Amenability Alan L. T. Paterson
American Mathematical Society
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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)
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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) <- Remo(Oo) - e
(1)
for all f E P(G). We can suppose (by (i)) that ¢o is real-valued. Let ko = esssupoo. Then we can find a compact, nonnull subset C of G such that 00 (z) VeG
ko - 'e for all z E C. Taking f = Xc/A(C) in (1) we obtain mo(Oo) ? ko + aE. This is impossible by (i). o An important special case of (0.1(111)) occurs when G is discrete. In this case and A is counting measure. Each x E G can LI(G) = 11(G), be regarded as an element of II (G) via the correspondence x -+ X{:} = bx. It is easy to see that P(G) is the norm closure of the convex hull coG in 1I(G) and we have cod is weak* dense in 9)1(G). (The same conclusion holds when G is replaced by any set X.) We have shown how a mean as a measure p can be switched to a mean m as a linear functional on Lo, (G). How does the invariance of p translate in the m-context? To answer this, if E E and u(x-'E) = µ(E) (x E G), then m(XEx) = m(XE), where X--x(y) = XE(xy) for all y E G (since XEx = X-This leads to considering the natural actions of G on La,(G) and its dual. First, let us establish some terminology. A set A is called a right G-set if for each x E G, we are given a transformation a -+ ax of A such that (ax)y = a(xy) for all a E A, x, y E G. (So G has a right action on A.) Left and two-sided Gsets are defined in the obvious way, and the same definitions apply for semigroup actions. For 0 E
by Oz(y) = O(xy). (A commonly x E G, define Ox E used notation for our ix is 20, but we prefer to use Ox since it makes explicit the right action involved and is consistent with the terminology of Banach G-
modules.) Then L (G) is a right G-set under the transformations 0 - ¢x. The action fits in well with the Banach space structure of L (G): indeed each map 0 .- ¢x is an isometry in (If Y is a Banach space, B(Y) is the Banach algebra of bounded linear operators on Y.) (For future use, we note that there is a natural left action 0 -+ xO of G on L.,, (G), where x¢(y) = ¢(yx).) The relevance of this G-action on L. (G) for amenability is as follows. If p, m are as earlier, then u is left invariant (that is, p(xE) = µ(E) for all x E G, E E..f(G)) if and only if
m(Oz) = m(4) (0 E L. (G), x E G). This equality can be reformulated as follows. The right action of G on L (G) gives rise to a dual left action on zn(-O) = n(Oz)
(x E G, n E L. (G)', 0 E L. (G)).
INTRODUCTION
5
So the above equality can be expressed n is left invariant for G. This leads us to the most important definition in the book. (0.2) DEFINITION. The locally compact group G is called (left) amenable if there exists a left invariant mean for G. The set of left invariant means on G is denoted by £(G). (A more commonly
used notation is Ml(G), but we have chosen to use £(G) in the interest of succinctness.) It is almost trivial to check that £(G) is a weak* compact, convex subset of +91(G). Every compact group is obviously amenable: in that case, the map 0 -+ f 0 d.1 is in £(G).
Before launching into the theory of amenable groups, it is clearly desirable to look at some special, well-understood groups to see if they are amenable or not and how invariant means can be produced. This will also help to motivate a number of the important questions discussed in the later chapters. The five groups we will look at now are (a) Z, (b) R, (c) the "ax + b" group, (d) F2, the free group on two generators, (e) the group SL(2, R) of real 2 x 2 matrices of determinant 1. The results below follow from more general theorems proved later.
(0.3) EXAMPLE: G = Z. Is there an invariant mean on Z? Since I.. (Z)' (= is inaccessible, we concentrate on exploiting P(G) with (0.1(iii)) in mind. A typical element of P(G) is an infinite sum of the form E° a,-4, where a, > 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
<
2xJ1011
2n
-+ 0
as n - 00.
Again we have that every weak* cluster point of {fn} is a (left) invariant mean and so R is amenable.
(0.5) EXAMPLE: G is the "ax + b" group. This is the affine group S2 of R (Appendix A). So S2 = R xo (0, oo) with multiplication given by (b, a) (b', a') = (b + ab', aa').
It is not so clear how to construct the fn's in this case since the multiplication
is getting "twisted" in the first component. The following construction was shown to the author by Paul Mimes. Let An be the shaded subset of S2 shown below: 0
(-n2, n)
(n2, n)
4 b
A Haar measure A on S2 is given by dk and direct integration shows that if we take fn = XA" f)'(An) E P(G), then we again have that every weak* cluster point of { jn} is left invariant, so that 82 is amenable. (See Problem 0-8.) (0.6) EXAMPLE: G = F2. Let F2 be free on the generators u, v (with, of course, the discrete topology). We will show that F2 is not amenabler
To this end, as usual, we identify F2 with the set of reduced words in For z E {u, v, u I, v 1 }, let E. be the set of elements in F2 beginning with x. If M E .C(F2), then, regarding m as a measure on F2, m(G), and m(G) = m(Eu)+ m(uEE-, ), m(G) = Now m(G) = 1 and a contradiction {u, v, u-1, v-1 }.
follows using the invariance of m.
(0.7) EXAMPLE: G = SL(2, R). This locally compact group is not amenable either. To show this, we can use the result (an easy consequence of (3.2)) that G = SL(2, R) contains a discrete subgroup H isomorphic to F2. (Discreteness
here means that there exists a neighbourhood U of the identity e in G with
INTRODUCTION
7
U n H = {e}. This implies that H is closed in G.) Using Appendix C, Corollary
(with G = X, G\H = Y), we can find a Borel cross-section B for the space G \ H of right H-cosets in G. So B C G is Borel, G = HB, and hB n h'B = 0 if h, h' E H, h h'. Since H is countable, EB is Borel for every subset E of H. If m E £(G), then we can define n r= £(H) by setting n(E) = m(EB). This is impossible by (0.6).
The above examples "point beyond themselves" and suggest a number of research directions that will be developed later. Most of the rest of this chapter will be devoted to discussing briefly some of these directions; these include FOlner conditions, topologically left invariant means, the sizes of sets of invariant means, polynomial growth, fixed-point theorems, the construction of new amenable groups from old, and the relationship between F2 and nonamenability. We concentrate first on Examples (0.3)-(0.5). If we look closely at how Z, R, and S2 were shown to be amenable, it becomes
clear that the arguments relied on constructing a sequence {Kn} of compact subsets of G with the property (3)
A(xKfOKn)/A(Kn)-.0 asn -+oo
for each x E G. Such a sequence is called a summing sequence for G. For each
n, the sets Kn of (0.3), (0.4), and (0.5) are [-n; n) (= {r E Z : - n < r < n}), [-n, n), and An respectively. We will show in Chapter 4 that a v-compact amenable group always admits such a sequence {Kn} and that amenability is equivalent to the "Folner condition": given e > 0 and C E F(G), the family of compact subsets of G, there exists a nonnull set K E '(G) such that (4)
\(xK,& K)/.1(K) < e
(x E C).
What this remarkable condition asserts is that, roughly, there is a nonnull compact subset of G which is "almost invariant" under left translates by elements of C. The condition illustrates two fundamental options in amenability theory. We can either have precise invariance and pay for it by having to work in the difficult space L,,. (G)', or we can work in a "down to earth" setting with compact subsets of G and pay for it by having only "almost" invariance. The Folner conditions are among the most useful in amenability. For example, as in (0.3)-(0.5), they can be used to produce invariant means with little effort. Deriving such conditions is, understandably, not easy. However, the first stage in such a derivation is a version of (4) for LI-functions known as Reiter's condition. To formulate this condition, we require a left action of G on LI (G) whose right dual action is just that given by the maps ¢ Ox discussed earlier. For x e G, f ELI (G), we define x * f ELI (G) by
x * f (y) = f
(x-I y)
(y E G).
This is clearly an isometric left action, and since, for ¢ E L". (G), we have
*x * f) = f Ax-IY)O(r) da(y) = f f(y)O(xy) dA(y) = (¢x, f)
INTRODUCTION
8
the above claim is justified. Of course, x t = (x * f)" (in L,, (G)'). There is a corresponding right action of G on L1 (G) given by
f * x(y) = f(yx-1)A(x-1)
(y E G),
where A is the modular function on G. Now if f = XK f.\(K), then
iix * f - f111= (A(K))-1 f IxK(x-'y) - xx(y)I da(y) _ .1(zK A K)/A(K). This suggests a weaker version of (4) in which xK/) (K) is replaced by a function in P(G), and such a version ("Reiter's condition") is proved in (4.4). A related, easier condition will now be proved (Day 141, Namioka [1)). The elegant argument depends on switching from L (G)' with the weak* topology to L1 (G) with the weak topology using the left invariance of an element of £(G).
(0.8) PROPOSITION. The group G is amenable if and only if there exists a net {f6} in P(G) such that IIx * f6 - f6)11 - 0
(5)
for all x E G. PROOF. Let {f6) be a net in P(G) such that 11x * f6 - (6111 -+ 0 for all x E G. Then (xfe - f6) -+ 0 weak* for each z, and every weak* cluster point of {f6} is in .C(G), so that G is left amenable. Conversely, suppose that G is left amenable. Let m E £(G) and find a net
{g,} (c E I) in P(G) such that g, -+ m weak*. Let z ,...,x,, E P(G). Since ((xi
g,),.
* g, - g,)) converges weakly to 0 in L1(G)", we can find
[DS, V.3.13J a net {h.,} with each h7 E co{g,: c E I} C P(G) and such that 0 (1 < i < n). The existence of the required net {f6} now 11x; * h7 - h1111 follows.
0
We now consider those elements m that arise as weak* cluster points of a net
{f6}, where { f6 ) is as in (0.8). It turns out that, for nondiscrete G, such an element m is rather special-it is topologically left invariant. Generally speaking, topologically left invariant means are much more manageable than means that are merely left invariant. Before defining topologically left invariant means, we recall that L1 (G) is a Banach algebra under convolution, where for f, g E Ll (G), the convolution product f * g is given by
f * g(x) = f f
(t)g(t_1x) dA(t).
c
The left action by multiplication of L1 (G) on itself dualizes to give a right action of LI (G) on Lo, (G) = Ll (G)': 0 * f (g) = -O(f * g)
(-0 E L. (G), f, g E L I (G)).
INTRODUCTION
9
(Our definition of 4,* f differs from that in [HR1, §20]. However, the two versions of ¢ * f are closely related, and ours is natural from a duality viewpoint and fits in well with our requirements.) Note that 110 * f II. <- 11011.11f 111. As a function, we have
¢ * f (x) = f ¢(yx) f (y) da (y) = (x4,) (f ).
This right action dualizes to give a left action of LI(G) on La,(G)' written n -+ f * n (or fn) (f E LI (G), n E L. (G)'). One readily checks that x(f * n) _
(x * f)*nforallxEG. (0.9) DEFINITION. A mean m on
is called topologically left invariant
(Hulanicki [4)) if fm = m for all f E P(G), that is, if (6)
M(0 * f) = m(¢)
(0 E L.(G), f e P(G)).
The set of topologically left invariant means is denoted by £t (G), and is clearly
a weak* compact, convex subset of Wt(G). Further, if m E £t(G) and x E G, then xm = x * (f * m) = (x * f) * m = m since x * f E P(G). So £t(G) C £(G). It is readily checked that if G is discrete, then every function in P(G) is a sum
of point masses so that £t(G) = £(G). However as we shall see in Chapter 7, the two sets are very different in general for nondiscrete locally compact groups. Why are means satisfying (6) called topologically left invariant? The reason
for this is that the means in (6) are P(C)-invariant, and P(G) depends on A which in turn is determined by the topology of G. The fact that P(G) C LI (G) whereas G % LI (G) in the nondiscrete case will be usefully exploited later. To illustrate the relevance of £t(G), we can ask the following question: what can one say about those means arising as weak* cluster points of the net {fs} in (0.8)? Can we get all of £(G) in this way? Given that we know that £(G) # Ct(G) in many cases ((7.21)) and that we assume (as we may) that the convergence in (5) is uniform on compacta, the
answer to the second question is no; all such weak* cluster points m are in £t(G)! Further, as we shall see (cf. (4.17)), in a certain sense, we obtain all of £t(G) from the weak* cluster points of {j6}. We now illustrate these comments in the case G = R: the same proof applies under much more general circumstances.
(0.10) PROPOSITION. Let fn = xl-n,ni(2n, and let m be a weak" cluster point of { fn} in L. (R). Then m C- Zt (G). PROOF. Let ¢ E L,, (R), f E P(R). Using the norm continuity of the map g - 0 * g (g E LI (G)), we can suppose that f vanishes outside a compact set C. Let N be such that C C [-N, NJ. Then for n > N, we have, using Fubini's
INTRODUCTION
10
Theorem and (2), (fn 11,J+' * f -0)1 =
)r
/
/
/
if fn (x) dx J[tti(v + x)f (y) - 0(x)f (y)] dyl II
=
If f(y)dy af fn(x)[0(y+x)-0(x)]dxI
< f 1 f(y)Jdx
x'11±
<
N1101111f111
_0
as n -> oo. It follows that m (4i * f) = m(O), and m E £t (G) as required.
We shall see in Chapter 1 that there always exist topologically left invariant means on an amenable locally compact group. Let us return again to the notion of a summing sequence {Kn} with Example
(0.3) in mind. In this example K,, = 1-n; n] and integrating ¢ E 1. (Z) over gives a Cesaro average. Since such averages appear in summability
theory, one might wonder if summability ideas can be developed for general amenable groups using the K,,'s. This is in fact the case; in particular, the theory of almost convergence has been developed in the amenability setting (see, for example, Problems 0-13, 4-14, 4-15, 5-8). Another context in which Ceshro averages occur is in the mean and pointwise ergodic theorems. We will show in Chapter 5 that a version of the mean ergodic theorem, involving the K,,, does hold for amenable locally compact groups. The question of a pointwise ergodic theorem for amenable groups is, understandably, rather more subtle, but using Lie group techniques, we will establish such a theorem in Chapter 5. Another interesting issue arising naturally from Examples (0.3)-(0.5) is the question of the cardinality J.C(G)J of .C(G) for an amenable group G. The fact that in all three examples, any weak* cluster point of { f,,} is in .C(G) suggests that .C(G) is probably rather large. For the present, let us give support to this suggestion by showing that .£(Z) is infinite (Mitchell [2]). (Our estimate will be greatly improved in Chapter 7.)
(0.11) PROPOSITION. There are infinitely many left invariant means on Z. PROOF. One might think of trying to show this by proving that there exist infinitely many weak* cluster points of {f,,} (where, of course, fn =
(1/(2n + 1)) . (,r=-,, ). This can be done, but requires some functional analytic machinery (cf. (7.28)). Instead we use the following elementary argument (which prefigures the theory of `left thick" sets developed in Chapters 1 and 7). The main idea here is one already operative in (0.3)-(0.5), viz., if In is a sequence of intervals of increasing length in Z, then, for fixed r, the r-translate of does not differ much from for large n. This gives that every weak* cluster point of {(X1" /JInJ)^} is in.C(Z), and to show that .C(Z) is infinite,
we only have to produce an infinite number of disjoint families {I,,). We now proceed with the proof. Divide P into "intervals" 111,121,122,131,132,133,141.... as follows: Il = {1 }, IZ = {2, 3}, 12 = {4, 5}, 131 = {6,778)7 13 = {9,10,11), 133 = {12,13,14}, 14 =
INTRODUCTION
11
{15,16,17, 18), etc. For each r, let m, be a weak* cluster point of {(XI., /n)n}
i n .C(Z). Let A, = U ,1 . Ifs 0 r, then As n A, = 0 and since m, (A,) = 1= m,(Z), we have m, # m, if r 8. So 12(Z)l is infinite. In fact JC(Z){ is considerably bigger than No. As we shall see in Chapter 7, 22' if G is infinite, discrete, and amenable, while, more generally, if G is amenable and noncompact, then j.Ct(G)j = 22"x, where m is the smallest possible cardinality of a cover of G by compact sets. There are three essential ingredients in the proofs of such results. The first involves delicate, transfinite induction arguments, which are used to construct certain special subsets of the group that, in a sense, generate "many" left invariant means. The second involves interpreting means as (genuine!) probability measures on a compact space. To motivate this, consider the argument of (0.11). Each mean in,. can be regarded as "supported" on the set A,. = Un°D , I,, in the sense J.C(G)J =
that m,(A,.) = I or, alternatively, that m,-(i0) = m4XA,) (0 E Since A, n A. = 0 if r # a, the m,'s are "disjointly supported" and are therefore different. The trouble is that any partition of Z has to be countable so that there is no hope of getting an uncountable set of invariant means by this method.
Another unsatisfactory feature is that there is no unique "support" of a mean m regarded as a finitely additive measure on Z. To overcome these difficulties, we can proceed as follows. For simplicity, let
us suppose that G is discrete and let 6G be the Stone-Cech compactification of G. The points of fG are the complex valued, continuous homomorphisms of and 8G is given the relative weak* topology which it inherits as a subset
1.(G)'. In the obvious way, G C 8G. In fact G is open in fG, and the closure
of any subset of G in pG is both open and closed. The map ¢ --* , where 4,(a) = a(4,) (a E fG), is an isometric isomorphism from lo,(G) onto C(fG), the algebra of continuous, complex-valued functions on 6G. (Of course, ¢ -' ¢ is just the Gelfand transform of the commutative C*-algebra 1.(G), and #G is its carrier space.) Why is it worth looking at 8G to obtain information about invariant means?
The crucial point here is that lm(G) is identified with C(/G) and we know what C(fG)' is: by the Riesz representation theorem, C(/3G)' is just M(/3G), the space of regular, complex-valued, Borel measures on /3G. Since 1o, (G)' "=" C(/G)' and £(G) C t (G)', it readily follows that every mean m can be regarded
as a "genuine" (that is, countably additive) probability measure m on #G. As such, in has a (unique) support-the smallest closed subset C of pG such that snI$G. c = 0. Further, G has a natural left action on ;BG, and m E £(G) a rfi. is a probability measure that is invariant under the action of G. The whole philosophy is simple: we shift from studying a bad (finitely additive) measure on a good set (G) to studying a good (that is, countably additive) measure on a complicated space ;6G. (In the latter connection, 6G is much bigger than G: if G is infinite, then W GI =
22101.)
12
INTRODUCTION
Reverting to the m,'s and A, 's above, we readily see that the fact that m,(A,) = 1 implies that the support of m, is contained in the closure A; of A, in 6G. The trouble is that Af is too "big"-it is actually an open set. What we must do is to try to construct invariant means which have much smaller supports in PG so that more means can be "squeezed in." We will see how this can be done in Chapter 7. The idea is to obtain "small" closed, invariant sets in 6G by forming large intersections of closed sets of the form A- (A C G) and then finding means in .C(G) supported inside these small closed sets.
The third essential ingredient is an amenability fixed-point theorem that can
be used to show that, for every closed invariant subset C of fG, there exists m c- ,C(G) with ih supported inside C. Then if we have a large, disjoint collection of closed invariant subsets of 8G, we automatically also have a large set of left invariant means.
The fixed-point theorems of amenability are of great importance and of wide application. (For this reason, we have included an early discussion of these theorems in Chapters 1 and 2.) A completely trivial (but significant) observation is that amenability is equivalent to the existence in 9R(G) of a G-fixed-point--such a fixed-point is, of course, a left invariant mean! Note that Wt(G) is a compact convex subset of a locally convex space (L. (G)' with the weak* topology) and that the elements of G act affinely on 9J1(G) (that is, preserve convex combinations). Not surprisingly, amenability fixed-point theorems work in the context of afilne actions on compact convex sets. The theorems are proved basically by lifting a C-action to an action of a set of means that possesses a natural' semigroup structure. Multiplying by a left invariant mean then gives G-fixed-points. It turns out that the fixed-point property for a reasonable class of group actions is equivalent to the existence of a left invariant mean on a certain space of functions on G. The algebraic structure behind this and other parts of amenability theory is related to the "Arens product construction" familiar in the theory of Banach algebras, and this is discussed in Chapter 2.
Reverting back to Examples (0.3)-(0.5), closer inspection shows that there exists a distinction between, on the one hand, (0.3) and (0.4), and, on the other hand, (0.5). In the former cases, the K"'s satisfying (3) are of the form C" for some compact neighbourhood of the identity. In the case of Z, C = [-1; 1], while in the case of R, C = [-I, I]. This is not the case in (0.5). We also notice that
in (0.3) and (0.4), respectively, we have )(C") = 2n + 1 and k(C") = 2n. In other words, t (C") grows in a polynomial way, and this is true for every compact
subset C of Z or R. The "ax + b" group does not share this property. This can be checked directly; a different approach is given in Chapter 6. It turns out that this group is a "fundamental" solvable, simply connected Lie group that does not have polynomial growth. We now formally define polynomial growth for a group. Let &,(G) be the family of compact neighbourhoods of e in a locally compact group G.
INTRODUCTION
13
(0.12) DEFINITION. A locally compact group G is said to have polynomial growth if, for every C E F.(G), there exists a polynomial p such that A(C") < p(n) for all n > 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/" <
and" < Ki/n(ni/")', so that an I/n - 1. An elementary result on convergent sequences gives liminfn_.,,(an+i/an) = 1. Let a > 0. Then there exists N Such that A(CN+I) !).(CN) < 1+e/2. So .\(CN+i ACN)/A(Cn) = () (CN+1) ),(CN))/h(CN) < a/2. For any i, xsCN w CN C CN+1 CN, and So A(x{CN CN)/,\(CN) < e/2. Similarly \(CN .., xtCN) = ),(x: 1CN "` CN) < !CA(CN), we have \(x,CN ACN)/A(CN) < E, that is, IIxi * f - f I` 1 < E (1 < i:5 n), where f = XcN/A(CN) E P(G). By (0.8), G is amenable. o We note that when G has polynomial growth, we can answer a natural question concerning the set K of the FOlner condition (4). The question is given C, can we construct a set K satisfying this condition? The way in which K is derived is highly nonconstructive, so that the method of proof is not helpful for this question. When G has polynomial growth, we can give a reasonable answer to this question: the set K can be taken to be CN for some N. Returning once again to (0.3)-(0.5), we note that two of the (amenable) groups (1, R) are abelian, while the other group S2 is solvable and so is "built up" out of abelian groups. (In fact, S2 is a semidirect product R x , R.) This raises the question is every abelian G amenable? The answer is "yes" and is a simple consequence of the Markov-Kakutani Theorem whose proof we now give ([DS]).
(0.14) PROPOSITION. Let K be a compact convex subset of a locally convex space E. Let S be an abelian semigroup of continuous transformations T : K -
K such that eachT is affine (in the sense that T(ak1+(1-a)k2) = aTki+(1a)Tk2 for all a E [0, 1), k1i k2 E K). Then the elements of S have a common fixed point.
PROOF. Let T E S and FT be the set of fixed points for T in K. We show
first that FT 9 0. Let k E K and Tn = (n + 1)-1 Ero r. Then Tn : K -+ K. Let kn = Tnk and ko be a cluster point of {kn} in K. We prove that Tko = ko,
that is, that ko E FT. To this end, we have, using the affineness of T, that
INTRODUCTION
14
(T - I)k = (n + 1)-1(Tn{1 - 1)ko E (n + 1)-1(K - K). Since K - K is compact in E and (T - I) is continuous, we have (T - I)ko = 0 and ko e F. Note that FT is a compact convex set.
Let Z = n{FT: T E S}. If T,T' E S, then T'(FT) C FT, and applying the preceding result to T'EFT, we obtain that FT fl FT. 76 0. Thus {FT: T E S} has the finite intersection property, and Z 0. Any element of Z is a common fixed point for S. (0.15) PROPOSITION. Every abelian locally compact group is amenable.
PROOF. Let G be an abelian locally compact group. For the purposes of applying (0.14), take E = La(G)' with the weak* topology, K = OR(G), and let S be the group of transformations m - xm (x re G) of 91(G). The common fixed-point set of S is just £(G). The group S2 is an extension of an amenable group by an amenable group, and (0.5) tells us that S2 is amenable. Further, the argument of (0.7) proves that a discrete subgroup of a (separable) amenable locally compact group is itself amenable. These two observations suggest the question: under which algebraic processes is amenability preserved? The topological version of this question requires some work to answer-see Chapter 1. However, von Neumann [1] showed that the question can be readily answered for discrete groups, and we give the proof here. For more examples of amenable groups see the Problems at the end of this chapter.
(0.16) PROPOSITION. Amenability for discrete groups is preserved by the following five processes: (1) taking subgroups;
(2) forming quotient groups; (3) forming group extensions; (4) forming upward directed unions of amenable groups; (5) forming a direct limit of amenable groups.
PROOF. (1) This follows as in (0.7)-merely delete any "Borel" requirements.
(2) Let G be amenable, H a normal subgroup, and 0: t 0(G/H) -+ l o(G) given by 0 (0)(x) = ¢(xH). If m E £(G), then mo11 E £(G/H), so that G/H is amenable.
(3) Let H be a normal subgroup of a group G such that both H and G/H are amenable. We have to show that G is amenable. Let 0 E la, (G) and p E £(H), q E £(G/H). Let 0,: G/H --+ C be given by ip(tH) = p((0t)1H) (t E G). Since Now p is left invariant, the function tPp is well-defined and belongs to check that (%Pto), = %PptoH (to E G) and the functional , -+ q(r(rp) belongs to
£(G). (4) Suppose that {H6: 6 E &} is an upwards directed family of amenable subgroups of G such that G = U6ea H5. (So A is a directed set and H., C H6 if -t < 6.) For each 6, choose ms E £(H6), and define n5 E 91l(G) by setting ns(4) = m6(O`H6). We can suppose that n6 - n weak* in 931(G). If x E G, then
INTRODUCTION
15
for some -y E A, z E H6 whenever 6 > ry, so that n6 (Ox) = m6(¢1Hbx) = n6(¢).
It follows that n E C(G) and G is amenable. (The proof of the corresponding result for semigroups is similar.)
(5) Let A be a directed set, let {G6: b E A} be a family of left amenable groups, and for each pair (o, 6) E A x A with o < 6, let p6,: G, -r G6 be a homomorphism such that whenever 11 < S S 0, we have pe,n = pe,s 0 PC"? The groups G6 and homomorphisms p6,, form a direct system of groups, and
if G is the direct limit of this system then, for each 6, there is a canonical homomorphism P6: G6 -+ G such that pn = ps o pt,n whenever n < S, and G = Ube& P5 (G6). The amenability of G follows from (2) and (4).
'0
The above results together with Example (0.6) suggest the following important questions. We introduce three classes of (discrete) groups. The first of these is EG ("elementary groups"): this is the smallest class of groups that contains all finite and all abelian groups and that is closed under the four processes of taking subgroups and of forming factor groups, group extensions, and upwards directed unions. The class AG ("amenable groups") is the class of amenable groups. The class NF ("no free") is the class of those groups not containing F2 as a subgroup. Since AG contains all finite and all abelian groups and is closed under the above four processes, we have EG C AG. Further, we have AG C NF ((0.6), (0.16(1))). We therefore have EG C AG C NF. The questions referred to above are: does EG = AG and does AG = NF? The conjecture that AG = NF (or, equivalently, that a discrete group is not amenable if and only if it contains F2 as a subgroup) is known as von Neumann's Conjecture. (It is sometimes referred to as Day's Conjecture.) It has been shown
(Ol'shanskii [4)) that, in fact, AG # NF. As might be expected, the proof is difficult. It is discussed in (4.30)-(4.33). R. I. Grigorchuk [Si] has recently
shown that EG # AG. We will show in (3.11) that EG # NF. Turning to (0.7), the nonamenability of SL(2, R) depends on its containing F2
as a discrete subgroup. This raises the speculation that a connected Lie group is nonamenable if and only if it contains F2 as a discrete subgroup. We will show (Chapter 3) that this result is true for almost connected groups (that is, for those groups that are extensions of a compact group by a connected group). The proof involves detailed Lie theory. Having discussed a number of the directions that (0.3)-(0.7) lead to, we now mention some extensions of the amenability notions of this chapter that often arise in practice.
(0.17) Other types of invariant mean. We start by commenting on the "left-wing" prejudice shown in our definition of amenability in (0.2). We can, after all, define a right invariant mean on G in a natural way. In fact, we define a right action n nx (x E G) of G on L. (G)' by requiring nz(') = n(z,), where x$(y) = -O(yx) (z, y E G, ¢ E L.(G)). A mean m on G is called right invariant if mx = m for all x E G, and the set of right invariant means on G is denoted
INTRODUCTION
16
by 9t(G). If m E £(G), then we can obtain n E 9t(G) by setting n(¢) = m(¢`), where ¢'(x) = 0(x -I). So left and right amenability for G are equivalent.
The set of means on G which are invariant (that is, both left and right invariant) is denoted by 3(G). We will see (Problem 1.26) that amenability for G is equivalent to the existence of an invariant mean on G. This is not immediately clear and in fact depends on the "Arens product" structure of We can also define the set N (G) of topologically right invariant means on G in the obvious way. Similarly, we can define 3t (G). Of course, results for left invariant means have right invariant versions. It is often important to study invariant means on subspaces of L,,. (G). Much
of the theory we have discussed extends trivially when L (G) is replaced by a subspace A of that contains 1. (In some cases, for example, in (0.1(1)), we also require A to be closed under complex conjugation.) We say that A is right invariant if ¢x E A whenever ¢ E A, x E G. A mean on A is left invariant case). The if it is invariant under the dual left action of G (just as in the set of left invariant means on A is denoted by £(A). So £(L (G)) =.C(G). The sets 91(A), 3(A) are defined in the obvious way for A with the appropriate kind of G-invariance. Similar results apply if A is invariant under the action of P(G).
For example, if A is right invariant for P(G) (that is, if ¢ * f E A if ¢ E A, f E P(G)), then £t(A) is defined in the obvious way. Similar considerations apply to 9it(A) and I(A). The advantage of studying invariant means on subspaces of L (G) is that these subspaces can have "nice" properties (e.g., continuity) not possessed by
(0.18) Semigroups and amenability. Let S be a discrete semigroup. Recall that 11(S) is the space of functions f : S -+ C such the E.es If (s) I < 00.
It is often convenient to write f = E8ES f(s)s. The norm II f III of f is, of course, just LsES If (s) 1. We make 11(S) into a convolution Banach algebra in the obvious way:
Bes
(Eass)(Ebes) seS
=uES 1 (Ea8b)tL (=Eaesist) sEStES st=u
a8s, EBES bes E 11(.9)). Of course S C 11(S), and S has a left action by is given as convolution on 11(S). The dual right action of S on 1(S) for groups: Os(t) = ¢(st) (s, t E S). As in the group case, S is left amenable if
there exists a mean on invariant under the dual left action of S on The set of left invariant means on S is denoted by £(S). Right amenability and (two-sided) amenability are defined in the obvious ways, and the corresponding sets of means are denoted by 9t(S) and 3(S), respectively. As in (0.15), it follows from (0.14) that every abelian semigroup is amenable. The study of left amenable semigroups goes back to the 1920s. In fact, Banach's well-known limits [DS, p. 73) are nothing other than invariant means on P, the additive semigroup of positive integers.
INTRODUCTION
17
There are a number of results in the theory of amenable semigroups that are straightforward generalisations of corresponding group results. However there are also a number of results for groups that do not readily extend to semigroups, the reason for this being that the algebraic structure of a semigroup is much more complicated, in general, than that of a group. At the same time, the semigroup situation allows for more variety than the group one; for example, no nontrivial group admits a multiplicative left invariant mean, yet many semigroups do ((2.29), Problem 2-36). The main interest in amenable semigroups, therefore, lies in the algebraic side. An example of an amenable group result that does not extend to the semigroup case is the F0lner condition of (3): it is not the case that a semigroup S is left amenable if and only if, for any e > 0 and finite subset C of S, there exists a finite, nonempty subset K of S such that )xK A KI JI KI < e. (Here, JAI is the cardinality of a set A.) The reason for this is the lack of cancellation in S in general. In this connection, see (4.9), (4.22).
Another example is that of the cardinality question. The question of the cardinality of £(S) is substantially more difficult than the corresponding group question. (See (7.22)-(7.27).) The theory of left amenable semigroups is often very elegant and has a combinatorial flavor to it. In our view, it does have a legitimate place in amenability theory.
There is a not inconsiderable literature on amenability for topological semigroups (cf. Problem 1-30, Solution). Invariant means on such semigroups do arise naturally in the context of weakly almost periodic compactifications. However, it has to be admitted that the study of amenable topological semigroups is very much in its infancy. One serious problem is the lack of good concrete examples
of such semigroups to serve as a basis for the theory. (There is a multitude of readily available examples of discrete amenable semigroups). For these reasons, we do not discuss amenable topological semigroups in detail.
(0.19) The scope of amenability. One of the interesting features of amenability is the wide range of mathematics with which it is involved. Another related feature is the way in which its ideas can be fruitfully modified to enable it to jump categories. These two features suggest that amenability is more than just a phenomenon in which analysts are interested-it does have a wider significance for mathematics as a whole. We now mention briefly some of the contexts involving amenability ideas that are discussed in this work and that we have not so far mentioned in this chapter. Amenability ideas arise in the following classical problem: when does a group
G acting on a set X not admit a Banach-Tarski paradox? The answer is: when X admits a G-invariant mean. In particular, G is amenable if and only if it does not admit a "paradoxical decomposition." This beautiful theorem by Tarski char-
acterises amenability purely in terms of translates of subsets of G-no mention
INTRODUCTION
1s
of means, measures, or functional analysis! The result is discussed in (3.12)(3.15). Problem 3-15 gives a topological version of the above characterisation of amenability.
Two major classical problems in amenability have been settled in the 1980s: these questions are the Banach-Ruziewicz Problem and von Neumann's Conjecture. Detailed discussions of these solutions are given in (4.27)-(4.29) and (4.30)-(4.33), respectively. The version of the Banach-Ruziewicz result that we discuss reads as follows: Let An be the natural normalised O(n + 1)-invariant measure on S" obtained from Lebesgue measure on R"+1. Then p, is the unique O(n + 1)-invariant, finitely additive, positive measure on S" of total mass I if and only if n > 2. The proof of this theorem uses the Kazhdan Property T as well as the theory of algebraic groups. A consequence of the theorem is that the only invariant mean on the orthogonal group On (n > 5) is Haar measure. We have already met von Neumann's conjecture in (0.16). The resolution of the conjecture falls into two parts. The first part is the construction of a group that is a candidate for a nonamenable group not containing F2 as a subgroup. The construction of the group, due to A. Yu. Ol'shanskii, involves complicated and lengthy combinatorial group theory, and we do not attempt it. The second part is of great interest from an amenability viewpoint and characterises amenable, finitely generated groups in terms of combinatorial group theory. This characterisation (due to Grigorchuk and Cohen) is used by Ol'shanskii to show that his group is not amenable, and it runs roughly as follows. Let G be a group on k generators u1, ... , uk. For each n, let Nn be the number of words of length n in u;, ui 1 whose product in G is e. Then (roughly) G is amenable if and only
if iN"I11"-'(2k-1). Amenability applies to the statistical theory of testing hypotheses. The HuntStein Theorem ensures the existence of invariant minimax tests when the group of an invariant test problem is amenable. This and other statistical applications of amenability are discussed in (4.24)-(4.26), Problem 4-19. Amenability also proves relevant to the study of transience/recurrence of random walks on locally compact groups and to Poisson spaces ((6.46), (6.47)). Amenability's impact on the representation theory of locally compact groups
is largely through the remarkable "weak containment property". Let G be a locally compact group. The representation theory of G is equivalent to that of its enveloping C'-algebra C' (G), the completion of L1(G) under its largest C-norm. Unfortunately, C' (G), though categorically "nice", is hard to get hold of in general. On the other hand, there is available a natural, accessible C'-algebra CJ (G) associated with G: CC (G) is the C'-subalgebra of B(L2(G)) generated
by F2(L1(G)), where 72 is the left regular representation of G. One can do calculations with Cj (G)! Unfortunately, the representation theory of Ct (G) picks up only some of the irreducible, unitary representations of G, viz., those which are "weakly contained" in 7r2. In the best of both worlds, we would have C' (G) = C1 (G). When is this true? The answer ((4.21)) is that Ci (G) = C' (G)
INTRODUCTION
18
if and only if G is amenable. Polynomial growth has, as might be expected, an even stronger impact on the representation theory of solvable Lie groups ((6.45)). Indeed, polynomial growth has a particularly rich range of applications. For example, if a complete Riemannian manifold M has nonnegative mean curvature, then its fundamental group Tri (M) has polynomial growth ((6.41)). Symmetry of the group algebra L, (G) is related to amenability: in fact, every almost connected, locally compact group with symmetric group algebra is amenable. The
converse is false. However, a beautiful theorem due to Ludwig ((6.50)) states if G is connected and has polynomial growth, then G is symmetric. A good example of how amenability ideas can cross mathematical categories is afforded by the theory of amenable Banach algebras. This theory, which is largely the creation of B. E. Johnson, is discussed in (1.30). A Banach algebra A is called amenable if the first cohomology group H' (A, X') = {0} for every dual Banach A-module X'. This means that every continuous derivation D: A -+ X' is inner. At first sight, there does not seem to be any connection between invariant means and derivations. However, Johnson proved that if G is a locally compact group, then G is amenable if and only if the Banach algebra LI(G) is amenable. This justifies the terminology amenable Banach algebra. Banach algebra amenability has proved very fruitful when specialized to operator algebras. Amenability for a C'-algebra A is equivalent to a strong "tensor product" property of A known as nuclearity. A brief discussion of amenable C*algebras is given in (1.31). Amenability for a von Neumann algebra, obtained by modifying Banach algebra amenability in a sensible way, is equivalent to a number of very important properties: injectivity, semidiscreteness, Property P, and approximate finite-dimensionality. A discussion of amenable von Neumann algebras is given in (2.35). It was not realistic to give a thorough account of amenable von Neumann algebras, and instead, we have tried to show how the various formulations of amenability in that context relate to one another and to group amenability.
In the study of the Fourier algebra A(G) of a locally compact group G, amenability of G translates into a familiar Banach algebra property. When G is abelian, A(G) can be identified with L, (G); more generally, the commutative Banach algebra A(G) is the predual of the von Neumann algebra VN(G) generated by the left regular representation of G with a natural "pointwise" multiplication. It turns out ((4.34)) that G is amenable if and only if A(G) contains a bounded approximate identity. A recent development of amenability is that of K-amenability. In Kasparov's KK-theory, a (countable discrete) group G is called K-amenable if K°(C' (G)) K°(Ct (G)), where K° denotes K-homology. So K-amenability is a homological version of the weak containment property. Remarkably, F2-the bete noire of amenability-is K-amenable! (References: Kasparov [S1), Cuntz [S21, Julg and Valette [S1].) We started this chapter with Lebesgue's question on the uniqueness of his integral. The reader will probably agree that in the last few pages we have
INTRODUCTION
20
"come a long way." We hope to have demonstrated the range of amenability ideas.
References Banach [1], [2), Banach and Tarski [1), Chou [9), Day [4), [9), Fslner [2), Greenleaf [2), Guivarc'h [1), Hulanicki [2), Lebesgue [1), Milnes [Si), Namioka [1], von Neumann [1), Reiter [R], and Tarski [2).
Problems 0 1. Let X be a set and for each m E 9R(X), define a set function p, by is a bijection from p,,,.(E) = m(XE) (E C X). Prove that the map m
93(X) onto the set of positive, finitely additive measures on X of total mass 1.
2. This is the topological version of Problem 1 above. Let X be a locally compact space and p be a positive, regular Bore] measure on X. Let . '(X) be the o-algebra of p-measurable subsets of X. Let m - p,, be the map defined as in Problem 1 from 93t(X), the set of means on La0 (X, p), into the set of measures
on 4'(X). Prove that this map is a bijection onto the set of positive, finitely that vanish on p-locally-null sets. additive measures of total mass 1 on Deduce that the set of left invariant means on a locally compact group can be identified with the set of left invariant, positive, finitely additive measures on X(G) of total mass 1 that vanish on .1-locally-null sets. 3.
Let A be a subspace of L (G) containing 1 and closed under complex
conjugation. Establish the analogue of (0.1) with A in place of L,o(G).
4. An element F E is called left invariant if xF = F for all x E G. The set of left invariant functionals F is denoted by 3,(G). Show that 31 (G) is an invariant subspace of Lfl, (G)' containing £(G). What is 3 (G)i n L.,, (G)^? 5.
Let G be a group and X be a left G-set with ex = x for all x E X.
Let A C X and suppose that there exists a G-invariant finitely additive, [0, 001valued measure p on X such that p(A) = 1 . Prove that if n > 1, sI, ... 1 8 n E G, Q i, ... , an E R, and E= I aiXs, A > 0, then F,", a; > 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
AMENABLE LOCALLY COMPACT GROUPS
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 <- I (z¢)(vx - vy)I 5 II0II Ilvx - vyll
AMENABLE LOCALLY COMPACT GROUPS
28
and I(AO)Z(Z) - (,U?P)v(z)I <- I(zp)^(Ox -1Pv)I <- 11J All 110X -1pvll.
(2)
The first and second assertions of the proposition follow using (1), (2), and (1.2)' Replacing t(i in the second assertion by ¢v, it follows that pqz E Ur(G). Similarly, pOv E Ul(G) and so belongs to U(G). Q (1.5) COROLLARY. Each of the spaces C(G), Ur(G), and U(G) is invariant
for P(G). In the next result, {p6} is a bounded, approximate identity for Ll (G) in P(G) such that the support C8 of each p6 is compact, and every neighbourhood of the identity e contains Ca eventually [HR1, (20.15)].
(1.6) LEMMA. If 0 E Ur(G), then Ilpa -Oil - 0. PROOF. For each 6, we have, using (1.1.(2)), 11 014 -
fl =
suG
lfc, (,(tx) - O(x)) dp6(t)1 < SUP Hot -,Oil,
and the last term tends to 0 since {C6} "contracts" to e and 0 E Ur(G). 0 The next result is of great importance. Recall ((0.17)) that .C(A) and £t(A) are the sets of left invariant and topologically left invariant means on a suitable subspace A of Loo(G). The next result is the key for relating invariant means on subspaces of Loo(G). References are Reiter [14], Greenleaf [2], Hulanicki [4], and Namioka (2). (1.7) PROPOSITION. Let Elo, vo E P(G). If m E £(U(G)), then m' E Lt (G), where m'(O) = m(poq5vo), and m' is independent of vo. If n E £(Ur(G)), then n' E .Ci(G), where n'(¢) = n(Ovo), and n' is independent of vo.
PROOF. Let po,vo E P(G) and m E £(U(G)). Let Oo > 0 be in Lm(G). Define F E L1(G)' by
F(v) = m(poOov'), where v` is defined in (1.1). Note that F is well defined by (1.4). Now for all
v E L1(G), z E G, we have (xv)` = v'x-1. Since m E .C(U(G)), it follows that F(xv) = F(v). Regarding the elements of Ll (G) as functions, we obtain F(x * f) = F(f) (f E Ll (G)). Now x * f = f x-1 for f E CM(G). So F, restricted to CC(G), is a positive, left invariant, linear functional on CC(G) and so is a multiple of the restriction of the left Haar measure A to Cc (G). Since CC (G) is dense in L, (G) and F is continuous, it follows that F is a multiple of A regarded i We also use the fact that if 0' E 1. (0) is such that the map x - 0'x is norm continuous, then 0' E Ur(G). Indeed, the continuity of 0' follows from I0'(x) - 0'(v)I = I0'x(e) - O'v(e)I <- II0'x - O'vII.
AMENABLE LOCALLY COMPACT GROUPS
29
as a linear functional on Ll (G) and so is constant on P(G). Since L. (G) is the span of such functions to and P(G) = {z/*: v E P(G)}, we obtain
m(po¢v) = m(po n/)
(¢ E L,, (G), v, v' E P(G)).
So m' does not change if vo is replaced by another element of P(G), and hence m'(¢v) = m(po¢vvo) = m(po¢vo) = m'(b). Since m' trivially belongs to i9)1(G), it is clear that m' E.Ct(G). The corresponding assertions for n' are proved similarly. The next result is generalized in Problem 2-15.
(1.8) COROLLARY. If B is Ur(G) or U(G), then £(B) = £t (B). PROOF. Let B = Ur(G), m E £(B), and {µ6} be as in (1.5). Then we obtain from (1.6) that µ6m -+ m weak* in B'. Now µ6m E £t(B) by (1.7). Since £t(B) is weak* closed, m E £t(B) and £(B) C £t(B). The reverse inclusion is givenby (0.9). The proof for the case B = U(G) is similar, with µ6mµ6 in place of µ6m. (1.9) COROLLARY. Let B be a subspace of L. (G) with 1 E B. If B D Ur(G) (so that B is right invariant for P(G) ((1.4))), then the mapping m mIB is a bijection from £t (G) onto £t (B). If B D U,(G) U U1 (G) (so that B is invariant for P(G)), then the mapping p -+ pls is a bijection from 3t(G) onto 3t (B).
PROOF. Suppose that B D Ur(G). Using (1.4), elements m, n E £t (G) coincide if and only if they coincide on Ur(G). It follows that the map m -» mIB
is an injection from £t (G) into £t (B). By (1.7), if p E £t (B) and n is the restriction of p to Ur(G), then n' E .Ct (G), and if 0 E B, then p(.0) = p(Ovo) = n(givo) = n'(0)
so that the map m -» mlB is a bijection as required. The other assertion of the corollary is proved similarly, noting that if m E 3(U(G)), then the mean m' of (1.7) belongs to 3t(G). As we shall see in (7.17), the map m -+ mlc(G) is not, in general, a bijection from .C(G) onto £(C(G)). (1.10) COROLLARY. Let B be a right invariant subspace of If B D U(G) [B Ur(G)J, then G is amenable if and only if £(B) # 0 12t(B) # 0].
In particular, Z(G) # 0 if and only if £t (G) 36 0. Our next objective is to show that every closed subgroup H of an amenable locally compact group G is itself amenable. To prove this, we require some results relating Haar measures AG and aH on G and H respectively. A good discussion of these results is given in [R., Chapter 81. An important concomitant of these results is Weil's formula, which we will need later and is given in (1) below.
AMENABLE LOCALLY COMPACT GROUPS
30
(1.11) Well's formula and Bruhat functions (R). Let AG, AH be fixed left Haar measures on G and H. If H is normal in G, then it is reasonable to expect, crudely, that there exists a left Haar measure AG/H on G/H such that "AG = AH X AG/H". For example, with G = R2, H = R x {O}, G/H"="{0} x R,
we do indeed obtain dx dy = dx x dy! Making this precise, it turns out that when H is normal in G (or more generally, when i H(h) = OG(h) for all h E H, where t1H, AG represent the modular functions on H and G respectively), we have Weil's formula: IG c1(x) dAG(x) =
(1)
dAG/H (xH) IC/N
r
JH
O(xh) dAH (h)
(.0 E C.(G)).
Here, AG/H is a positive, regular Borel measure on G/H that is left invariant under the action of G. There are a number of situations that arise in practice in which AH 56 OGIH so that (1) does not apply. However, in general, there exists (0, co) and a quasi-invariant measure AG/H such a continuous function q: G
that (2) f ¢(x)q(x) dAG(x) = IC/N dAG/H(xH) IN O(xh) dAH(h)
(0 E
(Quasi-invariance for AG/H means that AG/H (xE) = 0 whenever x E G and E is
AG/H-null.) If H is unimodular, then we can take q(x) = AG(x)-I. In general, the function q can be produced using Bruhat functions for H, and these are the functions which will concern us in our next result. Roughly, a Bruhat function for H is a function on G that behaves like a continuous function with compact support on left H-cosets and integrates "well" with respect to AH. We now give the precise definition. A Bruhat function for H is a function /3: G --+ R such that: (i) if C E F (G), then there exists l' > 0 in C,(G) such that I9ICH = '0{CH;
(ii) for all z E G, we have IH ;8(xh) dAH (h) = 1.
Such a function can be shown to exist. (Indeed, the construction uses an argument closely related to that of (4.11).) Given a Bruhat function /3, we can define the function q: G R of (2) by q(x)
fH $(xh)t1G(h)AH(h'I) dAH(h).
= For future use, we note that if H is also normal, then there is a natural *-homomorphism TH: M(G) -y M(G/H) given by THµ(E) = µ(Q-1(E)) (E E .l' (G/H)), where Q : G - G/H is the quotient map. It turns out that QH = THI (L, (G)) is a *-homomorphism from Ll (G) onto Ll (G/H) = Ll (G/H, AG/H ). (In function terms, we have QH f (xH) = fH f (xh) daH (h).) So the dual map QH is an isometric *-homomorphism from L.,(G/H) into L. (G). References for the next result are Rickert [2) and Greenleaf [2[. We follow the proof in [R, Chapter 81 using properties (i) and (ii) of Bruhat functions.
AMENABLE LOCALLY COMPACT GROUPS
31
(1.12) PROPOSITION. Every closed subgroup of an amenable locally compact group G is amenable.
PROOF. Let H be a closed subgroup of G, and let i be a Bruhat function for I.. (G) by setting H. Define I?: C(H)
4,(0)(x) = f 6(x 1h)O(h) d,\x(h).
x
Using (ii), 0 is a norm-decreasing, unit-preserving, linear map. We claim that the map x -» (/3x)IH is norm-continuous from C into C(H). For suppose otherwise. Then we can find e > 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
AMENABLE LOCALLY COMPACT GROUPS
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.)
AMENABLE LOCALLY COMPACT GROUPS
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 <
Oi < 1, {b,(R,) = {1}, O,(R,,) = {0} for i # j, and 0 < 01 + 02 < 1. Since Oixi = 1 (x, E R,), we have m(¢1) + m(02) = 2. A contradiction results. Conversely, suppose that T contains precisely one minimal right ideal R. Then
R is the kernel of T, and we can find an idempotent e in R with eT = eR = R. If a E T, then se E R, so that se = ese. Now eRe is a compact group and so has a Haar measure p. If 0 E C(T), s E T, and u E eRe, then (0s)(u) = 0(seu) _ .0IeRe(ese)(u).
It follows that the functional ¢
µ(¢[eRe) belongs to .C(C(T)).
(1.18) COROLLARY. There exists an invariant mean on C(T) if and only if the kernel of T is a group.
(1.19) COROLLARY. A finite semigroup is left amenable if and only if it contains precisely one minimal right ideal. In If S is finite, then ll (S) is finite-dimensional so that 9l(S) = particular, if S is also left amenable, then .C(S) fl P(S)' = £(S). This leads us to our second question which we now answer. (The proof uses the later P(S)n.
result (1.22).) References for the following result are Sorenson (2], Lau [2], and Granirer [2].
AMENABLE LOCALLY COMPACT GROUPS
34
(1.20) PROPOSITION. The following are equivalent for a left amenable semigroup S:
(i) £(S) n P(S)n # 0; (ii) S contains a finite left ideal; (iii) S contains a group which is a finite left ideal.
PROOF. Suppose that a =
E P(S) and that a E .C(S). Note
that a8 > 0 for all s and that E8ES a, = 1. Let k = max$Es a8 and L = {s E S: a8 = k}. Since & E ,C(S), we have &(s-'E) = &(E) for all s E S, E C S. Let so E L. Then k > 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) _
AMENABLE LOCALLY COMPACT GROUPS
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.
AMENABLE LOCALLY COMPACT GROUPS
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.
AMENABLE LOCALLY COMPACT GROUPS
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
38
AMENABLE LOCALLY COMPACT GROUPS
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.)
AMENABLE LOCALLY COMPACT GROUPS
39
(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.
40
AMENABLE LOCALLY COMPACT GROUPS
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
AMENABLE LOCALLY COMPACT GROUPS
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
<- ML a N 5 MIIaII, and similarly, IIaI 15
MIIRahI 5 MIIaII. So A can be identified with a closed ideal of &(A), and the &(A)-norm on A coincides with the original A-norm if M = 1. The latter happy situation occurs fairly often, in particular, when A is a C'-algebra, or a group algebra L1(G), or the Fourier algebra A(G) when G is amenable ((4.34)). The algebra A(A) can be concretely realised in the natural way in a number of cases: for example if A = C0(Y) (Y locally compact Hausdorff), then A(A) = C(Y), while if A is the C'-algebra of compact operators on a Hilbert space Sj, then
A(A) = B(5). Further, A(L1(G)) = M(G) (Wendel's Theorem) and, as we shall see in (4.34), A(A(G)) = B(G) when G is amenable. After the above digression, we now discuss how a neo-unital Banach A-module
X is naturally a Banach A(A)-module. Indeed, if T E A(A), then Te6, e6T E A
and {(Te6)e}, {e(e6T)} converge in norm for each e E X by the neo-unital property. We simply define TC = lim(Te6)e, T = limg(e6T), and X becomes a Banach A(A)-module which is unital in the sense that IC = C = CI (C E X), where I is the identity of LE(A). The actions of A(A) on X are point-norm continuous for the multiplier topology, that is, if T6a -» Ta for all a E A, then T61; T. for all is E X. A number of familiar properties of amenable locally compact groups have analogues for amenable Banach algebras. The following is an analogue of (1.13).
(iii) Let A be a Banach algebra and J a closed, two-sided ideal of A with a bounded approximate identity. Then A is amenable if and only if both J and
AfJ are amenable.
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AMENABLE LOCALLY COMPACT GROUPS
[We shall prove only one implication, the other being left to Problem 1-32. Suppose that both J and A/J are amenable. Let X be a Banach A-module and D: A -+ X' a derivation. Since J is amenable, D[a is inner, and by subtracting an inner derivation from D, we can suppose that D(J) = {0}. If a E A, b E J,
then 0 = D(ab) = (Da)b, 0 = D(ba) = bDa, and it follows that D(A) C Xf, where XJ is the closed linear subspace of X spanned by JX U XJ. Clearly, X/Xr is a Banach A/J-module in the natural way, and so Xa = (X/XJ)` is a dual Banach A/J-module. Now Di : A/J XJ is a derivation, where Di(a + J) = Da for all a E A. Since A/J is amenable, Di is inner, and it immediately follows that D is inner and A is amenable.)
Now let G be a locally compact group and A = L1(G). Then (neo-unital) Banach LI(G)-modules and LI(G)-derivations can be usefully regarded as G_ modules and G-derivations as follows. Let X be a neo-unital Banach LI (G)module. Since A(LI (G)) = M(G) [HR2, (35.5)], X is a unital Banach M(G)module. In the obvious way, G C M(G), and it follows that X is a G-set, g with each map e -+ gx a bounded, invertible element of B(X), and eg = g = ge for all e E X. Further, there exists K > 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.
AMENABLE LOCALLY COMPACT GROUPS
43
[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
AMENABLE LOCALLY COMPACT GROUPS
44
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
AMENABLE LOCALLY COMPACT GROUPS
45
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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AMENABLE LOCALLY COMPACT GROUPS
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.
AMENABLE LOCALLY COMPACT GROUPS
47
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.)
AMENABLE LOCALLY COMPACT GROUPS
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.
AMENABLE LOCALLY COMPACT GROUPS
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.
AMENABLE LOCALLY COMPACT GROUPS
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.
THE ALGEBRA OF INVARIANT MEANS
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 <- MIIFIIIICII
Let X(S) be the linear subspace of
spanned by {FC: F E X',C E X}.
Now let P E X(S)' and F E X'. Then we definePF:X -+Cby PF(C) = P(FC)
(£ E X).
It follows from (2) and the linearity of P and F that PF E X' and IIPFII < MIIPIIIIFII Let 31(X) = IF E X': sF = F for all s E S). 'The space X could also be called a right Banach S-module (cf. (1.30)).
THE ALGEBRA OF INVARIANT MEANS
54
(2.5) Some simple facts. We now list some elementary facts about Ft: and PF above, leaving the proofs to the reader. (i) The map (F, C) -- Fg is bilinear from X' x X into X(S). (ii) The map (P, F) - PF is bilinear from X (S)' x X' into X'.
(iii) Ifs E S, then (1)
s(FC) = (sF)C and (FC)s = F£s,
so that X(S) is an invariant subspace of (iv) For $ E S,
s(PF) = (sP)F and (PF)s = P(Fs). (v) Suppose that F E 31(X). Then FC = F(C)1, and if 1 E X(S), then
PF = P(1)F. (vi) If P E 5 (X(S)) then PF E 3,(X). FCo is continuous from X' into X(S) (vii) If Co E X, then the map F when X' is given the weak" topology and X(S) is given the topology of pointwise convergence on S.
(viii) If Fo E .V, then the map P
PFo is continuous from X(S)' into X',
both spaces being given the weak` topology.
(2.6) Discussion. Now suppose that X above is a right invariant subspace In favorable circumThen both X and X(S) are subspaces of stances, we might hope that forming X(S) does not take us outside X. This leads to the definition of introversion below. Four examples of left introverted spaces are given in (2.11). If G is a locally compact group, then C(G) is not in general, fails spectacunormally left introverted ((2.33)). The space is always larly to be left introverted ((7.29)). However, it is trivial that of
left introverted. As far as amenability theory is concerned, left invariant means are particularly well-behaved in the presence of left introversion. (For example, Problem 2-15 shows that for left introverted spaces, left invariance for S is the same as left
invariance for the semigroup PM(S) of probability measures on S, S being a locally compact, separately continuous, Hausdorff semigroup.) It is convenient to allow the P in the formula PF of (2.4) to be a continuous linear functional on any subspace Y of containing X(S); indeed, if P' is
the restriction of P to X(S), then we just take PF to be P'F. The following definitions are formulated for a locally compact group, but apply in the obvious way to the case of a discrete semigroup S. Let G be, as usual, a locally compact group, and let Y be a right invariant subspace of L,,. (G). We define Y to be left introverted if Y(G) C V. Note that
if Y C C(G) is left introverted, then it is also left invariant, since for ¢ E Y, xO = iO E Y (i(¢) = O(x) for 0 E Y).
THE ALGEBRA OF INVARIANT MEANS
55
The notion of a right introverted subspace Z of L.(G) is defined similarly. Here, Z is left invariant, and in place of Y(G), we put the subspace of 1. (G) spanned by functions of the form CF, where
CF(s) = F(sC).
An invariant subspace of L.(G) is said to be introverted if it is both left and right introverted. Normally, we will concentrate on the left introverted case. If Y is a left introverted subspace of Lo,(G), then, since Y(G) C Y, we can
define, as in the third paragraph of the present (2.6), the "product" PQ E Y' whenever P, Q E Y'. The next corollary shows that Y' is actually a Banach algebra under this product. In the next result, PQ E X(S)' is the product of P, Q E X(S)'-this makes sense since X(S) is left introverted. (2.7) PROPOSITION. Let S be a semi group and X be a right Banach S-space.
Then X(S) C l.(S) is left introverted, and (PQ)F = P(QF)
(1)
for all P,QEX(S)',FEX'. PROOF. It follows from (2.5(iii)) that X(S) is right invariant in 1,.(S). Now
if P E X(S)', FE X', CE X, and '= FeEX(S),then P¢(s) = P((F£)s) = P(F(Cs)) = (PF)C(s)
(2)
so that P¢ = (PF)g E X(S). Thus X(S) is left introverted. Further, using (2), with PQ in place of P, ((PQ)F)(C) = PQ(FC) = P(QO) = P((QF)C) = P(QF)(C) and (1) follows.
(2.8) COROLLARY. If Y is a left introverted subspace of L.(G) or l"(S), then Y' is a Banach algebra with product (P, Q) -+ PQ. The proof of the next proposition is routine.
(2.9) PROPOSITION. Let Y be a left introverted subspace of L,Q(G) with I E Y. Then (i) 1 E Y(G); (ii) if s E G, P, Q E Y', then
s(PQ) = (sP)Q,
P(sQ) = (Ps)Q,
and P(Qs) = (PQ)s;
(iii) 93t(Y) is a subsemigroup of Y';
(iv) if Q c-,31(Y) and P E Y', then PQ = P(I)Q. Before going any further, we need examples of left introverted subspaces.
THE ALGEBRA OF INVARIANT MEANS
56
(2.10) Definitions. A separately [jointly] continuous semigroup is a semigroup S which is also a topological space for which the multiplication map (x, y) - xy is separately [jointly) continuous. Let S be a separately continuous semigroup. As in (1.3), U,(S) is the space of functions Qi E C(S) such that the map s 4,s is norm continuous. (Note that C(S) is a right invariant subspace of I.,, (S).)
The set of continuous functions 0 for which the map s --> 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
THE ALGEBRA OF INVARIANT MEANS
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.
THE ALGEBRA OF INVARIANT MEANS
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
k
J>i(A6F'?)(£?) J=1
f=1
0
it follows that 11mli < 1. Clearly, m(1) = 1. From (2) and (ii), (a) we have m(Os) = m(¢) for all ¢ E B. Thus m E £(B), and (ii) implies (i). Now suppose that P satisfies (a) and (b) of (ii). From (a), T,(PF) = PF, so
that A6PF = PF. Use (b) to deduce that P(PF) = PF, so that P2 = P. Since IIA61151, we have 11PII < 1.
0
(2.16) Discussion. We now continue with a discussion of some semigroup fixed-point theorems related to invariant means. Two typical theorems of this kind will be proved. Day's Fixed-Point Theorem ((1.14)) can be regarded as a discrete version of these theorems. The procedure involved in proving all of these theorems is as follows. We are given a separately continuous semigroup S and are concerned with determining
THE ALGEBRA OF INVARIANT MEANS
60
under what circumstances there exists an S-fixed-point in a compact, convex subset K of a locally convex space when S acts affinely and in a suitably continuous fashion on K. (We require an S-fixed-point in every such K.) The Banach space X of continuous, complex-valued, affine functions on K is a right invariant
subspace of t (K). Let B be a left introverted subspace of C(S). If we know that X(S) C B and that £(B) 34 0, then it is easy to find an S-fixed-point in 931(X), viz., mp where m E .C(B) and p E 931(X). Since K can be identified canonically with +.111(X), we
have a fixed-point in K. The problem, then, is to find a suitable B such that X(S) C B whenever S acts in an appropriately continuous fashion on a K. A good help in resolving
this is to find a B so that the natural left action of S on 91i(B) satisfies the continuity condition. In the two cases with which we are concerned here, the B's associated with the joint and separate continuity conditions are Ur(S) and WUr(S) respectively. For other fixed-point theorems, see Problems 2-16, 2-17. References for Theorem (2.23) proved here are Furstenberg (1), Rickert (2), and Mitchell (6).
(2.17) Definitions. Let XI and X2 be left S-sets. A mapping 0: XI -+ X2 is called an S-morphism if 4(sx) = sO(x) for all s E S and x E X1. If 9 is a family of left S-sets, then an element Xo E . ' is called a source for `p (or a ,V-source) if for each X E °, there exists an S-morphism 0 mapping Xo into X. The significance of .-sources for semigroup fixed-point theorems lies in the following trivial, but important, proposition.
(2.18) PROPOSITION. An S-morphism carries an S-fixed-point into an Sfaxed-point. Further, if a.'-source contains an S-fixed-point, then every X E . ' contains an S-fixed-point.
(2.19) Affine maps. Let K be a (nonempty) compact, convex subset of a locally convex space E. Recall that A f (K) and affine left S-sets were defined in (1.13). Clearly, A f (K) is a closed, linear subspace of C(K) containing 1.
(2.20) PROPOSITION. The mapping T, where Tk is the restriction of k to Af(K), is an affine homeomorphism from K onto 9R(Af(K)).
PROOF. Since {fJX: f E E'} C Af(K) and E is locally convex, it follows that T is one-to-one. The map T is obviously affine and continuous. The equality T(K) = 93(Af(K)) follows from (0.1), and compactness yields that T is a homeomorphism.
(2.21) Notations. It is obvious that if K is an affine left S-set, then both C(K), Al (K) are right invariant subspaces of I.. (K). An important example of an affine left S-set is obtained as follows: if B is a right invariant subspace of i,o(S) with 1 E B, then 931(B), with the relative weak' topology, is an affine left S-set with respect to the maps m sm. The essence of each of the fixed-point theorems to be proved is that for a suitable family g' of affine left S-sets, there
THE ALGEBRA OF INVARIANT MEANS
61
is a W-source of the form Ti(B) for some B. We now introduce the two families of affine left S-sets to be considered.
The family of affine left S-sets K for which the map (s, k) - sk is jointly [separately] continuous is denoted by . (98]. Let Kl = MT(U,(S)) and K. = s911(WUr(S)).
(2.22) PROPOSITION. The affine left S-sets Kl and K. are sources for Aj and Ae respectively.
PROOF. We show first that Kl E . and K. EX. If m6 - m in Kl, 86 --' s in S, and 0 E Ur (S), then 1(s6md - sm)(4')[ < [[[Os6 - 0811
0
so that Kt EE.AjIt is trivial to check that K. E A.
It remains to show that Kl and K. are sources for . and .g respectively. Let K be an affine left S-set. It is sufficient to show that for some Fo E 9) (Af(K)) and all ¢ E Af(K), the function Fo¢ E l,o(S) is in the appropriate space of functions on S for appropriate K. For then, the map P -+ PF0, where PFo(q5) = P(Fo¢), is an S-morphism from Kl or K8 into K. Identifying Fo with
some ko E K ((2.20)), we have to show that ko¢ is in the appropriate space. Note that ko4'(s) = qS(sko) (s E S) and that obviously ko4' belongs to C(S) in both cases.
Suppose, first, that K E Aj. Since the map (s, k) -+ sk is continuous and K is compact, it follows easily that the maps s -+ Os are norm continuous from S into C(K), and as [{(koq5)s - (ko¢)t[J < 114s - 4't[[, it follows that ko¢ E Ur(S) as required. Now suppose that K E .YB. Let s6 --+ s in S and F E M(C(S)). Let k1 E K
be such that Fko = k1 in fl(A f(K)). Then F((ko4')s6) = (Fko)(4's6) = q5(s6ki) -.O(ski) = F((ko¢)s) so that ((0.1(ii))) ko¢ E WU,.(S) as required. (2.23) THEOREM. There exists an S-fixed-point in every member of [J if and only if there exists a left invariant mean on U, (S) (WU,(S)). PROOF. Use (2.22) and (2.18).
0
(2.24) THEOREM. The group G is amenable if and only if every affine left G-set K for which the map (x, k) -+ xk (x E G, k E K) is jointly (separately] continuous contains a fixed-point for G.
PROOF. The theorem follows from (1.10), with B = Ur(G) [WU,(G)], and (2.23).
0
is an abelian C'-algebra. Let G be a locally compact group. Then Let 4'(G) be the carrier space of L00(G). Then the map ¢ -y ¢ is an isometric *-isomorphism from L,o(G) onto C(4k(G)), and dualising this map and using
THE ALGEBRA OF INVARIANT MEANS
62
the Riesz Representation Theorem, we obtain a linear isometry F - F from Loo(G)' onto the measure space M(S(G)) = C(,P(G))'. Clearly
F(4) = F(¢) _ f {c)'(p) dfr(p) Similar considerations apply to the case of a semigroup S; in this case, 4(S) is just $S, the Stone-tech compactification of S, and, of course, Lc(S) = loo(S). One might hope that the Arens product on too (S)' can be realized by some reasonable convolution formula involving O(S). Unfortunately, this is rarely the case.
Let E be either a locally compact group or a discrete semigroup. Our next proposition collects some useful facts about invariant means when regarded as elements of M(O(E)) as above. Note in (i) below that 4'(E), as a set of linear functionals on Lao(E), is a subset of the Banach E-space Loo(E)', is the Borel algebra of O(E). In (iii), Ext K is the set and that in (ii), of extreme points of a compact convex subset K of a locally convex space.
(2.25) PROPOSITION. (i) t(E) is a E-invariant subset of Lc(E)' with the maps P sP, P -+ Ps continuous on O(E) for each s E E. Further, with C(O(E)) and M(ob(E)) = C((I(E))' given their natural Banach E-space structures, we have (1)
(¢s)^ = ¢s,
(sF)^ = at
for all,0 E Loo(E), s E E, and F E Loo(E)'.
(ii) If m E Tt(E), then m E Z(E) if and only if yn(s'IE) = m(E) for all s E r,, E E .3(O(E)). Further, if m E .fr(E), then the support S(fn) of in is a (compact) left invariant subset of O(E). (iii) Let E be left amenable with the discrete topology, and let X be a closed, left invariant subset of ib(E). Then there exists mo E Ext.C(E) with the support S(m,o) of rno contained in X.
PROOF. (i) Let P E 4(E) and s E E. Then for ¢ E Loo(E), (2)
sP(¢) = P(0s),
(Ps)(4) = P(so)-
Since (0s) (Vs) = (OO)s, (s0)(sO) = s(Otli) (0,0 E L. (E)), it follows that sP, Ps are multiplicative on Loo (E) and so belong to (E). Thus,P(E) is E-invariant. Noting that the topology of 4,(E) c Loo(E)' is just the relative weak' topolPs are continuous for each ogy, we see from (2) that the maps P sP, P
sEE. It follows that we have a natural Banach E-space structure on C(4)(E)) given by
iIis(P) = (sP),
(stbb)(P) = ty(Ps).
Of course, M(' (E)) is given its dual structure. The equalities in (1) follow immediately from the first equality of (2) and the dual version.
THE ALGEBRA OF INVARIANT MEANS
63
(ii) Let in E £(E). We will show that
m(s-'E) = in(E)
(3)
(8 E E, E E 3($(E))).
(I am grateful to John Pym for suggesting the following argument.) Suppose that C is an open and closed subset of 4(E). Then Xc E C(4?(E)) and so we can find 41 E L. (E) with 4) = xc. Using (i),
tn(s-IC) = in(Xcs) = m(4)s) = m(4)) = tn(C).
(4)
Now let U be an open subset of 4(E) and A = {Xc : C is an open and closed subset of 4)(E), C C U}. Since 4;(E) is zero-dimensional, we have that A is upwards directed, and
Xu = sup{Xc: Xc E Al.
Also, the set B = {Xcs: Xc E Al is upwards directed, and supB = X8-lu. Further, every element of A, B is lower semicontinuous, and so using [HR1, (11.13)) and (4), we have
rn(s 'U) = sup{in(s 1C): Xc E A} = sup{m(C): Xc E A) = rn(U). Since the family of open subsets of 0 (E) generates . (4;(E)), we see, using the Monotone Class Lemma, that (3) holds.
Conversely, suppose that m E'9)1(E) and (3) holds. Let F E .af(E) and let C be the open and closed subset of 4'(E) such that XF = Xc From (3) with E = C, we obtain m(XFS) = m(XF), and using the density of the span of the set of characteristic functions in L.,,(E), we obtain in E Z(E). Thus the first assertion of (ii) is established. Now suppose that m E .L(E) and let D be the support of tn. Let p E D and suppose that sp 0 D. Then we can find an open subset U of 4i(E) containing p and such that U fl D = 0. Then by the first assertion,
0 = ?h(U) = tn(a-1U) > 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<4)<1,¢(u)=1, and ¢(X) = {0}. Since X is left invariant, 4s(p) = 0 for p E X, and so sµ(ip) = 0.
It follows that S(su) c X. Obviously sµ E PM(X) = K. Now K is an affine left E-set under the maps y - sµ. Since E is left amenable with the discrete topology, Day's Fixed-Point Theorem yields an element v E K with sv = v. Clearly, v = n for some n E .C(E). Thus the convex set 91, where
IN = {m E C(E): S(m) C X},
THE ALGEBRA OF INVARIANT MEANS
64
is not empty. Use of the Tietze Extension Theorem shows that the weak' topol-
ogyontheset fl ={ne:mE9R}={pEK:sµ=ufor all sES}CM(X) corresponds with the weak' topology on 9t, and since the former set is obviously weak' compact, so also is 92.
By the Krein-Mil'man Theorem [DS, V.8.4], we can find mo E Ext9l. We claim that mo E Ext.C(E). For if mo is a convex combination am + (1 - a)m' with 0 < a < 1 and m, m' E Z(E), then, since mo > am., mo >- (1 - a)m', it follows that m, m' E K. But in that case, m, m' E 91, so that mo = m = m'.
Hence mo E Exte(E). o We can say more about 4P(E) (= 8E) when E is a discrete semigroup, basically
because L,o(E) is introverted in that case. Let S be a semigroup. Recall ((2.9)) that R(S) is a semigroup in a natural way. Recall also that S is a dense subset of 8S, an element s E S being identified with s E;8S, where s(¢) _ -O(s) (0 E l0,(S)). We thus have S C '8S C 931(S).
(2.26) PROPOSITION. The set PS is a subsemigroup of 9Jt(S) containing S as a subsemigroup, and for each q E ;BS, the map p pq is continuous on /3S.
PROOF. Let p, q E PS, 01, 02 E 1,o(S). Then for s E S, 9((01Sb2)8) = q((0js)(02s)) = q0I (s)g02(s).
Thus q(4i1¢2) = (9c1)(02), and so
Pq(0102) = P(q(0102)) = Pq(01)P9(02) Thus pq E ,8S, and 8S is a subsemigroup of 91(S). A routine check shows that S, with its given multiplication, is a subsemigroup of (3S. The last assertion of the proposition follows from (2.5(viii)). 17
We note that there is no conflict between sp, ps defined, on the one hand, as in (2.25(i)), and on the other, as semigroup products evaluated in fS. The two interpretations are obviously the same! Note also that, in general, the maps q -+ pq are not continuous (cf. Problem 2-21).
We are now in a position to tackle the sixth problem in the theory of amenable
semigroups posed in (1.15). The elegant solution is due to E. E. Granirer (5) after earlier work by Mitchell [3]. Mitchell introduced the following class of semigroups.
(2.27) DEFINITION. A semigroup S is called extremely left amenable (or ELA) if there exists a multiplicative left invariant mean on S. Similarly, S is called extremely right amenable (or ERA) if there exists a multiplicative right invariant mean on S. So S is ELA if and only if there exists m E .C(S) such that m = ba for some a E fS. The following lemma, which is of intrinsic interest, will enable us to characterise ELA semigroups.
THE ALGEBRA OF INVARIANT MEANS
65
(2.28) LEMMA. Let S be a right cancellative semigroup and x e S. If x" 6 x for all n > 1, then let N be any integer > 1; otherwise, let N be the smallest of the integers n > 1 for which x" = x. Then there exists a family {Ai: 1 < i < N - 1) of subsets of S such that
(i)AinA,=® ifi96 j; (ii) S = UNi-11 Ai;
(iii) for each I, xAi C Ai+1, where we take AN = AI.
PROOF. Using Zorn's Lemma, we can find (N -1) subsets VI,...,VN_I of S that are maximal with respect to the following properties: (1)
VinVj=Q ifig6j,
xVicV2+1
(1
where, of course, we take VN = V1. Let V = U iI V. It suffices to show that V = S, for then we can take Ai = Vi.
Suppose, on the contrary, that V 0 S. Let b e S - V, and, for 1 < i < (N - 1), Vi(b) = (xi+r(N-1)b: r > 0). We claim that
(a) Vi(b)nV,(b)=0ifi9i j; (b) xVi(b) C V+1 (b) (1 < i < (N -1)). As (b) is obviously true, it remains to prove (a).
Suppose that i ¢ j. If x" # x for all n > 1, then, as i+r(N-1) 56 j+s(N-1) for all r, $ > 0 and as S is right cancellative, we must have x'+r(N-1)b 0 = x, so that xj+B(N-1)b, and (a) follows. If, however, x' = x, then xi+r(N-I)b = xib xxN-I
and as N is the smallest integer n > 1 with x" = x and S is right cancellative, we again see that (a) is true. So (a) is true in general as required. We establish the lemma by contradicting the maximality of the Vi's. There are two cases to be considered. Suppose, first, that x"b V for all n, and let V' = Vi U Vi (b). Using (a), (b), and the fact that Vi(b) n V = 0 for all i, we see that (1) is true with Vi replaced by Vi'. Since V j4 V', the desired contradiction follows.
Now suppose that xf°b E V for some m > 1. Let m be the smallest such integer. For some unique j, xmb E Vj. If j > 1, set Vi' = V, if i ¢ (j - 1) and V?_1=Vj_1U{x"'-1b}.Ifj= 1, set Vi'=V2ifI 96 N, andVj; =VNU{x''-1b}. Then V, C Vi` for all i and VV # VV for some k. The existence of the sets V,' contradicts the maximality of the V's. 0 (2.29) THEOREM. A left amenable semigroup S is ELA if and only if every nonempty, finite [two-point] subset of S has a right zero.
PROOF. The following proof which deals with the "finite" version of the theorem trivially modifies to establish the "two-point" version.
Suppose that every finite subset of S has a right zero. Then for each F E
F(S), we can find sF E S such that FsF = {sF}. Now (sF) forms a net, where sF < sE if F C E. By taking a suitable subnet, we can suppose that
THE ALGEBRA OF INVARIANT MEANS
66
sp is continuous and as sF -+ p for some p E 6S. Since each of the maps p SSp = sp eventually for each s E S, it follows that Sp = {p}. Define m E X1(S) by setting rn = 5,. Using (1) of (2.25(i)), we see that m e .C(S). Since m is trivially multiplicative, it follows that S is ELA. Conversely, suppose that S is ELA, and let . be the congruence of (1.24). Let T = S/rd and Q: S - T be the canonical homomorphism. Then if p E .C(S) is multiplicative, so also is r e .C(T), where r(O) = p(b o Q)
(0 E l.(T))
Hence T is also ELA.
We now show that we can assume S = T. For suppose that we know that every finite subset of T has a right zero, and let F = {s2 i ... , s,,} E F(S). Let a E S be such that Q(a) is a right zero for Q(F). Then sia a for each i. Using the definition of -_ and the fact that it is a right congruence on S, we can define, recursively, elements b2 , . . . , bn in S such that siab2b2 ... bn = ab2b2 ... bn
Then F(abib2 . . bn) = abibz .
(1 < i < n).
bn and every finite subset of S has a right zero.
The upshot is that we can suppose that S = S/--, that is, that S is right cancellative. We shall show that S is trivial. Suppose not. Then we can find yi, yz E S, S, yI 0 Y2. Suppose that u2 = u for all u E S. Then for all
u,y E S, yu2 = yu, so that yu = y. But then y,S Cl y2S = 0 contradicting (1.23). So there exists x E S with x2 0 z. By (2.28), we can find N > 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).
THE ALGEBRA OF INVARIANT MEANS
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 <, is said to be an abstract L-space if it has the following properties: (i) < is transitive, reflexive and antisymmetric;
(ii) (X,:5) is a vector lattice, that is, for x, y, z E X and .1 > 0, we have
(a) if x <-y, then x+z
(c) I1x+Y11= Ilxll+llvll ifs, y> 0. Every abstract L-space is an Li-space "in disguise," and (obviously) every L2space is an abstract L-space. (1) (Namioka [3]). The space 31 (E, R) is an abstract L-8pace. [As in the proof of (2.2), we can identify 31 (E, R) with a closed subspace of
M(Z,R), the space of real-valued measures in M(Z), where Z =
v+, evaluated in M(Z,R), belongs to 31(E,R). Since 31(E,R) is a subspace of M(Z, R), v+ must also be the least upper bound v V 0 in 31(E, R). Finally, since (iii)(a) is true in M(Z), it is also true in 31(E,R).]
(2.32) Invariant measures and the M-anslate Property. (von Neumann [1], Dixmier [11, Greenleaf [2], Rosenblatt [1], (21,141, [HR1]). Let G be a group
and X a left G-set with ex = x for all x E X. A natural and interesting question is when does there exist an invariant, finitely additive, positive measure µ defined on the family (X) of all subsets of X? To make this question reasonable, we "normalise" t by specifying a nonempty subset A of X and requiring tz(A) = 1. Thus we say that an invariant measure for (G, X, A) is a finitely additive measure
p: °8(X) -, (0,oo] such that p(xB) = µ(B) for all x E G, B E .9(X), and 4(A) = 1. A characterisation of when there exists an invariant measure for (G, X, A) (involving paradoxical decompositions) will be proved in (3.15).
THE ALGEBRA OF INVARIANT MEANS
e8
When G is amenable, much more can be said about the existence of an invariant measure for (G, X, A). If (b) below holds, then (G, X, A) is said to have the 7lanslate Property. Groups for which every triple (G, X, A) has the Translate Property will be briefly considered in (6.42). (i) Let G be an amenable group and (G, X, A) be as above. Then the following two statements are equivalent. (a) there is an invariant measure for (G, X, A);
(b) if n > 1, 81, ... , Sn E G, al, ... , an E R and aiXs;A ? 0,
(1)
i=1
then El, ai !>0. (Suppose that (a) holds. Let si and ai be such that (1) holds. Let p be an invariant measure for (G, X, A) and B = U;`_1 siA. Note that p(B) > 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
THE ALGEBRA OF INVARIANT MEANS
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)[ <_ Q([f [) < 11f [[oJ(XE) < nil f [[,,. Since G is amenable, Day's Fixed-Point Theorem applies to give a G-invariant functional a E K. Now define an invariant measure !c for (G, X, A) by setting p(E) = a(XE) if E is bounded and 1A(E) = oo otherwise.]
(2.33) The nonintroversion of C(G). Our main objective here is to prove the theorem that if G is a locally compact group, then C(G) is left introverted if and only if G is either discrete or compact. The two main ingredients in the proof of this theorem are (a) WUr(G) = Ur(G), and (b) Ur(G) = C(G) if and only if G is either discrete or compact. The result of (a) was initially proved by Mitchell [6] using a theorem of Ellis (Problem 2-24). Rao [1) proved a version of (a) for metric groups. Following Milnes and Pym [1], we prove (a) (in both the locally compact and complete metric cases) using a remarkable theorem of Namioka dealing with points of joint continuity for a separately continuous function. Namioka's theorem requires the presence of a certain completeness condition satisfied by both locally compact and complete metric spaces. Some such completeness condition is required for the result to be true in view of the counterexample to the equality "Ur(G) = WUr(C)" given in Milnes and Pym [1]. Namioka proves his theorem in his paper [5]. Another proof in the context of game theory is given by Christensen [S). We will be content to state the theorem and use it to deal with the introversion question. A topological space X is (Frolik [11) called strongly countably complete or s.c.c. if there exists a sequence f.2 Q of open coverings of X such that if {F,,} is a decreasing sequence of nonempty closed subsets of X with Fn contained in
a member of fin for each n, then nn I Fn 0 0. Every locally compact space Y is s.c.c.: for all n, take Wn = {U: U is open in Y and contained in a compact subset of Y}. Every complete metric space (Z, d) is also s.c.c. In this case, take .9t to be the family of all open balls of radius
1/n. Both Y, Z are regular (in the sense that every neighbourhood of a point contains a closed neighbourhood).
Let X, Y, Z be topological spaces and f : X x Y -+ Z. Recall that f is said to be separately continuous if whenever x6 -+ x in X, y6 -+ y in Y and xo E X, yo E Y, then
f(xs,yo) -+f(x,yo),
f(xo,y6) - flxo,y).
We say that f is jointly continuous at (x, y) if f (x6, y6) -+ f (x, y) whenever
x6-+xinXandy6-+y in Y. We can now state Namioka's theorem. (i) (Namioka [5]). Let X be a strongly eountably complete, regular, topological
space, Y a a-compact, locally compact space, and (Z, d) a metric space. Let
THE ALGEBRA OF INVARIANT MEANS
70
f : X x Y -* Z be separately continuous. Then there exists a dense Gs-subset A of X such that f is jointly continuous at each point of A x Y. The following generalisation of Ellis's theorem is given by Namioka.
(ii) Let G be a group that is also an s.c.c., regular topological space, and let Y be a left G-set that is also a locally compact Hausdorff space. Suppose that ey = y for all y E Y, and (a) the map (s, y) -+ sy from G x Y into Y is separately continuous; (b) the map t -+ is from G into G is continuous for each s E G. Then the map (s, y) -+ sy is continuous. (Let Y., be the one-point compactification of Y. We can make Ym into a left G-set by defining $a =
fsa ifaEY, l0o ifa=oo.
A simple argument shows that condition (a) is true with Y replaced by Y., and we may suppose that Y is compact. Since Y is now compact Hausdorff, its topology is determined by C(Y). It therefore suffices to show that for each E C(Y), the function Qk : G x Y C is continuous, where Q0(s,y) = O(Sy).
Namioka's theorem obviously applies to the function Q0, and so there exists a dense, G6-subset A of G such that Q0 is jointly continuous at each point of A x Y. Let so E A and t6 -, tin G, y6 - yin Y. Then using the joint continuity of Q0 at (so, so'ty) and (a) and (b), Q0(t6, y6) _ 4'((tat-'so)(so'ty6)) = Q0(t6t-'so, so'ty6) Q'5 (so, So Ity) = Q0 (t, y),
so that Q b is jointly continuous on G x Y.] We note in passing that (ii) can be used to prove the interesting result of Ellis given in Problem 2-24. We now proceed with our discussion of the left introversion question for C(G).
The next result, which also depends on (ii), implies that if G is either a locally compact or a complete metric topological group, then the maximal, left introverted subspace WUr (G) of C(G) is the same as the (more manageable) space Ur (G).
(iii) (Rao [1], Mitchell (6], Milnes (1], Milnes and Pym (1], Berglund, Junghenn, and Milnes (11). Let C be a separately continuous semigroup which is algebraically a group and topologically is s. c. c. and regular. Then
WUr(G) = Ur(G). (We have to show that WUr(G) C Ur(G), the reverse inclusion being trivial.
Let ¢ E WUr(G) and let U be the closed unit ball of C(G)' with the weak' topology. Let
Y={f4': f EU},
THE ALGEBRA OF INVARIANT MEANS
Ti
and give Y the topology of pointwise convergence on G. Now U is (weak*) compact and by (2.5(viii)), the map f -+ f4 is continuous from U onto Y. Hence Y is a compact Hausdorff space. Clearly, Y is a left invariant subset of C(G) ((2.5(iii))), and, since ¢ E WUr(G), the maps s --+ sy are continuous from G into Y for all y E Y. Clearly, the maps y -+ sy are continuous from Y into Y, so that the map (s, y) -+ sy from G x Y into Y is separately continuous. By (ii), the latter map is in fact jointly continuous. Define F: G x U --+ C by F((s, f)) = s(f 0)(e) = f t(s). Then F is continuous, and so VV.Oss - .0-,It = sup IF(ss, f) - F(s, f)) -+ 0. JEU
Thus 0 E Ur(G).j
Before proving the theorem concerning left introversion on C(G), where G is locally compact, we need to know under what circumstances Ur(G) = C(G). The answer is given by Problem 1-3. (iv) Let G be a locally compact group. Then U, (G) = C(G) if and only if G is either discrete or compact.
(v) Let G be a locally compact group. Then C(G) is left introverted if and only if G is either discrete or compact. [The space C(G) is left introverted if and only if WUr(G) = C(G). Now use (iii) and (iv).)
(2.34) Left invariant means and non-Archimedean analysis. We discuss some of the results obtained by van Rooij [1] on the existence of left invariant
means on certain subspaces of the space l (P, K) of bounded functions on P with values in a complete non-Archimedean field K. (See Schikhof [1) for the study of invariant means on a locally compact group in the non-Archimedean context.) A good account of non-Archimedean analysis is given in van Rooij [2], to which the reader is referred for further details. For convenience, we discuss briefly the definitions and facts from this theory that we need. Throughout, (K, I 1) will be a non-Archimedean field. This means that K is a field and (a non-Archimedean valuation) is a function from K into R such
that (a) lxi > O for all x E K;
(b) JxJ=0 if and only ifx=0(xEK); (c) Ix + yj < max(Ixl, Lyd) for all x, y E K; (d) (xyl = jxI LyI for all x, y E K. The non-Archimedean nature of (K, I ) is expressed in (c), which is a particularly
strong form of the triangular inequality. Clearly, (d) and (b) give that Ill = 1, where 1 is the unit of K, and then (c) gives that In1I < 1
for all n E P. Thus (K, J j) does not satisfy a version of the well-known "Axiom of Archimedes."
THE ALGEBRA OF INVARIANT MEANS
72
Evidently,
I determines a metric d on K, where
d(x, y) = Ix - yl.
It easily follows from (c) that every ball in K is open and closed, so that K is zero-dimensional. (Not surprisingly, the zero-dimensional condition is often is complete if the required in non-Archimedean analysis.) We say that (K, metric d on K is complete. The most familiar examples of complete, non-Archimedean fields are the fields Q9 of p-adic numbers. Here p is a prime, and Q, can be thought of as the set of formal Laurent series 00
E anpn n=N
with N E 2 and an E (0, 1, ... , (p - 1)). Addition and multiplication are the obvious formal processes, and if a E Q., then we define
P N if a = En=N anp" with aN 0 0,
laly-{0 Then (Qp,
ifa=0.
jy) is a complete, non-Archimedean field.
Let (K,
) be a complete, non-Archimedean field. Let Y be a vector space over K and jj II : Y ---' [0, oo). Then Y is said to be a normed space (over K) if the following properties hold: (a) 111;f1 = 0 if and only if g = 0 (f E Y); E Y), (b) Ij.+njl 5 IInjj) (c) Ijaell = latheII (a E K, g E Y). Of course, Y is said to be a Banach space (over K) if Y is complete in the metric on Y induced by Ij jj. The dual Y' of Y is the space of continuous, K-valued, linear functionals on Y and is a Banach space over K in the obvious way. K) where X is a nonempty set An example of such a Banach space is and {45:
0 maps X into K and sup I0(x)I < oo I xEX
The norm is, of course, given by 11011 = supxEX 10(x)j. When on S is a semigroup, 1,,,(S, K) is a "Banach S-space" in the obvious way. containing the constant function Let B be a K-linear subspace of 1. The set Tt(B) of means on B is defined as earlier. However, the analogue of (0.1(i)) does not make sense in the present context since the order properties, which are so important for real analysis, are absent in K. If B is, in addition, right invariant for S, then £(B) is the set of left invariant means on S. It is convenient to concentrate on the case where S = P, the additive semigroup of positive integers. The Hahn-Banach Theorem holds for every Banach space over K if and only if K is spherically complete, that is, every decreasing sequence of closed balls in K has nonempty intersection. Every Q. is spherically complete.
THE ALGEBRA OF INVARIANT MEANS
73
K)) by T4(n) = 4,(n + 1). Then for all r E P, (n-1 n-1 n-1 ¢(r) - 4(r+n) = E(T'¢(r) -Ti+14,(x)) _ 4,i - T 4i
Define T E
i=0
i=0
lE
1=0
)) (r),
)
where 4i = T'4,. Thus if B is invariant, then the set {4' - T4,: 0 E B) is a norm dense subspace of D(B), the norm closure of Span{ (¢ - On): ¢ E B, n E P}. The proof of the next result is routine (cf. (2.1)). (i) Let K be spherically complete and B be an invariant subspace of t (P, K) containing 1. Then J(B) 0 if and only if
Ill-(0-T4,)11 >>-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<1-F (nEP). Thus
11 - (O(n+ 1) - 4,(n))l = 11(n+ 1)1 - ¢(n+ 1)] -,[nl max(l(n + 1)1 -4,(n + 1)I, lnl - 0(n)l)
<1-e.
THE ALGEBRA OF INVARIANT MEANS
74
Thus Ill + (0 - Tgi)JJ < I and (i) is contradicted (with -¢ in place of 0). So
j 1 B. Conversely, suppose that j 1 B and that C(B) = 0. By (i), we can find
JJl-(0-TL')JJ<1-17 forsome r7>0. Then foralln>2, I(n - 1)1 + ,b(n) -gi(l)l = i[l +,0(2) - tp(1)l + 11 +,0(3) - 0(2)l +
+ [l +
0(n - l)jJ
<max(J1+ b(2)-0(1)I,...,J1+0(n)- 0(n-1)[) <jJl-(0-TgG)JJ<1->7.
Thus JJj-1+V, -'(1)IIJ <max(I1-1+9'(1)I-?7) =1-27. Since 1-b+10(1)1 E B, we have contradicted the fact that j 1 B. Hence .C(B) i4 0] (iv) For an abelian semigroup S, van Rooij [1) also discusses invariant means on l,o (S, K), on the non-Archimedean versions of AP(S) and WP(S), and also on the space PC (S, K) of functions in i. (S, K) with precompact range. In this, as also in the work of Schikhof [1) on invariant means on locally compact groups, the characteristic of the residue class field F of K plays an important
role. The field F is the quotient of the subring {x E K: in < 1} of K by its maximal ideal {x E K: Ixi < 1}. (When K = Q p, the field F is just Z9.) Van Rooij characterises those semigroups S for which .C(S, K) 96 0 in terms of the nonexistence of an element in S with "period" a multiple of p, where p is the characteristic of F. Among other results, Schikhof shows that if G is an amenable, locally compact, zero-dimensional group and p = 0, then there exists a left invariant mean on the space PC(G, K) of functions in C(G, K) with precompact range. Van Rooij 12, p. 346] asks if the converse to this result is true.
(2.35) Amenable von Neumann algebras. It would require a book on its own to do justice to the theory of amenable von Neumann algebras. We will be content to aim for the more modest objectives of discussing how the various formulations of amenability for von Neumann algebras are related to one another and to classical group amenability, and how earlier results of this chapter relate to some of the (easier) results in the theory. In what follows, we will assume that the von Neumann algebras involved act on a separable Hilbert space. (Many of the results discussed, however, are valid without the separability requirement.) We discussed amenable C'-algebras in (1.31). The latter class of C'-aigebras is defined in cohomological terms, that is, in terms of "many" derivations being inner. It is natural, then, to define amenable von Neumann algebras in similar terms. Since such algebras are in general very "large," it is reasonable to require extra ultraweak continuity conditions to "keep things under control." The relationship between amenability for C-algebras and amenability for von Neumann algebras is expressed in the deep theorem due to Connes [31 and Choi and Effros 11J, [2) (cf. (1.31)): A C`-algebra B is amenable if and only if its second dual B" is amenable as a von Neumann algebra. (For the easier implication involved, cf. Problem 2-31.)
THE ALGEBRA OF INVARIANT MEANS
75
The amenability of a von Neumann algebra A is characterized by four other properties, the equivalences involved being far from obvious. These properties are injectivity, Property P, semidiscreteness, and approximate finite dimensionality (AFD). The proof of the equivalence of the five formulations of amenability involves deep mathematics-in particular, it relies heavily on Connes's contributions to von Neumann algebras. A discussion of these matters is given in Connes [S2]. The present author has found the introduction to [D1] by E. C. Lance particularly helpful. The proof that amenable injective is given by Cannes 141-a more elementary proof is given by Bunce and Paschke [S]. The equivalence of injectivity and Property P follows from the work of Schwartz [1] and Tomiyama 14J. Effros and Lance [1] showed that semidiscrete * injective; the converse follows from Cannes (3] and Choi and Effros [1], the proof being substantially simplified by Wasserman [1] and Connes [Si]. The really difficult implication is that injective a AFD. This was proved in Connes 13]; a new proof, in which many of the technicalities of Connes's proof are avoided, is given by Haagerup [6]. A short proof has been given by Pofa [S]. Johnson, Kadison, and Ringrose [1] showed that AFD amenable. We will discuss each of the five characterizations of amenability for von Neumann algebras in turn. We then state some facts about the class of amenable von Neumann algebras. The section will conclude by discussing two related topics, viz., first the relationship between amenable G-actions and amenable von Neumann algebras, and second, how the inner amenability of a locally compact group G relates to the amenability of VN(G). (A) Amenability. Let A be a von Neumann algebra (acting on a separable Hilbert space 15) and X a Banach A-module. The dual Banach A-module X' is called a dual normal Banach A-module if for each a E X', the maps a -+ aa, a -+ as are ultraweak-weak' continuous from A to X'. A simple example of a dual Banach A-module is given by X = A., X' = A under the obvious actions. The von Neumann algebra A is called amenable if every derivation from A to a dual normal Banach A-module is inner. There is a close connection between amenable C'-algebras and amenable von Neumann algebras: if a C'-algebra B C B(SS) is amenable, then the von Neumann algebra generated by B is amenable. (This follows since B amenable a B" amenable as a von Neumann algebra.) We now discuss injectivity and illustrate how amenability and injectivity are related. (B) Injectivity. The von Neumann algebra A is called injective if there exists
a linear, norm one projection map P from B(fj) onto A. As it stands, this definition depends on the particular Hilbert space f on which A is realised. However, it can be shown that if the injective property holds for one such 55, then it holds for all of them (Hakeda and Tomiyama [1]). (It is possible to define such von Neumann algebras as the injective objects of a category.) Let A° be the commutant of A: so A° = {T E B(ll): aT = Ta for all a E A). If A is standard on 55, then A is *-anti-isomorphic to its commutant A°, using the
76
THE ALGEBRA OF INVARIANT MEANS
J-map of the Tomita-Takesaki theory. It follows that in the standard case, A is injective a A° is injective. Tomiyama (4) shows that the latter equivalence is true in the general case. Tomiyama (4) also shows that such a projection
P is positive and satisfies P(aTb) = aP(T)b for all a, b E A, T E B(b). To relate amenability and injectivity, we need to show how the former gives rise to projection maps. The argument below shows how this can be done. This argument is taken from Bunce and Paschke (S), where it forms an important part of their proof that amenable injective. Note that the argument does not give immediately that A` is injective, since the projection Q need not be of norm 1. However, the map Q can be "modified" using a polar decomposition argument to produce a norm one projection. The injectivity of Ac, and therefore of A, then follows.
Recall that for every von Neumann algebra B, there exists a unique Banach space B. with (B.)' = B. The space B. is the predual of B, and when B acts on g, the elements of B. can be realised in the form E;° I w f;,g;, where fi, gi E h, E:1()Ifi((2 + 119i((2) < oo and wf,9(T) = (Tf,g) for all T E B, f,9 E b ((Dl), Part 1, Chapter 3). This realisation will be used in the following proof with
B=B(E). Let A be amenable. Then there exists a bounded, linear projection Q from B(5) onto A° such that Q(bTc) = bQ(T)c for all T E B(fj), b, c E K. [Let X = B(fj) (b B(55).. Then X is a Banach A-module under the actions: a(T (9 0) = T ® aO, (T 0 O)b = T 0 Ob (a, b E A). (Here, of course, R(b(S) = ¢(SR), cR(S) = ¢(RS) (R, S E B(15)).) Obviously, X' is a dual Banach Amodule. We claim that X is normal. To this end we can write ' E B($). in the form F,', w f;,g; as above. Note that aw f,g = waf,g, w f,ga = w Now if ab -+ a 'strongly in a bounded subset of A, then E7.=, II(ab - a)fi(I2 - 0, E 'l 11(as-a')gi112 --+ 0, and it follows that I)a6O-a¢(( -+ 0, ((cbab-¢a(( -+ 0. So
if a E X' and g E X, then the functionals a --+ aa(le), a -+ aa(t;) are "strongly continuous on bounded subsets of A, and so are ultraweakly continuous (cf. (Dl, Part 1, Chapter 3, Theorem 1)). Thus X' is normal. Now X' is canonically identified with B(B(fj)), an element ' E B(B($5))
corresponding to the functional T ® -+ 0(,P(T)). With this identification, the
A-actions on X' are given by 'a(T) _''(T)a, (aO)(T) = a('(T)). Let for all T E B(fj), b E A`}.
One readily checks that Y is a closed submodule of X' and can be identified with Z1 (= (X/Z)'), where Z is the closure of the submodule of X spanned by elements of the form (Tb ®¢ - T (9 b.b), (bT 0 . - T 0 b), b ® 0. So Y is a dual normal Banach A-module, and the derivation D: A -+ X', where D(a)(T) = Ta-aT, maps A into Y. Since A is amenable, there exists W E Y with D(a) ='I+a-at. The map Q that we require is given by Q = (I-W) E B(B($5)), where I is the identity on B(35).)
THE ALGEBRA OF INVARIANT MEANS
77
There is a striking analogy between the above result and (2.15). Now (2.15) does involve an S-invariant mean, and one might wonder if the amenability of A can be formulated in such terms. The natural candidate for S is H, the unitary group U(A) of A, and we shall see in (C) that the amenability of A is equivalent to the existence of a left H-invariant mean on a space of functions on H. This brings us to Property P, the only one of the five formulations of amenability for A which explicitly involves H.
(C) Property P. The von Neumann algebra A is said to have Property P (Schwartz 11]) if for all T E B(15), the weak operator closure of co{UTU' : U E H} contains an element of the commutant Ac. (Property P is discussed in Sakai [1, (4.4)].) Again, Property P is intrinsic to A and does not depend on the Hilbert space 35 on which A is realised (Hakeda [1]). We first show how Property P relates to injectivity.
For U E H, define to E B(B(15)) by'U(R) = URU', and let C = co{ltu: U E H}. Note that C forms a convex subsemigroup of the unit ball of B(B(S7)) and that C(T') = {T'} for all T' E Ac. (i) (Schwartz [1], Hakeda [1]). Suppose that A has Property P. Then there exists a net {Vb} in eo{'U: U E H} such that for each T E B($), {V5(T)} converges in the weak operator topology to an element of Ac. Further, Ac is injective.
[Let 1:1, ... , gn E 15, T1, ... , T,. E B (b), and e > 0. We claim that there exists V E C and T ... , T; E A° with 11T,'11 < [IT; [[ (1 < j < r) such that
jjV (Tj - T,)(tt)jj < e (1 < i < n, 1 < j < r). Indeed, since A has Property P and the weak and strong operator closures of a bounded convex subset of B(55) coincide, there exists W1 E C such that j[W1(T1-TI')1:1 [j < e. Arguing along the lines of the proof of (2.13(1)) produces W1i ... , W E C such that V = Wn W1 has the desired property. Allowing the et, T,,, e to vary, we see that the V's form a net {V;} in the natural way and that every ultraweak cluster point of {V,(T)} belongs to Ac for all T E B(15). Now use the ultraweak compactness of the unit ball of B(SS) together with Tychonoff's Theorem to produce the desired net {V, }. A norm one projection P from B(15) onto AC is given by P(T) = IimV5(T). (For use in (G), we note that P(SITS2) = SIP(T)S2 for all S1, S2 E A`.)]
We now discuss how Property P fits in with (2.13) and (2.15) (cf. Lau [8], [15), Miines [3], Yeadon [11). From the viewpoint of (2.15), we should take S = H = U(A) and X' = B(J5) (so that X = B(SS).). The dual left action of
U.T = UTU' (U E H, T E X'). This left action is Hon X' is given by T induced by the obvious right action of H on X: 8. U(T) = 8(U. T) (8 E X). It is easily checked that these actions are isometric. The space X (H) C l',. (H) is spanned by functions of the form TO, where T8(U) = T(8. U) = 8(U .T). The space 31(X) is readily identified: indeed, 31(X) = {T E B(35): UT = TU for all U E H) = A` (since H spans A). We now show how (2.13) and (2.15) can be used to give an invariant mean formulation of Property P.
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THE ALGEBRA OF INVARIANT MEANS
(ii) (de Is Harpe [2)). The von Neumann algebra A has Property P if and only if there exists a left invariant mean for H on X(H). [Suppose that A has Property P. For the purpose of applying (2.13(1)), we take
G = H (as discrete) and B = X(H). Note that B is left introverted ((2.7)) and 1 E B (since 1= IB for any state 0 E B(55)., where I is the identity on fj). Let Ti E X', 6, E X (1 < i <_ n). With V6 as in (i), V6 (Ti) -+ 7 ultraweakly for some we see that the constant function Ti E Ac, and noting that U(T;B,) =
(ELI T;(8s))1 belongs to the pointwise closure of co{U(ET,B;): U E H}. By (2.13(1)), £(X(H)) 0 0. Conversely, suppose that .C(X(H)) ,-b 0. Then 3j(X) = Ac, and A has Property P by (2.15).) Suppose, now, that G is an amenable locally compact group, Tr is a continuous,
unitary representation of G on a Hilbert space b, and B is the von Neumann algebra generated by ir(G). Noting that r(G) C U(B), we can define X(G), in the obvious way. It is readily checked that X (G) c C(G) and so carries a left invariant mean since G is amenable. It then follows as above that B has Property P, so that the von Neumann algebra generated by a continuous, unitary representation of an amenable locally compact group has Property P. In particular, if G is amenable, then the von Neumann algebra generated by the left regular representation Ira of G ((1.31)) has Property P. The preceding result (ii) does not use any topology on H. However, H is a jointly continuous, complete, separable metric group in the strong operator topology, and X(H) is a subspace of Ur(H) (Problem 2-26). De Is Harpe shows, using the AFD property, that A is amenable if and only if there exists a left invariant mean on Ur(H) (Problem 2-27). This remarkable result shows that amenability for von Neumann algebras
can be reformulated in familiar group terms and raises the speculation that the theories of amenable von Neumann algebras and amenable locally compact groups should be unifiable in an amenability theory for topological groups (cf. Paterson [8)). A deep result related to that of de la Harpe above and involving the semigroup of isometries in A has been established by Haagerup [3]. This theorem uses Haagerup's extension of the Grothendieck-Pisier inequality and is used to show amenable for C'-alggebras. that nuclear (D) Semidiscreteness. The semidiscrete property was introduced and studied by Effros and Lance [1]. The property can be defined using tensor products in a way analogous to the nuclear property for C`-algebras ((1.31)). From an amenable group viewpoint, however, semidiscreteness can be regarded as a von Neumann algebra version of Problem 4-28. The appropriate definition of semidis-
creteness involves the notion of completely positive maps. A map T E B(A) is called completely positive if for each n > 1, the map [a,j] [T(aij)) on Mn ®A is positive. The von Neumann algebra A is called semidiscrete if there exists a net {T6} of weak` continuous, completely positive, unit preserving, finite rank
THE ALGEBRA OF INVARIANT MEANS
79
operators in B(A) such that T6a -- a weak* in A for every a E A. (This property is clearly reminiscent of Grothendieck's metric approximation property.) if G is an amenable locally compact group, then a certain net {f6) of continuous, positive definite functions with compact support gives rise to such a net {T6) on VN(G) so that VN(G) is semidiscrete (Problem 4-43). (E) Approximate finite dimensionality. The AFD property is the most elementary and easily defined of the five formulations of amenability for A. The algebra A is called approximately finite dimensional (AFD) if A is generated by an increasing sequence of finite-dimensional "-subalgebras. (For an extension of this notion to the case where A cannot be realized on a separable Hilbert space, see Elliott [i], [21 and Elliott and Woods [1].) The AFD algebras form a very natural class of von Neumann algebras, and the determination of the AFD factors has been a major theme in operator algebras. This determination is completed in the work of Connes [1J, [3J and Haagerup [S3J. Paradoxically, of the five formulations of von Neumann algebra amenability, the AFD formulation is the one which seems furthest away from group amenability. It is certainly not true that an amenable group is always an increasing union of finite subgroups! Note, however, that if A is AFD, then U(A) is generated (in the strong operator topology) by an increasing sequence of compact subgroups. (F) The class of amenable von Neumann algebras. References for the following results are Arveson [1), Choda and Echigo [Si), [S2], Choi and Effros [2), Connes [3), [Si), Effros and Lance [1], Hakeda and Tomiyama [1), Schwartz [1), Tomiyama [4).
(a) Every Type 1 von Neumann algebra is amenable.
(b) The von Neumann algebra generated by an amenable C`-algebra on a Hilbert space f} is amenable. In particular, every continuous, unitary representation of a locally compact group in the class say of (1.31) generates an amenable von Neumann algebra. (c) The weak operator closure of the union of an upwards directed family of amenable von Neumann algebras on a Hilbert space is amenable. A similar result holds for the intersection of a downwards directed family of amenable von Neumann algebras. (d) A von Neumann algebra on a Hilbert space is amenable if and only if its commutant is amenable. (e) Let fj be a separable Hilbert space, X a standard Bore] space with probability measure µ, and x -+ Ax a measurable field of von Neumann subalgebras of B(b) (D1, Part 21. Let A = fx Ax dic(x). Then A is amenable if and only if A. is amenable for p-a.e, x E X. (f) If (A, p, G) is a von Neumann covariant system (Pedersen IS]), then the cross product von Neumann algebra A x,G is amenable if A and G are amenable, while A is amenable if A xr C is amenable (whether or not G is amenable). (g) The tensor product of two von Neumann algebras is amenable if and only if each of the original von Neumann algebras is amenable.
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THE ALGEBRA OF INVARIANT MEANS
(G) Amenable G-spaces. We saw in (f) above that a cross product A xv G is amenable if A and G are amenable. Now the cross product construction is important for producing factors. In particular, the Murray-von Neumann construction [D1, Part 1, Chapter 91 associates with a free ergodic action of a group on an appropriate measure space a cross product factor M. An important question is what conditions on the group and measure space will ensure that M is amenable? There is a substantial literature on this theme, a good reference for which is Moore [S). A related procedure involving ergodic equivalence relations
produces the Krieger factors, and a similar question to that above arises for such factors. Zimmer (1)-[7) has introduced the notion of an amenable measured equivalence relation that answers these questions. Further developments have been made by Anantharaman-Delaroche [1)-[3). See also Ocneanu [1), Hahn [S), and Bezuglyi and Golodets [S]. Connes, Feldman, and Weiss [S] formulate the above notion of amenability in terms of an operator that is invariant (in a suitable sense) under partial transformations. We shall be content to illustrate some of the ideas involved in this area by concentrating on the Murray-von Neumann construction and showing how the amenability of the resulting von Neumann algebra implies that the G-space involved is amenable. Our account is based on Zimmer [3). We require some preliminaries.
Let X be a standard Borel space: this means that X is equipped with a oalgebra .9 of subsets, where R is the Borel a-algebra generated by the open sets for some complete, separable metric on X. Let G be a countable, discrete group acting invertibly and measurably on the right on X. We further suppose that u is a probability measure on (X,R) which is quasi-invariant in the sense
that p(Ea) = 0 whenever p(E) = 0 (E E R, a E G). For a E G, we define the measure au by ap(E) = p(Ea). By quasi-invariance, ap < p, and by the Radon-Nikodym Theorem and the countability of G, there is a Borel function r: X x G .- 10, oo) such that (d(ap)/dp)(x, a) = r(x, a). We can take r(x, e) = I for all z E X. The quasi-invariance of p yields a natural left Banach G-space
structure on L.(X): a.(x) = O(xa). Note that (1)
f 0 d(a{1) = J (Q 1 O) du = J O(x)r(x, a) dµ(x),
from which it follows that r satisfies the cocycle identity: r(x, ab) = r(x, a)r(xa, b) a.e. x. There exist unitary representations U, V of G and isometric'-representations M, N of L. (X) on 55 = L2 (X x G) = L2 (X) ®12 (G) given by Ua f (x, b) = f (xa, ba)r(x, a) 112, (Mo f) (x, b) = O(x) f (x, b),
V. f (x, b) = f (x, a-1b),
(No f) (x, b) = -k(xb-1) f (x, b)
(a, b E G, X E X, ¢ E L. (X), f E 55). It is readily checked that (2)
UaMOUa-i = Mao,
U0FUU-, = aF
where F E L ,,,,(X x G) is identified as a multiplication operator on 55 and aF(x,b) = F(xa,ba). Note that M,," =110 ® 1 under this identification. The
THE ALGEBRA OF INVARIANT MEANS
81
von Neumann algebra generated the operators Va, N. is denoted by L, while that generated by the operators Ua, M,0 is denoted by R. If J E U(15) is given by Jf (x, b) = f (xb-1, b-I)r(x, b- 1)1/2, then j2 = I, JVaJ = U0, JNOJ = M0, so that J implements a spatial isomorphism between L and R. Obviously, therefore, L has Property P * R has Property P. It is easily checked that the elements of L commute with the elements of R, and indeed, by [D1, Part 1, Chapter 9, Exercise 1), R is the commutant Lc of L. (A tedious calculation shows that the representations considered by Dixmier are the "left action" versions of, and are equivalent to, those above.)
A simple, but important, example of the above situation occurs when X is a singleton {x0} and µ = c5 . In this case, L, R are the von Neumann algebras generated by the left and right regular representations of G [D2, (13.1.6)). We now introduce Zimmer's notion of an amenable G-space, motivation for which comes from Mackey's theory of virtual groups (Mackey [S)). Let E be a separable Banach space and I(E) the group of isometric, invertible linear operators on E equipped with the strong operator topology. (So T6 --' T in
I(E) a I1Taf - Tf11 -r 0 for all e E E.) With respect to its resulting Borel structure, the group operations in I(E) are measurable. Since E is separable, the unit ball El' of the dual space E' is a compact metric space in its weak* topology, and we give El' the resulting Borel structure. Let H(r1) be the group of weak' continuous homeomorphisms of Ei. A cocycle y: X x G I(E) is a Borel measurable map satisfying the cocycle identity: y(x, ab) = y(x, a)p(xa, b) (x E X, a, b E G) for a.e. x. Every such cocycle y induces a cocycle -y*: X x G -
H(E') given by y*(x,a) = (y(x,a)*)-'IE;. A set-valued function x - Ax on X is called a Borel field (of weak* compact, convex subsets of E1) if every A,,
is a weak* compact, convex subset of El' and A = {(x, a): x E X, a E Ax} is a Borel subset of X x El. With -y as above, the field x - Ax is called yinvariant if y'(x,a)Axa = A. for a.e. x. A Bore] function p: X El' is called a -y-invariant section if for all a, -y' (x, a)p(xa) = p(x) a.e. Finally X is called an amenable G-space if for every such E, every cocycle y: X x G -> I(E), and A. of weak* compact, convex subsets of Ell, every -y-invariant Borel field x there exists a y-invariant section p with p(x) E A. a.e.
The amenable G-space property can be interpreted as a sophisticated extension of amenability fixed-point theorems. To illustrate this, consider the simple case (above) in which X is a singleton. Then the cocycle y can be regarded as a mapping from G into I(E), and the cocycle equality translates into an isometric G-action on E: ae = -y(a) C. A -y-invariant Borel field is just a G-invariant, weak* compact, convex subset K of El, and a -y-invariant section is just a fixed-point of K. With Problem 2-14 in mind, we see that a singleton is an amenable G-space G is amenable. However note that in general, nonamenable groups can have amenable G-spaces (Zimmer [6)). -
THE ALGEBRA OF INVARIANT MEANS
82
We now discuss the theorem of Zimmer alluded to above (Zimmer (11, (31). The G-space X is amenable if and only if the von Neumann algebra L has Property P. (We shall give the proof of only one of the implications involved. Suppose that L (and therefore R) has Property P. We shall show that X is an amenable G-space.
Since L has Property P and R = Lc, there exists, by (C)(i), a norm one, linear, positive projection P from B(f)) onto R such that for all T E B(35), we have (a) P(T) is in the weak operator closure of co{UTU': U E U(L)}, and (b) P(S1TS2) = S1P(T)S2 for all S1i 82 E R. We will use P to produce a norm such that one, linear, positive, unit-preserving map a: LO,(X x G) (c) a(F(¢®1)) = a(F)O and (d) a(aF) = aa(F) for all F E L,p(X x G), a E G, The map a can be produced by disintegration. The following and 0 E argument produces a directly and applies more generally. Let D be the space of operators T E B(J5) such that whenever
gl,...,9nEL2(X)aresuch that E 1figi=0,thenE 1(T(fi®e),gi®e)=0.
Obviously, D is weak operator closed in B(f)). We claim that R C D. By (2), we need only show that each UaMM E D. Indeed, if fi, gi are as above, then 0 e)(x, b) =.0(xa) fi(xa)E,(ba)r(x, a)112, so that n
E(UaM,0(fi (9 e), gi 0 e) i=1
=
f O(xa)r(x, a)1/26.(a) E fi(xa)gi(x) dp(x) = 0, i=1
and UM, E D. So R C D. It follows that for T E R, if h E L,(X) and we write E L2 (X), then the complex number E' (T (fj(De), gj (9e} h is independent of ff, gJ. We define 8(T)(h) to be this common value. Clearly, 8(T)is alinearfunctional on LI(X)with (e)8(T)(fg)=(T(f(&e),g®e). Writing h = Ihja where a: X T is measurable, we have j8(T)(h)I = I(T(IhI1/2 0 e), IhI1"2a 0 e)I IITIIIIIhI1/2II2 = H 'IIIjhII1
LI(X)' and Ije(T)II S IITII. Define a(F) = 8(P(F)) So 8(T) E x G)). It is easy to check that a is a norm one, linear, positive, (F E unit-preserving map from x G) into L,o(X). The equality (c) follows using (b), the fact that ¢ ®1 = M,, E R, and (e). (Note that (P(F)M4,(f 0 e), g ® e) = (P(F)(¢ f (9 e), g (9 e).)
To prove (d), apply (2) and (e) to obtain a(aF)(fg) = (P(F)U4-, (f ®e), Ua-, (g (9 e)).
Now note that 2 Ua-, (k 0 e) = (a'k)r{®6a = Va((a 'k)raJ2®be),
TAE ALGEBRA OF INVARIANT MEANS
83
where r,-, (x) = r(x, a-'). Finally, use the fact that the unitary V. commutes with P(F) (E R), together with (1), to obtain (d). Now let -y be a cocycle and x -, A. a i-invariant Borel field as in the definition
of amenable G-space. We will use the map a above to construct a y-invariant section r. Since X x Ei is a standard Borel space and A is a Borel subset of X x E11, a well-known result [Dl, Appendix 51 shows that there exists a
measurable map c: X -» Ei such that c(x) E A,, a.e. Define S: X x G E1 by S (x, a) = ry' (x, a1)c(xa-1). Note that S (x, a) E A. a.e. Using the measurability of -y and c, we can define, for
E E, a function (g, S) E L (X x G)
by (,S)(x,a) = S(x,a)(g). Define 4i: E -- L,,(X) by 40(C) = a((E,S)). It is easily checked that II (£, S) II. < II gf 1, and that 4? is linear with 11011 :5 1. Then Y'' ,
restricted to LI (X) (or rather LI (X)^ ), maps LI (X) into E', and applying [DS, V1.8.6) there exists a map p: X -> 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
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THE ALGEBRA OF INVARIANT MEANS
{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
THE ALGEBRA OF INVARIANT MEANS
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
<- llx * f - fs * X1111
/2
0.
Let ws be the state on VN(G) given by w6 (T) = (Tg6, g6). Let H = U(VN(G)c),
X = B(fj)., and let X(H) be as in (C). Note that VN(G)o has Property P since VN(G) has Property P. (This is easy to see directly using the J-map of [D1, Part 1, Chapter 5J.) For each 6, define 1Y6: L.(G) -+ X(H) by %5(4)(U) = (UL,,U*g6i g6), where Lo is the multiplication operator on L2(G) associated with ¢. By (C),(ii) there exists a left invariant mean m for H on X(H). Then n6 = m o %jrs E 9)1(G). We can suppose that n6 -+ n weak* for some n C 271(G).
We claim that n is a right invariant mean on Loo(G) so that G is amenable. To this end, let x E G, c > 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 <- 211011c.
So Il%Y6(4x-1) - WY6(Q5)7rr(z'1)floo - 0, and since m is H-invariant and 7rr(G) C
H, we have n(¢x-1) = n(O) as required.)
(2.36) Invariant means and almost periodicity. Let G be a locally compact group. If G is not amenable, then there is no left invariant mean on U(G). When we go to a smaller space, however, there is always the chance that left invariant means might exist. An interesting example of this possibility occurs with the space WP(G) of weakly almost periodic functions on G-this
THE ALGEBRA OF INVARIANT MEANS
86
space always has a (unique) invariant mean. This result is a consequence of the Ryll-Nardzewski Fixed-Point Theorem, which goes as follows. (THE RYLL-NARDZEWSKI THEOREM). Let X be a locally convex space and K be a convex, weakly compact subset of X. Let E be a semigroup of weakly
continuous, affine maps from K into K such that E is distal in the sense that whenever ic,
it E K with f # rl, then 0 l {(Tg - Trl): T E E}-. Then there
exists fo E K such that TCo = go for all T E E.
A recent short proof of this theorem is given by Namioka (S]. Let S be a separately continuous semigroup. References for the following theorem are de Leeuw and Glicksberg (Si, Burckel [S], and Berglund, Junghenn, and Milnes (1]. Recall ((2.11)) that AP(S), WP(S) are invariant, left introverted; closed subspaces of C(S).
(i) The spaces WP(S) and AP(S) are C'-subalgebras of C(S). (ii) The carrier space Y (Z) of WP(S) (AP(S)J is a separately (jointly] continuous semigroup under the Arens product ((2.8)), and if 0 E WP(S) (AP(S)], then OS is relatively compact in the weak [norm] topology. (iii) If G is a locally compact group, then
E(WP(G)) = 0(WP(G)) = 91(WP(G)), and there is exactly one element in each. A similar result holds with AP(G) in place of WP(G).
[(i) Let 0, b E WP(S). In order to show that 00 E WP(S), it is sufficient, by the Eberlein-Smulian Theorem, to show that if {sn} is a sequence in S with
sno - 0' and snit) -,0' weakly in C(S), then sn(4 ') -+ 4'z0' weakly. Let X be the carrier space of C(S), and identify C(S) with C(X) and C(S)' with M(X). If x E X, then
(x) (x) _ sn(00)(x) _ so that sn (Ozl) -+ 0',0' pointwise on X. Now apply Lebesgue's Dominated Convergence Theorem to obtain that sn(4vl) -; O'zji' weakly as required. Since the product of two norm compact subsets of C(S) is also norm compact, it follows that AP(S) is an algebra. Trivially, both WP(S) and AP(S) are closed under complex conjugation so that they are C'-algebras. (ii) Let Y be the carrier space of the C'-algebra WP(S). If p, q E Y, then since q.00 = (q¢) (q') (4,0 E WP(S)), it follows that pq E Y. Thus Y is a seinigroup. If qa - q in Y and 0 E WP(S), then gscy -a q4' pointwise on S. But since the pointwise and weak topologies coincide on the (weakly compact) weak closure of co(SO), it follows that qa¢ -+ q¢ weakly. Hence the maps q - pq are continuous on Y. Of course, the maps p -» pq are also continuous, so that Y is a separately continuous semigroup (the weakly almost periodic compactification of S). It now follows that for each 0 E WP(S), the set A = {cbp: p E Y} is pointwise compact and bounded in C(Y). Now apply a result of Grothendieck
THE ALGEBRA OF INVARIANT MEANS
87
[DS, IV.6.141 to obtain that A is weakly compact in C(Y). Now for s e S, we haves E Y, and so (¢S)^ = {¢s: s E S} C A. It follows that 4,S is relatively compact in the weak topology of C(S) as required. In the corresponding proof for AP(S), Z turns out to be a jointly continuous semigroup, and the relative norm compactness of ¢S follows easily. In this case, Z is the almost periodic compactification of S. (iii) Let ¢ E WP(G), and let C be the (compact) weak closure of co(G4,). For
xEG,define Ls:C-+C by Lx4,=x+y. Let E={L.,:xEG}. ThenEisa semigroup of weakly continuous, a£Fine maps on C. Further, if Y+1,'42 E C with 01 0 02, then IILx,,bi -Lzi2[1 = 1101-+0211, so that 0 does not belong to the norm
closure of the set {T(41 - 4,2): T E E}. By the Ryll-Nardzewski Theorem, we can find 00 E C with x4o = qSo for all x E G. Obviously, ¢o E C1. Since 00 is also
in the pointwise closure (= C) of co(G4,) and WP(G) is left introverted, (2.13) applies (G discrete) to yield .C(WP(G)) 14 0. Let m E Z(WP(G)) and suppose that al E D, the weak closure of co(4,G) in C(G). (Interchanging C and D (using (ii)), we see that D contains a constant function.) Since m(co(4,G)) = {m(O)}, it follows that a = m(rb). Hence there exists exactly one constant function in
D. Now D is weakly compact (using (ii)), and interchanging left and right, C contains exactly one constant function. Now apply (2.13(ii)) to give that 1r(WP(G)) = {m). Further since m(q5)1 is the only constant function in C U D, the "91(B)" version of (2.13(ii)) gives 91(WP(G)) = {m}. The corresponding proofs for AP(G) are left to the reader.] Kovacs and Szucs [i), generalising the notion of a finite von Neumann algebra, introduced the notion of a G-finite von Neumann algebra. If A is a von Neumann algebra and G is a group of *-automorphisms on A, then A is called G-finite if for every a > 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
THE ALGEBRA OF INVARIANT MEANS
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].)
THE ALGEBRA OF INVARIANT MEANS
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 < i < n) in G such that n
Dori-oso>>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).
THE ALGEBRA OF INVARIANT MEANS
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.
THE ALGEBRA OF INVARIANT MEANS
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.
THE ALGEBRA OF INVARIANT MEANS
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
THE ALGEBRA OF INVARIANT MEANS
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
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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.
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FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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
For suppose that (1) holds and let 41: Aut G -+ Aut Ge be the continuous homomorphism given by 91(a) = a I G.. Clearly A n ker if = N. Let H = T-1(1(Ge)) nA. Then H is a closed subgroup of AutG. Since %P = I(Ge), we have NIG, (G) = H, and using (1) and an isomorphism theorem for groups, we have (2)
JH/JG.(G)J = J(NIG.(G))/IG.(G)J
=IN/(NnIG,(G))I
So IG, (G) is open in H, which, in turn, is open in A, which, in turn, is open in Aut G. Hence IG, (G) is a connected, open and closed subgroup of Aut G, so that IG, (G) = (Aut G); as required. (The openness of IG, (G) in Aut G will be used in Case 3.) It remains to establish (1).
Let a E N. Since N C A, we can write a(x) = xu(x), where u: G y E Ge and x E G, then since a E N and Ge is normal in G, we have xyu(x) = (xyx'1)(xu(x)) = a(xyx-1)a(x) = a(xyx-1x) = a(xy) = xu(x)y
G. If
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
103
so that u(x) E Z(Ge). Further, xyu(x) = xu(x)y = a(xy) = xyu(xy), so that u(x) = u(xy) and u is constant on each left Ge-coset. Also, if x1, x2 E G, then
u(xlx2) = x2 1x11a(xlx2) = x21x11X1u(z1)x2u(x2) _ (x 21u(x1)x2)u(x2).
Since G is compact and Ge is open, it follows that {G/Ge f = n < oo. Let {z1,. .. , xn} be a left transversal of G with respect to Ge, and let v = U(x1)u(x2)
. u(xn) E Z(Ge). It is easy to check that for each x E G, {x1x,... , xnx} is also a left transversal of G with respect to G. Using the earlier results of this paragraph, we have, for each x /E G, V = U(xjx)u(x2x) ./.. u(xnx) / = [(x-IU(x1)x)u(x))[(x_1u(x2)x)u(x)) ... [(x'IU(xn)x)u(x)) (3)
= (x-1vx)(u(x))'.
Now Z(Ge) is an abelian, compact Lie group, and so Z. = Z(Ge)e = T'n for some m. Clearly Ze is a divisible subgroup of Z(Ge), and so there is a subgroup Z1 of Z(Ge) such that Z(Ge) = Ze x Z1 [HR1, (AS)). The subgroup Z1 is finite since Z(Ge) is covered by a finite number of translates of its identity component Z. Let p = IZ1 I and Z2 = {x E Z(Ge) : xp = e}. Then Z2 is a finite, characteristic subgroup of Z(Ge), and since Z1 C Z2, we have Z(Ge)/Z2 = T9 for some q. So we can find w E Z(G.) such that wn,v 1 E Z2. Clearly aw E N n IG, (G). Now
define u': G -, Ge by awa(x) = zu'(x). Then awa E N, and as in the case of u, we have that u'(G) C Z(Ge) and u' is constant on each left Ge-coset. Using (3) and the fact that Z2 is characteristic, we have (tu'(x))n =
[(x_1wx)u(x)w-1)n = (x-1wx)nu(x)' w
'
= x-1w"xx 1V-1xVw n =
x-1(wnV`1)x(wnv-1)-1
E Z2.
So u' can be regarded as a function from the finite group G/Ge into the finite group {z E Z(Ge) : znp = e). Now the number of such functions u' is finite. This establishes (1). For u' determines an element of the right coset (N n IG, (G))a, and every right (NnIG, (G))-coset in N is so determined. So I N/(NnIG, (G)) I _ 1N\(N n IG, (G)) I < oo. Case 3: G is a compact group. Let R(G) be the class of finite-dimensional,
continuous, unitary representations of G. If it E R(G) and a E Aut G, then Ira E R(G) is defined by Ira(x) = Ir(a(x)). Let A.i _ {a E AutG: Ira is equivalent to it}. It follows from [HR2, (27.32)) that a E A if and only if Xer = X7111, where X, is the character of a representation a ER(G). We will show first that A,, is an open (and closed) subgroup of Aut G. By composing and inverting intertwining operators, we see that A,, is a subgroup of Aut G. Suppose that 0 E (Aut G) - A,,. Since Ir/3 is not equivalent to it, we
104
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
can find v in the dual G of G such that n, # m where n, [m,] is the number of times v occurs in it [Tri]. Using [HR2, (27.30)), we have
IIx, - mall'. ? J Ix. - x,Ql2dAc
=(x- x+rR,Xa - x) _ (n,-m,)2>_1. So the set
W = {a E Aut G: IIX, - X,,11,, < 1}
trlr(a(x)) = x,(a(x)). is contained in A,,. Now note that for x E G, Since G is compact, the continuous function X, is uniformly continuous, and we can find a neighbourhood U of e in G such that Ix,1(a) - x,(b)) < I whenever ab-1 E U. So the neighbourhood {fl E AutG: Q(x) E Ux for all x E G} of i is contained in W C A,,, and hence A is an open (and therefore closed) subgroup of Aut G. Now if a E A,,, then a(x) E ker 7r if and only if x E ker Tr. So for each a E A,,,
we can define an (algebraic) automorphism 4i (a) of H = ir(G) by setting
ir(a(z))
(z E G).
AutH is It is routine to check that 4', (a) is continuous and that 4),: A,, continuous. (The compactness of G entails that H can be identified with the quotient group G/(kerrr).) If x E G, then ir(x) is an intertwining operator for it and 1ra2, so that ax E A,,. Since 4in(a,.) = a,(,), we have IH, (,) (H) C 4p, (A,,). Now H is a compact Lie group, and from Case 2, IH (,) (H) is an open subgroup of Aut H. Hence An = 4?'-' (IH,,) (H)) is an open and closed subgroup of A., and so of Aut G. Let X = n{An : it E R(G)}. Then IG. (G) C (Aut G); C X, and Case 3 will be proved once we have established that X C IG. (G). Now So let 6 E X. For each it E R(G), let B = {x E Ge: ir: G -, ir(G) = H (= G/(kerir)) is an open mapping and, using IHRI, (7.12)), ir(Ge) = As 0 E A,, 4>,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<: G -, Aut K be given by T (x) (k) = xkx-1. Then, since IY is a continuous homomorphism, W(G) is a connected subgroup of Aut K and hence by (3.4), is contained in IK, (K). So if it E G, then there exists u E Ke such that
xkx'1 = uku-1 for all k E K. Hence u 1x E H, and x = u(u-1x) E KH. 0 References for the next result are Yosida [1) and Goto [1).
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
105
(3.6) PROPOSITION. Let G be a semisimple, Lie group and H a connected, semisimpte, Lie subgroup of G. Then H is closed in G.
PROOF. Suppose first that H is a Lie subgroup of GL(W), where W is a finite dimensional vector space over R. Suppose that H- # H. We will derive a contradiction. Note B14 that H- is a connected Lie subgroup of GL(W) with the relative topology (B16). By B15, the Lie algebra l1 of H is a proper subalgebra of the Lie algebra f of H-. Let Y E t - Fj and t E R. Find a sequence {xn} in H with x -+ exp(tY) in GL(W). Since the Lie group topology of His the relative topology induced by GL(W), and as Ad (on H-) is continuous, Ad x,, (F)) C l3, and exp(tadY) = Ad(exp(tY)) (B12), we have exp(tadY)(4) c b, so that ad Y(b) C b. Since b is semisimple, the derivation ad Y [ q is inner (B42). So by subtracting an element of b from Y, we can suppose that adY(4) = {0}. Identifying t in the canonical way with a Lie subalgebra of L(W), the space of linear transformations of W, we see that Y commutes with the elements of 4. Let WC[bC) be the complexifications of W and 4, and identify the complexification (L(W))C in the obvious way with the complex Lie algebra L(WC). If T E L(W), let TC be the element corresponding to T. So TC(v+iw) = T (v) + iT(w) (v, w E W). Of course, Yc commutes with all the elements of 4c. Since F)C is semisimple, the 4C-module WC can be written as a direct sum k
WC = (D W., i=1
where Wi is an bC-invariant, irreducible subspace of W (B41). Let i E [1;k]. Since (expX)C(Wi) C Wi for all X E I) and expb generates H, it follows that xC (Wi) C Wi for all x E H. An argument similar to the one above involving {x,,} shows that YC(Wi) C W. By Schur's Lemma, Y = (YC) [w, _ ,\;1j, where Ai E C and Ii is the identity of L(Wi). We now show that A, = 0. To this end, let hi = {A I wi : A E FbC}. Then the Lie algebra F)i is semisimple, and so [bi,F)i] = l)i. Hence trT = 0 for all T E l)i. This implies that for all such T, det(expT) = exp(trT) = 1. Thus for all X E F), det(exp(Xc)) = 1, and it follows that det((xc) [ w,) = 1 for all z E H. As earlier,
we obtain det(expt(Yc)) = 1 for all t E R, giving that A, = (1/ni)trYC = 0 as required, where ni = dim Wi. Hence Y = 0 (since YC = 0) and the resulting contradiction establishes H- _ H.
Now let H and G be as in the statement of the proposition and let b and g be the Lie algebras of H and G respectively. Let Ad: G GL(g) be the adjoint representation. By the first part of the proof, Ad H is a closed subgroup of GL(g). Let H' = (Ad)-1(AdH). Then H' is closed in G, and since kerAd C Z(G), we have H' = HZ(G). Since G is semisimple, Z(G) is discrete in G. Now Z(G) is countable (G is a-compact) and H is connected, so that H x Z(G) is a-compact. Hence the canonical homomorphism from H x Z(G) onto H' is open by [HR1, (5.25)], and it follows that H = (H'),. So H is closed in H', and hence also in
G. 0
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
106
We now generalize the well-known fact that there exists, in every Lie group G, a maximal, connected, normal, solvable subgroup, called the radical of G (cf. B35).
(3.7) PROPOSITION. Let G be a locally compact group. Then G contains a connected, closed, normal, solvable subgroup r(G) which contains every connected, normal, solvable subgroup of G. Further
(i) r(G/r(G)) is trivial; (ii) if G is connected, r(G) is trivial and L is a compact, normal subgroup of G, then r(G/L) is trivial. PROOF. Let 999 be the family of connected, normal, solvable subgroups of
G, and So the subgroup of G generated by U'. Then U5° is connected and invariant under the inner automorphisms of G, and so (Appendix A, (*)), So is a connected, normal subgroup of G. If we can show that So is solvable, then it will follow that Sa is solvable, and the first assertion of the proposition will
have been established with r(G) = ST (= So). To this end, we can suppose that G is connected, and we can find a compact, normal subgroup K of G such that G/K is a Lie group (B6). Let Q: G - G/K be the quotient map. If S E 5°, then Q(S) is a connected, normal, solvable subgroup of G/K and so is
contained in the radical R of G/K. Hence Q(So) C R. For any group H, let {DI (H), D2(H), ... } be the derived series of H; so D1(H) = H and D;+1 (H) is the commutator subgroup of Dt(H). Since R is solvable, we can find r so that Dr(Q(S0)) = {e}. So D,.(So) C K. Now if S1,. .. , S,,, belong to 51, then so does the subgroup of G generated by U!"_1 S, (Appendix A, (*)). It follows that D7(So) is the union of the upwards-directed family of subgroups {D7(S): S E .P). Now for each S E 5°, the compact subgroup D,.(S)- is solvable, and so abelian (B51). Hence Dr(So) is abelian, and So is solvable as required.
The result (1) follows by observing that if ' : G -+ G/r(G) is the quotient map and T is a connected, normal, solvable subgroup of G/r(G), then -V1(T) is a connected, normal, solvable subgroup of G.
Now suppose that G is connected and r(G) and L are as in (ii). Let H = {h E G: hl = lh for all I E L}. By (3.5), G = LeH. Let 1Y: G GIL be the
quotient map, R = r(G/L), and RI =-1(R) n H. Since G = LH, we have %P(H) = GIL, and it follows that RIL =1Y`I (R). Using the equality R1 n L = H n L, (1)
RI/(H n L) = Ri/(R1 n L) = (RIL)/L = *-1(R)/L = R.
Clearly, R1 is a closed subgroup of G. We claim that Ri is a normal subgroup of G. To prove this, it is sufficient to show that H is a normal subgroup of G. This follows from the equalities below, where z E G, h E H, and I E L:
(xhx-1)l =
zh(x-ilx)x_1
= z(z-1lx)hx 1 = l(zhx-1).
Now H nL is solvable (being abelian) and R is solvable. It follows from (1) that Ri is also solvable. Hence (RI), C r(G) = {e), and R1 is totally disconnected.
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107
It follows from [HR1, (5.29)] and [HRI, (7.12)] that RI /(H n L) = R = {e}. So (ii) is established.
Consonant with the terminology used in the theory of Lie groups, we shall call the subgroup r(G) above the radical of G. The group G is called semisimple if r(G) = {e}. The group G is called almost connected if G/Ge is compact. The main section of this chapter culminates in the two theorems below (Rickert [1), [2], Balcerzyk and Mycielski [S]). (3.8) THEOREM. The following three statements are equivalent for an almost connected, locally compact group G: (i) G is amenable;
(ii) 0 does not contain F2 as a discrete subgroup; (iii) G/r(G) is compact. PROOF. It follows from (1.12) that (i) implies (ii) and from (1.13) and Prob-
lem 1-6 that (iii) implies (i). Suppose that (ii) holds. It remains to show that (iii) holds. Since G/Ge is compact, it is sufficient to show that H = Ge/r(G) is compact.
Since H is connected, we can find a compact normal subgroup K of H such that H/K is a Lie group. The compactness of H will follow once we have shown
that H/K is compact. Suppose, on the contrary, that H/K is not compact. Now H is semisimple by (3.7(i)) and so therefore also is H/K by (3.7(ii)). Note
also that H/K is connected. Let 4 be the Lie algebra of H/K. If f) were a compact Lie algebra, then, by B52, the group H/K would be compact. Hence h is semisimple and noncompact, and so by (3.3(ii)), contains a subalgebra I isomorphic to sl(2, R). Let L be the connected, Lie subgroup of H/K with Lie algebra I. Then L is semisimple (since its Lie algebra is), and so, by (3.6), is closed
in H/K. Let W be the simply connected covering group of PSL(2,R) (and L). Then (B6) PSL(2, R) = W/Al and L = W/A2, where Ai is a subgroup of Z(W). Since PSL(2,R) has trivial centre ((3.2 (ii))), it follows that Al = Z(W) A2, and there exists a continuous homomorphism from L onto PSL(2,R). Hence by (3.2(i)) and (3.1), L contains F2 as a discrete subgroup. Since L is closed in H/K, its topology coincides with the relative topology induced by H/K (B16), and it follows that the group H/K also contains F2 as a discrete subgroup. Now use (3.1) twice to contradict (ii). So H/K is compact as required. (3.9) THEOREM. The following three statements are equivalent for a connected Lie group G: (i) G is amenable as a ,discrete group; (ii) G does not contain' F2 as a subgroup; (iii) G is solvable.
PROOF. Clearly, (iii) implies (i) and (i) implies (ii). Suppose that (ii) holds. Then G does not contain F2 as a discrete subgroup, and so by (3.8), G/r(G) is compact. Suppose that G/r(G) is nontrivial. Then by (3.3(i)), we can find a Lie subgroup H of G/r(G) whose Lie algebra is so(3,R). Now STJ(2) is the
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FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
simply connected Lie group with Lie algebra so(3, R) (B19), and so there exists a continuous homomorphism Q: SU(2) -+ H with kerQ C Z(SU(2)). By (3.2(iii)) and (3.1), H, and therefore G/r(G), and therefore G, contains F2 as a subgroup.
This contradiction of (ii) yields that G/r(G) is trivial. Hence G = r(G) is solvable, and (ii) implies (iii). 0 (3.10) A THEOREM OF TITS. Let V be a finite-dimensional vector space over a field F and G a subgroup of GL(V). Then G is amenable if and only if it does not contain the free group F2 on two generators as a subgroup. Now suppose that G is amenable. Then (a) if F has characteristic zero, then G contains a normal, solvable subgroup of finite index;
(b) if F has nonzero characteristic, then G contains a normal, solvable subgroup H with G/H locally finite. This remarkable theorem is proved by Tits [2]. A Lie group version of this result is proved above.
References Dekker [1], van Dijk [1), Goto [2], Greenleaf [2], Iwasawa [1], Milnor [4], Moore [2], von Neumann (1), [R], Chapter 8, Rickert [1], [2), Takenouchi [1), Tits [2), Wagon [S2], and Yosida [1).
Further Results (3.11) A group in NF but not in EG. The classes NF and EG were introduced in (0.16). We will show here that EG * NF, basing our account on Chou [14].
R. I. Grigorchuk [Si) has proved the stronger result that EG AG. The proof uses the existence of a periodic group A that is not locally finite. (So every element of A has finite order, and there exists a finite subset of A that generates an infinite subgroup.) The existence of such a group A is well known, and settles the Burnside problem negatively. The existence oT A follows from Golod [1) and Golod and Shafarevitch [1]. A short, lucid account of the construction of A is given in Chapter 8 of Herstein [21. See also Grigorchuk [1), who realises such a group in terms of measure preserving transformations of 10,I).
The group A is in NF but not in EG. [Obviously, A E NF. To prove that A V EG, it is sufficient to show that every periodic group in EG is locally finite. To this end, we define, by transfinite recursion, for each ordinal 1, a class EG., of groups as follows:
(i) EGo is the union of the class of finite groups and the class of abelian groups;
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109
(ii) if y is a limit ordinal, then EG., = U{EGp: 6 < -y};
(iii) if iy has a predecessor I, then G E EG., if and only if either G is an extension of H by K with H, K E EGO or G is the union of an upwards-directed family of members of EGO. Let EG' = U,, EG.,. We show first that EG' = EG. (The reason for working with EG' rather than with EG directly is that it is easier to check in EG' rather than in EG that every periodic group is locally finite, since the two processes in (0.16) of taking subgroups and forming factor groups need not be considered.)
Using transfinite induction, it is easily checked that EG, C EG for all -y. Hence EG' C EG. We now prove the reverse inclusion. Firstly, as EGo C EG', it follows that
EG' contains all finite and all abelian groups. It is an easy consequence of (iii) that EG' is closed under the process of forming group extensions. Now let
{H6} (6 E d) be a net in EG' with H6, C H6, if 61 <_ 62. For each 6, let H6 E EGQ(6) and let y = (U6EA a(6)) + 1. Then -y is an ordinal, and from (iii), U6EO H6 E EG., C EG'. Thus EG' is closed under the process of forming upwards directed unions. It remains to prove that EG' is closed under the process (I) of taking subgroups and (U) of forming factor groups. For then EG' is closed under all four processes of (0.16) and so must contain EG. It is clearly sufficient to show that, for each ordinal 7, EG., is closed under (I) and (II). This is proved by transfinite induction.
The result is obvious when -y = 0. Now suppose that a is an ordinal, and that EGO is closed under (I) and (II) for all Q < a. Let G E EGQ, and let B be a subgroup of G, C a normal subgroup of G, and D = G/C. We have to show that B, D E EGQ. If a is a limit ordinal, then G E EGO for some 6 < a, and it follows that B, D belong to EGO C EGQ. Suppose then that a has a predecessor e. Then either (a) there are groups L, M in EGE with L a normal subgroup of G and GIL = M or (b) G is the union of an upwards directed family {G,} of groups in EGE. Suppose that (a) holds. Then L r1 B and B/(L f1 B) = LB/L belong to EGE as they are subgroups of L and M respectively. Hence B E EGQ. Now (CL)/C = L/(L f1 C) E EGG, and so
D/(CL/C) = G/CL = (G/L)/(CL/L) = M/(CL/L) E EGE. Hence D E EGQ as required. Now suppose that (b) holds. Then
B = U(G, fl B) E EGQ r
and
D = U(CG,/C) E EGQ. r
So EGQ is closed under (I) and (II) and it follows that EG' = EG. To show that A % EG, it is sufficient to show that every periodic group in every EG., is locally finite. This is proved (yet again!) by transfinite induction. The desired result is easy when -y = 0. Suppose that a is an ordinal and that the result is true for all - < a. Let G be a periodic group in EGQ. If a is a limit ordinal, then C E EG., for some -y < a and so is locally finite. Suppose, then, that a has a predecessor 7. There are two cases to be considered.
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110
Suppose that H, K E EG.r with H a normal subgroup of C and G/H = K. Since G is periodic, so also are H and K. By assumption, both H and K are locally finite. By Problem 3-10, G is also locally finite. The other case is that in which G is the union of an upwards directed family {G7} of groups in EG7. Since every finite subset of G is contained in some Cr, it follows that G is locally finite. This completes the proof.]
(3.12) Three problems considered by Banach. In his paper [1], Banach considers three questions, all of which involve invariance of finitely additive measures. A commentary on Banach's paper is given by Mycielski [2]. Mycielski [6) and Wagon [82] are useful references for the discussion below.
The first problem originates with Lebesgue [1). Essentially, Lebesgue asked if the monotone convergence theorem, which was a property of his integral on R, could be dispensed with. Now the latter property is equivalent to countable additivity, and thus Lebesgue's question can be formulated: if µ is a finitely additive, positive, translation invariant measure on.4,(R), the family of bounded
Lebesgue measurable sets, such that µ(j0,1)) = 1, does µ = Al, where Al is Lebesgue measure (restricted to .4,(R)). Banach showed that the answer to this question is in the negative. He constructed a finitely additive, positive, translation invariant measure µ on 5ab(R), the family of bounded subsets of R, such that (i) µ(A) < oo for every bounded subset A of R (so that µ gives rise, in the obvious way, to an element µA of 10, (A)' for each such A); f Q ¢(x) dx for every Riemann integrable function on an interval [a, b);
(iii) there exists a Lebesgue integrable function zy on an interval [c, d] such
that J
dVG(x)dAI(x)
(A similar result to this for the circle group T, with continuous functions in place of Riemann integrable functions, follows from (7.17).) The other two questions considered by Banach involve isometry invariance
rather than translation invariance. (Indeed, in terms of the physical world, invariance under isometries (rigid motions) is more natural than just invariance under translations.) Let Gn be the group of isometries of W. Then G. is a semidirect product R's xy O(n, R). Further, Lebesgue measure An on R" is Gn-invariant on . '(R"). (Indeed, the equality A(TA) = An (A) follows from the "change of variables" formula for n-variables, since the Jacobian of the function x Tx is the determinant of an element of O(n,R) and so has modulus 1.)
The second problem considered by Banach arose out of results proved by Hausdorff 12]. Hausdorff raised the following question, which originated with
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
111
Lebesgue and which Banach calls "le problbme large de la mesure". It is often referred to as "the problem of measure." The question is (Hausdorff 12, p. 401]) does there exist an invariant measure for (Gn, R", [0, I)")? So ((2.32)) we are asking if there exists a Gn-invariant, finitely additive, positive measure µ on
R(R") such that µ((0,1J") = 1. Hausdorff showed (ibid., p. 469ff.) that there is no such measure when >_ 3. The crucial step of his argument was his construction of a partition n{P, S1, S2, S3} Of S2 = {(XI, x2, x3) E R3 : Es 1 xj = 1} C R3 and elements 0, iG E SO(3, R) such that P is countable and O(S1) = S2 US3,
'+'(S1) = S2,
'
2(S1) = S3.
This initiated a line of development leading to the Banach-Tarski Theorem of 1924 (Banach and Tarski 11]). (This theorem embodies the Banach-Tarski Paradox.) A short account of the theorem is given in (3.16). This leaves the cases n = 1, 2 to be dealt with. Banach showed in these cases that there does exist an invariant measure for (Gn, R", [0,1]"). This follows from (2.32(i)). Indeed, in the latter, we can take X = R", A = 10, 11", and (b) easily follows by applying a" to the left-hand side of (1). Since G1 and G2 are amenable as discrete (Problem 0-25), the desired result follows. Von Neumann [1) realised that the case n =1, 2 could be generalised to groups carrying a finitely additive, invariant positive measure of total mass one, that is, to amenable groups. For example, if H is an amenable subgroup of Gn for any n, then there exists an invariant measure for (H, R", [0,1]"). Von Neumann also showed that versions of the Banach-Tarski Theorem can be obtained in two dimensions by considering, instead of the amenable group G2, the (nonamenable) group A2 of all affine transformations of R2 that preserve area. (The nonamenability of A2 follows from the inclusions ((3.2)): F2 C SL(2, R) C A2, where the fact that T E SL(2, R) is area preserving follows by using the change of variables formula for multiple integrals and the equality detT = 1.) Von Neumann also realised the importance of F2 in the problem of measure: G" is not amenable if and only if Gn contains F2 as a subgroup. (For more in this direction see Wagon [2), [S2).) Indeed, if n > 3, we can regard SO(3, R) as a subgroup of G,, by allowing the elements of SO(3, R) to act on the first three coordinates, and then (3.9) gives F2 as a subgroup of SO(3, R). The reader is referred to S. Wagon's monograph [S2] on the Banach-Tarski Paradox for a comprehensive account of Banach's second problem and its concomitants. The general problem of measure, considered in (2.32), is concerned, of course, with the existence of an invariant measure p for a triple (G, X, A). We showed in (2.32) that if G is amenable, then such a measure µ exists if and only if (C, X, A) has the Translate Property. Tarski proved a remarkable theorem (which we shall call Tarski's Theorem) in which it is shown that a measure p exists if and only if A admits a "paradoxical decomposition." (One might hope for such a characterisation, given the
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FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
"paradoxes" inherent in Hausdorff's partition of S4 mentioned earlier and in the Banach-Tarski Theorem.) The set A is said to admit a paradoxical decomposition if there exists a parti-
tion Al, ... , A., B1, ... , Bn of A and elements x1, ... , xm, yl, .... Y. of G such that {x1A1,...,x.A,n} is a partition of A and {y1B1,...,y,,Bn} is a partition of A.
If such a decomposition exists, then there cannot exist an invariant measure for (G, X, A). For if µ were such a measure, then n
m
1 = µ(A) _
j=1
i=1
n
m
µ(Bj) _
µ(A,) +
A(xiAi) + i=1
µ(yjBj) j=1
+µ(
yjBj1 = 2 (t}. cl This is the easy implication in Tarski's Theorem. The converse implication is
A U xiAi) t.I
much more difficult, and is established, using (3.13) and (3.14), in (3.15). It remains to mention briefly the third problem considered by Banach. This was raised by Lebesgue and Ruziewicz and is often referred to as the BanachRuziewicz Problem.
Recall that An is Lebesgue measure on Rn and that 4b(R") is the algebra of bounded Lebesgue measurable sets. The Banach-Ruziewicz Problem then is does there exist a finitely additive, Gn-invariant, positive measure µ on4"b(R") such that µ([O,1)") = I and µ ¢ An. (There is a version of this problem for the n-sphere S".) Banach showed that the answer to this question is affirmative if n = I or 2, but left the cases n > 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).
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
(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
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FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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?
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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.)
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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.]
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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) <_ F(t) whenever s, t E T and
s
from B' into [0, oo), T C B', F' [ T = F, and F'(s) < F'(t) whenever s, t e B'
and $ < t (in S). Clearly, (F,T) E 9, so that 9 0 0. Then 3 becomes partially ordered under <, where (F', B') < (F", B") if B' C B" and F" I B' =
118
FREE GROUPS AND THE -AMENABILITY OF LIE GROUPS
F. Let sad be a linearly ordered subset of R, and let S1 = U{B' : (F', B') E sW). Define F1: Sl 10, oo) by F1(s) = F'(s), where (F', B') E .s9, and s E B'. Clearly (F1, SI) is an upper bound for d in a. Thus Zorn's Lemma applies to give a maximal element (g, W) of R. We need to show that W = S. Suppose, on the contrary, that W 96 S, and let a E S - W. Let
W'={w+na:wEW, nEP}UP6. Obviously, W' is a subsemigroup of S. Let x E W' and define (1)
9'(x) = sup{(g(w1) - g(w2))/n: n E P, WI, w2 E W, and w1 < w2 + nx}.
Since for w E W, w < w + lx, g'(x) is well defined and > 0. Further, e e T C W and, for some k E P, x < ke, so that if w1 < w2 + nx, then w1, (w2 + nke) E W and w1 < (w2 + nke) in S. Since (g, W) E 3, (g(wl) - g(w2))/n < g(ke) = F(ke) = k.
Thus g'(x) < oo, and g': W' [0, oo). We will contradict the maximality of (9, W) by showing that (g', W') E 3. To this end, it is sufficient to show that if x = w+ka (w E W, k E N) and n E P, then (2)
g(x) = g(w) + kg'(a), g'(na) = ng'(a).
Suppose that x = w + ka as above and let wl, w2 E W and m E P be such that w1 < w2 + ma.
(3)
Then
mw+kwl <mw+kw2+kma=kw2+m(w+ka) = kw2 + mx.
Referring to (1), we obtain (4)
g' (x) > 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).
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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).
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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.
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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.
FREE GROUPS AND THE AMENABILITY OF LIE GROUPS
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 <- p < oo and G a locally compact group, we define PP(G) = {f E LP(G) : f >- 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 <_ llisllilf lip for all p E M(G), f E LP(G).
(4.3) PROPOSITION. Let {g6} be a net in PP(G) such that llt*g6-9611, - 0 (t E G) uniformly on compacta. Then his * g6 - g6 lip - 0 for all p E PM(G). PROOF. Since lip * 911, _< llpll1191IP (g E LP(G)) and 119611, =1 for all 6, we
can suppose that p has compact support C. For any g E LP(G) and k E Lq(G), where p-' +q-1 = 1, we have, using the theorems of Fubini and Holder (which are easily seen to apply) and the fact that A E PM(G),
if (p * g - g)(x)k(x) dA(x)= I
<
dg(t) f (t * g - g)(x)k(x) dA(x){
f
It * g - 911P11kll, dp(t)
5 llkllq
Fp
lit * 9 - g11,.
Hence 1!s *
l
- 9119 = sup {f(is * g - g)(x)k(x) da(x)I : k E Lq(G), llkllq < 1 } 111
II
S Epllt*9-gl1P. Since suptEC Ilt * 96 - 96lIP -0, the required result follows.
Q
Our next result will require the use of the following inequalities: if f, g E P(G) and p E [1, oo), then (1)
(11f
1/P - g1/Pllp)P <- 11f - g111 G p2P_111f 1/P - g1/PIIp.
This is trivial for p = 1. One can prove (1) for p > 1 by first establishing, by elementary differentiation, that b)P-1
lb - all' < IbP - apl < plb - al(a + for all a, b > 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 < p < oo. Then the following statements are equivalent: (i) G is amenable;
(ii) if C E ''(G) and e > 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
lie*h - hlli < Ilzi*u*9*go-xi*g*9oli1+IIxc*g*9o-g*90111 <- Ilu*9-gill+Ilxi*9*go-9oli1+lIg*9o-9olli <377 by (2), (3), and the fact that e is an xi. Using (4.3(1)) and noting that (c*h)1/P = c * h1"P, we have (4)
IIc * h"P - h1"Pllp < (3,7)3/P
Taking f = h1/P and (3,7)1/P < s, we have that (i) implies (ii). Now suppose that (ii) holds. Then there exists a net {g6} in PP(G) such that lit * 96 - 96 lip - 0 uniformly on compacta, and using (4.3), (ii) implies (iii). Finally, suppose that (iii) holds. Then a fortiori, IIx * g6 - g6IIp 0 for all x E G. It follows from (4.3(1)) that IIx * gs - gb III -+ 0 for all x E G and G is amenable ((0.8)). (4.5) Notes. Obvious modifications of the preceding proof give other characterisations of amenability involving a right-handed version of (ii) of (4.4). Indeed,
let G be amenable, C E F(G), and s > 0. From (4.4), there exists f E P(G) such that lix * f - f 11, < e (x E C'1). Since the involution k k- on L1(G) is isometric, we have (1)
IV- * c - f-lli <S
(c E C).
Note that for g E LP(G) (15 p < oo) and p E M(G) with compact support, we have
g * !s(x) = f g(xy-1)d(y-l) dp(y),
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128
and g *,u E LP(G) with Ilg * iUIIP < HIgIIP
J
A(y)-1/9dlAl (y)
[HR1, §20].
In particular, g * c(x) = g * bc(x) = g(xc 1)A(c-1). Due to the complications caused by the modular function A, care has to be taken in obtaining the analogue ofr (4.4(ii)) when p > 1. Indeed, for h E P(G), (h * c)11'(yy) = ih(2/c-1)©(c-1)J1/P = A(c)1/9[h(yc 1)1/P,&(c
1)j
_ A(c)1/9(h1/P * c)(y) (14.13(1)),
Taking h = f and using (1) and
we readily obtain the result: G is amenable if and only if, given C E 9(G) and e > 0, there exists h E Pp(G) such that IlA(x)1' h * x - hIIP < e (x E C) (cf. Skudlarek (1j).
One easily checks that if C, e, and f are as in (4.4(ii)), if p = 1, and if g E P(G) is such that IIg * x - gill < e for all x E C, then k = f * g belongs to P(G) and satisfies the two-sided condition (2)
IIx*k - klll < e,
Ilk*x - kill < e
(xEC).
The amenability of G is obviously equivalent to such a two-sided condition. By taking g = f- as above and replacing C in (4.4(ii)) by C U C-1, we can take k
in (2) to be of the form f * f"' with f E P(G). Note also that in the preceding results of this chapter, we can replace P(G) [PP(G)J by P(G) fl C,, (G) [Pp(G) fl C.(G)j since the latter set is norm dense in the former. Here, Cc(G) is the space of functions in C(G) vanishing at infinity. Our next aim is to improve the characterisation (4.4(ii)) of amenability in the case p = 1. It will be shown ((4.10)) that the function f of (4.4(ii)) can be taken to be of the form Xx/A(K) for some nonnull K E £'(G). Recall that for sets A,
B, the symmetric difference AA B is defined to be (A - B) U (B - A). Note that IXA - XBI = XA-B + XB-A = XAOB Since
fix * f - fill = IIXzK - XKII1/A(K) = A(xK 0 K)/A(K) for such a function f, it follows that the inequality (4.4(1)) becomes: A(xK L1 K)/A(K) < e
(x E C).
We start by establishing some simple, but useful, results on symmetric differences in G and Haar measure.
(4.6) LEMMA. Let C, K E '(G).
(i) IfxEG, thenA(xKflK)=,1(K)- 2.1(xK©K). (ii) If x E C and C1 = C U C-1, then
) (xK 1 K) < 2A(C1 K / K).
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129
(iii) If C 96 0, C2 E ir. (G), and C C C2, then A(CK A K) < 2A(C2K Li K). PROOF. (i)
2A(xK n K) = [A(xK) - A(xK - K)) + [A(K) - X(K - xK)} = 2A(K) - A(xK 0 K). (ii) If x E C, then A(xK L1 K) = A(xK
K) +,\(x-'K - K) < 2A(CIK A K).
(iii) We have A(CK) >- 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))
K) = 2A(C2K L1 K)
(since C2K D K as e E C2). References for (4.7)-(4.13) are Namioka [1), Emerson and Greenleaf [1), Folner [2), and Frey [1).
(4.7) LEMMA. Let G be amenable, C E '(G) nonnult, and e, b > 0. Then there exists nonnuIlK E '(G) and a compact subset N of C such that A(N) < b, and
A(xK L1 K)/A(K) < e
(I}
(x E C
N).
PROOF. Let q > 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
for each i, then for each i, we can find (by inner regularity of A) an element Di
of 9'(G) with Di c E; and A(E1 - D,) as small as desired. It follows that we can suppose that each gn is of the form =I ryiXD, with ryi > 0, Di E W(G), A(D=) > 0, and Di n D, = 0 for i 94 j. Since gn -+ f in LI (G), it follows that lignin --' llfIii, and we can suppose that lignin = 1, that is, that gn E P(G). Finally, since
ll(x*gn-gn)-(x*f -f)l11-- 0 it follows that we can find g E P(G) with (2)
llx*g - gill < q
(xEC)
such that g = E 1 aiXc,, where Ci E'(G) are such that A(C1) > 0, CinCj = 0 if i 96 j, and the order of summation is arranged so that a1 > a2 > >
FOLNER CONDITIONS
130
an>0. Let Br=Uf=1 C, (10for all r, and xc, = (xB, - XB._1) SO, With Bo = 0, we have n
n
9=EaiXc, _Eai(XB; -XB,_,) s=1
i=1
n-1
n
= anXB,. + E(ar - ar+l)XB, _
Qr(XB,./A(Br)) r=1
r=1
Note that
(1
Qr>0
(3)
r=1
For xEGand 1
{Qr/l{Br)}{Xxa, .n. - XB,-xB,) Ir_1
= H \r
1
(QrP(Br))XzBr_.B)
(4)
-
\rE{Qrl
A{Br))XBr...xB) 1
1
rt
E(Qr/),(Br)),\(zBr A Br). r=1
Integrating the continuous function x -+ IIx * g - g111 over C and using (2) and (4), we obtain
Qr (C(A(xBr L Br)/A(B,)) d) (x) < rtA(C). r=1
Using (3), we can find s such that (5)
tc(A(xB, A B9)/\(B8)) dA(x) < 27A(C).
Then (1) is satisfied with K = Bs, 27 = (e6)/A(C), and N = {x E C: A(xB, A B,)/A(B,) > e}.
(4.8) Notes. If G is unimodular, then, by (4.5), we can suppose f in the above proof is such that f = f. Modifying the proof and replacing g by 1 + 9 ), we can further suppose that g" = g. Then the sets C are symmetric {g and we conclude if G is unimodular, then the set K of (4.7) can be taken to be symmetric.
FOLNER CONDITIONS
131
The analogue of (4.7) for a discrete left amenable semigroup S is false in (In such an analogue, the sets C and K are finite and we can take N = 0.) Following the argument of (4.7) in this case and using the semigroup version of (4.1) instead of (4.4(ii)), we obtain the results of the first paragraph of the above proof for S. If S is left cancellative, then the rest of the argument is valid. But if S is not left cancellative, then x * XB, may not equal XzB, so that the step "x * XB, - XB, = XZB, - XB," in the argument fails. general ((4.22)).
Recalling the formula for the convolution in 11(S) ((0.18)), we see that if x E S and E E JT(S), we have x * XE(s) = 0 if s oo xE, and that
x*XE(s)=I{eEE:xe=s}I>1 if s E xE.
Thus, replacing E by B it follows that x * XB, ? XxB,. We still have
(xB,-B,)fl(B.-xB,)=0ifs96 r. So ifA=Ur 1(xB,-B,),then (X.-B, - XB,)I A = (Xx*B, - XB,nxB, - XB,-xB,)i A 2 and so
A,
n
jx*g - gh ? E(,6r/A(Br))(x * XB,
- XB, )I A
r=1
1
n
> E(,a,/i1(B,))A(xB, - Be). r=1
Recalling that, in this case, A is counting measure and taking b = 1(!), the rest of the argument of (4.7) applies mutatis mutandis, and we have the following result. (4.9) PROPOSITION. Let S be a left amenable semigroup, C a finite subset of S, and s > 0. Then there exists a nonempty, finite subset K of S such that
(xK--KI/lKt < r
(xEC).
See (4.22) for further discussion of Folner conditions for left amenable semigroups. The next result is the fundamental Folner condition for a locally compact group. See Problem 4-10 for a symmetric version.
(4.10) THEOREM. A locally compact group G is amenable if and only if whenever e > 0 and C E '(G), then there exists a nonnull, compact subset K of G such that (1)
.1(xK A K)/A(K) < e
(x E C).
PROOF. If, given e and C, we can find nonnull K E '(G) such that (1) is satisfied, then G is amenable ((4.5)). Conversely, suppose that G is amenable, e > 0, and C E P(G). Replacing C by a larger compact subset of G if necessary, we can suppose that X(C) > 0. Let
D = C U C2. Applying (4.7) with 6 = z.\(C), we can find nonnull K E '(G)
FOLNER CONDITIONS
132
and a Borel set N C D such that A(N) < 6 and a(xK © K)/A(K) < as for all
xED-N.Nowifx,yEDN, then (2)
a(xy-1K II K) = )+(y-IK A i 'K):5 A(y-1K A K) + A(x 1K L K) = A(yK A K) + A(xK © K) < sa(K).
Further, if x E C, then xC C D and
xD n D C [x(D - N) n (D - N)] U [x(D - N) n NJ U xN, so that
2b = A(C) = A(xC) = a(xC n D):5 a(xD n D)
C C (D
0 so that x e (D . N)(D - N)'1. Thus
N)(D - N)-1, and (1) follows using (2).
13
Note that by (4.6(i)), (1) is equivalent to the inequality
) (xK n K)/A(K) > 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
Take N1 = 1 and t1 = e. Let m E P and suppose that N" (1 < n < m) and tj (1 < j < N,,,) have been constructed so that (a)-(e) are valid whenever they make sense. If V"' C U;_I tiV, then we take Nm+1 = N. Suppose that VI ¢ UN i tiV. Define recursively a (finite) sequence tNm+1, tNm+2, with ti E V- [U tkVJ for i > N,,, + 1. If ti is defined with i > N,,, + 1, then, using (b), t;Z n tkZ = 0 (1:5j < k < i), and, using (c), U;,1 tkZ C V nZ, so
FOLNER CONDITIONS
133
that i < A(V-Z)/.1(Z). So the sequence tNm+1, ... has a last member tNm}, One readily checks that this completes the recursion process. From (d), G tiV. If x E G and x E t1V for some i, then t; E
xV-1
=
xV and t,Z C xVZ. Since t,Z f1 t1Z = 0 if i 0 j, we have J{i: x E t1V}] < A(VZ)/A(Z). So the lemma is established in this case, with T = {ti: i E P} and R any integer > .1(V Z)/. (Z). We now deal with the case where H = U 1 V" is no longer assumed to be G. By the earlier part of the proof, we can find a subset S of H and an integer
R > 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
FOLNER CONDITIONS
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'(6)) = A(K') - A(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)) <.(K') + A
(U(CC'1t-'
- K': t E 1(6)))
),(K') + E A(C'yt ^- K') tEI(6)
< A (K) + E 0(yt)A(C' - K'yt 1) vEI(6)
tEI(6)
Clearly the required result (ii) will follow once we have established
E
(4)
A(t_1)
< RA(K')/(A(C-1) - 6M).
tEI(6)
Now if t i , .. ,
E T, then, by (4.11(ii)), each x E G is contained in at most
R of the sets C-Iti1, so that
_1 Xc_,t_1 SR. Hence a(C_1t= 1
s=1
n K').
icI
'It is straightforward to show using (4.11(ii)) that if D E "(G), then D lT is finite. It therefore follows that I(6) is finite.
FOLNER CONDITIONS
135
Hence
RA(K') > E \(C-'t-1 fl K') tEl(6)
A(C-1t-1
K'))
tEl(6)
[A(C''t-') - A(C-1Cyt - K')) tel(6)
['\(C-1t-1) - A(C'yt - K')) tEl(6)
E IA(C-1) & (t-') - 6MA(t-1)) tEl(6)
_ (},(C-') - 6M) E A(t-1). tEl(6)
The inequality (4) now follows.
O
(4.13) THEOREM. The following statements are equivalent: (i) G is amenable; (ii) if C E'W(G) with C 96 0 and e > 0, then there exists nonnull K E W(G) such that
A(CK L K)/A(K) < e;
(iii) if C,L E I'(G) with C # 0 and e > 0, then there exists nonnull K E W(G) with L C K such that (1)
A(CK L1 K)/A(K) < e:.
PROOF. Trivially, (iii) implies (ii). That (ii) implies (i) follows from (4.6(ii)) and (4.10). We now prove that (i) implies (iii). Suppose, then, that G is amenable, and
let C, L E'(G) with C 0 0, and e > 0. By (4.6(iii)), we can suppose that C E P(G). Let e > 0 and b, M, and R be as in (4.12). Choose 6 > 0 so that 6 < A(C-1)/M and 6MR/(A(C-') - 6M) < Now choose rl > 0 so that 2e.
nb < 6 and [I - (rlb)/6))-1(1+ 2e) < 1 +e. Then with K' as in (4.12), we have A(CK'(6) 4, K'(6))
- A(CK'(6)) - a(K'(6)) A(CK'(6)) A(K')
A(K') A(K'(6))
< 1 + {6MR/(A(C-') - 6M)) 1- (r1b)/6 < C.
So given e > 0, there exists nonnull KE E '(G) such that (2)
A(CKE A K.)/.A(K.) < e.
-1
FOLNER CONDITIONS
136
It remains to be shown that KE can be chosen so that L C K. Again we can suppose that e E C. If G is compact then we can take KE = G. Suppose that G is not compact. We construct by recursion a sequence {K,} in W(G) such that for all r
(a) )(K,+1) _2.1(K,)>0; (b) A(CK, AK,) < ZE,1(K,). By (2), we choose KI so that (b) is satisfied. Suppose that K1,... , Kn have been constructed so that (a) and (b) are satisfied whenever they make sense. Let A = {x E G: L1(z) > 1}. Since G is not compact and G = A U A-1, the set A is also not compact, and we can find yn E A - Kn 1C-1CKn. Let Kn+I = K. U Knyn. Now Knyn n K. = 0, and so A(Kn+l) = A(KK) + A(Knyn) = (1 + L(yn))A(KK) > 2A(Kn).
Further, CKn f1 CKnyn = 0, and so (CKn+I 0 K,,.+ 1) = A([CKK U CKnyn] - [Kn U Knyn])
(CKn . Kn) + \(CKnyn - Knyn) < 1 £a(Kn)+©(yn)2e.1(Ka) = 2
This completes the construction of the sets K. Note that from (a), \(K,,) 00. Let Ln = Kn U L. Then CL, A Ln = CL,, . Ln C_ (CKn A Kn) U CL. So A(CLn l& Ln)/A(L,,) <_ [A(CK,, A K,) + A(CL)]IA(Kn),
and using (b), we can take the set K of (iii) to be Ln for large enough n. (4.14) COROLLARY. Let
I(G) =
sup {inf{A(CK)/.1(K): K E'(G), .(K) > 0}}. CE5'(G)
Then I(G) = 1 if and only if G is amenable. PROOF. We can obviously restrict attention to the case C 54 0 in the definition of I(G). Since A(CK)/.(K) > 1, we have I(G) = 1 if and only if
inf{.1(CK)/.1(K): K C '(G), .1(K) > 0} = 1
(1) for each C.
Suppose that G is amenable and let C (96 0) be in '(G). Find a set C1 E W.(G) with C C C1. By (4.13), we can find a sequence {Kn} of nonnull elements of '(G) with A(C1Kn & Kn)/A(K,,) 0. Since
A(C1Kn © K,,) _ .\(CjK.) - A(Kn),
it follows that .\(C1K,,)/A(K,) - 1. As 1 < A(CK&)/A(K.) <- A(C1Kn)/A(Kn),
we see that A(CK,)/.1(K,,)
1, and (1) follows.
FOLNER CONDITIONS
137
Conversely, if (1) is true with C replaced by C1, then we can find a sequence {K;,} of nonnull elements of'(G) with .k(C1K,',)/a(K,,) - 1. Now use (4.6(iii)) again to deduce that
,1(CK A K,)/A(K,) < 2[i\(C.1K,) so that G is amenable by (4.13).
0
13
It is easy to prove that if G is not amenable, then I(G) = oo (Problem 4-11). An application of (4.13) to transference theory is given in Problem 4-44. (4.15) DEFINITION. A net {Kb} (b E A) of nonnull, compact subsets of G is called a summing net for G if the following conditions are satisfied:
(1) K8CK1,if b
and {[-n; n]} are summing sequences for R and Z respectively. References for the result below are Emerson [2j and Bondar and Milnes [1]. (4.16) THEOREM. A locally compact group G is amenable if and only if there
exists a summing net for G. If G is a-compact, then G is amenable if and only if there exists a summing sequence for G. PROOF. Let G be locally compact. If there exists a summing net for G, then G is amenable by (4.10).
Conversely, suppose that G is amenable. Let {H.,: y E r} be the set of acompact, open subgroups of G. Every compact subset of G is contained in some
H.,. Further ((1.12)) each H1 is amenable. For each y, find a sequence {Cn} in F,(H1) such that C,-,t C (C,',,+1)°, (Cn)-1 = C,' and H, = U°=1 C,','. Let
A = P x r. For each b = (n, y) in A, find, using (4.13) and (4.6(ii)), a set Kb E F(H1) such that Kb
C,7, and
A(xKb 0 Kb) < A(K5)/n
(x E C, ).
The set A is made into a net by defining (n, y) < (n', y') if n < n', H,, C H.,,, C,7, C C,,,, and K(,,,.7) C
To prove this, let b1 = (n1, y1), b2 = (n2, -f2)
be elements of A. For i = 1, 2, let ij be a countable family of symmetric, compact subsets of G, such that H1 = U £°. Define
H.,=U{Ai...An: n> 1, Aj E91UW2}. Then H., is a a-compact, open subgroup of G containing H1, U H.72. Now find N > n1, n2 such that CN ' C.1'1 U C,7,? U Kb, U Ks,.
Then (N, -y) > (ni, y1), (n2i 12), so that A is a net as required.
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138
It is obvious that (1) and (2) of (4.15) are satisfied. We check (4.15(3)). Let
CEi'(G)ands>0. Find bo=(no,7o)such that CCCg,and 1/no<e. Then for x E C and b > (no, -yo), we have
A(xKb A K6)/A(K8) < e and (4.15(3)) is established. Obvious modifications of the above argument deal with the case of a summing sequence for a-compact G.
The next result shows that all topologically left invariant means on G can be obtained from a given summing net. Let G be an amenable locally compact group and {K6} a summing net for G. For each 6, let {16 = XK+/A(K6) E P(G).
For each net {x6} in G, let C({xe}) be the set of weak' cluster points of the net {(µs * x6)^} of means in 9 Z(G). (One readily checks that p6 * x6 = Xxa=+/A(K6x6).) Let £ = U{e({xb}): {x6} is a net in G}. The theorem below is due to Chou [4).
(4.17) THEOREM. The setZt(G) is the weak` closure cbC of L. PROOF. Let {xb } be a net in G and m be a weak* cluster point of { (µ6 *x6)^}. Then m E!m(G). By (4.15(3)) and (4.3), for p E P(G), 11µ*u6*x6 -IA6 *x6111 5 jju * µ6 - µ6[j1 - - + 0. Hence µm = m and m E £c (G). So 3E C C (G).
Conversely, let m1 E .G(G). Suppose that m1 ¢
C. Separating m1 and
2ro.C, we can find a real-valued function 0 E L,o(G) and positive numbers c, e such that (1)
.m(V,)
(mEco,E).
For each 6, let M6 = supxEG Oy6(x). Fix n E P and for each b, choose x6(n) E G such that (2)
[M6 -V}16(x6(n)}1 < n-1
Let m2 be a weak' cluster point of the net {(µb * x6(n))^} in 9A(G). Then m2 E C. Let t7 > 0. Then we can find v such that Im2(') - OA.,(xo(n)j < 77, (3) where we have used the equality: (µ * x)^(O) = &p(x) ((1.1)). So using (1), (2), and (3),
m2('O)+e<ml(10 =m1(i,b )5Mo 5duo{xb{n))+n'1
<m2(0)+7+ By choosing n large enough and then r/ small enough, we contradict (1). So m1 E c3.C and the proof is complete. We conclude this chapter with some applications of Reiter and Folner conditions. The first is due to Glicksberg [21 and Reiter 1141, [R).
For f E L1(G), let Cf = co{f * x: x E G} and d(0,C1) = inf{jjgjjj: g E Cf}.
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139
(4.18) THEOREM. The locally compact group G is amenable if and only if, for all f E LI (G),
d(O,Cf)=I f f dAl
(1)
PROOF. Suppose that G is amenable. Since lI f *xf ll > I Xc(f *x)l = I f f dAI, we have
-fdAI. if
d(O,Cf)>
(2)
To show equality in (2), we assume first that f f d.1 = 0. Suppose that d(0, Cf) > 6 > 0. Separating the convex sets Of and {g E L1(G): u ghh1 < 6} [DS, V.2.8), we can find 06 E L1(G)` = LA(G) and 7 E R such that Re 06 (g) 5 7 5 Re 06 (h)
(3)
for all h E Cf and g E L1(G) with IIgII1 < 6. We can suppose that 110611 = 1. Since sup{Re ¢6 (g) : 11g111 = 6} = 6, we can take 'y = 6 in (3). Let D = {¢ E L00(G): 11.011 = 1, Req5(h) > d(0,Cf) for all h E Cf}. By taking a weak` cluster point of {06}, we see that D is not empty. Clearly, D is a weak` compact
subset of L0,(G). If 0 E L.(G) and Ret/i(h) > d(O,Cf) for all h r= Cf, then IIV511 > 1. It follows that D is convex. Since h * x E Cf if h E Cf, x E G, we have that D is left invariant for G, and since each map (h, x) h * x is norm continuous (cf. (1.2)), it follows that the map (x, q5) -{ x¢ is continuous on D. Applying the fixed-point theorem (2.24), we can find 4 E D with x¢ = ¢ for all x E G. Hence ¢ is constant (cf. (1.7)), and so =f al where a E T. So
d(0, Cf) < Re 0(f) < fa
( f dAl = 0
(
giving a contradiction. So (1) is proved for functions f with f f dA = 0. Now suppose that f is general, and let h E P(G), a = f f dA. Applying the preceding result to (f - ah), we obtain (4)
n
ai(f *xi-ah*xi)
:n>1, a,>0, Eai=1, xiEG
sc l
i=1
Now
a,(f *xi-ah*x,) i=1
I
(tah/1*) Ri=1
> 0, and (1) follows from (2) and (4).
dA
FOLNER CONDITIONS
140
Conversely, suppose that (1) holds for all f E L1(G). Let e > 0, x1i...,xn CG, f E P(G). Since d(0, Cz,. f_ f) = 0, there exists a convex combination g1 of f * y's such that Iin 1 * 91- 91 Ill < s. Similarly, there exists a convex combination 92 of g1 * y's such that IIx2 * 92 - 92111 < E. Since Ilx1 * gi - 91111 < s, we also have 11x1 * 92 - 92111 < E. Proceeding in this way, we produce gn E P(G) such
that 11xi*9n-9nl11 <s
(1<
and G is amenable ((0.8)). Our next result concerns the norms of operators associated with the left regular representation of M(G) on Lp(G) (1 < p < oo). We prove a more general result which is the key to the "Lp-conjecture" for amenable locally compact groups (Problem 4-17). References for the result are Dieudonne [2], (R] and Leptin [1].
(4.19) THEOREM. Let p be a positive, regular Borel measure on an amenable
locally compact group G such that p * LI, (G) C Lp(G). Define Tp: Lp(G) Lp(G) by T u(f) = p * f . Then Tp E B(Lp(G)), p E M(G), and IITpII = Ilpll
PROOF. Of course, the assertion that p * Lp(G) C Lp(G) means that if f (y 1 x) is in L1(p) for p-a.e. x E G, and that
f E Lp (G), then the function y the function p * f given by
p
p*f(x)= fGf(i 1x)dp(y) belongs to Lp(G). We show first that Tp is bounded.
Let p' be the measure on .q(G) given by p'(E) = p(E-1). It is easily checked that p' is a positive, regular Borel measure on G, and that if k E L, (y), then the function k*, where k* (x) = k(i t1), belongs to LI (,u*), and
J k du = J k` du'.
(1)
Let f E L9(G), g E Lq (G). We write (f, g) = f f g dl. Since
f dA(x)
J If(y-ix)9(x)I dp(y) = (p * If1,191) < oc,
Fubini's Theorem applies, and we have, using (1),
(p * f, g) = f dp(y) f f (y-1x)g(x) dA(x)
= f du(y) f f(x)9(yx)d.(x) (2)
r
= J dA(x) f f (x)g(yx) dp(y)
f
= .J dA(x) f f (x)9(y-1 x) dju' (y). Define h: G
C by h(x) = f g(ymx) dp*(y) whenever the latter integral exists, and h(x) = 0 otherwise. Then from (2), fh E L1(G). Since this is true for all
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141
f E L9(G), it follows that h E LQ(G). So Sp(g) = h belongs to Lq(G). From (2),
(Tp(.f), g) = (f, Sp(9)) so that Sp is an adjoint for Tp. By the Closed-Graph Theorem, Tp is bounded. It remains to show that p is bounded and JIpJJ = IJTpJI.
Let C E F(G) and e E (0,1). By (4.10(3)), we can find nonnull K E "(G) such that (3)
.1(cK n K)
(1 - e).1 (K)
(c E C).
Let f = .1(K)-'/ XK and g = . (K)-'19XK. Then f E L9(G), g E Le(G), and JJ f JJp = 1 = II9II9 Recalling that the function y -+ A(yK n K) is measurable
(indeed continuous) a n d t h a t p 1+q-1 = 1, and using (3) and Flibini's Theorem,
p(C)A(K)(1- e) < p(C)
A(cK n K) Cinnff
< f A(yK n K) dp(y) =f dp(y) f XK(1 1x)XK(x) dA(x) c
=)(K) f dp(y) f f(y-1x)g(x) dA(x) c
=.A(K) f (p * f) (x)g(x) dA(x)
= A(K)(p * f, g) < A(K)J(TpJIIIf IIp(IgII9 = A(K)JITpO
Hence p(C) :5 IITpIII(1 - e), and letting a -+ 0 and using the inner regularity of p, it follows that p is bounded and IIpII 5 IITpII The reverse inequality follows since p E M(G). It follows from the preceding result that the amenability of G implies that for all p E M(G), p > 0, we have IIirp(p)II = IIpII = p(G), where lrp(p): Lp(G) -+ Lp (G) is given by irp (p) f = p * f. The converse is also true. It is sometimes impractical to use this as a test for amenability, and our next result is a great improvement in this direction: under very modest assumptions, we need only check that IIirp(p)II = IIpII for one measure p. This will prove useful in our discussion of von Neumann's Conjecture. References for the following result are Kesten (1), Derriennic and Guivarc'h I1) and Berg and Christensen (2). See also Day [6), (8), Dieudonne (2), Eymard (1), Gilbert (1), and Leptin (11, I5). We deal with the case p = 2, the extension to general p being left to Problem 4-25. Let A E M(G), p >- 0. Recall that the support S(p) of p is the set of elements
x E G such that p(U) > 0 whenever U is an open neighbourhood of x in G. If the property IIir2(s)II = IIpII is to give us the amenability of G, it is reasonable to
require that S(p)-the set on which p "lives"-should be algebraically "large" in G. To make this precise, we could require that the subgroup K,. of G generated by S(p) is dense in G. (Such a measure p is sometimes called aperiodic or
142
FOLNER CONDITIONS
adapted.) This is the condition used in (ii) below. In (i), we require the subgroup Hi, generated by S(µ)';S(µ} to be dense in G. (Such a measure u is sometimes called strictly aperiodic.) If v = u- * p, where ^' is the natural involution on m(G) ((1.1)), then S(v) = (S(/x)-1S(lc}}', so that Hr, is dense in G if and only if K is dense in G. Notice that in general H; will be a proper subgroup of K; (so that requiring Hu = G is stronger than requiring K; = G). For example, if G is the free group on two generators x, y and p = bx + by, then K = G while Hr, is the (abelian) subgroup of G generated by y-lx. The results (i) and (ii) are obviously similar in kind: is there a theorem which unifies (i) and (ii)?
The author is grateful to J. M. Rosenblatt for pointing out an error in an earlier version of (4.20), and for suggesting the elegant proof of (ii).
(4.20) THEOREM. Let A E M(G), p > 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 <- gn < 1 and gnlE CM(G). Further,
(ir2(v)fn, fn) =
f
gn(x) dv(x) -, 1.
Since v(G) = 1 and 0 < gn < 1, we have gn --t. i pointwise v-a.e. Let L = {z E G: gn(z) --> 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
F$LNER CONDITIONS
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).
FOLNER CONDITIONS
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 <
JW 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,,(¢)
sup lnf ,,,op) = inf PW(.0).
,PET., s+EO'
.EO,
The Hunt-Stein Theorem is then if G is amenable, then there exists an invariant, minimax test in T.1 for all -y E [0, 1]. The proof falls into two parts. The
first part (a) is the result of Problem 2-17 for measurable actions. Note that is a left Banach G-space, where we define x.ifi(z) = O(x'1z) (x E G,
z6Z,+jlEL,o(Z)). (a) Let K be a nonempty, weak' compact, convex, G-invariant subset of Loo(Z), and let G be amenable. Then K contains a G-fixed-point. (b) Let G be amenable and y E [0, 1]. Then there exists an invariant, minimax test in T.1.
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150
[Since x.XE = x.-r, the equality of (3) gives PA,,,(xXE) = P,,(XE), and it follows that PP",(x.4) = P",(¢) for all 0 E L.(Z). Hence T, is G-invariant. Define F: T, [O,1) by F(¢) _ wiEnnflP.(O).
Since each of the functions ¢ -' PW(gs) is (weak') continuous on T,r, it follows that F is upper semicontinuous on T,. Since T,r is weak* compact in L. (Z), it follows that A.r, where
A, = { ¢ E T,: F(¢) = sup F(0) WET.,
is nonvoid and weak' compact in
, JJJ
It is easy to check that F(x.40) = F(¢)
and F(a45 + (I - a)?i) > 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}.)
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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 <_ 4IIAII
so that T E B(A,B). Now consider the adjoint T*: L0,(Y) - L,,. (X). Since ¢ > 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
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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.
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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 <- 6(l+, v) + e,
and it easily follows that KE is C-invariant. Applying the fixed-point theorem ((2.24)), we can find T E KE with xT = T for all x E G. From (5), we see that T is a right multiplier in the sense that T(xC) = x[T(£)]
(6)
for allxEG,eELi(G). Define T' E B(L1(G), M(G)) and p E L3 (G)' by (7)
T(e) = T' (f) + p(e)6o,,
where we use the direct sum of (4).
Now z5 = 6 for all z, M(G) is G-invariant, and (6) holds. So (8)
T'(z) = zT'(f),
p(ze) = P(C).
Our objective now is to show that (9) below holds.
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154
We observe that p, being left invariant, is represented by a constant function
c in L.(G). So p(g) = c for all
E P(G). Since T E T(L1(G),M(G.)), we
must have 0
Clearly, 7" (g) = xT"(e) and T" E T (L1(G), M(G)). Also 11 xv - T (xy)11= lizv - T'(xu)ll + c =11xv - (1 - c)T" (x.)Il + c11T" (xu) II
lixv - [(1 - c)T"(zp.) +cT"(xµ))II = lixv - T"(xu)II. So T" E K, fl T (L1(G), M(G)) as asserted. Let {v&} be a,bounded, approximate identity in P(G) and let Tb': L1(G) L1(G) be given by Tb'(C)
Then Tb E T(L1(G)) and TT(x£) =
=T"(E)vs.
for all z E G,
E L1(G). By
Wendel's Theorem, there exists vb E PM(G) such that TT(Y) = vs
(g E L1(G)).
Then vbva E P(G) and
llv-uvbvsll = liv-T"(A)v611 11v- w6211 +liv-T"(u)1I
< llv- vvali+b(s,v)+e since T" E Ke. Since 11v - vv6 li -- 0, it follows that
inf{llv - µ£lI : f E P(G)} < b(p, v).
(9)
To prove the reverse inequality; let e E P(G) and define TT E T(L1(G)) by Tf(rl) _ ?7C. Then
b(p, v) < sup llxv - T{ (xp)11= 11V -.41. xEG
This completes the proof.)
(4.27) Uniqueness of invariant means. The burden of Chapter 7 is that, normally, if there exists one invariant mean on a semigroup or group, then there
are many other such means. Much less seems to be known about the corresponding question for a group acting on a set. Perhaps the most outstanding particular case of this question which has been studied is the Banach-Ruziewicz
Problem ((3.12)). The latter problem is concerned with whether or not normalised Lebesgue measure A,, is the unique finitely additive, positive, normalised measure on the ring ..sFb(R") of bounded, measurable subsets of R", which is in-
variant under isometries. Banach showed that an is not unique when n = 1, 2.
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155
For n > 3, the problem had to wait till the 1980s, when Margulis [3) demonstrated the uniqueness of an (n > 3). We will discuss the solution of the Sn version of the problem in (4.28) and (4.29). An essential ingredient in the solution of the Banach-Ruziewicz Problem is the uniqueness theorem below. This result is stated in del Junco and Rosenblatt [1), and a more general version is proved by Rindler [S11). (Other relevant references are Losert and Rindler [2) and Schmidt [1), [2].) The proof we give involves careful adjusting of Namioka's proof ((4.7)).
Let (X,.3, u) be a measure space with u a probability measure. Let G be a group and X a left G-set, such that ex = x for all x E X, and, for each s E G, the map x ax from X to X is measurable. We also suppose that u is G-invariant in the sense that u(8A) = p(A) for all s E G, A E R. The group G is said to be ergodic (on X) if whenever A E 3 and
IL(sAAA)=0 for all s E G, then u(A) is 0 or 1. (Sometimes, either of the expressions "metrically ergodic" or "metrically transitive" is used where we have used "ergodic.") The set X is said to admit arbitrarily small, almost invariant sets if there exists a sequence {An} of nonnull, measurable subsets of X such that p(An) 0 and
u(sA. A An)/u(An) -- 0
(1)
for all s E G.
As in the case where X = G, the space L. (X) is a right Banach G-space under the maps 0 4s. The set of means on LA(X) is denoted by 93 (X), and the set of G-invariant means in'9JI(X) is denoted by .0(X). Obviously, µ E .0(X). We now state and prove the uniqueness theorem alluded to above.
(i) Let G be countable and ergodic on X. Then .0(X) # {µ} if and only if there exist arbitrarily small, almost invariant sets in X. [Suppose that X admits arbitrarily small, almost invariant sets, and let {An}
be a sequence as in (1). We can suppose that u(An) < 2-n-1. Let C = X - (U°ZI An). Then C E 3 and /u(C) >/ z . Define mn E 9R(X) by Mn = (XAn/u(Aa))^,
and m be a weak" cluster point of the sequence {mn}. Let
E L,o(X) and
x E G. Then using the invariance of u, 1mn(.Os - 0)1=
u(An)-1
if XAns-1(st)fi(st) du(t) - fXA(t)Q5(t)dlL(t)
=.u(An)-1 if [XAns-1(t) - XAn (t)10(t) du(t)
5 u(An)-' (sA, L A41011, and it follows from (1) that m E .0(X). Now mn(C) = 0 since An f1 C = 0, so that m(C) = 0. But u(C) > 2 so that µ 0 m. Hence .0(X) 0 {Ii).
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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 <_ k-1p. Hence m <
and since p is countably additive and finite,
m(En) - 0 for every decreasing sequence {En} in 3 with empty intersection. So m is countably additive on .9. Since p is a finite measure, we can apply the Radon-Nikodym Theorem to obtain a function f > 0 in Li(p) such that dm = f dp. Let A = {x E X : f (x) < 1}. Let s E G, and suppose that p(sA - A) > 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
and we contradict the fact that m is a mean. Thus p(A) = 0. A similar argument shows that m({x E X: f (x) > 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
Ijxi * $n - gn1j1 < n-2
(1 < i:5 n),
µ(CP) > pUn(CP)
(n >_ 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 )) <
2n_1}, and V.
= {1__ Nn}
U,,. Then n-1 > , 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
Monotone Class Lemma.) It is easily checked that we obtain a probability measure µn_1i defined on the Bore] subsets of S"-1, by defining
An-i(E) _ An(E")/A (Bn) where Bn = {z E R" : En 1 x? < 1}. It is left as a pleasant, measure-theoretic exercise for the reader to check that µn_1 is a regular Borel measure on S11-1 and that if E is a µn_1-measurable subset of S"-1, then E' is An-measurable in R". Let be the algebra of µn_1-measurable sets. Clearly, if T E 0(n), then (T(E))' = T (E), and since 1n is 0(n)-invariant, it follows that µn_1 is also 0(n)-invariant. For convenience, we now replace n by (n+ 1) (so that An, µn_1 i S"-1 are replaced by .1n+1, p,,, and S" respectively). The B-R problem for S" is the following: is An the only 0(n + 1)-invariant, finitely additive, positive measure on 11(S") of total mass 1?
FOLNER CONDITIONS
160
The solution to the B-R problem for S" is negative if n = 1 and affirmative for n >- 2. This solution is discussed in (4.29). In order to allow the use of functional analytic techniques, it is convenient
to formulate the B-R problem for S" in terms of invariant means on L..(S")
(= Lo,(S'a, µn)). Let H. = O(n + 1). Then as in (4.27), Lo,(Sn) is a right Banach Hn-space. The set of left invariant means on L.(S") for H is denoted by £(S"). Clearly [n E ,C(S"), and every m E £(S") is associated with an Hn-invariant, finitely additive, positive measure on _V(S") of total mass 1. The following result
shows that the B-R problem for S" can be reformulated for n > 2: (*)
doesC(S") = {fcn}?
The result is due to Tarski; the R"-version is discussed in Rosenblatt [14] and Wagon [3).
Let n > 2 and µ be an Hn-invariant, finitely additive, positive measure on .4f (S") of total mass 1. Then every µn-null subset of S" is also p-null (so that µ can be identified with an element of £(S")). [We will deal with the case n = 2, the proof for general n > 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 <_p(Ao)=m 1. Since this is true for all m E P, we have p(E) =//0 as required.)
(4.29) The Banach-Ruziewicz Problem and the Kazhdan Property (T). We now examine the B-R Problem for S", in the form of (*) of (4.28). The notations of (4.28) will be used without comment. We start by dealing with the case n = 1. A result which will be proved later is used in the proof. A useful fact is that S1 is the circle group T, and so is a compact, abelian group (and hence amenable as a discrete group).
(i) (Banach [1j). The answer to the B-R Problem when n = 1 is negative, that is, .C(S1) 34 {µ1}. [The following result follows from (7.19). Let G be a non-
discrete compact group that is amenable as a discrete group. Then there exists an element m E .C(G) such that the probability measures th, AG on the maximal ideal space 4?(G) have disjoint supports. Let G = T (= S1). Clearly, multiplication by an element of S' corresponds to a rotation (E H1 = 0(2)), and the uniqueness of Haar measure gives p1 = AG. Now H1 is generated by the group R1 of rotations of S1 and the reflection
T, where T(w) = w. Let m E ,C(T) be such that nn and (#,)-have disjoint supports. Then (1)
for each a > 0, it is not the case that am < (A,)-in L,o(G)'(= M(4'(G))). Define T(m) E M(S1) by T(m)(0) = m(OT), where ¢T(w) = ¢(T(w)) for all Now for ¢ E L,o(S1), we have T(m)(0) = w E S1, 0 E where ¢(w) = 0(w), and if z E S1, then
T(m)(Oz) = m(( z)) =
T(m)(0)
so that T(m) E .C(T). Let n = I (m+T(m)) E .C(T). Since T(n) = n, we haven E C(S1). If n = }i1, {µ1}.J then 27h 5,h = (,41)-, and this contradicts (1). So n 96 µ1 and C(S1)
When n > 1, the answer to the B-R Problem is affirmative, that is, .C(S") _ {µ"}. This is a uniqueness theorem, and so it is not surprising that the uniqueness result (4.27(i)) is used in the proof. Obviously, the latter result cannot be applied directly-for a start, H. is uncountable! However, it turns out that
FOLNER CONDITIONS
162
(4.27(1)) can be applied to a certain subgroup of O(n + 1), using the Property (T) of Kazhdan (Kajdan). This Property, which we will now briefly discuss, has been found useful for other questions (such as that of the finiteness of the number of faces of a fundamental polyhedron). (ii) Kazhdan's Property (T). The Property was introduced and studied by Kazhdan [1). Detailed proofs of Kazhdan's results are given in the indispensable paper of Delaroche and Kirillov (1]. (Unfortunately, the volume in the Seminaire Bourbaki series containing this paper seems to be out of print.) We start by discussing briefly some basic ideas from the representation theory of locally compact groups. (See, for example, (D2].) Let G be a separable, locally compact group. The class of all continuous, unitary representations of G is denoted by Rep G. For 7r E Rep G, b,r stands for the Hilbert space of the representation 7r. We will denote by G# the set of equivalence classes of representations it E Rep G for which h, is a separable, Hilbert space (that is, dim 15, < No). The dual of G is denoted by G. Thus G is the set of equivalence classes of representations 7r E Rep G that are irreducible. Since G is separable, we have that G c G#. The notion of weak containment of representations is important in discussing
the topology on C. To define this notion, let us say that a positive definite function
E C(G), is associated with it E RepG if there exists i; E 57 such that
for allzEG, O(z) = (ir(x)g, 6 .
We write 0 = g5f. If S C RepG, we say that 7r is weakly contained in S if, there exists a net {¢.}, where each OC is a sum of positive definite functions associated with elements of S, such that qie -+ gb uniformly on comgiven t: E
pacta. The set d is topologised by requiring S C G to be closed if and only if it E S whenever it E C is weakly contained in S. Of course, if we identify C with then weak containment is formulated in terms of weak* convergence of positive functionals.
It is tempting to use the weak containment property as above to define a topology on G#. Unfortunately, we cannot define a topology in this way: as Fell (3] points out, if it, 7r' E G# are such that neither is weakly contained in the other, then 7r a a' is weakly contained in S = {1r,7r'} but not in {,r} or in {7r'}. Instead, (Fell (3], Delaroche and Kirillov [1]) we define a base of sets V (e, C, S1) .... &,7r) for a topology on G#, where e > 0, C E '(G), it E G# and g, E .fj,,, aJg; fl = 1, and where 7r' E G# is in V (c, C, if there exist r), E 55, , , I1Ili HH = 1 (1 < i < n) such that I(a'(x)rt=,17,) - (1r(x)1:,, £,)I < e
7r) if and only
(i < i, j < n, x E C).
The separability of G ensures that G# is "first countable", that is, each point in G# has a countable base of neighbourhoods. The relative topology inherited by C from G# coincides with the "weak containment" topology above. it
The following simple result will be useful. Let. iro be the trivial representation 1 of G on C. Of course, Iro E G.
FOLNER CONDITIONS
163
(a) Let it E G. Then it,, - iro in G#, where it,, = it for all n > 1, if, given e > 0 and C E "(G), there exists t; E 55n with [[ t; [[ = 1 such that (2)
llir(x)e - ell < e
(x E C).
[If (2) holds, then, for each e > 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 < i < n, n > 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 <,Ylfj1.
So (9) is true for n = 0,1. Now suppose that (9) is true for some n > 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
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
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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 <- k).
As in (4.31(i)), 1(x) is the length of the reduced word in {xk 1, ... , xf 1, xis...,xk} corresponding to x E Fk. For n > 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) <
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177
Letting p oo and then 77 -. 0 gives a < p. Hence a = 6 and {bn/n} converges. Thus y = limn_,p((12n)1/2n) exists.
It remains to be shown that (2k -1)1J2 < 7 < (2k - 1). From (1), 72(n+no) >- (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 <- an(s), it is obvious that p, < p. Let 00
fs(z) =
00
Ea. (S)Zn, (jzI) < pa)
g(z) = E ao,nzn (jzI < p)
n=0
n=0
(iii) (a) For n > 1, (13)
an+1(s) = (2k)-1ao,n+1 - ((2k - 1)/8)ao,n + (s + ((2k - 1)/s))an(s).
(b) If jzj < p then 2k[1- (s + (2k - 1)/s)z] f,(z) = (2k - 1) + [1- (2k(2k -1)z/s)jg(z). (14) [(a) Let n > 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
an+1 - an = bn
Since jbnj < Ln eventually and L < 1, the series E7=0 jbnj converges. Let e > 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-
After the "classical analysis" excursion above, we are now in a position to evaluate II7r2(9)ll. Recall that g E 11(G) and -y are defined in (i). (vi) 111r2(9)f1 _ ('7 + (2k -1)/y).
[Let s E (0, oo) - {(2k - 1)1J2}, and re = (s + (2k - 1)1s)-1. From (13),
an+l(s) = bn+re 1an(s), where
bn = f (2k)-1ao,rz+1 - ((2k -1) f s)ao,,,,].
Since s 0 (2k - 1)1/2, r-1 > 2(2k - 1)1/2. Let L E (2(2k - 1)1/2,r, -I). From (4.31(iv)) and the fact that ao,r = 0 if r is odd, {bnI < Ln
eventually. Now using (v) and the "lim sup" formula for the radius of convergence of a power series, we have
p.-' = r,-'.
rim sup a,, (s)
n-00
From (iv), an(s)1/n -+ r.-' = s + (2k - 1)/s.
Now let e > 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
where A =
Fio 1 a,,2ny,
L ai,2n(y+71)i,
i=2N
i=O
So we can find B, C E R such that for all n > N, 1/2n
2n
(B+a2n(y- 7))112n <
(Eainii)
< (C + a2n(y + 17))1/2n
ii=OO
that is, (a2n(y-)))I/2n
1\
(E2n
1/2n
B
! 1+
<
a2n (y - 17) JJJ
C < (a2n('I + 7))1/2n I 1 + \\
a2n(y+r1)
i=o 1/2n
\j1/2n ai,2nyi J 1
FOLNER. CONDITIONS
162
Since 1 < y - rl, y + rl, it follows that a2n (^7 - rl), a2n (y + n) --+ oo as n So
1/2n
(Eai.2n.li)2n
'f-17
ft-00
oo.
E s=0
Now let rl -» 0 and use the continuity of the function s , r;-1 to obtain Jim
( 2n
1/2n si,2n 1t)
_ (7 + (2k -1)/7)
i=O
Finally, the desired result follows from (ii).]
We can now prove the amenability criterion for G for which the above has been the preparation. (vii) The group G is amenable if and only if 7 = 2k - 1. [By (4.20(ii)), C is amenable if and only if 117r2(9)11=119111 = 2k.
Now, by (vi), 117r2(g)1f = ry + (2k - 1)/-y. Hence G is amenable if and only if
7+(2k-1)/7 = 2k, that is, -y2-2k-y+(2k-i) = 0, that is, (-y-(2k-i))(7-1) = 0. Since -y > (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 <
Pn/a).
It then follows that ry = lim (,y2,) 1/2n < 33/4 < 2.2 -1, n-.oo
so that Go is not amenable. OV'shanskii comments that an example of a periodic, nonamenable group is given in his paper [3]. Is there an easier counterexample to von Neumann's Conjecture? What about the Golod-Shafarevitch groups ((3.11))?
(4.34) Amenability and the Fourier algebra of a locally compact group. Duality theory for a locally compact group G can be investigated in two different ways. The first way is that of representation theory in which one tries to determine G and the Plancherel measure (for G of Type 1). The latter approach uses extensively the theory of induced representations, and when combined with Lie theory, has produced the remarkable representation theory of Lie groups. (For a related approach in the non-Type 1 case, see Pukanzsky [Ni].) Another approach to duality, developed by P. Eymard [N1], is to study certain Banach algebras of continuous functions on G, which, in a sense, contain duality
information about G. These algebras are the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of G. For abelian G, A(G) and B(G) are just L1(G) and M(G) "in disguise." Indeed in this case, the Fourier and FourierStieltjes transforms, applied to L1 (C) and M(G), produce two subspaces A(G), B(G) of Co (G) and C(G) respectively. The convolution product on L1(C), M(G) goes over to the pointwise product of functions on G, and A(G), B(G) become commutative Banach algebras under the LI(G), M(C) norms.
To obtain versions of A(G), B(G) for G nonabelian, we need to reformulate A(G), B(G) above in a way that only uses concepts present for general G. These concepts are C'(G) (the enveloping C'-algebra of G) and VN(G), the von Neumann algebra generated by the left regular representation 1r2 of G. To see how this is possible, let us suppose, once again, that G is abelian. Then LI(C) is the predual of L.., (0). Now the Plancherel Theorem asserts that the I is a linear isometry from L2(G) onto L2(G), and hence induces a map f *-isomorphism'' from B (L2 (O)) onto B (L2 (G) ). Regard Loo (C) as a *-algebra of multiplication operators on L2(C). What is 4>(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 <_ 1101111f 11. <_ 110111IfIII, so that 11011. < 11011 By considering functions
f * gt, where f, g have suitably chosen compact supports, and using (i) below, we see that the Stone-Weierstrass Theorem applies to give that CC(G) f1 A(G) is 11 11. dense in Co(G). So A(G) C C0(G).) (i) (Eymard [Ni]) (a) B(G) is a Banach algebra with pointwise product, and A(G) is an ideal in B(G). (b) The map f 0 g -F f * g' extends to a continuous linear map 4>: L2(G)6L2(G) C(G), and (L2(G)6L2(G))/ker(1 is isometrically isomorphic with A(G).
[(a) Let 0 e B(G). We show first that if Tr is the universal representation of C'(G) on the Hilbert space 9j, then there exist e,rl E f5 such that ¢(a) = (rr(a)l;,rl) (a E C'(G)), 11011 = IICIIIIi711. To this end, since 0 E (C'(G)")., the polar decomposition theory applies [D2, 12.2.4): so there exists a partial isometry u in C' (G)" and a positive functional p E C* (G)' such that IImH = IIp11
and g(¢) = (ug)(p) for all g E C'(G)".
FOLNER CONDITIONS
185
Now C'(G)" can be identified with the weak operator closure of 7r(C'(G)), and there exists E 35 such that p(a) = (7r(a)t;,C) (a E C* (G)) [D2, 12.1.31. Then 0(a) = (7r(a)t;,77), where n = Further lIC112 = Ilpll = (1011:5 11C11111711 :5 110112, so that II011= IICIIIl7711 as required.
Now let zfi E B(G) and 1:1,771 E 15 be such that V'(x) _ (ir(x)e1ir71) and 11ifilI = IICIIIIIi?1lJ. Then (.Ot')(a) = ((a (&7r)(a)(C (&g1),?7 ®772), and so II0'11 5 I1E(9llllln®mll = (IIe11I11711)(11e111lI'iiII) _ ((0111liPp. So B(G) is aBanach algebra.
We now show that A(G) is an ideal in B(G). Let h E A(G) and k E B(G). We show that hk E A(G). We can suppose that h and k are positive definite. If h E CC(G), then hk E A(G) by Godement's Theorem ((4.20)). It is therefore sufficient to show that C0(G) n A(G) is dense in A(G). Let p = Z°O_1 fi * g, E A(G) as above. Note that as a member of VN(G)., p = E,_° 1 wg, j,, and also that, since Jw9,, f (T)J < IIT11119 Il2llfill2 (T E VN(G)), we have 119J211f:112 Then the Cauchy-Schwarz inequality gives
Il <
(1)
N
p-A*9st
L: w9.,l:
i=1
e=N+ 1
1/2
t _E 119;112 {
i=N+1 co
0 l1/2 r -E Ilfill2 ) = \iN+i 1
JJJ
JJJ
while if Fi, Gi E C, (G), then by the triangular inequality, 11fi*gi -Fi *GiII <- 1lIfi-F1II21I9tII2+UJFi112I19i-Gi1121-
It follows that CM(G) n A(G) is dense in A(G) as required.
(b) Regard VN(G). c VN(G)', B(L2(G)). C B(L2(G))'. The restriction map j: B(L2(G)). - VN(G). is clearly norm continuous and surjective. Now B(L2(G)). is identified with the Banach space L2(G)(bL2(G) of trace class operators, the duality being given by (T, f (9 g) = (Tg, f) (T E B(L2(G)), B(X, Y') with X = Y = Y' = L2(G) f, g E L2(G)). (Use the result a9 identifies L2(G) with L2(G)', where a9(f) _ (g,7). So the map g VN(G). = (L2(G)(U2(G))/ker j. Now VN(G). is identified with A(G) C C(G) by the map wh,k --> k * h', and composing j with this map gives the map it of the statement of (b).1 As we shall see, (b) above gives a useful device for estimating norms in A(G). A simple illustration of this is the inequality 1(f * 9* 11 5 IIf 112119112
The following two results explore the consequences of the amenability of G for A(G) and B(G). (ii) (Leptin (41). The Banach algebra A(G) has a bounded approximate identity if and only if G is amenable.
(Suppose that G is amenable. For each C E 9,(G) and each e > 0, let K = K(C, e) be such that the inequality of (4.13 (ii)) holds. Let nc,c = (A(CK)A(K))112,
FOLNER CONDITIONS
186
and uc,E = (XCK * Xit)(ac,e E A(G). The set of pairs (C, e) is a net under the natural ordering (C increasing, a decreasing). We claim that the net {uC,E} is a bounded approximate identity for A(G). First, lluc,Ell < (A(CK)),(K))-1/211XcKI1211XK112 = 1. Second, let f E A(G)n
C0(G) and let f vanish outside some C E P (G). Let e, K be as above. Then
(uc,ef - f)(x) = f (x)(uC,E - 1)(x). Now direct calculation shows that uc,E(x) = (\(xKf1CK))/aC,E, and obviously, A(zK nCK) = A(K) if z E C. Since f vanishes outside C, we have uc,Ef - f = (A(K)1(A(CK)A(K))X12 -1) f. So IIuc,Ef - f11 = l(A(K)/A(CK))112 -1111f11-j 0
by (4.13(ii)). From the proof of (i)(a), A(G) nCC(G) is norm dense in A(G). So {uC,E} is a bounded approximate identity for A(G). Conversely suppose that there exists a bounded approximate identity {v5} in A(G). Let M = sup6 Ilv611. Using (4.20) and Problem 1-7, the amenability of G will follow once we have shown that llull = 117r2(u)II for all u > 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 <_ M.
To prove this, let W = U l I K : f E A(G)}. Using the Stone-Weierstrass Theorem (as in the proof that A(G) n C,(G) is II 1100-dense in Co(G)), we see that W is dense in C(K). So given 77 E (0,1), there exists h E A(G) with lh(x) - 11 < 77
for all x E K. Now for some a, llhvv - hll.. 5 Ilhvo - hll < 77. If x E K, then
Iv,(x) - 11 = Ih(z)vo(x) - h(x)I/Ih(x)I <- n/(1 -?l) For (2), choose 77 so that r7/(1 - 77) = e and then take 0 = vo. Now let p E A(G). Suppose that p fc * gL (fl, g= E L2(G), < oO, L. 1 llgtll2 < oo). Let p E M(G), p > 0. Then using (1),
f
J
_i ilf:112
00
p dul
fcl
f (ff * gi)(x) du(x)
00
00
(fi,1r2(u)gi) i=1
dp(x)
:=1
5 11-72 (u)11 E IIf=11211 IM 12 . 4c F
Taking the inf of Et'-1 IIf=11211ge 112 over all such representations F,"01 f t * gi for p
and using (i)(b), we obtain that I f pdpl 5 lllr2(u)IIIIPII (p E A(G)). Let p be as above with compact support K and ¢ be as in (2). Then applying the preceding inequality with p = ¢,
hull = u(K) <- if 0 dal + E11pll <-11k2(u)II IIOII + E1Ipll S M11ir2 (p)ll + Ellpll. Letting c -» 0, we have lljAll < Mllir2(u)ll. Now l1,0II = llull" (n >- 1), and replacing At by un in the preceding inequality, taking nth roots, and then letting
FOLNER CONDITIONS
n
187
oo gives [lµii < 117r2(p)1I. It is elementary that 117r2(A)11 :5 JIM11, and so
117r2(&)11 = 114 as required.)
From the above, when G is amenable, A(G) is a commutative Banach algebra with bounded approximate identity {u6} with [[u6 [[ = 1 for all 6. We will now determine the multiplier algebra A(A(G)) in that case. Multiplier algebras were briefly discussed in (1.30). Since A(G) is commutative and has an approximate identity, one readily checks that if (T1,T2) E A(A(G)), then T2 = T1 so that A(A(G)) can be identified with the left multiplier algebra {T E B(A(G)): T(fg) = (Tf)g for all f,g E A(G)}. (iii) (Cowling [3], Herz [6), McKennon [S), Renaud 12)). Let G be amenable. Then the multiplier algebra A(A(G)) of A(G) is canonically isometrically isomorphic to B(G). [Since A(G) is an ideal in the Banach algebra B(G), the map q5 -. LP, where L4,(f) = Of (f E A(G)), is a norm-decreasing homomorphism from B(G) into A(A(G)). It remains to show that this map is isometric and onto. To this end,
let T E tl(A(G)). We claim that there exists 0 E C(G) such that Tub 0 uniformly on compacta. Indeed, from (2) in the proof of (ii), if C E '(G) and e > 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)[ <- li(Tu6)h - (Tu,)hii. + 2eiIT1i < [l(Th)u6 - (Th)u,[[ + 2e[lT[l, and so {Tu6} converges uniformly on compacts to some 0 E C(G). If f E A(G), then { (Tu6) f } converges to T f in the A (G)-norm, and hence also in the topology
of uniform convergence on compacts. So T f = (kf. It remains to show that 0 E B(G) and [[ThI ? 11011. Let f E CM(G) have compact support C, and let h be as above. Let ([ f [[< be the C' (G)-norm of f. Then [[(kh[l = iITh[[ < [IT[[, and fGf0d,\l
If
fohd,\l+If
fO(l - h)da
lifp.Il4hll +I elifii' 11 .011. <-11111.11211-f llf11111011,
0 and using the density of CC (G) in C' (C) we see that ¢ E C' (G)` _ Letting e B(G) and I1¢[[ < [[T[[. This completes the proof.)
The converse to (iii) is also true (Losert [S6]). Herz introduced and studied L, ,-versions of A(G), B(G) called A,(G), B,(G). In the latter terminology, A(G) = A2(G) and B(G) = B2(G), and much of the A(G)-theory extends to the A, (G) case. An account of A,(G) when G is abelian appears in Larsen [S]. There is a large literature on A(G), B(G)-indeed, more generally, on A,,(G), BB(G). This includes papers by Akemann and Walter, Arsac, Bozejko, Carling, Cecchini, Cowling, Derighetti, Eymard, Figa-Talamanca and Picardello, Flory,
Forrest, Gilbert, Granirer, Haagerup, Herz, Kugler, Lai, Lai and Chen, Lau, Leinert, Losert, de Michele and Soardi, Nebbia, Renaud, and Walter.
FOLNER CONDITIONS
188
Problems 4 Unless otherwise specified, G is a locally compact group and S is a discrete semigroup.
1. Let E be either a a-compact, amenable, locally compact group or a countable, left amenable semigroup. Prove that there exists a sequence {f,a} in P(E)
such that llx * f - All, - 0 for all x E E. 2. Let G be a locally compact group and inn{
G)
0
N
Prove
(a) D(G) is a closed subset of R) and ess infyEG4,(y) > 0, then 0 D(G); (b) if 0 E (c) if ¢ E D(G) and a E C, then a¢ E D(G); (d) (Ox -.0) E D(G) for all 0 E x e G. (ii) G is amenable if and only if D(G) is a subspace of L,o(G). Give another proof that G is amenable if G is abelian. (i)
3. The group G is said to be contractible if for every U E Sl,(G) and F E .r(G), there exists a E Aut G such that a(F) C U. Prove that every contractible group is amenable. Give examples of contractible groups and of noncontractible amenable groups.
4. Suppose that G is not amenable and let p E (1, oo). (i) Show that there exist z1,... , xn E G and c1, ... , c,, 0, ET 1 ci = 1, such l{irp(n-1 E that 1 ci8z;)II < 1, where 9rp is defined in (4.19). (ii) Prove that there does not exist a nonzero, left invariant linear functional
on LP(G)-
5. Show that if C, D E '(G) then the map x
A(Cx n D) is continuous.
(This was used in (4.12).)
6. Let p E (1, oo) and p I + q-1 = 1. Prove that G is amenable if and only if, given e > 0 and C E F(G), there exists f E Pp(G) such that IIx * f - flip < E, 7.
Give an example of a summing sequence Let
) (zK,, 6
(x E C).
in R for which
0 for all C E x°(R).
A(C + K,, A 8.
IIA(x)I/°f * x - flip < e
be a sequence of nonnull, compact subsets of G such that 0 for all x E G. Prove that .1(xK,, 6 K,,)/a(K,,) 0
uniformly on compacta. Show also that the smallest closed subgroup of G con-
taining u° 1 K is G itself.
FOLNER CONDITIONS
9.
189
Suppose that U E ..(G) and {VV} is a sequence in R(G) such that
A (U) < co, 0 < .1(V,) < oo for each n, and .1(uVn ,& Vn)/.1(Vn) -+ 0 for all u E U. Prove that lim n a(V,a 1) >'\(U).
Deduce that if G is noncompact and unimodular and {Kn} is a sequence of nonnull, compact subsets of G such that A(xKn AKn)/.1(K,,) -+ 0 for all x E G,
then A(K,,) - oo. 10. Show that G is amenable if and only if, given C, L E 9(G) and e > 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<2sup llx-1*f-[112 zEc (ii) Prove that there exists a net {06} in P(A(G)) such that 110¢6 - 0,511 0 for all 0 E P(A(G)). (iii) Deduce that there exists a state m on VN(G) that is "topologically in-
variant" in the sense that m(cT) = m(T) for all 0 E P(A(G)), T E VN(G). [Here, VN(G) = A(G)' and 0T(') = T(0,0) for all 0 E A(G).] 43. Prove that if G is an amenable, discrete group, then VN(G) is semidiscrete ((2.35)).
44. Let G be a a-compact, amenable, locally compact group, 1 < p < oo and k E L1(G) have compact support C. Let Tk E B(Lp(G)) be the convolution operator
TkF(v) = J k(u)F(uv) da(u). G
Now let (X,E,p) be a a-finite measure space and R: G
B(L9(X)) be a
strongly continuous antirepresentation of G on Lp (X) such that the map (u, x) -*
R,j(x) is A x p-measurable from G x X into C for all f e Lp(X), and such that IIR-11 = M < co. "Transfer" Tk to an operator Rk E Re = I and B(Lp(X)), where Rkf (x) = f k(u)R f (x) da(u). G
Show that pRk11 < M2IITk11. [Hint: use (4.13).]
CHAPTER 5
Ergodic Theorems for Amenable Locally Compact Groups (5.0) Introduction. The classical Mean Ergodic and Pointwise Ergodic Theorems are concerned with the convergence (in the appropriate topology) of
the sequence of Cesaro means {n-1 =' T'}, where T is a bounded linear operator on a suitable Banach space. Our primary concern in this chapter is to prove versions of these theorems for an amenable locally compact group G. (We shall also establish ((5.2)) a Mean Ergodic Theorem for amenable semigroups.) Both theorems involve a (suitably measurable) representation z T. of C on a Banach space X. The Cesaro means referred to above are replaced by "operator averages" of the form A (Kb)-'
J KS
T,,-, da(x),
where {K6} is a summing net for G. The Mean Ergodic Theorem is established in this general context ((5.7)). Not surprisingly, the Pointwise Ergodic Theorem requires care, but it will be shown ((5.20)) that if G is, in addition, connected, then we can construct a summing sequence for G so that a natural version of the Pointwise Ergodic Theorem holds. The proof uses the Dunford-SchwartzZygmund Theorem and Lie group arguments. Other aspects of the relationship between amenability and ergodic theory are mentioned in Problems 5.
(5.1) Preliminaries. Let E be either a locally compact group or a discrete semigroup. If E is discrete, then, of course, L1 (E) = l1 (E).
Let p -+ S,, be an antirepresentation of the Banach algebra L1 (E) on a S, is a norm continuous, linear map from L3 (E) Banach space X. Thus u into B(X) such that Si, , = SvSa, for all p, v E L1(E). Let M > 0 be such that (jS,,jj <- Mjjpjj for all li E L1(E). The antirepresentation u --. S,, is said to be ergodic if there exists a net {B6}
in B(X) such that (a) for all p E P(E),
the convergence being in the strong operator topology of B(X); 195
ERGODIC THEOREMS
196
(b) for each f E X and each 6, B6 (£) E CC, the norm closure of the convex
set {S (£): k E P(E)}. The notion of an ergodic representation of E is defined similarly. We now define three important subspaces Xf, Xo, XE of X:
Xf={CEX:S,,(C)=fforallpEP(E)}, Xo = [Span{(I - S )g: £ E X, la E P(E)}]-,
XX=Xf+Xo. By definition, X0 is a closed subspace of X. It is obvious that X f is also a closed subspace of X. Note that, since P(E) spans L1 (E), an element g is in Xf if and only if, for all v E Li (E),
The justification of the use of the adjective "ergodic" above lies in the following version of the Mean Ergodic Theorem. References for (5.2), (5.3) are Eberlein [1] and Day [9].
(5.2) THEOREM. Let v S be ergodic with {B6} satisfying (a) and (b) of (5.1). Then (i) JIB611 < M for all 6, and if C E XE, then {B6(g)} converges in norm to an element of Xf n CC and lim B6 (C) = 0 if £ E Xo;
(ii) XE is invariant under every S, (v E Li (E)) and every B6 and is the vector space direct sum Xf 0} XD; further, Cc C XE for all t' E XE; (iii) the net {B6}, where B6 is the restriction of B6 to XE, converges in the
strong operator topology of B(XE) to the projection P: XE -+ Xf associated with the direct sum decomposition of (ii);
(iv) for g E XE, {PC} = XI n C£. PROOF. (i) For C E X, p E P(E), we have IISF,(e)[H -< M11g[I, and using (5.1(b)), we see that JIBE[[ < M for all 6. Now let E E X. Write C = t f + Co, where Cf E'Xf, Co E Xo. By definition, S,(Cf) = £f for all p E P(E), so that Cef = {Cf}. Since each B6(Cf) E Ctf = {i f}, it follows that B6(Cf) = gf for
all 6. Now by (5.1(a)), if 71 E X and p E P(E), we have IIB6(I - SM)(rl)MM -+ 0. So 14B6(S)IC
0 for all S E Span{(I - S,,)(il): Y7 E X}. Now Xo is the norm
closure of this span, and since JIB611 < M for all 6, it follows that IIB6(c)EU - 0 for all S E Xo. In particular, [[B6(Co)II - 0.
Hence R6 (C) = [B6(t:f) +B6(Co)] - Cf E Xf. Since Bs(C) E Ct and Ce is norm closed in X, it follows that Cf E Xf nCt. This establishes (i). (ii) Let k E P(E). For a' E P(E) and rt E X,
S, (I - S,.,)(0) = [(I - Su,,.)(n) - (I - Sp)(n)] E Xo, and it follows that S,.(Xo) C Xo.
ERGODIC THEOREMS
197
Now let E = C f + o E Xr as in (i). Then
So(e)=S
a)EXf+X0=XE:
so that XE is invariant under each S. Since P(E) spans L, (Z), Xr is invariant under every S (v E L1(E)). We now prove that Cg C XE. Indeed, recalling that Xo is closed in X,
C£ - Cf = {EEf+SS(W)-ff pEP(E)}={S,.(to): pEP(E)}- C Xo C XE so that Ct C XE. Thus B6(e) E CE C XE, and XE is B6-invariant for all 6. Finally, if C E x f fl Xo, then, using (i), B6 (e) -+ C on the one hand (since e E Xf), and on the other hand, B6(e) 0 (since E Xo). Thus = 0, and the asserted direct sum decomposition follows. (iii) This follows as in the preceding paragraph. (iv) Let C E XE and 17 E Xf n C. By approximating i by elements of {S,(C) : p E P(E)} and using the fact that Xo is closed in X, we have C-77 E Xo. Hence
0=lim B6(f-77)=P(6-P(17)=PM-17 so that P(C) =77. By (i), P(g) E Xr flCt, and so (iv) is established.
D
There are two obvious deficiencies in the preceding result. In the first place, it does not give that XE is closed in X. (Is there a counterexample available?) Secondly, it does not provide information about when XE is "large" in X. To remedy these deficiencies, we can impose an additional condition on the {B6}: (C)
l bm(I - SS)B6 = 0,
the convergence being in the strong operator topology of B(X). The weak compactness condition of (ii) below is obviously satisfied if X is reflexive.
(5.3) PROPOSITION. Suppose that the net {B6} of (5.2) satisfies the additional property (c) above. Then (i) XE is closed in X; (ii) if Cf is weakly compact for all e E X, then XE = X. PROOF. Let £ E X and suppose that B6 - 77 weakly in X for some 11 E X. Then for p e P(E), S, (t7) is, using (c), the weak limit 1T of the net
{-(I - Sg)B6
B5(£)}.
So 12 E X f. Now Ct is weakly closed in X, since it is convex and norm closed in X. Since B6 (e) E Ce for all 6, it follows that 77 E Cc n X f. Further,
(C - r7) E {(I - SOe: M E P(E)}- C Xo. , So C E XE.
ERGODIC THEOREMS
198
Using (5.2(i)), XE is the closed subspace {t; E X : {B6 g} is norm convergent in X}
of X. This establishes (i). Suppose, now, that C{ is weakly compact in X for all E X. Let S E X. Then we can find a subnet {Ba(o)} (o E A') of {B6} such that (B.(,,) (5)} is weakly convergent in C. Replace {B6} by {B,,,(o)} in the first paragraph of the present proof, and observe that Xr, depends only on the antirepresentation v S,,. Then S E X. So X = XE, giving (ii). 0 We now show that when E is left amenable, then a net {v6} in P(E) satisfying a Fminer-type condition of (4.1) gives rise to a net {Ba} satisfying (a) and (b) of (5.1).
(5.4) PROPOSITION. (i) Let E be left amenable and ((4.1)) {v6} be a net in
P(E) such that
I116-p*v611-'0 for all p E P(E). Let v S, be an antirepresentation of L1(E) on a Banach space X, and let B6 = S,. Then {B6} satisfies (a) and (b) of (5.1) (so that (1)
v -+ S. is ergodic). (ii) E is left amenable if and only if every antirepresentation of L1 (E) on a Banach space is ergodic.
PROOF. (i) For µ E P(E),
1IB6(r-SN)1I=11S, -Sp..,11:5 M1Iv6-µ*v6ll--+0 so that (a) is satisfied. Trivially, (b) is satisfied. (ii) Suppose that E is left amenable. By (i), every antirepresentation of L1 (E) on a Banach space is ergodic. Conversely, suppose that every antirepresentation of L1 (E) on a Banach space is ergodic. Let v Sv be the antirepresentation of L1 (E) on L.,(E) given by
(0 E L. (E)). S, (0) = 0S, is ergodic. Clearly lp = 1 for all p E P(E), so that, in
By hypothesis, v the notation of (5.2), 1 E (LS(E)) f. By (5.2(h)),
10 (L.(E))o Hence (cf. (2.1) and (2.2)), .ln (E) # 0. So, appealing to (1.10) in the nondiscrete case, we see that E is left amenable. 0
(5.5) Discussion. If E is amenable, then, from (4.2), the net {v6} of (5.4(i)) can be chosen so that we also have
11v6-v6*All-'0 for all p E P(E). In this case, with B5 as in (5.4(i)), the third condition (c) of (5.2) is also satisfied.
ERGODIC THEOREMS
199
When E is discrete, then an antirepresentation µ Sµ of E determines, and is determined by, the antirepresentation x --y SS (= Ss.) of E. In this case, the preceding results can be reformulated in terms of x S. in the obvious way. (For example, Ct is defined as the norm closure of co{S.(t;): x E E} in X.) We now concentrate on the Mean and Pointwise Ergodic Theorems for a locally compact group. In this case, of course, the relationship between antirepresentations of L1(E) and antirepresentations of E is more complicated. Let G be a locally compact group and x - S. an antirepresentation of G on a Banach space X. Thus S,, E B(X) and S., = S.S. for all x, y E G. Further, for some M > 0, IISxII < M for all x E G. No measurability conditions on x --> SS are assumed at the moment. Of course, X is a right Banach G-space (G regarded as discrete), where the right action is given by
x=ST(ia)
(taEX, xEG).
As in (2.4), for C E X, f E X', we define fU E lm(G) by f e(x) = f (ex)
(x E G).
We define the antirepresentation x - S,, to be weakly measurable if f E Lc(G) for all f E X', Ta E X. So in the notation of (2.4), the antirepresentation is weakly measurable if and only if X(G) C L,(G). Weak measurability for a representation of G is similarly defined. Our next result shows that, under certain conditions, the antirepresentation
x - SZ can be "integrated up" to give an antirepresentation of the Banach algebra L1 (G) on X. A familiar, special case of this result occurs in the theory of unitary representations of locally compact groups. (Compare, also, (1.30).) For E X, let D4 be the norm closure of co{S.., : x E G} in X.
(5.6) PROPOSITION. Let the antirepresentation x
Sz be weakly measur-
able on G, and suppose that each Df ( E X) is weakly compact in X. Then there exists an antirepresentation v - A of the Banach algebra L1 (G) on X such that (i) IIA,II <_ MIIvfl for all v E L1(G); (ii) if f E X', v E L1(G), and E X, then f (A, (C)) =
ff(S:e)dv(z)
(= v(f ));
Ay.,, and SPA, = A,,.y for all y E G, v E L1(G); (iv) if p E P(G) and E X, then A,, (C) E Dt. (iii)
PROOF. Recall ((2.7)) that X(G) is left introverted and is a Banach algebra with multiplication given by (1)
pq(') = P(qO)
(p,q E X(G)', ' E X(G)),
where (2)
ggs(x) = q(,kx)
(x E G).
ERGODIC THEOREMS
200
Further, again from (2.7), there is a representation p -+ 4,, of the Banach algebra X(G)' on X', where
' p(f)=pf
(3)
with pf(i;)=p(fd). Recalling that X (G) C L. (G), we see that the map µ µ (strictly iII x(G) ) from LI (G) into X(G)' is a norm decreasing homomorphism. Let B, = 45u. Then µ Bµ is a representation of LI(G) on X' with IIBJ < MIIu(I for all µ E LI (G). Hence µ -+ B, is an antirepresentation of LI (E) on X", and IIB;,II <_ MIIpII for all p ELI (E).
Let µ E P(G). Then we can find a mean m E TZ(Z00(G)) such that mIx(G) _ A. Let C E X. Then DC is weakly compact in X, and by the right Banach G-space
version of Problem 2-12, m E (DC)^. (Of course, for F E X", p E f E X', we have Fp(f) = F(p f ).) But for f E X', using (3), &m(f) = (mf) = m(fg) =
so that Define
µf (f) = E D4. Since P(G) spans LI(G), E X for all v e LI(G). E X by B;
Clearly v -+ A is an antirepresentation of LI (G) satisfying (i) and (iv). As for (ii),
f(A,.(C)) = £(B,,f) =
=f The first equality of (iii) is a consequence of the following argument, which uses (ii):
(y * v)"(fe) = v((fe)y) = U(f y) = f(A.,(Cy)) = f(A,.Sp(C)) The second equality of (iii) is proved similarly. In the circumstances of (5.6), we shall sometimes write
A =) S2 dv(x). G
If E is a measurable subset of G and dv = XfE d A, then we write ffS.,
da(x)=fGS., dv.
Let x -+ T. be a weakly measurable representation of G on a Banach space X. Let Sx = Tx-, for all x E G. Then x -+ S. is a weakly measurable antirepresentation of G on X. Suppose that, in the notation of (5.5), Dt is weakly compact
for all C E X. Then by (5.6), we can form the antirepresentation v -> A of LI(G) on X. Now let {K6} be a summing net for G. Then it follows from (4.15(3)) and (4.3) that if µ6 = XK8/) (K6), then {µ6} satisfies (1) of (5.4). The following result can be referred to as "the Mean Ergodic Theorem" for amenable, locally compact groups (Greenleaf 14)).
ERGODIC THEOREMS
201
(5.7) THEOREM. Let G be an amenable, locally compact group and {K6} a summing net for G. Let x -+ T. be a weakly measurable representation of G on a Banach space X such that DF = (co{Tzf: x E G})- is weakly compact for all l; E X. For each 6, let A6 E B(X) be given by ((5.6)):
A6 = a{K6}-1 J
{1)
Tz-i dA(z)
xb
and define
Xf={CEX:Tzg= for allxEG}, Xo = (Span{{I - Tz)1:: e E X, x E G}l-. Then
(i) X is the direct sum X f ®Xo of X f and Xo; X f is the projection onto X f associated with the direct sum (ii) if P : X decomposition of (i), then P E B(X), and X f n D for all E X; (a) P in the strong operator topology of B(X). (b) A6
PROOF. Applying (5.6) with Sz = Tz-i, we obtain an antirepresentation v -» A of G on X with the properties (i)-(iv) of (5.6). By (5.6(iv)), for each I E X,
E P(E)}- C D. CE = Since DE is weakly compact and CE is norm, and therefore weakly, closed in Dc,
it follows that C£ is also weakly compact. From (5.5) and (5.3), we see that X = Yf + Yo, where
Yf={SEX: Aµ(£)=g for allµEP(G)}, Yo = (Span{(1- A,)(e) : g E X, µ E P(G)})-. Applying (5.2) to the antirepresentation v - A, with B6 = Aua = A6, the proof will be complete once we have shown
Yf = Xf,
(2)
Yo = X0.
For then (i) and (ii)(b) follow immediately, and (ii)(a) follows by a simple adaptation of the proof of (5.2(iv)). If E Yf, x E G, and .i E P(G), then, by (5.6(iii)),
Sz-,A,,(e) = AN-2-' since p * x-1 E P(G). So Yf C X f. Conversely, if e E X f, then, by (5.6(ii)), for
all v E P(G), f E X', f(Av(e)) = f (e)
so that EYf. Thus Xf=Yf We now turn to the proof that Xo = Yo. Let f E X and x E G. Then
IIA6(I - Tz)II = <M11µ6 -x-I *>h6II1 IIII = M(A(xK6 O K6)1a(K6))IIflI
0.
ERGODIC THEOREMS
202
It follows from (5.2(iii)) that P((I -T4)E) = 0, so that (I -Tx)E E Yo. Hence XoCYo. Conversely, let p E P(G), E E X. By (0.1), we can find a net {a,}, where each a, is a convex combination of elements t (z E G) in (X(G) + Cl)', with a, it weak*. But then, using (5.6(i)), for f E X',
[f(E) - a,(fE)]
[f (C) - A(fE)] = f[(I - Ap)E]-
Now if (E) - a, (f E)] is of the form f (K, E) with K E E Xo and K, independent
of f. Clearly K,E - (I - A,)E weakly. Thus (I - A,) C is in the weak closure Xo of X0. Hence Yo C Xo. So Xo = Yo and (2) is established. Q
Before turning to the Pointwise Ergodic Theorem, we require some preliminary results on vector-valued measurability and integration, which will be used, without comment, in the sequel. A good reference for these results is Hille and Phillips [1, pp. 71-85]. See also [DS] for a more extensive treatment and Diestel and Llhl [S].
(5.8) Preliminaries. Let X be a Banach space and (N, Sat, p) be a complete, a-finite, measure space. A function g: N X is said to be a measurable, countably valued function if g is of the form EO°_I in X and {En} is a disjoint sequence in 91.
where {En} is a sequence
A function f : N X is called p-measurable if there exists a sequence (fn) of measurable, countably valued functions from N into X such that II fn(x) - f (x)II -+ 0 almost everywhere on N. It is obvious that if f : N X is p-measurable, then the real-valued function x -+ ]I f (x)II is measurable. The following useful result characterises p-measurable functions: a function
X is p-measurable if and only if f is weakly measurable (that is, the f:N complex-valued function x -* F(f (x)) is measurable for all F E X') and there exists a null subset D' of N such that f (N D') is norm separable. A function f : N -+ X is said to be p-integrable (or Bochner integrable) if there exists a sequence {f.} of measurable, countably valued functions from N into X such that f II fn(x)[I dp(x) < oo for all n, and I I fn(x) - f (x) II -s 0
almost everywhere,
([I f (x) - fn (z)II dg(x) - 0.
X
A necessary and sufficient condition for f : N - to be p-integrable is that f be p-measurable and the function x - I If(x)II be in LI(N). The space B(N;X; p) of equivalence classes of p-integrable functions from N to X is a Banach space, the norm being given by
IIfll = f IIf(x)[E dp(x). The integral for functions in B(N; X; p) is defined first in the obvious way on the (dense) subspace of countably valued functions in B(N; X; p) and then extended
ERGODIC THEOREMS
203
to the whole of B(N; X; p) by continuity. Further, if f E B(N; X; p), then 11( f(x)dp(x)lf
<_
f IEf(x)fldp(x),
and if T E B(X), then T o f E B(N;X; p), and
T (f f f (x) dp(x)) =
f
T (f (z)) dp(x).
Also, the natural analogue of Fubini's Theorem holds for a p-integrable function of several variables. We note that if g: N -+ X is p-measurable, bounded, and of finite measure, a measurable set then vanishes outside g E B(N; X; A). For the remainder of this section, the group G will be assumed to be o-compact. A representation z -+ T. of G on X is said to be strongly measurable if each of the maps z -+ Tx (le r= X) is A-measurable (for the measure space of A-measurable sets). Similarly, a representation (G) t - Rt of the additive semigroup [0, oo) on X is said to be strongly measurable if each of the maps t -+ Rt C (C E X) is p-measurable, where p is the restriction of Lebesgue measure to 10, oo). The next result is well known but for the sake of completeness, a proof is given. (The corresponding result for representations of [0, oo) is given in [DS, VIII.I.3).)
(5.9) PROPOSITION. A representation x - T2 of G on X is strongly measurable if and only if each of the maps x -+ TTg (l; E X) is continuous.
PROOF. Suppose that x -+ Tx is strongly measurable, and for rt E X, let F,7(x) = Tzvl (x E G). Then Fn is A-measurable for all t? E X. Let C E X. We have to show that F£ is continuous. Since each Tx,, is continuous (x° E G), it is sufficient to show that FF is continuous at e. Fix C E % (G) and xo E G. Then FEIc, FTT,EIc belong to B(G;X; A), and so, with M = supzec IITzII, A(C)II(Txo -Te)e1I = VC T.-, (Tx=, -Tx)edA(x)11 (I)
< M fc I I (Txxo - Tx) eI I dA(x).
Fix D E F(G) such that C C D°. Now let c > 0. First approximate FEIc by a finitely valued function g E B (G; X; .t) supported in C and then, using the regularity of A and Urysohn's Lemma, approximate g by continuous functions on D to obtain a continuous function f E B(G; X; A) with support in D such
that (2)
f IIFE(x) - f (x)II da(x) < e. D
Since f is uniformly continuous on D, we can find a neighbourhood U of e in C
such that CU C D and [[f(xxo) - f(x)II < E for all z E C, xo E U. Using (I)
ERGODIC THEOREMS
204
and (2), we have, for xo E U, 11M. - Te)eII <- (M/A(G)) VG xczo(xxo)IIFF(xxo) - f(xxo)11 dA(x)
+f IIf(xxo) - f(x)II dA(x) + f IIf(z) - FF (x)I[ dA(x)] C
C
< (M/A(C))[M1 + A(C) + I)e
where M1 = sup{d(y-1): y E D}. The continuity of FF now follows. Conversely, suppose that (in the above notation) every FF is continuous. Then for every g, F{(G) is a-compact (since G is) and so is norm separable in X. Since FF is trivially weakly measurable, we see that FF is A-measurable, so that x T. is strongly measurable. 0 (5.10) COROLLARY. Let X be a Banach space, E X, Gi (1 < i < n) be a or-compact, locally compact group, D be a compact subset of Gl x . . x On and, for each i, Ti be a strongly measurable representation of G, on X. Then
the function (t1,...,tn) -. XD(t1,...Itn)T1(t1)...Tn(tn)e from GI x ... X On into X is AG, x
x AG,,-integrable.
PROOF. Let F(t1, ... , tn) = Ti (t1) Tn(tn)e. It follows, using (5.9) and the joint continuity of multiplication on a bounded subset of B(X) for the strong x operator topology, that F is continuous, and so AG, x Now use the fact that D has finite measure. 11
(5.11) The Dunford-Schwartz-Zygmund Theorem. This theorem, stated below, will provide a basis for the Pointwise Ergodic Theorem for connected amenable locally compact groups. The theorem is proved in [DS, VIII.7.10].
Let (Z,-",µ) be a measure space. Denote the operator norm on B(Li(Z)) by II [Jr. Let W = LI (Z) n L.. (Z). Then W is a dense subspace of L,(Z) for I < p < oo. (Indeed, if 0 E W, and I < p < oc, then
f If[Pdp= f so that 0 E Lp(Z). Consideration of simple functions shows that W is dense in Lp(Z).) For a measurable function f: Z - C define IIfllp < oo in the obvious way. For T E B(L1(Z)) and p E [1,00), let
IITIIp =sup{ItTflip: f E W, Ilfllp =1} (where the sup of an unbounded subset of [0, oo] is taken to be oo). Note that if p E [1, oo) and IITIIp < oo, then TI w extends uniquely to an element of B(Lp(Z)) of norm IITIIp. Of particular importance is the case where 11T111 < 1 and IITII., < 1. In this case, IITIIp < I for all p E [I, oo) by the Riesz-Thorin Convexity Theorem [HR2, (E18)], so that T can be regarded as a bounded linear operator on Lp (Z) (p E [i, oo)) with norm IITIIp < 1. The Dunford-Schwartz-Zygmund Theorem is the following result.
ERGODIC THEOREMS
205
Let (Z,X",,u) be a measure space, n e P, and let t --* Ti(t) (1 < i < n) be strongly measurable representations of [0, oo) on LI (Z) such that [[Ti(t)][1 <
1, ][Ti(t)[[0 < 1 for 1 < i <- n and t > 0. Let p E (1,00) and for R = (R1,...,Rn) E (0,00)" define A(R) E B(Lp(Z)) by
RnR,
A(R)9=(RI...Rn)-1...j 0
TI(tI)...T(tn)gdt1...dtn
0
for g E Lp(Z). If f E Lp(Z) (1 < p < oo), then there exist functions f', fl in Lp(Z) such that (i) A(R) f - f' u-a.e. as R1,. .. , Rn - oo; (ii) A(R) f -+ f' in Lp (Z) as R1, ... , Rn oo; (iii) [A(R) f ] < If, [ µ-a. e. for all R E (0, 0o)n. It is convenient to have a version of this theorem involving representations of R.
(5.12) PROPOSITION. Let (Z,.", p) be a measure space, K a compact group,
and t - Ti(t) (1 < i < n) and k - Tk strongly measurable representations of R and K respectively on L1(Z) such that [[Ti(t)[j1 < 1, [[Ti(t))[,o < 1 (1 < i < n,
t E R), and ][Tk[]1 < 1, []Tk[[. < 1 (k E K). Let p E (1,00) and for R = (R1,.... R,,) E (0,00)' , define A'(R) E B(Lp(Z)) by (1)
A'(R)g = (2nR1 ...
Rn)-1
R
xJKJ -R ...
,
Ti(tl)...Tf(tn)Tk-igdtl...dtnd),K(k). f-,
R,
Then (i), (ii), and (iii) of (5.11) hold with A(R) replaced by A'(R).
PROOF. The integral in (1) exists by (5.10). Using Fubini's Theorem and replacing g by fK Tk-, 9 d)K (k), we may delete the terms " fK" "Tk-,", and "AK (k)" in (1). Now write [-R,, Ri] = [-Ri, 0) U 10, Ri] and expand the expression for A'(R)g into a sum of 2n repeated integrals I, the range of each ti in each I being either [-F.,, 0) or 10, Ri]. Whenever the range of ti in some I is [-Ri, 0], change the variable t, to s, = -ti. (The appropriate "change of variables" formula is obviously valid for countably valued, integrable functions, and its truth for general integrable functions follows by a routine approximation argument.) Now apply the Dunford-Schwartz-Zygmund Theorem to each I. 0 The proof of the Pointwise Ergodic Theorem (5.20) (due to Greenleaf and Emerson [11) involves first a reduction to the Lie group case and then to a situation in which we are dealing with a group that is obtained by forming a finite sequence of semidirect products of copies of R, a compact group K entering at the- last stage, so that (5.12) can be applied.
In the following two lemmas, G is a semidirect product N x, H. Recall (Appendix A) that p(h)(n) = hnh-' (_ (e, h) (n, e) (e, h- 1)) in G (n E N, h E
H). Define j : N x H - G by j (n, h) = (hnh-1, h). Observe that j (n, h) = (e, h) (n, e), or simply hn. The group H is identified with GIN in the obvious
ERGODIC THEOREMS
206
way, and the Haar measures AN, AN and AG(= A) are chosen so that Weil's formula holds.
(5.13) LEMMA. The map j is a homeomorphism from N x H onto G, and for all f e LI (G), E E R (G), we have
f f(x)dA(x)=ff f (j(n, h)) dAH (h) dAN(n), G
AN X \H(j-'(E)) _.1(E).
NxH
PROOF. Trivially, j is a homeomorphism with j-1 (n, h) = (h-Inh, h). If f E LI(G), then by Weil's formula ((1.11)), f f (x) dA(x) = fH dAH(h) f f (hn) dAN(n) N
(1)
=f
f
f (j(n, h)) d(AN x AH)(n, h). NxH Now (1) is also true when f is replaced by XU for some open subset U of G. But
then AN x.1H(j-I(U)) = A(U), and the second equality of the lemma follows by outer regularity.
Q
A more general result than the above holds. Indeed, let N1.... , Nm be locally compact groups, and let G be obtained by forming semidirect products of Nz: thus
G = (... ((Ni xp, N2) xp2 N3) xpa ...) xp_ Nm.
Let AN, be a left Haar measure on Ni and AG the left Haar measure on G determined by the AN,'s through Weil's formula. Let j : NI x
x Nm --+ G be
given by (2)
9 (xi, ... , xm) = xmxm-I ... X1,
The result alluded to above is as follows. The map j is a homeomorphism from NI x x Nm onto G, and for all f E LI (G), E E .SA(G), (3)
f f(x)dAG(x) =
f
f(j(x1,...,xm))dAN,(xl)...dANm{xm)>
AN, x ... x ANm (j-' (E)) = AG (E). Using the above lemma and Fubini's Theorem, this result easily follows by induction.
We now introduce some notation. Let C be a semidirect product N x, H as in the above lemma. For x E G, write x = (xy, xN), where xN E N, xH E H. If A C G, then we set AN = {xN : x E A},
AH = {xH : x E A}.
The set AN [AH) is sometimes denoted (A)N [(A)H).
Note that, given C E c(G), the existence of sets KI, K2 in the next lemma is assured by (4.13). The lemma enables us to find, for given C E V(G), a set K E r'(G) satisfying the inequality of (4.13(ii)), given that the corresponding problem for N and H can be solved.
ERGODIC THEOREMS
207
(5.14) LEMMA. Let G = N xo H be amenable, e > 0, and C E 91,,(G). Let K1 E '(H) be nonnull and (such that / AH(CHK1 AKi)/AH(Ki)
(1)
Define C.1 E F(N) by CN = {kr 1cH1CNCHk1: c E C, ki E Ki},
and let K2 E F(N) be nonnull and such that AN(CNK2 A K2)/AN(K2) <
(2)
Then if K = j(K2 x K1) (= K1K2), we have A(CK A K)/A(K) < E(2 + E).
PROOF. Let C E C, k1 E K1, k2 E K2, and k = &2, k1). Then ck = (cN,cH)(klk2kr 1,k1) = (CNcHklk2ki 1cH1,cHk1) _ (cxki(kI 1c H1cNcHklk2)(cHk1)-1,cEk1)
Hence CK C j(C.,K2 x CHK1), and using (5.13), together with the elementary set inclusion
(A x B) - (C x D) C (A x (B - D))U((A--C)xD) and (1) and (2), we have
A(CK AK) < A(j(CNK2 x CHK1) j(K2 x K1)) = AN X AH((CNK2 x CHKI) ^- (K2 x K1)) < A+N (CNK2 )AH (CHK1 - K1) + A(CNK2 - K2)AH (K1)
= [(1+E)E+e]AN(K2)AE(Kl) =e(2+e)A(K). We now embark on the reduction procedure described in the second last paragraph of (5.12). The B references in the sequel refer to Appendix B.
(5.15) PROPOSITION. Let G be a connected Lie group. Then there exists a maximal, compact, normal subgroup K(G) of G. The subgroup K(G) contains every compact, normal subgroup of G, and if S is a closed, connected, normal subgroup of G, then K(S) is a normal subgroup of G contained in K(G).
PROOF. Let sad be the set of maximal compact subgroups of G. By B62, .sad is not empty. If K E ..V and K' is a compact normal subgroup of G, then K'K is a compact subgroup containing K, so that K' C K'K = K. Observing that E say whenever x E G, K E .say, it readily follows that fl.' is the xKx_I
maximal, compact, normal subgroup of G. Let S be a closed, connected, normal subgroup of G. Then xK(S)x 1 is a normal subgroup of S for all x E G, and the
maximality of K(S) yields that xK(S)x-1 C K(S). Hence, K(S) is a closed, normal subgroup of G and is thus contained in K(G).
208
ERGODIC THEOREMS
(5.16) PROPOSITION. Let G be a connected, amenable Lie group such that K(G) is trivial. Then there exists n E P, closed subgroups of G each of which is isomorphic to R, and a compact subgroup K of G such that (i) the map j : L1 x x Ln x K -+ G, where j (ll, ... , ln, k) = kln l1 is a homeomorphism onto G (as in (5.13(2))); (ii) the subsets Hi = j(L1 x x L,) (1 < i < n) are closed subgroups of G; (iii) with H,,+1 = G and Ln+1 = K, we have that each Hi+1 (1:5 i< n) is a semidirect product Hi x,,, Li+1. In particular, Hi is a normal subgroup of Hi+1.
PROOF. Let N be the nil-radical of G. (So N is the maximal (closed), connected, nilpotent, normal subgroup of G (B35).) We show first that N is simply connected.
Let (N, r) be the simply connected, covering group of N (B6). Since N is nilpotent and simply connected, its centre Z(N) is simply connected (B34, 44) and hence of the form R' for some m > 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 < i < n) with 0 < R;(m) < oo for all i, m and R;(m) --> 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<1 for all aEG'.
Now if x E G and 0: ,T(Ko)
.3(K0) is given by the formula
O(E) = xEx-I, then the uniqueness of normalised Haar measure on K0 yields
that \K,(¢-I(E)) = AK.(E). Thus 'K0 XE(xkx-I) dAKO (k) =
f
X-- (k) d'K.(k),
Ko
and it follows that (3)
for f E LI(Ko).
f f (k) dAK0(k) = f f(xkx-I) d.AKO(k) Ko
Ko
ERGODIC THEOREMS
212
Let 1 < p < oo, and form E P, g E LP(Z), define A,, A;,, E B(Lp (Z)) by Am9 =
(A(Km))-1
Jm
Ts-. g d.1 (x),
A'' g= (AG'(K;,,))-1 f Sa-I9d1G,(n). K,,,
Recalling that K, = Q-I(K',) and using Weil's formula again, (3), (2), and the weak measurability of the representations involved, we have, for f E (LP(Z))', p
f (Am9) = f dAG,(xKo) J
L. f (T.-.Tak-3x-19) dAxo (k)
K
=
J x'
dAG, (xKo)
J xo
f (Tx-=Tk-, g) dAK. (k)
=/f(Tz-1Po9)dAG,(xKo)=f (fS4-1dAo(a)). x,
x'
Thus A, = A' for all m. Let Em = [-R1(m),R1(m)] x ... x
x K, so that K,,, _
j(E,). Then using the first equality of (5.13(3)), f (Am9) = f (Am9)
_ (A (K.))-' LG Xj(E,)(a)f(Sa-i9)dAG'(a)
_
(A(Km))-1
/ f ...
R (m)
_ (A (Km))-f
Xj(Em) (j (t1, ... , k))f (S(j(t...... t,,,k))-' g)
K 9?
X -R n (m)
R
pR,(m)
... J
S_t1 ... S_t,Sk-I g dtl ... dt, d)K(k)
-R, (m)
Now A (K,) = A'(K,,,) = 2'nR1(m) ... -,,(m), using the second equality of (5.13(3)). The proof is completed by applying (5.12). 0
References Alaoglu and Birkoff [1], Aribaud [1), [2], Chatard [1], Conze and Dang Ngoc [1), [2), Day [1), 12],14), [9], Eberlein [1), Greenleaf (4), Greenleaf and Emerson [1], Greenleaf, Moskowitz, and Rothschild (1], R. T. Moore (1), Peck (1), Renaud [1], and Witz (1].
Further Results (5.21) Other ergodic theorems. We now state another Mean and Pointwise Ergodic Theorem for amenable groups. References for this theorem are Calderon [1), [2) Templeman [1], Chatard [1], Bewley [1], and Emerson [4].
ERGODIC THEOREMS
2i3
Let G be a-compact and amenable. Suppose that there exists a summing sequence {K,,} such that sup(.1(Kn 1Kn)JA(Kn)) < oe.
(1)
nEP
Let (Z,2', p) be a o' finite measure space such that Z is a left G-set, the map (x, z) -+ xz is measurable from G x Z into Z, and p is G-invariant. Let I < p < oo, f E L9(Z), and x -i TT be the representation of G on LD(Z) given by Tz f (z) = f (x-1 z). If {An } is defined as in (5.17(1)), then the sequence {An f } converges both in the Ly-norm and pointwise p-a.e. to a G-invariant function in Ln (Z).
An obviously important question, in connection with the preceding theorem, is the following. Which groups G admit summing sequences {K,,} satisfying
(1)? On the positive side, Auslander and Brezin (1) prove that every simply connected, nilpotent Lie group admits a summing sequence satisfying (1), and' they use this fact to obtain results on uniform distribution on nilmanifolds and almost periodicity on nilpotent groups. We shall see in Problem 6-15 that if G is abelian or if G is discrete nilpotent, then G admits a summing sequence {Kn} satisfying (1).
On the negative side, Greenleaf and Emerson [1) assert that the condition (1) and the summing sequence condition (4.15(3)) "are actually incompatible in many amenable Lie groups." In particular, there are difficulties for nonunimod-
ular groups. In this connection it is very easy to see (Chatard [1J) that if the summing sequence {Kn) satisfies the related condition nEP
1)Ja(Kn}) < oo,
then G is unimodular. Is it true that if {K,,} satisfies (1), then G is unimodular? We now turn to another problem in ergodic theory-the existence of equivalent G-invariant measures.
Let (Z,-'r,p) be a measure space, and G a group of invertible, measurable transformations of Z. In view of the importance of measure-preserving transformations, the following question is of interest: under what conditions can one find a (positive) measure v on (Z,2") which is C-invariant and (in a suitable sense), is equivalent to p? A complete solution to this question when (Z, 2', p) is o-finite and nonatomic and v is finite is given in Hajian and Ito [i): if p is a -finite and nonatomic, then there exists a finite, G-invariant measure v on (Z, _T) equivalent to p if and only if there does not exist E E 2" with p(E) > 0 which is weakly wandering under G. (Here, a set W E ' is called weakly wandering (under G) if there exists a sequence {xn} in G such that p(x=W f1 xjW) = 0 if i 0 j.) Less is known about the above question when p [v) is no longer required to be a-finite [finite]. For results in this general case see Friedman [1] (with its references), Millet and Sucheston [1), and Rosenblatt [3). We now state a result
due to Rosenblatt [12) that solves the question when G is amenable. It is not known if the result is true without the hypothesis of amenability. (It is of interest
ERGODIC THEOREMS
214
that the amenable case of the Hajian-Ito result was proved in the (earlier) work of Natarajan [1], [2]. See also Granirer [12].) We require a stronger version of equivalence of measures. Recall that if (Z, 27,,u) and (Z,Z, v) are measure spaces, then p and v are said to be equivalent if they have the same null sets. In order to cope with sets of infinite measure in the non-a-finite context, we say that u and v are strongly equivalent if p and v have the same null sets and the same sets of infinite measure, that is, if E E Z,
then u(E) = 0 if and only if v(E) = 0, and p(E) = oo if and only if v(E) = co. Rosenblatt's result is the following. Let G be amenable and p be nonatomic. Then (1) and (ii) are equivalent: (i) there exists a G-invariant measure v on (Z, Z) that is strongly equivalent to U;
(ii) if E E 2, then infXEG p(xE) > 0 if µ(E) > 0 and supzEG µ(xE) < co if u(E) < oo.
Problems 5 Throughout, G is a locally compact group.
1. Let G, {K6} be as in (5.7) and x Tz be a weakly measurable representation of G on a reflexive Banach space X. Show that the conclusions of (5.7) hold.
2. What does (5.7) say when X = WP(G), the space of weakly almost periodic functions on G under the natural left G-action? identify X f, X0, and P. 3. By following the proofs of (5.19) and (5.20), construct a summing sequence for the "ax + b"-group that is admissible for the Pointwise Ergodic Theorem.
4. Let Z = [0, 1) and F1 be the restriction of Lebesgue measure to [0,1). Let a be irrational and n -» Tn the representation of Z on L1 (Z) given by Tn9(x) = 9(n.x)
where n.x is the unique element of [0,1) (1 (x + not + Z). Let an = E n-1 r(1) Show that [[Tn[[1 < 1, [[TT[[oo < 1.
(ii) Show that if m > 0 and x E [0,1) then Az,,n = {r E 7L: r.x E [m'1, 2m-1)} is infinite.
(iii) Define inductively a sequence {K,} in F(Z) by setting K3 = [-1; 1] and,
for m > 2, K,n = [-(m!);m!] U K,,,I U D,n, where D,n C Aa,,,,n is such that (m!)m-1f4 - 1 < [D ,n[ < (m!)m-1/4. Show that {K,n} is a summing sequence for Z.
(iv) Show that {K,n} is not admissible for the Pointwise Ergodic Theorem. [Hint: show that {An f } does not converge pointwise, where f E L, (Z) (1 < p < 2) is given by f(x) = x1/2,]
5. A sequence {p,, } of probability measures in M(G) is called ergodic if, whenever r is a continuous unitary representation of G on a HiIbert space 55,
£RGODIC THEOREMS
215
then r(pn) - P' , in the weak operator topology of B(. ), where P" is the orthogonal projection of 15 onto the subspace 17f = {g E 15: r(x) _
for all
x E G}.
(i) Show that if {K } is a summing sequence for G and pn, = XK, /.1(K.), then {pn} is ergodic.
(ii) For each r E G let B,T be the set of functions on G of the form x - a (r(x)i;,rl) for some 4,17 E AT, the Hilbert space of r. Let BI(G) be the closed subspace of C(G) spanned by U{B,,: r E G}. Prove that if G is abelian, then BI(G) = AP(G). Show that BI(G) C WP(G). (iii) Prove that the following are equivalent for separable G: (a) {A,,) is an ergodic sequence; (b) r(pn) 0 in the weak operator topology for all r E G ^ {1};
(c) }1n - m weak* on BI (G), where m is the (unique) invariant mean on BI(G) (obtained by restricting the invariant mean on WP(G)); (iv) Show that if C is abelian, then {pn} is ergodic if and only if the sequence {pn}, regarded as a sequence of probability measures on the Bohr compactification J3G of G, converges weak* to the Haar measure on /3G. (v) If H is the Heisenberg group (Problem 4-29) show that BI(H) = (APo(R)6Co(R2)) @ (C (9 AP(R2))
where APo(R) = {0 E AP(R): m(O) = 0, m the unique invariant mean on AP(R)} and - denotes the injective tensor product completion. 6. Let S be an amenable semigroup of bounded linear operators on some C(X) (X compact Hausdorff). Suppose that (1) (co{Tq5: T E S})- contains an S-invariant function for each 0 E C(X); (ii) each T E S is Markov (in the sense that T(1) = 1 and TO > 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)
<_
(.\(C))-IA(C-+1)A(Cn+P).
Substituting n = 2 in (3) yields (4) \(C-+2):5 (,\(C))-1A(Cm+1)A(C2+p)
and substituting n = p + 1, m = r - 1 (r > 2) in (3) yields (5)
A(CT+P) < (A(C))-IA(CT)A(C2P+1).
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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) <_ kcn'. (6.5) PROPOSITION. If G is connected, then G is exponentially bounded [has polynomial growth) if and only if there exists Co E 9, (G) such that .1(Co )'1' -+ 1 as n oo [for some polynomial p, A(CQ) < p(n) for all n > 1). PROOF. The proof of the case involving polynomial growth follows from (6.3).
The other case is dealt with by a simple modification of the proof of (6.3).
(6.6) PROPOSITION. If G has polynomial growth, then G is exponentially bounded.
PROOF. Let C E
(G) and G have polynomial growth. Then for some k > 0
andrEP, A(Ct)
noo
n-+oo
(6.7) PROPOSITION. Let G be exponentially bounded [have polynomial growth) as a discrete group. Then G is exponentially bounded [has polynomial growth).
PROOF. Let C E 9,(G). A simple compactness argument shows that there exists a finite subset F of G such that C2 C FC. The desired result follows from the inequalities A(Cn) <_ A(Fn-1C) < jFn-1[A(C). The preceding proof is due to Hulanicki [6), who attributes it to Greenleaf and Emerson. An improvement of the next result for certain groups is given in (6.43). A reference for the next result is Glasner [S). (6.8) PROPOSITION. Let G be exponentially bounded and C E Fe(G), e > 0. Then there exists arbitrarily large n such that A(C11+1 AC")fA(C") < e. Further, G is amenable.
PROOF. From an elementary result on convergent sequences (Rudin [5, p. 681),
1 < lim nf(A(C"+t)/A(C")) s lim f(A(C")l/n) =1. n-oo n-00
Now observe that A(C"+1)/A(C") = 1 + (A(C C" L C")/A(Cn)), and use (4.13).
(6.9) PROPOSITION. Every exponentially bounded group is unimodular.
PROOF. Let G be exponentially bounded and x E G, C E 9,(G) with x E C. Then for all n > 11 A(C)A(,)n-1
= A(Cxn-1) < A(Cn),
where A is the modular function on G, and taking nth roots and letting n - 00, we have A(x) < 1. Hence G is unimodular.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
221
Our next aim is to show that every nilpotent group has polynomial growth, and to determine the degree ((6.4)) of a finitely generated nilpotent group. The arguments involved in this determination can be regarded as an apotheosis of induction! References for the results are Wolf [1], Bass [1], Tits [4], Guivarc'b [1], and Hulanicki [S]. Our account is an exegesis of Tits [4].
(6.10) Preliminaries. Throughout (6.10) -(6.17), G will be a finitely generated nilpotent group. For x, y E G, we define the commutator Ix, y] to be X-1 y- Ixy. The lower central series G = GI rG2C> ...>Gcc> Gc+1 = {e}
is defined by setting G;+1 = [G;, G], the subgroup generated by elements of the form [x, y], x E Gi, y E G. So in the notation of Appendix A, G; = CG. When G = {e} we take c = 0. Otherwise, we require G, 0 {e}. So c is the class of G. Whenever convenient in the sequel, we can suppose that c > 0. Recall that [Gi, Gi] C Gi+?
A generating subset for G is a subset E of G such that UT_1(EUE-I)' = G, and in such a case, E is said to generate G. We choose a finite, generating subset
E for G so that for each i (1 < i < c + 1), E; = E fl G; generates Gi. (We take E,+1 = 0. It follows from Appendix A that each G; is finitely generated.) Let
Fi = E - E;+I (0:5 i:5 c). Since E;+1 C E,, it follows that F, C F;+1. Note
that Fo=0. For each word w = (w],._
in E U E-1, let a(w) = wl w2 w E G and
ni(w) be the number of w3's in F, UFO 1. Observe that every element of G is of
the form a(w), and that n;(w) < n,+i (w). For r = (rl, ... , r,), where ri > 0, let
AE _ {a(w): n;(w) < ri (I < i
of H is denoted by r'(H). Further, if L is a normal subgroup of H, then IM, Chapter 5] L is also finitely generated, and we have
r'(H) = r'(L) + r'(H/L).
(1)
In particular, each of the quotient groups G;/G,+1 is a finitely generated, abelian
group. Let di = r'(G;/G;+1) and d = F,`=I id,. We shall prove in (6.17) that d is the degree of G.
(6.11) LEMMA. (i) For each k E P, (E U E-1)k c AE(k). (ii) Let E' be a finite, generating subset of G with E,` = E` fl Gi generating G; (1 < i < c) and EE+1 = 0. Let Ft, n,(w'), a'(w'),AE' be defined with respect to
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
222
E' in the same way as F,., ni(w), a(w), ArE were defined with respect to E. Then there exist a, b > 0 such that for all r = (r1i ... , rc), we have
lAa < IAEI < {Ab ,
(1)
where ar = (arl,...,arc), br = (brl,...,brc).
PROOF. (i) Let z E (E U E-1)k. Then x = a(w), where w = {w,... , w,.} is a word in E U E. Clearly, n1(w) < k < k1, so that (i) follows.
(ii) For 1 < i < c, let Hi = (Fi - Fi_1) U (Fi -
F,_1)-1 and H'
=
(F; -r F;_1) U (F, - F=_1)-1. Using a simple recursion argument, the fact that the canonical image of each of the (finite) sets Hs (= (E; E,+1) U (E; Es+1)-') generates the group Gi/Gi+1, and arguing as in the proof of (6.3), we can find finite subsets Ci of Gi and integers mi > 0 such that C1 = 0,
(2)
Let w = (w1.....
Hi UCi C (Ha )"',C,+1.
be a word in E U E-1 and x = a(w). We construct a
E'U(E')-1 with x = a'(w') as follows. Let pi(w) be the number of wj word w' in in Hi. Replace each of the p1(w) elements in the expansion x = w1 W2 . w" by an element of (H')', C2. This produces a new product expression for x, involving m1 p1(w) elements from H,, pl (w) elements from C2, and pi (w) elements from
Hi (2 < i < c). The same procedure enables us to replace the elements of H2 U C2 by elements of (H2')12 Q. Repeating this process. we obtain a word w' (E')-1 with x = /a(w') and, in an obvious notation, in E' U pl (lw') = mlpi (w), ps (w') = m2 (p2 (w) + PI (W)),
pc (w') = mc(pc(w) + ... + p1(w)). Let m = max{mi : 1 < i < c}. Then
ni(w') = Epc(w') < me i -Pr(w) I = mcni(w). rol r-3 f It follows that (AEj < !AE I, where b = mc. Reversing the roles of E and E' gives (1).
(6.12) LEMMA. Let E be such that (i) IF, { is smallest possible;
(ii) (u, v) E E whenever u, v E E U E-1. Let y E F1 and X E AE(k). Then there exists s E Z with Isi < k and a word to, with wi E (E {y}) U (E - {y})-' and ni(w) < (c+1)k' - Isi (1 < i < c), such that (1)
x = y°a(w).
PROOF. Noting that every commutator [u, v) in G is contained in G2 C G F1, it is easily checked that E can be chosen so that (i) and (ii) are satisfied.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
223
Note that, using (i), y-1 f E, so that the word w in (1) does not involve y or Let x = a(w'), where w' = (w',,..., w'' ) is a word in EUE-1 and ni(w') < ki. < i4) be those wi's which are either y or y -I.
Let wi',,...,wt9 (i1 < i2 < Using the equality (2)
wj'wt! = WL wj, [wj" wtiI
we bring the elements wt, , ... , w;o to the left, one at a time, in the expression for x. Note that by (ii), [wj, wi,j in (2) belongs to E. When wt,, ... , wily (1 < p:5 q) have been brought to the left, we have
x = lllii ... wiya(wv), where wP is some word in E U E-1. Take wo = w'. We now prove by induction on p that for all i < c, nt(wP) :5
(3)
ni-j, 9=0
where nt = ni(w°) (12: 1) and nj = 0 for l < 1. The result is obviously true when p = 0. Suppose that it is true for p = m, 0 < m < q. Applying (2) with l = (m + 1), we see that if w? E F1 U Fi 1, then w,' still appears on the right-hand side of (2), while a new element [wj', w=, ] appears. Now if wj' E Ek,
then z = [wj', wt,1 E Ek+1. Hence if k > i, then z 0 Fi. So if z E Ft, then wj' E Fi-1. Recalling that ni_I (w') = 0 if i < 1, we have
ni(wm+1)
E (7)ni.i+
(7) n 9
-9
=nf.+, ({7)+// + .m1) Jni-i+ni-(m+1)
=r:(m+ll 1=o
Thus (3) is established.
Putting p = q in (3) and setting w = w9, we have x = y'a(w), where Isi < q:5 n1 < k (since z E AE(k)). Clearly wi E (E now use (3) with p = q. If i < q, then ni(w) <
E-o
kiki-i
{y}) U (E -r {y})'1. We giki-j < Ej' --'
= iki = (i + 1)ki - ki < (c + 1)0 - Isl. If i > q, then ni(w)
(s)ni-i < (q + 1)k' < iki < (c + 1)ki - Is,. This completes the proof.
O
(6.13) LEMMA. Let H be a subgroup of G of class d and such that H D G2. Let {Hi} be the lower central series for H and si = r'(H,1H,+1). Then, in the notation of (6.10), (1)
1
d=1
isi < d - r' (G/H).
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
224
PROOF. Note that H is normal in G since G/G2 is abelian. Since H C G, we have Hi C Gi. Hence 1 < c. We can obviously replace I by c in (1), since si = 0
fori>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
<
idi + cr' (Gc) = d.
(6.14) LEMMA. There exists ko > 0 such that, for all k > 1, 4AE(k)E < ko(kd).
PROOF. The proof involves an induction argument within an induction argument. The result is trivally true if c (the class of G) is 0. Let p > 0 and suppose that the result is true for all groups of class < p. Suppose that c = (p + I). By (6.11(1)), E can be chosen as in (6.12).
Let .' be the set of subgroups H of G generated by a subset of the form FH U E2, where FH C Fi. For such a subgroup H, let m(H) = {FHI. Note that H D G2, and that G2 and G belong to 5°. We now prove, by induction on m(H), that the result is true for all H E .P (and in particular, for G itself). When m(H) = 0, then H = G2, and since G2 is of class < (p+ 1), the result
is true for G2. Now let 0 < m < m(G), and suppose that the result is true
for all H such that m(H) < in, Let H E , be such that m(H) = (m + 1) and let A = FH U E2. Let y E FH and H' be the subgroup of G generated by
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
225
A' = (Fm -r {y}) U E2. Then H' E S° and m(H) < m. By hypothesis and (6.13) (with H, H' in place of G, H), there exists k, > 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 <
(2)
k,k(d.-I).
By (6.12), every element x of A' (k) is of the form y'x', where x' = a(w) with ni(w) < (1 + 1)k' - jsl < ((1 + 1)k)', and 1s1 < k. Since there w, E A' U are (2k + 1) possibilities for s and as w E A" ((1 + 1)k), we have, using (2), IAA(k)I < k1(2k+ 1)((1 + 1)k)d"-I < k2(kdx) for some k2 independent of k.
(ii) Suppose now that IH/H'] = t < oo. So r'(H/H') = 0. Then there exists a smallest positive integer p, with 1 < p < t, such that yP E H'. Using (6.11(1)), we can suppose that (1) is true with A' replaced by ©" = A' U {y'}. Let x E A' (k) and write x = y'x', x' = a(w) as in (6.12). Now set s = np+s1, where n E Z and 0 < sl < p < t. So (3)
x = 1i
j ((?)"x')
Clearly, (yP)"x' = a(v), where v = (vl,... , vQ) is a word in p' U and ni(v) < (1 + 1)k' - !s] + ]nj (1 + 1)k', since (]al - ]nj) > 0. Hence a(v) E A"' ((I + 1)k), and from (1), there exists k' > 0 such that
IA°"'((1+1)k)1
(k> 1).
0 For z E G, let IG,E(z) be the smallest integer s such that for some w =
Using (3), ]A°(k)J < pk'(kdH), and the proof is complete. (wj , ... , w5 ), wi E E U E-1, we have z = a(w).
(6.15) LEMMA. If z E G, then there exists k' > 0 such that (1)
1G,E(Z") < k(nl/c)
(n > 1).
PROOF. The proof proceeds by induction on c. The case c = 1 is almost trivial since then G is abelian. Suppose that p > 1 and the result is true for all groups H of class p. Let G have class c = (p + 1). As [G, GP] = GP+1i we can write Z = [XI, Y11" [x2, y2)e2 . . . [xr, yr]E'.
where xi E G, yi E Gp, and e, E {-1,1}. Let z, _ [xi, y,]E-. Since GP+1 is abelian, we have z" = zI z2
z,". Obviously, IG,E(zn) = 1G,E(zz "), and zr) < E._I IG,E(zi ). Thus to establish (1), we can suppose z = [x, y] for some x E G, y E GP. Now it is readily checked that if x E (E U E-1)'", then
1G,E(zt
'G,-- (u) < m1G,EU(x)(u)
(u E G).
226
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
Since E generates G, we can suppose that x E E. For each n E P, find ni E P such that nl/(v+l) < n, < nl/(v+l) + 1
(2)
and write n = a, nl +a2, where a1, a2 E Nl, a2 < nl. It follows from (2) that n < niv+1) = n1 ni, so that Cl < n,. Since n > (n, - 1)P+l = nl+l(1 - (nl)-1)P+1
and n1 -* oo as n -, oo, we see that a, > 0 eventually. For the purposes of (1), we can suppose that a1 > 0. Now H = G/Gv+l is nilpotent of class p, so that by hypothesis, there exists k1 > 0 (independent of n) such that if y' [E'] is the image of y [E] in H, then IH,E'((y')YP) < kin1,
lx,E'((y')°2) C kl(a2)1/D < kin,.
So there exist u,, u2 E Gv+1, yi, yz E Gp such that y' = ulyli y°2 = u2y2, and
IG,E(y2) < kin,.
IG,E(Vi) <_ kln,,
(3)
Now for v E G, [v, y] E Gv+1 C Z(G), and so, for all n > 1,
(x-ny-lx' y)(x-ly-ixy) = x n(y-lxnyx n)(xn-1y-lxy) = x-n(xn-1y-lxy)(y-ixnyx-n) = x-ly-Sxn+lyx-n
[xn,y][x, ,) =
-1-1 = x-1(y-Sxn+lyx )X = x-lx(y n-ly-I xn+1[y-1
= y-1(xn+lyz
=
y[xn+1 y]y-1
)y =
=
[xn+1,
lxn+lyx-n-1)
xn+1]x-n-1
y]
Thus for all m > 1, [x, y]m = [xm, y]. Similarly, we also have [x, In = Ix ym] More generally, it readily follows that [x, y]m,m2 = [xm, ym2] for all m1, m2 > 1. Recalling that yn° = u,y,, y°2 = U2V2 for u; E Gv+1, we have, for a2 > 0, zn = (x, y]n = ([x,
y]°, )n, [x, y]a2 = [x°j , yn' ] [x, ya2 ]
= [x°j,uly]][x,u2y2] = [x°',y]][x,y2] since u1, U2 E Z(G). Of course, if a2 = 0, then z' = [x°' , yi]. Hence, using (3) and (2), IG,E(z') < (2a1 + 2k1n1) + (2 + 2kin1)
= 2a1 + 4k1n1 + 2 <- 2(1 + 2k1)n1 + 2
<- 2(1 + 2k1)(nl/(v+1) + 1) + 2.
The induction step readily follows.
D
The following lemma is left as an exercise to the reader (Problem 6-4).
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
227
(6.16) LEMMA. For m,r E P, let m
Am,r = l (nl,... , nm) : ne E N, E ny = r} . t q=1 r' _ \(r+m-1) (_ (r+m-1)) Then IAm i
m-1
r
(6.17) THEOREM. The nilpotent group G has polynomial growth with degree d.
PROOF. By (6.14) and (6.11(i)), there exists k1 > 0 such that JEkj < k1(kd) (k > 1). It remains to prove that there exists k2 > 0 such that (1)
IEkI > k2(kd)
(k > 1).
The proof proceeds by induction on c. Trivially, the result is true when c = 0. Suppose that the result is true for all groups H of class m > 0. Let c = (m + 1).
Let d' = d - cdt, G' = GIG,, Q: G
G' be the quotient map, and E =
Q(E). Then G' is of class m and r'(G2/G;+1) = r'(G,/G,+1) (1 < i < m). By hypothesis, there exists k3 > 0 such that
(k> 1). I(E')kI > 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 < i < p).
If, in the notation of (6.16), (n1i...,np) E A,,,,, and T = {i: ni 54 0}, then, for c > 1,
1/<
IG,E(2i' ... z' P) < E 1G,E(zi') :5 k4 E(n=f c iET
1)
iET
c /2E Iq k4 (zETE ni
)
JJJ
where c-1 + q-1 = 1 and Holder's inequality has been used. So for c > 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.
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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)")
<.(B(AI)")
aH (x-1 B(Al )" fl H) dAG/H (ZH), c1H
and since G/H is compact, and, since, using the properties of B given in the first paragraph of the present proof, for each x E G, x-IB(A1)" flH = h2Ai for some hz E H, the inequality (2) would follow. So it suffices to prove (3). We will be content to give the proof of (3) for n = 3, the proof for general n being similar.
Let b3, b2, b3 E B and al, a2, a3 E Ao. Find b4, b5 E B and h, h' E H such that b2b3 = b4h,
blb2b3 = blb4h = b5h'h.
Then both h, h' belong to/ B-' (B2) n H C D. So (bla1)(b2a2)(b3a3) = (bib2b3)1(b2b3)-'a1(b2b3)J[b3 1a2b3)a3
b5h'h(h-'b4 lalb4h)(b3 1a2b3)a3
= b5(h'ba 1a1b4)(hb31a2b3)(eea3e 1) E B(A1)3. (6.22) PROPOSITION. (i) Let G be a discrete group containing a nilpotent subgroup of finite index. Then G has polynomial growth. (ii) Let G be a semidirect product of the form H xgK, where H has polynomial growth and K is compact. Then G has polynomial growth.
PROOF. (i) It is an elementary fact of group theory that if N is a subgroup of G of finite index, then the normal subgroup N' = n{xNx-1: x E G} C N is also of finite index. Now apply (6.17) and (6.21). (ii) Use (6.21). (The proof of the latter applies without the separability hypothesis by taking B = K.)
The converse to (6.22(i)) is true when G is finitely generated. This result is much harder to prove than (6.22(i)) and is briefly discussed in (6.40). A locally compact group of the form H x , K, with H abelian and K compact, is called a motion group. (An example of such a group is the isometry group R" x,, O(n, R) of R".) It follows from (6.22) and (6.19) that every motion group has polynomial growth.
230
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
For the remainder of this section, it will be assumed that G is connected. References for the work below are Jenkins [6), [9) and Auslander and Moore [!). We now examine the properties of Type R groups ((6.0(iii)). The first major
result to be proved is ((6.30)) that if G is of Type R, then G has polynomial growth. We start by stating an elementary result from linear algebra; the reader is invited to provide the easy proof. Of course, for a linear transformation T on a complex, finite-dimensional vector space, the spectrum Sp(T) of T is just the set of eigenvalues of T. (6.23) LEMMA. Let W be a linear subspace of a complex, finite-dimensional
vector space V, and let T E L(V) be such that T(W) C W. Let Tw = Tiw and
Tvjw E L(V/W) be given byTv/w(e+W) =TI;+W. Then Sp(T) = Sp(Tw) U Sp(Tv/w).
All B references in the following are references to Appendix B.
A real Lie algebra 9 is said to be of Type R if Sp(ad X) C iR for all X E 9.
(6.24) PROPOSITION. Let G be of Type R, and let L be a compact, normal subgroup of G such that G/L is a Lie group. Then Sp(ad X) C iR for all X in the Lie algebra 9 of GIL.
PROOF. By definition, there exists a compact, normal subgroup K of G as in (6.0(iii)). In particular, the Lie algebra 9K of G/K is of Type R. Now the Lie algebra of G/(LK) is canonically identified with a quotient of 9K, and so ((6.23)) is also of Type R. We can thus suppose that L is trivial. In particular, G is now a Lie group, and since the Type R condition is formulated in Lie algebra terms, we can also take K to be connected. Let 9 and t be the Lie algebras of G and K respectively, and let X E 9. Since G/K is of Type R and has Type R Lie algebra 9/t (B18), it is sufficient to show
((6.23)) that Sp(D) C iR, where D = (adX)t. Now the radical r(K) of K is a connected, compact, normal subgroup of G, and so by B51, B21, r(K) C Z(G). Let t be the radical of t. Then [X, Y) = 0 for y E t, so that D(t) _ {0}.
Thus Sp(D) = {0} C iR. It remains to show that if s = t/r and D' = D5, then Sp(D') C iR. Note that s is a semisimple, compact Lie algebra. Let B be the Killing form on s. Then -B is an inner product on s with respect to which D is skew symmetric (B49, B39). Hence Sp(D') C iR and the result follows. 0 (6.25) COROLLARY. If G is a Lie group, then G is of Type R if and only if Sp(adX) C iR for all X in the Lie algebra of G. The next result relates the possession of purely imaginary eigenvalues by T E L(R') with the polynomial growth of the norm of eeT (t E R).
(6.26) PROPOSITION. Let r E P and T E L(R') be such that Sp(T) C iR. Let II' (I be a norm on L(R'). Then there exists an (even) polynomial p such that {I etT [l < p(t) for all t E R.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
231
PROOF. Extend 11.11 in any way to a norm, also denoted 11.11, on L(C*). In the
obvious way, we have T E L(C). Let S and N be respectively the semisimple
and nilpotent components of T in L(Cr) (B31). So T = S + N, SN = NS, N' = 0, and for some basis for CT, S can be realised as a diagonal matrix with entries in iR. Using the equivalence of norms on L(Cr), we can find C > 0 such that llets II < C for all t E R. Then for some K > 0, IIetTII = Ilets . e"vll c KlletslllletNll 5 KC, Itl8(11 Neiilsl) < p(t), 8=o
where p is some even polynomial.
0
(6.27) PROPOSITION. Every simply connected, Type R, solvable Lie group has polynomial growth.
PROOF. Let G be a simply connected, Type R, solvable Lie group with Lie algebra g. The proof proceeds by induction on the vector space dimension dimg of g. If dim g = 1, then G = R, and the result is obviously true. Let n > 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") <- q(n) (n > 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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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,)
PROOF. Both H and G/H are Lie groups and the Lie algebra of H is an ideal of the Lie algebra 9 of G, while g/I7 is the Lie algebra of G/H. Now use (6.25) and (6.23).
Recall ((3.7)) that r(G) is the radical of G. (6.29) PROPOSITION. Let G be a Type R Lie group. Then G/r(G) is compact.
PROOF. By (6.28), the (Lie) group H = G/r(G) is of Type R. Suppose that G/r(G) is not compact. Then the (semisimple) Lie algebra h of H is not compact. So ((3.3 (ii))) Il contains sl(2, R) as a subalgebra. Let
Y= Then X, Y E sl(2, R), and ad X (Y) = XY - YX = -2Y, and a contradiction of the fact that H is of Type R results. (6.30) PROPOSITION. Every (connected) locally compact, Type R group has polynomial growth. PROOF. Suppose that G is of Type R. Appealing to (6.20(i)), we can suppose
that G is a Lie group. By (6.29) and (6.21), we can further suppose that G is G the solvable. Let C be the simply connected covering group of G and r: G canonical homomorphism. As kerr is a discrete subgroup of G, it follows from (6.27) and (6.20(ii)) that G has polynomial growth. Our next objective (6.32) reduces the study of polynomial growth for G to the solvable case.
(6.31) LEMMA. Let W be a finite-dimensional vector space over R and f a compact, semisimple subalgebra of gl(W). Then Sp(X) c iR for all X E f. PROOF. Let K be the connected Lie subgroup of GL(W) whose Lie algebra is t. Then K is compact (B52) in GL(W) and so the eigenvalues of its elements have modulus 1. Since ex E K for X E t, the lemma is established.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
233
(6.32) PROPOSITION. Let G be a Lie group. Then G is of Type R if and only if r(G) is of Type R and G/r(G) is compact. PROOF. If G is of Type R, then the required implication follows from (6.28) and (6.29).
Conversely, suppose that r(G) is of Type R and G/r(G) is compact. Let g, r, and t be the Lie algebras of G, r(G), and G/r(G) respectively. Observe that t = g/t and is a compact, semisimple Lie algebra. Let X E g. We have to show that Sp(adX) C iR. Now (ad X )t is a derivation on t and so belongs to ad t (B42). By (6.31), Sp((adX)t) C iR. So ((6.23)) it is sufficient to show that Sp((adX),) is contained in iR. Let n be the nil-radical of g. Then ad X(t) C n (B36), and so it is sufficient C M. Let {n(i) } be the lower central series for n and to prove that Sp(ad r the class of n. So n(I) = n, n(i+1) = [n,nW?i] (1 < j < r - 1), n(T) 0 {0}, and
mi'+'l = {0}. Let T(j) _ We will show, by downwards induction on j, that Sp(T(j)) C iR for all j. The result is trivial if j = r + 1. Suppose that 1 < j < r + 1 and that Sp(T(j)) C iR. Now by the Levi-Malcev Theorem (B43) we can find a Lie subalgebra t of g isomorphic to t such that g is the vector space is a Lie homomorphism direct sum t®t'. Further, the map Y (ad Y).(,from g into gl(n(i-1)/n(i)). Since r is of Type R and t' is compact, we can write
T(j - 1)
A + B, where A, B E gl(n(i-')/n(i)), Sp(A) C iR, and
B belongs to a compact, semisimple Lie subalgebra of gl(n(i-1)/n(i)). Further, since [g, tj C n and ad W (nii -' l) C n(i) (W E n), we see that (A, BI = 0 so that
AB = BA. Also by (6.31), Sp(B) C iR. Hence Sp(T(j - 1) (,-,).(j)/=Sp(A+ = Sp(T(j)) C B) C Sp(A) + Sp(B) C iR. Since, by hypothesis, Sp(T(j iR, we have Sp(T(j - 1)) C iR. This completes the induction argument, and Sp(adX) = Sp(T(1)) C iR as required. 0 The preceding result, in a sense, reduces the study of Type R Lie groups to the solvable case. We now investigate the Type R condition for solvable Lie algebras. We start by discussing some results from the theory of nilpotent Lie algebras. We use the results and terminology of B31f. Our objective is (6.37).
(6.33) Preliminaries. Let V be a finite-dimensional vector space over R and I a nilpotent Lie subalgebra of L(V). In the natural way, we can regard k as a nilpotent Lie subalgebra of L(Vc). Let W' be the set of weights of IC with respect to VV, and, for each 0' in 'I", let V,y, be the weight space associated with b'. Then Vc = ®O,Ck, V,p'. Let W = {011: 0' E tiIY'}. Let us call the elements of ' the weights of I (with respect to V). A simple exercise in complexifications shows that E >' 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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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)
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.)
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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<-2,{w-jl<e 2}. Let T be the (open) subsemigroup of G generated by UI U U_I and let I = UIT,
J = U- IT. Clearly, I and J are right ideals of T. We show that I fl J = 0. this end, let n > 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 <[tI-lE+(1+e 2)2: (e-2)r r-1
< e-2 + ((1 + e 2)e 2)/(1 - e2) =
2e-2
< 1.
1 - e-2 Similarly, if (wi, z1) E U_1i then kw + 11 < 1. Hence I n J = 0 as required.
Suppose, now, that G = S3 for some a 0 0. Then G is identified, in the natural way, with the semidirect product C x , R, where (w, t)(w', t') = (w + e(6-`Itw`. t + t').
We shall assume that v > 0, the argument for the case c < 0 being very similar.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
239
Let 1, k > 0. (The numbers 1, k will be specified later.) For j E {-i,1}, let VI = {(w, t) : t < -1, 1w, - j I < k}.
Let T be the (open) subsemigroup of G generated by VI U V_1. Arguing as in
the S2, S4 cases, we see that if j E {-1,1} and (w,t) E V?T, then Jw - j( < (k + e-Iff)/(1- a-I°). Now choose k = 1/2. Then for large enough 1, we can take I = VIT, J = V_IT. (ii) Suppose that G is not of Type R. Let G be the simply connected covering
group of G and Q: G G the canonical homomorphism. Since G and G have the same Lie algebra, G is also not of Type R. Let Q' be a continuous homomorphism from G onto one of the groups S2, S3, S4 ((6.37)). Let T be an open subsemigroup of Q'(G) such that (i) applies, and let ' = (Q')-1(T). Then T' is an open subsemigroup of d and there exist disjoint, right ideals
I', J' of T'. Now G is amenable (since it is solvable), and, by hypothesis, 0. Using Problem 2-10, Q(I') f1 Q(J') 0. Hence we can find x E I', y E J' such that x-Iy E kerQ c Z(G), the centre of G. Hence xy = x2(x-Iy) = x(x-Iy)x = yx, and as I', J' are right ideals of G, xy E I'nJ',
C(U(Q(T')))
yielding a contradiction. So G is of Type R. We now state and prove the theorem for which much of this chapter has been a preparation.
(6.39) THEOREM. Let G be a connected, locally compact group. Then the statements (i), (ii), (iii), and (iv) below are equivalent, and each implies (v). If G is, in addition, a solvable Lie group, then all five statements below are equivalent. (i) G has polynomial growth. (ii) G is exponentially bounded. (iii) C is of Type R. (iv) G does not contain a free, uniformly discrete semigroup in two generators ((6.0(iv))). (v) If S is an open subsemigroup of G, then £(U(S)) ,-E 0. PROOF. It is proved in (6.30) that (iii) implies (i), and in (6.6) that (i) implies (ii). Further, if G is a solvable Lie group, then (6.38(ii)) yields that (v) implies (iii). So to establish the theorem it is sufficient to show that (ii) implies (iv), (iv) implies (v), and (iv) implies (iii). Suppose that (ii) is true, and that G contains a free, uniformly discrete semigroup T in two generators a, b. Let U E F, (G) be such that sU f1 tU = 0 whenever s, t E T with s # t. Since G is connected and U is compact, there exists k E P such that aU U bU C Uk. For each n, let
Tn = {SIS2 sn: si E {a,b},1 A(TnU) = 2n)(U), so that
lIm \(Un),/n > 21/k > 1, n-oo thus contradicting (ii). So (ii) implies (iv).
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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].
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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}.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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,
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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) <_ 11xcn11211f(n1112
(1)
< A(Cn)1/211r2(f(n-1))1111f112 to
Taking nth roots, letting
oo, and using the exponential boundedness of G,
we obtain
vif)=n M11fln1111/n
< lim
7
f
{[()(C'))I/n)1/2[1i'a(f{n-1))111fn11f)11/n}
= v(1r2(f)) The reverse inequality follows since ))7r211 < 1, and we have v(f) = v(7r2(f)).
Let A f be the (commutative *-) subalgebra of LI (G) generated by f. Let X f be the closure of A f in LI (G), and Yf the closure of 7r2 (X f) in B (L2 (G) ).
Let F be a multiplicative linear functional on X f, and define P: 7r2 (X f) -; C by F'(72(g)) = F(g) (g E X f). (The map F is well defined since vr2 is a monomorphism.) If g, h are hermitian elements of Af, then, since A f is commutative,
1F'(7r2(9 + ih))1 < IF'(ar2(g))1 + 1F'(72(h))) = 1F(g)1 + )F(h)1 < v(g) + v(h)
=v(r2(g))+v(r2(h)) <2v(7r2(g+ih)), and so F' is continuous on 7r2 (A f ). Extending F' by continuity to Yf, it follows that SpX1 (f) \ {0} C SpY1 (are (f )) \ {0}. The reverse conclusion is obvious, and since the preceding two spectra equal their own boundaries, we have [Ri, (1.6.12))
SpX1(f)U{0}=Sp(f)U{0},
SPY, (f)U{0)=Sp(1r2(f))U{0}
so that {0} U Sp(f) = {0} U Sp(ar2 (f )) as required.)
A much more general result than this is proved by Jenkins 111).
(6.50) Polynomial growth and the symmetry of L1(G). Our main objective in this section is to establish the symmetry result (v) of Ludwig. (Losert [S7) generalises (v).) The following result (i) is valid for all connected groups of polynomial growth.
However, we shall be content to deal only with the simply connected, Lie case.
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255
(i) (Guivarc'h [1]). Let G be a simply connected, Lie group of polynomial growth. Then there exists a series of closed, normal subgroups G; of G such that for each k > 1 and x E Gk/Gk_1,
there exists C E 9(Gk/Gk_1) with x E C and yCy'I C C for ally E G/Gk_1. [Let S be the radical of G, and s, g the Lie algebras of S and G respectively. Then S is of Type R and using Lie's Theorem (B25) inductively, we can find real linear functions -yq : s ---> iR (15 i:5 r) and distinct subspaces Wk (0< k < r) of
sc such that
{0} = W0CWIC...CW,=sc and, for each k >_ 1, WklWk-1 = {Z E sc/Wk_1: (adX)SC/ k_, (Z) = yk(X)Z for all X E s}. We claim that Wk is an ideal in gc. The proof will be given for k = 1, the general case proceeding by induction and involving a very similar argument.
Let Y E g, Z E WI, and X E s. Noting that 71(X)[Y,ZJ = [Y,71(X)ZJ = [[Z,X],Y], we have, using Jacobi's identity, '71([X, YJ)Z - [X, [1', Z]] +'71(X) [1', Z) = [[X, YJ, ZJ + [[Y, ZJ, XI + [[Z, XJ, Y) = 0
Now [X, YJ belongs to the nil-radical of g (B37), so that T = ad([X, YJ) E L(sc) is nilpotent. Since 71([X,YJ) E Sp(T), it follows that 7i{[X,YJ) = 0, so that by (*), ad X ([Y, Zj) = 71(X) [Y, Z).
Thus [Y, Zj E W1 and W1 is an ideal in 9c as required. Since exp(s) is dense in S (Appendix E), it follows that, for each s E S. Wk
is (Ads)-invariant, and (Ad s)wk/wk_, is of the form zI, where z E T and I is the identity of L(Wk/Wk_1). Let Vk be the subspace of s consisting of the "real parts" of elements of Wk and Yk = VklVk_1. Then Vk is an ideal in g and
{0}=VoCV1 We now claim that each Lie algebra Yk is abelian. For let Xy (1 < j < 4) in Vk be such that X1 + iX2, X3 + iX4 E Wk. Let 7k (X1) = ia, 7k (X3) = ib (a,b E R). If a = 0, then [X1,X3 + iX4] E Wk-1 so that [X1,X3J E Vk_1. Suppose that a # 0. Then i[X1,X2J = [X1,X1 +iX2] E ia(X1 + iX2) + Wk_1, whence, on taking real parts, X2 E Vk_1. Thus [X3,X1j E [X3,X1 +iX2j + iVk_1 C ib(X1 +iX2) +Wk_1 +iVk_1, whence [X3i X1j E Vk_1. So in all cases, [X1,X3) E Vk_1i and Yk = Vk/Vk_1 is abelian. Now as {(Ads)wklwk_, : s E S} C TI and G/S is compact ((6.32), (6.39)), it follows that the group {(Adu)wk/wk_, : u E G} is contained in a compact subgroup of GL(Wk/Wk_1). It readily follows that H , where Hk = { (Ad u)Yy : u E G}, is compact. Let p ilk be a norm on Yk, relative to which the elements of Hk are isometries [HR1, (22.23)J.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
256
Let Gk be the connected, normal. Lie subgroup of G with Vk as Lie algebra. Then Gk is closed in G since G is simply connected (B17). Let n = r + 1 and G. = G. The result of (i) is obviously true when k = n, since Gn/Gn_1 = G/S is compact. Suppose that k < n. Then Yk is the Lie algebra of Gk/Gk-1, and since Yk is abelian, exp(Yk) _ Gk/Gk-1. Let x E Gk/Gk-1 and X E Yk be such that expX = x. Let U = (Y E Yk: [[Y[[k <- [[X[[k + 1},
and C=exp(U). Then CE?(Gk/Gk-1),xE C, and ifuEGandy=uGk_I, then (B12)
yCy-1 = exp((Adu)yk(U)) c exp(U) = C.] (ii) (Jenkins [101). Let e > 0, let G be an exponentially bounded, locally compact group, C E F(G), and let H be the closed subgroup of C generated by C.
Then there exists p E Li(G) such that p(t) > 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`,
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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).
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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) <- Ilf III 2 (f if (x) Ils(x * 9, f * g)12 dA(x) }
1/2
\ 1/2
S lif lli 2 (f If (x)l s(x * g, x * g)s(f * 9, f * 9) dA(x) r
Dividing both sides of the preceding inequality by (s(f * g, f * g))1/2 (when the latter is nonzero!) and squaring, it follows that (6)
s(f * 9,f * 9) :5 1lfIII f if(x)I(sx)(9,9)dh(x).
Similarly, one can show that (7)
s(f * 9, f * g) <-119111
f [9(x)I(xs)(f, f) dA(x). G
Further, if u is a right modular unit for I and s E .9'i, then, using (6),
s(f,f)=s(f *u,f *u)+s(f - f *u,f *u)+s(f,f - f*u) =s(f *u,f *u) (8)
Ilf[I' f If(x)i(sx)(u,u)dA(x) <-
Ilf[I'sup(sx)(u,u).
zEG
Suppose that 9r 0 0. Let L = GIN. By defining s(xN) (= s(Nx)) to be sx (s E .9r"), we see that 9N is a right L-set. Let K E F(H/N) and e > 0. We show that there exists 9 E gN such that (a) 9(u, u) > 2,
(b) hill 51, (c) 1(3k-s)(f,f)l:5 Ellf112 (kEK, f EL1(G)). If z E H/N, then, by hypothesis, there exists C E %, (H/N) such that z E C and ICI-1 C C for all t E L. Replacing C by C2 if necessary, we can suppose that z E Co. A simple compactness argument now shows that there exists U E 9,(H/AT) with K C U and lUl'1 C U (t E L). Let V be the subgroup of H/N generated by U. Then V is an open and closed, normal subgroup of L. Since H/N is an (FC]- group, it is of polynomial growth (Problem 6-7) and so exponentially bounded. Hence by (ii), there exists p E LI(V) such that p(v) > 0 for all v E V and (9)
Ip(u'v) - p(v) I < 6p(V)
(u' E U, V E V).
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LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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 <_ rtCfllI
for allkEK,f ELI(G)}. compact in the topology of pointwise convergence on
Further AK,E is LI (G) x LI (G). Allowing K to vary over 9'(H/N) and e over (0, oo), the sets AK,E have the finite intersection property. So we can find r' E fl{AK,E : K E
T(HIN), e > 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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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:
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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, Sp(a) C 1-1, 1]. Let Tn be the manic Cbebyshev polynomial of degree n; so T,(x) = 2-n+1 cos(n cos- 1 (x))
(-1 < x < 1).
Since IT,,(x)I < 2-n+1 it follows that v(Tn(a)) < 2-n+1. Now for fixed n, (Tn(x))m is manic of degree mn, so that cap(a) :5
lm
I](TT(a))m]li/mn
=
Hence cap(a) < 1/2, giving a contradiction.] (iii) The Banach '-algebra 11(G) is not symmetric.
(v(Tn(a)))I/n < 2-I+I/n
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
264
[From (ii), it is sufficient to construct f E 11(G) with f - = f , II f II 1 = 1, and
cap(f)>1/2. Leta,b>Oandr,NEpbesuch that r
w
f =aExi+bE2-'xj+N
(2)
i=1
j=1
Using (i)(a), f- = f and 11f ill = Na + b = I. Define recursively a sequence {Yn} of subsets of W by
Yi = {(xi,, ... , x,,) : 1 < ij < N, ij :0 ik if j # k}, Yn+1 = {w3 xn+Nw2: w1i w2 E Yn}.
Let Zn = {aw: w E Yn}. We claim that jZnj = jY, I. By (i)(h), this is true for n = 1. Suppose that m > 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).
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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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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.
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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
LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
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:
SIZES OF SETS OF INVARIANT MEANS
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.
SIZES OF SETS OF INVARIANT MEANS
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<, go., is well defined and satisfies all the required conditions except possibly that of right uniform continuity. We now prove that ¢p E Ur (G). This will follow once we have shown (1)
Il0pu-0a11S11fu-fII
(uEV)
since f E QG) C Ur(G). Let u E V and suppose that t E G is such that 0p(ut) - Op (t) # 0. If ¢s(ut) # 0, then ut E V2K.,xp., for some unique 7, giving t E u-'V2K1xp., C UK.,xp.,. So ut, t E UK.,xp1. Similarly, if 0,9 (t) 56 0, then again, ut and t are in UK.,xp1 for some unique 7. So in either case, l0p(ut) - 0p(t)l = Igp1r(ut) - gp1(t)I = Id(ut, K,.xp,.) - d(t, K.,xp.,)I
< d(ut, t) = Ilfu - f1I,
SIZES OF SETS OF INVARIANT MEANS
273
establishing (1) and the first assertion of the proposition. For K 0 0, K C a, let ¢K = E EK 019 If u E V, t E G, 8 E K, and either bp(ut) 54 0, Op(t) 0 0, then gsp'(ut) = 0 = gsp'(t) for Q' # 0, and it follows that IIOKU - 40KII = max(EK IIOOU - 0011. From (1), #K E Ur(G). 0 Now ((2.22)) the action (x,F) -, xF of G on Ur(G)' is jointly continuous for the weak* topology. The maximal ideal space X of U,(G) is, of course, a subset
of U, (G)' and is clearly (left) invariant for G. It is routine that G is a dense subset of X in the obvious way. Recall that m = Ial.
(7.4) PROPOSITION. There exists a disjoint family {Cp: p E P} of invariant, nonempty, compact subsets of X, where IPI = 2". PROOF. By (7.1), we can find a family {N.7: -Y E r} of subsets of a, where Iri = 21, such that if "y1, ... , -y'm are distinct elements of r and ei E {1, c}, then
ni `I N.' 4 0. Let P = {I, c}r so that. IPI = 22'". Let Zp, ¢p be as in (7.3), and for each y E r, let E.7 = U{Zp: 6 E N.7}. For A C G let A (or A') be the closure of A in X and A' _ (UOE,, Zp) A. For p E P, let
Cp = n{(xE?(,))^: x E G, 7 E r}. Clearly Cp is compact. We show that Cp is invariant and nonempty. If u E Cp, xo E C, then for y E G, -y E r, we have xou E xo[(xoIyE7(T))^J = (yE7(7))^
since the map v -+ xpv is a homeomorphism on X. So xou E Cp and Cp is invariant. Now let xI, ... , x,,, E G, yi, ... , -y,,,, E r, where the -y,'s are distinct.
Let E nL I g(-O. If p(-Ii) = 1, then Z,,. C EyIif P(-,) = c, then, since the Zp's are disjoint ((7.3)), we have Z,9,, C (UpEa Zp) - E.I; So for each i. Since ZsQ is left thick, there exists x' E G L = n? I x,,Zpo C
such that xi Ix' E Zpo (I < i S n) and so x' E L. So nsn 1 D L^ # 0 and the sets {(xE.p,(1))^: x E G, -y E r} have the finite intersection property. Since X is compact, it follows that Cp 0 0. It remains to show that if p, q E {1, c}r with p # q, then Cp fl C4 = 0. To prove this, let 'yo E r be such that p(^yo) j4 q(-yo). Without loss of generality, we can suppose that p(-yo) = 1, q(.io) = c. Then C. C E.',., Cq C (E 0)^. Let Q E N.ro. Since E7o = Up,15rvTO Zpy, we see that Op(E0) = {0}. Let Qi = ON,. as in (7.3). Then ¢(E.,,,) = {1}, ¢(EE,) = (0) so that (E4,,) = {1}, ¢((E7 o)^) = {0}; where is the Gelfand transform of ¢. So is I on Cp and 0 on CQ giving Cp fl C. = 0 as required. 0
As in (2.25), for m E M(U,(G)), let rig E M(X) be given by tn(¢) = m(O) (0 E U1(G)). Then the map m - fn is an affine weak* continuous bijection from 1st(Ur(G)) onto PM(X). It is easy to check that m E.C(Ur(G)) if and only if r"n m({b) for all 0 E U,(G). Recall that S(fc) is the is invariant, that is, if support of a measure ,u E M(X).
(7.5) PROPOSITION. There exists a subset E of .t(G) and a disjoint family of compact, left invariant subsets of X such that
SIZES OF SETS OF INVARIANT MEANS
274 22°'
= IFI; (i) IEI = (ii) S(m) is contained in a member of F for each m E E. PROOF. Let £' be the set of Cp's of (7.4). For each p, let Kp = PM(Cp). As in the proof of (2.25), Kp can be regarded as a subset of M(X) = C(X)' and is an affine left G-set in the standard way. From the comments preceding (7.4), the G-action is jointly continuous, and since G is amenable, there exists a G-fixed point up E Kp. Let mp E M(Ur(G)) be given by mp = µp. Then mp E Z(Ur(G)) = .Ct(Ur(G)) by (1.8). Identifying .Z;t(UU(G)) with .t(G) ((1.10)), we take E = {mp : p E P}. We now determine the cardinality of .Ct(G).
(7.6) THEOREM. IC(G)I =
22"
(G noncompact).
PROOF. From (7.5), IZt(G)l >_ 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<_ IXI, then IC(X)I = 22'"'. An interesting open question is what is the cardinal IC(X)l when
SIZES OF SETS OF INVARIANT MEANS
275
IGj > 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
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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
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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 <_ 1/2.
(2)
Then £(B) - £ (B) is not empty when B = C(G) or when B = L,p(G). PROOF. Let E be the interior of the set {x e G: ¢(x) = 1). Let x1, ... , xn E G and choose V so that (1) is satisfied. Then
h(flxg1E)7.A(V)>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) <
11gll-00
? Since G is not discrete, the family V is an open cover for G. Note that: given x1 t ... , x,, E G, there exists W E V such that xi E W (2) (1 < i < n). We first cut down the size of S/, replacing it by a subset ?i.
' ).
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Choose a family . of compact subsets of G with U 9 = G, where the cardinal number m of 9' is smallest possible. Since G is not compact, the cardinal m is infinite. Let n E P. If Zi E Sr (1 < i < n), then, using (2), the compact set ZI x x Z can be covered by a finite number of sets of the form W(') = W x . . x W (n copies), where W E St. Since the sets ZI x . x Zn cover GI and as m is infinite, we can find a family .i &n of sets of the form W(n) (W E d) such t h a t U s a f e = G" and t. 1 = m.
Now define .9" = {W E d: W(n) E sfn for some n}. Then (2) is satisfied with d replaced by 9, and !.9'E = as. Well-ordering 9', we can write 0 _ {W,6:,8 E a}, where re is the first ordinal such that at = as, Let 8 E a. Using (1), the regularity of A, and Urysohn's Lemma, there exists ¢Pp E C(G) with compact support such that d i p_ > 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<_ 2
We now translate the functions 0,9 so that the resulting functions have disjoint support. To this end, let Sp be the support of qrp, and let U be a relatively compact, open neighbourhood of e that contains the support of g. Using transfinite
recursion, we now construct, for each /3, an element ap e G such that the sets {U-'US,Ba5: 6 E a) are disjoint.
Let f3o E a and suppose that a.1 has been defined for ^y < Q0 such that the family {U-' US.,a, : -y < (30) is disjoint. If there is no x E C for which U-'USpox fl U'IUS-1a., = 0 for all - < io, then the family .7 = ((Sa IU''UU-'US.,a.,)-: -y < f3o) is a covering of G by compact sets, and since 191 < If8ol < tat = as, the definition of m is contradicted. So the existence of the desired element apo is assured, and by recursion, the required elements ap (f3 E a) exist. The function 0 that will satisfy (i) and (ii) of (7.12) is defined by (5)
0=Fa-'(6p. aEa
Since the support of each ao'0p is contained in U-IUSpap and since
U-'US0apf)U''US.,a,=0 iff3O^y,
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279
it follows that 0 is well defined, and is bounded on G. We now prove that ' is continuous on G. Let x E G. If U-IUx does not intersect any of the sets Soap, then 0 is 0 on U-'Ux and so is trivially continuous at x. If, however, U'lUx fl Spap 0- 0 for some p, then x E U'1USpap, and 0 coincides with the continuous function a010# on U'1USpap. We now check (i) of (7.12). Let x1,. .. , xn E G. Recalling that.0 satisfies (2), we can find 6 E a such that xi E Wp (1 < i < n). Let V = In 1 x= 1Ws)ap. Then V is an open neighbourhood of ap, and using (3), we have Wp C So, and q5(xiV) c 4'(Wpap) = ap'4'p(Wpap) = Oe(Wp) _ {1} as required.
To prove (ii) of (7.12), let x E G. If x E USpap for some j3, then U''x C U''USpap, and as g vanishes outside U, I
(4'i }{x)I = I fu 4'(t-'x)g(t) da(t) ' =
using (4). If, however, x 0 USpap for ail f3, then for all t E U,
EaFi00(t-'x) = 0. A
So in this case, 4µ-(x) = 0. It follows that 0,u- o < 1/2, and (ii) of (7.12) is established with i, = ,u-. 0 (7.14) THEOREM. Let G be a locally compact group that is amenable as a discrete group. Then L(C(G)) = Lt(C(G)) if and only if G is either compact or discrete. Further, if G is noncompact and nondiscrete, then £(G) }i Lt(G). PROOF. The first assertion follows using (7.12) and (7.13). (It is obvious that L(C(G)) = Z (C(G)) if G is compact or discrete.) Now suppose that G is noncompact and nondiscrete, and find, by the first assertion of the theorem, an element mo E L(C(G)) -r Lt(C(G)). By Problem 1-13, we can find no E L(G) with nolc(c) = mo. Clearly no E L(G) - Lt(G). O
Note that (7.14) does not give any information about L(G) - Lt(G) when G is compact. This will be remedied later ((7.21)). (Of course, when G is discrete, then L(G) = Lt(G).) Our next concern is to obtain an estimate for {L(G) -r Lt(G)I. To this end, the following technical proposition gives, for a large class of groups, a family
{E,: -y E t} of subsets of G that will produce elements of L(G) - Lt(G). The proposition uses Mycielski's Theorem. References for the results (7.15)-(7.21) are Rosenblatt 151, (7), (8), 191 and Wells (1).
(7.15) PROPOSITION. Let G be a nondiscrete, a-compact, locally compact metric group. Then there exists a family {E.,: ry E t} of measurable subsets of G, each of which has finite A-measure, such that (i) and (ii) below hold: (i) 1f71 = c, the cardinality of the continuum;
280
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(ii) given n, rn E P and xij, yij (1 < i < n, 1 < j < m) in G such that for each i, the points xil,... , xim, Vii, . . , ydm are distinct, then we have a{{njxijE,, n yijE1,j} n V) # 0
{1)
i,j
for everylnonvoid, open subset V of G and for every n-tuple (ryl, with ^y,, T 'y12 for i1
.
. , in) E rn
# i2.
PROOF. We derive the result by applying Mycielski's Theorem (MT) (Appendix D). The metric space X of the application is the set {E E 4'(C): .\(E) < oo}, where, as usual, El and E2 are identified if A(E1 A E2) = 0. The metric d on X is defined by d(E1, E2) = .1(El 0 E2). Then jDS, 11I.7.1) (X, d) is a complete, separable metric space. Further, since G is not discrete and .1 is regular, the space X contains elements of arbitrarily small measure, and as a result, (X, d) has no isolated points. So (X, d) satisfies the conditions of MT. Since G is locally compact, a-compact, and metric, it is separable, and so we
can find a (countable) family'' = {C,: r E P} of compact subsets of G such that {CO: r E P} is a base for the topology of G and C, = (CO)-. (So we need only consider in (1) the case when V is one of the Cr.) Let V,, be the set of all 2p-tuples (K1,... , K2q), where Ki n Kj = 0 if i # j
and each Ki E F. Let
. ' _ U5(). A typical element S of S° is of the
form (A,,. .., Aq) with A, = (Ail, ..., Aip, Bil, ...1 Bip) E Ap. Note that S° is countable. For such an S and r E P let U(r, s) c X9 be the set of all q-tuples (El, ... , Eq) with the following property: there exist 2p-tuples (xil, xi2, ... , xtp, yil, ... , yip) E Ail x ... x (2) that A{{njxijEi n y,jE; j} n Cr) = 0.
For the application of MT we will take the R, to be the set U(r, S), (r, p > 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 < i < q, 1 < j < p) be be elements of G with ail,... , lip, yil, .... yip distinct for each i. Let distinct in r and r E P. We can find S' =(A 'I, ... , A'q) E .9' such that xi, E A;3 , yij E since {C°: r > 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).
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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°.
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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
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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
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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
SIZES OF SETS OF INVARIANT MEANS
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
A,s=s,nsiIT Suppose that Aij n A,, 0. Since {S1,. .. , S,,) is disjoint, we have i = k, and since si 'Tins, ITI = s, i (T, n Tt) 96 O and {TI , ... , Tm } is disjoint, we have j = 1. Further, ifs E S, then s E Si, for some i', and si,s E Ti, for some j' (since si,Si, c A C T), so that s E A,'j,. It follows that {A,,r: 1 < i < n, 1 < j < m} is a partition of S, and so m
m < Qt?siA,j S I=.j
UtjTj
j=1
286
SIZES OF SETS OF INVARIANT MEANS
It follows that m(T) = m = ITl and T is a.l.c. (ii) Suppose that m E'C(S). Since Uz 1 s= 1A = S and m(S) > 0, we have
m(A) > 0 and so m(T) > 0. By Problem 1-21, mT = (mIT)/m(T) belongs to '(T) We now show that the map m -+ mT is one-to-one from E(S) into e(T). Suppose that m, n E L(S) is such that mT = nT. Let p E 3, (S) be given by P = n(T)m - m(T)n. Then pIT = 0. By (2.31), IpI, evaluated in the L-space l,o(S)'"="M(flS), belongs to 31(S). We claim that IpI IT = 0. To see this, observe that p(l,,,(T)) = {0}. One can argue, using L-space theory (Sch, p. 721, that for 0 > 0 in 1,,. (T), Ipl(O) = sup{Ip(')l: -0 E 100(S), 0:5101 S 0} = 0,
so that, as required, lpI IT = 0. (Alternatively, one can argue directly in terms of measures on ,QS.) If IPI ,E 0, then IpI is a positive multiple of some m' E .C(S), and we know that
m' (T) > 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.
SIZES OF SETS OF INVARIANT MEANS
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 s`I(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 <_ Q <'61; (iv) rl{s-I(A ): s E F0) 0 whenever e :5,6.
Suppose that -y E a and that sets &R have been constructed so that (i)-(iv) are satisfied for 3, 01 < 't. Let
K=U{Ao:e5/3<^y}. If 7 is finite, then K is finite by (ii), so that (K( < in. If -y is infinite, then, again using (ii), (K( < (y(3 = (7( < m. So in both cases, (K( < m. Let F.. _ {sI...... n}. We have to construct the AE.
288
SIZES OF SETS OF INVARIANT MEANS
To this end, we first construct, by transfinite recursion, a family {r* : e < y}
of subsets of S - K such that (a) r7 = {s71, ... , s",, } for some sE
(b) rE n r,,= 0 if e
rl;
(c) for each e, there exists sE E S with sisE = s, i (1 < i < n). Suppose that b < -y and that sets {r,,: e < 6} have been constructed so that (a), (b), and (c) are valid for e, ty < 6. Let E = U{r7 : s < 6} and L = K U E. Then JLJ < m since both jEJ < m, jKJ < m, and m is infinite. Now define recursively a disjoint family {S1,. .. , S"} of subsets of S as follows:
SI = si 1(L) and Si = [s,-'(L)] - (U'-i S,) for i > 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<
(e<7),
oy=r, We now check that the conditions (i)-(iv) are satisfied for ,6,,81 < y. Clearly, we need only check (i), (ii) and (iv) when fj = -y. Suppose that c,?? < y with e 01i. We can assume that e < t1. If e S flu < y, '! «I < y, and p' = max{$o, Qi }, then by (iii) and (i), Af- nOn' C AOE 'fl' = 0. Noting that rE Urn C X - K, we have, using (b), 77
©EnAl_ [(U{t1o
77 < flu < y})
urn] = 0.
So (i) follows for ji = y. When we note that rF is finite and that (c) holds, it readily follows that (ii) and (iv) hold for fj = y. This completes the construction of the sets O For each e E a, let
9E = U{AA: 6:50, '8 E a}. Since d'9 n &0, = 0 if e 0 ri, it follows that {8E : e E a} is a disjoint family. It remains to show that each 8E is left thick.
Let F E -9-(S) and A={flEc:FCFp}. Then JAI=m=jSJ. Since lei < jSj, there exists fi E A with e S p. Then by (iv),
n{s-18E: s E FF}
s E Fp} ¢ 0.
Let z E n{s-18E: s E Fe). Then Fx c 8E, and so eE is left thick in S. We now obtain the precise cardinality of £(S) when S is left amenable with m infinite.
SIZES OF SETS OF INVARIANT MEANS
289
(7.26) THEOREM. Let S be a left amenable semigroup with m infinite. Then 22' I£(S)I =
PROOF. Let A and T be as in (7.24). Then T is a.l.c. with m(T) = m and so by (7.25), there exists a disjoint family {eE : e E A} of left thick subsets of T. If F E .9 (S), then, as T is left thick in S, there exists s r: S with Fs C T, and since Fs E .$(T) and AE is left thick in T, we can find t r= T with Fst C ee. Thus 9E is also left thick in S. Now apply an argument similar to that in the proof of (7.4). We take A = a and form .,'s, P = {1, c}', and E., = U{ec : e r= N.,). We do have to be careful with the Cp's, since multiplication on S is not necessarily left cancellative. We define ={[x-1(E.Pf7))J^:
Cp
x E S}.
(Here, A^ is the closure of A C S in QS.) One easily checks that the Cp's form a disjoint family of nonempty, compact, invariant subsets of /3S. Now apply Day's
Fixed-Point Theorem to give that I£(S)I > 2". The reverse inequality follows from (7.23(ii)): IC(S)I <_ I£(T)I < Il0,(T)`I =
22"T1
=
22"
(cf. Problem 7-1). C
It readily follows that in the situation of the above theorem, dim3i(S) = 22'. It remains to determine I£(S)I and dim3l(S) when S is left amenable and m is finite. In this case, 31(S) is finite-dimensional so that dim3r(S) is a more appropriate cardinal than I£(S)I. (7.27) THEOREM. Let S be a left amenable semigroup with m finite. Then S contains a finite ideal, and if A(S) is the set of finite, left ideal groups in S, then
dim,1t(S) = IA(S)I < oo.
PROOF. Let {S,. .. , S,,} be a partition of S and sI ..... s E S be such that A = U"1 siS= is finite. Let m E £(S). Then m({a}) > 0 for some a E A, and since m({sa}) > m({a}) for a l s E S and m(S) < oo, we have that Sa is a finite left ideal of S. By (1.20), S contains a finite left ideal group L. Further ML belongs to £(S), where ML (E) = IE n LI/ILI
(E C S).
Since mL(A) > 0, L n A 54 0. Since A is finite, the set A(S) of finite, left ideal
groups in S is finite. Let L1,...,L,,, be the elements of A(S). Clearly, L1 is a minimal left ideal of S, and every minimal left ideal of S is of the form L1x (x E S). Now each L1x is also a group, where, if el is the identity of LI, then the identity of L1x is (xel)-Ix (with (xe1)-I the inverse of xe1 in L1). It follows that the kernel K of S ((1.16)) is the finite ideal UmI Li. Now every n E £(S)
can be regarded as a mean on K, since n(K) = 1. Noting that L, f1 Lf = 0 if i j, we readily see that
n=
n(Li)mL, i=I
SIZES OF SETS OF INVARIANT MEANS
290
Since .C(S) spans 31(S) (cf. (2.2)) and {my; : 1 < i < m} is a linearly independent
subset of 3t (S), we see that dim3,(S) = in.
0
References Banach [1), Chou [2], (41, (9), Day (4), Granirer [1), [2), 13), [4), (6), [13), [15], Greenleaf (21, Johnson [4), Klawe 11], [3), Liu and van Rooij (1], Luthar [1], [2), Rosenblatt [5], [7], 181, [91, Rudin 12], [41, wells [1), and Yang (Si].
Further Results (7.28) Exposed points and left invariant means. Let X be a Banach space and K a convex subset of X'. A point ao of K is said to be exposed (cf. Klee (2)) if there exists g E X such that
Rea(i;)
theorems. For prototypes see Fairchild [1, 21, Chou 151, and Granirer (6). As usual, E is either a locally compact group or a discrete semigroup. (i) Let E be noncompact, u-compact, left amenable, and with no finite, left ideals. Let be a sequence in LS(E) and H a compact, G6-subset of '(E). Let
S={m E.L't(E):S(rh)CH, m(On)=0 for alln>1}. Then Exp. = 0, and if .fi $ 0, then . is not norm separable. [Suppose that Exp. ¢ 0 and let m E Exp.. Find 00 E L00(r) with Ren(Oo) < Re m(r¢o)
(1}
(n E S - {m}).
Since H is a G6-set, we can find a sequence {U,} of open subsets of 4)(r) such that U,-,+1 C U,, and H= f °°_1 U,,. Now as (P(E) is the maximal ideal space of an L.-space, it is zero-dimensional. So we can suppose that each U is open and closed. Let -0, E L,, (E) be such that 1Gn = Xun
Let {K,,} be an increasing sequence in F(E) with U' 1 Kn = E. From the proof of (4.4) in the group case, we have: {#)
given 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.);
SIZES OF SETS OF INVARIANT MEANS
291
(b) I(fn - m)(4r)I < n 1 (1 < r < n); (c) I(fn - m)(0o)I < n-1, (d) Ifn(4P,)-1I - I. Further, from (d), p(Un) = p(lp,,) = 1, and since Un = Un, it follows that S(p) C n 1 Un = H. So p E A From (c), p(4o) = m(ho), and using (1), p = m. Hence fn m weak*. It follows that (fn I is a weak Cauchy sequence in LI (E), and since the latter space is weakly sequentially complete IDS, IV.8.6), m = f for some f E P(E). If E is a discrete semigroup, then, by (1.20), there exists a finite, left ideal in E, and a contradiction results. If, on the other hand, E is a locally compact group, then m can be regarded as a left, Haar measure of total mass I on E, and this implies that G is compact, so that a contradiction results. Thus Exp St = 0 as required. Now suppose that A:0 0 is norm separable, and let (m,,: n E P} be a normdense subset of £ Let m E A. Then for each n, we can find a sequence {4 ,k} in
with 110n,klloo = 1 and k limo 1(mn -
(2)
117n. - mI1.
Now find a sequence { fn} in P(E) satisfying (a), (b), (d), and the condition (e) I(fn - m)(0r,s)1 < n-1 (1 < r, s < n). Let {fats)} be a subnet of {J,2} weak* convergent to some p. As earlier, p E .A From (e), m(¢,,,) = p(>pr,s) for all r, s. Using (2), for each n, I1mn - p1I 2! elixn K-1. - p)(0n,s)l =11mn - m[I. 00
So any sequence in the set R = {mn: n E P} that converges in norm to p also converges in norm to m. The norm density of R in A now implies that in = p. The rest of the argument proceeds as in the above proof that Exp. = 0.1 We now look at the case where E is a discrete semigroup and the requirement that H be a Gs-set is removed. The conditions on the semigroup are strengthened.
(ii) Let S be a countable, left amenable semigroup that is not n-ELA for any n ((2.30)). Let {(b,, } be a sequence in lo,(S), H a compact subset of 8S, and
iZ = {m E C(S): S(fn) C H, m(4n) = 0 for all n > 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)
(n E 6
{m}).
Replacing H by S(m), we can suppose that SH C H ((2.25(ii))). Let {Kn} be as in the proof of (i) and r E P. Find g, E P(S) such that IIx * g, - g,II1 < r-1 (x E Kr). Let {a6} be a net in coH (C M(3S)) with as -+ in weak*. Now each g, is a convex combination of points s E S. Using (2.25(i)) and the
292
SIZES OF SETS OF INVARIANT MEANS
fact that m E s(S), we have limb (9,.a6) = m and IIx(9ra6) - 9ra6II < r-1 (z E Kr). By choosing suitable elements grab, we can find a sequence in co H such that (a') IIx$n -13,]] < n-I (x E K,,), (b') I(N, - m)(0,) I < n-1 (0 < r < n). A version of the argument in (i) now gives that Q -» m weak* in (Note that since S(;8,,) C H, it follows that if m is a weak* cluster point of in then S(tia) C H.) A theorem of Grothendieck ISch, p. 1311 entails that Qn -, m weakly in 1,.(S)'. Now if al, ... , a, are distinct in H and Ai,
. , A, E C, then, identifying 1,o(S) with C($S) and using Tietze's Extension Theorem, we obtain
A4at
liE
=, 1A
1
so that 11(H) is isometrically embedded in Since $, E i (H) and 11 (H) is weakly sequentially complete, we have m E 11(H). So m is not continuous, and by (2.37), S is n-ELA for some n. This is a contradiction. To prove that Q is not norm separable (if it is nonempty), the reader is invited to combine the above construction with the corresponding proof given in (i).] A natural question (raised in Granirer 1131) involves a locally compact group analogue of (ii). Talagrand 161 proves the following result. Let G be an infinite, locally compact group that is amenable as a discrete group, and let F be a nonvoid,
compact, left invariant subset of 4,(G). Then the set {m E .C(G): S(th) C F}
has cardinality not less than 2' and is not norm separable. A substantial amount of information is known about the extreme points of sets of invariant means. References are Chou [1], Nillsen [1], [2], and Talagrand [1), [4), [7). See Yang [S1] for recent work on exposed points.
(7.29) Nonmeasurability of some
(Wells 11],
Rudin [4]). Let C be a locally compact group and * E p E 4)(G). Let 0 = p0. Then ¢ E and 11011 = suP.,EG [0(x)[ < II''[Ioo In the striking words of Rudin: "What else can one say about 0?... the answer is: Nothing!" The Paper by Wells deals with the case of the circle group. Rudin proves that if G is infinite and metric, then there exists 0 E L,o(G) with 1[,PII = I. and such that every 0 E lo,(G) with Im(x)I < 1 for all x E G is of the form pO for some p E '(G). In particular, like C(G) ((2.33)); is not, in general, left introverted. We now prove Rudin's result when C is metric, nondiscrete, and a-compact. The proof below is due to 3. M. Rosenblatt. Let G be a nondiscrete, a-compact, locally compact metric group. Then there E L,,,(G) with I]'GIIoo =1 such that
exists
{PO: p E 41(G)} = {O E
1[4[I. <- 11.
SIZES OF SETS OF INVARIANT MEANS
293
[Let {Ey: y E I'} be the family of subsets of G given by (7.15). Let (y,) be a sequence in r with y, 0 yn if n 0 m. Let An = Ey,,,, Bn = Ey,n+,. Define li1,iP2 E L. (G) by setting co
Co
2-nXAn,
01 =
02 =
n=1
L
2'°XBn.
n=1
= 01 exp(21ri'02). Then 0 E L,o(G). For each N, X(nn 1 An) > 0 (by (1) of (7.15)), so that if x E An (1 < n < N), then
Let
N
1 > 111100 > 01(x) > E 2-n. n=1
It follows that 11011Co = 1.
Let ¢ E 1.(G) with 110110 < 1. Let g'1 = 101, and find 02: G --> (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
(N E P).
n=0
(Indeed, we can take EN = {z E G: (¢1 EO = 0). Clearly
n=o
2-nXEn)(x) > 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
SIZES OF SETS OF INVARIANT MEANS
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 <- II0(I.lliAub - µ6(!i - 0 using (3) of (4.15). It follows that given e > 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
SIZES OF SETS OF INVARIANT MEANS
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
SIZES OF SETS OF INVARIANT MEANS
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.)
SIZES OF SETS OF INVARIANT MEANS
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).
SIZES OF SETS OF INVARIANT MEANS
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 < i < k) such that e=e, = e, for all i, j, and G is a finite group.
APPENDIX A
Nilpotent, Solvable, and Semidirect Product Groups Let G be a group. A normal series of G is a finite sequence (1)
G=G1>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
The set of real-valued, differentiable functions on M is denoted by COO(M). The set COO(M) is an algebra over R under the usual pointwise operations. Since M is a separable, locally compact Hausdorff space, it is paracompact, that is, if .9 is an open cover of M, then there exists an open cover {Ua} with the following properties: (i) each U,, is contained in some member of 3; (ii) {U4} is locally finite in the sense that each m E M has a neighbourhood that intersects only finitely many of the sets UQ.
Further M admits partitions of unity in the sense that if {UO': a E A} is a locally finite, open cover of M with U; compact for all a, then there exists a set {&a: a E A} in C°O(M) such that (i) d has compact support contained in UQ (a E A); (ii) 0rra > 0 (a E A); (iii) LC,EA 0- = 1.
B2. Tangent spaces and vector fields. Let M be an n-dimensional manifold, and m E M. The tangent space TM,n at m is the real vector space of linear functionals F: COO(M) R such that (1)
F(zJi) =
¢(m)F(z/')
for all 0, b E COO (M).
If a < 0 < b in R and u: (a,b) -+ M is a C'-path with u(0) = m, then we can define an element F E TMm by setting
Fu (0) = dt MOM]
t=o
It can be shown that every element of TMm is of the form F for some C°°-path U.
Going into local coordinates at in, so that an open neighbourhood of m is identified with an open subset of R", we can define (c1/ax,),., E TM,n by (a/axi)m(0) = a¢(m)/ax,. It turns out that the set {((9/axi}rn: 1 < i < n} is a basis for the vector space TM,, so that the latter is n-dimensional.
APPENDIX B
$07
Let M and N be manifolds and 4): M -+ N a C°O-map. Then 4' can be "dif-
ferentiated" at a point m E M to give a linear map it. (m): TM. - 7Mep(m} where, for F E TM,,,,
1.(m)F(O) = F(O o l,)
(0 E Coo (N)).
The map . (m) is called the differential of fi at m.
If u: (a,b) --r M is as above, then F,, = u.(0)((d/dt)o). If to E (a,b), then we sometimes identify u. (to) with u.(to)((d/dt)t,). If P is also a manifold and
q b: M N and T : N -- P are CO°-maps, then the following "chain rule" is easily established: (1k o 4').(m) _ W:(cb(m)) 0 4b. (M).
A vector field on M is a derivation X on COO (M), that is, a map X : COO (M) COO (M) that is linear, and satisfies the equality
X(OO) = (X(46))'+O(X('))
(0,V) E Coo (M)).
Obviously, the set A(M) of vector fields on M is a linear subspace of L(COO(M)), the space of linear transformations on COO(M). Further, A(M) is a Lie algebra
in the sense that if ]X, Y] is defined to be XY - YX, then [X, Y) E A(M) whenever X, Y E A(M), and the following properties hold: (i) (X, X] = 0 (X E A(M)); (ii) (Jacobi's identity)
[X,[y,z]1+[Y,[Z,X11+[z,[X,Y11=0 for all X, Y, Z E A (M).
The space A(M) is a "C' (M)-module" under the map (o5, X) for (k E C' (M), X E A(M), we define ¢X E A(M) by
¢,X, where,
(0 E COO (M)).
(ox)(LG) = OWN,))
(Thus, (46X)(0) is the pointwise product of the functions 0 and X(tb).] R by For m E M, X E A(M), define Xm,: COO(M)
Xm(0) = (X0))(m) One readily checks that the map X - X,,, is a linear map from A(M) into Mm. The range of this map is actually the whole of for a local chart U, each map In terms of coordinate functions 8/8x,, where (8/8x4)(q5) = 8¢/8xt, belongs to A(U), and this (8/8xt)m is the same as the (8/8x4),,, introduced earlier. It is sometimes convenient to identify X E A(M) with the function m -- X,,, from M into UpEM M. In terms of local coordinates for a local chart (U, 4s), there exist functions 0, E C°°(U) such that
X. =
Sli(m) s=1
C3x,
}
(m E U). ^s
APPENDIX B
308
B3. Covering spaces and the fundamental group (Massey 11), Spanier [1], [SWJ). We will discuss the theory of covering spaces for manifolds only, although it applies more generally under suitable local connectedness hypotheses.
Let M be a connected manifold. A covering space of M is a pair (Pp), where P is a connected manifold and p: P --+ M is continuous, surjective, and satisfies the following: if m E M, there exists a neighbourhood U of m such that p I (U) is the union of a disjoint family of open subsets, each of which is mapped homeomorphically onto U by p. The map p is called a covering projection. Sometimes, P itself is referred to as a covering space for M, reference to p being left implicit.
Clearly, if (Pp) is a covering space of M, then p is open and surjective. Equally obvious is the fact that p is a local homeomorphism in the sense that if x E P. then there exists an open neighbourhood of x mapped homeomorphically onto an open neighbourhood of p(x). In the natural way, we can define a manifold structure on P so that p is a C°°-mapping. Of course, P is an analytic manifold and p an analytic map if M is analytic. The study of covering spaces essentially involves the fundamental group, which we now briefly discuss.
Let mo, mI E M and a, 8 be paths joining mo to rnl. The paths a, f0 are said to be homotopie (written a -. 6) if there exists a continuous map h: [0,1] x [0,1] M such that h(0,5) = mo, h(1, s) = ml, h(t, 0) = a(t), and h(t, 1) = 6(t) for all s, t E 10, 1). Homotopy is an equivalence relation on the set of paths joining mo to ml. The equivalence class containing a is denoted by [a].
Let a be a path in M joining mo to ml and y be a path in M joining mI to m2 E M. The product a-y is the path in M defined by:
o-40 -
-y(2t-1) (2
Now define a path a' by setting or* (t) = a(l - t) (0 < t < 1). If a - a', then ^- (a')*, so that it makes sense to define [a]` = [a']. The set vrl (M, mo) of homotopy equivalence classes of paths starting and finishing at mo is a group, with product and inversion given by ([a),18)) , [a,8]
and [a) - [a]'. Recall that every connected manifold is pathwise connected. Hence, if ml E
M, we can join up mo to ml by a path in M, and using this path to relate iri (M, mo) to art (M, ml), we see that these two groups are naturally isomorphic.
Thus 7ri (M, mo) is essentially independent of the "base point" mo, and so is usually written arj (M), reference to the base point mo being left implicit. The group ir1(M) is called the fundamental group of M. The manifold M is called simply connected if irj (M) is trivial. Let (P, p) be a covering space of M. Then (P, p) has unique path lifting in the
sense that, whenever xo E P and a is a path in M with a(0) = p(xo), then there
APPENDIX 8
309
exists a unique path a' in P with a'(0) = xo and p o a' = a. A related result is the following: if a', f3' are paths in P with a'(0) = 8'(0) and p o a' - p o 0, then a' 6' (so that, in particular, a'(1) = ft'(1)). The proofs involve a fairly routine compactness argument. The simply connected covering space (M, ir) is constructed as follows. Two
paths a, 0 in M starting at the base point mo are said to be equivalent if a(1) =,6(i) and a and ,6 are homotopic. This gives an equivalence relation on
the set of paths in M starting at mo. We take M to be the resulting set of equivalence classes, and define Tr([aj) = a(l), where [aj is the equivalence class containing a path a. If U is an open neighbourhood of a(1) in M, let Ua be the set of equivalence classes [afi], where ft is a path in U starting at a(1). It is easy to check that the family of all such sets Ua is a base for a topology on M and that (M, ir) is a covering space of M that is simply connected. Let (P, p) be a covering space for M, and fix xo E P such that p(xo) = mo.
Let r`n E M and a E m.. So a is a path in M. We can lift a to a unique path a' in P starting at z0. By a result above, a'(1) is independent of the choice of a. The map An ---' a'(I) is a continuous map F: M P such that the following diagram commutes:
M,P
>.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<j
, x,, u Note that the topology of a submanifold M of N need not be the relative
topology induced by N. An example of this is given by the case where N = T X T and M = {(le2atat , e2nitt): t E R}
where a, b E R -r {0}, a/b is irrational, and the topology of M is that of R, a point t E R being associated with the point (e2>r'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
d°(n) = {A E A(n): Aii = 0 (1 < i < n)}. This latter result is true in greater generality (B35). B24. Let G be a connected Lie group with Lie algebra 9. Then (i) for each i, C'G [DIG) is a Lie subgroup of G with Fig [-?i9] as Lie algebra: (ii) G is nilpotent [solvable] if and only if 9 is nilpotent [solvable].
APPENDIX B
319
We now state some of the properties of solvable and nilpotent Lie algebras.
For the rest of this Appendix, every Lie algebra is assumed to be finite dimensional.
The next result shows that, under reasonable circumstances, solvable Lie algebras can be realised as algebras of upper triangular matrices.
B25 (Lie's Theorem). Let V be a nontrivial, finite-dimensional vector space over C, and let g be a solvable Lie subalgebra of L(V). Then (i) there exists v E V that is an eigenvector for all T E g; (ii) there exists a basis for V with respect to which every T E g is an upper triangular matrix.
Using the exponential map, the Lie group version of Lie's Theorem, stated below, easily follows.
B26. Let G be a connected, solvable Lie subgroup of GL(V) where V is as in B25. Then there exists a basis for V with respect to which every element of G is an upper triangular matrix.
The next result gives a version of B25 for nilpotent Lie algebras. Recall that T E L(V) is nilpotent if Tk = 0 for some k E P.
B27 (Engel's Theorem). Let V be a nontrivial, finite-dimensional vector space over F (= R or C) and g a Lie subalgebra of L(V) whose elements are all nilpotent. Then (i) there exists v 0 0 in V such that Tv = 0 for all T E g;
(ii) there exists a basis for V with respect to which every T E g is upper triangular with all diagonal elements equal to 0; (iii) g is a nilpotent Lie algebra and generates a nilpotent associative subalgebra of L(V). We omit the obvious Lie group version of Engel's Theorem. The next two results are easy corollaries to Engel's Theorem.
B28. A Lie algebra g over F is nilpotent if and only if ad X is nilpotent for
allXEg. B29. Let g be a Lie algebra over F. Suppose that 1) is an ideal of g such that g/ is nilpotent and, for each X E g, adX1g is nilpotent. Then g is nilpotent. The next result is simple to prove.
B30. Let g be a Lie algebra and h be an ideal of g. Then g is solvable if and only if ll and g/l7 are solvable.
B31. Weight spaces and the Jordan canonical form. Let W be a finite-dimensional vector space over C, and let g be a Lie subalgebra of L(W).
APPENDIX B
320
A weight of g is a (linear) functional b: g - C such that the weight space WO, where W,, = {v E W : for each T E g, (T - i (T)I)' v = 0 for some n E P}, is not {0}.
When T E L(W) and g = CT, then the weights of g can be identified, in the obvious way, with the eigenvalues A1, .... .lk of T. The theory of the Jordan canonical form (e.g., Herstein [1]) shows that if Wi is the weight space associated with Ai, then Wi is T-invariant, W is the direct sum of the weight spaces Wi, and, for each i, a basis can be chosen for Wi such that, with respect to this basis, TJw, is an upper triangular matrix of the form:
(1)
L 0
...
0 Ad
The semisimple part of T is the element S E L(W) given by SE, = A,Ci (Za, E W
i < i < k). The nilpotent part of T is the transformation N = T - S. From (1), we see that N is nilpotent; further
T = N + S, N is nilpotent, and NS = SN.
(2)
There exists JSW, p. 237J a polynomial p in one variable such that p(O) = 0
and p(T) = S. We note that the semisimple (nilpotent) part of ad T (E L(L(W))) is ad S (ad NJ.
We will have occasion to use the weight spaces associated with D, where D is
a derivation on a Lie algebra g over C. Let al,...,a, be the eigenvalues of D, and let g, be the weight space associated with a,. Then we have the following result.
For each pair i, j, we have (g g, ] C gk if ai + al = ak, and (v g3] = {0} if
ai + a' 0 jai, ...,a,). The results above on weight spaces leading up to the matrix representation (1) extend to the context of nilpotent Lie subalgebras of L(W). B32. Let g be a nilpotent Lie subalgebra of L(W), and let 0I , ... , ib,, be the weights of g with corresponding weight spaces 14'1,... , W,. Then W is the direct sum of the subspaces Wi, and each W, is g-invariant and has a basis with respect to which the matrix of TJw, (T E g) has the upper triangular form: (T) ihi(T) 0 L
0
...
0
Oi (T) J
The above theorem is still valid if W is a finite-dimensional vector space over R and the eigenvalues of the every ad X (X E g) are assumed to be real. Similarly, the "real" version of the following result holds.
APPENDIX B
321
B33. Let b be a nilpotent Lie algebra of derivations on a Lie algebra g over C. Let 41 be the set of weights of t) (C L(g)), and, for Eli E T, let go be the weight
space associated with ib. Then each go is !)-invariant, g = ®{p ,: , E'P}, and there exists a basis for g,, such that the matrix of TJ$,y (T E 4) is upper triangular with diagonal entries equal to (T). Finally, if 0, i,b E T, then [00,k] c 90+w if 0 +,0 E IF and is {0} otherwise. The following interesting result on the centre of nilpotent Lie groups appears in (Ho, p. 188].
B34. Let G be a connected, nilpotent Lie group. Then the centre Z(G) of G is connected.
B35. Radicals of Lie algebras and Lie groups. Let g be a Lie algebra over F. The following result is of fundamental importance. There exists a largest solvable [nilpotent] ideal t [n] of g. The ideal r [n] is called the radical [nil-radical) of g. Let G be a Lie group with g as Lie algebra. Then the radical [nil-radical] of G is the connected Lie subgroup R [NJ of G with r [n] as Lie algebra. Note that R [N] is a solvable [nilpotent] subgroup of G (B24). By the maximality property
oft [n], we see that R- = R [N- = NJ. Thus R [NJ is the maximal (closed) connected, solvable [nilpotent] subgroup of G, and, of course, is normal in G. More generally, r, n [R, N] are characteristic in the sense of being invariant for every element of Autg (or Derg) [AutG].
B36. Let g be a finite-dimensional Lie algebra over F, and let D E Derg. Then Dg fl r C n. In particular, Dr C n.
B37. In the above notations, (g, tJ c n.
B38. Semisimplicity and the Killing form. Let g be a (finitedimensional) Lie algebra over F. The Lie algebra g is called semisimple if its radical is {0). It is easy to check that g is semisimple if and only if it contains no nontrivial abelian ideals. In particular, if g is semisimple, then Z(g) = {0}, and so ad, is one-to-one. Also, if b is a Lie algebra with radical r. then h/r is semisimple (B30).
A Lie group G is called semisimple if its radical is the trivial group {e}. Clearly, G is semisimple if and only if its Lie algebra is semisimple (B35). Further, Z(G)e = {e), so that Z(G) is a discrete subgroup of G if G is semisimple. Recall that a bilinear form on a vector space V over a field F is a bilinear map 4i: V xV F. A bilinear form (I on V is called symmetric if 4?(X,Y) = 4 (Y, X) (X, Y E V) and is called nondegenerate if it satisfies
{0}={XE V:(I(X,Y)=0for all YEV}. The Killing form of the Lie algebra g is the bilinear form B: g x g .- F defined as follows:
B(X, Y) = tr(ad X ad Y).
APPENDIX B
322
If a C 9, define
01 ={XEg:B(X,A)=0forall AEa}. Clearly, a-L is a subspace of g.
Obviously, B is symmetric. We now state some simple results involving B. The radical of g is denoted by r.
B39. (i) If T E Autg and D E Derg, then
B(TX,TY) = B(X,Y),
B(DX,Y) _ -B(X,DY)
(X,Y E g);
(ii) B(X,Y) = B(Y,X) and B(X, (Y, Zj) = B(Y, (Z, Xj) = B(Z, [X.Yj) for all X, Y, Z E g; (iii) if a is an ideal of g, then a-L is an ideal of g; (iv) (g, 91, = r: (v) g is semisimple if and only if B is nondegenerate; (vi) if g is semisimple and a is an ideal of g, then
g=a6?al,
(al)--=a.
A Lie algebra b is called simple if i7 is nonabelian and the only ideals of h are {0} and h. The next result is a consequence of (vi) above.
B40. Let g be semisimple. Then g is the direct sum gI @ ®g,,, where gI, ... , gn are the simple ideals in g. Further, every ideal of g is a direct sum of g, 's.
The above result can be expressed as asserting the "complete reducibility" of g as a 9-module under the adjoint representation. The following much more general result holds (B, Chapter 1, §61, (SW, Chapter 12, §41. B41. Let g be semisimple, and let V be a finite-dimensional g-module. Then V is a direct sum EDi=i V, of irreducible g-submodules Vi.
Let g be a Lie algebra. Since ad g is a Lie subalgebra of L(g), there exists a unique connected Lie subgroup of GL(g) with ad g as Lie algebra. This Lie subgroup is denoted by Intg. Note that if G is a connected Lie group with g as Lie algebra, then (B12) we have Intg = Ad G (= G/ ker Ad). B42. Let g be semisimple. Then (i) (g, 91 = 9;
(ii) every ideal and every quotient algebra of g is semisimple;
(iii) Derg = adg, and if I is the identity map on 9, then Int9 = (Autg)1; (iv) if G is a connected, semisimple Lie group with g as Lie algebra, then (AutG), = I(G) ((3.3)). The next result is known as the Levi-Malcev Theorem. Proofs of this important theorem are given in (J, Chapter 3, §91 and (B, Chapter 1, §6.8(. The subalgebra s below can be taken to be any maximal, semisimple subalgebra of g. The decomposition s e w of the theorem is called the Levi-Malcev Decomposition.
APPENDIX B
323
B43. Let r be the radical of the Lie algebra g. Then there exists a (semisimple) subalgebra s of g such that g is the vector space direct sum s ® r. (Thus g is a semidirect product r x b s in the canonical way (B5).)
It follows from the above result and B22 that if G is a simply connected Lie group with radical R, then G is a semidirect product R x, S. B44. Let G be a simply connected Lie group, H a closed, connected, normal subgroup of G, and Q : G --'. G/H the quotient map. Then there exists an analytic
map p: G/H - G such that Q o p is the identity map on G/H. For any such map p, the map (h, m) - hp(m) is an analytic diffeomorphism from H x G/H onto G. Finally, both H and G/H are simply connected. The next result is a corollary of the preceding theorem.
B45. Let G be a connected Lie group containing a closed, simply connected normal subgroup H with G/H simply connected. Then G is simply connected. We now turn to two other results involving semidirect products. The first of these essentially appears in [Ho, Chapter 12, Theorem 2.2). B46. Let S be a simply connected, solvable Lie group with s as Lie algebra. Then there exists a basis {X1,. .. , Xn } for s such that if si = Span{X1 s ... , Xi }, then st is an ideal in si+1, and if Hi is the connected Lie subgroup of G with Si as Lie algebra, then Hi is closed in G and
(1) for each i, the map EJ_I a3Xj - (expa1X1)
exp(a,X,) (where a, E
R) is an analytic diffeomorphism from si onto Ht;
(ii) Ht+I is a semidirect product Hi xp, Li+1, where Li+i = exp(RXi+I)
(0
The next result is proved in [Ho, Chapter 12, Theorem 3.2J.
B47. Let G be a locally compact group, and suppose that S is a simply connected, solvable, closed, normal subgroup of G with G/S compact. Then there exists a compact subgroup K of G such that G is a semidirect product S x, K.
B48. Compact Lie algebras. Let g be a Lie algebra over R, and recall (B41) that Intg is the connected Lie subgroup of GL(g) with adg as Lie algebra. The algebra g is said to be compact if lntg is compact (in its Lie group topology). More generally, a subalgebra t of g is said to be compactly embedded in g if the connected Lie subgroup K of GL(g) with adg I as Lie algebra is compact. It is easily checked that t is compactly embedded in g if and only if K is compact in the relative topology induced by GL(g), and that if t is compactly embedded in g, then t is a compact Lie algebra.
We shall see that the Lie algebra g is compact if and only if it is the Lie algebra of a compact Lie group.
B49. Let 9 be over the field R with Killing form B (B38). Then
APPENDIX B
324
(i) if g is compact, then g is the direct sum Z(g) ED [g, g), and [g, g] is compact and semisimple;
(ii) if g is semisimple, then g is compact if and only if B(X, X) < 0 for all nonzero X E g; (iii) if g is semisimple and t is a compactly embedded subalgebra of g, then
B(X, X) < 0 for all nonzero X E t. Here are two easy consequences of the preceding result.
B50. Let g be a Lie algebra over R. Then g is compact if and only if it is the Lie algebra of a compact (connected) Lie group.
A more general version of B51 is proved in [HR2, (29.44)]: the Lie group requirement is unnecessary.
B51. Let G be a compact, connected, solvable Lie group. Then G is abelian. The next result is proved in [He, Chapter 2, Theorem 6.9]. See also [HS, pp. 221-224].
B52. The simply connected covering group G of a compact, connected, semisimple Lie group G is itself compact.
It immediately follows using B50 that if G is a connected, semisimple Lie group with compact Lie algebra, then G is compact.
For the remainder of this Appendix, we will be discussing the theory of semisimple Lie algebras. Our account is primarily based on the treatment of this topic given by Helgason [He, Chapter 3 and Chapter 6], though [Ho) and [SW) have also influenced our treatment.
We shall discuss briefly the theory of Cartan subalgebras, roots and root spaces, complexifications and compact forms, the Cartan decomposition, the Cartan involution, and the Iwasawa decomposition associated with a semisimple Lie algebra go over R. For the remainder of this Appendix, g will be a (finite-dimensional) semisimple Lie algebra over C.
B53. Cartan subalgebras and roots. A subalgebra h of g is said to be a Cartan subalgebra if (i) l is nilpotent; (ii) if Y E9 and IY,bl C I, then YE F).
A fundamental result [He, Chapter 3, §3] asserts that there exists a Cartan subalgebra h for g. Using property (i), ade 4 is a nilpotent subalgebra of L(9). The weights (B31) of adg h are called the roots of g, and the corresponding weight
spaces are called the root spaces of g. (It can be shown that the roots and root spaces are essentially independent of the choice of the Cartan subalgebra F .) The root space associated with a root a is denoted by g°. Using (i) and (ii), it turns out that 0 is a root and 9o = F) (and h is abelian).
APPENDIX B
325
The set of roots of g is denoted by R, and we set A = R
{0}. If a E R,
H E 1), we write a(H) in place of a(adg H). We now give some elementary properties of the roots and root spaces of g (cf. B33). The Killing form of g is, as usual, denoted by B.
B54. (i) g is the vector space direct sum (ED 9a) ED 3;
aEa (ii) [ga,9"1 C 9a+Q if a + 8 E R and is {0} otherwise;
(iii) i f a,,8 E R and a+R $ 0, then B(X,Y) = 0 for X E ga, Y E gd; (iv) B14),4 is nondepenerate;
(v) -a is a root if a is a root.
Now each a E A is a nonzero, linear functional on h, and since B]4,,4 is nondegenerate, there exists a unique element Ha E i) such that
a(H) = B(Ha, H)
(H E i)).
Clearly, H. = -H-,,, and Ha+p = Ha + Hp if a,,8, a +,8 E A.
B55. (i) Let a E A. Then (a) if X E ga, Y E g-a, then [X,Y) = B(X,Y)Ha; (b) A spans 1)', the dual space of i); (c) ga is one-dimensional; (d) a(H0) > 0 and 13(Ha) E R for all ,6 E A.
(ii) There exists a subset IX,,: a E Al of g and a subset {Na,0 : a,)3 E A} of R such that (a) 9a = CXa;
(b) [Xa,X_a) = Ha, B(Xa,X_a) = 1, and [H,Xa] = a(H)Xa (a E A, HEb); 0 if a+,8 OR, (c) [X a,XFJ = t Nu,0Xa+0 if a+Q E A; (d) Na,s = -N-a,-B B56. Complexifications and real forms. Let go be a Lie algebra over R 1)/
and g = (ga)c. Every element of g is uniquely of the form A + iB (A, B E go), and the map a: g g, where
a(A+ iB) = A - iB
(A, B E go)
has the following properties: (1}
a(X + Y) = a(X) + a(Y), a(aX) = aa(X), a([X,Y]) = [a(X),a(Y)), a2 = I,
for all a E C, X, Y E g. Clearly
go = {X E g: a(X) = X}.
APPENDIX B
326
Conversely, suppose that g is a Lie algebra over C and that r: g -+ g is a conjuIf we set gation, that is, a map satisfying the conditions of (1).
go = {X E g: r(X) = X}, then g - (go)c in the obvious way, and r(A+iB) _ A - iB (A, B E90). It is easy to prove that if r is a conjugation of g, then B(rX,rY) = B(X,Y) (X,Y E g). A real form of g is a Lie algebra over R whose complexification is isomorphic to g. From the above, the real forms of g axe in a natural, one-to-one correspondence with the conjugations of g.
A compact real form of a Lie algebra g over C is, of course, a real form of g that is a compact Lie algebra (B48). The next result enables us to associate with a semisimple Lie algebra go (over R) a compact semisimple Lie algebra a over R through the mediation of (go)c. B57. Let g be a semisimple Lie algebra over C. Then g has a compact real form.
Indeed, using B54 and B55, a is a compact real form of g, where a is the set of elements of the form
va(i(X0 + X_a)),
A0(iHa) + E µa(Xa - X_a) + QEA
aEA
aEA
where for each a E A, a, fla, La E R. If go is a semisimple Lie algebra over R, then a direct sum decomposition to fl po of go, with the properties given in (ii) of the next result, is called a Carton decomposition of go. To motivate the Cartan decomposition, suppose that Go is a closed subgroup
of GL(n, R) and that the transpose U' E Go whenever U E Go. Then the map U --> (U')'' is an involutory automorphism e of Go. Let go be the Lie algebra of Go. Then 8 = dO(I) is an involutary automorphism of go C L(R'), with 8(X) = -X'. We obtain a Cartan decomposition of go by setting to = {X Cr go: 8(X) = X}, Po = {X E go: 8(X) _ -X}. Note that to is the Lie algebra of the maximal compact subgroup Ko = O(n. R)nGo of Go. (See Knapp (S2, p. 3f1,.) The result below shows that a similar result holds for general go.
B58. Let go be a semisimple Lie algebra over R, g its complexification, and a the conjugation of g associated with go. Let B be the Killing form of g. Then (i) if a is a compact real form of g, then there exists an automorphism ¢ of g such that o(4(a)) = g5(a); (ii) there exist sets to, po and gk, where to is a subalgebra of go, Po is a vector subspace of go, and 9k is a compact real form of g such that: (a) to = 9o n 9k, po = 9o n (igk);
(b) o(9k) = 9k; (c) go = to (B po;
(d) Ito,po) C po, (po,po) C to; (e) if X E gk {0}, Y E Po {0}, then B(X,X) < 0, B(Y,Y) > 0;
APPENDIX B
327
(f) to is a maximal, compactly embedded subalgebra of go; (iii) any two compact real forms of g are isomorphic. A simple example of the Iwasawa decomposition below is provided by go = sl(m, R). Here, to = so(m, R), ao is the algebra of diagonal matrices in go, and no is the algebra of upper triangular matrices with zeros on the diagonal.
B59. The Iwasawa Decomposition Theorem. There exist an abelian subalgebra no of go and a nilpotent subalgebra no of go such that no + no is a solvable subalgebra of go, and go is the vector space direct sum to 6 no E) no.
B60. More on the Iwasawa decomposition. (i) If go is not compact, then ao # {0} and no 0 {0}. (ii) Let X E no. Then ads,) X is nilpotent. We now state the Lie group version of the Iwasawa decomposition [He, Chapter 6, §5].
B61. Let G be a connected, semisimple Lie group with go as Lie algebra. Let Ko, A0, and No be the connected Lie subgroups of G with to, ao, and no as Lie algebras respectively. Then Ao, No are simply connected, and the map ': Ko x Ao x No -+ G, where 4) (k, a, n) = kan
is an analytic diffeomorphism from Ko x Ao x No onto G.
Of particular importance (especially for representation theory) is the case where Z(G) is finite. Indeed, AdG Ko, as a group, equals the Lie subgroup of GL(go) with adso to as Lie algebra (since both groups are generated by Ad0(expto)) and so is compact (B58(ii)(f)). The canonical bijection from Ko/(Ko n Z(G)) onto AdG Ko is bicontinuous, noting that both groups have isomorphic Lie algebras. Hence Ko/(Ko n Z(G)) is compact, and since Z(G) is finite, Ko is also compact. The maximality property of to yields that KOZ(G) is a maximal compact subgroup of G. (In fact, K0Z(G) = Ko [He, Chapter 6, §11.) The following more general result is proved in [Ho, Chapter 151. B62. Every connected Lie group contains a maximal, compact subgroup. The final result of this section appears in [J, Chapter 3, Theorem 171 and [He,
Chapter 9, §7]. Note that if g in the result is noncompact, then (B60) ensures that there exists a nonzero element X E g with ad X nilpotent. B63. Let g be a semisimple Lie algebra over R, and let X E g be nonzero, and
such that adX is nilpotent in L(g). Then there exist elements H', E', F E g with E' = X such that [H'; E'] = 2E', [H', F'1= -2F', [E', F'1 = H' (so that the subalgebra (RH' + RE' + RF) of g is canonically isomorphic to sl(2, R)-cf.(3.2)).
APPENDIX C
Existence of Borel Cross-Sections The following result is due to Federer and Morse [1J. See also Mackey fl) and Feldman and Greenleaf [1].
THEOREM. Let X, Y be Hausdorff spaces with X separable. Let g: X
Y
be continuous and C E ?(X). Then there exists a Borel set B C C such that g(B) = g(C) and 91B is one-to-one.
PROOF. The argument of (e), p. 166 of Kelley [2J shows that there is a continuous map h from the Cantor set K onto C. For each n E P, let (1)
Kn = {t e K: g(h(t)) ¢ g(h([O,t - n-'J n K))}.
(Here, of course, K is identified with the Cantor discontinuum.) Clearly, K is an open subset of K. Let B = h(K,,). Since Kn (and so B,,) is a-compact, it follows that Bn is a Borel subset of X, and so B= nn l Bn is a Borel subset of X. We now check that B satisfies the conclusions of the theorem. Let bi,b2 E B with bl # b2. Since h is continuous, there exists n r: P such that n-' < jtl - t2i whenever t3, t2 E K are such that h(ti) = bi. Choosing t1, t2 with h(ti) = bi and t, E K, , we see that g(bl) g(b2). So 91B is one-to-one. Now let yo E g(C) and define
co = inf{t E K: g(h(t)) = yo}. It is easy to check that co E Kn for all n. Hence h(co) E B and g(h(co)) = yo. This concludes the proof.
COROLLARY. If X is also a-compact, then there exists a Borel set B' C X such that g(B') = g(X) and g!R, is one-to-one.
329
APPENDIX D
Myclelski's Theorem We state below a very useful category theorem due to Mycielski. The theorem
is fundamental for (7.15); it can also be used to construct "large" free groups in SO(n, R) (Wagon [S21). References for the theorem are Mycielski [3), [S2] and Kuratowski [1). The reader is referred to (6.5) of Wagon )S2) for a short, ingenious proof. The theorem is true without the separability requirement, but then requires the Axiom of Choice. THEOREM (MYCIELSKI). Let (X, d) be a complete, separable, metric space with no isolated points. Let {mq) be a sequence of positive integers, and for each i, let R; be a closed subset of Xm, with empty interior. Then there exists a subset
A of X such that JAI = c and if i > I and a,.... , a,,,; are distinct points of A, then (a1:... , ami) R,.
331
APPENDIX E
On the Density of the Exponential Map The following result seems to have been proved first by Dixmier [2]. The proof we give is based on Hofman and Mukherjea [1].
THEOREM. Let S be a solvable, connected Lie group with Lie algebras. Then
exps is dense in S. PROOF. The proof proceeds by induction on n = dim S. The result is trivially true if n = 0. Let n > 1. Suppose that the result is true for all solvable, connected Lie groups of dimension < n. By Lie's Theorem (B25) we can find Z E sc - {0} and a function a:s such that (A E s). adA(Z) = a(A)Z
Write Z = X + iY, where X, Y E s, and let ro = RX + RY. Then m is either one- or two-dimensional, and is an ideal in s. Let (S, p) be the simply connected covering group of S. Since poexps = exps, we can (and shall) suppose that S = S. Let W be the connected Lie subgroup of S with m as Lie algebra. By B17(iii) and B44, W is a simply connected, closed, normal subgroup of S. Let U be a nonempty, open subset of S. It is sufficient to show that (1)
U f1 exps (s) 0 0.
Since dim(S/W) < n and S/W is solvable, we have, by hypothesis, that exps/w (s/ro) is dense in S/W. Let Q: S --> S/W, ol>: s s/ro be the canonical maps. Since Q is open, we can find K E s/ro such that exps/w(K) E Q(U). Since the following diagram commutes (B11(i)): exps
S
--
5I
Q ll--
S/W
exps/w
s/ro,
we can find A E s such that exps A E UW. Let H = (exps RA)W. Then H is a (solvable) connected Lie subgroup of G with b = RA + m as Lie algebra. 333
APPENDIX E
334
Now u n H is open in H (with its Lie subgroup topology), and u n H D U n (exps A)W'I # 0. So (1) will follow if we can show that expH 4 (= exps 41 is dense in H.
(2)
As earlier, we can take H to be simply connected. Since 4 is either m (if A E m) or a semidirect product m x .18, we see (B22) that H is either W or a semidirect product W xP R.
We can rule out the cases H= W = R, H = W = 182, since in those cases, H is exponential, that is, exp is surjective. Suppose that W is two-dimensional (solvable) and not 182. Then we can find a basis {X, Y} for m such that [X,Yj =
Y. So m is s2 ((6.35)) and W = S2, the "ax + b" group. Thus ((6.36)) the multiplication of W is given by (s, t) (s', t') = (s + ets', t + t'),
where s, s', t, t' E R. We thus have to consider the following (semidirect product) possibilities for H, where M is R or 182:
S2x,18.
MxP18, We consider these cases in turn.
(1) Suppose that H = M x p R. Then p: R -+ Aut M is a one-parameter subgroup in Aut M = GL(M), and so we can find A E L(M) such that p(t) _ etA (t E R). Thus the product in H is given by (m, t)(m', t') = (m + etAm', t + t').
(3)
The Lie algebra 4 of H is (cf. B22) M xe, R, where f(s) = sA. We now deal with four subcases. (a) Suppose that A = 0. Then, trivially, H is R2 or R3 and exp is surjective.
(b) Suppose that M = R2, A 0 0, and A is nilpotent. Using the Jordan canonical form, we can take A = (oo 'j. Then etA = [o t 1, and (3) becomes
((ml,m2),t)((mi,m2),t') = ((ml +m' +tm2,m2+m2),t+t'), where mF, m2, mi, m2 E R. Identifying ((ml, m2), t) with the 3 x 3 matrix
B= 0
1
-ml m2
0
0
1
,
we see that H is the "Heisenberg group" (Problem 4-29). Since
B = exp((B - I) - 2 (B - I)2) (using the power series expansion for log(i + x)), we see that H is exponential. (More generally, every connected, nilpotent Lie group is exponential [Ho, Chapter 12, §2.01.)
APPENDIX E
335
(c) Suppose that M = R2, A is not nilpotent and not invertible. Then the {0}. So we can take A = [a a]. An
eigenvalues of A are 0, k, where k E R elementary calculation shows that etA
ekt o l
-
0
1,
so that (3) becomes
((MI, m2), t)//((mi , m2), t) = ((mi + ektmi,'m2 + m2), t + t`).
Writing ((MI,t),m2) in place of ((MI,m2),t), we see that M is isomorphic to S2 x R. Since R is exponential, we can take H = S2. This case is covered by (d) below.
(d) Suppose that A is invertible. Suppose that M = R2. Let s (m(s),t(s)) be a one-parameter subgroup of H. The homomorphism property satisfied by the preceding map gives (using (3)): (4)
m(r + s) = m(r) + et(r}rtm(s),
(5)
t(r + s) = t(r) + t(s)
for r, s E R. From (5), there exists k E R such that t(r) = rk. If k = 0, then, using (4), we obtain c E M such that (m(r), t(r)) = (rc, 0)
(r E R).
Suppose that k qF 0. Then from (4) and noting that m(0) = 0,
m`(r) = Ii a(m(r + s) - m(r))/s = Jim ek'A (M (S) - m(0))/s = ek'Am`(0). 8-0 It follows that if a = (kA)-1(m`(0)), then m(r) = ek"Aa - a. The one-parameter subgroups of H are thus of two kinds: those of the form r -+ (rc, 0) (c E M) and those of the form r - (ek'Aa - a, kr) (k E R - {0}, a E M). Observing that (ek,A - I) is invertible whenever 2niri is not an eigenvalue of krA for any n E Z, we see that (2) holds. The case M = R (H = S2) is an easier version of the above.
(ii) Now suppose that H is of the form 82 x, R. Then the Lie algebra f? E Ders2. Thus (B22) D(A) = [Z, A] (A E 52). Let {X. Y} be a basis for 52 such that IX, Y] = Y. Since [s2,52] = RY, we have DY = by for some b E R. Thus by = DY = D([X,Y]) = [DX,Y] + bY, so that [DX,Y] = 0 and DX = aY for some a E R. Let T = Z + aY - bX. Then IT, X] = 0 = [T, Y], and since
of His of the form 52 x0 R. Let Z = (0,1) E E? and D =
{X, Y, T) is a basis for h, we see that H is of the form S2 xR. This case obviously
reduces to the S2 case of (d) above. 0
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Sketched Solutions to Problems
Problems 0 0-1. To show that the map is onto, given µ, define m on simple functions = L a=XE, ({E,} disjoint) by m(4) a,µ(E,). The space of such functions is norm dense in I,o(X). 0-2. This is a variation on Problem 0-1; cf. [DS, IV.8.16). 0-4. E
3 (C)-' n x E G} in
A", where A is the closure of Span{4k - ¢x: ((2.1)).
0-6. 2i1(xAn n An) = 2.\(An) - )) (zAn L An).
0-7. Let m be symmetrically invariant. For n > 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
SKETCHED SOLUTIONS TO PROBLEMS
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
SKETCHED SOLUTIONS TO PROBLEMS
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 <
1- (e/a(C)). For c, modify the first part of the proof of (1.21) to produce mo E 9Ji(Ur(G)) in the set K = {n E 9JI(Ur(G)): n(XEP) = 1 for all p E P(G)}. (Note that if C C D E F(G), then XEPC ? (1 - e) if XEUD(x) [1 - -(A(C)IA(D))) for x E G.) Apply the Fixed Point Theorem (2.24) to find no E .c(Ur(G)) n K. Finally use (1.7) to extend no to the desired m E.CC(G).
-
1-28. Use 1.27.
SKETCHED SOLUTIONS TO PROBLEMS
417
1-29 (Klawe [21). Let U = a x Z, where p(n)(Z) = {0} for all n. Show that U is not left reversible. 1-30. Use the natural maps j : C(G) --> 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
SKETCHED SOLUTIONS TO PROBLEMS
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.
SKETCHED SOLUTIONS TO PROBLEMS
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 < i < n) and /i ,v; E P(G), then
_ i (0:µ4 - d jv,) is of the form '7{ ,o - Ov) ' for some g. v E P(G), 7 > 0.
SKETCHED SOLUTIONS TO PROBLEMS
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.
SKETCHED SOLUTIONS TO PROBLEMS
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
SKETCHED SOLUTIONS TO PROBLEMS
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))).
SKETCHED SOLUTIONS TO PROBLEMS
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).)
SKETCHED SOLUTIONS TO PROBLEMS
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.)
SKETCHED SOLUTIONS TO PROBLEMS
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
SKETCHED SOLUTIONS TO PROBLEMS
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.
SKETCHED SOLUTIONS TO PROBLEMS
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 y`IC_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)).
SKETCHED SOLUTIONS TO PROBLEMS
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,
SKETCHED SOLUTIONS TO PROBLEMS
429
4-17. Let g E LP(G) g >- 0. Define a measure µg onR (G) by u,(E) = fE gd)1. By (4.19), µg(G) < oo. This gives LP(G) C L1(G). Show that this forces G to be compact. (References: Leptin [1], Lohoue [1], Rajagopalan [1]-[5], Rajagopalan and Zelazko [1], Rickert [31, Urbanik [1], Zelazko [11-[31, [HR2, pp. 469-4721}.
4-18 (Heath and Sudderth [1], Bondar). Suppose that m E £ (G). Let µ6, x6, and £({x6}) be as in (4.16)-(4.17). We can suppose that m E .£({x6}). Let v E P(G), v with compact support K. Show that I (A6 * X6)' (v °2 4 - v- 010) I
= (A(K6x6))-i 1 f dv(x) fc [XKaxa W08, xs)
x
-Xx-, K6x6 (5)0(s, xs)] da(s)
< sup(X(xK616K6)/a(K6))ll(bll-. xEK
4-19 (Heath and Sudderth [11, Bondar). Show that m, = m, P(s)(8O) = µ o2 ¢(s), Q(x)(.Ox) = u of O(x), and use Problem 4-18. (m1 is called the marginal of (m, P).)
4-20 ([R, p. 175], Derighetti [11. Glicksberg 12]). Adapt the proof of (4.18).
(The converse of the result is also true, and the normality condition can be removed.) For the last part, note that the map g g" is isometric from Cf onto co{x * f-: x E G}. 4-21 (cf. Greenleaf 12, p. 79)). Note that f *µ6 E LI (G) and µ6 = XK, /)H (K6 ) E LI (H) C M(G). The LHS of (1) is just IjQHf 111, and the RHS is lim IIf * µ61l1. By Problem 4-20, (1) will follow once we have shown d(O,Cf(H)) = lim Jl f *µ6l[1.
Suppose first that f E C,(G), with support C. If v E coH, then
III*v*µS-f*µ6Ill
IlfIlillu*µ6-µ6[[i-+0,
so that lim sup II f *,u6111 -< d(0, C f (H)). For the reverse inequality, it is sufficient
to show that IIf * µjji > d(0,Cf(H)) for µ E P(H) with compact support K. Show that for r > 0 there exists a measurable partition Ei,... , E of K and elements hi E E, such that if F(x) = F,", f * hi(x)}t(E1), then If * µ(x) F(x)l < c (x E CK). Now approximate Il f * µll I = fcK If * µ(x)l d\(x) by fcK IF(x)l d)(x), giving Il f * µ1l, > d(O,Cf(H)) as required. The equality (1) for general f E L1 (G) follows by approximation.
For the last part, take G = R, H = Z, and KN = [-N; NJ. 4-22 (Rindler [11-13J. [Sijff.). (i) Use (4.18).
Let Lo = (f E LI (G) : f f d.1 = 0}, and let {f,,: n E P} be a norm (ii) dense subset of Lo. There exist elements z;` E G (1 < r < M,,,) such that JIM;I Z ; ? ; fk -z , '1 11 < m-I for I < k < m. Construct sequences N K, such
that N1 = 1,Ai{I-N, =K,Mi,M,/N, -+0. FornEPwrite n=Nn+jMm+i
430
SKETCHED SOLUTIONS TO PROBLEMS
(j < Km, i < Mm), and set a = Zr. Then {a,,} is a u.d. sequence for G. First show this for f E L°, and deduce the general case as in the second paragraph of the proof of (4.18). - Show that if {z,,,} is a u.d. sequence in G, then G is the closure of the subgroup of G generated by [x,,: n > 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 <- t <- 1).
4-27. G abelian: The map ir2(f) -> f from 7T2 (C,* (G)) into C0(G) = C* (G) is isometric by Plancherel's Theorem. G compact: Use the Peter-Weyl Theorem.
4-29. The support of p is, by 02, the same as the set A = {p E G: p is weakly contained in r2}. Now r E G is in A 4* every state associated with is a weak" limit of states associated with r2 [D21 (3.4.10)]. Now show that A = G C* (G) = Ci (G). When G is abclian, µ is a Haar measure on G and so has dense support in G. Now let G be the Heisenberg group. For information about this group, see, for example, Warner [1], Kirillov (1], and Nielsen (S2]. As a set, G can be identified with (R - {O}) U R2. Both R - {0}, R2 have their normal topologies, but the closure of R - {0} in G is the whole of G. .
The Plancherel measure can be identified with 1.51 de on R -- {0}. (Nielsen gives
Plancherel measures for the low-dimensional, simply connected, nilpotent Lie groups. See also Paterson 1521.) 4-30 (Paterson 12]). Show that xH E S(G) - co{SJ(G) U {1}} ((HR2, (32.43)]). The argument proceeds by contradiction using xH 0 S!(H). This requires showing that 0I0 E S1(H) if 0 E St(G). To prove this, approximate as by functions E CM(G) n P(G) and then apply Godement's Theorem ((4.20)) to zlil x
4-31. Since G is amenable, ro is weakly contained in r2, and by Property (T) there exists e E L2(G), 11 11 = 1, such that I = (r2(x)e, ) (z E G). But then 1 E C°(G), forcing G to be compact.
4-32. (Argabright and Wilde (11). Suppose that S satisfies (SFC). If C E .` (S), s > 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.
SKETCHED SOLUTIONS TO PROBLEMS
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
<¢
J(t'_b)dA<E.
Then (Problem 0-12) lm(¢) -' ¢ d\[ < e for all m E C(G), giving m(o) = f 0 d\. 4-36. G has to be unimodular.
4-37. The map f -+ f f d A extends to a multiplicative linear functional on CC (G). The kernel of the functional is a proper, closed, two-sided ideal in CC (G).
[Note: C(F2) is simple (Powers [S]).] 4-38. (Reiter [17], Johnson [2), Rindler [S2], [S9]). Suppose that G is amen-
able. We can suppose that {f6}, {jub} are nets in P(G) as in (4.1), (1.5). By expressing f E L°(G) as a member of SpanP(G), we have [l f * fbll, --, 0. Then {f16 - f5) is a bounded right approximate identity for LO(G). Conversely, if {u6 }
is a bounded right approximate identity for L°(G) and we fix v e P(G); then 0, where 6 = vu6 - v. for all IL E P(G), (µv - v) E L°(G), and [)f46 - bbl{I Since e(1) _ -1, any weak' cluster point of {eb} is a nonzero element of,ij(G), and (2.2) applies. [Note: Johnson [4) showed that L°(F2) does not even possess an (unbounded) right approximate identity. Liu, van Rooij, and Wang (S] have shown that if G is amenable, then every cofinite, closed ideal in LI (G) has a bounded approximate identity. Willis [1), [2] has shown that if LI (G) contains a cofinite, closed, left ideal that has a bounded right approximate identity, then G is amenable.]
4-39. If A(G) = B(G), then 1 E A(G) C C0(G), giving G compact. Conversely, if G is compact, then I = 1 * 11 E A(G), giving B(G) = B(G)I C A(G).
4-40. (Boidal et al [1]). (i) Follow the C'-algebra version in [D2, (3.1))-the obvious version of [D2, (2.11.4)) holds. So a set Y C Prim. A is closed if and
only if S° _ {I E Prim. A: I D nY}. (ii) The bijectiveness of WYG follows from the natural correspondence of LI (G)^
with C`(G)^ (D2, (2.7.4)]. (iii) Suppose that G E 1,P), Y _ {I E Prim C" (G) : I D ker 7r2}. Then 5° is closed in PrimC`(G). So WG(5°) is closed in Prim. LI(G). Now (}'G(5°) = n LI (G)) _ {0} since r2 is faithful on LI (G), and so 4'G(.") _ {J E t
Prim. LI(G): J D {0}} = Prim. LI(G). Hence n,7 = {0} and C'(G) CI (G), giving G amenable.
SKETCHED SOLUTIONS TO PROBLEMS
432
4-41 (Greenleaf [3]). Suppose that G satisfies the weak Frobenius property. With N = {e}, it E C, we have U° unitarily equivalent to ire. Thus ire weakly contains it for all it E C and so C' (G) = C; (G) and G is amenable. Conversely, suppose that G is amenable, and let N be a closed normal subgroup of G, it E C and o = 1r[N. Let 55 be the Hilbert space of it, and let s: G\N - G be a Borel function such that s(Ng) E Ng for all g c- G. Then U° is realised on Y = L2 (G\N, 55) = L2 (G\N) 0 i5, and for all f E Y, g E G, we have U°(g)f (x) = o(s(x)gs(xg)-1) f (xg)
(x E G\N).
(Kirillov [2, §13)). Now the map or is an isometry of Y, where a(f)(x) _ ir(s(x))(f (x)), and implements a unitary equivalence between U° and U3 ® it, where J is the trivial one-dimensional representation of N. (This is a special case of a result of Mackey [1].) If I is the trivial one-dimensional representation
of G, then 10 ir = ir, and it follows that we can take 7r = 1, u = J. Now U' is the "quasi-regular representation" of G on L2(G\N); to prove that U3 weakly contains I, use the fact that GIN has the weak containment property. [,Note: Greenleaf actually proves a more general "weak Frobenius" characterisation of amenability, in which the "separability" and "normal" requirements above are removed. Weak Frobenius properties were first discussed by Fell (4). Gootman 11) examines the Frobenius Reciprocity Theorem for simply connected Lie groups. See also Henrichs [1], [2) and Nielsen [Si).]
4-42 (Renaud [4]). The function t -. (tf)g ® (tf)g is continuous and of compact support from G into Hence its Bochner integral h exists and belongs to L2(G)®L2(G), and a direct calculation with an application of Fubini's Theorem gives '(h)(x) = (f * ft)(x)g * gt(x). It follows that 11(f * ft)(g * g') - g * g}[l <
l f (tf - f(t))g ®(t f)g da(t) +
J
f(t)g ® It f - f(t)]gda(t)e
(r 11(tf)g112 d. (t))
(f (1(tf - f(t))g112 (f +
1/2
1/2
1h91121f(t)I2 dA(t)
) Y Il (tf - f(t))g112 da(t) )
Now argue
f
11(tf - f(t))g(12 da(t) =
J
da(x)
J
l f (xt) - f(t)121g(x)12 da(t)
<-Sup llx-1*f-f112. XEC
An easy Fubini-type argument gives f [[(tf)g[[2 d-\(t) = 1 and (i) follows. [Note:
This argument is a special case of the use of certain maps F, M that play a useful role in the Ap(G) theory-these maps have their origin in the work of Varopoulos.]
SKETCHED SOLUTIONS TO PROBLEMS
433
Using (i) we can take each 06 to be of the form g * gt ("C e") to obtain --+ 0 for each 0 E CM(G) fl P(A(G)). The latter fact is true for all 0 E P(A(G)): consider 4500, where ¢o is a fixed element of CM(G) fl P(A(G)). A topologically invariant state m on VN(G) can be obtained by taking a weak" cluster point of {46} in VN(G)'. [Note: Renaud shows that G is discrete if and only if there exists exactly one topologically invariant state on VN(G). Granirer [16] shows that the set of topologically invariant states on VN(G), where G is nondiscrete, is not norm separable; cf. (7.28).) 1100s -.06 11
4-43 (Lance (1), Effros and Lance [11). By replacing uc,c of the proof of (4.34(ii)) by (XK * Xir)/.A(K), we see that there exists a bounded approximate identity of positive-definite functions {¢6} for A(G), with ¢6(e) = 1, 06 E Cc(G) for all b. Define D6 E B(VN(G)) by b6T. Then {'6} is a net of weak*
continuous, unit preserving maps, and D6(T) - T weak` for all T E VN(G). Since 4'6 (ire (x)) = 06 (x)7r2 (x) and 06 has finite support, the range of 4D6 is the finite-dimensional space Span{ir2(x): 06 (x) 0 0}. To show that'16 is completely positive, realise Mn 0 VN(G) = Mn(VN(G)) on L2(G) 9 ED L2 (G). Let
fij E Span?r2(G), A = [fsj] E Mn.(VN(G)), and h = (h1....;hn) E Cc,(G)n. It is sufficient to show that (ms(A'A)h,h) > 0. In fact (
6(A`A)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
SKETCHED SOLUTIONS TO PROBLEMS
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°].
SKETCHED SOLUTIONS TO PROBLEMS
435
5-6 (Takahashi (2]). 11T11 <- 1 for all T E S. Let 0 E CR(X), the set of realvalued functions in C(X), and e > 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 <- v < p, where dv/dp = 46. Then m(vzli) = n(vib) for all m, n E .C(S). [Note: Another remarkable ergodic mixing theorem is proved by Dye [1].]
Problems 6 6-1. The limit is 3. 6-2.
If k = limn.. )(C")I"n, then k' = limn, A((C')n)1I". What else
can be said about A? What if G is the "ax + b" group?
SKETCHED SOLUTIONS TO PROBLEMS
436
6-3 (Milnor 11)). The class of HZ is 2. With G = Ha, GI/G2 = Z x Z, G2 /G3 = Z so that ((6.17)) the degree d = I - 2 + 2. 1 = 4. 6-4. Use induction. 6-5. Take G = S2.
6-6 (Jenkins [6)). (ii) The freeness of S follows by showing that the (1, 2) component w of the matrix u E S satisfies 1w - I I < 2 if u E aS and )w + I [ < 2
if u E bS. (iii) If for some X E g, 9 E Sp(adX) n (R - {0}) and Y is a 9-eigenvector of ad X, then Span{X, Y} = ge. Otherwise let XI, X2, X3 E g, 91, 92 E R, be such that [X1,X2 + iX3j = (91 - i92)(X2 + iX3), X, 0 0, 9i 0 0. Now use Jacobi to obtain go = Span{Xi, X2, X3}. 6-7. Use (6.20(i)).
6-8 (Auslander and Moore [1)). If a is irrational, then M(a) is a Mautner group and so is not of Type 1. 6-9. Leptin 181-
&-11 (Guivarc'h [1)). G Suppose not. Then ((6.37)) G has a homomorphic image of the form H/Z(H), where H E {S2, S3 , S4}. Show that Z(H) is discrete, and (using B13) that H/Z(H) is not unimodular. [Note: the solvability requirement can be removed.)
6-12 (Milnor 111). The groups rI(S2) = {e}, IrI(P2) = Z2, and ri(S1 x S') _ Z x Z clearly have polynomial growth. Now G = ri (K) is the group generated by two elements a, b with the one relation a2b2 = e. The subgroup H of G generated by {a2, ab} is abelian and normal, and G/H is finite. So G has polynomial growth by (6.22(1)). [Question: For which Riemannian manifolds M is r1(M) amenable?)
6-13 (Jenkins [9), [10), [11]).
C) = 0, c6xc 0 0 in
Let C = C-i E F,(G) be such that
Let a,,= fc q5 d)). Then anon Let {ank} be a subsequence such that ank+2/ank -+ 1. Let Ak(s) _ (f(J+nk+l0(sz)dA(x))/ank. Show that Ak(s) I uniformly on C, and using )µ) (G
(G).
1.
Fubini, that p(G) = Ili m J Ak(s) du(s) = kli mo
g5µ(x) dk(x)/// ! / ank > 0 .
Jc
6-14 (J enkins 111)). Define P(Ox) = 1. Use Problem 6-13 with y =Li a2b2, to show that P extends linearly to X.
6-15. For (i), use (3) of (6.43) and the proof of (6.17). For (ii), if C = C1 E S', (G) generates G. then we can take Kn = Ca(n) for some subsequence {a(n)} using (6.8) and (i).
SKETCHED SOLUTIONS TO PROBLEMS
437
6-16 (Wulbert [1], Lloyd (2], Seever (1]). (i) We can identify H with C(X/ -r),
and, by the Stone-Weierstrass Theorem, H is the closed subalgebra of C(X) generated by B. The first assertion is true if 41, E Ext{v1H : v E C2} ("=" {fi: u E X/ ti}), and true for general p E Cr by the Krein-Milman Theorem. For the second assertion, if yo E S(µ) - {y E X: y - x}, obtain a contradiction
by considering f > 0 in H, f (yo) = 1, f (y) = 0 if y - x. (ii) Note that P,i, (PO)(Pzli) E H; show, using the second assertion of (i), that
i o P(OPr(i) = P¢(x)P0(x) =:i o P((P¢) (P))). [Note: a C'-algebra version of this result is proved in Choi and Effros (2].] 6-17 (Milnes [Si]). Take K to be the polygon connecting, in order, the points
(n, 1/n), (n,1), (n2, n), (-n2, n), (-n, 1), (-n,1/n), and back to (n,1/n) (n > 2). Show that there exists E > 0 such that A((0, 2)K'" n K'")/A(Km) < 1 - e
for all m
and hence contradict the localisation conjecture.
6-18 (Rosenblatt [10)). * If G is not of Type R, we can find y e expg (g the Lie algebra of G), YI, Y2 E g, YI + iY2 ,-E 0, and A E C, [a] < 1, such that
Ady(YI+iY2)=A(YI+iY2). Then(Ady)"(X) 0forsomeX00in{YI,Y2}. If expX = x, show that y"xy " -y e, contradicting the distality of G. C Suppose that G is not distal. Using (6.32), show that the radical S of G is also not distal. There exists a basis for sc (s the Lie algebra of S) such that Ad S C L, the Lie group of upper triangular, complex matrices with diagonal entries of modulus 1. Then L is not distal, and neither is the subgroup N of L with diagonal entries all 1. Now N is nilpotent, and an induction argument based on the central series for L gives a contradiction. [See Moore [1) for more on the distal property.] 6-19 (Auslander and Moore [1], Jenkins [6]).
6-20 (Greenleaf, Moskowitz, and Rothschild [2]). G Suppose that for some
M > 0, [ Tr(Ad x) [ < M for all x E G. Suppose that x E G and Sp(Ad x) = a"} ¢ T. Let r = maxi<:<" [A4]. We can take \I..... Ak to be those eigenvalues of modulus r. We can suppose r > 1. Let Ai = reili (1 < j < k). 0 as N -+ oo. Now use the Kronecker Show that for N E P, [ Ej_I ectve, [ Approximation Theorem [HR1, (26.15)] to infer that for arbitrarily large AT, k - 1/2. This gives a contradiction. [ Zk eir'9' 6-22 (Dixmier [4), Hulanicki [8], Kahane [2]). (i) Let a > 0 be such that [-a, a] contains Sp f and the support of 0. Define 40: Z --. C by ra(n) _
(2a)_I
( a 0(s) exp(-urns/a) ds. 11J
SKETCHED SOLUTIONS TO PROBLEMS
438
Show, using integration by parts, that Vi(n) = O(Inl-('11)) (n - oo). Then take
Co
9 =
E
(n)(e:nnfla - 1) E A.
_00
(ii) Use the fact that if f, g E L, (G, w), then If (t)g(t-'z)Iw(x) <- I (fw)(t)I I(gw)(t-'x)I. (iii) By (6.49), it makes sense to talk of q5 o f. It is sufficient from (i) to show
that lle'1 - ill, = O(nrl) as n --' oo for each ,(i E R. We can suppose that = 1. Define a, w: G - 11, oo) by a(z) = min{n: x E Cn}, w = exp a, where C E Fe (G) is such that f vanishes outside C and U' I C' = G. Let M > 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).
SKETCHED SOLUTIONS TO PROBLEMS
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'.
SKETCHED SOLUTIONS TO PROBLEMS
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 <_
be a partition of S, 81..... sn E S. Find u E R=1 Ssi. Then I Ui1 siSil = maxi<:
from (i). Suppose, now, that S is infinite. Trivially.. S is a.l.c. if (iii) holds. Now suppose that (iv) holds, and S is infinite. It is sufficient to show m = ISI.
Let m E .C(S). With Si. si as above, show m(B) > 0, where B = U 1 siS1. If IBI < ISI; then m(B) = 0 and a contradiction follows. To show the former, construct a sequence {yn} in S such that {y,,B} is a disjoint family-the right cancellative property is used here. Now suppose that (v) holds. With S,, si as above, let Z = {s E S: sis = s (1 < i < n)}. Then ((2.29)) Z is a right ideal of S. Then siS Si n Z and P:5 IZI < I Us 1 siSil. So. p = m. For the final part,
SKETCHED SOLUTIONS TO PROBLEMS
441
if G is a finite group, then p = [G[, m = 1 so that p # m in general. What about infinite semigroups? 7-13. Use (7.26) and Problem 7-12 above.
7-14 (Klawe [31). There exists so E S such that if To = soSso, then [To[ = n.
Every m E 3(S) is supported on To. Suppose that n is infinite. Let .9r(To) = {Ffi: $ E a}, where a is the smallest ordinal of cardinality n. Construct subsets A, of To such that (i), (ii), and (iii) of (7.25) are satisfied, and (iv) of (7.25) is replaced by n { (x 1 dF y 1) : x, y E F,9} # 0 for each pair -,,6 with e < . In place of the rF of (7.25), construct pairwise disjoint sets TE = {xE, i : I < i, j < n) C To - K such that for each e, there exists zE E To such that xiz£xj = xE,i,j (1 < i, j < n). Modify the rest of (7.25) to produce a disjoint family {Dp: p E P) of compact (two-sided) invariant subsets of 1S, each supporting some my, my E 3(S), with [P{ = 22" . If n is finite, show that To is the unique left (right) ideal group in S. 7-15 (Chou (91). Follow the proof of Problem 7-14 with S = G = To, it = [G[. We require that the sets L1,6 satisfy the additional condition: (v) (A$)-1 = AO
7-16-17 (Douglas [2], Kaufman [1], Rajagopalan and Witz [1), Paterson [41, Rosenblatt and Talagrand [1)). (i) Show that if F E .9(G). then OF is left thick in G and so has cardinality [G[ (Problem 1-18).
(ii) (b)'(c). If (c) does not hold, then there exists Fo E .(G) such that (FoxFo) n (FoxFo)-1 0 0 for all x E G. For a, b, c, d E F0, let Aa,b,,,d = (cxd)-1}. If E C A,,,b,,,d, m(E) = m(E-1) for m E 3(G) and {x E G: azb = 3(G) = T (G). Proceed as in Problem 7-15, replacing (v) of its solution by ©" n 0 whenever e < $ < a, 77 < 61 < a. This produces a subset 1' of 3(G) - 3'(G) with Ilk [ = 22w' For Problem 7-17, use Problem 7-16 and the observation (FxF) n (FxF)-1
0forallxEG#* BCF-4e* Bisfinite. 7-18. Let A C S be such that A C S, m(A) = 1 for all m E Z(S), and JAI = r. Then 22° = [Z(S)[ <_ [100(A)'[ < 22' ', giving p < s. Obviously, r < p. [Note: Z. Yang [S4) has shown that m = z in general.)
7-19 (Rajagopalan and Ramakrishnan [2), Klawe [31). c If G is a finite left ideal group in S, then every right thick subset of S D the right thick subset Let a E S be such that T = Sa has smallest possible cardinality. Then GS. show that if T1, T2 are right thick subsets of T, then T1 n T2 34 0. Next show that T is finite, by assuming the contrary and constructing (using transfinite induction) two disjoint right thick subsets of T. This gives the first assertion of the Problem.
442
SKETCHED SOLUTIONS TO PROBLEMS
7-20 (Civin and Yood 11]). Let JH(G) = {F E l,,(G)": hF = F for all h E H, F(x.H) = 0 for all x E G}. Show that JH(G) is a left ideal with square {0} and containing J(G), and that JH(G)/J(G) contains a copy of J(H). This follows using the map F - FF"+J(G), where F E J(H) and FR E J(G) is given by: F1 (0) = F(0IX) (0 E 1.(G)). 7-21 (Granirer [3j..[131). Using (7.28(i)) and (1.20), S contains a finite left ideal group. Now use the left cancellative property and the theory of the kernel ((1.16)).
Index of Terms
and statistics, 4.21 and translation experiments, 4.26 and von Neumann algebras, 2.35 criterion for finitely generated groups.
References for terms in this index are section numbers, except for references to the problems (found at the end of each chapter), which begin with the abbreviation Pr, followed by the chapter number and the problem number. Capital letters A, B, C, D, E refer to the appendices.
4.32
fixed subspace theorem, Pr 2-39 amenable extremely left (right) (ELA, BRA), 2.27
inner, 2.35 K-, 0,19 left, 0.2 n extremely left (n-ELA). 2.30 right, 0.17 amenable as discrete, 7.30 amenable algebras, groups, etc. Banach algebra, 1.30
A-module, Banach, 1.30 "ax + b" group (S2), A abelian Lie groups, connected, B21 abstract L-space, 2.31 action, 0.1
adapted (measure) 4.19 adjoint representation, B12
C'-algebra, 1.31
for Lie algebras and groups, B4, B12
Ado's Theorem, B4
G-space, 2.35 left-semigroup, 0.18
affine left S-set. 0.13
affine map, 1.13
Lie algebra, Pr 3-7 locally compact group, 0.2 right-semigroup, 0.18 semigroup, 0.18 subgroup, maximal, Pr 0-26, 6,46 von Neumann algebra, 2.35 analysis, non-Archimedean, 2.34 analytic maps, BS analytic structure, B1
algebra
Lie, See Lie algebra multiplier, 1.30 symmetric Banach, 6.48 von Neumann, See von Neumann algebra almost connected group, 3.7 almost convergence, Pr 0-13, 0.19, Pr
4-141, Pr 5-8
aperiodic measure 4.19
almost invariant sets, arbitrarily small,
approximately finite-dimensional (AFB) von Neumann algebras, 2.35 arbitrarily small almost invariant sets,
4.27
almost left cancellative (a.1.c.) semigroup, 7.22
4.27
almost periodic functions, 2.10 Bohr, Pr 4-16 amenability
asymptotically right invariant sequence of probability measures, 4.24 automorphism group, 3.3 443
INDEX OF TERMS
automorphism Lie group, B4 Banach A-module, 1.30 amenable, 1.30 dual normal, 2.35(A) element, capacity of, 6.51 symmetric, 6.48 Banach G-module, 1.30 Banach limits, 0.18 Banach-Ruzlewicz Theorem, 0.19, 3.12, 4.27, 4.28, 4.29 Banach S-space, right, Pr 1-15, 2.4 Banach-Tarski Theorem (Paradox), 0.19, 3.12, 3.16 bilinear form, B38 bipartite graph, k-regular, 3.14 Bohr almost periodic functions, Pr 4-16 Borel cross-section, 0.7, C Bore) paradoxical decomposition, Pr 3-15 boundedness, exponential, 6. Bruhat function, 1.11
C`-algebra amenable, 1.31 nuclear, 1.31 reduced, of a locally compact group, 0.19, 4.20 strongly amenable, 1.31 Cl* (G), operator norms on, 4.31 COO maps, B1
C°°-structure, BI Cantor-Bernstein theorem for paradoxical decompositions, 3.13, Pr 3-13 capacity of a Banach algebra element,
compactly generated Lie algebra, B48 completely positive map, 2.35 completeness spherical, 2.34 strongly countable, 2.33 complexification of a real Lie algebra, B56 Conjectures Day's 0.16 Sorenson's, 1.27, 1.29 von Neuman's, 0.16, 0.19, Pr 3-4, 4.30 conjugation, B56 connected Lie groups, Haar measure for, B13 connected abelian Lie groups, B21 connected group, See almost connected group containment, See weak containment continuity joint, 2.33 separate, 2.33 continuous actions, fixed-point theorem for, 2.24 contractible grup, Pr 4-3 convergence, almost, Pr 0-13, 0.19, Pr 4-14f, Pr 5-8 coordinate functions for a chart, BI coordinates, local, B1 cover, locally finite, B1 covering space, B3 simply connected, B3 Criterion of Weyl, Pr 4-23 cross-section Borel, 0.7, C local, B18
6.51
Cartan decomposition, B57 Cartan subalgebra, B53 Clech-Propfsil theorem, Pr 7-16 central series, A. B23 chart, B1 coordinate functions for, BI local, B1 class of a nilpotent group, A Clifford semigroup, 4.21 cofinite volume, 4.29 Cohen's Factorisation Theorem, 1.30 cohomology, Hochschild, 1.30 compact Lie algebra, B48 compact normal subgroup, maximal, 5.15 compact real norms, B38 compact subgroup, maximal, B61, B62 compactification, Stone-tech, 0.11
Day's Fixed-Point Theorem, 1.14 Day's Conjecture, 0.16 degree of a compactly generated, locally compact group of polynomial growth, 6.4 derivation, B2, B5, 1.30 G-, 1.30 inner, B5, 1.30 derived group, A derived series, A, B23 diagonal, virtual, Pr 1-36 differential of a map, B2 distal group, Pr 6.18 Division Theorem, 3.14 dual normal Banach A-module, 2.35 Dumford-Schwartz-Zygmund Theorem, 5.11
INDEX OF TERMS edge of a graph, 3.14
elementary groups (EG), 0.16, 3.11 end of a graph, 3.14 Engel's Theorem B27 equicontinuous actions, fixed-point
theorem for, Pr 2-16
equidecomposable for S0(n + 1, R), Pr 3-14
equidecomposable, G-, 3.13 ergodic group action, Pr 5-8 ergodic mixing theorem, Pr 5-8 ergodic representation (antirepresentation), 5.1
ergodic sequence of probability measures,
Pr 5-5 ergodic theorems Mean, 5.7 Pointwise, 5.20; admissible for, 5.17 Sine's Mean, Pr 5-7 other 5.21
ass inf, ess sup, 0.0 essential, See neo-unital etal4e, See spread out experiment, translation, 4.26 exponential growth, 6.1 exponential map denisty of the range of, E for a Lie group, B9 exponentially bounded, 6.0 exponentially bounded groups, Pr 0-20, 7.0
Folner conditions for, 6.43 Translate Property for, 6.42 exposed point, 7.28 extension, group,
extremely left (right) amenable (ELA), (ERA), 2.27
[FcJ- group, 4.23 filter property for right thick sets, Pr 7-19 finitely generated groups, amenability criterion for, 4.32 fixed subspace theorem, amenability, Pr 2-39 fixed-point property, nonzero, and polynomial growth, 6.44 fixed-point theorems Day's, 1.4, 1.14 for equicontinuous actions, Pr 2-16 for jointly (separately) continuous actions, 2.24
Ryll-Nardzewski, 2.36
445
For strongly amenable C'-algebras, 1.31
Folner conditions for exponentially bounded groups, 6.43 for locally compact groups, 0.7, 4.0f for semigroups, 4.8f, 4.22 strong, 4.22, Pr 4-52 Folner sequence, 4.27 form (compact) real, B38 bilinear, B38 Killing, B38 nondegenerate, B38 symmetric, B38 Fourier algebra, 4.34 Fourier-Stieltjes algebra, 4.34 free abelian group on two generators, Pr 0-19
free group on two generators, 0.6, 3.0f free semigroup on two generators, Pr 0-28 functions
p-integrable, 5.8 p-measurable, 5.8 rapidly decreasing, 6.49 Riemann measurable, 4.27, Pr 4-34 fundamental groups. B3 and polynomial growth, 6.41 G-derivation, 1.30
G-equidecomposable, 3.13
G-finite von Neumann algebra, 2.36 G-module, Banach, 1.30 G-set, left (right, two-sided), 0.1 G-space, amenable, 2.35 Generalized Translate Property, 6.42 Golod-Shafarevitch group, 3.11, 4.33 graph bipartite. k-regular, 3.14 edge of a, 3.14 end of a, 3.14 group action, ergodic, 4.27 group extension, group(s) "az + b" (S2), A almost connected, 3.7 amenable locally compact, 0.2 automorphism, 3.3 contractible, Pr 4-3 derived, A distal, Pr 6-18 elementary (EG), 0.16. 3.11
446
INDEX OF TERMS
exponentially bounded, See exponentially bounded group(s)
[PC]-, 4.23 free abelian, on two generators, Pr 0-19 free abelian, on two generators, Pr 0.19 free, on two generators, 0.6, 3.0f fundamental, B3; and polynomial growth, 6.41 Golod-Shafarevitch, 3.11, 4.33 Heisenberg, Pr 4-29 isometry, of R", pr 0-25, 3.12 Lie quotient, B18 Lie, See Lie group(s) lotlaly compact, See locally compact group(s) locally finite, Pr 0-21, 3.11, Pr 3-10 matrix, B19 motion. 6.22 nilpotent, A periodic, 3.11 polycyclic, 6.40 semidirect product, A semisimple locally compact, 3.7 solvable, A
supersolvable, Pr 0-27 symmetric locally compact, 6.48 Type Rm, Pr 3-8 Type T, 6.46 uniformly distributed, Pr 4-26 weakly Wiener, 6.48 growth polynomial, 6.0 sube,cponential, 6.41
Haar measure for connected Lie groups, B13
Heisenberg group, Pr 4-29 Hochschild cohomology, 1.30 homeomorphism, local, B3 homomorphism, Lie, B4 homotopic paths, B3 Hunt-Stein Theorem, 4.24, 4.25
right. 0.17 topologically left, 0.9 topologically right, 0.17 unique left, 4.27 invariant measure (for (G, X, A)), 2.32, 3.15
invariant minimax test, 4.25 invariant state, topological, on VN(G), Pr 4-42 invariantly complemented subspace, Pr 2-20
inverse semigroup, Pr 1-25, 4.21 inversion invariant mean, Pr 2-7, Pr 7-15 irreducible Lie algebra module, B4 isometry group of R", Pr 0-25, 3.12 isomorphism, Lie, B4 Iwasawa Decomposition Theorem, B59
jointly continuous, 2.33 jointly continuous semigroup, 2.10 K-amenable, 0.19 Kazhdan Property (T), 4.29
kernel, 1.16 Poisson, 6.46 Killing form, B38 Konig's Theorem, 3.14
LI(G)", LI(S)", radical of, 7.31 convolutions, nonmeasurability of, 7.29 L-space, abstract, 2.31
left amenable, 0.2 left(right) amenable, extremely (ELA, ERA), 2.27 left (right, two-sided) G-set , 0.1 left (right) introverted, 2.6 left invariant mean, 0.1 unique, 4.27 left invariant subspace, 0.17 left measurable semigroup, 1.29 left regular representation, 1.31, 4.19 left (right) reversible, 1.22
ideal, Lie, 54 injective von Neumann algebra, 2.35 inner amenable, 2.35 inner derivation, B5, 1.30 inner invariant mean, 2.35 introverted, left (right), 2.6
invariant mean, 0.17
inner. 2.35 inversion, Pr 2-6, Pr 7-15 left, 0.1
left S-set, 0.1
left-semigroup, amenable, 0.18 left thick (right thick) subset, 1.20 topological, 1.30, Sol. Levi-Malcev Decomposition, B42. Pr 3-7 Levi-Malcev Theorem, B42 Lie algebra(s), B4 amenable, Pr 3-7 compact, B48 compactly generated, B48
INDEX OF TERMS
radical of, B35 real, complexification of, B56 semidirect product, B5 semisimple, B38 semisimple, root of, B53 simple, B39 Type R, 6.23 weight space of, B31 and groups, disjoint representation for, B4, B12 nilpotent or solvable, B23 Lie automorphism, B4 Lie group(s), B6 adjoint representation for, B4, B12
automorphism, B4 connected abelian, B21 connected, Haar measure for, B13 exponential map for, B9 nilpotent or solvable, B23 radical of, B35 semidirect product, B22 semisimple, B38 Lie homomorphism, B4 Lie ideal, B4 Lie isomorphism, B4 Lie product, B4 Lie representation, B4 Lie subalgebra, B4 Lie subgroup, B17 Lie's Theorem, B25 linear functional, translation invariant, Pr 3-16
linear transformation nilpotent part of, B31 semisimple part of, B31 local chart, B1 local coordinates, B1 local cross-section, B18 local homommorphism, B3 locally compact group(s) amenable, 0.2
Felner conditions for, 0.7, 4.0f of polynomial growth, 6.0 radical of, 3.7, B35 reduced C'-algebra of, 0.19, 4.20 semidirect product of, A semisimple, 3.7 symmetric, 6.48 Type 1, Pr 2-31 Type R, 6.0 locally finite cover, B1 locally finite group, Pr 0-21. 3.11, Pr 3-10
447
locally null, 0.0 lower central series, A, B23 manifold, B1; See also submanifold map(s) affine, 1.13
anayltic, Bl C°o, B1 completely positive, 2.35 differential of, B2 exponential, density of the range of, E exponential, for a Lie group, B9 nonexpansive, Pr 2-18 Markov-Kakutani Theorem, 0.14 Marriage Theorem, 3.14 matrix group, B19 maximal amenable subgroup, Pr 0-26, 6.46
maximal compact normal subgroup, 5.15 maximal compact subgroup, B61, B62 Mean Ergodic Theorem, 5.7 Sine's, Pr 5-7 mean(s), 0.0 inner invariant, 2.35 invariant, 0.17 left invariant, 0.1 mutually singular, 7.8 posterior for, Pr 4-19 right invariant, 0.17 singularity for, 7.8 symmetrically invariant, Pr 0-7 topologically left invariant, 0.9 topologically right invariant, 0.17 unique left invariant, 4.27 measure(s), See also probability measures adapted, 4.19 aperiodic, 4.19 Haar, for connected Lie groups, B13 invariant, (for (C, X, A)), 2.32, 3.15 Plancherel, Pr 4-29 problem of, 3.12 recurrent, 6.46, 6.47 spread out (etal4e), 6.46 strictly aperiodic, 4.19 strongly equivalent, 5.1 transient, 6.46, 6.47 mixing theorem, ergodic, Pr 5-8 motion group, 6.22 µ-integrable function, 5.8 p-measurable function, 5.8 multiplier algebra, 1.30 Multiplier Theorem, Wendel's, 4.26
INDEX OF TERMS
448
multiplier topology, 1.30 mutually singular means, 7.8 Mycielski's Theorem, 7.15, D
n extremely left amenable (n-ELA), 2.30 aeo-unital (= essential), 1.30 nilpotent group, A class of, A
nilpotent or solvable Lie groups and algebras, B23 nilpotent part of a linear transformation,
symmetry, and the Wiener Property, 6.48f
pointwise for a mean, Pr 4-19 probability measures asymptotically right invariant sequence of, 4.24 ergodic sequence of, Pr 5-5 problem of measure, 3.12 Property (T), Kazhdan, 4.29 Property P for von Neumann algebras, 2.35
B31
non-Archimedean analysis, 2.34 nondegenerate form, B38 nonexpansive maps, Pr-18 nonmeasurability of Lec (G) convolutions, 7.29
nonzero fixed-point property, and polynomial growth, 6.44 normal series, A, B23 normal subgroup, maximal compact, 5.15 nuclear C'-algebra, 1.31 null, See locally null
Ol'shanskii's counter example to von Neumann's Conjecture, 4.33 one-parameter subgroup, B9 operator norms on Ci (G), 4.31 Ore's Theorem, 1.25 paradoxical decomposition (p.d.), 3.12, 3.16
Borel, Pr 3-15 Cantor-Bernstein Theorem, 3.13, Pr 3-13
for F2, Pr 3-12 for Cn, Pr 3-18 partition of unity, Bi paths, homotopic, B3 periodic functions almost, 2.10
Bohr almost. Pr 4-16 periodic group, 3.11 Plancherel measure, Pr 4-29 Pofntwise Ergodic Theorem, 5.20 admissible for, 5.17 Poisson spaces and kernels, 6.46 polycyclic group. 6.40 polynomial growth, 6.0 and fundamental groups, 6.41 and the nonzero fixed-point property,
6.44
quotient group, BIB radical of a Lie algebra (Lie group), B35 of a locally compact group, B35
of Li(G)", LI(S)", 7.31 random walk of law u on G, 6.47 rapidly decreasing function, 6.49 real form, compact, B38 real Lie algebra, complexification of, B56 recurrent measure, 6.46, 6.47 reduced C'-algebra of a locally compact group, 0.19, 4.20 Reiter's condition, 0.7, PR 0-11, 4.If representation adjoint, B12 ergodic, 5.1 left regular, 1.31, 4.19 Lie, B4 right regular, 2.35(H)
strongly measurable, 5.8 weakly measurable, 5.5 representation theory, type R, 6.45 reversible, left (right), 1.22
Riemann measurable function, 4.27, Pr 4-34
right regular representation, 2.35(H)
Note: For right amenable, right invariant, etc., see also left amenable, left invariant, etc. right amenable, 0.17 right Banach S-space, Pr 1-15, 2.4 right invariant mean, 0.17 right invariant sequence of probability measures, asymptoticall, 4.24 right invariant subspace, 0.17 right S-set, See S-set right thick sets, filter property, Pr 7-19
root of a semisimpie Lie algebra, B53
INDEX OF TERMS
root space, B53 Ryll-Nardzewski Fixed-Point Theorem, 2.36
S-set of ine left, 0.13 left (right, two-sided), 0.1 S-space, right Banach, Pr 1-16, 2.4 semidirect product Lie algebra, B5
Lie group, B22 of locally compact groups, A of semigroups, 1.29 semidiscrete von Neumann algebra, 2.35 semigroup(s) almost left cancellative, 7.22 amenable, 0.18 Clifford, 4.21
F0lner conditions for, 4.8f, 4.22 free, on two generators, Pr 0-28 inverse, Pr 1-25, 4.21 jointly continuous, 2.10 left amenable, 0.18 left measurable, 1.29 semidirect product of, 1.29 separately continuous, 2.10 uniformly discrete, in two generators, 6.0
semisimple Lie algebra (Lie group), B38 root of, B53 semisimple locally compact group, 3.7 semisimple part of a linear transformation, B31 separtely continuous, 2.33 separately continuous semigmup, 2.10 sequence of probability measures asymptotically right invariant, 4.24 ergodic, Pr 5-5 series, central (lower central, upper central, derived, normal), A, B23 similarity problem, Pr 0-34, Pr 1-40 simple Lie algebra, B39 Sine's Mean Ergodic Theorem, Pr 5-7 singularity for means, 7.8 solvable group, A solvable or nilpotent Lie groups and
algebras, B23 Sorenson's Conjecture, 1.27, 1.29 spectral radius, 6.48 spectrum, 6.48 spherically complete, 2.34 spread out (etal4e) measure, 6.46
449
statistics, amenability and, 4.24f Stone tech compactiScation, 0.11 strictly aperiodic measure, 4.19 strong Fr lner conditions, 4.22, Pr 4-52 strongly amenable C'-algebra, 1.31 strongly countably complete, 2.33 strongly equivalent measures, 5.1 strongly measurable representation, 5.8 subalgebra Cartan, B53 Lie, B4 subexponential growth, 6.41 subgroup Lie, B17 maximal amenable, Pr 0-26, 6.46 maximal compact, B61, B62 maximal compact normal, 5.15 one-parameter, B9 submanifold, B14 subset, left thick (right thick), 1.20 subspace invariantly complemented, Pr 2-20 left invariant (right invariant, two-sided invariant), 0.17 summing net (sequence), 4.15f supersolvable group, Pr 0-27 symmetric Banach algebra, 6.48 symmetric form, B38 symmetric locally compact group, 6.48 symetrically invariant mean, Pr 0-7 symmetry, polynomial growth, and the Wiener Property, 6.48f tangent space, B2 Tarski's Theorem, 3.15 test, invariant minimax, 4.25 thick, See left thick (right thick) topological invariant state on VN(G), Pr 4-42
topological left thick, 1.30, Sol. topologically left invariant mean. 0.9 topologically right invariant mean, 0.17 topology, multiplier, 1.30 transference, Pr 4.44 transient measure, 6.46, 6.47 transition, 4.26 Translate Property, 2.32 for exponentially bounded groups, 6.42 generalised, 6.42 translation experiments, 4.26 amenability and, 4.26
450
INDEX OF TERMS
translation invariant linear functional, Pr 3-26 two-sided S-set, 0.1
two-sided invariant subspace. 0.17 Type 1 locally compact group, Pr 2-31 Type R group, Pr 3-8 Type R Lie algebra, 6.23 Type R locally compact group, 6.0 Type R representation theory, 6.45 Type T group, 6.46 uniformly discrete semigroup in two generators, 6.0 uniformly distributed group, PR 4-26 unique left invariant mean, 4.27 upper central series, A, B23 vector field, B2 virtual diagonal, Pr 1-36 Vitali-Hahn-Saks Theorem 4.27, 7.9 volume, cofinite, 4.29 von Neumann algebras amenable, 2.35 approximately finite-dimensional (AFD), 2.35 G-finite, 2.36
injective, 2.35
Property P for, 2.35 semidiscrete, 2.35 von Neuman's Conjecture, 0.16, 0.19, Pr 3-4, 4.30 O1'shanskii's counterexample, 4.33
wandering set, See weakly wandering set weak containment, 4.29 for inverse(and Clifford) semigroups, 4.21
weak containment property, 0.19, 4.20 "weak Frobenius" property, Pr 4-41 weakly Wiener group, 8.48 weakly measurable representation (antirepresentation), 5.5 weakly wandering set, 5.21 weight space of a Lie algebra, B31 Weil's formula, 1.11 Wendel's Multiplier Theorem, 4.26 Weyl's Criterion, Pr 4-23 Wiener Property, 6.48 Wiener Property, symmetry, and polynomial growth, 6.48f
Index of Symbols
R(X) (Borel algebra) B(Y) (bounded operators on a Banach
A fl B, A U B (intersection, union of sets A, B)
A-B(={xEA:xVB})
space), 0.1
AAB(_(A-.B)U(B'-A))
C (complex numbers) Cc(X) (continuous functions with compact support)
JAI (cardinality of A)
A- (closure of A) B (A equidecomposable to a subset A of B), 3.13 A y B (A equidecomposable to B), 3.13 A' (complement of A, commutant of a von Neumann algebra A)
C1, C f (H), 4.17, Pr. 4-20
CI(G) (lower central series), App. A Cl(x) (conjugate class) C* (G) (C'-algebra of G), 0.19, 4.20 Cj (G) (reduced C'-algebra of G), 4.20 Co(X) (continuous functions vanishing at
A (identity adjoined to a Banach algebra
A), 6.48 A. (prednal of a von Neumann algebra A) (2.35(B)) AC (almost convergent sequences), Pr.
oo)
C(X) (continuous bounded functions)
''g (lower central series), B23 '(X) (compact subsets of X) ire(G) (compact neighbourhoods of e) XA (characteristic function of A)
0-13
ACI(G) (almost convergent functions), Pr. 4-14 Ad, AdG (adjoin representation), B12 ad,adp (adjoins representation), B4 Af (K) (affine functions), 1.13 AG(amenable groups), 0.16 A(G) (Fourier algebra), 4.34 AP(S) (almost periodic functions), 2.10 Aut G (automorphism group), 3.3, B20 Auto (automorphism group), B5 Q' K, a= (inner automorphisms), 3.3 .a< (class of G with C' (G) amenable),
Der o (derivations on g), B5
D(G) (Span of the (0 - ¢s)), 2.1 gig (derived series), B23 d(01 Cf), 4.17
AG, G (modular function), 1.11 11(A) (multiplier algebra), 1.30
EE'' 7.8 EG (elementary groups), 0.16 Exp K (exposed points), 7.28 Ext K (extreme points), 2.24
1.31
ePG,exp,epg (exponential map), B9
BAP(G) (Bohr almost periodic functions), Pr. 4.16 B(G) (Fourier-Stieltjes algebra), 4.34 /3S (Stone-tech compactification), 0.11 B(N; X; p) (p-integrable vector-valued functions), 5.8
F (field)
PC, [PC]-, 4.23
f(fix) = f(x-1)o(x-1)),f *g, 1.1 F2 (free group on two generators)
FS2 (free semigroup on two generators) 451
INDEX OF SYMBOLS
452
F(X) (finite subsets of X) FC, 2.4 !P(E), 2.24
G (simply connected covering group) G#, 4.29 G (dual of G), 4.29
Ge (identity component) GL(n, F) (invertible n x n matrices) G. (isometry group of R") gt, 4.20
P(G), F(S), Pp(G), 0.0, 4.2 PM(X) (probability measures) PF (Arens-type product), 2.4 PSL(2,R), 3.1 11(P) (finite permutations of P), 3.15 l iaEA Go (Cartesian product), Pr. 3.1 xp (left regular representation), 1.31, 4.19 Fr (right regular representation), 2.35(H) mi (M) (fundamental group), B3
.9(X) (subsets of X), 2.32 (W), Pr. 4-40, Pr. 6-25
9t (complexification of 9), B4 2.21, Pr/ 2-16
Q (rationale)
Rp (p-harmonic functions), 6.46 H xp K (semidirect product), App. A
R (real numbers) r(G) (radical of G), 3.7 rad A (radical of a Baaach algebra), 7.31 r'(G) (rank of abelian G), 6.10 9t(G), DD(S), Stt(A) (right invariant means), 0.17
1(G),IA(G) (inner automorphisms), 3.3 I(G), 4.14 Int. 9, B41
3(A), 3t (A), 3(G), 3t (G) (invariant means), 0.17 J(E), 7.31
3(G), 31(G), It (0), 31(X), 2.0, 2.4 31 (E, R) (invariant functionals), 2.31
K(G) (maximal compact normal subgroup), 5.15 L(V) (linear transformations), B4 L1(G), Lco(G), 11(S), loo(S), Loo(G,R), loo(X,K), 0.0, 0.1, 2.3, 2.34 Z(G), Z(S), L(A), .t(G),.Ct(A); 0.2, 0.17 .C(S"), 4.28
A (left Haar measure)
a" (left Haar measure on R") .C({x6)),. C, 4.16
1rn;n1 (={rEZ:m
M. (F) (n x n matrices) M(X) (regular Bore] measures) A*,µ",A-L" 1.1 m(G), m(S), 7.1, 7.22 031(G),931(A) (means), 0.0
4'(G) (A-measurable sets), 0.0 4C4(R), 3.12
N (natural numbers) NF. 0.16 11
11n (Lp-norm)
O(n,R), O(n), B19 P (positive integers)
QX (: Li(G) -- L1(G/H)), 1.11
Rep G, 4.29
S" (n-sphere), 3.16 S2, Si' , Sa, 6.36
SL(n,F), SO(n,R), SU(n), B19 sl(n,R), ao(n,R), 3.2, B19 Sp(a) (spectrum of a), 6.48 S(A) (support of measure {t) *oeA Go (free product) Tv, 6.23 TH(: M(G) -» M(G/H)), 1.11 T (circle group) U(G), Uj(G). Ur(G), 1.3 Ur(S) (uniformly continuous functions), 2.10
U(A) (unitary group of A), 1.31 U(n), B19 VN(G) (von Neumann algebra generated by tr2(G)), 2.35(c) WLir(S), WP(S), 2.10 X1, X0, XE, 5.1 X(S), 2.4 XM (Poisson space), 6.46
X' (dual of a locally convex space)
z, f, 0.0
x°f,f'z,x+µ,IA »x,op,,uo, 0.7, 1.1 [z, y] (commutator), App. A xO, ex, xn, nz, 0.1 Z (integers) Z(G), Z(9), 3.1, B4
