is the Lie algebra of DG, and since d is simply connected, DG is a closed, connected subgroup of G (B17). Clearly, R can be identified with the radical of G, and R D S. Now the Lie algebra of DG fl R is 2g fl r (1335), and

by B36,DGf1RCN=N. SoifxEGand sES, then xsx-IS 1 = (xsx-1)s-1 E DGf1R C N. Thus xsx-1 E NS = S. Thus S is a normal subgroup of d, and it follows that S = r'(S) is a normal subgroup of G = Clearly, R/S is compact, since it is isomorphic to P. Further, since G is amenable, it follows from (3.8) that G/R is compact. So G/S is also compact. By B47, G is a semidirect product S xv K with K a compact group. By B46, there exists a basis {XI, ... , Xn) for the Lie algebra s of S such that

ERGODIC THEOREMS

209

(a) the map %P: s --y S, where >y

:_ I

aiXi) = exp(aiXi) ... exp(anX,) J

(ai E R)

is a homeomorphism from s onto S; ($) if si = Span{XI, ... , Xi} (1 < i < n), then W(si) is a closed subgroup Hi of G;

(-i) Hi+I is a semidirect product Hi x p, Li+l (1 < i < (n - 1)), where Li+I = exp(RXi+i) - R. This gives (ii) and (iii) of the Proposition, recalling that

G = S xP

K = Hn x p K. Note now that G is formed by a sequence of semidirect products

(... (L1 xp, L2) xps ... xpn-, Ln) x9 K. Finally, (i) follows by (5.13) and the equalities

j (exp(a1X1), ... , exp(a,X,), k) = k exp(anXX)

exp(a1X1)

= j([ (- cx,Xi)]-',k

)

with an obvious abuse of the j-notation. 0 (5.17) DEFINITION. Let G be a a-compact, amenable, locally compact group.

A summing sequence {Kn} for G is said to be admissible for the Pointwise Ergodic Theorem if whenever (Z,-2', p) is a measure space, x -} T. is a strongly measurable representation of G on LI(Z) such that 1ITTIjj _< 1, IITTI1OO < 1 for all x ((5.11)), p E (1, oo), f E Lp(Z), and, for each n, An E B(Lp(Z)) is defined by ((5.7(1))

f

An = A (K,,)-' f TT-, d.)(x),

(1)

then

(a) there exists f' E Lp (Z) such that T. f = f for all x E G, and A. f -+ f' both in L,,(Z) and pointwise u-a.e.; (b) there exists fl E Lp(Z) such that IAn f 1 < If, p-a.e. for all n.

(5.18) Note. If {Kn} is any summing sequence for G and {An}, x -+ TT are as above, then the conclusions of the Mean Ergodic Theorem (5.7) hold with {An} in place of {Aa} and L,(Z) (p E (1, oo)) in place of X. (Of course, Lp(Z) is reflexive.) So there exists f' in L9(Z) such that TT f' = f (x E G) and An f --> f'

in Lp(Z). (Indeed, in the notation of (5.7), f' = Pf.) However, the other two conditions in (a) and (b) of (5.17) are not always satisfied (Problem 5-4). As we shall see, the next two results give a procedure for constructing summing sequences that are admissible for the Pointwise Ergodic Theorem when G is a connected, amenable, Lie group (cf. Problem 5-3).

ERGODIC THEOREMS

210

(5.19) LEMMA. Let G be a connected, amenable Lie group such that K(G) is trivial, and let L1, ... , Ln, K, j be as in (5.16). Then there exist sequences

(Rj(m)) (1 oo as m --j oo for each i, and such that if (1)

K' =j([-Ri(m),R1(m)] x ... X t-Rn(m),Rn(m)] x K),

then A(CK;,, © K,i)/a(K',) -+ 0 for all C E F(C). PROOF. We see from (5.16) that G is obtained by forming a sequence of n group semidirect products involving R or K. It is clearly sufficient to show that if G is a semidirect product of (c-compact) locally compact groups N, H where H is R [K] and if {L,, } is a sequence of nonnull, compact subsets of N such that A(C'L;,,

(2)

7n)"' 0

(C' E 9,(N)),

then there is a sequence {Xm}, where Xm = [-Rm, Rn,], Rm > 0 [Xm = K] for all m and a subsequence {L'Q( m)} of {L;,,} such that if K,',', = j(L''( ,,,) x Xn,) _ Xf,L'(,m), then

(3)

A(CK;;, L K",)/A(K,')

0

(C E 9, (G)).

Let G, N, H, {L71} be as above.

Suppose first that H = R. Find a sequence {Cn, } in T, (G) such that C,, C C°,+1 for all m and G = Um-1 Cn,. Observing that for D E F(R), AH ((D + [-p, pl),6 [-p, p])/aH ([-p, p])

0

as p - oo, we can find a sequence {Rn,} in (0, oo) such that Rn, the notation of (5.14),

oo and, using

AH(((Cm)H + [-Rm:Rm]) A [-R,,,,, Rm]}/AH([-Rm,Rm]) - 0.

(4)

We define (Cn,)jy as in (5.14), with C, K1 replaced by Cn3, [-Rm,Rm]. Now choose a subsequence {LQ{n,)} of {L' } such that )'N(((C,n)NLacm))

(5)

0.

The required result (3) now follows using (4), (5), (5.14), and (4.13). Suppose now that H = K. Then (4) is valid with [-Rm. R,n] replaced by K and (5) follows, yielding the required result. o

We can now state and prove the Pointwise Ergodic Theorem for amenable groups.

(5.20) THEOREM. Let C be a connected, amenable, locally compact group. Then there exists a summing sequence {K,,,} for G which is admissible for the Pointwise Ergodic Theorem. PROOF. Find (B6) a compact, normal subgroup Ko of C such that G/Ko is a G' be the quotient map. Replacing

Lie group, and let G' = G/Ko and Q: G

ERGODIC THEOREMS

211

Ko by Q'I (K(G')), we can suppose that K(G') = {e}. Applying (5.19) with G' in place of G, we can find Rj(m), K, and K' such that (1)

Let

(C E'(G')).

AQ'(CK;,, t Km)/AG'(K,.) -+ 0

Q-I(K,,,). A simple application of Weil's formula yields that

,\ (Q-'(D)) = \G, (D)

for all D E F(G'), and the latter equality together with (4.6(ii)) enables us to deduce that {K,,,} is a summing sequence for G. By (5.18), the first two assertions of (5.17(a)) follow. The rest of the proof is devoted to showing that {K,,,} is admissible for the Pointwise Ergodic Theorem. Let z - T. be a strongly measurable representation of G on some LI (Z) such S 1 for all x E G. Using (5.9) and Mazur's Theorem, that I ITx I f I < 1, I ITx I I

we see that for each f E LI (Z), the set (co{Tk f : k E Ko})- is norm, and hence weakly, compact in LI(Z). Further the representation k

Tk is strongly

measurable (using (5.9)) and so, a fortiori, the antirepresentation k - Tk-, is weakly measurable. Hence the Mean Ergodic Theorem (5.7) applies to the antirepresentation k - Tk-3 of the compact group K0. So if Po = f Tk-, dAKo(k),

(2)

Ko

then Po E B(LI(Z)) is the projection from LI(Z) onto the subspace LI (Z) = f f E LI (Z) : Tkf = f for all k E Ko}. Clearly IIPoIII < 1, and if f E W ((5.11)), then the function k - Tk-, f is AK,,-integrable and tIPoflloo _< f IITk-,fIIoodAKo(k) 5 f Ilf11OOd)Ko(k) = ]If II..

Hence IIPoIII S 1, IlPoll,. 5 1. If x E G, k E K0, then TTTkPo = TPo, and

if F E LI(Z), then Tk(TTF) = TTTx-,kxF = TZF, so that TxF E LI(Z). It follows that we can define a representation a - S. of G' on LI (Z) by (x E G).

SSK0 = TTPo

Using (5.9), the representation a -+ S. is strongly measurable, and since IIPoIII < 1, IlPolloo 5 1, IITxfli < 1, and IITxII,o < 1 for all x E G, we have IISalfi flSC)JOO 0 whenever

0 _ 0inC(X)). Show that there exists a positive projection P E B(C(X)) that maps onto the fixed-point subspace of C(X) and is such that PT = TP = P for all T E S. 7. (Sine's Mean Ergodic Theorem). Let X be a Banach space and T E B(X), JJTJJ < 1. Show that the sequence {(n + 1)-1 E7_0 T'} converges in the strong operator topology if and only if the fixed points of T separate the fixed points of T" E B(X').

8, (An ergodic mixing theorem). Let S be a left amenable semigroup with a continuous left action on a compact Hausdorff space X. (So C(X) is a right Banach S-space and 1(S)' has its "Arens product" left action onM(X) = C(X)' as in (2.5), (2.6).) Let Y be the carrier space of C(X)" so that M(X)" = M(Y). For

m E .C(S) define P,n: M(X) - M(X) by P,,,v = mv. Let Q,n = P;,, : M(Y) M(Y). (i) Show that Q,,, is a positive, linear, weak* continuous projection with JjQ,n(C)il = JJCJJ for l; >_ 0 in M(Y). Show also that F = Q,n(M(Y)) is weak* closed in M(Y), and is the same set for all m E £(S).

ERGODIC THEOREMS

216

(ii) Show that if

r= F, 17 E M(Y), and 0:5 q< C, then Qn(q)

all m, n E .C(S). (iii) Show that if µ E PM(X) is S-invariant (Problem 1-11) and

Q,,(77) for

E C(X), then the function s -» fx ¢(x),i(sx) dg(x) is left almost convergent on S (Problem 4-15).

CHAPTER 6

Locally Compact Groups of Polynomial Growth (6.0) Introduction. In this chapter, we shall investigate the properties of a remarkable class of amenable, locally compact groups. Consider the following properties that such a group G might possess:

(i) G has polynomial growth, that is, if C E (G), then there exists a real polynomial p such that for all n > 1, ) (C") < p(n). (Note that we can take p(n) to be of the form kn'r for some k > 0, r E N.) (ii) G is exponentially bounded, that is, if C E We (G), then A(C' )'1'

1 as

n-yoo. (iii) G is of Type R, that is, there exists a compact, normal subgroup K of G

such that GfK is a Lie group with Sp(adX) C iR for all X in the Lie algebra of GJK. (iv) G does not contain a free, uniformly discrete semigroup in two generators, that is, there is no subsemigroup T of G which is a copy of FS2 (Problem 0-28) and is such that for some U E %(G), sU n tU = 0 whenever s, t E T, s # t.

(v) ,C(U(S)) # 0 for every open subsemigroup S of G (where U(S) is the space of bounded, uniformly continuous, complex-valued functions on S).

We shall prove the theorem ((6.39)) that when G is connected, the first four of the above conditions are all equivalent, and that each implies (v). (All five properties are equivalent if G is assumed to be a connected, solvable, Lie group.) The class of groups satisfying the "growth" conditions (i) and (ii) is large and includes all nilpotent and many solvable groups. (Complete information about growth in discrete nilpotent groups is given in (6.17).) Every group G of polynomial growth is amenable, and the amenability properties of such a group are strikingly sharp. Consider, for example, the property (iv). There are amenable groups (such as the "ax + b" group) which contain FS2 as a subsemigroup. Which amenable (discrete) groups do not contain FS2 as a subsemigroup? The equivalence of (i) and (iv) shows that in the connected case the answer to the topological version of this question is groups of polynomial growth. Property (v) is concerned with those locally compact groups whose open subsemigroups are (in a suitable sense) left amenable. Further, a group G of polynomial growth satisfies a very strong Folner condition: (6.8) asserts that if C E F (G), then the set K of the Folner condition (4.13 (ii)) can be taken to be 217

LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH

218

one of the sets CN (see also (6.43)). A nonzero fixed-point theorem is discussed briefly in (6.44). Groups of polynomial growth have a particularly strong "weak containment" property ((4.20))-see Problem 6-25.

The scope of the theory presented in this chapter is wide. There are, for example, applications to Riemannian manifolds, the representation theory of connected, solvable Lie groups, the theory of Poisson spaces, a C(T)-functional calculus for Ll (G), and the much studied symmetry problem for Li (G). We shall be primarily concerned with the study of polynomial growth for connected locally compact groups. The corresponding theory for discrete groups is discussed in (6.40). However, the polynomial growth of discrete nilpotent groups is examined in detail in (6.10)-(6.17). We start by establishing some elementary results on polynomial growth and exponential boundedness. Throughout the chapter, C will be a locally compact group with left Haar measure A. For the next result, see Milnor [1), Guivarc'h [1).

(6.1) PROPOSITION. Let C E 9,(G) be such that U' l C" = G. Then A(CI)III exists and is greater than or equal to 1. PROOF. For m, n > 1,

A(Cm+')'`(e-1C") =

//

XC'"+, (Z)A(ZC-1C") dA(z)

(

G

=

(1)

f

Xcm+3 (z) dA(z) f XC-,c. (z-'x) dA(x)

JJ/(

= ( d,\(x) J// Xcm+i(z)XC-i0 (z-1x)dA(z)

f

G

G

A(Cm+' n xC-"C) d A(,). "'+n

Now if x e Cm+", then IT = c1 c2

.

c,n+n (ci E C) and

Cm+' fl xC "C D cl ... CmC fl (c' ... cm+n) (cm+n ... cm-a-1)C = (c1 ... Cm )C.

Hence A(Cm+1 fl xC-"C) > A(C), and from (1), (2)

A(Cm+1)A(C-1Cn) > A(Cm+")A(C).

Since C E E,(C) and U°°_l C" = G, there exists p > 1 such that C-1 C C. So from (2), (3)

A(Cm+n)

2) in (3) yields (5)

A(CT+P) < (A(C))-IA(CT)A(C2P+1).


219

Using (3), (4), and (5), we have for m, n > 2, \(Crn+n) < (A(C)) 1.\(Cm+l)A(Cn+p) < (A(C))-2A(Cm))(C2+pJ1)(Cn+p) :5 flf_(C))-3,(C2+p)A(C2p+l)1A(Cm)A(Cn).

L

Setting 7(m) =llog(a(Cm)), we see that there exists a E R such that -y(m + n) < a + -y(m) + -t(n)

(m, n > 2).

To establish the existence of limn_o,.1(Cn)1/n it is sufficient to show that {-f(n)/n} converges. Let 5(n) = ry(n+ 1). Then since A (C') < A(Cr+1)

(6) b(m+n) = -y(m+n+l) < ry(m+n+2) < a+b(m)+6(n)

(m, n > 1).

It is clearly sufficient to show that {b(n)/n} converges (c.f. [DSI, VIII.1.4). Let N > 1, and for n > 1, write n = qnN+rn, where qn E N and 0 < rn < N. From (6) with n > N,

5(n) < a + b(gnN) + b(rn) < a + ((qn -1)a+ gnb(N)l + b(rn) = qna + qn.b (N) + 6(rn ).

So b(n)/n < (qn/n)[a+b(N)]+b(rn)/n, and since qn/n -+ 1/N and b(rn)/n as n oo, we have

0

lim sup(b(n)/n) < b(N)/N + a/N. n-oo So

f(b(n)/n). n-oo It follows that {b(n)/n}, and hence {A(Cn)1/n}, converge. Since A(Cn)1/n > A(C)I/n, we have limn-,, \(Cn)1/n > 1. 13 limss op(b(n)/n) < lim

Note that if G is exponentially bounded and D E F(G) is nonnull, then, by putting D inside some C E $ (G), it readily follows that limn-.(A(Dn))1/n exists and equals 1. If there exists a nonnuil, compact subset C of G such that

(Cn)1/n > 1, then G is said to have exponential growth. It is easy to see that either G is exponentially bounded or has exponential growth. (Exponential boundedness and polynomial growth are defined in (6.0).) The proof of the next proposition is trivial.

(6.2) PROPOSITION. Every compact group has polynomial growth. (6.3) PROPOSITION. Suppose that C1, C2 E 9' (G) are such that 00

00

UCI=G=UC. n=1 n=1 Let r E N. Then {a(C1)/nr} is bounded if and only if {A(C2)/n'} is bounded. PROOF. Find p, q E P such that C1 C C2, C2 C C'. Then A(Cl) < A(C2°), A(C2) < A(C ), and the desired result follows. 13 The preceding proposition justifies the following definition.


220

(6.4) DEFINITION. The degree of a compactly generated, locally compact group G of polynomial growth is the smallest integer r > 0 such that for each

C E F'e(G) with (,J°D_I Cn = G, there exists kc > 0 such that for all n, A(Cn) 1. We establish (1) by induction on c. The result is trivial if c = 0. Suppose that N E P and the result is true for groups of class < N. Suppose that c = N and let G' = G/Gc. Applying the induction hypothesis to G', we have c-1

C-1

i=1

c-I

ir' (Gz/G,+i) >

id,

(2)

is; + r' (G/H),

iel

i=1

where si = r'((H1Gc)/(Hi+1Gc)) Now (H(Gc)/(Hs+IGc) = (Hi+1Gc)Hi/(Hi+1Go) = Hil(Hi n (Hi+1Gc)) For 1 < i < (c - 1), we have, using (6.10(1)),

si = r'(Hil (H, n (Hi+1Gc))) +r'((HH n (H,+IGc))/H,+1) = s; + r' (Hi+1(Hi nGc) /Hi+1) = sj' + r'((H, n Gc)/(Hi+l n G,)). So using (2) and (6.10(1)) again, c-1

c

isi+r'(G/H) _ Eis=+r'(G/H) =1

=1

C-1

ir'((Hi n Gc)/(Hi+i nGc)) + cr'(Hc)

+ i=3 C-1

c

< idi+Er'(HinGc) C-1

0 such that for k > 1, (1)

[A4'(k)[ 5 k,k(d"-r(H!H')),

where dH = Ef-1 ir'(H=/H;+I) with 1 the class of H. There are now two cases to be considered. (i) Suppose that H/H' is infinite. Then r'(H/H') = 1, and from (1) JA°'(k)1
ks(kd') Let p = d, and Gc = RP x H, where H is a finite group, and let z1, ... , z , be the standard basis for RP. We can find ((6.15)) k4 > 0 such that (2)

(3).

IG,E(z,) < k4n1"c

(n > 1,1 1,

1/
1, there exists k' > 0 such that (4)

IG,E(zi'

z'P) < k'n1/

(n > 1).

For convenience, we now suppose that e E E. From (4) and (6.16), if m > k'n1/c, then (5)

1 jGcnE'"{>jfzl,...,zp}"l=IAP,"I_

(n+P_ 1)

>k5n9,

where k5 > 0 is independent of n. Replacing n by nc in (5), we have (6)

(n > 1, m > k'n).

)Gc n Em I >- k5ncP

From (2) and (6), if N > (i + k')n, then IE'vj > :(E')"uuGc nEN-"I > k3k5nd. The inequality (1) now readily follows.

O


228

(6.18) COROLLARY. Every locally compact nilpotent group has polynomial growth. In particular, every abelian, locally compact group has polynomial growth. PROOF. Use (6.7).

0

(6.19) COROLLARY. Every abelian, locally compact group has polynomial growth.

The next three results discuss polynomial growth for subgroups and quotient groups of a locally compact group G.

(6.20) PROPOSITION. Let H be a closed, normal subgroup of G. Then (i) if H is compact and G/H has polynomial growth, then G has polynomial growth;

(ii) if H is discrete and G has polynomial growth, then G/H has polynomial growth.

PROOF. Let 7r: G - G/H be the quotient map and V E W,(G). By Weil's formula,

AG(V) = f

G/H

dAG/H(xH) f Xv(xh)dAH(h) H

(1)

,r(v)

AH (x-1V n H) dAG/H(xH).

So if H is compact, then AG(V) S .H(H)AG/H(r(V)) = )G/H(r(V)), and (i) follows;

Now suppose that H is discrete, so that AH can be taken to be counting measure. If x E VH, then AH(x-1V n H) > 1, and so from (1), .G (V) >

AG/H(r(V)). Thus (ii) follows, noting that every element of F,(G/H) is of the

form r(V). 0 The next result is due to Hulanicki [6). (6.21) PROPOSITION. Let H be a closed, normal subgroup of G. Let H have polynomial growth and be such that either G/H is finite or G is separable with G/H compact. Then G has polynomial growth.

PROOF. Let Q: G -+ G/H be the quotient map. Since G/H is compact and Q is open, we can find D E F(G) with Q(D) = G/H. Using Appendix C to deal with the separable case, we can find a Borel subset B of D such that QIB

is one-to-one and Q(B) = G/H. Than C = B- is compact. Replacing B by ba 1B, where bo E B is such that Q(bo) is the identity of G/H, we can suppose that e E B. Let A E 9, (G) and Ao = H n C- 'A. Then A0 E %(H), and since G = BH,

AcBAo.

(1)

We define the following subsets D, Do, and Al of H as follows:

D = C-1C2 n H,

Do = U (z-'Aox), zEC

AI = DDo.


229

Clearly Do, and hence Al, belong to %(H). Since, by hypothesis, H has polynomial growth, the proposition will be established once we have shown that for all n, (2)

A(A") < A (A').

Since GJH is compact, we can scale A so that AGIH(G/H) = 1. Now if we know

that (BAo)" C B(A1)",

(3)

then, using (1) and Weil's formula, we would have (cf. (6.20)) ))(A") < A((BA0)")

1 with dim g = n and suppose that the result is true for every group of dimension less than n. Then (B46) we can write G as a semidirect product H xv L, where L = R. Let tl, l be the Lie algebras of H and L respectively. If X E [), then Sp(adfi(X)) C Sp(ads(X)) C iR, so that H is of Type R and hence, by hypothesis, has polynomial growth. Let X0 E [ -r {0}. Then [ = RXo. Note that I) is an ideal of g and [ a Lie subalgebra of g.

Let h(t) = exp(tXo) (t E R) and adXo = ad5Xo. Now Ad0(h(t)) _ exp(tad(Xo)) (B12) and Sp((adXo)i) C iR. Identify b as a linear space with R"-1 and let 11 11,2 be the Euclidean norm on ll. Applying (6.26) with T = (adXo)q and r = (n - 1), there exists a nonzero, even polynomial p, which we can take to have nonnegative integer coefficients, such that II{AdG(h(t)))bII < p(t)

(1)

(t E R),

112. Let W = Ii is the operator norm on L(4) corresponding to 11 {X E h: IIXI12 < 1), V = expW, and C = h([-1,1))V. Then V E ?,(H), and using B46, B9(4), it follows that C E %(G). It remains to find a polynomial q such that )(C") 1) ((6.5)).

where II

Let t1, t2 E [-1, 11, X1, X2 E W, and v, = exp(X,) (i = 1, 2). Then

h(t1)v3h(t2)v2 = h(t1 +t2)[h(-t2)vih(t2))v2 = h(t1 + t2) exp(AdG h(-t2) (X1)) exp X2, and using (1), we have C2 C h{1-2, 21} exp(p(1)W) exp W. Now p(1) E P and if Z E p(1)W, then expZ = (exp(Z/p(1)))P(1) E (expW)P(1). Hence

C2 C h([-2,21)(expW)+P(1) = h([-2,2I)V1+P(1).

An easy induction argument shows that if a, = I + (2)

CT C h([-r,r])V°

(r > 1).

p(i), then

232


Since H has polynomial growth, we can find a polynomial P with coefficients

in N such that )H(V*) < P(r) (r > 1). Choose .1L so that the isomorphism h: R -+ L takes Lebesgue measure on R into \,,. Then using (2) and (5.13), with j : H x L -. G given by j(h, l) = lh, we have ,\(C'):5 .\H x .1L,(9-I(j(V°' x h((-r,r])))) =2rAH(Va,) ' if E W. Let V,1 be Vw,, where 0'1i =1y. There is a basis for VC such that the matrix of each T E I relative to this basis is of the form [Bi

B2

0

0

Bn

234


where Bi is of the form 'Vb (T)

ib (T)

0

with

0n the weights of I.

Now let g be a real, finite-dimensional, Lie algebra and X E S. Applying the results of the preceding paragraph with I = R(ad X), W = 9, we see that, in the obvious way, IF = Sp(adX), and 9c is the direct sum of weight spaces ox (p E Sp(adX)). Further the subspace [gx ,gam ] is contained in g,,,+,,) if (PI + 92) E Sp(adX) and is {0} otherwise. For any Lie algebra El, let {E)(i)} be the lower central series for l (B23). In the next three results, g is a solvable, real, finite-dimensional Lie algebra, and n is its nil-radical.

(6.34) LEMMA. Let b be a subalgebra of g. If 1?+n(2) = g, then I) = g. PROOF. Suppose that b+n(2) = S. It is sufficient to show that n C 4 (for then 4+n(2) C [}). Let r E P and ZI, ... , Zr E n. Write Zi = X,+Yi, where Xi E bnn and Yi E n(2). Substituting for Zi in [ZI, [Z2, [..., Zr]] ... ] and using the fact that (by Jacobi's identity) [n(2), n(P)] C n(P+2), we see that n(r) C h + Since n(') = {0} for some m, it follows that n C b. n(r+I).

(6.35) COROLLARY. Let t be an ideal of g contained in n(2) such that g/t is of Type R. Then g is also of Type R.

PROOF. Let X E 9 and E = Sp(adX) f iR. Obviously, if p, if E E and (lt + p') E Sp(ad X), then (A +,u) E E. Sob = GrIEE g is a subalgebra of gc. Let Ft E E' = Sp(ad X) - E. Then, since g/1 is of Type R, p is not an eigenvalue of ad(X + t). However, if e E 9c is such that for some n, (ad X - lt)' (C) = 0,

then it follows that (ad(X + f) - µ)r`( + tc) = 0 in (g/t)c (= gc/tc). Hence E E fc and

9c=(ed) MEE

cc[j+n)C9c. µEE'

So b+n$2) = gc. By (6.34) (applied to 9c), we have h = gc, so that E = Sp(ad X) and g is of Type R. 0 The solvable Lie algebras of (i), (ii), and (iii) below are the "building blocks" (in a suitable sense) of solvable Lie algebras which are not of Type R. (i) The algebra s2 is the two-dimensional Lie algebra for which there is a basis

{XI,X2) such that [XI,X2] = X2.


235

(ii) Let a E R - {0}. Then 4 is the three-dimensional Lie algebra for which there is a basis {X1,X2iX3} such that [X3,X1] = oX1 - X2, [X3,X2] = X1 -I-OX2,

[X1,X2] = 0.

(iii) The algebra s4 is the four-dimensional Lie algebra for which there is a basis {X1iX2,X3,X4} such that IX1, X2] = 0 = [X3, X4J,

IXl, X3J = X2 = -[X2, X4], [X1,X4J = -X1 = [X2,X3J.

The simply connected Lie groups yielding these Lie algebras are given after the following result, which is due to Auslander and Moore [1]. (6.36) PROPOSITION. If g is not of Type R, then it has (at least) one of the algebras $2, s3, s4 as a homomorphic image.

PROOF. Suppose that g is not of Type R. We shall construct Lie algebras go = g, 91, 92, 93, and 94, each 9i+1 being a quotient algebra of gi, every gi not

of Type R, and g4 E {12,s3,54}. For each i, set mi = g=2), and let n be the nil-radical of gi. Let a, = (mi )e, and ii = { (ad X).,: X E g,). Note that mi C ni (B37).

Since mo c no, we have m(2} C By (6.35), 91 is (i) Let 91 = not of Type R. Trivially, m(2) = {0}. If Y, Z E 91 and A E at, then by Jacobi's identity, we have (1)

[Y, [Z, A]) - [Z, [Y, All = [Y, [Z, A]] + [Z,

JA,Y]J + [A, [Y, Z]] = 0

since m1'1 = {0} and A, [Y, Z] E a1. Hence Cl is an abelian (and so nilpotent) Lie subalgebra of gl(a1). Let W be the set of weights of I. If Y E 91, A E Sp(ad Y) {0}, and Z is an eigenvector for adY associated with A in (g1)c, then

.1Z=adY(Z)=[Y,ZJ so that Z = \-'[Y,. Z] E a1. and \ E Sp((adY),, ). Hence, since 91 is not of Type R, there exists a E T such that a([1) is not contained in M. With W,,, the weight space for t.b E T, let

b=

{W,y: iEW-{a,a}}

and a' = b fl g1. Then a, is an ideal in 91, and (a! ,)c = b. (ii) Let 92 = g1/al. Then a2 = ai/b. Clearly a2 can be identified with W,, if a = U, and with W. a W-& if a U. Suppose that a = a, and (B32)

let {e1,...,e,} be a basis for m2 such that for all X E 92, (adX)m2 is upper triangular with all diagonal entries equal to a(X) for this basis. Let q2 be the span of {el i ... , eb_ 1 } . Note that q2 is a n ideal in 02. Suppose now that a # a.

236


This time, we choose (fl, ... , f,} to be a basis for We such that relative to this

basis, (adX)w, is upper triangular for all X E 92 Note that {fl,..., f,} is a basis for W. Then c = Span{ fl, ... , f,_1 } is an ideal of (92)c. Let q2 be the ideal of 92 spanned by Re c (in an obvious notation). Clearly c+c is the complex span of q2 in (22)c. (iii) Let 93 = g2/q2. Then 9s is not of Type R (since a, a induce nonimaginary weights of 13). We prove that n321 = {0}. If Or = a, then a3 = CX for some X E m3, so that m3 is one-dimensional. Now n32) C MS. If n32) = m3, then trivially, g3/n32) is abelian and so of Type R, contradicting (6.35). It follows that n32) = {0} in this case.

Suppose, then, that a 0- ZV. Then a3 = CZ 6) CZ, where Z is the image of f, in (93)C. So Re Z and Im Z are linearly independent over R. (For otherwise, { f f,} would be linearly dependent over C.) Hence m3 is two-dimensional. By 56 m3. So if n321 # {0}, the argument at the end of the preceding paragraph, then it is one-dimensional. If X is a nonzero element of n(2) , then there is a real weight -y of 13. This is impossible. Hence nt3 ) = {0} in both cases. Now let h = {(ad Y E 93). Arguing as in (i), h is an abelian Lie subalgebra of gl((n3)c). If Q is a nonzero weight of 4, then, from the definition of "weight space" and with W = (n3)C, we have Wp c as. So (n3)C = a3 ® Wo, where Wo is the weight space associated with 0 if 0 is a weight of 4, and is {0} otherwise. Let to = Wo f1 113, and note that Wo = (to)c.

(iv) Let 94 = g3/to. Let p be the inverse image of n4 in 93. Then p is a nilpotent ideal in 93 by B29. Since p obviously contains n3, we have p = n3, and n4 = n3/to = m4. But m3 has dimension one or two. So m4 = n4 has dimension one or two. Also, n92) = {0}. Write n = n4, and, abusing notation, write 9 = $4. Suppose, first, that n is one-dimensional. If X E g is such that [X, n] = {0},

then (RX + n) is a nilpotent ideal in g, and so X E n. It follows that n is the kernel of the map X -. (adX),, (X E g), and since n is one-dimensional and g # n (as g is not of Type R), we must have g two-dimensional. Let {X1,X2} be a basis for g with X2 E it. Since g 54 it, [X1, X21 = kX2 where k 54 0. Scaling X1, it follows that g = 52. Suppose, then, that it is two-dimensional. Then consideration of the weights

b. T of I4 and their relationship to a, a shows that there are two cases to be considered. Let bl = Re b, b2 = Im6, and {X1,X2} be a basis for it such that [X, X1 + iX2] = 6 (X)(X1 + iX2) (X E 9). in the first of the cases, bl and b2 are linearly dependent. Since both bl, b2 are nonzero, there exists o E R - {0} such that 61 = ob2. Choose X3 E g such that 62(X3) = 1. Then with respect to the basis {X1,X2},

(adX)n = 62(X) [ -1 Q] (X E 9). So ad(X-62(X)X3)(n) = {0} for all X E g, and, as in the one-dimensional case, it = {X E 9: [X, n] = {0}}. So {X1,X2,X3} is a basis for g, and using (2) and the fact that n(2) = {0}, we have g = >3. (2)


237

In the second case, we suppose that 61 and b2 are linearly independent. Choose X3i X4 E g such that 61(X3) = 0, 62 (X3) = 1, 61(X4) = 1, and 62 (X4) = 0. Then with respect to the basis {X1,X2} of n, 1-10

(3)

1],

(adX4)_

{1

0].

As above,

n = {X E g: (ad X) = 0}.

(4)

Replacing X3 by (X3 + [X3,X4J) and using (3), we can suppose that [X3, X4) = 0. Further, by (3) and (4), for all X E 9,

(X - (6i(X)X4+62(X)X3)) E n so that {X3 , X2, X3, X4 } is a basis for g. It readily follows that g = s4. The simply connected Lie groups with Lie algebras 52, s3, and s4 respectively are the groups S2, S3, and S4 defined below: (i) S2 is the "ax + b" group; so S2 is R2 with multiplication given by

(s,t)(s',t') = (s+eis',t+t'); (ii) S3 is the semidirect product R2 xD R, with multiplication given by

(a,t)(a',t) = (a+Ao(t)a',t+t'), where

(iii)

() cost `4d t - Cal I - sin t

S4

sin t

cost is the semidirect product R2 X , R2, with multiplication given by

(a, I3) (a', 0') = (a + B(6)a', 0 +0'), where

-cos s B(0) = e' I sins sins 1 cos s

(Q = (s, t)).

[To check this, note that each of these groups is of the form G xv H with G, H E (R, R2), and, by B22. the Lie algebra s of G x, H is, in an obvious notation, g xy. 11, where p'(X) = (adX)e (X E 4). For example, in the case of S4, we have G = H = R2 and p = B. Since expj6 = $ ()0 E g) and Ad(exp(ufl)) = e" as R, we have, for ;8 = (sit), (Cu, t 6 us sinus l P, (Q) = du [ -sinus csus, u_o - -s t j The multiplication in s = R2 xo R2 is given by B5:

(a', /3')) _ ([a, a') + p' (/3)(a') - p%/3')(a), _ (p' (0) (a') - p' (Q') (a), 0) and it readily follows that s =s4, with {-X1,-X2,X3,X4} the standard basis for s.)


238

(6.37) PROPOSITION. Let G be a simply connected, solvable Lie group that is not of Type R. Then there is a continuous homomorphism from G onto (at least) one of the groups S2i S31, S4.

PROOF. Use (6.36), Bit, and B7. 0 The next two results involve the (so far neglected) conditions (iv) and (v) of (6.0). The results (6.38), (6.39) are due to Jenkins [6), [9). (6.38) LEMMA. (i) Let G be one of the groups S2, Ss", S4. Then G contains an open subsemigroup T with disjoint right ideals I, J. (ii) Let G be a solvable Lie group with the property that there exists a left

invariant mean on U(S) for every open subsemigroup S of G. Then G is of Type R. PROOF.

(i)

It is elementary to check that the map ((a, b), (s, t))

(a + ib, t - is) is an isomorphism from S4 onto the complex version C xp C of S2, multiplication in C x, C being given by (w, z) (w', z') = (w + ezw', z + z').

We thus identify S4 with C x, C. Suppose that G is S2 or S4. For j E {-1,1}, define

Ul={(w,z)EG: Re z 1 and (w,,z,) E UI U U_1 (1 < i < n). Let (w, z) _ To (wl,z1)(w2,z2)...(wn,zn) Then

(w, z) _ (WI + ez' w2, zl + Z2) (W3, z3) ... (wn, zn) _

=

n

(w1 s=1

If (wI, z1) E U1, then (w, z) is a typical element of I, and since lez' I = eRez, < e2 and 1w,l < 1 + e-2, we have n-1

Ito -If 1, n-oo thus contradicting (ii). So (ii) implies (iv).

240


Now assume that (iv) holds. Then, a fortiori, G does not contain F2 as a discrete subgroup, so that by (3.8), G is amenable. Suppose that there exists an open subsemigroup S of G such that .f(U(S)) = 0. By Problem 2-10, there exist disjoint, open, right ideals I, J of S. Let a E I, b E J, and let T be the subsemigroup of S generated by (a, b). Since I and J are open in G, we can find C E Fe(G) such that aC c I, bC C J. A simple argument shows that T is a free, uniformly discrete semigroup in G in the two generators a, b, with sC fl tC = 0 whenever s, t in T are different. This contradicts (iv), so that (iv) implies (v). Still assuming (iv), let K be a compact normal subgroup of G such that G/K is a Lie group. If G/K contains a free, uniformly discrete semigroup in two generators, then so does G (cf. (3.1)), and (iv) is contradicted. So G/K does not contain a free, uniformly discrete semigroup in two generators. One easily shows that such a semigroup cannot be contained in any closed subgroup

of G/K, in particular, in the radical H of G/K. Since (iv) implies (v), we see, using (6.38(u)), that H is of Type R. Now G/K is amenable (since G is). Applying (3.8) and (6.32), G/K is of Type R, and hence so also is G. Thus (iv) implies (iii).

a References

Adelson-Welsky and Sreider [1), Dixmier [4), Guivarc'h [1), Jenkins [6), [9), Milnor [1), Hulanicki [3), [6), Greenleaf [2), Wolf [1], Bass [1), Tits [4), and Auslander and Moore [1).

Further Results (6.40) Discrete groups and polynomial growth. Theorem (6.17) asserts that every nilpotent group has polynomial growth, and, in the finitely generated case, gives the degree. What can be said about finitely generated, solvable groups? The question is examined by Wolf [1] and Milnor [2). (See also Bass [1).)

A group G is said to be polycyclic if there exists a normal series

G=A,>A2D...t> A,={e} such that every quotient AL/A,+1 is cyclic. The following theorem follows from the work of Wolf and Milnor. Let G be a finitely generated, solvable group. Then (a) if G has polynomial growth, then it is polycyclic. (b) The following three statements are equivalent: (i) G has polynomial growth; (ii) G contains a nilpotent subgroup N of finite index; (iii) G is exponentially bounded.


241

It follows using (6.21(2)) that the degrees of G and N are equal. The latter, of course, is given by (6.17). A natural question (raised by Milnor, Wolf and Bass) is the following: is the preceding theorem true if G is no longer required to be solvable? Major progress on this problem has been made by M. Gromov 111, who proved the deep theorem that a finitely generated group has polynomial growth if and only if it contains a nilpotent subgroup of finite index. An account of the proof of this theorem is given in the useful paper of Tits 141. Milnor 12l essentially raised the following natural question: is every exponentially bounded group of polynomial growth? R. I. Grigorchuk ISll has shown that this is not the case. Property (iv) of (6.39) for a discrete group G is investigated by Rosenblatt [1l, [4l. (Note that for discrete amenable groups, properties (iv) and (v) of (6.39) are equivalent (Problem 1-23).) Rosenblatt shows that if G is solvable, then G has polynomial growth if and only if G does not contain FS2 as a subsemigroup. Rosenblatt defines a group G to be supramenable if, in the notation of (2.32), there exists an invariant measure for every triple (G, X, A). By considering (G, G, G), we see that every supramenable group is amenable. Further, by (2.32), G is supramenable if and only if every triple (G, X, A) sat-

isfies the translate property. He also shows that every exponentially bounded group is supramenable (cf. (6.42)), and that no group containing FS2 as a subsemigroup is supramenable. A natural conjecture is that an amenable group G is supramenable if and only if G does not contain FS2 as a subsemigroup. This conjecture is established when G is solvable.

(6.41) Polynomial growth and fundamental groups. It is interesting to record that groups of polynomial growth arise, somewhat unexpectedly, in the work of Milnor on the relationship between curvature in a Riemannian manifold and the fundamental group of the manifold. The book by Bishop and Crittenden ill is a good reference for the differential geometry required. Let (M, d) be a complete, connected, Riemannian manifold that is assumed to have nonnegative mean curvature. Let (M, d) be the simply connected covering

Riemannian manifold of M with covering map p: M --> M. Then (M, d) is also complete, and by relating the Riemannian connections on M and k in the natural way, we see that k also has nonnegative mean curvature. An important "comparison" theorem is the following: if wn is the volume of the unit ball in Rn, then f warn, U(B.(r)) where, for x E M, r > 0, By(r) is the ball with centre x and radius r in M, µ is the volume measure on M, and n is the dimension of M. Recall (B3) that ir1(M) can be realised as the group of covering transformations of M. Also each T E ir1(M) is isometric on k, that is,

(1)

(2)

d(Ta,Tb) = d(a, b)

(a, b E M).

The following beautiful result is proved in Milnor [1].

242


Let (M, d) be a complete, n-dimensional, Riemannian manifold with nonnegative mean curvature. Let F be a finite subset of G = rI (M). Then there exists C > 0 such that JFi' < Cr" (r E P). In particular, G has polynomial growth.

[Let xo E M and k = maXTEFd(xo,Txo). Using (2) and the triangular inequality, we have (3)

F'(xo) C Bxu (kr)

(r E P).

It follows from (3) that for any e > 0, (4)

F'(Bx0(e)) C Bxo(kr+E)

(r E P).

Find an open neighbourhood U of p(zo) such that p I(U) is a disjoint union of open sets V. (a E A), with plv a diffeomorphism from V. onto U and such that

if T, S E G with T 0 S, then T(V,) f1 S(V0) = 0 for all a. (See B3.) Hence we can find e > 0 such that the sets T(Bxo(e)) (T E G) are pairwise disjoint. Noting that T(Bxo(e)) = BTxo(e), it follows using (1) and (4) that 1FJ'p(Bxo(e)) = g(F'(Bxa(e)))

lt(Bxfl(kr+e)) <W"(kr+E)"

and the desired result follows.]

Milnor also showed that if a Riemannian manifold M is compact and is such that all of its sectional curvatures are less than 0, then rI (M) has exponential growth ((6.1)). He conjectures that a similar conclusion is true if some of the sectional curvatures are allowed to be zero. P. Eberlein )1] and Chen 13] have made some progress in this direction. J. A. Wolf [1) showed that if M is compact with every sectional curvature nonpositive, then r1(M) contains a nilpotent subgroup of finite index if and only if M is flat (that is, every sectional curvature is 0). S. T. Yau 11) shows that, with the same conditions on M, if rl (M) is solvable, then M is fiat. Other papers in this area are Preissman [1], Byers 11], Myers 11], Chen 121, [1], Wood 11), Gromoll and Wolf 111, and Lawson and Yau [1].

The study of the relationship between a manifold M and the polynomial growth or amenability of its fundamental group is continued in the paper of Hirsch and Thurston 11). Let 9 be the smallest class of groups that contains all amenable groups, is such that the free product G * H E 9 whenever G,

H E ', and contains every group K with a subgroup L E 9 of finite index in K. They show that if M is a compact, Riemannian manifold with negative sectional curvatures and rl (M) E ', then the Euler characteristic X(M) of M is 0. Chen 12) shows that if 7rl (M) E ', then rrl (M) is actually free, and any amenable subgroup of rI (M) is infinite cyclic. Milnor (4) studies the question: which groups occur as fundamental groups of complete affinely fiat manifolds? He shows that every torsion free group with a polycyclic subgroup of finite index is such a fundamental group. The converse seems to be unresolved.


243

A discrete group G is said to have subexponential growth if there exists a finite,

generating set F for G, with F-I = F, such that

lim nf(jF"}1j/jF^j) = 1. This is a Fminer-type condition, and is closely related to conditions considered in (6.8) and (6.43). Indeed, it follows from (6.8) that every finitely generated, exponentially bounded group has subexponential growth. There is a close connection between "subexponential growth" of volume in a Riemannian covering space and subexponential growth of a related fundamental group (Plante (11, (21, 131)

(6.42) Exponentially bounded groups and the Translate Property

(Rosenblatt (11, 141; Jenkins (91, (101, (111). Recall ((2.32)) that if G is discrete,

then G has the Translate Property if whenever xl,... , x,, E G, al, ... , a E R,

and z 1 aiXx,A > 0, then £i 1 a; > 0. We note that if µ = E7--, a;bx-J E

M(G), then Z%-1 aiXX;A = XAU, and that F; ai = (En ai6x -1 )(G) _ 1

µ(G). This motivates the version of the Translate Property for general locally compact groups in (i) below. The proof of (i) is left as Problem 6-13. (i) Let G be an exponentially bounded, locally compact group. Then G has the

Generalised Translate Property, that is, if y E M(G) has compact support and E L. (G) is nonzero such that ¢ > 0, op >- 0, then µ(G) > 0. (ii) Let G be an amenable locally compact group having the Generalised Translate Property. Then .1r(U(S)) 34 0 for every open subsemigroup S of G. [Suppose that there is an open subsemigroup S of G with .B(U(S)) = 0. By Problem 2-10, we can find open, disjoint, right ideals I, J of S. Note that S, I,

J are not {e}. Let a E I - {e} and b E J -r {e}. Choose C E Te(G) such that C = C-1, CaC Cl, and CbC C J. Let 0 = XS and let v E L1(G) be given by v = XaC - XabC - Xa2C-

Let µ = V. We will contradict the Translate Property by showing that µ(G) < 0 while Op > 0.

First, u(G) = v(G) = (.l(aC) - A(abC) - A(a2C)) = -A(C) < 0. Second, let t E G. Then (1)

00) = ,(Sr') = v(tS-') = (A(tS-1 n aC) - .1(tS-1 n abC) - A(tS-1 n a2C)).

There are three cases to consider. Suppose that tS-1 n abC = 0 = tS-1 n a2C. Then, trivially, 4p(t) > 0. Suppose now that tS-1 n abC 0 0. Then t = abcs = S-1 and for some c E C, s E S. If cl E C, then, noting that SS-' D s(Ss)-1

cilbeECbCCJCS, tS'1 =

= {acl(cr'bc)(cilbe)`1} = {acl}. abcsS-1

abcS-1

aci(ci'bc)S-1


244

Hence ,l(tS-1 fl aC) = .1(aC) = A(C). Now if z E b-Ia-ltS-I n a 2tS-1 fl C, then t E a(bzS fl azS) C a(bCS fl aCS) C a(I n J). This is impossible since

InJ=0. Hence A(tS-I n abC) +.1(tS-I fl a2C) = A(b-la-ItS-1 n C) + A(a 2tS-I f 1 C)

< A(C) < A(tS-I ft aC), and from (1), 45µ(t) > 0. A similar argument deals with the case where we suppose that tS-1 fl a2C o 0. Hence Op ? 0 as required.]

(6.43) A FOlner condition for exponentially bounded groups (Jenkins If G is amenable, C E '(G), C 0 0, and E > 0, then it follows from (4.13(ii)) that there exists nonnull K E F(G) such that A(CK K)/A(K) < s. In general, it is not clear how to construct such a set K. We will show that (91).

if G is an exponentially bounded, connected, solvable Lie group and C° 34 0, then very strong information about K is available: K can be taken to be one of the sets C' ! It seems plausible that, in general, if G is a connected, locally compact group, then G is exponentially bounded if and only if it satisfies the Folner condition (1) below. (This result is asserted in Jenkins [9), but there are gaps in the proof.) The Folner condition we will be considering is: given C E'(G) with Co gE 0 and e > 0, there exists N E P such that (1)

,(CN+1 d CN)I.'(CN) < e.

(i) Let G be exponentially bounded and C E F(G) with C nonnull. Suppose that for some p E P, there exists x E C such that xP+l E Cp. Then given e > 0, there exists N E P such that the inequality (1) is true. (Let r > 1. Since xP+l E CP+1 fl CP, we have Cr+P+l fl Cr+P+2 D xP+1 Cr+l so that A(Cr+P+1 0 Cr+p+2)/A(Cr+p+1) 1V

< A(Cr+p+l ,L Cr+p+2)/A(Cr+l)

(2)

(,\ (Cr+p+ 1) + A (Cr+p+2) -2.\(Cr+p+l flCr+P+2)1[A(Cr+i) < 2(A(Cr+P+2) - A(Cr+1)]/A(Cr+1) 1

21(A(Cr+P+2)/A(Cr+I))

=

1

- 1).

Let a, = A(C7). Since G is exponentially bounded, a,/r --p i as r -+ oo, and 1 < Jim inf(a,+P+2/ar+l ) r-.oo = lim inf(a,+p+2ar+p+1 r-+oo

lim (a,+P+l

.

ar+2)/(ar+p+lar+p ... ar+l) ...a,+I)1/r = 1.

00

The required result now follows.]

Note that by taking C E 9,(G) and x = e we obtain (6.8). In the discrete case, it is hard for (1) to be satisfied for all C. Indeed, if we take G = Z and C to be a finite, nonempty, set of odd integers, then, for


245

all n, [C"+l A C"(/lC"[ >- 2! We therefore concentrate on the case when G is connected.

(ii) Let G be a connected, solvable Lie group. Then G is exponentially bounded

if and only if (1) is true. [Suppose that G is exponentially bounded, and let C E F(G) with C° 0 -F 0. Let g be the Lie algebra of G. By Appendix E, exp(g) is dense in G. Hence we

can find X E g and r > 1, r E P such that both expX and exp((1 - r-1)X) are in Co. Let x = exp((1 - r-1)X). Then x' _ (expX)'-I E C"1. Then the inequality of (1) follows using (i). Conversely, suppose that (1) is true. Let S be an open subsemigroup of G. By

(6.39), it is sufficient to show that £(U(S)) 0 0. Suppose that £(U(S)) = 0. By Problem 2-10, there exist disjoint, open, right ideals 1, J of S. Let CI E @'(I)

and DI E '(J) with CIO # 0, DO # 0. Let C = CI U DI, and write Cn =

C"UDn,where C"=C"f1I,Dn=C"nJ. Then then Cn+1 D (cCn) U (cDn), so that .1(C"+,) > .A(Cn) + A(Dn). Similarly, A(Dn+I) >- A(CC)+A(D,,), so that A(C"+1) >- 2(A(Cn)+A(D,,)) = 2A(C"). So A(Cn+1,L C")/A(C") >- 1 for all n, and (1) is contradicted.] (iii) Emerson and Greenleaf (2) obtain sharp estimates on the growth of,\(Un), where U is an open, relatively compact subset of a locally compact abelian group

G. They show that there exists a constant A > 0 and k E N such that

A(U) = Ank +

(3)

O(nk-I log n)

(n - oo).

It immediately follows that A(U"+1)/A(U") - 1 as n -> oo. It also follows from

their work that if C E '(G) has nonempty interior, then A(C"+1)/A(C") - 1 as n

oo. Similar issues for a nilpotent, simply connected Lie group are treated

in Porada 111.

(6.44) Polynomial growth and the nonzero fixed-point property (Jenkins (121). Recall ((2.24)) that the amenability of a locally compact group G is equivalent to the existence of a fixed-point in every affine left G-set K on which G acts in a jointly continuous manner. Now such a set K is, by definition, a compact convex subset of some locally convex space Z. One can thus enquire what properties of an amenable, locally compact group G are necessary for there to exist a nonzero fixed-point in every such set K? (Of course, by the above fixed-point theorem, such a question is relevant only when 0 E K.) Particular examples show that we need to qualify the question as follows. The group G is said.to have the nonzero fixed-point property if whenever Z is a locally convex space, x T. is a homomorphism from G into the algebra of linear

operators on Z, with Te the identity operator, such that the map (x, e) - T. (e) is continuous from G x Z into Z, and K 54 {0} is a compact convex subset of Z, with TT(K) = K for all x E G, for which there exists a E Z' such that for all e E K {0}, a(C) > 0 and e/(supZEG a(xe)) E K, then K has a nonzero fixed-point for G.


246

Jenkins proves that if G is either a connected, locally compact group or a discrete, finitely generated, solvable group, then G has the nonzero fixed-point property if and only if G has polynomial growth.

(6.45) The representation theory of solvable Lie groups and the Type R condition. Recall that a locally compact group G is liminal (OCR) if ir(f) is a compact operator whenever f E Li (G) and it is an irreducible, unitary representation of G. Every liminal group is of Type 1 (postliminal, GCR). If G is nilpotent, then G is liminal (D±nier (3], Kirillov 11], Fell 121). Not every Type R group is liminal. Indeed the Mautner group (Problem 6-8)

is not even of Type 1. C. C. Moore, in Auslander and Moore (11, proved the following remarkable theorem. Let G be a Type 1, simply connected, solvable Lie group. Then G is liminal if and only if G is of Type R. (We shall be content to outline the proof of the easier of the two implications involved. Let G be a liminal, simply connected, solvable Lie group. Suppose that G is

not of Type It. Then we can find a closed, normal subgroup H of G such that G/H is one of the groups S2, S31, S4. Since the map QH: Li (G) , Li (G/H) of (1.11) is a continuous '-epimorphism, it follows that G/H is also liminal. A contradiction is obtained by showing that none of the groups S2, S3, S4 is liminal. (This is well known for S2 (Nelson and Stinespring (1]).) Each of the groups is

a semidirect product of the form R" xn R"'', so that a well-known theorem of Mackey (Warner 11, Vol. 1, pp. 439-440]), together with general results of Fell on induced representations, can be used to establish the contradiction (Problem 6-10).]

See p. 129 of the above memoir of Auslander and Moore for a discussion of the importance of the Type R condition for applying the "Mackey machine." The question of characterising Type 1, simply connected, solvable Lie groups is investigated by Auslander and Konstant 11]. They develop a theory in the spirit of Kirillov's work on nilpotent Lie groups. In the work of Pukanszky 121,

(1] it turns out that, for a locally compact group G, the primitive ideal space Prim(e), rather than the dual space G, is the appropriate object of study. He proves that if G is a simply connected, solvable Lie group, then Prim(G) is a Ti-topological space (or equivalently, every element of Prim(G) is maximal in 0 *(G)) if and only if G is of Type R. (Note that this generalises Moore's result

above since if G is of Type 1, then Prim(G) can be identified with G and G is a Ti-topological space if and only if G is liminal.) Moore and Rosenberg 121 and Pukanszky 141 have generalised this theorem to cover the case of almost connected, locally compact groups.

(6.46) Poisson spaces, amenability and the Type R condition. Let G be a locally compact group and u E PM(G). We define H,., the space of p-harmonic functions on G, by

Hp_{0EUr(G):µg5=4}.


247

Recall that U,(G) is the space of bounded, right uniformly continuous, complexvalued functions on G, and that for 0 E U,(G), po(x) = fG O(xy) dp(y)

(x E G).

Clearly, 1 E H,, and 45xo E H, if ¢ E Hµ, xo E G, so that H is a right invariant subspace of U,(G). Obviously, Hµ is closed in U,(G). We now formulate the notions of Poisson space and Poisson kernel of M. A Poisson space for µ is a pair (X, v) such that (i) X is a compact, Hausdorff space;

(ii) X is a left G-set with ee = e for all E E X, and the map (x, continuous from G x X into X; (iii) v E PM(X), and the map T,,: C(X) - Ur(G), where (1)

x is

T,,f(x) = fx f(xf) dv(e)

is an isometry from C(X) onto H.. It is immediate from (1) that T,, is "equivariant", that is, (T f)xo = for all f E C(X), xo E G. The measure v is the Poisson kernel of the space. Sometimes we just refer to X as a Poisson space, reference to v being left implicit. If (X, v), (Y, v`) are Poisson spaces of µ, then T,-,,'T, is a linear isometry from C(X) onto C(Y) that preserves the action of G, and the Banach-Stone Theorem (DS, (V.8.8)] then shows that (X, v) and (Y, v`) can be identified. So provided there exists a Poisson space for µ, then we can talk of the Poisson space for µ. The above notions were essentially introduced and studied in the important paper of Furstenberg (1]. We now establish the existence of a Poisson space in general. This was proved, using probabilistic techniques, by Furstenberg when G is separable. (See Furstenberg (1), (3], (4], Cartier (1], Zimmer (6], and Azencott 11].) We will follow the

proof in Paterson [6]; this proof does not require G to be separable and even applies to locally compact semigroups.

(i) There exists a Poisson space for every u E PM(G). (Let p E PM(G). Let L be the weak' closure of

co{;?: n> 1} in MG)`. Now U,(G) is left introverted ((2.11)), and by (1.4) and Problem 2-15 (Solution), cv E Ur(G) and v(p(k) = p(¢v) for all v E PM(G), p E Ur(G)` and 0 E U,(G). Hence if p6 -+ p weak* in Ur(G)`, then vp6 -+ Pp weak` in the Banach algebra Ur(G)`. For n > 1, we thus have a weak` continuous affine map T. on L given by Tn(p) = A'p. Applying Day's Fixed-Point Theorem (or even the Markov-Kakutani Theorem) we can find po E L such that Apo = Po. If vo -+ Po weak' with v6 E co{µn: n E P}, then DQpo = po for all c, and since lgpo -+ pogo weak`, it follows that (po)2 = po. We can therefore define a positive, norm one, unit preserving, linear projection P on Ti,. (G) by setting P0 = PoO. We claim that P(U,(G)) = H,. Indeed, if 0 E Ur(G) and po5 = 0,

248


then µ45 = (µpo)cl = poo = 0, so that 4) E H,,. Conversely, if 0 E H and v, - po as above, then ¢ = v,0 --+ pod, pointwise, so that P¢ = 0. By Problem 6-16 H is a commutative, unital C'-algebra with 1 as the identity and multiplication and involution given by 46 x 0 = P(OO),

(0' _ 5

(0, ?k E H,).

Let X be the maximal ideal space of H,,. As the map (¢, x) - 4x is jointly continuous from H,M x G into H, (since H, C Ur(G)), it follows, after dualising, that X is a left G-space with ee = tz for all 1; E X, and that the map (x, £) --+ xi

is continuous. Let T : H, -* C(X) be the Gelfand transform, and define v E PM(X) by setting v(f) = T-1(f)(e) (f E C(X)). Define T,, as in (1). Then for

0 EH,,,zEG.. T, 4(x) = v(qx) = v(ex) = T-I (x)(e) _ Ox(e) _ O(x)

Thus Tv 1 = T so that T is a linear isometry. Thus (X, v) is a Poisson space of µ.)

(ii) Furstenberg investigates the deeper issue of determining the Poisson spaces that can arise when G is a connected, semisimple, Lie group with finite centre. An excellent account of this, and of Poisson spaces in general, is given in Azencott [1}.

Let G be a connected, semisimple, Lie group with finite centre, and let KAN be an Iwasawa decomposition of G (B61). Then K is a maximal, compact subgroup of G and S = AN is a closed, solvable subgroup of G. Let M be the centraliser of A in K; so

M= {xEK: xc = ax for all aEA}. Then xSx-1 c S for all x E M, and it follows that H(G) = MS is a closed subgroup of G. The group H(G) is called a minimal, parabolic subgroup of G and is of importance in the representation theory of semisimple Lie groups. Clearly,

S is a normal subgroup of H(G) and M is compact, so that H(G) is amenable. The work of Furstenberg, together with a result of C. C. Moore, shows that there are only a finite number of possibilities for the Poisson spaces X. as fL

ranges over P(G): such an XM has to be of the form G/(PS), where P, is a (compact) subgroup of G such that

McCP, CM. (Of course, there is only a finite number of such subgroups since M/Me is finite.) The subgroup P,, depends only on the semigroup generated by the support of A. In particular, G acts transitively on every Poisson space. We will not give the proofs of these remarkable results. However, we note that the amenability of H(G) is used in the proof in an application of the fixed-point theorem (2.24) and that G/H(G) is the "maximal boundary" of G.


249

It is convenient at this point to mention some results of C. C. Moore [2] on the maximal (closed) amenable subgroups of G. Two examples of such subgroups are

the maximal compact subgroup K and the minimal parabolic subgroup H(G). Moore proves that all such subgroups are obtained as follows. Let X = G/H(G). For each p E PM(X), let

G, ={xEG:XU =p}, where, of course, the action of G on M(X) is that induced by the action of G on the homogeneous space X. Moore proves, using techniques from the theory of algebraic groups, that G, is always an amenable subgroup of G. (This result is also proved by Guivarc'h 12].) Of particular importance are those subgroups of the form G, where p is in the weak' closure E of the set of measures v E PM(X) that are invariant under the actions of some maximal compact subgroup of G and the maximal, normal, amenable subgroup of G. It turns out that, with a certain connectivity condition assumed, the set say of maximal amenable subgroups of G is precisely the set {G,, : p E E}. The set E is G-invariant, and by examining the orbits of G in E, the elements of .a' can be written down explicitly and the cardinality of the set of conjugate classes of d determined. We now proceed with our discussion of Poisson spaces. Let G be a locally compact group. A measure p E PM(G) is said to be spread out (etalee) if for some n E P, p' is not singular with respect to Haar measure

A. The group G is said to be of Type (T) if whenever p is spread out on G, then G acts transitively on the Poisson space X,. for p. (So X, can be identified with some compact quotient space G/H, where H is a closed subgroup of G.) Azencott [1] generalises Furstenberg's results above to the case of a spread out measure on a group of Type (T). Paterson [6] shows that if S is a compact, jointly continuous semigroup and p C PM(S) is such that its support S(p) generates a dense subsemigroup of S, then the Poisson space of p can be identified with X, where the kernel of S is a "Rees product" X x G x Y.

(iii) What can be said about Hu, X,, and the amenability of G? The space X, is said to be trivial if X,, is a singleton, or equivalently H, = Cl. If there exists p E PM(G) such that X,, is trivial then G is amenable. [Suppose that H, = Cl for some p, and let P: UT(G) H, be the projection in the proof of (i). Then m E £(Ur(G)), where PO = m(o) 1, and G is amenable.) Furstenberg [3, p. 213] gives a simple example of a measure p on the (amenable) "ax + b" group with X. nontrivial. However, if G is a locally compact abelian group and p E PM(G) is such that S(p) generates a dense subgroup of G, then Xµ is trivial. An elegant proof of this has been given by Choquet and Deny, and appears in Furstenberg [4]. See also Revuz [1, Chapter 5, §11. A related result for nilpotent groups is given in Furstenberg [3]. As we shall see in (6.47), if p is "recurrent," then X. is trivial.

250


If G is o-compact, then there is an important criterion for determining whether

or not X. is trivial. This involves the augmentation ideal L°(G) = (f E LI (G) : f f dA = 0} of L1(G). (Recall (Problem 4-38) that G is amenable if and only if L°(G) has a bounded approximate identity.) We consider two related conditions for u E PM(G):

(A) for all f E LO(G), N-1'[ f * n I nl[I --* 0 as N -+ oo, (B) for all f E L° (G), 11f * n 1 1 11 0 as n oo. It is elementary that (B) (A). The criterion alluded to above is (cf. Rosenblatt

[Si)): X, is trivial if (B) holds. [Suppose that (B) holds and let 0 E H,., so that µn¢ = ¢ for all n. Then for f E L°(G), lim

N-I

N-oo

45

f * n=1 µn)

< lim I N-1 N-co

*

00

1) 110H. = 0. I

So f *0= 0. If g E L1(G) and x E G, then putting f = (g * x - g) gives so that (x4. - 4.)(L1(G)g) = {0}. Since L1(G)2 = L1(G), we have x¢ ¢ E CI.) The converse to this result also holds. Rosenblatt [Si) shows that if G is or-compact and amenable, then there exists (A), = µ"' E P(G) with support equal to G such that (B) holds. Since (B) this gives a positive answer to a question of Furstenberg [3]: a a-compact, locally compact group G is amenable if and only if there exists µ = µ" E P(G) with S(p) = G and X, trivial. An earlier proof of the last result (when G is discrete) was given by Kaimanovich and Vershik, and their paper [S) contains a useful detailed discussion of how amenability relates to boundaries and entropy. See also Birge and Raugi [1]. (iv) Polynomial growth, in its Type R form, turns out to be relevant to the problem of determining when a locally compact group G is of Type (T). The following theorem is the result of work by Azencott [1], Brown and Guivarc'h [1), and Moore and Rosenberg [i]. Let G be a connected Lie group with radical S and g and s their respective Lie algebras. If G/S has finite centre, then G is of Type (T) if and only if Sp(ad(X)S) C iR for all X E g. This result, combined with Azencott [1, Theoreme V.31, yields that the following three statements are equivalent for a connected, solvable Lie group G: (a) G has polynomial growth;

(b) G is of Type (T); (c) Xµ is finite for every spread out measure µ E PM(G).

(6.47) Random walks and amenability. Let G be a separable, locally compact group and µ E PM(G). Let Q'= G', 11= GP. Let µj = µ (i E N), and let P be Sl' = GP, the infinite product. jj xI µ, on Q. Then P is a probability measure on 0'. For each i, let X,: 11 -+ G be the canonical projection onto the


251

ith coordinate space G, (= G). For x E G. let Tx: f2' --y f2 be given by

T,(xi, x2, ...) _ (x, xx1, xxix2, ... ) and P. be the 11-probability measure P o TZ-1. Then the triple {f2, {PZ}, {Xn}) is a random walk of law p on G. Now suppose that the subgroup of G generated by S(p) is dense in G. Consider the following two properties: (i) Pe({w E ft: Xn(w) E C for infinitely many n}) = 0 for every C E E°(G);

(ii) Pe({w E f2: Xn(w) E U for infinitely many n}) = 1 for every open neighborhood U of e.

If (i) [(ii)] holds, then p is said to be transient [recurrent]. It can be shown that p is either transient or recurrent. Furstenberg [3] shows that if G is not amenable, then p is necessarily transient. (This result is also a consequence of (6.46(iii)) and the Dacunha-Castelle et al. result below.) On the other hand (Dacunha-Castelle et at. [1, p. 286]) p is recurrent if and only if Hu = Cl (so that G is amenable) ((6.46)). However, there exist amenable groups with transient measures p (cf. (6.4(iii))). Indeed, it is proved in Brunel, Crepe] et al. [1] that if G is not unimodular, then every p is transient. Thus, every p on the "ax+b" group is transient. Even more striking, if G = Rd or Zd with d > 3, then every p is transient (Revuz [1, p. 100]). It is shown in Dacunha-Castelle et al. [1, p. 294] that if there exists a recurrent p on G, where G is a connected, Lie group, then G has polynomial growth.

The measure µ, where 71 = ==o pn is defined on R(G), is of importance. It can be shown (Revuz [1, p. 89]) that p is recurrent if and only if )7(U) = 00 for every open neighborhood U of e. If p is transient, then µ is a Radon measure in the sense that µ(C) < 00 for all C E '(G) and µ >- 0. Suppose that p is transient. The vague topology on the set R(G) of Radon measures on G is the weakest topology for which the maps v --> f f dv (f E C, (G)) are continuous. Let K = {by * µ: x E G}. The transience of p entails that K is relatively compact in the vague topology. Consider the closure K of K in R(G). It can be shown (ibid, p. 140) that either 'K -r K = {0} or there exists c > 0 such that K --- K = {0, ca}, where A = AG. In the first case, p is said to be of Type I; in the second case, p is said to be of Type U. Guivarc'h [2] shows that if G is not amenable, then p is necessarily of Type I. The amenable case is discussed in Elie [1], who proves the following theorem. Let G be amenable and almost connected. If G is unimodular, then there exists

a transient, spread out measure of Type II on G if and only if G = R x K for some compact group K. Elie also completely analyses the (more complicated) nonunimodular case.

(6.48) Symmetry, the Wiener property, and polynomial growth. Recall ([Ri], [BD]) that the spectrum Sp4(a) (or Sp(a)) of an element a in a Banach algebra A with identity 1 is the set of complex numbers a such that (a - al) is not invertible in A. If A does not have an identity-for example, if A = LI (G) with G nondiscrete-then Sp(a) is the spectrum of a in the algebra

252


A = A + Cl obtained by adjoining an identity to A. The set Sp(a) is nonempty and compact in C, and the spectral radius v(a) of a is defined to be ((an((1/n sup{(a(: a E Sp(a)}, and equals A Banach *-algebra A is said to be symmetric if Sp(z*x) C [0, oo) for all z E A (RI, p. 233]. It is known that A is symmetric if and only if Sp(h) C R whenever h e A is selfadjoint [BD].

Let G be a locally compact group. The group G is said to be symmetric if the Banach *-algebra L1(G) is symmetric. The class of symmetric groups is denoted by [S). An interesting, and topical, question is that of determining which groups G are in [S) (Bonic [11). This question will also be referred to as "[S]." The question has amenability overtones: every almost connected locally compact group in (S] is amenable. This result, due to Jenkins (8], uses the representation theory of semisimple Lie groups. Is there an easier proof available? As Leptin and Poguntke (1] comment, "the history of [S) is a line of destroyed hopes and wrong conjectures." For example, at one time it was hoped (Hulanicki (1]) that every amenable group was symmetric. However, Jenkins [1] produced an amenable group not in [S). Indeed, Jenkins [2) showed that if a discrete group G contains FS2 as a subsemigroup, then G 0 [S] (Problem 6-24). Thus,

for example, the (amenable) "ax + b" group S2, as a discrete group, is not symmetric ((6.39)). (However, with its usual connected Lie group topology, S2 E [S]. Indeed, it is shown by Leptin and Poguntke that there exists exactly one simply connected, solvable Lie group of dimension less than or equal to 4 that is not symmetric.) Does every group G of polynomial growth belong to (S] (Gangoli [1])? Evidence in favour of this is provided by the following: G E (S) if one of (a), (b), (c), (d), or (e) holds: (a) G is compact; (b) G is discrete and nilpotent (Hulanicki (51); (c) G is a connected, nilpotent Lie group (Poguntke (1]); (d) G E [FCJ- ((4.23) (Anusiak (1]); (e) G is a motion group (Gangoli [1]). However, it is shown in Fountain, Ramsay and Williamson [1) that there exists a discrete, nonsymmetric group of polynomial growth. This example is discussed in (6.51).

Perhaps the most remarkable result in recent years on [S] is the following beautiful theorem of Ludwig (1): G E (S) if G is connected and of polynomial growth. Ludwig's theorem is discussed in (6.50). He also establishes the symmetry of compact extensions of nilpotent groups. These results have recently been generalised by Losert (S7]. Jenkins (8] shows that every connected, reductive Lie group with noncompact, semisimple component is nonsymmetric.

Let G be exponentially bounded. Then there exists an important dense, *-subalgebra B of L1(G). References for B are Jenkins [2), Hulanicki (7), (8], (9], and Pytlik (1). The elements of B are the "rapidly decreasing" functions on


253

G. The algebra B is "symmetric" in the sense that SpL, (C) (f * f-) C (0, oo) for all f E B, and is relevant to the problem of determining when SpL, (C) (f) = Sp(ir2(f)) for f = f in LI(G). (Here, 72 is the left regular representation of G.) In this connection, see (6.49). The algebra B is also relevant to two other issues.

The first of these is the elegant "Q(')-functional calculus theorem" (Kahane (2], Dixmier (4], Hulanicki (8])-see Problem 6-22. The second of these concerns the study (Hulanicki [81) of the algebra generated by the fundamental solution {pt} of the heat equation on a connected Lie group

G. The functions pt form a commuting, approximate identity for B. (This is generalised in Hulanicki and Pytlik [1).) Hulanicki obtains a Wiener-Tauberian type theorem when G has polynomial growth, using the above functional calculus theorem. Another important property for a group, related to symmetry, is the Wiener Property (Leptin (7), (8]). A locally compact group G is said to have the Wiener Property (or, simply, is Wiener) if every closed, proper ideal of LI (C) is contained in the kernel of a nondegenerate, continuous, *-representation of LI (G) on a Hilbert space. (The classical Wiener theorem [R, Chapter 1, (4.4)1 can be reformulated: R is Wiener.) A related condition is introduced in Leptin [8): G is said to be weakly Wiener if every closed, proper ideal of LI(G) is contained in a primitive ideal of LI (G). It is proved in Leptin [7] that if G is symmetric, then G is Wiener if G is weakly Wiener. From Leptin (8] and Ludwig [1], we have that G is weakly Wiener if G has polynomial growth. It follows from Ludwig's theorem on symmetry that every connected group of polynomial growth is Wiener. (See also Hulanicki, Jenkins et al. (1).) Ludwig shows that discrete exponentially bounded groups and discrete solvable groups are Wiener groups. Does polynomial growth imply Wiener (Hulanicki, Jenkins et al. (11)? Does there exist a discrete group which is not Wiener (Leptin (8))? Using a theorem of M. Duffio, Leptin shows that nonamenable, connected Lie groups are never Wiener. However, not every amenable group is Wiener; Leptin and Poguntke (1) show that a certain 4-dimensional, solvable Lie group is not Wiener. Other relevant papers are Hauenschild and Kaniuth [1], Ludwig [2], and Weit [1].

(6.49) A symmetric algebra associated with an exponentially bounded group (Hulanicki [7), (8); Jenkins [11]). Let G be a locally compact group. A *-subalgebra A of LI (G) is said to be symmetric if SpL, (G) (f * f-) C [0, oo) for all f E A. A function f E LI (G) is said to be rapidly decreasing if there exists C E 9, (G) such that for all r > 0,

of

[ f (x) [ dA(x) - 0 C _C^

an-,,.

If G is exponentially bounded, it turns out that the set B of rapidly decreasing functions on G is a symmetric *-subalgebra of L, (G). We shall be content to


254

prove that if G is exponentially bounded then {0} U Spy, (G) (f) = {0} U Sp(7r2 (f))

for all hermitian f e C,(G) (so that the subalgebra CC(G) of B is symmetric). [Suppose that G is exponentially bounded. For a E LI(G) [B(L2(G))], let v(a) and Sp(a) be the spectral radius and spectrum respectively of a in LI(G) [B(L2(G))). Let f E CM(G) be such that f = f^'. Let C E %(G) contain the support of f , and let f (n) be the nth convolution power of f in CC (G). Then Cn contains the support of f lnl, and using the Cauchy-Schwarz inequality; for

n>1, 11f(n) 111= fa lfinl(x)Ixc^(x)dx(x) 0 for all t E H and

[p(ct)-p(t)[ < ep(t) fort r= H, c E C. [Find U E F,(G) such that C c U = U-1.

>0. With U°=0,set

Then HCU1 Uk. Let

00

P= k=°

7(1 + 17)-kXUk+,-Uk.

Then p(t) > 0 for all t E H, and for c E C U C-1, Uk D c-1Uk-1. Hence if t E Uk+1 _ Uk, then ct Uk-1, so that ct E Ur+1 - Ur for some r > (k - 1). Hence, for this r, (1)

p(ct) _ (1 +rt)-r 5 (1 +,7)-(k-1) = (1 + T7)p(t)

Replacing t, c by ct, c_1 in (1), we have p(t) < (1+77)p(ct), so that [p(t)-p(ct)[ < t7(1 + r7)p(t). Now choose 97 so that r7(1 + 77) < e.

It remains to show that p E LI (G). To this end, 00

fG [p(t)[ da(t) _

00

k=o

fu

k

p(t) dA(t) = E(i

+rl)-kA(Uk+l

Uk)

k=0

00

< E(1 +77)-kA(Uk+1) < 00 kk-0

since ,\(Uk+1)1/k -. 1 as k -+ oo.]

Jenkins obtains a converse to the above result. We now introduce some notation. Recall [R!, p. 42] that a left ideal I in an algebra A is called modular if for some u E A, A(1 - u) C I. Such an element u is said to be a right modular unit for I. Let G be a locally compact group and 2' the set of proper, modular left ideals of L1(G). A bounded, sesquilinear, positive, hermitian functional of L1 (G) is a mapping

s: L1(G) x L1(G)

C for which there exists M > 0 such that for all g, g`,


257

h c- LI (G) and all a E C, we have s(g, h) = s(h, g), s(g, g) > 0, slag + g`, h) = as(g, h) + s(g`, h), and Es(g, h)(< M[[g[(I [[hl[I .

The set of such functionals is denoted by R. Of course, every s E .°s satisfies the Cauchy-Schwarz inequality: ts(f, g)1 5 s(f, f)I/2s(g, g)I/2.

(2)

Each positive, linear functional 0 on LI (G) yields an element s, E F, where so (g, h) = ¢(h^' * g) and h ---, h- is the involution on LI (G). Let P be the set of positive, linear functionals on LI (G). For s E 9, define [[all by 1[sll = sup{ls(g, h)l: g, h E LI (G), llg[li = 11hlll = 1}.

Clearly .9 is a G-set with action given by (sx)(g, h) = s(x * g,x * h),

(xs) (g, h) = s(g * x, h * x).

Clearly, llsxll = Ilsll = llxs[I. Now let s E .9 and f , g, h E LI (pG) . We claim that

s(f * g, h) = (f (x) s(x * g, h) dA(x), (3)

s(f * g, h) =

f J

g(x)s(f * x, h) dA(x).

Indeed, if we define z4i E LI (G)` by iP(k) = s(k, h) (k E LI (G)), then

Vf(f * g) = (gb)(f) = f 1(x)(0)(x)d.1(x)

s(f * g, h)

_

f

f (x)V,(x * g) da(x) = / f (x)s(x * g, h) dA(x),

giving the first equality of (3). The second equality follows similarly. and a subgroup H of G, define subspaces of 5a: For I E

.F1 ={sE.9 : s(f, g) = 0 for all f EI,gELI(G)},

RH={sE8j: sx=sforaIlxEH}. (iii) (Leptin [7(, Ludwig (1(). (a) Let I E _°. If so E .G, then so(fi * f, g) so(f, fi * g) (fl, f, g E LI (G)). (b) .°1'j 4 {0} for all I E Y. (c) The following three statements are equivalent:

(a) LI (C) is symmetric; (ff) if I E _, then there exists nonzero m E P such that ¢(I) = {0};

(-t) if I E 2', then 91G # {0}. [(a) Let So E.9 . If x E G, then so(x* f, g) = sox-'(x* f, g) = so(f,x'I *g).

258


Define 09, i f E LI(G)' by 08(fi) = so(fi,g), 'f(gi) = so(f,gi). Then O9(x * f) = so(x * f, g) = so(f, x i * g) = V)f(x-I *g), so that so(fi * f, g) _ O9(fi * f) = (f09)(fi), and using (1.1(3)),

so(fi*f,g)= f fi(x)(fO9)(x)da(x)ffi(z)g(z*f)d(x) =

fi(x)Of(x_1

J

* g) dA(x) = (gOf)*(ff)

Of(fi- *g)=so(f,fi *g) as required.

(b) If I E 2° and 45 E LI (G)' is nonzero and such that 0(I) = {0}, then p o E.IR1 --- {0}, where p ,(f,9) =

(c) Suppose that (a) holds, and let I E -F. We prove that (p) holds. We will suppose that LI (G) does not have an identity, the argument when LI (G) does have an identity being similar. Let A be the Banacb algebra obtained by adjoining an identity 1 to LI (G). Let J = I+C(1-u), where u is a right, modular

unit in LI(G) for I. Then I V J, since otherwise u E I. So J is a proper, left ideal of A. Using {R!, (4.7.9), (4.7.11)], A is also symmetric, and there exists a

nonzero positive functional 0 on A with 0(J) = {0}. Since A = LI(G) + J, it follows that 0 _ OIL, (G) # 0. So (a) implies (J3). If I E -F and 0 is as in (J3) then s,, E "'01G since

vanishes on the left ideal

I, and ¢((x * h)- (x * g)) = 4(h-g). Also, so

0 since LI(G)2 = LI(G) and 0 0 0. So (f3) implies (-y). Conversely, suppose that (-y) holds. Let I E 2° and a E -91G ^- {0}. Let u be a right modular unit for I. Then if f, g E LI (G), we have, using (a), (2), and the fact that LI (G)(1 - u) C I, (4)

s(u,u)i12s(f- * g, f- * g)if2. Since s # 0, it follows that s(u, u) > 0.

(5)

(Note that this inequality is true with s replaced by any s' E.PI

{0}.) Define

0 E LI (G)' by 0(f) = s(f *u,u). Then by (a), 0(f- * f) = s(f * u, f * u) >0. Thus E P. Now let f E 1. Then ,(f) = -s(f - f * u, u) + s(f, u) = 0 so that 0(I) _ {0}. Finally, if {es} is a bounded approximate identity for LI(G), then using (5), O(C5) = s(eb * u, u)

s(u, u) > 0

so that ¢ j4 0. So (^y) implies (p).

It remains to show that (0) implies (a). Let z E LI(G) and h = z- * z. Suppose that -h is quasisingular, and let I be the proper, modular, left ideal LI(G)(1 + h). Let 0 be as in (j3). For all f E L1(G), f + f * h E I, so

that ¢(f) _ -O(f * h). With f = h, we obtain ¢(h) = ->p(h2), and since ¢(h),&2) > 0, we have 0(h) = 0 = ,(h2). The Cauchy-Schwarz inequality yields that 0(f * h) = 0 for all f ELI(G)sothat 0(f)=rp(f+f * h - f *h)=0.


259

Since 0 ¢ 0, a contradiction results. Hence -h is quasiregular and [Ri, (4.7.5)] LI (G) is symmetric.] (iv) (Ludwig [1]). Let H and N be closed, normal subgroups of C and be such

that N C H and for each z E H/N, there exists C E 9,(H/N) with z E C

and yCy-I C C for ally E GIN. Let I E Y. Then ,9f # 0 if 91N # 0. [We first establish the three inequalities (6), (7), (8) below for s E .9. For f, g E LI(G), we have, using (3) and (2), writing If (x)[ = ([f(x)I1/2)2 and using the Cauchy-Schwarz inequality twice,

s(f * 9, f * 9) = f f (x)s(x * 9, f * 9) di(x) 0 for all v E V and (9)

Ip(u'v) - p(v) I < 6p(V)

(u' E U, V E V).

260


Now let s e .9N - {0} and, for f , g E LI (G), define

s'(f,g) =

(10)

P(v)(sv I)(f,g)dAv(v)

Jv It is easily checked that s' E .91N. Since the map v - sv-I (u, u) is continuous, p(v) > 0 for all v, and s(u,u) > 0 (cf. (5)), we have s'(u,u) > 0, so that S' ¢ 0. Let b = SUPXEG s'x(u, u), and find y E G such that s'y(u, u) ? b. Finally, put s` = b-I (s'y). We check that (a), (b) and (c) are satisfied. The 2inequality (a) is immediate. Further, using (8) with s replaced by s-,

li(f,g)l 5 (S(f,f))I/2(S(g,g))I/2 5 llfIIIIIgIII sup(sx)(u,u) aEG

= llfIII llglll sup(b-I(s'yx)(u,u)) = Ilflllilghhi. zEG

So (b) is true. It remains to prove (c). Let k E K, and let 'y = yN. Using the substitution v = yky` 1 w, the fact that C U, and (9), yU-y-I

I(sk - i)(f,f)I = b-I L[P b

sv"yk - sv-'V) (f,f)dav(v)

-I fV p(yky Iw)sw'Iv(f, f) dAv(w) f p(v)sv-Iy(f,f)dav(v)j

v < b-I !v p(yky-Iw) - p(w) Isw-'V(f, f) day (w) ('

5 eb-' f p(w)sw-Iy(f, f) day (w) = Es(f, f) 5 EBfi 112. v Thus (c) is satisfied. So s belongs to the set AK,E, where AK,E = {r E

': Ifrfl < 1, r(u, u) >_ Z, I(rk - r)(f, f)l 0}. Clearly, r'x = r' for all x E H and r' j6 0. So .11'

10}.)

(v) (Ludwig [1)) The algebra LI (G) is symmetric if either (a), (b), or (c) holds:

(a) G is a connected, Lie group of polynomial growth; (b) G contains a closed, nilpotent, normal subgroup N with GIN compact; (c) G is discrete, finitely-generated and has polynomial growth. [Suppose that

(a) holds. Suppose first that G is simply connected. Let {G,} be the sequence

of (i). From (iv), we see that for I E Y, gG;+ # 0 if 9G` 96 0. By (iii)(b), 91G' = .9 j 0 0. So .91' -A 0 and LI(G) is symmetric by (iii)(c). Now drop


261

the requirement that G be simply connected and let G be the simply connected covering group of G. Then G is of polynomial growth (since it is of Type R), and by the above, LI (G) is symmetric. Then for some closed subgroup H of G, G = G/H. Now ((1.11)) the map QH is a continuous, '-homomorphism from L1(G) onto L1(G). Since Sp(QH(f)) C Sp(f) U {0} for all f E L1(G), it follows that L1(G) is also symmetric. Now suppose that (b) holds, and let Z be the centre of N. Then Z is a normal subgroup of G. For T E G, let as E Aut Z be given by ax (z) = xzx-1. Since ax is the identity if x E N, and as GIN is compact, it readily follows that if z E Z,

there exists C E %(Z) with z E C and xCx-1 C C for all x E G. Using the subgroups {e} and Z, we apply (iv) to deduce that for I E.°, -,Pr-' 0 0. An easy induction argument involving the terms of the upper central series (Appendix A) for N and, finally, G, gives the symmetry of L1(G). From Gromov's theorem ((6.40)) and [M. Theorem 4.16], we see that if (c) holds, then G has a normal, nilpotent subgroup of finite index. Now apply (b)].

We shall see below that the requirement in (c) that G be finitely-generated cannot be dropped.

(6.51) A discrete, nonsymmetric group of polynomial growth (Fountain et al. 11)). We shall produce a nonsymmetric, discrete group G that is locally finite (and so, a fortiori, is of polynomial growth). (Another example of a nonsymmetric group of polynomial growth is given by Hulanicki [12J.) The group G is also an example of an amenable, nonsymmetric group (cf. Jenkins [1J). The author is grateful to John Williamson for a very helpful communication. Let P be the group of permutations of P that leave all but a finite number of integers fixed. Let x11 E P be the finite product of transpositions: x11 = (1,2" + 1)(2,2" +

Let C be the subgroup of P generated by {x11: n > 1}. (The group G will turn out to be locally finite and nonsymmetric.) For each n, let Gn be the subgroup of G generated by A finite sequence w = (z1,,. .. , z,,,) is said to be a word in G. If ij < n for all j, then w is said to be a word in Gn. We also allow w = 0 to be a word in G (and in Gn). The set of words in G [Gn] is denoted by W [W"J. If w = (z , ... , x!k ), then the length l(w) of w is defined to be k. (Of course, 1(0) = 0.) If w' = (z, ..... x,,) E W, then ww' E W is defined by ww' = (xq, , ... , x;k , xj, , ... , xj, ). Associated with w is an element a,, of G, where aw = x xik. (We take A nonempty word w is said to be irreducible in W ao = e.) So aww' = [WnJ if whenever w' E W [Wn] is such that aw = aw', then 1(w) < 1(w'). We now establish a number of useful properties of G. (i)(a) For all n, x11 = e and x,1 1 0 Gn; (b) xjxkx,xk = xkx=xkxj whenever i, j < k; (c) each G,, is finite; (d) every element of Gn is of one of the forms:


262

(a) a,,,, for some w1 E Wn_1i (0) aw, xn awe for some wi, w2 E W,,- I;

0) awlxnawxn for some wl, w2 E Wn-1; (e) if w1i w2 E Wn_I are irreducible in W,,- 1, then WIxnw2 is irreducible in Wn;

(f) if w E Wn is irreducible in W,,, then w is irreducible in W; (g) i f i i , . . . , ik are distinct, positive integers, then w , xik) is irreducible in W; (h) if w1 = (xi, , ... , and w2 = (z,, , ... , X j_) are such that i1 i ... , in are all distinct, j1, ... , jn are all distinct and w1 # w2, then a,,,, # awe . [(a) Trivially, x,? = e. The element xn+i f Gn since every element of Gn fixes 2n+I + 1. (b) Let o', r E P with r2 = e. Then for each n,

crrr 1(c(n)) = o(r(n)),

arc-1(c(r(n))) = c(n)

so that crc-1 is the product of transpositions (c(n), c(r(n))). Applying this result with c = xjxk, r = x,, we have, using (a) and the fact that i, j < k, xJxkxaxkxj = (c(1),c(2i + 1)) ... (0(21), (xk(1), xk(2t + 1)) ... (xk(21), xk(2i+i )) = xkxixk 1.

(c) The group G. is finite since it can be regarded as a group of permutations on the finite set (1; 272+I[

(d) Since xJ = xi 1 for all j ((a)), every element of Gn_1 is of the form a,,,,

for some wI E W, 1. Let z E Gn - G, 1. It remains to show that x is of the form (f3) or (y). Let wj = (x....... x,,,,), w2 = (x,...... xjk) belong to W,,_1. Using (b), aw;xnawe=xil...xs,

,(2i,,,xnxj,)xJ2...xjk

= xi 3

)xJ2 ... xjk

= x ...

...xjk

= xi, ... (xtm_22nx7)

xn)xt,,,_,2i,,,xnx32...xjk

xnz,, xn (aw, xnz,2 ... xjk )

Repeating the above process,

aw,xnaw2=

(xny,ixn)(2nx72xn)...(xnXjkzn)aw;xn

= xn aw2 xn awl xn

so that (1)

awl xnaw2xn = xnbrW2xnaw,.

Since every element of Gn is of the form a,,,

use of (1) shows that x is of the form (8) or (y).

a,,,xn, where vi E Wn_1


263

and let w be irreducible in W such that aw = u. (e) Let u = Let 1 = 1(w). Noting that the reduction to forms (J3) and (-y) in (d) does not increase the number of elements involved, we can suppose that w is of the or w is the latter, then maws = form (aw,)_Iawixnawzxn = xnaw2xn(awi)-Iawi (by (1)) and we obtain z,, E Gn_1. Sow is of the form w'xnw'2. Suppose that w,xnw2 is not irreducible in W,,. Then

l(wlznw2) > l(wiznw2) so that either 1(wi) > l(wl) or l(w2) > l(w2). Without loss of generality, we can suppose 1(w2) > l(w2). Since w2 is irreducible in Wn_I,

awe 0 awl. From the equality aw,znaw2 = aw,,xnawz, we deduce that there exist u3, u4 E Gn_I with U3xnu4 = xn and u4 # e. Since u4 # e and u4(r) = r if r > 21, there exists N E 11; 2n1 with u4(N) # N. So u3xnu4(N) = u4(N) + 2n, while xn (N) = N + 2n. This is a contradiction. (f) An argument along the same lines as the proof of (e) shows that if w is irreducible in W,,, then w is irreducible in W,,+I. Now use induction.

(g) Use (e) and (f) and induction on max{il,...,ik}. (b) The proof proceeds by induction on n. If n = 1 and aw, = awz, then, since both w, and w2 are irreducible ((g)), we have m = 1, so that w, = w2. Hence the desired result is true when n = 1. Now let r > 1 and suppose that the result is true for n < (r - 1). Suppose

that aw, = awz. Then n = m. Write w1 = zlx=kz2, w2 = z3x3,z4, where ik = max{ip: 1 < p < n}, j1 = max{jp: i < p < m}. Let u2 = a.;. If ik j1, then a contradiction results from (a). So let N = ik = j,. Note that ui E GN_1. Then (u31uI)xN(u2u4I) = ZN, where u31U1iu2u41 E GN_I. Using (e), it follows that u31u1 = e = u2U41, that is, u3 = u1, u4 = u2. By the induction hypothesis z1 = Z3 and z2 = z4, so that WI = w2. This is a contradiction.] The nonsymmetry of 11 (G) is proved using the notion of capacity in Banach algebras [BD, §45]. The spectral radius [spectrum] of an element a of a Banach algebra A is denoted by v(a) [Sp(a)). The capacity cap(a) of a is defined to be cap(a) = limn_,,,,,{inf{jIp(a)IIII': p is a monic polynomial of degree n}}. (A manic polynomial of degree n is a polynomial Fn=, A,z' with A, E C, an = 1.) The existence of the above limit follows from [BD, p. 251]. (ii) Let A be a Banach *-algebra and a be a selfadjoint element of A such that

v(a) < 1 and cap(a) > 1/2. Then A is not symmetric. [Suppose that Sp(a) is real. Since v(a) 1 and the result is true for n = m. Suppose that u, v, u', v' E Zm are such that UX,n+NV = u'xm+NV' Then (u-IU')xm-+ 4(V'V-1) = xm+N, and it follows from (i)(e) that u-1u' = e = v'v-1. Thus u' = u, v' = v, and by hypothesis the map to --y aw is bijective from Ym+1 onto Zm+i So IYnI = 1Z-n1 for all n. It also follows from (e) and (f) of (i) that the elements of every Yn are irreducible in W.

Let In be the (common) length of the elements of Yn. Then 11 = r and In+1 = 21n + 1, and it follows that

In=2n-1(r+1)-1.

(3)

Let f ("_) = f *

* f be the mth convolution power of f . Suppose that x E Zn+1

and that x = z1zz2 with f (1) (zi) # 0, f (z) # 0. Now x = aw, where to = wixn+Nw2 is irreducible and w1, W2 E Yn. From (2), each zi E {x1,x2,...Y", and z = xp for some p. As wixn+Nw2 is irreducible, we can find irreducible v, E W with a,,, = zi and I(v,) = In. Let k be the largest integer such that xk occurs in the word vlxpv2. Now vlxpv2 is irreducible in W. So ((i)(d)) xk occurs either aw, xn+Naw2 once or twice in v1 xpv2. Considering the equation a,,, and allowing both sides to act on a suitable integer (cf. the proof of (i)(e)), we see that k = n + N and that Xk occurs only once. We see further that z = xn+N EzEz, P-) (z)), and z1 = CYw, : z2 = awe. It follows that if mn = f{!")(Zn) then, using (2)

mn+1 = f") * f * f")(Zn+1)

= E{

(4)

f(t")(Zi)f(z)f('")(z2): z1,z,z2 E G,zizz2 E Z.+1}

= E { f(I") (zl) f (xn+N) f(!") (Z2):

zi, z2 E Zn }

= f(t")(Z.)f(xn+N)f('")(Zn) = b2-n(mn)2. Now if (x,, , ... , xi,) E Y1 and x = zi, xj,, then, using (2) and arguing as above, f (') (x) = a'. So (5)

m1=

f(')(Zi)=a'IZ1I=a'IYII=a'N(N-1)...(N-r+1).


265

From (4) and (5), mn = b2-(n-1) (mn-1)2 = (6)

b2-(n-1)b22-2'(n-2)

(mn-2)4

= b(2"-'-1)2-12"-n-11 [arN(N - 1) ... (N - r + 1)]2"-').

Since f(m) is supported by the set {xn: n > 1}-, we have f(T)(Yn) = 0 if m < ln. Hence, if p is a manic polynomial of degree In, we have [[P(f)1[ > f(1")(Zn) = mn.

(7)

Using (3), (7), and (6), (8)

00

We now have to choose a, b, r, N so that the right-hand side of (8) is > 1/2. Recall that the only constraints on a, b, r, N are

r < N, and Na + b = 1. For each r E P, r > 1 choose N = r2, a = 2/(3r2), and b = 1/3. Then (9) is (9)

a, b > 0,

r, N E P,

satisfied, and substituting these values in (8), we have

cap(f) > Tli mo{(1/12) r2a (r2 - 1)a . . . (r2 - r + 1)a} 1/(r+1) > rlim {(1/12)[(r2 - r + 1)a]r}1f('+1)

= rlim {(1/12)(2/3)'[1 - ((r - 1)/r2)]'}1/(r+1) = 2/3 > 1/2.] 00

Other References Barnes [1), Brooks [1), Hulanicki [13), Losert [2], Margulis [1), Tits [3].

Problems 6 Throughout, C is a locally compact group.

1. Determine limn. 2.

A(C")1/n when G = F2 and C = {e, x, y, x-1, y-1 }.

Let A = {limn_wa(C")1/n: C E °(G),Un°_1Cn = G}. Show that

k'EAifkEA,rEP.

3. Let HZ be the discrete group consisting of all matrices in the Heisenberg group with integer entries. (See Problem 4-29.) Show that Ha is a finitelygenerated, discrete nilpotent group. What is the degree of Hz? 4. Prove Lemma (6.16).

5. Give an example to show that a locally compact group G need not have polynomial growth if it contains a closed normal subgroup H with both H, G/H of polynomial growth.

266

6.


Let 81i 82 E R, 81 # 0 and 8 = 81 + i82. If 82 = 0, let g8 be the 2-

dimensional, real Lie algebra with basis {X1, X2} and multiplication determined by [XI,X2J = 8X2. If 82 # 0, let go = Span{X1,X2, X3} be the 3-dimensional, real Lie algebra with multiplication determined by [X1.,X21 = 81X2 + 02X3,

[X1,XaJ = 81X3 - 92X2i [X2,X3] = 0. Let GB be the simply connected Lie group with ge as Lie algebra.

(i) Show that if 82 = 0, then Go is the subgroup of GL(2,R) consisting of matrices of the form

s,1

feet I

0

1

(s , t E R) ,

while if 62 # 0, G8 is the subgroup of GL(2, C) consisting of matrices of the form eet

z

(tER, zEC)

1j (ii) Let to E R be such that [e$t0[ < 1/3 and 0

eeto

a= O

1

11

b_

.

{eeto 0

1

Show that a, b generates a free, uniformly discrete semigroup S in two generators. (iii) Let g be a real Lie algebra, and suppose that for some X E 9, Sp(adX) ¢

M. Show that g contains o as a subalgebra for some 8. (iv) Hence give another proof (not using (6.36)) to show that a connected Lie group G has polynomial growth if and only if it does not contain a free, uniformly discrete semigroup in two generators (cf. (6.39)).

7. Let G E [FCJ- ((4.23)). Prove that G has polynomial growth. (Hint: use the following result of Grosser and Moskowitz [2): if G is compactly generated, then the commutator subgroup of G has compact closure.)

8. For a E R, let M(o:) be the semidirect product C2 xp R, where P(t)(zl,z2) = (e21ritz7

e2wxat22).

Show that M(a) has polynomial growth. Deduce that a simply connected, solvable Lie group of polynomial growth need not be of Type 1.

9. Let H be the locally compact group that, as a set, is C x R, and that has multiplication (z, r)(w, s) = (z + w, Im(zw) + r + s). The "diamond group" D is defined to be the semidirect product H xp R, where p(r)(w, s) = (e2-r w, s). Show that D is a solvable Lie group with polynomial growth. 10. Show that S2. S3, and S4 are not liminal (cf. (6.45)).

11. Show that S2, S. and S4 are not unimodular, and deduce that a connected, solvable Lie group G has polynomial growth if and only if G/H is unimodular for every closed, normal subgroup H of G. 12. It is known that the sphere S2, the projective plane P2i the torus Si x 5i, and the Klein bottle K each admits a complete 2-dimensional Riemannian structure with nonnegative mean curvature. Check Milnor's theorem

((6.41)) directly for each of these four surfaces.


267

13. Prove (6.42(i)).

14. Let G be exponentially bounded and 0 > 0 be nonzero in LA(G). Let X4, be the linear subspace of LA(G) spanned by {Ox: x e G}. Show that there exists a linear map P: X0 -+ C such that P(O) = 1, and for 1P E X4,, x E G, P(hi) >_ 0 if vl, > 0 and P(Ox) = P(hi). [Hint: use Problem 13 above.] 15. Let G have polynomial growth and r be the degree of G. (i) Show that if G is abelian or discrete nilpotent, then for each C E %(G), there exists me > 0 such that

mcnr < A(C")

(1)

(n > 1).

(ii) Suppose that (1) holds for all C E 9,(G) and that G is compactly generated. Show that there exists a summing sequence {.1(Kn 1Kn)/)(Kn)} is bounded (cf. (5.21)).

for G such that

16. ((6.46.(i)) Let X be a compact Hausdorff space and P: C(X) -> C(X) be a unit preserving, positive, linear projection from C(X) onto a (closed) subspace B of C(X). Let - be the equivalence relation on X given by: x - y a f (x) = f (y)

for all f e B. Let H = {0 E C(X): O(x) = 0(y) whenever x - y in X}. For X E X, let C. = {p E PM(X): µ(f) = f (x) for all f E B}. (i) Show that if .i E Cx, then }t(O) = ¢(x) for all 0 E H, and the support S(µ) of y is contained in l y E X: y x}. (ii) Show that for x e X, we have i o P E Cx, and deduce that

P(4Plp) = P((Pb)(Piy))

*,i E C(X)).

(iii) Prove that B is a commutative, unital C'-algebra with multiplication

"x", where f x g = P(fg) and f' = f (f, g E B). 17. The "localisation conjecture" (Greenleaf [2, p. 69]) runs as follows: let G be a connected, separable, amenable locally compact group and K = K-1 E

%,(G). Is it always true that givens > 0 and C E 9,(G), there exists mo E P such that (m >_ mo, x E C)? A(xK"' A K"`)/)(K"`) < e (A weaker version of this is true if G is exponentially bounded ((6.8)).) By

considering the "ax + b" group, show that the conjecture is false (c.f. (0.5)). 18. The locally compact group G is said to be distal if, whenever x E G - {e}, then e (C1(x))- (Cl(x) = conjugate class of x). Suppose that G is a connected Lie group. Show that G is distal if and only if G has polynomial growth.

19. Let G be a connected Lie group. Show that Sp(Ad x) C T if and only if G is of Type R. 20. Let G be a connected Lie group. Show that G is of Type R if and only if {[Tr(Adx)[: x E G} is bounded. 21. Show that if G is abelian or compact, then G E [S) ((6.48)).

268


22. (i) Let A be a commutative Banach algebra with maximal ideal space X. Suppose that f E A is such that f is real-valued on X and that for some k E P, lle:"vt

- ill = O(nk) as n - oo for every 6 e R. Let CS(R) be the space of

functions 0 E Cc(R) such that, for all n < r, D"¢ exists and is continuous. Show that Ck+2(R) "acts" on f in the sense that if l E Ck+2(R) and ¢(0) = 0, then

there exists g E A such that g=¢ o f. (ii) Let G be a locally compact group, and let w: G - [1, oo) be measurable and such that w(xy) < w(x)w(y) for all x, y E LI (G, w) = I f E LI (G) :

Il f llw

0. Let

if (x) [w (x) dx < oo }

.

Show that with norm Ilw, involution f --t fff and convolution multiplication, LI(G,w) is a Banach *-algebra. (iii) Let G be compactly generated and of polynomial growth. Let k be the degree of G and p E P be such that f'k+1 < p < i k+2. Let f = f- E CC (G), and let Af be the closed subalgebra of LI (G) generated by f. Show that CCP+2) (R) "acts" on fin the sense of (i) (with A = A1).

23. Let G E IS], and let H be a closed normal subgroup of G. Show thatG/H E [S), and that H E [S] is H is open in G. a and identity 24. (i) Let A be a Banach *-algebra with involution a element 1. Let P(A) be the set of positive functionals F on A with F(1) = 1. Show that A is symmetric if and only if, for all a E A, the spectrum Sp(a) of a is contained in {F(a): F E P(A)}. [Hint: use the result ([Ri], (4.7.11)): if A is symmetric and B is a closed *-subalgebra of A containing 1, then the map F FIB maps P(A) onto P(B).]

(ii) Let G be a discrete group containing FS2 as a subsemigroup with generab. For f E tI (G) let v(f) be the spectral radius of f. Let fo = ba3 + ba + i(ba4 + b + 2ba2).

tors a,

(a) Show that v(fo) = Ilfolll = 6. (b) Show that if G is symmetric and if f E lI (G) is such that f- * f = f * f-, then v(t * f) < v(f) (t E G). (c) Let go = a + a-I + i(a + a_I)2 (= (ba2)-I * fo). Show that if G is symmetric, then v(go) < 2f. Deduce that G is not symmetric. 25. Let [T] be the class of locally compact groups G of Problem 4-40.

(i) Show that G E [SP] if and only if, whenever ar, p are nondegenerate *-representations of LI (G) such that ker7r c kerp, then lIp(f) II S IIT(f) lI for all f of the form h * h^' with h E C,(G). (ii) Show that C E [91) if G has polynomial growth. [Hint: use Problem 22 above.] 26.

Let C be a locally compact group and p E P(G). Prove that FI sat-

isfies (B) of (6.46(iii)) if and only if the following Reiter-type condition holds:

llx*u"-p"III -,0 for allxEG.

CHAPTER 7

Sizes of Sets of Invariant Means (7.0) Introduction. In this chapter, we are concerned with determining the sizes (cardinalities) of sets of invariant means on semigroups and locally compact groups. The general conclusion is remarkably simple: the size of such a set is "biggest possible" unless the semigroup or group possesses some strong property which clearly limits it. For example, if a discrete group G is infinite and amenable, then ((7.8)) IC(G)I = 221" (= if G is finite, then 12(G) I =1.

However, a word of caution is in order. Recall that, as a consequence of the results involved in the solution of the Banach-Ruziewicz Problem ((4.27)-

(4.29).), there is an infinite, compact group G (e.g. G = 0(5)) with I,C(G)I smallest possible! However if we restrict attention to compact groups which are amenable as discrete, then the above general conclusion still applies. There are four natural cardinality questions that arise in connection with left invariant means. Let G be a locally compact amenable group. (1) What is the cardinality I.C(G)I of the set of left invariant means on G? (2) What is the cardinality I.Ct(G)I of topologically left invariant means on G? (3) What is f,C(G) - .Ct(G)I, that is, how many (if any) left invariant means on G are not topologically left invariant? (4) What is the cardinality Ia(S)I of the set of left invariant means on a left amenable semigroup S? A related question is: what is dim3t(S), the dimension of the space of left invariant continuous linear junctionals on These are the four questions which we will study in this chapter. We now briefly discuss these questions in turn. The first question is unsolved in general and seems to be very difficult. For example, some (nonfinite) compact groups admit exactly one left invariant mean, while others admit many. The more tractable second problem, however, does provide a lower bound for I.C(G)I since ,C(G) D 4(G). The second question has been completely settled recently. It has been shown that if G is noncompact and m is the smallest possible cardinality for a covering of G by compact subsets, then 12,(G)I = 22". 269

SIZES OF SETS OF INVARIANT MEANS

270

(Of course, when C is compact, then .. (G) = {A}.) The main idea of the proof

is to construct "many" left invariant, compact subsets of the maximal ideal space of U, (G) and then to use a fixed-point theorem to construct left invariant means supported on these subsets. The result is proved in (7.6). A remarkable consequence of the result is the following ((7.8)): if G is an infinite, amenable discrete group, then IC(G)I = 22101 .

The third question, like the first, is unsolved in general. Progress has been made in the case where G is amenable as a discrete group. The reason for this is that we want to realise invariant means as probability measures on the maximal

ideal space Z(G) of Lo,(G), each supported on a compact invariant subset of fi(G), and in order to ensure that such an invariant subset actually supports some m E e(G), we require the use of Day's Fixed-Point Theorem (cf. (2.25)). The latter result, of course, only works if G is amenable as discrete. We will show ((7.21)) that if G is amenable as discrete, then Z(G) = £. (G) if and only if G is discrete and that if, in addition, G is o-compact and nondiscrete, then ((7.20)) )Z(G) - .(G) I >_ 2', where c is the cardinality of the continuum. The fourth problem is an example of a semigroup problem which is substantially harder than the group problem. Indeed, the exact cardinality of c(S) was determined only recently. The determination uses the following cardinal m(S), where n

m(S) = min j UsiSi : n> 1, {S1,...,Sn} is a partition of S, i=i

$Ii...,3n E SI. If m(S) is infinite, then I.ff(S)I = 22'"(5) = dim3t(S). If m(S) is finite, then 31(S) is finite-dimensional, and its dimension is the number of finite left ideal groups in S ((7.26), (7.27)). In many cases, m(S) equals the more accessible cardinal min{IsSI: s e S}. The central, rather technical, construction involved is effected in (7.25). The construction produces a "large" disjoint family of left thick subsets of S, and these in turn can be used to produce an even larger disjoint family of compact invariant subsets of OS, each "supporting" a left invariant mean. A number of other cardinalities associated with invariant means can be determined by modifying the construction. (See Problems 7.) We start by proving a useful set-theoretic result. The result is closely related to [HR1, (16.8)1.

(7.1) PROPOSITION. Let A be an infinite set. For B C A, let BI = B and B` = A -r B. Then there exists a family IN,: 'y E r} of subsets of A such that (i) Irl = 21AI:


271

(ii) if y1, ... , ym are distinct elements of 1' and Ei E {1, c} for 1 < i < m, then

m

fl i 7.

0.

i=1

PROOF. Let A' be a set disjoint from and equipotent with A, and let r be a

bijection from A onto A'. Let r = .9(A) (so that in = 21AI), and for y E r, let B(y) = r(y)° U y E .9'(A' U A), where r(-y)' = A' - r(-y). If y1, y2 E r and y1 0 y2, then, since B(y1) ~ B(-y2) = (yi y2) U (r(-y1)` (y2)`) _ (11 ~ y2) U 'r(12 {1)

y1), it follows that

B(-1)~B(-2)#0

(y1 #y2).

Let SO, = -9-(A' U A), . _ .`9'(B(-1)), and M., = 9' (.j ~ F,). (Recall that .9(X) is the family of finite subsets of a set X.) Since .9() is equipotent with A, we can find a bijection a from l ( ) onto A. We set N.r = a(M7). To prove (ii) it is sufficient to show that if y1, ... , yn, b1, .... bn are distinct elements of r, then (2)

My, n...nM;nnMs,

o 0,

where M. = (91) M.r, . For 1 < i < n, 1 < j < m, choose (using (1)) xi, E B(%)

B(6,). Set

Ai = {x23: 1 < j < m} and consider 9 = {A,,. - - , An} E

For each i,

Ai it B(6) so that 0 E Men. Further, for each i, Ai E F, so that 8 f M,y,. Thus 8 E Mry, n ... n M7" n Mb, n ... n Mam as required.

We now turn to our second problem. Let G be a locally compact amenable group. We will determine the cardinal J2t(G)J. References for the results discussed here are Chou 141, [91, Granirer [131, Lau [S81, and Lau and Paterson [s11. By (1.9), every m E £e (G) can be regarded as a member of Ur (G)'. Now UT (G)

is a commutative unital C'-algebra and so can be identified with C(X), where X is the maximal ideal space of Ur(G). The idea of the proof is to construct a "large disjoint" family of subsets of G each supporting a suitable function of Ur(G). Now G is a dense subset of X in the natural way, and using the above family of sets, we can produce a large disjoint family of compact invariant subsets of X. Each such set supports an invariant probability measure and this "belongs"

to Zt(G). (Here the left introversion of Ur(G) is important in order to apply the fixed-point theorem of (2.24). This would not work if we had used L,,. (G) or C(G) in place of Ur(G).) This gives a lower bound for J.Ct(G)J, and we then show that it is also an upper bound for J.£,i(G)I. Suppose that G is noncompact and let m (or m(G)) be the smallest possible cardinality of a covering of G by compact subsets of G. Note that in is infinite since G is noncompact. Let a be the smallest ordinal of cardinality in. Let {Kp : 0 E a} be a family of compact subsets of G such that UpEa Kp = G. We can suppose that the family is closed under finite unions.


272

(7.2) PROPOSITION. Let U E Z(G). Then there exists a subset {xp.,: 0 < ,6:5,7 < a) of G such that the family {UK.,xp., : 0 :5,6:5 7 < a} is disjoint.

PROOF. Let 70 E a and suppose that elements xp.,(0 < /3 < 7 < 7o) have been constructed so that UK.,xp., n UK.,,xp,.p = 0 whenever (9,7) #

Let 9 = {UK.,xp.,: 0 < p < 7 < 7o} and W = U9. We now construct by transfinite recursion the elements xp.,o. Suppose that the xp.,o have

been constructed for J < 00 < 7o, and let Z = W U ({UK,,,x090: /3 < Po}).

Then K; U'1Z admits a covering 9' of compact sets, where I9'I < in. So K;1 U-IZ # G, and we can pick xp0.,,, in G -- K; U-1 Z. This completes the recursion. 0 Recall that a subset E of G is called left thick ((1.20)) if whenever F E F(G), there exists x E G such that Fx C E. (7.3) PROPOSITION. There exist a family {Zp: Q E a} of left thick subsets of

G and a set {0p: $ E a} of functions in U,(G) such that for each 0, 00 (4) = {1} and 00 (U,0,6 Z.,) = {0}. Further, if K C a is nonempty, then 1'K = Ep5K 0,6 E Ur(G).

PROOF. Let U and xp., be as in (7.2). For each 8, let Zp = U{K.,xp.,: 6:5 -y < a). Let F E 9"(G). Since the family _V = {K,6,:,6' E a) covers G and is closed under finite unions, there exists -y such that F C K.,. The family {K1 U Kg,: Q' E a} c .say also covers G, and from the definition of in, has cardinality = in. So we can take -y > Q. Then Fxp., C Zp, that is, Zp is left thick. It remains to define the functions dip. To this end, let V E °(G) be such that V3 C U and V-1 = V. By Urysohn's Lemma, we can find f E CM(G) with 0 < f < 1, f (e) = 1, f (G - V) = {0}. Let d be the pseudo-metric on G given by d(x,y) = I If x - f y1I , and, for each p, 7 with fi < 7, let gp.y(x) = 1 - d(x, K.,xp1) (x E G). Clearly, gp, E C(G), 0 < gp1 < 1, and go-,(x) = 1 for all x E K.,xp1. Now if gp.,(x) > 0, then d(x, y) < 1 for some

y E K.,xpy. This implies that VxfVy # 0; for if VxfV y = 0, then yx-1 V, and f x(x-') = 1, f y(x-1) = 0 giving d(x, y) = 1 and a contradiction. So if gp.,(x) > 0, then x E V-1VK1xp., = V2K.,xp1 C UK.,xp.,. It follows, using (7.2), that gyp, where dip = Ep_ 22'".

It remains to establish the reverse

inequality.

To this end, let H be a noncompact, v-compact open subgroup of G. Then there exists a compact normal subgroup K of H such that H/K is separable (cf, [HRI, (8.7)) or Problem 7-7). Let p be the normalised Haar measure on K regarded as a probability measure on G. Let v E P(G) and 8 = v * p. Then

8EP(G),andif0EU7(G),xEG,kEK,then 0 * 8(kx) _ (kxO)^(v * p) = (kxq5 * v)^(p) _ (x4 * v)^ (pk) = (xo * v)^ (p) _ 0 * 8(x).

Thus * 8 can be regarded as a continuous function on the right coset space G\K. Clearly, each m E Zt(G) is determined by its values on Ur(G) * 8, and so £t(G) can be regarded as a set of continuous linear functionals on some subspace

A of C(G\K). As K is compact, the smallest possible covering of G\K by compact sets has

cardinality m. Further, since H\K is a separable, open subset of G\K, every compact subset of G\K is separable. Hence, there exists a dense subset T of G\K of cardinality Kom = m. Since every function in C(G\K) is determined by

its values on T, we have IC(G\K)I < C' = 2". So Iet(G)I < IA'I < IC(G\K)'I < C2" = 22". This establishes the theorem.

n

(7.7) COROLLARY. IC(G)l > 22m.

(7.8) COROLLARY. If G is an infinite, amenable, discrete group, then I2(G)I = 221".

The preceding corollary extends to the case where C has a left action on an infinite set X and £(G) is replaced by the set of G-invariant means on X. Indeed, Rosenblatt and Talagrand 111 show that if G is amenable and Cl I JXi? Recent progress on this issue has been made by Z. Yang [S3), who shows that under the assumption of the Continuum Hypothesis, there exists a locally finite (and so amenable) group G with IGI = c and a denumerable left G-set X such that l.£(X)i = I. We now turn to our third question: what is the cardinality of £(G) - Zt(G)? The next result provides a criterion for determining when a mean m E .£(G) is not in .£t(G). References for the result are Granirer [15), Rudin [2), and Rosenblatt [5). We use the notations of (2.24). In particular, 4'(G) is the carrier

space of L,(G). Let G be a locally compact group. If means m,n E 9X(G) are such that the probability measures rn, n on iP(G) are mutually singular, then we say that m and n are mutually singular or that m is singular to n (rim. 1 n). If A c M(G) and m is singular to every element of A, then we will say that m is singular to A.

In our discussion of the cardinal I.ft (G) 1, it was useful to use the closure t of a subset E of G in X, the maximal ideal space of U7 (G). When we consider this does not work so well since if G is not the corresponding issue for The considerations discrete, G ¢ the maximal ideal space of below show that this difficulty can be readily overcome. Let (G) be the c-algebra of A-measurable subsets of G. If E E%'(G), then XE is an idempotent in L,,,(G) and so XE is of the form XE, where t is an open and closed subset of (When G is discrete, then t is just the closure of E in ,OG (= 4'(G)).) Some simple facts about E - E are given in (7.11). ,f,

(7.9) PROPOSITION. Suppose that D is a measurable subset of G for which

A(G - D-1) < 1 and that m E £(G) is such that m(XD) = 0. Then m is singular to .£t (G).

PROOF. The compact and noncompact cases are treated separately. (i) Suppose that G is compact. Since C.t(G) = {A}, we must show that m 1 A.

Let e = S(m) n S(.1) and v = Ale. Now for E E .9('(G)), x E G, we have, using the left invariance of e (2.25(ii)),

A(En9) _ 55(x(Ene)) _ A(xEne) so that v(xE) = v(E). If 33 (e) = 0, then m 1 A. Suppose, then, that .1(e) 0 0. Then for some c > 0, v = cµ, where /t E P(4'(G)). By (2.25(h)), there exists n E .£(G) with n = µ. Clearly 0 < n < c-1 A. Now n can be regarded as a finitely additive measure on R(G) by setting n(A) = n(XA) (A ER(G)). It follows as in the proof of (a) of (4.27(i)) that n is countably additive. The uniqueness of Haar measure then yields that n = A so that v = ca. So S(a) = 9. Now since

fn(D) = m(D) = 0, the open set b cannot intersect S(rn). Thus D n e = 0, and since A vanishes off e, we have A(D) = 0. This is impossible since

A(D)=A(D-m)=1-A(G-D-1)>0. (ii) Now suppose that G is not compact. Let m1 E £. (G). Noting that m is D)", it suffices to prove that if B = G D, then ml (B) = 0.

supported on (G

276


To this end, let {C,,} be a sequence in `d'(G) such that 0 < A(C,,) -+ oo, and defuse ,u,, e P(G) by setting dµ.. =

dX.

Then µ;, E P(G), and for each x E G, we have, using (1.1(3)),

xaun (x) = f (xxB)(t) dun (t) = f (ZXB)(t-') dun (t)

= (A(Cn))-' fc XB(t-1x) da(t)

= \(xB-1 nCn)/ (CC) s x(B-1)/x(Cn)Since A (B-1) = A(G

D-1) < 1, we have jJXBu;, {loo < (.(CC))-1.

As m1(B) = m1(XB) = mI(XBun), Jim, 11 = 1, and (.1(Cn))-' - 0, it follows that ml(B) = 0 as required. The preceding result will be used in (7.17) with D = (G - V)-', where V is as in the next proposition. (7.10) PROPOSITION. Let G be a nondiscrete, cr-compact, locally compact group, and let e > 0. Then there exists a dense, open subset V of G such that A(V) <e.

PROOF. Let N, {Un} be as in Problem 7-7, where A(Un) < c2-". Let {xn} be a sequence in G such that {x,,N: n E P} is a dense subset of GIN. Then Un__1(xnN) is a dense subset of G and we can take V = UO°_1(xnLU,).

We now state some simple facts about the map E

13

E (introduced in (7.8)).

(7.11) PROPOSITION. (i) The map E -- E preserves finite intersections, finite unions, and complementation.

(ii) If E E.'(G) and x E G, then (x-1E)^ = x-'(E). (iii) If {E.,: ry E T} C .,C(G), then n7erE., is empty if and only if there exists a finite number of elements 'y1..... yn of r for which

A(E.,,n...nE.,,,)=0. PROOF. (i) Let E1, E2 E . (G). The equality (E1 n E2)A = E1 n E2 follows since XE, XEz = XE,nE, To prove that (E1 U E2)' = E1 U E2, use the equality XE,UE2 = XE, + XE2 - XE, XE2

From the equality 1 = XE, + XG-E we deduce that (G - E1)^ (ii) Use the equalities XEx = Xx-,E and (XEx)^(p) = XE(xp) (p E

(iii) Since O(G) is compact, n-rEr E7 = 0 if and only if there exist

7i, ... ,'7n E 1' such that t,, n ... n k,, = 0. Now k.,, n ... n t,. = 0 if and only if XE n...nE,,,, = 0 in L,,,(G), that is, A(E1i n ... n E.,,,) = 0.

D


277

We now show that when C is amenable as a discrete group, then, in general, ,C(G) 0 ,Ct(C) and .C(C(G)) ¢ .Ct(C(G)). We require two preliminary propositions. The results (7.12)-(7.14) are due to Liu and van Rooij [1], although the assertion "C(G) 56 .Ct (G)" in (7.14) was proved earlier in the work of Stafney [S), Granier [15) and Rudin [2).

(7.12) PROPOSITION. Let G be a locally compact group that is amenable as a discrete group. Suppose that there exists 0 E C(G) satisfying the following conditions:

(i) given x1, ... , x E G, there exists a nonvoid open subset V of G such that

¢(xiV) _ {1}

(1)

(1 < i < n);

(ii) there exists v E P(G) such that 110vlloo 0. So (cf. (7.4)) the set i; = f,EC(xE)^ is a closed, left invariant subset of 4(G), and there exists m E .C(G) such that S(th) C 9. The fact that OXE = XE implies that q5(a) = 1 for a E E, and so, since %I C E, QSEy = 1. Hence m(O) = m(¢) = 1. But from (2), we have [m(¢v)1 < 1/2, so that m E .C(G) - .Ct(G). Finally, since ¢ E C(G), it follows that mIc(c) E ,C(C(G)) .Ct(C(G)). O

We now construct a function 0 satisfying (i) and (ii) of (7.12) when G is nondiscrete and noncompact. The cardinal in of (7.1) will prove useful again. (7.13) PROPOSITION. Let G be a nondiscrete, noncompact, locally compact group. Then there exists an element 0 of C(G) satisfying conditions (i) and (ii) of (7.12).

PROOF. The idea of the proof is to construct (by transfinite recursion) a certain disjoint family of relatively compact, open subsets of G (viz. the sets U-IUSaac, below) and build up ¢ from simpler functions defined on these subsets.

Let ji E P(G) be such that dp/dA = g, where g is continuous with compact support. Define (1)

sad = {W: W is open in G, W- is compact, and A[(W-)-1)
0, t(rp(x) = 1 if x E (WR )-', (3)

f iGa(t) dA(t) < 2JJgJJ00

Define 0,6 (z) = V50 (x-') (x E G). Then

J(kau'"(x)I = If Os(t-x)g(t)dA(t)

= If V5a(x-t)g(t) dA(t)

I

=

if 00(t)g(xt) dA(t) I

5 JJ9JJoozJJ9Jtoo1 ` 2

So (4)

II

I3L

IJoo 1, SE the point about (2) is that we are picking out those q-tuples (E1, ... , Eq) that violate (1). Suppose that we can show that each U(r, S) is closed with empty interior in X. Then MT applies, and we have a subset A of X as in the statement of MT. Indexing A, we write A = {Ery : 'y E r}. Let xi,, y,? (1 1} is a base for X. Since (E.,,, ... , E.,,) 0 U(r, S') (by MT), (ii) of the present proposition is established. it therefore only remains to show that every U(r, S) is closed and has empty

interior in X. As above, let S = (A,,...,Aq). For the purposes of this paragraph, a qp-tuple z E Gqp will be indexed:

X = lxll,...,xlp,x21,...,x2p,x31,...,xgl,...,xgp).


281

Let C = U,j[Aij U Bi,J E W(G), and for E E X, let K(E) = C-1C, -- E. Let

El,-, Eq E X.

Define maps 1P, l from GQF x GQP x X4 to R by

,,D(x, y,

Ei, ... , Eq) = A[[n(xijEi n yijE= )J n C,[, ,,J

T (x,y,E1,...,Eq)=\ [n(z1E1nyijK(Ei)) nC,

.

The map E - K(E) is continuous on X since K(E) A K(F) C EL F, and applying Problem 7-10 with n = 2qp, it follows that 91 is continuous. Let

H=A11 x...xAl.xA21 x...xA.,xB11 x...xB., and D = H x X. Now if yij E C, then yijE= n C, = y:jK(Ei) n Cr,

and as Aij U Bi, C C for all i, j,

and %P coincide on D. It follows that L = ,vI ({o}) n D is closed in D. Since D = H x Xq with H compact, it follows that P(L) is closed in Xq, where P is the projection map from D onto X. So U(r, S) = P(L) is closed as required. We now show that U(r, S) has empty interior. Let e > 0 and choose, by (7.10), a dense, open subset V' of G such that \(V') < e. Let (E1, ... , Eq) E U (r, S). It is sufficient to construct elements F1,..., Fq of X such that (F1, ... , Fq) U(r, S) and \(E, A F1) < E. We can suppose that e E C°. By the definition of .5', for each i, the family (Ail,..., Ai,) B11..... BiP} C i'(G) is disjoint. So we can find an open neighbourhood W of e such that for each i, the family {Ai1W,... , BW } is disjoint. Let P

G, _ U

[(Aikw)-I

n v'J

k=I and

Fi = (Ei n (V')c) u G,.

Then Fi E X, and since Ei L F, C V', we have A(E, L F,) < e.

It remains to check that (F1,... ,

0 U(r, S). For each pair (i,j), let

xij E A,J,yij E Bij. Now (3)

x,JF, D x,jG, D xi,[(A,1W)-IJ nxijV',

(4)

yt, Fc D yij (Gi W) D y,j [(BijW)-I J n y,jV'

since AikW n BijW = 0. Let

z = (flIz1I(A1wy.h] nyij[(BijW)-1]1} nC°.


252

Then Z is an open neighbourhood of e, and as every translate of V' is dense and open in G, we have x,jV' n Z,1yijV' n Z dense and open in Z. From (3) and (4),

n

F') I C° D f l(z n xv')

%J

io

Ill

(z n yjv'))

and by the Baire Category Theorem for G, the finite intersection (fi,j(xijFi n yijF')) n c, contains a nonvoid, open set and so has positive .t-measure. Thus (F1,. .., 0 U(r, S) as required. 0 (7.16) PROPOSITION. Let G be a nondiscrete, v-compact, locally compact metric group and let {E7: 7 E I'} be a family of measurable subsets of G of finite .1-measure satisfying properties (i) and (ii) of (7.15). Let B and R be the linear subspaces of L,,(G) spanned by the sets {X:-r,,: x E G, 7 E I'} and {XE: E E .4f(G), (E-)° is empty} respectively. Let 0 E C(G), b E B, and r E R. Write b = Ez ,r b(x, where, for x E G, 7 E T, b(z, 7) belongs to C and is nonzero for only a finite number of pairs (x,7). Then the following assertions are true: (i) '+b +r = 0 in Lo,(G) if and only if 0 = 0, b(x, -y) = 0 for all pairs (x, 7) and r = 0 almost everywhere. (ii) Let M be the sum of all the negative b(x, 7). If 0 + b + r > 0 in Lm(G), then 0 is real-valued, and 0 + M > 0 on G. PROOF. (i) Suppose that 0 + b + r = 0 in LOO(G). If E E 4l (G) and (E') is empty, then E` contains a dense open subset of G. Since the open, dense property is preserved under finite intersections, there exists a dense, open subset U of G such that r(U) = {0}. Let V be any nonvoid, open subset of U. By (7.15), the set

E=

[n{xEfl: b(x,-y) 0 0}] n V

is not A-null. Since b(E) = {0), 0j E = (4 + b + r) jE = 0

almost everywhere.

Since E is not null, we have 0 E O(V). Since V was arbitrary and d is continuous,

vanishes on U and so on G. Thus ¢ = 0. It readily follows that b = 0 almost everywhere on U. Let xo E G, 7o E r. Then the set

El = x0E,0 n

in{zEE: b(x,7)

0, (X0' -to) ` (x, ;')}} n U

is not null, and b(El) = {b(xo, 7o)}. Since b = 0 almost everywhere on U, b(xo, 70) = 0. So b = 0, and the nontrivial implication of (i) is proved. (ii) Suppose that d' + b + r > 0 in L,o(G). Using arguments similar to those of (i), the functions 0 and b are real-valued on G. By considering the set In{xE1: b(x, 7) < 0}] n

[n{xEc : b(x, 7) > 0)] n V,

where V is as in the proof of (i), it follows that (6 + M > 0 on C. 0


283

(7.17) THEOREM. Let G be a nondiscrete, c-compact, locally compact metric group that is amenable as a discrete group. Let m E,ir(C(G)). Then there exists a set {m6: b E &} C Z(G) with the following properties. (i) JAI = 21;

(ii) S(m6) ft S(mo) = 0 when 6 96 or in A; (iii) if V is a dense open subset of G with A(V) < 1, then S(tne) C for all b;

(V_1)n

(iv) 'n'ib 1C(G) = m for all 6;

(v) ms is singular to .(G) for all 6.

PROOF. Let {E,: 7 E r}, B, and R be as in (7.16). Then t1 = {0,1}r satisfies (i). Let A = C(G) + B + R and let 6 E A. Using (7.16), we can define a linear functional mb on A, where (1)

ma(0 + b + r) = m(O) + Z b(x, 7)b(7) x,7

for 0 E C(G), b E B, and r E R. If 0 + b + r > 0 in L.(G), then, by (7.16),

m6(¢+b+r) _> inf{O(x)+M: x E G} > 0. Noting that mb(1) = 1, it follows that ma E 9R(A). It is easily checked that B and R are right invariant subspaces of L,o(G) and that

(Ebx'1xxE1)xo = (Eb(xox-iXzz.,). x,ry

x,ry

It follows using (1) that m'6 E £(A). By Problem 1-13, there exists m6 E Z(G) for which m6IA = ma. It remains to prove (ii), (iii), and (v). Let V be an open, dense subset of G with A(V) < I. (The existence of such

a V is assured by (7.10).) Let D = (G

V)-1. Since XD E R. we have

M6 (XD) = M6' (XD) = 0, and applying (7.9), m6 is singular to £2(G). This gives

(v) and (iii), noting that m6(V-1) = M6 (G - D) = 1. Finally, if a E A and b or, we can, without loss of generality, find 7o E r

such that 6(70) = 1 and a(-yo) = 0. Let E = E.,o. From (1), it follows that m6(XE) = 1 while mo(XE) = 0. So S(?h6) C E and S(mto) C (G_- E) _ ,D(G) - E, and (ii) is proved.

O

(7.18) COROLLARY. Let G be a nondiscrete, c-compact, locally compact metric group that is amenable as a discrete group. Then the map m m(c(G) from Z(G) onto C(C(G)) is not one-to-one. The next result shows that much of (7.18) is still valid with the metric condition removed. (7.19) THEOREM. Let G be a nondiscrete, or-compact, locally compact group

that is amenable as a discrete group. Then there exists a subset P of .>r(G) of


284

cardinality 2`, such that S(m) fl S(n) = 0 whenever m :An in P. Further, every element of P is singular to C (G). PROOF. From Problem 7-7, there exists a compact normal subgroup H of G

such that G/H is metrisable and nondiscrete. Further, G/H is amenable as a discrete group and is or-compact.

Let Q: G

G/H be the quotient map. Now ((1.11)) the mapping

Q*: L,o(G/H) -+ L.(G) is an isometric *-homomorphism, where (Q`V))(x) = V,(Q(x)) for 0 E L.(G/H) and x E G. It is routine to verify that Qo = Q"'j"(c) is a continuous map from

Z'(G) into t(G/H) for which Qo(xa) = Q(x)Qo(a) for all x E G, a E c4(G). Further Qo(' (G)) ='(G/H), since, with L.(G/H) identified by means of Q* with a C'-subalgebra of L. (G), every element of (D(G/H) extends to an element of >(G).

Now G/H satisfies the hypotheses of (7.17) (with G replaced by G/H). So we can find a set {ma: 6 E A} C Z(G/H) such that the conclusions of (7.17) hold. For each 6, let 916 = Qo'(S(mm6)). Then {Ws: 6 E A} is a disjoint family of closed, left invariant subsets of For each 6, find (2.25(iii)) an element n6 E Z(G) such that S(h6) C W6. Let P = {n6: 6 E A }. It remains to prove that n6 is singular to £t (G) for all b. To this end, let Ac/H be the canonical left Haar measure on G/H, determined, through Weil's formula, by A and the normalised Haar measure on H. By (7.10), we can find a dense, open subset V of G/H with .1c/H (V) < 1. Applying Weil's formula to the lower semicontinuous function Xq-,(v), we obtain A(Q(V)) =

Ac/H (V) < 1. Since Q is open, Q-1(V) is dense in G. Trivially, E = V-1 E .Il (G/H) and F = Q-1(E) E . f(G). As Q* (XF) = XE, it follows that F = Qo 1(E). Further, as S(m6) C (V-1)n = E, it follows that S(hs) C T6 C F = ((Q-1(V))-1)1. Applying (7.9) with D = G (Q-1(V))-1 we obtain that n6 is singular to ,ice (G).

0

(7.20) COROLLARY. Let G be a nondiscrete, group that is amenable as a discrete group. Then

locally compact

J.rr(G) - Lt(G)I > 2`.

We can now give the promised improvement to (7.14).

(7.21) THEOREM. Let G be a locally compact group that is amenable as a discrete group. Then .C(G) = .(G) if and only if G is discrete. PROOF. Use (7.20) and (7.14).

We finally turn to our fourth problem. Let S be a left amenable semigroup. What is the cardinal JZ(S)J? We know, of course, that if S is an (infinite) group, 221 ". then J£(S)I = As semigroups are so much more complicated than groups, we are not going to get such a simple result for them. We introduce the cardinal m (or m(S)) which will replace JSJ in the group case. In many cases, m equals


285

the more natural cardinal p(S), where p(S) = min{IsSI : a E S}. References for the work discussed here are Day [4), Granirer [1)-161, Chou [2), [9), Klawe [1), (3), Paterson [9), and Yang [Si].

(7.22) Definitions. Let S be a semigroup. The cardinal m(S) (or simply m) is defined by: r n

m(S) = min { 111

V siSi

: n > 1, {S1,.. . , Sn} is a partition of S,

i=I

sl,...,Sn E S I

m(S)) -

The significance of this cardinal for us is that if S is left amenable and m (= is infinite, then IC(S)I = 22".

The semigroup S is called almost left cancellative (a.l.c.) if m = ISI, For justification of this nomenclature, observe that if S is a.l.c. and s E S, then, applying the above definition with n = 1, sI = s, we have IsSI = ISI, so that, in a rough sense, S is "close" to being left cancellative inasmuch as multiplication

by s does not "collapse" S too much. If S is left cancellative and infinite, and {S1..... Sn } is a partition of S, then IS[ = IS= l for some i, and it easily follows that S is a.l.c. We now show that if m is infinite, there exists an a.l.c. subsemigroup T of S with ITI = m(S) and .C(T) closely related to .C(S). Note that

in the statement of the following result, we can take T to be the subsemigroup of S generated by A (since JAI is infinite).

(7.23) PROPOSITION. Let m be infinite, {Si, ... , S,,} be a partition of S and si, ... , sn E S be such that A = U', s;SS has cardinality in. Let T be a subsemigroup of S such that A C T and JAI = ITI (= m). Then (i) T is a.1.c. with m(T) = m; (ii) m(A) > 0 for all m E .C(S), and the map in -+ (rIT)fm(T) is one-to-one from .C(S) into .C(T); (iii) IC(S)J< IC(T)I.

PROOF. (i) Let {T1,... 1 Tm} be a partition of T and ti, ... , t,,, belong to T.

Forl 0. So IpI = 0. Since IpI = p+ + p , we have p+ = 0 = p-, and hence

P:= P+ - P = 0. Thus n(T)m = m(T)n, and evaluating at S gives m = n. So (ii) is proved, and (iii) is an immediate consequence of (ii).

Our next result asserts that if S is left amenable with m infinite, then T of the above proposition can be chosen to be left thick ((1.20)). (7.24) PROPOSITION. Let S be left amenable with m infinite, and let A be as in (7.23). Then there exists a left thick subsemigroup T of S with A C T and 1T1=m. PROOF. Let m E .E(S). Let

k = sup{m(RA): R is a countable subset of S}. For each n E P, we can find a countable subset Rn of S such that m(R,,A) >-

k-n`1 Let R

U{Rn: n > i}. Then R is countable, and for each n,

k - n'1 < m(RnA) < m(RA) < k. So m(RA) = k. Since m is infinite, IRAI < in. Now let T be the semigroup generated by Y = (A U RA). Then IT1 = m and A C T. The proposition will be proved once we have shown that m(Y) = 1. (For then m(T) = I and T is left thick in S by (1.21).) Suppose, on the contrary, that m(Y) < 1. Let Z = S -, Y. Then m(Z) > 0. So n E Zoo (S)',. where

n(E) = m(E fl z), is nonzero and > 0. It is sufficient to show that n E 3, (S). For then n/m(Z) CZ(S) with 0 = n(Y) = n(A), and (7.23(ii)) is contradicted. Now for E C S,sES,

n(s-1E)=m(s'1EflZ)=m(s 'E) -m(s 1Ef1Y) = m(E) - m(s-1E fl Y) = n(E) + [m(E fl Y) - m(s-1E fl Y)1.


287

We therefore have to show

m(EnY) = m(s-IEnY).

(1)

Now

s-IE n s-'Y c (s-'E n Y) U (s-IY - Y), and

s-'E n Y c (s-'E n s-IY) u (Y - s-IY), and, applying m to both of the preceding inclusions and noting that =m(s-I(EnY)) =m(EnY), m(s-'E ns IY) we see that (1) will follow once we have shown

m(s-'Y d Y) = 0.

(2)

To prove (2), observe first that m(Y) = k = m(sY), since

k = m(RA) < m(Y) < m(sY) = m(({s} U sR)A) < k. Similarly, m(sY U Y) = m(Y). Thus

k = m(sY) + m(Y

sY) = m(Y) + m(sY - Y),

and as k=m(sY)=m(Y),wehave m(YAsY)=0. Now 0 = m(Y © sY) = m(s I(Y A sY)) = m(s IY A sÌ(sY)),

while m(s-'(sY) A Y) = m(sI(sY)) - m(Y) = k - k = 0. The equality (2) now follows.

We now state and prove the following key result that will enable us to determine (£(S)( when S is left amenable with m(S) infinite.

(7.25) PROPOSITION. Let S be an a.l.c. semigroup with m (_ (S() infinite. Let a be the smallest ordinal of cardinality in. Then there exists a disjoint family {®E: e E a} of left thick subsets of S.

PROOF. The set F (S) of finite subsets of S has cardinality m since m is infinite. We can thus well-order -9-(S): F (S) = {F,6:,6 E a}. Using transfinite

recursion, we will construct a family {A f: C,,6 E a, e < p} of subsets of S having the following properties: (ii) (©Q1 S 1f3E if 0 is infinite and A8 is finite if 6 is finite (e < p); (iii) A-6 c A0' whenever e 1. Note that UtI si 1(L) _ < m, and we U 1 Si. Suppose that U I S i = S. Then J U 1 siSiJ < JLJ (Un contradict the fact that S is a.l.c. So I Si), and 1 Si 0 S. Let s' E S S6 i = siss (1 < i < n). Clearly, s6 i 0 L, and it is obvious that (a), (b), and (c) are true for s, 77 < 6. This completes the construction of the sets rE. Now set

11"_{U{A :e
0 and n E P, there exists a net {96} in P(E) with g6

m weak* and, for all 6, [[x * g6 - 96((1 < c (x E K,,).

Using (s) and the fact that m(zj,,) = 1, we can find a sequence of functions (fn} in P(E) such that (a) ((x * fn - fn((1 < n-1 (x E K.);


291

(b) I(fn - m)(4r)I < n 1 (1 < r < n); (c) I(fn - m)(0o)I < n-1, (d) Ifn(4P,)-1I 1}. Then ExpiD = 0, and if iZ 96 0, then iZ is not norm separable. [Suppose that be such that m E Exp 0 and let qo E

Ren(Oo) (0,11 such that 0 _ 01 exp(21.-i02). If p E (P(G) then p is a multiplicative linear functional

on L0(G), and as abx = (0ix)exp(2iri('02x)) (x E G) and pip(x) = p('bx), we obtain p1,b = (pol) exp(2lri(ptb2)). It therefore suffices to show that we can find p so that pOi = Oz (i = 1, 2). To achieve this we first express each ¢1 in the form Zoo, 2-nxc . In fact, a simple recursion argument shows that we can construct a sequence E0, El.... of subsets of G such that N

0 < {61 -

2 nXEn
2'N) with

00

01 = E 2-nXEn. n=1

Similarly, we can find a sequence {Fn} of subsets of G such that 00

02 = E 2-nXF n=1

Using (1) of (7.15), we take p to be a point in the set

n

[

n>1

LL

7

I

SEE,

(z-1An )AJ n

(x-IA`n )A [,n EE

n[Znn(x-IBn)A I n [.n

JJ

)n]

If x E En, then, as p E (x-IAn)^, we hJave PXA,(x) = (X.-3A. )A(p) = 1. If x E En, then we havePXAn (x) = (X.-IAn)^(p) = (XG-X.-3An)''(P) = 1-1 = 0. So PXAn = XEn, and 00

co

Po1 = E 2-n(pXA,) = E 2-nXEn = 1 n=i

n=1


294

Similarly, pzb2 = 02 as required.]

(7.30) Locally compact groups that are amenable as discrete. In view of theorems such as (7.14) and (7.19), locally compact groups that are amenable as discrete groups are of special interest. The following result is based on Rosenblatt (7] and Chou (11]. The condition (iii) is a fixed-point theorem for the groups G under consideration, and the fact that (iii) implies (i) is of note in view of the essential use of "(i) implies (iii)" in the chapter. Let G be a nondiscrete, c--compact, locally compact group. Then the following three statements are equivalent: (i) 0 is amenable as a discrete group:

(ii) G is amenable and for each 0 E L.(G,R), we have sup{m(O): m E Z(G)} =

(1)

AE9(ri

[esssup(IA(_1

1 0y),;

yEA

(iii) if C is a left invariant, nonempty, compact subset of fi(G), then there exists m E .C(G) with S(fn) C C. (We first prove that (i) implies (ii). Suppose that (i) holds. Let 45 E L.(G,R), and let a, b be respectively the left- and right-hand sides of (1). Obviously, a < b. We now prove that b :S a. Let {A6} be a summing net for G regarded as a

discrete group ((4.15)). For A E 3(G), let µA = ((A()-' (E y) E 1, (G) C M(G). yEA

Let µ6 = µA6. Then

esssup((4'pA)µ6] < (A6(-1 (E esssup((0µA)x] l = esssup(OpA),

I

xEA6

and

II(OPA96 - u6)II 0, there exists bo such that for all b

_ 60, b < ess sup ¢µ6 < (ess sup(OµAµ6 )) + E < ess sup Oµ6 + 2e

and from the definition of b, 1 6im(esssup(Op6)] = b.

Note that by replacing 0 by -0 we deduce the existence of limb less inf (Oµ6 )].

Let B be the linear span in L. (G, R) of 1 and the functions Ox (x E G). Let E B. Then for some 8, ,8, E R. xq E G, n

G= 61+1: #:(45x:)s=1


295

Since

,fi{;6 - (81 +

0,

0ft6) 00

II IIxs/26 - /26 f[1 - 0,

icl

it follows that lim[ess sup(i 5 )j = p + 16m [ess sup

([

ai) OFi6) }

,

(2)

li6m[ess inf (V'Ec6 )[ = (t + Jim f ess inf

(( s) #{16)]

If 1b = y1+F`m` 1 '7i(0yi) (1',-Y,E R, y{ E G), then 0 = (3-'7)1+E-1 (¢yi), and by taking either "ess sup" or "ess inf" and using (2), we obtain n m

--1+ EQi-E-h b=0. x=I

s=I

It follows that we can define a linear functional n on B by setting

n(O) = P+

(Es)b. 4=1

Clearly n(¢) = b, n(Ox) = n(V)), and [[nIH = I = n(1). By Day's Theorem, we can find mo E .C(G) with mo[B = n. Then b = mo (q) < a

as required, and (i) implies (ii). We now prove that (ii) implies (iii). Suppose that (ii) holds. Let C be a left invariant, nonempty, compact subset of Let

E° = {E: E is a Bore] subset of G and C C E}. Since C is zero-dimensional and every open and closed subset of '(G) is of the form E for some Borel set E C C, it follows that every open subset of 4P(G)

containing C also contains some E with E E 9. Let E E i', A E .9(G), and W=

I `XEA

x-1 E. Since

C C In (x-1E)^ = FV, xEA

it follows that A(W) > 0, and if w E W, then [JAI-' E1EA XEx)(w) = 1, and 1. Using (ii), we can find mE E .C(G) such that so esssup(IAI'I

mE(E) = 1. Now since EI n E2 E 9 if E1, E2 E ', it follows that ,' is a net (under intersection). Thus {mE} (E E F) becomes a net, and if m is a weak cluster point of this net in ,C(G), then, using regularity, th(C) = 1. So S(m) C C and (ii) implies (iii). It remains to show that (iii) implies (i). Suppose that (iii) holds. Assume for

the moment that C is metric. Let ¢ E I.,(G) be real-valued, and find ((7.29)) V) E L. (G, i8) and p E 4) (G) such that

Pile=0

296


Let C be the left invariant, compact subset (Gp)- of 4(G). By (iii), we can find m E .C(G) with S(tn.) C C. Let p = m. E PM(C). Then we can find a net {vb}, where each vb is a convex combination of point masses b£ (g E C), such

that v6 --r p weak* in PM(C). Now if {x,} is a net in G with x,p - C, then bz,p --+ b weak* in PM(C). It follows that we can find a net {a6}, with each a6 a convex combination of elements of G, such that a6p -. p weak* in PM(C). Then for x E G, in an obvious notation, a6O(x) = a6(pib)(x) = (asp)('4z) -+ µ( x) = µ(+G)

The "real" version of (2.13), with B, L,,,, (G) both replaced by the space 1. (G, R) of real-valued, bounded functions on G, then yields an element of 1.(G. R)' that

extends in the usual way to a left invariant mean on l ,(G). So G is amenable as a discrete group. Now remove the requirement that G be metric. To show that G is amenable as a discrete group, it is sufficient to show that every countable subgroup H of G is amenable as discrete. Let H = {e} U {h,,: n E P} be a (countable) subgroup

of G with h 0 e for all n. By choosing U,, in Problem 7-7 so that h,, f U,,, the resulting compact subgroup N is such that N n H = {e}. Let L = GIN. Then GIN is nondiscrete, a-compact, and metric, and contains a copy of H as a subgroup. The amenability of H will follow once we have shown that L satisfies (iii); for then, by the above, L will be amenable as discrete, and so also will be its subgroup H. Let D be a left invariant, nonempty, compact subset of iP(L). Let Qo : 4'(G)

(L) be as in the proof of (7.19). Then C = Q1(D) is left invariant in P(G), and if p E PM(C) is left invariant for G, then p o Qo I E PM(D) and is left invariant for L. Thus L satisfies (iii) as required. So (iii) implies (i).)

(7.31) Amenability and the radicals of certain second dual Banach algebras. (Day [4), Civin and Yood [1), Granirer [2), [3), [4), [6), [13), [14), Granirer and Rajagopalan [1), Klawe [1), Duncan and Hosseiniun [1)). Let E be either a discrete semigroup or a locally compact group. The second dual L, (E)" is a Banach algebra under the Arens product. Let the radical of a Banach algebra A be denoted by rad(A).

As we shall see, left invariant means can be used to produce elements in rad(LI (E)"). Recall that 31t(E) is the space of topologically left invariant, linear functionals on LI (E)' = L. (E). Define

J(E) = {f E31t(E): f(1) =0}. (i) J(E) is a closed, linear subspace of rad(LI(E)").

[Obviously, J(E) is a closed subspace of LI(E)". To show that J(E) C rad(LI (E)"), it is sufficient to prove that J (Z) is a left ideal in LI (E)" and

3(L)2 = {0}. Let p E L,,(E)' and f E 31t(E). Then pf = p(1)f. Clearly pf(1) = 0 if f(1) = 0, so that J(E) is a left ideal in LI(E)". Since p(l) = 0 if p E J(E), we also have J(E)2 = {0} as required.)


297

(ii) Let S be a left amenable semigroup.

(a) If m(S) is infinite, then dim(rad(l1(S)")) >

22",cS)

(b) If m(S) is finite and A(S) is the set of finite, left ideal groups in S, then 1 + dim(rad(l1(S)")) > IA(S)I.

[Fix mo E £(S). Since (m - mo) E J(S) for each m E £(S), we have dim J(S) > dim[Span{(m - mo) : m E £(S)}). (a) now follows from (7.26) and (b) from (7.27).)

(iii) Let G be an amenable, noncompact, locally compact group, and let m be

as in (7.0). Then dim(rad LI (G)") >

22m

[J (G) is of codimension 1 in 31t (G), and by (7.6) and (2.2), dim31t(G) = 22".)

It readily follows that if G is as in (iii) then rad L1 (G)" is not norm separable. Similar results (Granirer [6], 113)) hold for the radical of 'U, (G). (The latter space is a Banach algebra from (2.8).) (iv) Granirer [14), developing ideas of S. Gulick [1), shows that if G is locally compact and nondiscrete, then rad(L1 (G)") is not norm-separable. The case in which G is discrete and nonamenable still seems to be open.

Problems 7 1. Let X be an infinite set. Prove that I$XI = 22"' =

l(X)`.

2. Construct explicitly sets Zfl and functions Op satisfying (7.3) when G = R.

3. Let G be a noncompact, locally compact group and X the maximal ideal space of U,.(G). Show that there exists a family 5' of minimal, compact, left invariant subsets of X with [.'[ = 22"

4. Let G be as in Problem 3 above. Suppose, further, that G is amenable. 22m extreme points. Show that £t (G) contains at least 5. Let G be a a-compact, noncompact, locally compact amenable group. This problem gives another proof that [£t(G)I > 2r ((7.6)). (i) Show that there exists a summing sequence for G in W,(G) with

)(U +1) ? (n+ 1))(U.); (ii) Define r: L,.(G)

tar by

r(0)(n) = (A(UU+1

UU))-1 fu- i

U

¢(x) dA(x).


298

Show that r is linear, surjective, and of norm 1. (iii) Let Mc = {m E 9R(P) : m(q5) = b(n) for every convergent se-

quence 0 E l,o}. Show that r'(fiic) C £(G). (iv) Deduce that j. (G)i 2:2' (a special case of (7.6)). 6. Let E, . be as in (7.28(i)). Assume the Continuum Hypothesis. Show that J.fi` > 2'. (Use the tech-Pospisil Theorem [HR2, (28.58)]. It runs as follows: if X is an infinite compact Hausdorfspace, if u(z) is the smallest possible cardinal that a family of open subsets of X with intersection {x} can have and if m is a cardinal such that u(x) > in for all x E X, then !Xi > 2'.)

7 (cf. [HR1, (8.7))). Let G be nondiscrete and a-compact, and let {U,,) be a sequence of neighbourhoods of e in G. Show that there exists a compact normal subgroup N of G such that N c fO°1 U,,, and GIN is a nondiscrete, separable metric group.

8. Construct a dense open subset V of R such that )(V) < e (cf. (7.10)). 9. Let G be a nondiscrete, a-compact locally compact metric group. Show that there exists a family .Y of closed, right invariant ideals of L.(G) with 1- 9'1 = 2`.

10. Let (X, d) be as in the proof of (7.15). Let E E X, n E P, and define

T": C"xX" - R by T"(x1,...,x",EI,...,E") = A(x1E1 fl . . f1x"E" f1E). .

Show that T,, is continuous. 11. Give examples of a.I.c. semigroups.

12. Let S be a left amenable semigroup and

p (= p(S)) = minflsSj: s E S). (So m < p < ISI where m is as in (7.22).) Prove that if S satisfies (v) or if p is infinite and S satisfies either of the conditions (i), (ii) then m = p. Prove also that if S is infinite and S satisfies either of the conditions (iii), (iv), then S is a.l.c. (and so p = m = ES{). (i) S is right reversible; (ii) S is amenable; (iii) S is left cancellative; (iv) S is right cancellative and left amenable; (v) S is ELA ((2.27)). Does p = m in general?

13. Show that if S is an infinite, left amenable semigroup that is either left or right cancellative, then 221sl I'"(S)I = The next three problems can all be solved by adapting the proof of (7.25).

SIZES OF SETS OF iNVARIANT MEANS

299

14. Let S be an amenable semigroup and n = min{IsSsI : s E S}.

Recall that 3(S) is the set of invariant means on S. Show that if n is infinite, then I3(S)I = 22", and that if n is finite then S contains a finite ideal and I3(S)I =1. 15. Let G be an amenable discrete group and Y (G) the set of inversion invariant means on G (Pr. 2-7). Show that if G is infinite, then

17(C)I = 22101.

16. Let G be an infinite amenable discrete group. For F E,9-(G), let

CF = {x E G: FxF fl (FxF)-' = 0}. (i) Show that if, for each F E .T(G), there exists x E G such that FxF fl (FxF)-' = 0, then for all F, ICFI = IGI

(ii) Show that (a), (b), and (c) are equivalent: (a) I3(C) 3'(G)I = 22101; (b) 3(G) 3`(G); (c) for each F E .9- (G), there exists x E G such that FxF fl (FxF)-' = 0.

17. Let G be an infinite abelian group. Show that J(G) ,-f T (G) if and only if the set B = (x2: x E G) is infinite.

18. Suppose that m = p, where in, p are as in Problem 12 above. Assume GCH, the Generalised Continuum Hypothesis. Prove that if

r=min{IAI:A CS, then p=r. 19. Let S be a semigroup and .GO,(S) be the family of right thick subsets of S

((1.20)). Show that 9,(S) is a filter if and only if S contains a finite, left ideal group. Show that . ,.(S) is an ultrafilter in S if and only if S has a zero. 20. Let G be an infinite, amenable group containing a proper, normal subgroup H of finite index. Let J(G) be as in (7.31). Show that 22"'.

dim(rad(lj(G)")/J(G)) >

(So J(G) is not all of the radical of l1(G)".)

21. Let S be a countable, left amenable, left cancellative sernigroup. Prove that Z(S) has an exposed point if and only if S = Sk x G for some k E {2, ... , oo},

where Sk is the semigroup with elements ej (1 G2>...>Gr={e}

of subgroups G, of G, each G=+r being a normal subgroup of G. A normal series (1) for G is said to be a central series of G if, for each i, G= is a normal subgroup of G and, for 1 < i < r, G=/G,+, C Z(G/G,rt.1), or, equivalently, [G, G;] C GZ+1 (in the notation below). The group G is said to be nilpotent if it has a central series. The group G is said to be solvable if it has a normal series Gr r G2 C> ... I> Gr = {e}

such that every GZ/GZ+1 is abelian. Obviously, every nilpotent group is solvable. Commutators are often useful,

particularly when discussing nilpotent and solvable groups. For x, y E G, the commutator [x, y] is defined [x, y] = x-1 y-1 xy. If A, B C G, then [A, B] is the subgroup of G generated by {[a, b]: a E A, b E B). Let G be a nilpotent group. Then G has two central series of particular importance, the upper central series and the lower central series. The upper central series for G is the series {e} = where the ZZG are defined recursively: ZOG = {e} and Z,+1G D ZZG is specified by Z,+1G/ZZG = Z(G/ZZC). The lower central series is the series

where the subgroups CZG are defined recursively: C'G = G and Czi'G = [C°G, G]. The subgroup C2G is the derived group of G and is the subgroup of G generated by the set of commutators {[x, y]: x,y E G}. The class of G is the smallest integer r for which Cr+1(G) = {e}. A useful fact about the lower central series for G is that for all i, j, [CiG, C-1 G) C CZ+?G (M. Hall [1, Corollary 10.3.5]). The following result is given in [M, Theorem 9.16]. 301

302

APPENDIX A

Let G be a finitely generated, nilpotent group, and H a subgroup of G. Then H is also finitely generated. For solvable groups, the derived series plays a fundamental role. The derived

series {DiG} of a group G is defined recursively: DIG = G and Dt+1G = [D=G, D;G). It is readily checked that each DAG is a normal subgroup of G and

that DjG/D +1G is abelian. The group G is solvable if and only if DG = {e} for some r. Solvable groups behave well under extensions [M, Theorem 10.07]. Indeed (*) if G is a group and N is a normal subgroup of G, then G is solvable if and only if both N and GIN are solvable.

Let G, H and K be groups. The group G is said to be an extension of K by H if H is a normal subgroup of G and G/H = K. The semidirect product construction discussed below is a particularly important source of group extensions for our purposes. Let H, K be groups and p: K -+ Aut H a homomorphism, were Aut H is the automorphism group of H. We define a product on the set H x K as follows:

(h, k) (h', k') = (hp(k)(h'), kk'). With this product, the set H x K becomes a group, which is called the semidirect product H xP K of H and K [HR1, (2.6)). It is easily checked that G = H x, K is an extension of K by H.

The identity of H x , K is, with an obvious abuse of notation, the element (e, e), and the inverse of an element (h, k) E H xp K is (p(k-1)(h-1), k-1). Semidirect products also occur in the topological setting. Let H and K be locally compact groups and Aut H the group of topological group automorphisms

of H. Then Aut H is itself a topological group in the natural way ((3.3)). Let p: K Aut H be a homomorphism. The map p is continuous if and only if the map (k, h) p(k)(h) is continuous from K x H into H [Ho, Chapter 3, g31. Suppose that p is continuous. The semidirect product H x, K is then a locally compact group under the product topology.

Let G = H x, K. Then H and K are canonically identified with closed subgroups, also denoted H, K, of G, where H is normal, H fl K = {e}, and G = HK. Conversely, if a locally compact group G contains closed subgroups H, K with these three properties, then G is canonically isomorphic to a semidirect product H xP K: here, p is given by

p(k)(h) = khk-1. A number of naturally occurring locally compact (Lie) groups are semidirect products of locally compact groups. An important example for us is the "ax+b" group S2, the affine group of R. Here, S2 is the group of transformations of R

of the form x -» ax + b, where a, b E R with a > 0, the product being that of composition. Associating the transformation x -' ax + b with the pair (b, a), we identify S2 with the locally compact group R x (0, oo) with group product given by

(b, a) (b, a') = (b + ab', aa').

APPENDIX A

303

(Note that (0, co) is a locally compact abelian group with multiplication as product.) It is clear that S2 is the semidirect product R xp (0, oo), where p(a) (b) = ab. The map t --+ e= is an isomorphism from R onto (0, oo), so that S2 can be regarded as the semidirect product R xa R, where o(t)(s) = eis.

APPENDIX B

Lie Groups The purpose of this appendix is to describe briefly the concepts and results from Lie theory needed in the text. Proofs are almost entirely omitted. Our main sources for the theory here are [He] and [B], and results for which no references are given will normally be found in one or other of these texts. On a number of occasions, we will also require results from [Ho], [J], [HS], and [SW]. The book (SW] gives an excellent introduction to the subject.

Bi. Charts and manifolds. Let M be a topological space and m E M. An n-dimensional chart is a pair (U, 0), where U is an open subset of M and ¢ is a homeomorphism from U onto an open subset of R1. Sometimes, a chart (U, 4) will simply be referred to as U, reference to 0 being left implicit. If (U, 0) is an n-dimensional chart and, for p E U, 0(p) = (xi (p)..... the the numbers x=(p) are called the local coordinates of p.

x

A C°O-structure on M of dimension n is a family s1 = (U O,1 qQ): ce E A} of n-dimensional charts in M such that

(i) UQEA U = M; (ii) if a,,8 E A, then the map ¢Q o 0. I is a C°°-map from 0,8 (UQ n Up) onto 45Q(UQ n Up);

(iii) the family sad is a maximal family of n-dimensional charts in M satisfying (i) and (ii). We note that if sag is a family of n-dimensional charts in M such that (i) and (ii) are satisfied, then, by including all those n-dimensional charts in M that intersect the members of V in a "CO°-way," we obtain a COI-structure on M (containing.). A COO-manifold (or simply a manifold) is a separable, Hausdorff space M equipped with a C°O-structure sag. In such a case, we normally refer to M as a manifold, the structure 5f involved being left implicit. A local chart on a manifold M is just a member of the C°°-structure d. A topological space is said to be 0-dimensional if it is discrete. An analytic structure on M of dimension n is defined as in (i), (ii), and (iii) above, the maps 0Q o ¢i 1 being required to be analytic (in the sense that their coordinate functions are given, locally, by power series expansions in real variables). An 305

APPENDIX B

306

analytic manifold is defined in the obvious way, Every analytic manifold is also a C°O-manifold in the natural way. Every manifold M is obviously a locally compact Hausdorff space. It is routine to check that M is connected if and only if M is pathwise connected. If M, N are (analytic) manifolds, then M x N is also an [analytic) manifold in the obvious way. Trivially, every open subset of an n-dimensional [analytic] manifold is itself an n-dimensional [analytic] manifold.

Let M and N be manifolds and': M - N. The map 4 is said to be a COO-map if, whenever m E M, there exist in M, N local charts (U, q5), (V, V,) at

m and t(m) respectively such that .r M

If P is itself simply connected, then the procedure leading to the definition of F is reversible, and F is a homeomorphism. Thus, in the obvious sense, M is the unique simply connected covering space of M. A covering transformation of M is a homeomorphism T of M such that iroT = it. The set G(M) of covering transformations of M obviously forms a group of transformations of M. Using the unique path-lifting property of (M,7r), we see that if T1, T2 E G(M) and a E M is such that Ti (a) = T2(a), then TI = T2. The group G(M) is isomorphic to iri (M). Indeed, let f be a path in M

starting and ending at mo. Then the map Ti fl, where Ti fI (Joe]) = if a) is a covering transformation of M, and the map Ti fi is an isomorphism from irj (M) onto G(M).

B4. Lie algebras. A vector space g over a field F, where F is R or C, is called a Lie algebra if there is given a bilinear map (X, Y) -, [X, Y) on g such

that (i) [X, Al = 0 for all X E g; (ii) (Jacobi's identity) [X, [Y, Zjj+ [Y, [Z, Xjj + [Z, [X, Yjj = 0 for all X, Y, Z E

Let g be a Lie algebra. If X, Y E g, then [X. Yj is the (Lie) product of X and Y. It easily follows from (i) that [X, Yj = -[Y, Xj for all X, Y E g.

APPENDIX B

310

If a, b are subspaces of g, then [a, b] is the subspace of g spanned by the elements [X, Yj (X E a, Y E b). A subalgebra f idea4 of 9 is a subspace s of g such that Is, sj C s [Is, s) C s). The Lie algebra g is said to be abelian if Is, gj = {0}.

Let b also be a Lie algebra over F. A linear map P: g --+ 17 is called a (Lie)

homomorphism if [P(X),'(Y)j = -P([X,Yj) for all X,Y E g. The map (I is called a (Lie) isomorphism if it is a bijective homomorphism. An isomorphism from 9 onto g is called an automorphism of Z. Ifs is an ideal in g, then 9/s is also a Lie algebra in the natural way, and the

quotient map X X + s is a homomorphism from 9 onto g/s. The direct sum g ® h of g and ll is a Lie algebra in the obvious way. If F = R, then the complexifi cation gc of g is also a Lie algebra over C in the obvious way.

Now let V be a vector space over F. Then the space L(V) of F-linear transformations on V is a Lie algebra, the Lie product being given by

[A,B)=AB-BA. An important theorem of Ado asserts that if g is a finite-dimensional Lie algebra over F, then there exists a finite-dimensional vector space W over F such that g is isomorphic to a Lie subalgebra of L(W). Proofs of Ado's Theorem are given in [B, Chapter 1, §7.31 and [J, Chapter 6).

A representation of the Lie algebra g on an lc-vector space V is a Lie homomorphism Q: g --> L(V). The space V is called a g-module, and we often write Xv for Q(X)(v) (X E g, v E V). Direct sums of 9-modules are themselves g-modules in the obvious way.

An important representation of g on g is the adjoznt representation adg (or simply ad), where adg X(Y) = [X,YJ (Y E g). (That ad is a representation of g follows easily using Jacobi's identity.) We note that ker ad is the center Z(g) of g, where Z(g) _ {Z E g: [X, Z) = 0 for all X E g}. It is sometimes convenient to regard ad as a representation on gc: simply set ad X (Y + iZ) = ad X (Y) + i ad X (Z). The context will always make clear if ad is being regarded as a representation on g or gc. A nonzero 9-module V is called irreducible if the only 9-invariant subspaces

of V are (0) and V.

B5. Derivations, automorphisms, and semidirect products. A derivation of a Lie algebra g is a map D E L(g) such that D(]X,Y)) _ [DX,Y]+ IX, DY) for all X, Y E g. The set of derivations on g is denoted by Der(g). It is easy to check that Derg is a subalgebra of the Lie algebra L(g). Jacobi's identity shows that adX E Derg for all X E g, so that ad is a homomorphism from g into Derg. An element D of Derg is called an inner derivation if D E ad 9. If X E g, D E Derg, then ID, ad X) = ad(DX), so that adg is an ideal of Derg.

The set of automorphisms of g is denoted by Aut p. Clearly, Autg is a subgroup of GL(g), the group of invertible elements of L(g). If 9 has finite

APPENDIX B

311

dimension n, then GL(g) is a locally compact group under the relative topology that it inherits as a subset of L(9) ("=" Mn fl), and Aut g is a closed subgroup of GL(g).

Let D E Derg. By induction, Leibniz's rule holds for D: (1)

(')iDT(X),Dn-(Y)].

Dn([X,Y]) r=o

It follows that eD = E,°,°_o D"/n! E Auto. If T E L(g) is such that etT E Aut g

for all t E R, then T = limt,o(etT - 1)Jt E Derg. Thus if T E L(g), then T E Derg if and only if e1T E Autg for all t E R. Now let b be a Lie algebra over F, and 1P: b -+ Der g a homomorphism. The semidirect product g xa b is the Lie algebra that, as a vector space, is g x b, and whose product is given by [(X, Y), (X', Y')] = ([X, X'] + 4'(Y) (X') - 4'(Y') (X ), [Y,Y'))

(X,X' E g, Y,Y' E b). Canonically, g[b] is obviously an ideal [subalgebra] of

gx.,b. Conversely, if 91 is a Lie algebra containing g as an ideal and b as a subalgebra such that 91 is the vector space direct sum geb, then 91 is canonically isomorphic to the semidirect product g x 4, b, where (1) (Y) (X) = [Y, X] for Y E b, X E g.

B6. Lie groups. A Lie group is a group G that is an analytic manifold such that the map (x, y) -+ xy-' from G x G into G is analytic. Thus the product map (z, y) -+ xy and the inversion map x -+ x-' are analytic. Every Lie group G is, of course, a locally compact group. Since G is a manifold, the identity e of G has a connected neighbourhood, so that Ge is an open (and closed) subgroup of G.

We now state the "approximation" theorem, which effectively reduces the study of connected, locally compact groups to that of connected Lie groups. (Montgomery and Zippin 11, (4.6))). Let G be a connected, locally compact group and U a neighbourhood of e. Then there exists a compact, normal subgroup K of G such that K C U and GJK is a (connected) Lie group.

Let G be a connected Lie group, and let (G, r) be the simply connected covering space of G (B3).

The manifold G can be made into a Lie group as follows. Let a, ,6 be paths

in G starting at e and define paths a p and a-' in G by a 13(t) = a(t)8(t),

a-' (t) = (a(t))-'

(t E [0,1]).

Then G, regarded as the set of equivalence classes of paths starting at e in G, becomes a group, with product and inversion given by

[a]-'

_[a_1].

From the definition of r, it is clear that r is a homomorphism. By considering how the analytic structure on G is induced by that on G, one readily checks that

312

APPENDIX B

G is a Lie group. The group C is called the simply connected covering group of G [SW, Chapter 8]. A useful observation is that the kernel ker it of it is a discrete subgroup of Z(G). (Since it is a local homeomorphism, ker it is discrete. The fact that ker it C Z(G) is a consequence of the fact that for each n E ker7r, the set {xnx-I : x E G} is a connected subset of kerir and so has to be the singleton {n}.J Let G, H be Lie groups and U a neighbourhood of a in G. A map qi: U - H is called a local homomorphism if there exists a neighbourhood V of e in G such that V2 C U and 4>(xy) = 4>(x)gi(y) for all x, y E V. For the following result, see, for example, [HS, Chapter 1, §21.

B7. Let G and H be Lie groups with G simply connected. Let U be a neighbourhood of e in G and 0: U H a continuous local homomorphism. Then there exists a continuous homomorphism ¢': G - H such that 0 and qY coincide on a neighbourhood of e.

B8. The Lie algebra of a Lie group (Kirillov (2, (6.3)); (SW, Chapter 5)). Let C be a Lie group of dimension n and 9 the tangent space at e for the manifold G. For each x E G, define a linear map R2 : Coo (G) -+ C°° (G) by R24>(y) = 4>(xy)

(y E G).

Recall (B2) that A(G) is the Lie algebra of vector fields on G. Let

gL={XEA(G):R2X=XR2for all xEG). The elements of gL are known as the left invariant vector fields of G. It is elementary to check that 9L is a Lie subalgebra of A(G). For A E 9, define P(A) E L(C°°(G)) as follows: ,P(A)O(x) = A(R2¢)

(0 E C' (G), x E G).

(The fact that 4 (A)4> E C°O(G) follows from the fact that, in terms of coordinate functions y1, ... , V. about e, the map (z, y) - R4>(y) is C°O and A is of the form

a;(8/8yj)e with a1 E R.) It is straightforward to check that (I is a linear isomorphism from g onto pL, the being the map that associates X E 9L with the element Xe of g. Defining [A, B) = (1-1(['(A), 4,(B))) for A. B E 9, the space 9 becomes a Lie algebra. inverse-1

Kirillov gives three other ways of defining the Lie product on 9, each useful in the appropriate context. However, the above way is adequate for our purposes. It is straightforward to check that the Lie algebra of the simply connected covering space G of G is canonically isomorphic to g.

B9. The exponential map for Lie groups. A one-parameter subgroup of the Lie group G is an analytic homomorphism a: R -+ G. Thus such a map a satisfies the equalities: a(O) = e, a(t + s) = a(t)a(s) (t, s E 6$). A fundamental result is that for every B E 9, there exists a unique one-parameter subgroup'aB of G such that (aB). (0) = B.

APPENDIX B

313

We can now define the exponential map expG : 9 -+ G by exp(BtB) = aB(1). We sometimes write exp or exp5 for expG when the group G is understood. Here are some basic properties of exp.

(1) exp(tB) = aB(t), so that t - exptB is a one-parameter subgroup of G; (2) exp: g -+ G is analytic;

(3) exp. (0) = I, the identity map on 9, the tangent space at 0 to g being canonically identified with g; (4) there exist neighbourhoods Ua and Uo of 0 and e in g and G respectively such that exp [U., is a bijection onto Uo and the inverse log of exp [U., is analytic;

(5) if A, B E g, then there exists n > 0 such that for all t E (-g, r)), we have (a) exp to exp tB = exp(t(A + B) + tz [A, B] + a3 (t) ), 2 (b) exp to exp tB exp -tA = exp(tB + tz [A, B] + b3 (t)), (c) exp -tA exp -tB exp to exp tB = exp(tz [A, B] + c3 (t) ) 0. where a3(t), b3(t), CO) E 9, and [!a3(t)IE, f[b3(t)II, [[c3(t)[I are 0(t3) as t (Here 11

11 is some norm on the finite-dimensional vector space g.)

1310. Suppose that g is the vector space direct sum ®{gi: 1 < i < k} of subspaces g, and define V : g - G by k

X,

_ (expXi)(expXz) ... (expXk)

(Xi E g). Then V is analytic, and E(0) = I, the identity map on g. (So (V is an analytic diffeomorphism from an open neighbourhood of 0 in 9 onto an open neighbourhood of e in G.)

Bit. Let G, H be Lie groups with Lie algebras g, h respectively. (i) If T : G - H is an analytic homomorphism, then T. (e) : g -+ b is a Lie algebra homomorphism, and the following diagram commutes: T.(e)

h

g

expi

GH

(1)

I -P

T

(ii) If I: g -, f) is a homomorphism, then there exists a neighbourhood Uo of e in G as in (4) of B9 and an analytic local homomorphism S: Uo - H given by the formula: S = expb otl? o (exp5)-i.

Further, S. (e) = 4>.

(iii) Every continuous homomorphism from G into H is analytic.

B12. The adjoint representation. Let G be a Lie group with Lie algebra g. For x E G, let ax be the inner automorphism of G given by ax(y) = xyx-1.

Clearly, ax is analytic. Let Ad x = (a). (e) E GL(g). We sometimes write

APPENDIX B

314

AdG in place of Ad. By considering matrix entries, it is obvious that GL(g)

is a Lie group in a natural way. Since as o at = aab (a, b E G), the chain rule applies to give Ad(ab) = (Ad a)(Ad b). It is readily checked that Ad is an analytic homomorphism. Since each a. is a homomorphism, it follows that AdG C Autg. We now give three useful properties of Ad. (i) a(expX)a 1 = exp(Ad a(X)) (a E G, X E g);

(ii) Ad(exp Y) = e' ' (Y E g); (iii) if G is connected, then kerAd = Z(G).

B13. Haar measure of G. Let G be a connected Lie group. We require the following expression for the modular function © on G [B, Chapter 3, §3.161:

AG(a) = Idet Adal-1

(a E G).

B14. Lie subgroups. Let M, N be analytic manifolds with M C N, and

I: M -+ N the identity map: I(p) = p (p E M). Then M is said to be a submanifold of N if I is analytic, and, for each p E M, the map I. (p) : TMp -+ TNp is one-to-one. One readily shows, using the Inverse Function Theorem, that if M, N have dimensions m, n respectively and M is a submanifold of N, then, given p E M, there exists a local chart V in N with coordinate functions x I , ... , xn for which p E V and xt (p) = 0 (1 < i < n), and such that

U={gEV:xy(q)=0 for m+1<jr'at, e2vribt) E M. Now let G be a Lie group and H a Lie group that algebraically is a subgroup of G. The Lie group H is called a Lie subgroup of G if H is a submanifold of G.

In the above example, M is a Lie subgroup of the Lie group N = T x T. Thus the topology of a Lie subgroup H of a Lie group G need not be the relative topology and H need not be closed in G. The next result relates Lie subgroups of G to the subalgebras of the Lie algebra

gofG. B15. Let G be a Lie group with Lie algebra g. If H is a Lie subgroup of G with Lie algebra h, then the subset

fl` = {X E g: the map t ---* expg(tX) is continuous from R into H} is a subalgebra of g that is canonically isomorphic to ll, and, with h, ll` identified,

we have exph = (exp8)1q. Further, the map H - b is a bijection from the set of connected, Lie subgroups of G onto the set of subalgebras of g.

APPENDIX B

315

B16. Let H be a closed subgroup of a Lie group G. Then there exists a unique analytic manifold structure on H such that H is a Lie subgroup of G; further, the manifold topology on H is the relative topology induced by G.

We note that if G1, G2 are closed subgroups of G with Lie algebras 91, 92 respectively, then Gi fl G2 is also a closed subgroup of G, and, using B15, the Lie algebra of Cl fl G2 is gl ft g2.

B17. More on Lie subgroups. Let G be a Lie group and H a Lie subgroup of G. Let g, b be the Lie algebras of G and H respectively. Then

(i) b={XEg: exptXEH foralltER}; (ii) if G and H are connected, then H is a normal subgroup of G if and only if h is an ideal in g; (iii) if G is simply connected and H is connected and is a normal subgroup of G, then H is closed in G.

B18. Quotient Lie groups. Let G be a Lie group and H a closed, normal subgroup of G. Let g, b be the Lie algebras of G and H respectively. Since He is a Lie subgroup of Ge, it follows from B17(ii) that b is an ideal of g. Let ir: G -+ G/H be the quotient map and m a complementary subspace for in S. Observing that the Lie group topology of H is the relative topology and using B10, we can find an open neighbourhood U of 0 in m such that Qu = r o (exp JU) is a homeomorphism onto an open neighbourhood of the identity in G/H. Then (U, QU') is a chart, and by "translating" this chart by members of G/H, we obtain an analytic structure on G/H for which G/H is a Lie group. Clearly T = expoQU1 is a local cross section for V = Qu(U) in the sense that T : V - G is analytic and r, o T (v) = v (v E V). Further the homomorphism 7r is analytic, and, using B11(i) and B17(i), one readily shows that kerir,(e) = i) and that the Lie algebra of G/H is, canonically, 9/ll. Now let L be a Lie group and 4 : G -. L a continuous (and therefore analytic)

homomorphism. As in the case of iP = it above, the Lie algebra of kerl' is ker (I. (e). By considering the connected Lie subgroup of L whose Lie algebra is

,(e}(g), we see that when G is connected, '(G) is a Lie subgroup of L with '. (e) (g) as Lie algebra, and the canonical group isomorphism from G/ ker 1 onto the Lie subgroup 4'(G) is actually analytic.

B19. Matrix groups. Let F be either R or C and n E P. Let -,,(F) be the algebra of n x n matrices with entries in F and GL(n, F) the group of invertible elements of M,, (F). The determinant, trace, and transpose of a matrix A E M (F) are denoted by det A, tr A, and A' respectively. If F = C. we define

A- =A. The group GL(n, R) is an open subset of M,,(R) (= R'2) and is clearly a Lie group in the relative topology. It is standard (and easy to prove) that the one-parameter subgroups of GL(n, R) are all of the form t etA (A E M (R)) so that the Lie algebra of GL(n, R) as a set is just M (R) with exp A = eA (A E M,, (R)). Using the power series expansion for exp in M.(R), together with

316

APPENDIX B

(5) of B9, we see that the Lie algebra vector space structure on M (R) is the usual one, and that the Lie product on M (R) is given by [A, B] = AB - BA. Similarly, GL(n, C) is a Lie group with M,, (C) as Lie algebra. Every closed subgroup of GL(n, R) (or GL(n, C)) is also a Lie group (B16) and using (i) of B17,

the Lie algebras of the (closed) subgroups discussed below can be determined. For the connectivity properties of these subgroups see [He, Chapter 10, §2]. The group SL(n, R) is defined SL(n, R) = {A E GL(n, R): det A = 1}. The group is connected, and its Lie algebra is sl(n, R), where sl(n,1R) _ {A E Mn (R) : tr A = 0}.

The quotient group SL(2, R)/{-I, I} is denoted by PSL(2, R). Since {-I, I} is discrete, the Lie algebra of PSL(2,R) is the same as that of SL(2,R), viz., sl(2, R). The group PSL(2, R) is discussed in more detail in Chapter 3. The group SL(n. C) is defined in the obvious way. The (compact) orthogonal group O(n,R) (or 0(n)) is defined

O(n, R) = {A E GL(n, R): A' = A-1). The identity component of O(n, R) is SO(n, R), where SO(n, R) = {A E O(n,1R) : det A = 1} = O(n, R) fl SL(n, M).

The Lie algebra of SO(n, R) is so(n, R), where

so(n, R) _ {A E MM(R): A' = -A). The unitary group U(n) is defined

U(n) = {A E GL(n,C): AA" = I = A*A}. The group U(n) is connected. The subgroup SU(n) of U(n) is defined

SU(n) = {A E U(n): det A = 1) = U(n) fl SL(n,C). The Lie algebra of SU(n) is su(n), where

su(n) = {X E M,,(C): X' = -X, trX = 0). Of particular importance for us is SU(2). It is readily checked that the map

(x1,z2,z3,z4) -

X1 + 222

-23 T ix4

X3 -'t' 224

21 - 222

is a homeomorphisul from S3 onto SU(2), so that SU(2) is simply connected. Further, so(3, R) is isomorphic to su(2), so that SU(2) can be identified with the simply connected covering group of SO(3, R). (The canonical homomorphism from SU(2) onto SO(3, R) is given explicitly in [HR.2, (29.35), (29.36)].)

APPENDIX B

317

B20. The Lie group AutG. Let G be a connected Lie group. Suppose, first, that G is also simply connected. If E Aut G, then is an analytic diffeomorphism on G and 4).(e) E Aut g. Conversely, if T E Aut g, then, using B7, (ii) of BIi, and the invertibility of T, we can find a unique 4> E Ant G with 1. (e) = T. Using the chain rule, the map 49, -+ 4. (e) is an isomorphism from Aut G onto Ant g, where Ant G is given its canonical topology and Aut g is given

the relative topology it inherits as a subset of GL(g). Since Autg is a closed subgroup of GL(g), it is a Lie group, and so the above isomorphism enables us to impose a Lie group structure on the topological group Ant G. Now suppose that G is not necessarily simply connected, and let (G, v) be the simply connected covering group of G. Each 4> E Aut G induces a continuous local homomorphism between neighbourhoods of e in G, and since G is simply connected, this local homomorphism extends to a (unique) element of Aut G. The map ' ' is an isomorphism of Ant G onto the closed subgroup K = {41 E AutG: %P (kerar) = kerar},

of the Lie group Aut G, and Aut G inherits the Lie group structure of K.

Since the Lie algebra of Autg is Derg (B5) and AutG is isomorphic to a Lie subgroup of Aut G - Aut 9, it follows that the Lie algebra of Ant G can be identified with a subalgebra of Der g, and can be equated with Derg if G is simply connected.

B21. Connected abelian Lie groups. Let G be a connected, abelian Lie group. Then G is of the form W x V. [The Lie algebra of G is abelian (B9) and so is the Lie algebra of the abelian Lie group L for some n. Thus Rn is the simply connected covering group of G, so that G is a quotient R"/D with D a discrete subgroup of R. The group D is isomorphic to some V, giving the desired result.)

If G = T", then the covering projection p: Rn

T" can be taken to be the

obvious one: p(x1:...:xn) _ (e

27r%x1

...,e 21rix

It follows from B20 that Aut T" can be identified with the group

{T E GL(n,R): T(Z") = Zn},

which is readily shown to be the group of n x n, integer-valued matrices of determinant ±1 [HR1, (26.18)]. In particular, Aut Tn is discrete.

B22. Lie semidirect products. Let H x p K be a semidirect product of connected Lie groups H, K. Since Aut H is a Lie group and p: K --+ Aut H is continuous, the map p is actually analytic, and it follows that the map (x, y) p(y)x from H x K into H is analytic. Further, G= H x p K is a Lie group when given the product manifold stucture. Let Il, t, g be the Lie algebras of H, K, H xp K respectively. Then b is an ideal and t a subalgebra of g, and using B17, 4 n t = {0}. From dimensional considerations, 9 = l 9 t. Thus (B5) 9 is a semidirect product b x, t.

318

APPENDIX B

It remains to specify t: t - DerF in terms of p. Recalling that p(y)(x) _ yxy-1 = ay(x), we have, for Y E t, using B12,

(p(exptY)),(e) = (apiy),(e)]h = Ad(exptY)[q = etady], so that p.(e)(Y) _ (adY)[q = b(Y). The following gives a partial converse to the above result. A related result is given in B44. Let 0 be a simply connected Lie group with Lie algebra 9. Suppose that 9 is a semidirect product h xO t, and let H, K be the connected Lie subgroups of G with Lie algebras ti, 1 respectively. Then H, K are simply connected, closed subgroups

of G, and G is a semidirect product H x P K.

B23. Nilpotent and solvable Lie groups and Lie algebras. A Lie group G is nilpotent [solvable] if it is nilpotent [solvable] in the sense of Appendix A. Let G be a nilpotent Lie group with lower central series {C'G}. Clearly

[G, (CIG)-] C (CI+1G)- so that {(CIG)-} is a central series of closed subgroups of G. Similarly, if H is a solvable Lie group and {DiH) is the derived series for H, then {(DIH)-} is a normal series with (D;H)- normal in H and (D;H)-/(Di+1H)- abelian. Now let g be a Lie algebra. The lower central series {Fig} is the decreasing sequence of ideals in g defined inductively as follows: F'1g = 9, Fi+19 = [Fig, 9]

The derived series {gig} of 9 is the decreasing sequence of ideals in g defined inductively as follows: Q19 = 9, 2i+1S = [Oi9,_?i9]. Each Fig, ?iig is characteristic in the sense of being invariant for every automorphism of g. The Lie algebra g is said to be nilpotent (solvable] if, for some p, "'g = {0} [Op9 = {0}]. Since 2'i9 C F'9, every nilpotent Lie algebra is solvable. The class of a nilpotent Lie algebra 9 is the smallest integer p > 0 such that Fp+19 = {0}. An important source for nilpotent and solvable Lie groups and Lie algebras is the space A(n) of upper triangular matrices in M, (R). For example, let

G1={AE©(n):forsome k 0,Aii=k(1:5 G2= {AEA(n): Ali 00 (1 1, we have, for t E G, t2"m+

t-2nm = 2m, and combining with ten-lm + t-2"Tlm = 2m gives ten+lm = ten' 1m + 2(tm - m) whence t2" rn = 2n(tm - m). Now use the boundedness of m to infer tm = m. (Such means arise in connection with positive linear operators on C"-algebras (G. Robertson).)

0-8 (Milnes (S11). A(A,) = fn , y-2 dy f"ny dx = 4n log n. Let u = (b, a) E S2. Deal with different cases of u depending on how uAn intersects An. For example, if 0 < a < 1 and 0 < b < 2a, then for large enough n, uAn nAn is the trapezium, with vertices (b - 1, n'1), (1, n '), (an2, an), and (b - ant, an), so that .1(uAn n An) = [2nlogy+ b/y]nT` A(uAn nAn)/A(An) - 1, and Problem 0-6 applies.

0-9 (Mitchell [2j). Choose ga so that 96 -y m E £(G). (See (2.13) for the converse.)

0-10. If m(Cc(G)) 0 {0}, show that m(G) = oo i4 1! 0-11. J6(X)(X(z) - 1)

0 for each X E r. If X 0 1, then X(x) ,-f 1 for some

xEG. 0-12. Use the uniqueness of normalised Haar measure on G. 413

414

SKETCHED SOLUTIONS TO PROBLEMS

If 4b E AC, then 0-13 (Lorentz [1j). Let Or(n) = ¢(r + n). [Jp I Fp1 Or - l111a --y 0. Now apply m E r(G). Examples of elements of AC are z" (z E C, (z( 5 1), all almost periodic functions. If B = {k2: k E P}, then A = {{an} : a" = 0 if n B, a9, E {0, 1} if n E B} is a nonseparable subset of AC.

0-15. Construct {x" } in G with x"E f'. x.,,,E = 0 if n 0 m.

0-16. If D E [HRI, (7.9)].

(G), then there exist N1. N2 such that C C D", D C C242

0-18. Let G be abelian and F = {a1, ... , ak} C G. A typical element of F' is of the form a}' ak", r, > 0, Zk 1 r4 = n. There are p(n) = ("'n-1) ways of partitioning n into a sum of k nonnegative integers-see (6.16); Problem 6-4. For the free abelian group on two generators, JE'1 = (s + i)2 (Milnor [1]). 0-19. Let C = [-2,2] x [ 2 , 2] e ),(C") > 4[2" Use Problem 0-16.

Show C" D 1-2,21 x 12-",21 so that

References for many of the results below are Day [4] , [9].

0-21. G is an upwards directed union of finite (and so amenable) groups. 0-22. The group is locally finite. 0-23. An extension of an amenable group by an amenable group is amenable. For the solvable case, use the derived series (Appendix A).

0-24 (Dunkl-Ramirez [7]). Let H be a finite group containing every Gi as a subgroup. Show that every finite subset of G generates a subgroup of some H' (m E P). Now apply Problem 0-21 (cf. Problem 3-1). 0-25. G2 is a semidirect product R2 x o O(2, R), D,G2 C R2 x Di(O(2, 68) and [O(2, V8), O(2, 6R)] C SO(2,R) is abelian. So G2 is solvable, and G1 is a subgroup of G2. 0-26. Use Zorn's Lemma and (0.16).

0-27. Consider G/N, where N is as in Problem 0-26.

0-29. No-S2 is amenable and yet contains FS2 (cf. (6.38)) (Hochster [1], Appel and Djorup [1], Milnes [Si]).

0-31. Try a right zero semigroup (ef = f for all e, f ).

0-33. X ` 1 =SifxE1. 0-34 (Sz. Nagy [1), Dixmier 11]). This is the simplest version of the "similarity problem'', and the proof is close to the corresponding proof for compact groups

[HR1, (22-23)]. Let m E M(G), and define (e,77) = m(x - (r(x) ,r(x)rl)). Then (,) is "equivalent" to (,) in the sense that for some invertible, (, positive A E B(fj), we have (,i7) = (Ae,Ari). Take r,'(x) = Air(x)A-1. See


415

Bunce (4], Christensen (1) and Problem 1-40 for the C`-version of the similarity problem.

Problems 1 1-1. E.g. Ou(v) = 0(u * v) = ff q5(xy) dp(x) dv(y) = f dv(y) f O(xy) du(x) _ f 4u(y) dv(y). 1-2.

If 0 E CC(G), then there exist C, D E '(G), C C D°, 4, vanishing

e in G, then xS 1C C D eventually and compactness outside C in G. If xa gives eventually Ii6x6 - 011 = sup{j(4,x6 - 4,)(y)(: y E D} 0. So C0(G) _ C,(G)- C U,(G). 1-3 (Kister [ 1], Itzkowitz (1)). Suppose that G is neither discrete nor compact.

Let C E 9,(G), and let be a sequence in G with Cx,, fl Cx,,, = 0 if n#m. Find W,(G)such that Vi=Vi (n > 1), 0. Let ¢,, E C(G) be such that 0 < 1, (G --

{0}. Then

n 1 4,n E C(G) ^- U,(G).

1-4. Any m E CL (G) is determined by its values on C(G), and if G is compact,

then £(C(G)) = {a} by uniqueness of Haar measure. 1-5. Use (1.13).

1-7. Adapt (0.16(4)) and use (1.12).

1-8 (Chou (41). If R is a transversal for the left H-cosets in G and R4(rh) _ ¢(h) (r E R, h E H), then j3(p)(R4,) = p(¢), and 8 is one-to-one. The map -Y is defined in the same way as 0. Then j.C(C(H))i < j,C(C(G))l, and when the uniform structures are equivalent, j,C(U(H))l S j.C(U(G))(.

1-10. Suppose that G is not compact and that u is a countably additive left invariant mean (c.a.l.i.m.) on 9(G). If C ES' (G), then u(C) = 0 (cf. Problem 0-15). Since u(G) = 1, G cannot be a-compact. Construct a sequence {G,.} of open, compactly generated subgroups of G with G C G,,. H admits a c.a.l.i.m., then since it is a-compact and noncompact, we have a contradiction. We can obtain a c.a.l.i.m. v on H by setting v(E) = ,f AGE du, where E E 3(H), 4,E(x) = fE J3(x-1 h)d)H(h), and j3 is a Bruhat function for H ((1.11)) c.f. the proof of (1.12). This gives a contradiction. 1-11. Apply Day's Fixed-Point Theorem to the natural action of Son PM(X), the (weak' compact, convex) set of probability measures on X. 1-12. For (i) consider the natural left action of G on 931(G). .C(C(G)) 0 0 since C(G) C l(G).

For (ii),

1-13. For m E °(B), apply Day's Fixed-Point Theorem to {m' E'J1I(G): m'IB = m}.

416


1-14. Apply the R.H. version of Day's Theorem to the set of left fixed-points in K. 1-15. Silverman [1)-[4); cf. Problem 1-13.

1-16. The "Reel product" theory of the kernel (M. Rosenblatt [1, Chapter 5)) suggests the following class of compact semigroups. Let Y be a compact Hausdorff space, G be a compact group, and cp : Y --t G be continuous. Let T = G X Y with multiplication (g, y)(g', y') = (90(y)g', y'). Now apply (1.17). 1-17 (Mitchell [21).

A C N is (left) thick

.

A = Un° 1[an;bn]. where

supn(bn - an) = 001-18. Use (1.21) and Problem 0-15.

1-19. Suppose that L is a left amenable, left ideal of S. Using Day's FixedPoint Theorem, find m E 9Jl(S) with lm = m (1 E L). Then m E C(S). 1-20 (Klawe [2)). T is a homomorphic image of S and so T is left amenable. For the case of U, let K = {Mo E 9R(U): mo((k) = mo((r¢uo)p(to)) for all 0 E lo,(U), uo E U, to E T}. If m E Z(S), then the functional c¢ -+ m((u,t) -+ Ou)) is in K. A fixed-point for the left action t --T p(t)' of T on K gives an element of 2(u). 1-21. Day [4]. 1-22. Dixmier [1).

1-23 (Trey [1)). Using (1.27) and (1.28), a subsemigroup T of S is left amenable rs T is left reversible. If T is not left amenable, find s, t E T with sT fl tT = 0. Then s, t generate FS2 in T. Conversely use Problem 0-28.

1-24 (Day [9)). We can suppose that S generates G. Show that G = SS-1, and deduce that S is left thick in G and so is left amenable (Problem (1-21)). 1-25. (a) and (b), (i), (ii) are standard: e.g., see Howie [1). Use (ii) to prove (iii) and (iv) of (b). (c), due to Duncan and Namioka [1), follows from (b) and (1.25).

1-26 (Day [41). The functional mn E 3(G) if m E .C(G), n E 91(G). 1-27 (Day [9)). Modify the last paragraph of (1.21). If C E F(G), C not null. Ac = Xc/A(C), and A (C - Et-1) > e for all t E G, then )[XEµc[[oo
M(G)', k: M(G)' L1(G)' = L.(G), where j(¢)(µ) = f sbdp. (µ E M(G)) and k restricts elements of M(G)' to L1(G)'. Then j*(C(M(G)')) c Z(C(G)), k*(Z(G)) c £(M(G)').

[Note: The preceding suggests how amenability can be defined for a separately

continuous, locally compact, Hausdorff semigroup S. We define S to be left amenable if there exists a left S-invariant state on M(S)' (cf. Namioka [3), Jenkins [3), Lau 161). Another approach to amenability for S, based on an analogue of L1(G) studied by Baker and Baker [1), [2), [3) and Sleijpen [1), [2), [3) is developed by Paterson [1). See also Wong [10) and Kinzl [1). Versions of topological left thickness for Borel subsets of S are developed in Day [9), [11), [13], Junghenn (3), Kharaghani [2), and Wong [2), [4). [6), [7), [9), and [12). This depends on M(S) being a Banach algebra under convolution, a substantial fact following from the work of Glicksberg (1) and Johnson [1]. (See also Moran [1) and Wong [11].)) See Johnson [2] for Problems 31-34 below.

1-31. Every Banach B-module X is a Banacb A-module in the natural way, and if D: B - X' is a derivation, then D o 4 is a derivation on A.

1-32. Suppose that A is amenable. Then AJJ is amenable by Problem 131. To prove that J is amenable, we need only consider a neo-unital Banach

J-module X and a derivation D: J -> X' ((1.30(ii))). Then X is a Banach A(J)-module, D extends to a A(J)-derivation, and we can use the canonical homomorphism from A into A(J) together with the amenability of A to deduce that D is inner.

1-33. A&B has a bounded approximate identity (using (1.30(i))). To show that a derivation D: A&B X' (X neo-unital) is inner, extend D to derivation, also denoted D, on A(A6B) D (A ® 1) U (1 (9 B). By subtracting an inner derivation from D, we can suppose that D(1 ® B) = {0}. Using the fact that

(a (9 1)(1 (9 b) = a ® b = (1 (9 b)(1 ® a), we have D(A (9 1) C Yi, where E X, b E B). Now subtract (using the amenability of Y= A) an inner derivation from D so that we can take D((A ®1) U (19) B)) = {0). Then D = 0. What about the converse? 1-35 (Bunce [2), Lau [11)). Let h' E X' be any extension of h. The derivation (ah' - h'a) is Y1-valued and so is of the form a -+ (ah" - h"a) (a E A) for h" E Y-. Take k = h' - h". [Note: The converse of the result is also true.] a

1-36 (Johnson [3]). (i) Suppose that N is a virtual diagonal for A. Then there exists a bounded net {a,} in A®A, with each a, of the form Z 3 a; ®bq. such that &, -} N weak*. Since r`* (so ((a(9 b)')) = ab and r" is weak* continuous,

418


the second virtual diagonal condition gives (f,a-a)^ 0, (af,-a)^ -.0 weak* A routine argument (c.f. (0.8)) gives a bounded in A", where f, = approximate identity {e6} for A. (ii) Let N, a f e6 be as above, Each e6 is a convex combination of f,'s. Let 86 be the corresponding combination of a,'s. We can suppose that f3 M weak* in (A®A)". Now check that M is a virtual diagonal for A. Using the neo-unital condition, D(e6)(e) - 0 for all e E X. Since f36 (a®b D(ab)()) = D(e6)(e) and f36 M weak`, we have M(a®b D(ab)(e)) = 0. (iii) Let {e6} be a bounded approximate identity for A. We can suppose that (e6 ® e,5)^ -> n weak* in (A®A)". The map a -> (an - na) is a derivation from A into the dual module ker 7r*", and since A is amenable, there

exists m E kerf such that an-na = am-ma (a E A). Then (n-m) is a virtual diagonal for A. G Let X be a Banach A-module and D: A X' be a derivation. By (i), we can suppose that X is neo-unital. Let M be a virtual diagonal for A as in (ii). The virtual diagonal conditions translate to: (a) for all g E A', ao E A, g(aoab)), and (b) for all we have M(a (9 b -> g(abao)) = g(ao) = M(a 0 b h e (A' A)", ao E A, we have M(a®b h(a®bao)) = M(a®b h(aoa(&b). Let ao E A, C E X. Applying (b) with h(a(&b) = (Da)b(ie) and then (a) with g(a) =

Da(e) gives M(a 0 b -, (Da)bao(C)) = M(a ® b - (D(aoab) - aoaDb)(g)) = Dao(la)-M{a®b - (D(ab) - (Da)b(eao)). Now use (ii) (with Sao in place of g)

to obtain Dao = aoa - aao where a(e) = M(a 0 b -+ -(Da)b(l;)). [Note: The 0 for all a E A. Such a net is called by Johnson an approximate diagonal and is an analogue of the "Reiter type" net of (4.1). Virtual diagonals are analogous to invariant means, net {f36} of (i) can also be arranged so that 1Jaf36 -,66aH;

while virtual diagonals arising as weak* cluster points of approximate diagonals are analogous to topologically invariant means.]

1-37 (Lau and Paterson [S3j). The case H = {e} is (1.30(iv)). For general H, modify that proof. On needs: G/H amenable t* there exists a left invariant mean on the space Ur(G/H) of functions in C(G/H) which are right uniformly continuous under the natural G-action. To this end. modify (1.7). (See Eymard [2)).

1-38 (Johnson [21, Bunce [3], cf. (2.35)). Suppose that A = CI (G) is amenable (G discrete). Let Y = A, X = B(12(G)), and h E A' be the tracial state: h(T) = (Tbe,6e). By Problem 1-35, there exists an extension k E B(12(G))' of h with ak = ka for all a E A. Then 0 k(L,), where L,, is the multiplication operator associated with d' E lw(G), is a nonzero, right invariant functional, and G is amenable.

1-39 (Bunce [2], Rosenberg [21). Let S(A) be the state space of A and g E S(A). 'Applying strong amenability to the derivation D: A A' given by D(a) = ag - ga, we can find h in the weak" closure of co{-D(u)u' : u E U(A)} such that D(a) = ah - ha (a E A), and so as7 = rla, where rl = (g - h). Now if u E U(A), then g - (-(Du)u") = ugu" e S(A), so that n E S(A). So sl is a tracial state.


419

Note: Rosenberg proves that t, (n >- 2) is not strongly amenable by showing that it does not admit a tracial state. 1-40 (Bunce [1], 121). B(A) is a dual Banach A-module with operations aT =

r(a)T, Ta = Tr(a')' (cf. (2.35)). If D: A -> B(S) is the derivation given by D(a) = (r(a)-r(a')'), then we can find To in the weak' closure of co{-D(u)u' :

u E U(A)} with D(a) = r(a)To - Tor(a')* (a E A). Note that D(u)u' = ,r(u)ir(u)` - I, so that R = (I - To) is in the weak* closure of co{r(u)r(u)' : u e U(A)}. Now show that for all u E U(A), ,r(u)T(u)*, and hence R, is > []r11 `1, so that R is positive and invertible. Since 7r(a) - r(a')' = ,r(a)To - To7r(a')', we have ,r(a)R = Rr(a' )'. Then R112 implements the similarity between r and a '-representation of A. 1-41 (Green [S], Lau and Paterson [S3], Paterson (Si]). Let X be a neo-unital Banach C* (G)-module and D: C' (G) X' a derivation. Then D extends to A(C' (G)). There is a canonical '-homomorphism from C'(H) onto a C'-algebra A C ©(C' (G)). If H E.V, then DI A is inner. So regarding D as a G-derivation, DJs is inner. So D is inner by Problem 1-37.

1-42 (Lau and Paterson [S3], Paterson [Si]). Suppose that H E V. Let B be a C'-algebra. We have to show that Amax = C' (H) ®max B equals Amin = C' (H) ®min B. Realise Amax on the Hilbert space S of its universal representation, and Amin on a Hilbert space tensor product A, ®A2. Associated with these realisations are unitary representations r, r' of H on S and Sti ®.2 respectively, and each gives a faithful representation of C' (H). The representa-

tions r, r' induce representations (1, 4' of G, and C = 4'(C'(G)) = 4''(C'(G)) (Fell [3]). Note that fi is realised on L2(G\H,S) and is given by (4'(x)f)(u) = r(s(u)xs(ux)-I) f (uz) (x E G, u E G\H), where s: G\H G is a Bore] crosssection for the quotient map (Kirillov [2)). Now C is nuclear (since G E say ), and

one uses this to show that the two C'-algebras generated by 4'(C'(G))(1(D B), 4`(C' (G))(1 ® B) are isometrically *-isomorphic. The isomorphism extends to their multiplier algebras, and using the formulae for fi and 4' and the normality of H, we see that Am,,, Amin are canonically embedded in the multiplier algebras and are isomorphic.

Problems 2 2-1. Use (2.1), (2.2), and the fact that D(G) p E e(G)}.

E L. (G): p(O) = 0 for all

2-2 (Emerson [7], J. C. S. Wong [1]). Suppose that for all 0 E the set BR of real-valued functions in B, we have infXEG ('p - rnv) < 0 (µ, v E P(G)). For 0 E BR, let B , be the subspace of BR spanned by functions tµ

(p E P(G)) and 1. Show that if 0= E B, (1 0.


420

Deduce that supxEC F,", (¢,µi - Oivi)(x) > 0 and apply versions of (2.3) and (2.13) to obtain .t(B) 0. 2-4 (Emerson [8]). Suppose that the "ess inf" condition holds. Show that if

z(i =4 i(XZt , - Xy-FE.), then ess infxEG i(x) < 0. Show that the same holds if 0 is replaced by deduce that G is amenable.

i ri(Xx-' E, -

E) (r; E Q) and apply (2.3) to

2-5 (Emerson 17)). Adapt the proof of Problem 2-4, and use Problem 2-2 above.

2-6. For the second part, the two interpretations of F f coincide on S and hence on li (S). For the third part, take S to be the unit ball of A and B = {f is: f E A'}. 2-7. Take m = 2 (n + n*) where n is an invariant mean on G (Problem 1-26). 2-9. Show first that the norm and pointwise topologies coincide on the norm closure of OS (0 E AP(S)). 2-10 (Jenkins [3), Lau 161). For the first part, follow the proof of (1.27) using

U(G), U(S) in place of l.(G), 1,,O(S) to obtain mo E C(U(G)). If µ E P(G), S(p) C S, then the map d' - mo(pd p) is in .t(G) ((1.7)), and is 1 on S. For the second part, use the first part and adapt the proof of (1.28), using (2.23) in place of Day's Fixed-Point Theorem.

2-11. Use (2.15) to obtain an S-fixed-point in the weak* closure of coSyo. This weak' closure is (c90)" Let sf be the set of separable, unital right invariant 2-14 (Furstenberg 13)). C"-subalgebras of Ur(C). Since {tax: x E C) is norm separable for 0 E Ur(C), Ur(G) is the (upwards directed) union of sf. Every A Ed is of the form C(X), where X is a compact metric left G-space. So £(A) P. This gives G amenable (cf. (0.16(4))).

2-15 (Milnes [I], Kharaghani [1)). Let ' E B and, for a > O. S,, = { f E B': 11 ffl < a) with the weak*-topology. The map (t, f) -+ f ea(t) is separately continuous on the locally compact space S x Se, and by using a well-known result of Glicksberg [1), if u E PM(S), then the linear functional F,, where (1)

F,(f) =

1

f6(t) dp(t),

(f E B')

is continuous in the bounded B-topology and so weak' continuous [DS, V.5.61. So F,, = v for some tk E B. Now show that ib = Oµ, so that B is right invariant for PM(S). Taking f E £(B) and d'p for 0 in (1), we obtain f (tap) = f (ta). 2-16 (Lau [5)). (i) Show that the map (p,m.) -+ pm is jointly continuous on Ke = 9Tt(AP(S)) (cf. (2.36(ii))). It follows that the set of maps m pm (p E K,) is equicontinuous.


421

(ii) G Let K E fir, ¢ E Af(K). We claim (OS is equicontinuous. Indeed, let W be the uniformity on K, k E K and e > 0. Let W = {(k1i k2) E K x K_ : Ib(k1) - m(k2)I < e}. By equicontinuity, there exists an open neighbourhood U of k such that (sk, su) E

W whenever s E S, u E U. So

sup, ,,,,u I#(su) - 0(sk)f < E. So ¢S is

equicontinuous in C(K). Now use the Arzela-Ascoli Theorem [DS, (IV.6.7)] to give ¢S conditionally compact in C(K). So (with ko as in (2.22)), ko¢ E AP(S). 2-17. This fixed-point theorem will be used in proving the Hunt-Stein Theorem in (4.25). Proof of (b): let F E A f(K), p E TZ(A f(K)). It is sufficient to show that pF E L,o(G). For some 0 E K, we have pF(x) = F(x4'). Prove that pF E L,. (G) first when F is of the form Ff + al, where a E C, f E L1(Z) and Ff = f [K. For the general case, the set of such functions Ff + a1 is norm dense in Af(K) (Phelps [2, p. 31)). 2-18 (Takahashi [1] , De Marr [11). (i) Find xo, x1 E M with Ilxo - x111 = a = diam M. Let Mo D {xo, x1 } (Mo C M) be maximal for the property: ]]x-yUI = 0 or a if x, y E Mo. Show Mo = {xo, x1, ... , xm} for some m E P (x, distinct) and take u = (m+ 1)-1 Eq"_o x4.

(ii) Let X1 be a minimal, nonempty, S-invariant, compact convex subset of K and M a minimal, nonempty, S-invariant compact subset of X. Show that

sM = M for all s E S. If IMI > 1, then there exists u in coM such that sup,mEM IIu - mll = p < diam M. Then X o = {y E X1 : supem IIy - m1(< p} contradicts the minimality of X1. So IMI = 1.

2-19 (Lau (5]). Let Z be the a.p. compactification of S so that AP(S)^ _ C(Z). Considering the canonical homomorphism from S into Z, the left reversibility of S implies the same for Z. Now apply (1.17).

2-20. (i) Take P to be the projection of B ® C onto the first component. P is continuous by the Closed Graph Theorem or Banach's Isomorphism Theorem.

(ii) (c) The subgroup N is closed (using the "ii * x" version of (1.2)). Let

A = {O E L.,(G) : x0 = ' for all x E N} D B. Then A is a right invariant von Neumann subalgebra of L.(G). Suppose that B A. Then we can find p E L1(G) of compact support K such that ]]41811 < z, I(. Ir > 2. Let B0 = {0lK : (p E C(G) ; t B}, Ao

E C(G) ft A}. Using (a), {IFcis Ii =

IIAleo II, I141a II = IIAiao II. Derive a contradiction by showing that Bo, Ao have the

same norm closures in C(K). To this end, if k1, k2 E K and 4(k1) _ (k2) for all ' E Bo then, since Bo is right invariant, k14(x) = k20(x) (x E G, 0 E Bo); then k2 1k1 E N and k10 = k20 (, E AO). The Stone-Weierstrass Theorem then gives that the norm closures of Bo, Ao are the same. We can suppose B # {0}. Let N be as in (ii). Let X = L1(G), S = N (iii)

and apply (2.15) to obtain a projection P: L. (G) - 3z (X) = B.

Consider B = C1. [References: Lau [14), Crombez and Govaerts [1], Pathak and Shapiro [1], Rosenthal (1], Takesaki and Tatsuuma [11.]

622


2-21 (Milnes [1), Ruppert (S11). Suppose that p E OZ ^- Z and the map q -- pq p is continuous on 13Z. We can suppose that there is a net nb -+ oo in Z with nb

in 07, and that -nb

po for some Po E OZ. Then p(-nb)(X(-..-I)) = 0 for

limbpo(X(_,-llnb) = 1, giving a contradiction. all 6, while (Ruppert has a much more general theorem. See also Lau [S3j.) 2-22 (Granirer [13)). . The unit ball of 3,(S) is weakly compact. Apply the Dunford-Pettis Property f IDS, VI.8.13) to the identity operator on the L-space .31(G) to obtain 3, (G) finite-dimensional.

2-23 (von Neumann [1], cf. [HR1, (17.22)]). Adapt the proof of (2.32). 2-24 (Ellis [1], [2)). By (2.33(ii)), the multiplication in H is jointly continuous. For inversion, show first that if xb - x in H and {x61 } is contained in a compact set, then x6 1 --+ x-1 So C-1 is closed if C E KP(H). If H is separable and V E F, (H), show by the Baire Category Theorem that (V-1)0 0. Show then

that C-1 E ;,'(H) if C E S'(H). Deduce that inversion is continuous in this case. The general case can be reduced to this case. 2-25 (van Rooij 11]). (a) Use Zorn's Lemma and (2.34(iii)).

(b) Let n E Z(B,,, ), and let C be the set of functions ¢ E 1,(P, X) such that ¢ - T¢ E B,,,, n(¢ - T¢) = 0. Show (using [[I - T[J < 1 and (2.34(iii))) ¢(r)) - kn(¢) (k E P). Then that C = B. For ¢ E B. let ¢;,(k)

¢;, - T¢n = -T¢ + n(¢)1, and ¢n E B, If p E £(B,), th E Bn, show ¢n - ¢'r, = (p(¢) - n(o))j E B,,,, implying n = p (using (2.34(iii))). 2-26. (ii) Let T E B(Sj) and 0 be as in (2.35(C)). Show that U T8(U) is in Ur(H). (We can take 8(R) = (Re, n) (R E B(b)) for some %,n ES5.)

2-27 (de la Harpe [21). Let M = Un°_1 Mn. Since £(U(MM) ,'£ 0, we have

.(U,(U(M))) ? 0 (cf. (0.16(4))). Now show that U(M) is strongly dense in U(A) = H so that Z(U,(H)) 74 0. So £(X(H)) - 0 (preceding problem) and A has Property P. (Assuming the result that Property P= AFD and using (2.35(C)(ii)), we have a remarkable result due to de la Harpe: A has Property P .

L(U,(H)) # 0.) If m E .C(C(H)) define t: B()) --+ B(55) by = m.(U -+ (U°TUe, ii))

(Note that U -+ (U"TUe, ri) is of the form U - T8(U') and E C(H).) Then 4? is continuous, takes I into 1, and vanishes on commutators. But I is a sum of commutators.

2-28 (Tomiyama [21). If A is ahelian, then H is amenable as a discrete group

and (2.35(C)) applies. Now suppose that A is of Type 1. Realise A so that A` is abelian. So A' has Property P, implying that A also has Property P (cf. (2.35(F)(d))).


423

2-29. Let G be discrete and of Type 1. By a result of Schwartz ((2.35 (G), (H))) and Problem 2-28, G is amenable. (Thoma [1] gives much more: G is an abelian extension of a finite group.) 2-30 (Zimmer 161). For each n E P there exists a finite subset F. of E2 such that every point of Ei is of distance < 1/n from some point of F for a metric d on E; giving the relative weak* topology. For each Q E Fn let

Wg = X x {a E Ei : d(a,,6) < 1/n}. Let p: Wp f1A --; X be the projection map. Then Yp = p(Ws fl A) is analytic, and by the von Neumann Selection Theorem (Auslander and Moore 11, Proposition 2.15]: Rogers et al. [S]), there is a Borel subset Xp of Yp with ic(Yp - Xp) = 0 and a Bore] cross-section r' for p on X0. Now piece together the rfl's to find the desired Borel map b with (z., b,, (x)) in

some WpflAa.e. (0EFn). 2-31 (cf. Lance [S]). Using the nuclearity of A, we obtain a representation r of A ®min Al on 55 given by ir(a (9 b) = ab. There exists a representation r` of A (&min B(5)) on a Hilbert space K D Sj such that ir'(a 0 b)(r, = 7r(a (& b) for all a E A, b E A`. (To show this, we can assume that r is cyclic and so comes via the GNS construction from a state on A ®min K. Extend this state to one on A ®min B(.) and consider the resulting GNS construction.) If P is (Pv'(1 0T))(r, is the orthogonal projection from K onto S5, then the map T a norm one projection from B(S)) onto A`. For the last part, realise A through its universal representation and use AK injective . A" (= A") is injective. 2-32. (i) (Effros [1]). Adapt (0.6). (ii) (Paschke (2])

Let m be a nontrivial inner invariant mean on G. Find

{fb)inP(G)such that f6- m,[Jx*f6*x-1-fs11i-0 forallxEG. By subtracting f6(e)b, from f and scaling, we can suppose that f6 (e) = 0 for all

b. By considering (((P, -T)e((2 for T E Cr(G) and e = fb12,e, show that d(Pe,Cr(G)) > 1/2. G For all T E C, (G), we have PeT = TPe = (Te,e)Pe. The functional 0 on Cr(G) + CPe given by a(T + -tPP) = (Te, e) (-Y E C) is a state. Extend a to a state 6 on B(I2(G)). Since 6(r(x)) = 1 = ((r(x)((, we have 8(,r(x)W7, (x)`) =,9(W) for all W E B(12(G)). (To see this. use the GNS construction for p.) Noting that r(x)L0r(x)-1 = Lx&x-, (0 E 1.(G)) and Lxr, = Pe, the map 0 - /3(L,) is a nontrivial inner invariant mean on I.,, (G). 2-33 (Effros 11]). The operators in VN(G) can be identified with 12(G) functions by the map T Te. and the 1,1,(G)-norm corresponds to the [, 112-norm in VN(G). Let xl,... , x,, E G. Apply the Property 1: definition with Tj = r2(xj).

One checks (using rr(G) C VN(G)`) that if g = Use, then g(e) = 0 and ((r2(xj)g - rr(x? 1)g112 < C. Show that h = (g(2 E P(G) and use the h's to produce an inner invariant mean be in the obvious way. (Note: The converse to this result is important, and seems to be open. The main use of Property r and its refinements is to distinguish between III factors ([DI), [Pt.3, Ch. 7, 7).)


424

2-34 (Lau and Paterson [S2]). If m is an inner invariant mean on L.(G), m(4.)1 belongs to A. Conversely, suppose that T then the operator p is a nonzero, compact operator in Ate. Then [Sch, Chapter IV, §1j T has a (compact) modulus [Ti, where for 0 > 0, [T)(0) = sup{[T'[: yi' E L.(G), 101 < m}. Then )T[ E A' and f = [T[(1) is positive, ; 0, and inner invariant. Lets > 0 be such that X = f([e,oo)) is not locally null, and let P: LA(X) be the restriction map. Show that L,.(X) is a right Banach G-module E Lw (G), = P()T)(0)) E A, the space of functions ' E C(X) which are almost periodic for this action. Then 0

under conjugation action, and for all

is inner invariant on G, where n is a G-invariant mean on A. To show

that such an n exists, let r be the (locally convex) topology on A' determined by seminorms p o (0 E A) where po(a) = SUPZEG la(ox)l. Then Tt(A) is weakly compact for r (by the Mackey-Arens Theorem (Kothe [1), §21, 4)) and the RyllNardzewski Theorem ((2.36)) yields a G-fixed element n E TZ(A). 2-35. (i) cf. (1.20).

Let G = {-to, ... , 7n_I } be a finite left ideal group in QS (ii) (Lau 12)) with ryo as identity. Let A; _ {s E S: s'yo = - }. Then S = UL 1 Ai (disjoint),

A{Aj C Ak ('yi-tj = -Ik) and Ao is ELA. Let si E Ai and W = {so.sn_I}. Show that if F E.°(S), then there exists to E Ao such that F(Wto) = Wto. (iii) Sorenson [2), Granirer [13), Lau [2).

(iv) Try S = G x T, G a finite group, T ELA. 2-36 (Mitchell [3), Granirer [11), Ljapin 11)).

2-37 (Sorenson [2), Granirer [10), Lau [2)). Show that PM(K) = C(S)^, where K = {p E 6S: Sp = {p}}. Clearly, PM(K) C .0(S)^. Conversely, if a E S, then R,, = {t E S: at = t} is a right ideal of S and so if m E Z(S), m(R9) = 1, and son) C n,,3 F, = K. 2-38 (Wilde and Witz [1)).

2-39 (Ky Fan [S)). Let 3n be the set of n-dimensional linear subspaces of X contained in Y. For each Z E Vn, show that (C + H) 1 Z is a singleton {PZ(e)}. Then Pa is a projection in B(X) with range Z and kernel H. Let L(X) be the locally convex space of linear transformations on X with the pointwise convergence topology. Show that C = {Pz: Z E X} is a norm compact, convex subset of L(X) that is invariant under the group action (x E G, T E L(X)). Apply Day's Fixed-Point Theorem to obtain an invariant PL. [The above result has semigroup versions. Ky Fan gives an example to which his semigroup result applies with X a Hilbert space. Natural, geometric examples are readily constructed with X a Euclidean space Ri'. Lau, Paterson and Wong [S) have removed the finite-dimensional requirements from the result, and this paper, together with Lau [10). [S6), Lau and Wong [S) , characterizes amenability in terms of the "fixed-subspace" property.)


425

Problems 3 3-1. For each z E F2 - {e}, there exists a homomorphism Q2: F2 --i S, where S, is a finite permutation group and QZ(z) 96 e. Take the Se's to be the Ga's. Then FLEA Ga contains F2 and so is not amenable.

3-2 (Rosenblatt ]6]). Let S be the group of permutations of n elements and G = fInEP S,,. Then the direct limit H of the system of finite groups SI x x S is amenable as discrete and is dense in the compact group G. As in the preceding solution, the group G is not amenable as discrete, 3-3 (cf. Dixmier (1]). Suppose that (A] > 1. Let : G = *CEAZna -+ H = eaEA Zna be the canonical epimorphism. Then by the Kurosh Subgroup Theorem, K = ker4i is of the form F * (*iEJL,), where L; is conjugate in G to a subgroup of one of the free factors Z, and F is a free group. Using the fact that H is abelian, show that K = F. Every commutator of G is in F, and we can show in the appropriate circumstances that F is nonabelian by exhibiting two noncommuting commutators in G. 3-4 (Cole and Swierczkowski (S]). Use Problem 3-3. If I = {1, 2} = J and

x1 = x2 = e, show that H = {(xlx2)': n E Z} is a normal subgroup of G with abelian quotient, so that G is amenable. 3-5. SU(2) is the simply connected covering group of SO(3,R), and we can apply (3.1). A nice example of a pair of elements 0, in SO(3,R) generating F2 was developed by Swierczkowski and is presented in Wagon (S2, Theorem 2.1): rp, z0 are the anticlockwise rotations through cos-' (1/3) about the z- and x-axes respectively.

3-6. Aut T' is the group of n x n integer-valued matrices of determinant ±1 (HR1, (26.18)] and so is discrete. Thus (Aut G)i = IG, (G) = trivial group. 3-7 (Granirer (221). G G/ rad G has a compact, semisimple Lie algebra and so is compact (Appendix B52).

3-8. If G is nilpotent, then it is of Type R (Sp(adX) = {0}). Every compact G is of Type R ({(( exp t ad X((: t E R} is bounded). For more examples see Problems 6-7-6-9. For the "ax + b" group case, see (6.36). Suppose that G is of Type R and not amenable. Then using the Levi-Malcev Theorem and (3.3(ii)), 9 D sl(2, R). Finally SL(2, R) is not of Type R ((6.29)), giving a contradiction. 3-9. Use (3.9), (0.13).

3-10. G Let A = A-' E -9'(G), and let L be the subgroup of G generated by G/H be the quotient map. Let x1...... x, be a left transversal A. Let Q: G

of L with respect to L fl H. Write axe = xi(a)h,j (a), a E A, h;; (a) E H. Let M be the (finite) subgroup of H generated by the hj(a)'s, and show that

L C {xi,...,x,}Mxi', which is finite.

426


3-11. We get an easy negative answer by defining A1(E)

if E E 4lb(R),

00

otherwise.

3-12. Let F2 be free on x, y. For U E {z, y, z-1, y-1) let E,, be the set of elements in F2 whose reduced form begins with u. Let C = {y-1, y-2.'... } and obtain, in the notation of (3.12), the p.d.:

B1=E5, B2=E,-1 -C, B3=CU{e},

AI =Ex, A2=Ex xl = x-1,

X2=e, V1 = y-1,

Y2=e, Y3= y_1

3-13. Proof of (3.13(ii)) (cf. Jech [1]'s proof of the Cantor-Bernstein Theorem).

There exist subsets E of B and D of A such that A - E, D = B. Find partitions A Ez (1 < i < m), Dj , By (1 < j < n) of A, E, D, B and elements sq, t,7 such D by that s,A, = E,, t,DJ = B, Define 1-1 functions f : A -+ E, g: B

LetF=gof:A---A,anddefine Bo' = D, B;,..1 = recursively sets A',,B;, by Ao = A, Then g': A --+ D is a bijection, where g'(x) = F(z) if x E A,: B,, for some n and is x otherwise. Composing with g-1 gives a partition (A. A") of A and a bijection h: A -> B, where h(a) = f (a) if a E A' and h(a) = g-1(a) if a E A". Use the sets A' n A,, All n D3, and elements s;, t, to implement A = B. 3-14 (Wagon [S2, p. 1130. In the notation of (3.15), let a = r (A), ,8 = r(B). Show, using (3.15(v)) and (3.13(ii)), that na = no for some n E P, and then apply the Division Theorem. For the last part, note that if B C Sn, B° T O. then S' is contained in the union of a finite number of translates of B. 3-15 (Paterson [1011). (i) If A=, B,, s2. t, give a Borel p.d. for H, then AT, B;T, s t, give a Bore] p.d. for G, where T is a right transversal for H in G. (ii) Let H be a nonamenable, a-compact, open subgroup of G. Then C has a Borel p.d. if H has. Let K be a compact normal subgroup of H with H/K separable. It is sufficient to show that M = H/K has a Borel p.d. Let L be a

compact open subgroup of MIME, and let M' be the inverse image of L in M. If M' is not amenable, use (3.8) and the fact that F2 has a p.d. to produce a p.d. for M. If M' is amenable, there does not exist an M-invariant mean on M/M' and Tarski's Theorem applies. 3-16. Use Problem 3-15.

3-18. The existence of a p.d. for S3 follows from the n = 2 case by identifying SO(3, R) as a subgroup of SO(4, R) by allowing action on the first three coordinates only. The general case follows by induction.


427

Problems 4 n OX 4-2 (Ranch [1], Emerson [71). (i) (d) If A E P(G)s AN = Let cb, 0 E D(G). Let {A6} be a net in P(G) then ll(¢z - O)AN ll -> 0. (ii) such that 1[E * A6 - µ6[l2 - 0 for all £ E P(G). Show llq5µbll,o,11OA611v ,. -+ 0 so (N+1)-I

"A'

0. For G, cf. Problem 2-1 and use (b).

that

4-3 (Miiller-Romer [S]). Let F E .9(G), A E P(G), e > 0. Find C E Fe(G) such that l[x * A - All < e (z E C). If a E AutG is such that a(F) C C, then, if 1(z * AQ - Aa}^(¢)[ = 1(a(x) * A- A)^(0 o

a_')[

< ellO[[

(z E F).

Thus G satisfies a Reiter condition. R and T are contractible; any nontrivial discrete amenable group is not contractible.

4-4 (Day [6), Willis [S]). (i) Suppose not and let xo, xl t ... , x, E G, xo = e, A = (n + 1)-I o bx . From llarp(A)ll = 1 deduce that there exists a sequence

F,

{ fk} in PP(G) such that

hrnk 11(n+1)-I

xi * fk[[p = 1. i=0

Now show using uniform convexity that llx,* fk- fk11p -» 0 so that G is amenable.

(ii) Let a: Lp(G) -, C be linear and left invariant. Let x: (1 < i < n) be such that 1k7rp(A)ll < 1, where u = n"' E I bz;. Then (I - 7r, (A)) is invertible so that if f E Lp(G), there exists g E Lp(G) such that f = n I (Ei I (g - zq * g)). Now apply a to obtain a = 0. [There are always discontinuous invariant linear functionals on LI (G) (G infinite). See Willis [S) for references and more information.)

4-5. Let f (x) = A(Cx fl D). Then using f g(z-I) d)(z) = f A (z-I)g(z) d t(z) (g E LI (G)), rP

[f (x) - f (y)1 = I t (XCznD - XCynD)(z) dA(z)I sup Q &-IC_I). 4

.EDAlso

A(x-IC-I 'L yÌC_I)

©(z_')) .t(z-IC-I

= [lvx-I * Xc- - Xc-t 11I --, 0 as z -+ Y.

4-6 (Skudlarek [1]). cf. (4.5).

4-7. Try K,, of the form [-a,,, a"] + C,,, where a,, -, oo and C,, is a Cantor type set in an interval large enough and far enough away from [-a,,. a,,]. 4-8 (Emerson [2]). If cb" = XK, /A(Kn)I/2 then .1(xK" fl K")/.(K,,) _ ¢," * dn(x). Further {0" * 0n} is a sequence of positive-definite functions on G converging to 1 pointwise. The convergence is uniform on compacts [HR2, (32.42)).


428

4-9 (Emerson (21). Show, using dominated convergence, that A(VV)-I

/ A(uV, f1

) dh(u) - 1(U)

U

and that, by Fubini, the preceding integral is < A(V,T1 ).

4-10 (Chou [4), Emerson 131). - (G not unimodular). Let e > 0, U E F(G), L C U, and A(xU A U)/A(U) < 2e (x E C). For y E G let Uv = (Uy) U (Uy)--. Show that A(xUy

le+2(.\(U-1)/A(U))A(y)-1,

and choose y so that K = U1, U L U L-1 "works." For the unimodular case, the symmetric version of (4.10) holds (using (4.8)). Now apply Problem 4-9 (last part). 4-11 (Leptin [11, 121, [51}. Take the "inf" over K of each side of the equality:

)(C2K)/.(K) = (;,(C(CK))/1(CK))(.(CK)/a(K)). 4-12 (Emerson [71). The converse is open.

4-13 (Bondar

[1J).

.

'(fcEC c(C-1K))/\(C-1K)

is

close to

1

if

.l(C1K)/A(K) is. 4-14 (Lorentz (1), Chou (41). (ii) ' Use (4.17). (iii) If m E C(ACI(G)), show that m = n1AC,(G) for some n E c(U,(G)), and then use (ii). (Notes: In his paper [8], C. Chou determines the "multiplier" algebra {O E U(G): 6AC1(G) C ACI(G)}: the latter equals Ci+ACI,o(G), where ACI,o(G) = {O E U(G): m([d,j) = 0 for all m E r(G)} (C ACI(G)). The space ACr0o(G) is defined in the obvious way, and Chou shows that if AC0(G) = AC1,O(G) fl ACr,o(G), then AC0(G)/(ACo(G)nWP(G)) contains a linear, isometric copy of l,,,; for a very large class of a-compact groups G. So ACI(G) is, in general, very much larger than WP(G). Also, U(G) is, in general, much larger than ACI(G) (Granirer [131, Chou [81). Almost convergent functions also arise naturally in the study of the Ergodic Mixing Theorem for dynamical systems (Dye 11]). See also Problem 5-8.)

4-16 (Wu [S), Knapp [Si), Milnes (S21-[S5]}. Let e > 0. Suppose that 0 is in UI(G)r1BAP(G). Find V E 9,(G) such that [1v¢-¢j{ < e (v E V). Let C E S°(G)

be such that (1) holds, and cl, .... cn E G be such that U,'=, c,V D C. Then GO C the union of the bails. centre c,b, radius 2e, and Go is totally bounded. So +D E AP(G). Trivially. U(G) fl BAP(G) D AP(G). Now let tG E BAP(G). Choose an increasing sequence {Cn} in '(G) such that

for each x E G, 3cn E C such that [jx7p - n7pj[ < n-1. Find {vn} in P(G) such

thatjjv,, *x-vn11, 1}.

4-23. n-i Z:; i 4, * z, (y) - .* f rbda for all y E G. Now use the Dominated Convergence Theorem to obtain that {x,} is u.d. Property (1) is the classical definition of a u.d. sequence in a compact group-see Kuipers and Niederreiter 11). For the second part, follows by taking 0 to be a suitable exponential. For -, (1) is trivial if 0 is a trigonometric polynomial, and the general case follows by approximating 0 by such a polynomial. Using Weyl's criterion, one can easily show that if 0 is irrational, then {n8} (mod 1) is u.d. in the sense of (1).

4-25 (References as for (4.20)). Apply the Riesz-Thorin Convexity Theorem (DS, VI.10.111 to the function t log 11r1/3(p)(( (0 0, and non-void K E Sr(S) is such that 1K -r xK1 f 1 KI < Ze (x E C), show that the Reiter condition (1 x* f -f 1(i < s (x E C) holds, where f = XK J(K(.

For the second part, if S is left amenable, then 1K - xK1 = 1xK - K( in the left cancellative case and (4.9) applies, while (1.19) applies in the finite case.


431

4-33. (Granirer 131). Let {x1, x2.... } be an enumeration of T°. Using the Reiter condition, construct inductively a sequence {f,,) in P(S), where each

f,. has finite support A, such that llx * f,, - All, < n-1 for x belonging to {x1, ... , x,,} U AI U U countable subsemigroup.

Then T o U (U

1

generates a left amenable

4-34. (Talagrand [5]). Let 0 be Riemann measurable on G. Construct, for e > 0, functions ', b' E C(G) such that

0. In fact (

6(AÀ)h, h) =

lL

70

{ftzmhi(x2xmu))

Ix

by the positive-definiteness of 06, where for some f,ik E C, x%jk E G, we have f 2 = Ek fijklT2 (xtjk). (Note: What happens if G is locally compact?)

4-44 (Coifman and Weiss [1), [2), [3)). Let e > 0 and find ((4.13)) nonnull V E '(G) such that .1(CV)/.\(V) < 1 +e. For v r= V, show, using R,-,R, = I, that IlRkfllp < MIIR,,Rkflip, and deduce, using Fubini's Theorem, that

IIRkf11

-< (M'/A(V))

X

du(x) f IPwRkf (x)II d.\(v). v

Next show that for a.e. x,

fv

v

(Tkgz(v)Ipd1(v),

where gx(u) = Rf (x)xcv (u), and deduce, using Fubini's Theorem, that IIRkf HP < (MP/a(V)) fx d"(Z)IITkiiP f [M29ilTkilP.(CV)1 A(V)

((flip

IRu f(x)Ip dA(u)

v


434

Problems 5 5-2 Let m be the unique invariant mean on WP(G). Then X f = Cl, Xo = E WP(G): m(¢) = 0}, and P¢ = m(¢)1. 5-3 (Greenleaf [2], [4], Greenleaf and Emerson [1]). In the notation of (5.19),

(5.20), G = G' = N xa H, where N = H = R. Take C,,,, _ [-m, m] x [-m, m]. Then (C,n)H = [-m, m]. Choose R, to satisfy m/R,,, -> 0. Then (G',n) N = [-m exp(m + R,,, ), m exp(m + R,,, )].

The subsequence {L' (m) } = [-m, m]. (m exp(m + R,,,))/a(m) -+ 0. Then

Put L;,,

has

to satisfy

K,,, _ ? (LQ(,n) x [-un, Rm])

_ {(exy,x): - R , n $ x< Rm, -a(m) :5 y : 5 5-4 (Emerson [4]). (iv) Let f1 ([a,,,a,+1] +ml) and show that each x E [0,1) belongs to J,, for infinitely many n. Then show that if x E J., r E Dm,

then r. x < 3m-1, and obtain Am f (x) > m1/4/(6f) for large enough m. 5-5 (Blum and Eisenberg [1), Blum and Cogburn [1], Blum, Eisenberg, and Hahn [1], Milnes and Paterson [S]). (ii) If G is abelian, then B1(G) = (Span{-1: -y E G})- = AP(G). (iii) (c)

(a). Let r E G, e, rI E Jj,,. The function x --+

r)) is in B1(G)

and so

m(x - (r(x)£,r?)) some T E B(S)") in the weak operator topology. The invariance of m yields that T E rr(G)° and irreducibility gives T = P" E {0, I}. Now use disintegration theory to obtain P" for every continuous representation r' of G. (Note that the separability requirement can be dropped if G is either abelian or compact.) So

(iv) Use (iii)(c). (v) The finite-dimensional representations in if are the characters Ua,b (a, b E R), where Ua,b(x) = exp(27ri(ax2 -- bx3)), and the infinite-dimensional ones are of the form Ua (a E R {0}) acting on L2(R), wherep

(Ua(x)f)(t) = exp(2Tii(x1 - x2t)a)J (t - x3). In an obvious way, AP(H) = C 0 AP(R2). By considering functions of the form x -- (Ua(x)f,g) we obtain the APO(R)&Co(R2) part. For related results, see the Milnes-Paterson paper. Blum and Cogburn [1] and Niederreiter [S] show that if G = Z, then there are many nonsumming sequences For of finite subsets of G such that is ergodic, where µ = example, we could take A,, = {xl, ..., where, for some fixed a E (1, oo) 1,

x, = [r°].


435

5-6 (Takahashi (2]). 11T11 0. There exists 0 E CR(X), Txy = 0 (T E S), and a convex combination EL, aiT; in 11(S) such that 110 - E , aiTiO11 < s. Let hz(T) = TO(x). Then hz E l.,(S). Also 11' - E_ , o;TT4b11 < e (T E S) so that 11'(x)1 - E , aiTih.11 < e. Applying m E 3(S) gives 1tb(x) - m(h2)1 < e.

So m(hx) _ fi(x). Set PO(x) = m(h.). [More references: Derriennic and Lin [1), Foguel 111,Fong [1], Fong and Sucheston 11], Hiai and Sato [1], Lin [1), Nagel 11], Sachdeva (1], Sato [l]-[5], Sucheston [2], [3], Takahashi [31, [4), Wolff 111.]

5-7 (Sine [1]).

Let f E X', A,, (T) = (n + 1)-1 Z',07 . Every weak*

cluster point of {A,,(T)' f } is T'-invariant, and there is only one such by the separation condition. So for some T-invariant Q(f), A,(T)' f Q(f) weak'. Let B* be the unit ball of X'. Show that the topology on Q(B') generated by the T-invariant elements of X coincides with the weak` topology. Show then that for x E X, the functional f -+ Qf (x) is weak` continuous and so is of P in the weak (and indeed strong) operator the form (Px)". Then An(T) topology.

Not surprisingly, this delightful result has been extended to amenable semigroups of operators-Lloyd 14], Nagel [1), and Sato [3). 5-8 (Namioka 13], Lloyd 121, Sucheston [1]). (i) For the last part, use the facts that for m, n E .C(S), we have nm = m, mn = n (so that Q,nQn = Qn, QnQr = Q.) and F = ker(I - Q,,,.). (ii) K = F f' PM(Y) is a weak* compact, convex subset of M(Y). We can suppose that l E K. If C E ExtK, then Q,,,(r7) E F, 0 < Qr('7) < , and the extremeness of gives Qm(77) = ?7(1)e = Q,,(r7). Now suppose that C E

L = co(ExtK), and write e as a convex combination E , ai&, , E ExtK. By regarding the &,77 as functions in L1(c), one easily shows that there exist r), E M(Y) such that 0 < 77, < aiCi. 77 = 177,. Since Qm(77i) = Qn(77i), we have Q,,,(77) = Q. (77). For general E K, we can suppose, by the norm density of C(Y) in L1 (c), that dv/de = f E C(Y). By the Krein-Milman Theorem, there exists a net {;E} in L with Cs t; weak'. If v6 is such that dv6/dE = f, then vb -+ v weak' and Qm(v5) = Q,, (vb). By weak* continuity, Qm(v) = Q, (m) as required. (iii) We can suppose that 0 < 0 < 1. So 0 0 be such that A(Cm) < Mmk (m > 1). Then show that for n,m > 1, Ileinf -1111 < M112nk/2IITII Ilf 112 + e-m exp(nllf 11.) 5 C1n9

for some C, > 0, where

T = in E(inr2(f))k/(k + 1)! E B(L2(G)). k=0

Now show IITII < n and choose m so that I m - nll f 11.1 < 1 to obtain the desired result.

6-23 (cf. Palmer 111). L,(G/H) is a homomorphic image of L,(G) ((1.11)) and so G E IS) = G/H E IS). If H is an open subgroup of G, then L, (H) can be regarded as a closed *-subalgebra of LI(G), and (G E 1S) from [RI, (4.7.7)].

. H E IS]) follows

6-24 (Jenkins [4), 15j, [8]). (i) It is sufficient to show that if a E A and a does not have a left inverse in A, then Fo(a) = 0 for some Fo E P(A). For such an a, 0 E Sp(a-a), and since IF(a)12 < F(a"'a), it is sufficient to obtain Fo E P(A) with Fo(a"a) = 0. Let M be a maximal, closed *-subalgebra of A 0 for some ' in the carrier space of M, and containing a-a. Then

' E P(M) by symmetry of M and so extends to the desired Fo E P(A). (ii) (a) n Straight calculation shows that Ilfolli = 61 so that v(fo) = lllo Iii _ 6.

(b) Let A = 11(G). For F E P(A), IF(t * f )12 < F((t * f )- * (t * f )) =

F(f^' * f) < v(f" * f) < v(f)2. By (i), v(t * f) < v(f). (c) For F E P(A), IF(9o)I 5 1v(a + a-1)2 + v(a + a-I)4]'/2 < 2f. By (i), v(go) 5 2v/5- if G is symmetric. This is impossible by (b) and (a), since go * go = 9o * go and 6 = v(fo) = v(ba2 * go) 5 -(go). 6-25 (Boidal et al. 11)). (i) G If r,p are as in (i) and r., p. are the extensions of r, p to C' (G), then Ilp(f )I1 < llr(f ) 11 for all f E L, (G), and it follows that ker r. C ker p.. Let .5" be a closed subset of Prim C' (G), and let r' = ED,, S, rr, where rl E C' (G)^ has kernel I. If p E G has kernel D (ft9) f1 L, (G), then {J E Prim. L, (G): J D (r%5') n Ilp(f)11 5 1Ir'(f)Ii. It follows that L, (G)} is closed in Prim. L, (G).


439

(ii) Let r, p be nondegenerate, *-representations of L1 (G) with ker r C ker p, and let h E CM(G), f = h * h"'. Suppose that [I p(f )[[ > [[hr(f) [[. We can suppose

that G is compactly generated. Let ¢ E C,'°(R) with O([0,[[vr(f)[[J) _ {0}, ¢([[p(f )[E) = I. By Problem 6-22, there exists g E Af such that g = ty o f. Now show that g E ker r ker p, and derive a contradiction. It follows from (i) that GE[%1. 6-26 (Rosenblatt [S11). f I (x *,u - rz)p"[I1 --+ 0. G If f E L°(G) is of the form (g * x - g) for some g E LI (G), then 11f * n 1{ I -+ 0. Let X be the span of the set of such functions f. By examining X1 C show that X is dense in LO(G).

Problems 7 7-1.

With X = A and N., as in (7.1), the set T = {a E 13X: for some

E E {l, }r, a E f 7(N ')-} has cardinality > 22'XI . The inequality [1,o(X)I[ < 22'"1 follows as in (7.6). [For another proof that 1,6X[ = 221"4, see Gilman and Jerison [1].] 7-2. Chop up [0, oo) into intervals 111, , where Ill = [0,3], 121,122 are 13,71, 17,111, I31 = 111,161,.... On each It, = [aia, btJ[, let Otj be the tent function

atj aj + 1

b=j - 1

btj

'

maxzER Otj (x) = 1 and ¢{j vanishes outside 1t?. Take 4j

Oij, Zj =

U jJaij+1,b{j -1). 7-3.4. Let ', E be as in (7.5). Each member of 9' contains a minimal, compact, left invariant subset. If C E F, then Ext{m E £ (G): en(C) = 1) C ExtCt(G). [Note: the amenability of G is not used in the construction of ' _ {Cp: p E P) in (7.4).1 7-5 (Chou [4)). (iii) Let m r= 9310, f = XU/A(U) (U E F(G), U not null). Show that

/ Un)-111011oo sup/ A(t(UU+1 - U,,)) 0/ (U.+1 - Un))

/

I

r(Of - 0)(n)f G

//

'Cu [[0lhoo[((n+1)/n)sup[A(U.+1 AtUn+1)/A(Un+1)[+2/n) tEU

n-+oo. So for mE

(iv) By Gilman and Jerison 11), RP is homeomorphic to a compact subset E of 3P P. Since E C [Ct(G)[ ? hiV I > IE[ = 2'.


440

7-6 (Granirer [13]). Take X = A It is sufficient to show that u(m) > No for each m E K Suppose that the contrary holds for some m, and construct a sequence if,,} in P(E) such that fn m weak`. Derive a contradiction (cf. (7.28(i))).

7-7. Let {Cn} be a sequence in F (G) such that U°°1 Cn = G, C. C Cn+1 We can suppose A(U,,) - 0. Let {Vn} be a sequence in 9e(G) such that Vn =

Vn1,V,?cVR_1nUn,and xVnx-1cV,,_1(xECn)for n>1. Take N =

n`fn00-1Vn

7-8. Let {rn : n E P} be an enumeration of Q and 00

V=

V (rn - E2-n-1, rn + E2-n-1).

n=1

7-9 (Rudin [21). By forming intersections of translates of the sets E1, k, of (7.15), we obtain a disjoint family d of compact, nonempty, left invariant subsets

of 0(G) with is/ I = 2`. For C E.', let Ic = {¢ E L00(G): (C) = {0}}. Take

-F _{IC:CEd}. 7-10.

We can suppose E = G. It is sufficient to show that SS, where is continuous from Gn x Xn

L1(G).Ifn=1, IIS1(zi,El)-Si(y1,Fi)Il1

Ilxi *XE, -Vi *XE,III+IIXE, -XF,II1,

and S1 is continuous. Now use induction. 7-11. Let.V be the class of a.l.c. semigroups. If S is a left cancellative, infinite

semigroup and F is a semigroup with (FI < ISI. then S x F E .sag. If {Sn} is a sequence of finite semigroups with Sn a subsemigroup of Sn+1 and (KnF -+ oo. where Kn is the kernel of S,,, then Un 1 S,, is a.1.c. (Use (1.19).) If V, W are semigroups with V E .sag, V infinite and such that there exists an epimorphism

Q: V - W and a cardinal m such that IQ-1({w})(< m < IVI for all w E W. then W is also a.l.c.

7-12 (Klawe [3), Paterson [91). Suppose that (i) holds and p is infinite. Let {S1, ... , Sn } IuSI

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