The Analogue of the Group Algebra for Topological Semigroups (Chapman & Hall CRC Research Notes in Mathematics Series)

Research Notes in Mathematics

98

H A M Dzinotyiw ¥i

The analogue of the group algebra for topological semigroups

Pitman Advanced Publishing Program BOSTON · LONDON · MELBOURNE

H A M Dzinotyiweyi University of Zimbabwe

The analogue of the group algebra for topological semigroups

Pitman Advanced Publishing Program BOS'ION ·LONDON· MELBOURNE

PllMAN PUBLISHING LIMITED 128 Long Acre, London WC2E 9AN PI1MAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto

© H A M Dzinotyiweyi 1984 First published 1984 AMS Subject Classifications: (main) 43A20, 43Al0, 4302,4602 (subsidiary) 22A20 Library of Congress Cataloging in Publication Data Dzinotyiweyi, H. A.M. The analogue of the group algebra for topological semigroups. Bibliography: p. Includes indexes. 1. Topological semigroups. 2. Group algebras. I. Title 512'.55 83-25003 QA387.D95 1984 ISBN 0-273-08610-3 British Library Cataloguing in Publication Data Dzinotyiweyi, H. A.M. The analogue of the group algebra for topological semigroups.-(Research notes in mathematics; 98) 1. Topological algebras I. Title II. Series 512' .55 QA326 ISBN 0-273-08610-3 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers. Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford

Preface

A great deal of abstract harmonic analysis on a locally compact topological group G is pivoted at the object L1 (G) - the so called group algebra of G. The purpose of this book is to present analogues of L1 (G) for topological semigroups and, in particular, study those results whose success is largely due to the existence of such analogues on the underlying topological semigroup.

Accordingly, this book is primarily addressed to research workers

and post-graduate students in the field of abstract harmonic analysis on topological semigroups.

Also, research workers in the fields of

(theoretical) probability theory and functional analysis may find some useful information. In Chapter 1 we collect together some preliminary results needed throughout this book.

Most of the results are of an elementary nature.

We also

present the set of all bounded Radon measures on a topological semigroup as a convolution measure algebra. In the absence of a Haar measure on a topological semigroup, one analogue of the group algebra is the set of all absolutely continuous bounded Radon measures, (i.e. bounded Radon measures whose translates are continuous with respect to compact sets). a~gebra

This analogue of the group

is studied in Chapter 2, mainly for topological semigroups admitting

enough real-valued continuous functions to separate points.

For many cases,

it is shown that translates of absolutely continuous measures are weakly (and even norm) continuous.

Another analogue of the group algebra for

topological semigroups, namely the set of all bounded Radon measures which are equi-regular, is also studied in Chapter 2;

in particular, we examine

how far the latter differ-s from the set of all absolutely continuous measures.

In general, the theme of Chapter 2 says that among the various

analogues of the group algebra for topological semigroups, the set of all absolutely continuous measures seems to be the richest in structure. In Chapter 3 various results on the topological structure of certain topological semigroups (which are generalizations of those) admitting "many" For instance, for such semiabsolutely continuous measures are studied.

groups, we show how tar Lranslates of neighbourhoods remain neighbourhoods. At the end of the Chapter, we characterize continuity of semigroup actions on normed linear spaces in terms of separable orbits, and the equivalence of weak and norm continuity of such actions is established. In Chapter 4 various results on the space of weakly uniformly continuous functions WUC(S), the space of uniformly continuous functions UC(S), the space of weakly almost periodic functions WAP(S) and other spaces of functions on a topological semigroup S, admitting a non-zero absolutely continuous measure, are studied.

Functions with separable orbits, which

are also measurable with respect to every absolutely continuous measure, are shown to be "almost" uniformly continuous.

For a large class of topological

semigroups S we show that the existence of an invariant mean on A is equivalent to the existence of a topological invariant mean on B, for various spaces of functional& A and B on S where UC(S) is a subspace of A n B. Further, we show that the set of all invariant means on WUC(S) can be greater or equal to 2c, where c denotes the cardinal of continuum.

The

difference of the spaces WAP(S) and WUC(S), for many topological semigroups S that include all noncompact locally compact topological groups, is shown to be large in a drammatic manner - in fact we show that the quotient space WUC(S)/wAP(S) may contain a linear isometric copy of 1~.

We also examine

the prospect of turning the Stone-Cech compactification of a topological semigroup S into a semitopological semigroup under an operation extending that of S.

The Chapter ends with a study of the regularity of multiplication

in measure algebras on semigroups and the size of the radical of the second dual of such algebras. Chapter 5 is a natural continuation of Chapters 2 and 3.

Here various

characterizations of an absolutely continuous measure are given for the case of a topological semigroup studied in Chapter 3. For a discrete semigroup S the analogues of the group algebra studied in this book, of course, coincide with the algebra t 1 (s). The latter algebra is still interesting, particularly as it often exhibits results full of algebraic content.

We study some aspects of this algebra in Chapter 6.

particular we show that:

In

there are many infinite semigroups S and weight

functions w such that the corresponding weighted algebra t 1 (S,w) has regular multiplication, the algebra t 1 (S) may be factorizable without an approximate identity, the existence of an approximate identity for t 1 (S) may

be characterized in terms of an ordering type of property on the set of all idempotent elements of S. We also study the amenability of t 1 (s). Strictly speaking, all our studies, from Chapter 1 to 6, fall under the field of Archimedean harmonic analysis, as all functions and measures involved pick their values in the Archimedean fields 1R or C.

In Chapter 7

we replace 1R and C by a non-Archimedean field K and present a Fourier theory for the resultant measure algebras. Various open problems and detailed references are cited throughout the book.

Some results employed, that do not strictly fall under the subject

of abstract harmonic analysis, are assembled in Appendices A and B. We have discussed and communicated with many colleagues on many matters presented in this book.

In particular we would like to mention J.W. Baker,

Ching Chou, Paul Milnes, A.L.T. Paterson, G.L.G. Sleijpen and A.C.M. van Rooij;

whose company and correspondence we have always found to be very

inspiring.

Finally I wish to thank the referee for his invaluable comments

and the Publishers for their cooperation. University of Zimbabwe November, 1983

H.A.M.D.

groups, we show how tar Lranslates of neighbourhoods remain neighbourhoods. At the end of the Chapter, we characterize continuity of semigroup actions on normed linear spaces in terms of separable orbits, and the equivalence of weak and norm continuity of such actions is established. In Chapter 4 various results on the space of weakly uniformly continuous functions WUC(S), the space of uniformly continuous functions UC(S), the space of weakly almost periodic functions WAP(S) and other spaces of functions on a topological semigroup S, admitting a non-zero absolutely continuous measure, are studied.

Functions with separable orbits, which

are also measurable with respect to every absolutely continuous measure, are shown to be "almost" uniformly continuous.

For a large class of topological

semigroups S we show that the existence of an invariant mean on A is equivalent to the existence of a topological invariant mean on B, for various spaces of functionals A and B on S where UC(S) is a subspace of A n B. Further, we show that the set of all invariant means on WUC(S) can be greater or equal to 2c, where c denotes the cardinal of continuum.

The

difference of the spaces WAP(S) and WUC(S), for many topological semigroups S that include all noncompact locally compact topological groups, is shown to be large in a drammatic manner - in fact we show that the quotient space WUC(S)/wAP(S) may contain a linear isometric copy of ~~.

We also examine

the prospect of turning the Stone-Cech compactification of a topological semigroup S into a semitopological semigroup under an operation extending that of S.

The Chapter ends with a study of the regularity of multiplication

in measure algebras on semigroups and the size of the radical of the second dual of such algebras. Chapter 5 is a natural continu'ltion of Chapters 2 and 3.

Here various

characterizations of an absolutely continuous measure are given for the case of a topological semigroup studied in Chapter 3. For a discrete semigroup S the

analog~es

of the group algebra studied in

this book, of course, coincide with the algebra ~ 1 (s).

The latter algebra

is still interesting, particularly as it often exhibits results full of algebraic content.

We study some aspects of this algebra in Chapter 6.

particular we show that:

In

there are many infinite semigroups S and weight

functions w such that the corresponding weighted algebra

~

1

(S,w) has

regular multiplication, the algebra ~ 1 (s) may be factorizable without an approximate identity, the exjstence of an approximate identity for ~ 1 (s) may

be characterized in terms of an ordering type of property on the set of all idempotent elements of s. We also study the amenability of t 1 (s). Strictly speaking, all our studies, from Chapter 1 to 6, fall under the field of Archimedean harmonic analysis, as all functions and measures involved pick their values in the Archimedean fields IR or C.

In Chapter 7

we replace IR and C by a non-Archimedean field K and present a Fourier theory for the resultant measure algebras. Various open problems and detailed references are cited throughout the book.

Some results employed, that do not strictly fall under the subject

of abstract harmonic analysis, are assembled in Appendices A and B. We have discussed and communicated with many colleagues on many matters presented in this book.

In particular we would like to mention J.W. Baker,

Ching Chou, Paul Milnes, A.L.T. Paterson, G.L.G. Sleijpen and A.C.M. van Rooij;

whose company and correspondence we have always found to be very

inspiring.

Finally I wish to thank the referee for his invaluable comments

and the Publishers for their cooperation. University of Zimbabwe November, 1983

H.A.M.D.

Contents

PREFACE Chapter 1 : PRELIMINARIES 1. Semigroups 2. Topological semigroups 3. The algebra of bounded Radon measures M(S) 4. Some continuity properties of Radon measures Chapter 2 : ANALOGUES OF THE GROUP ALGEBRA FOR SEMIGROUPS 1. Absolutely continuous measures 2. Weak and norm continuous translations of measures 3. Equi-regular measures 4. Quasi-invariant measures 5. Multiplicative linear functional& and semicharacters 6. Notes on references Chapter 3 : FOUNDATION SEMIGROUPS AND THEIR GENERALIZATION 1. Introducti-on 2. The structure of stips 3. The role of idempotent& in a stip 4. Actions on normed linear spaces

1 1

2 3 9

11 11

20 26 35 38 43

45 45 46

51 57

Chapter 4 : ALGEBRAS OF FUNCTIONS 1. Uniformly continuous functions 2. Invariant means 3. The size of the difference WUC(S) \ WAP(S) 4. The Stone-Cech compactification of a semigroup 5. Regularity of multiplication in semigroup algebras

101

Chapter 5 : CHARACTERIZATIONS OF ABSOLUTELY CONTINUOUS MEASURES 1. Continuity of measures under translation on a stip 2. Measures with separable orbits 3. Measures vanishing on emaciated sets 4. Locally quasi-invariant measures

111 111 115 120 127

63 63

71

83

93

5. Continuity over direct translates of compact sets 6. Some further notes Chapter 6 : THE CONVOLUTION ALGEBRA t 1 {S) 1. 2. 3. 4.

Regularity of multiplication in weighted semigroup algebras Existence of approximate identities A factorizable Banach algebra without an approximate identity Amenability of the algebra t 1 {s)

129 131 132 132 139 143 144

Chapter 7 : NON-ARCHIMEDEAN FOURIER THEORY 1. Prerequisites 2. Semicharacters 3. The non-Archimedean measure algebra M{S) 4. The non-Archimedean semigroup algebra Mn{S) 5. Notes

152 152 156 161 165 172

Appendix A : THE STRICT TOPOLOGY AND DUALITY THEORY 1. The strict topology 2. A generalized Stone-Weierstrass Theorem 3. A generalization of the Riesz-representation Theorem

174 174

Appendix B : 1. Weakly 2. Weakly 3. Weakly

178 178 178 179

WEAKLY COMPACT SETS compact subsets of a Banach space compact subsets of M{X) compact subsets of C{X)

174

176

Bibliography

182

Index of symbols

191

Subject index

195

1 Preliminaries

In this chapter we collect together some of the notation and terminology used throughout this book and include some standard and elementary results for ease of reference.

We adopt the following nomenclature for references:

The symbol [n] refers to reference number n in the bibliography list; notations of the form "Theorem n.m.k." refer to a theorem given in section m of chapter n under the label "m.k. THEOREM", and of the form "Theorem m.k." refer to a theorem given within the same chapter under the label "m.k. THEOREM".

A similar convention is adopted for references to lemmas,

propositions and other items. 1. SEMIGROUPS 1.1 DEFINITIONS. called a

A nonempty set with an associative binary operation is

semig~up.

Let S be a semigroup.

For all A, B ~ S and x

£

S we

write AB := {ab : a £ A and b £ B}, A-lB := {y £ S : ay £ B for some a £ A}, A- 1x := A- 1 {x} and x- 1B := {x}-1B. By symmetry our definitions for BA-1 , xA-l and Bx-l must be clear to the reader.

We say S is left (or right)

cancellative if xy = xz (or yx = zx) for all x,y,z £ S implies that y = z. S is said to be cancellative if it is both left and right cancellative. A point x in S is called a left (or right) cancellation element if it has the property that xy

= xz

(or yx

zx) implies y

= z,

for all y,z in S.

The

centre Z(E) of a subset E of S is the set Z(E) := {s £ S : sx ~ xs for all X £ E}. A non-empty subset H of s is called a subsemigroup if HH = H, a left (or right) ideal if SH C H (or HS ~H) and an ideal if it is both a left and right ideal. Let e be an element of if ex

xe

s.

We say e is an idempotent if e 2

e, a

ze~

= xe = e

= x)

for all x in S, a left (or right) identity if ex x (or for all x in S and an identity (or unity) if it is both a left and

right identity. By a

g~up

The set of all idempotent elements inS is denoted by E5 • we mean a semigroup G with an identity element 1 such that

corresponding to each x in G there is a unique y in G with the property xy

= yx = 1.

A non-empty subset of a semigroup S that is itself a group 1

under the operation of S is called a subgPOup of 1.2.

s.

Our reference for results on (algebraic) semigroups that are not

proved or explicitly stated in this book is Clifford and Preston [20]. 2. TOPOLOGICAL SEMIGROUPS

Att topotogies

aFe

2.1 DEFINITIONS. called a

te~

assumed to be HausdoPff thPOughout this book. Let S be a semigroup endowed with a topology.

(or Pight) semitopotogicat semigPOup if the map x

Then S is +

xy (or

yx) of S into S is continuous, for all y in s. We say S is a semitopotogicat semigPOup if S is both a left and right semitopological semi-

x

+

group.

If the mapping (x,y)

+

xy of S x S into S is jointly continuous S is

called a topotogicat semigPoup. A group G is called a topotogicat gPOup if G is· endowed with a topology with respect to which G is a topological semigroup and the inversion map x + x-l of G into G is also continuous. -1

-1

Let A1 , ••• ,An be subsets of s. We define A1e A2 := {A1A2 ,A1 A2 ,A1A2 }, Al 8 A2 8 A3 := U{Al 8 B : BE A2 8 A3 } u U{B 8 A3 : BE Al 8 A2 } and hence inductively similarly define A1 8 A2 8 ••• 8 An. A subset E of a topological semigroup S is said to be Petativety neo-compact if E is contained in a union of sets in A1 8 ••• 8 An for some compact subsets A1 , ••• ,An of S. We say S is tocatty Petativety neo-compact if every point of S has a relatively nee-compact neighbourhood. The reader should be in a position to easily solve the following exercises. 2.2 EXERCISES (a) Every locally compact topological semigroup is locally relatively neo-compact. (b) There exist locally relatively nee-compact topological semigroups that are not locally compact. ~

~

(c) Let S be a topological semigroup such that C D and DC are compact whenever C and D are compact subsets of S. Then a subset E of S is relatively nee-compact if and only if E is relatively compact. (In particular, relatively nee-compact subsets of a topological group are precisely the relatively compact subsets.) 2

We say H is a minimal ideal of a semigroup S if H is an ideal contained in every other ideal of S. result without proof.

With this in mind we mention the following

(For a proof - see e.g. Hofmann and Mostert [62,

pages 15 and 16].) 2.3 THEOREM. Let S be a compact topological semigroup. Then there e:xists a minimal ideal I of A. Further>~ I n Es is non-empty and eie is a topological subgroup of s with identity e whenever- e E I n Es. One of the most important results in the theory of topological semigroups is the following.

2.4 THEOREM (Ellis [42, Theorem 2]). LetS be a locally compact semitopological semigroup. If S is algebroaically a group~ then S is a locally compact topological gr-oup. 3. THE ALGEBRA OF BOUNDED RADON MEASURES M(S)

3.1 SOME GENERAL NOTATION ON FUNCTIONS. are assumed to be Hausdorff.

Recall that all topological spaces

Let X and Y be any topological spaces and S

any topological semigroup. We denote by B(X,Y) the set of all functions from X into Y.

If E is a

subset of X and fa function in B(X,Y), we take fiE to be the restriction of the function f to the set E. Let CK(X,Y) := {f E B(X,Y) : fiK is continuous (on K) for all compact K S X} and C(X,Y) := {f E B(X,Y) : f is continuous}. The topological

sp~ce

X is said to be a k-space if CK(X,Y)

= C(X,Y)

for

every topological space Y and a k 0 -space i f CK(X,1R) = C(X,1R). (As illustrated in Chapter 4, k 0 -spaces are not necessarily k-spaces.) For all f in B(S,Y) and x inS we define the functions xf and fx in B(S, Y) by xf(y) := f(xy) Let m(X) := {f

E

and

fx(y) := f(yx),

for ally inS.

B(X,C) : f is bounded}, C(X) := m(X) n C(X,C),

C00 (X) := {f E C(X) : f vanishes outside a compact set} and C (X) :• {f E C(X) : f is arbitrarily small outside a compact set}. 0

We note

that m(X) is a Banach space under the norm II II X given by II f II X : = sup{ If(x) I : x E X}. Whenever C(X) is treated as a normed space, it will be understood that

II II X is

the norm, unless otherwise stated.

For

3

any subset E of X we have the if

X

if

X E

E

ch~cteristic

jUnction, XE• of E given by

E X'

E.

By the support of a jUnction f in m(X) we mean the closure of the set {x E X : f(x) ~ 0}. The closure of a subset E in X is denoted by

E.

If the real-valued

functions in C(X) separate points of X, we say X is C-distinguished. 3.2 SOME GENERAL NOTATION ON MEASURES.

Let X be any topological space and

M(X) the space of all bounded complex-valued (inner and outer) regular Borel

II .

measures with the usual total variation norm II the space of bounded Radon measures.)

(Equivalently M(X) is

Corresponding to each

~

in M(X) we

have 1~1 the measure arising from the total variation of ~ and if E is a Borel subset of X then ~IE is the me~sure given by ~IE(B) := ~(B n E) for all Borel B C X. If x is in X then x denotes the point mass at x. We also denote the completion of a measure ~ by ~ (where the distinction from the notation for a point mass will lie in the context). Let~

and v be measures in M(X) and A a subset of M(X).

of~ we mean the set supp(~) := {x EX:

By the support

I~I(V) > 0 whenever Vis an open

neighbourhood of x}.

We define the set function 1~1* on subsets of X by

I~I.<E) :=sup{ I~! (C)

: Cis a compact subset of E} and say~ is con-

centrated on a subset B of X if I~ I* (B) = II~ II

.

Thus if ~ is concentrated

on a Borel subset B then ~.IB = ~. We say ~ and v are orthogonal, in short denoted ~ L v, if ~ and v are concentrated on disjoint Borel sets. Whenever~

is orthogonal to every measure in A we may denote this

by~

LA·

A subset E of X is A-measurabZe if E is measurable with respect to every measure in A and A-negZigibZe if E is negligible with respect to every measure in A. By the notation v

0.

supp(v*~).

Let A be a subset of M(S). A is called a convolution or an L-subaZgeb~ if A is a norm-closed sub~lgebra of M(S)

that is solid. If A is an ideal of M(S) (i.e. v*~• ~*v € A for all v € A and ~ € M(S)) which is also an L-subalgebra, we call A an L-ideaZ. (For a more general theory of convolution measure algebras and L-ideals we refer the reader to e.g. J.L. Taylor [101].) We say A is Zeft (or nght) tronsZation invanant i f x*11 (or ~*x) is in A for all ~ in A and x in S. Analogously a subset B of m(S) is Zeft (or

nght)

invanant if xf (or fx) is in B for all f in B and x in A set is t~Zation invanant if it is both left and right translation

s.

t~Zation

invariant. The closure of the set 8

u{supp(~)

:

~ €

A} is called the foundation of A.

3.10 EXERCISES.

Let A be a subset of M(S).

(i) The foundation of A coincides with the set dA(S). (ii) If A is a (non-zero) subalgebra of M(S) then the foundation of A is a closed subsemigroup of S. (iii) There is a C-distinguished topological group G such that C0 (G) = {0}. (iv) There are_ locally compact topological semigroups such that neither C0 (S) nor C00 (S) is translation invariant. 3.11 NOTATION.

Let v and

functional&~

We define

~eh(n)

:= h(~*n),

for all n in M(S).

~

be measures in M(S) and h a functional in M(S)*.

e h, h e he~(n)

~

:= h(n*~)

ho~(x) := he~(x)

for all x in S. If A is a subset of M(S)*, let and

vehe~

in M(S)* by and

We define the functions

~oh(x) := ~oh(x),

Ae~, ~eA

and

~0A

:=

~oh,

and

{~oh

vehe~(n) ho~

:= veh(n*~)

and

voho~

on S by

voho~(x) := vehe~(x)

hEA} and similarly define

Ae~.

3.12 NOTES. Proposition 3.8 is proved, for the case where S is locally compact, in the paper of J.S. Pym, "Idempotent measures on semigroups", Pac. J. Math. 12 (1962), 685-698. The notion of a foundation of a subset A of M(S) is taken from A.C. and J.W. Baker [5]. In fact following [5] we shall define foundation semigroups in Chapter 2. The other results in this section are taken from Dzinotyiweyi [30].

4. SOME CONTINUITY PROPERTIES OF RADON MEASURES Throughout this section s denotes any topoZogicaZ semigzaoup. some zaesuZts mainZy needed in Chapteza 2. 4.1 LEMMA.

Let

~

We prove

be a measuzte in M(S) and E a BozaeZ subset of s.

The map

x-+ l~l<x- 1 E) of s into lR is (i) Zoweza semicontinuous when E is open~ (ii) uppeza semicontinuous when E is cZosed. Proof. £

Let E be open and (xa) a net converging to a point x inS. > 0, we can choose a compact set K C x- 1E such that l~l<x- 1 E\K)
I ~ l~l<x~ 1 E6x- 1 E), it is sufficient to To this end, given show that I ~ I (xa-1 E6x-1 E) + 0 if I ~ I (xa-1 E)+ I ~ I (x-1 E). £ > 0, we have that eventually Proof,

and, by Lemma 4.l(ii),

Hence -1 I~ I (x-1 E6x E) a .

I ~ I (x -1 E\X -1 E) + I ~ I (x -1 E\X-1 E) a

a

2 I ~ I (x -1 E\x -1 E) + I ~ I (x -1 E) a

< 2l~l<x- 1 E\x- 1 E) + a


l = 1J- Y*IPICx- 1C))dlviI

~

J1 IPIldlv1Cy)

~ JKI IPI ~(x

and hence

-1

X

E) - E.

-1 (xC) Ca

X

-1 E. a

Consequently

lim

(1)

a

Hence if E were closed. (i) and Lemma 1.4.l(ii) would give the result. Then by complementation. our lemma also follows for open E. But 1 by the outer regularity of x*~• we can choose an open set U ~ E with

So with (1) in mind we get x*~(E) -


0}

is a neighbourhood of s and clearly O(s)

£ {x

£

S : x- 1 (sc 1) n y0 C21 ~ +}

-1 -1 "' (scl)(yoc2 >

·s

-1 (sclc2)yo •

(s

£

S)

Now suppose that f € CK(S,Y) for some Hausdorff topological space Y, and let (sa) be a net converging to s in S. Then since (sc 1 c 2 )y~ 1 is a neighbourhood of s we have that eventually (s a yo ) is a net converging to syo in the compact set sc1c 2 • Hence f(s ao y) ~ f(sy) o or f y (s) a ~ f y (s). Hence o_

f

€

y0

C(S,Y) and so y 0

€

S n u. r

It follows that z

€

o 2

_

S and so H c S • r - r

From Lemma 1.4 and Theorem 1.12 we have the following result.

IfH is the foundation of M!(s) then H2 is contained in the closure of the set {x € S : ti*p is a ~eakly continuous mapping of S into M(S), foF all p € Ma(S)}. 1.13 COROLLARY.

1.14 THEOREM. (a) Sr

is a

We have that le~

ideal of

s.

(b) If s is the foundation of Ma(S) and p £ Ma(S), then p*x is concentmted on Sr' foF aZZ x € s 2 • In pazoticulaF, if in r:.ddition S has an identity element, then Ma(S) is concentmted on Sr. Proof. That Sr is a left ideal follows trivially from our definitions. We now prove item (b). Let p € Ma(S) be fixed, D := supp(p) and K be any compact subset.of S \Sr. sr' since sr Then, for all X € sr we have Dx is a left ideal and hence

s

IPI*x ..

o.

Now S2 S Sr' by Theorem 1.12. So, by the definition of Ma(S), we must have IPI*x(K) - 0 for all X € s 2 • Since p*x 0 be given.

€

a

We can find a 6 > 0 such that

for every Borel E S S with m(E) < 6 we have lvi(E) < £. Let (xa) be a net converging to x in S and K be a compact subset of S. We can find an open neighbourhood 0 of K such that m(O\K) < 6. Then m(s- 1 (0\K)) < 6 and hence lvl<s- 1 (0\K)) x*lvl x*lvi(O)

- 2£

> x*lvi(K)

- 2£

by Lemma 1.4.l(i)

Recalling Lemma 1.4.l(ii), we conclude that L(S,m) c M1 (S). -

a

(c) Let S be a topoZ.ogioaZ. group. Then either M!(s) is aero or S is Z.ooaZ.Z.y oompaot and M!(s) = M~(S) = L(S,m) ~here m is a Z.eft Haar measure on S. (cf. [48] and [59]). Proof. That if M!(s) is non-zero then S is locally compact follows from Corollary 1.10. Let m1 be a right Haar measure on G. Item (b) above implies that L(S,m) S M!(s) and L(S,~) S M~(S) •. Let p € M!(s) and V be a compact neighbourhood of the identity of S. Then clearly p IK (B)= 0 for all n,p p n

£

E, then (x*IPI> IK (B) =

o for all x

£

(*)

s.

p

Setting m :• IPI +

E

l~n,p<m

2-(n+p)i *IPI we have that m £ Me(S)+ since n,p

Me(S) is an L-ideal (by Theorem 3.7), and p «·m. So it remains only to 1 (S,H). show that m £ Meq . Let B be a Borel set contained in Hand such that m(B) = 0. Then x *IPI (B n K ) • 0 for all n £ E and hence n,p P x*IPI(B n K) - 0 for all X € s, by(*). Hence x*IPI(B) = 0 for all X € s, p since B c

m

u K • p=l p

In particular we have that

x*(xn,p*IPI>CB) for all x

£

S and n,p

m(x-lB)

• -x*

=Xin,p*IPI £

Ip I (B)

E.

=

o

Consequently

+

- o. This completes our proof of item (i). Item (ii) follows from item (i) by taking H • S. In a related manner one can prove the following result. 4. 3 THEOREM. . If S is a-compact and Fa (S) • S then Ma(S) • u{L(S,m) : m € Ma(S)+ n ~(S)}. (We omit the details of the proof as they are also contained in the proof of Theorem 5.4 .2.)

36

4.4 REMARK.

It is interesting to note that Theorem 4.2(i) does not hold

if we replace Fe(S) by Fa(S) and Me(S) by Ma(S), even when S is also locally compact and commutative. For an example we take S to be the topological semigroup constructed in item 3.4. Now let p be a lebesgue measure with supp{p) = {(x,l) : x € [2,3]} and let H be the compact subset {(x, y) : x € [7 ,8] and y € [1, 2]}. Then IPI € Ma(S). Suppose (on the contrary) there is a measure min Ma{S) n ~q(S,H) with IPI 0 an~ so m(z- B) ~ 0. By this contradiction our y y assertion follows. 4.5. One may ask whether Mteq (S,H) is contained in Me (S) or Ma (S) when H has non-empty interior. Unfortunately this is not the case even when S is a subsemigroup of a group. (For example let S be the additive subsemigroup [o,oo) of 1R and take m to be the point mass at 4 and H := [0,3]. Then m € M!q(S,H) but m. Ma(S).) However we have the following exercise for the reader. EXERCISE. Let S be a locally compact topological group with a Haar measure m. If H is a Borel subset of S with non-empty interior and p is a measure in M!q(S,H), then p


,. h 0 be given andy

E

So

S be fixed.

Wen~

llson - yonll .,:: £} E

O(y).

Setting

II" -

A :

yonll

< £},

we note that the Hahn-Banach theorem implies that xon

E

weak-cl(O(yon)) = norm-cl(O(yon)) = O(yon).

Consequently llxon- yonll .,:: £ and·x

E

O(y).

So O(y) • O(y) and O(y) is

closed. 1

Next we note that Mn(S,A) :=

{\I E

A : x

+

xo\1 is a norm continuous map

of S into A} is norm and so weakly closed in A. Let A be a compact neighbourhood of 1 and suppose (for the moment) that 1 is ~-isolated.

Then (the right handed version of) Proposition 3.8(i) say!

that A=

BG

for some countable B ~A and compact subgroup G of S. that {xon : x

E

By Lemma 4.5 we have

G} is norm compact and so we can find a countable subset D

of G such that {xon : x

E

D} is norm dense in {xon : x

E

G}.

Now for each u in A we have that uon

E

weak-cl({bxon : b

B and x

E

E

G})

since A

- weak-cl({bo(xon)

b

E

B and x

E

G})

= weak-cl({bo(don)

b

E

B and d

E

D}) since n

weak-cl({xon : x

E

= BG E

M!(G,A)

BD}).

Since BD is countable we thus have that n has a weakly separable left orbit over A. 1 So by Theorem 4.4 we have xon E Mn(S,A) for all x E dense in S and M!(s,A) is weakly closed, we thus have

n

E

weak-cl({xon : x

E

s1 •

Since

s1

is

1 s1 }) S Mn(S,A).

Thus our theorem follows for the case where 1 is

~-isolated.

61

Finally we suppose that 1 is not necessarily 6-isolated. Let U be any fixed compact neighbourhood of 1 and x £ s 1 • Lemma 3.4 says that we can find a 6-isolated idempotent e in S such that Ux c eSe. Thus eSe is a stip with a 6-isolated identity e; so it follows that-n £ M!(eSe,A) and hence that {yon : y £ Ux} is a norm compact subset of A. Hence {uo(xon) : u £ U} contains a countable norm (and so weakly) dense subset. By Theorem 4.4, we thus have yo(xon) £ M1 (s,A) for ally£ s 1 • Since x was arbitrarily chosen n t in s 1 and s 1 • s 1s 1 we have son£ Mn(S,A) for all s £ s 1 • Now 51 = S, so item (i) and the weak-closure of M!(s,A) imply that n £ weak-cl({son : s £ s 1 }) c M1 (s,A). n Thus (ii) holds and our proof is complete.

4.7 REMARK. Recalling the proof of Lemma 4.5 and the inner regularity of Radon measures, the reader should now be in a position to prove the following generalization of Lemma 2.1.6(ii) : Let S be any topological semig~up acting on a norn~ed linea:!' space A such that the map x -+ xon of S into A is weakly continuous fozo some n £ A. Then fozo all + £ A* we have +(von) =

whezoe von :•

62

J +(xon)dv(x)

J xondv(x),

fozo all v £ M(S).

4 Algebras of functions

Various results on algebras of functions on a topological semigroup S can be found in the literature.

These include results on the so called algebras

of weakly almost periodic functions, uniformly continuous functions and other subalgebras of C(S).

It is not our intention in this chapter to give

a complete study of such function algebras;

instead, we extract some of

those results whose success is largely due to the existence of absolutely continuous measures or a related property on the underlying semigroup.

Thus

to a large extent this chapter deals with some applications of our studies in Chapters 2 and 3. 1. UNIFORMLY CONTINUOUS FUNCTIONS First we collect together some definitions and notations. 1.1 DEFINITIONS.

Let S be a topological semigroup.

A function f in C(S) is said to be left uniformly (or left weakly

uniformly) continuous if the mapping x

~

xf of S into C(S) is norm (or

weakly respectively) continuous. We denote the space of all left uniformly continuous functions on S by LUC(S) and that of left weakly uniformly continuous functions by Similarly one defines the spaces RUC(S) and

Rl~C(S)

Ll~C(S).

of right uniformly

(respectively) continuous functions on S. We say a function is uniformly continuous on S if it belongs to the space UC(S) := LUC(S) n RUC(S), and weakly uniformly continuous if it belongs to the space WUC(S) := LWUC(S) n RWUC(S). A function f in C(S) is said to be weakly almost periodic if the set { f : x X

€

S} is relatively weakly compact.

We denote the set of all weakly

almost periodic functions on S by WAP(S). Although we have defined weakly almost periodicity of a function in terms of its left translates an equivalent definition in terms of right translates is also acceptable.

This follows easily from the following result which is

an immediate consequence of Grothendieck's Theorem (see Appendix B.7). 63

LetS be a topological semigroup and items are equivalent:

1.2 PROPOSITION. foll~ing

f

Then the

C(S).

£

(i) the set {X f : x £ S} is relatively weakly compact; (ii) ~henever {xn} and {ym} are sequences in S the intersection of the closures of the sets {f(xnm y ) : n < m} and {f(x y ) : n > m} is non-empty. nm (iii) the set {fx : x

£

S} is relatively

~eakly

compact.

As an application of Theorem 3.4.4 we have the following characterization of uniformly continuous functions in terms of separable orbits.

First we

note that a topological semigroup S has a natural left anti-action on C(S) give by the map (s,f)

~sf

of S x C(S) into C(S).

Lets be a stip and f (ii) and (iii) are equivalent~ ~here

1.3 THEOREM.

(i) f has a

£

With this in mind we have

Then item

C(S).

(i)

implies

(ii)

and

left orbit over a neighbourhood of 1; (ii) xf is left unifol'171ly continuous~ for all x £ s1 ; (iii) xf has a ~eakly sep~le left orbit over a neighbourhood of 1~ for all x £ s1 . Proof.

Let y

~eakly sepa~le

S and £ > 0 be given.

£

Then this Theorem will follow from

Theorem 3.4.4 i f we can show that the set {s £ S : llsf - yfll S ~ d closed. So suppose (s a ) is a net converging to s in S with

lis a f

-

lll s ~

£.

Hence lf(s a z) - f(yz)l 0 f(x) := { 0 for x ~ 0, (x e S) has a separable left orbit over S.

II xn f-

o

fll

s

+1

andso

! then xn + 0 while

But if xn := - 2

f.LUC(S).

1.5. For every topological semigroup S one easily notes that So for a stip S, taking A := C(S) in Theorem 3.4.6, we have THEOREM.

If S is a

(i) LUC(S) (ii) WAP(S)

stip~

WAP(S~WUC(S).

we have that

LWUC(S);

S UC(S).

1.6. The preceding results are taken from Dzinotyiweyi ([33] and [34]). When S i~ a locally compact group, Theorem 1.5(ii) has been proved in a different way by Burckel [13]. We now turn to the situation where the underlying topological semigroup Sis assumed to be the foundation of Ma(S). Hence we remind the reader that the term "foundation semigroup" mentioned hereafter is as defined in item 2.3.3. The results given in the remainder of this section are taken from Dzinotyiweyi and Milnes [38] •

any topoLogicaL semigroup S wheneve~ we taLk of a jUnction f in m(S) as having a (weakLy) separabLe Le~ o~bit~ this wiLL be done in te~s of the Le~ anti-action of S on m(S) given by the mapping (s,f) + sf of S x m(S) into m(S), fo~ aLL s e Sand f e m(S). FUrthe~~ f is said to be measurabLe i f f is p~asurabLe~ fo~ aLL P e Ma(S). Fo~

65

1.7. The following remarks will enable us to simplify the proof of our next lemma. Let S be a foundation semigroup with identity element 1. Let B be the space of all bounded linear operators from Ma(S) into LUC(S) with the norm operator topology and note that S has a left action on B given by the mapping (s,P) + soP of S x B into B. Here soP(v) := P(s*v)

:= {P

Let B

€

for all" in Ma (S).

B : the mapping s

note :hat the integral voP := P

€

Ba (see e.g.·

(v,P)

+

II s

voP.

[27]).

Given e

+

soP of S into B is continuous} and hence

f soPdv(s)

exists for all v in Ma(S) and

Thus Ba is an Ma(S)-module under the map >

II soP - P II

0 and P in Ba we have that

d is an open neighbourhood of 1· and hence, for v in Ma(S) with v(W) = llvll = 1, we have

W :=

S :

€

llvoP - Pll

=




~

s.

However the functions

fare continuous on Hand so we sn 0 such that II tf - s fll H ~ c5 for all n in lN. Hence n

II tf

- s f n

II

5

~

II tf

- s f n

II H ~

c5 > 0,

for all n

This inequality contradicts the fact that {snf : n {sf : s

€

U}.

€

lN}

€

lN.

is dense in

By this conflict our lemma follows.

We recall that for a stip Sand f

€

C(S), Theorem 1.3 says that f is

"very close" to being left uniformly continuous as soon as f has a weakly separable left orbit over a neighbourhood of 1. Our proof there is entirely independent of measure theory and relies more on topological techniques.

If

in addition S is a foundation semigroup then a more general result can be achieved via measure theoretic techniques. Theorem.

This is the message of our next

LetS be a foundation semi~up with identity element 1 and a fUnation with a weakly left sepaPable oPbit oveP a neighbouPhood of 1. If f is Ma (S) -measumble then sf is left uniformly aontinuous, foP all x in s 1 • 1.10 THEOREM. f € m(S)

Proof.

Lemma 3.4.3 says that f has a separable left orbit over some

neighbourhood of 1. LUC(S), for all x,y conclude that xf

€

Lemmas 1.8 and 1.9 then imply that xy f • y ( x f) is in s 1 • Since s1s 1 • s1 (see Corollary 3.2.5), we LUC(S), for all x € s 1 • €

69

If we visualize Theorem 1.10 as the corresponding measure theoretic generalization of Theorem 1.3, then we may take the following result to be the corresponding measure theoretic generalization of Theorem 3.4.4.

Let S be a foundation semigroup with identity element 1 and let S have a left anti-action on a normed linear space A. Let n e A have a ~eakly separable left orbit over a neighbourhood of 1 and suppose the functions s ~ ~(son) of S into Care Ma(S)-measurable, for all ~ e A*. Then s ~ so(xon) is a norm continuous rmpping of S into A for all x e s1 . 1.11 COROLLARY.

Proof.

Fix

~

e A* and consider the function f : s ~ ~(son) of S into C.

By Lemma 3.4.3 we may assume that n has a (norm) separable left orbit over a neighbourhood of 1.

Now for all

~

e C(S)* and x,y e S we have that

and so f has a weakly separable left orbit over a neighbourhood of 1. Theorem 1.10, we have that xf e LUC(S), for all x in xf e C(S) and so the mapping s Sl,

~

s1 •

By

In particular

so(xon) of S into A is weakly continuous,

Recalling Theorem 3.4.6 our result follows.

for all

X E

1.12.

In a general sense a function f e m(S) is said to be almost periodic

on the semigroup S if the set { f : x X

E

S} is relatively norm compact.

A

well known result of von Neumann says that a Haar measurable almost periodic function on a locally compact topological group is continuous.

Now if S is

a locally compact topological group and xf e C(S) for some x in S then f

C(S), and a function is Haar measurable if and only if it is Ma(S) -

E

measurable.

Noting that every almost periodic function on a topological

semigroup S has a separable left orbit over S, by Theorem 1.10 we obtain (in an elementary manner) the following generalization of von Neumann's result. (Davis

[24]

also proved von Neumann's result, but the proof is rather long

and technical.)

Let S be a foundation semigroup with identity element 1 and f e m(S) an almost periodic function.

COROLLARY.

Iff is Ma(S)-measurable, then xf is a continuous almost periodic function. (i)

70

If S is a group and atmost periodic jUnction. (ii)

1. 13

REMARK.

f

is Haazo measumbte, then

f is

a continuous

Since there are non-continuous characters on some locally

compact topological groups and characters are almost periodic functions, the measurability condition cannot be completely dropped in Corollary 1.12 and hence in Theorem 1.10.

[see e.g. Hewitt and Ross [59, page 405] for

the existence of discontinuous characters on the additive group 1R~ 1.14.

We recall that on a locally compact topological space countable

unions of meager sets are meager.

Now the reader can easily note that;

if

in the proof of LeiiDJia 1.9, we replace "Ma(S)-negligible" by "meager", use Theorem 1.3 in place of LeiiDJia 1.8 and recall LeiiDJia 3.4.3, we have the following extension of Theorem 1.3.

(We leave the details as an exercise

for the reader.)

Let S be a stip and f E m(S) a jUnction that has a teft separable orbit over a neighbourhood of 1 and such that the set {s E S : f(s) ~ h(s)} is meager, for some hE C(S). Then xf E LUC(S), for aU X E Sl •

PROPOSITION. ~eakty

2. INVARIANT MEANS

Throughout this seJtion let S be any topological semigroup with an identity element 1 and such that s .is a k-space coinciding ~ith the foundation of Ma(S) unless othel'wise specified. In particular, recalling Theorem 2.1.8 and Corollary 2.2.7 we have that S is

C-distinguished and Ma(S)

2.1 TERMINOLOGIES.

~

Mn(S).

Let A be a closed subspace of M(S)* such that xeh E A

for all x E S and h E A, and e E A where e is the functional which maps to

~(S)

for

all~

~

E M(S).

An element m of A* is a mean on A if llmll = m(e) = 1

and

m(f) ~ 0

for every non-negative functional f in A. A left invariant mean (LIM) on A is a mean m E A* such that

71

m(ieh)

= m(h)

for all

x £ S and h £ A.

H of M(S), let P(H) := {p £ H : p

For any subset

~ 0 and

!lull

= 1}. Now

if A is such that peh £ A for all p £ H and hE A, a P(H) -Left invaY'iant mean

(P(H) - LIM) on A is a mean m £ A* such that m(peh)

= m(h)

for all

p £ P(H)

and h £ A.

In particular a P(M(S)) - LIM on A is called a topoLogicaL left invaY'iant

mean (TLIM).

(Trivially a TLIM is always a LIM.)

If A is a closed translation invariant subspace of m(S) containing all constant functions, then we naturally take a LIM (or (TLIM) on A to be a mean min A* such that m(xf)

= m(f)

(or m(pof)

= m(f),

respectively) for

all x £ S, f £A and p £ P(M(S)). Let H be any locally compact topological semigroup.

AmenabiLity of H

has been defined to mean the existence of a LIM on (a) LUC(H) by e.g. Namioka [74] and Jenkins [63];

(b) MB(H) -the measurable functions in m(H)

where His a subsemigroup of a group, by e.g. Lau [71]; e.g. Wong [104]. In the case where H is a group, Greenleaf

[53]

and (c) M(H)* by

showed that all such

definitions are equivalent and also are equivalent to related definitions for a TLIM. We extend Greenleaf's result to topological semigroups in Theorem 2.4.

Our Theorem could also be deduced from Theorems 6.6 and 6.7 of

Paterson

[so]

if S is assumed to be locally compact.

The actual presentation

given here is taken from Dzinotyiweyi [31]. For a comprehensive account on the subject of invariant means on locally compact topological groups we refer the reader to Greenleaf

[53}.

2.2 LEMMA. Let f £A and p £ M(S),whePe A is any of the spaces UC(S), LUC(S), RUC(S), C(S) and MB(S) := {g £ m(S) : g is a Borel function}. Then (i) uof

£

A;

(ii) if p,v £ M (S) and h £ M(S)*, then poh £ LUC(S) and uohov £ UC(S). a

Proof.

(i) Note that for all x,y £ S we have uofx(y)

llyll

72

=

lluofx- uofyll 5 ~!lull

= (uof)x(y)

llfx- fyll·

and so

Hence item (i) holds for A= RUC(S). Also item (i) holds for A = C(S) by Lemma 1.3.4. Next we show that f £ LUC(S) implies that pof £ LUC(S). Evidently LUC(S) is norm and hence weakly closed. If on the contrary pof. LUC(S), the Hahn-Banach Theorem requires the existence of a + £ LUC(S)* such that +(pof) I 0 and+= 0 on LUC(S). Then recalling Remark 3.4.7 we would get the ridiculous situation

oI

+(pof) =

J+<xof)dp(x)

=

o.

By this conflict we conclude that pof £ LUC(S). Consequently item (i) also follows for A= UC(S), and it remains to show that f £ MB(S) implies pof £ MB(S). To the latter end, we first note that we may assume p to be a positive measure without losing generality. Now for any closed set F C S we have that the function poxF is upper semicontinuous, by Lemma 1.4.l(ii), so poxF is Borel measurable. Next if {F } is an increasing (or decreasing) sequence of Borel sets with n

poxF measurable (n £ 1N) then, for all s £ S, n

POXF (s) :• n co

f XFn (xs)dp(x)

-+

lJOXF(s),

co

where F := u Fn (or F := n Fn• respectively). Recalling the monotone n=l n=l class lemma and the fact that closed subsets generate the Borel subsets of S we have poxB £ MB(S) for every Borel set B C S. Finally if {fn} is a bounded sequence of simple Borel functions converging pointwise to f, then lJofn(s) + pof(s) for all s £Sand so pof £ MB(S). (ii) For all x,y £ S we have llx(poh) - y(poh) II S = sup{ lh(p*xs) - h(p*ys) I ~

II h II

s

£

S}

llp*x - P*Y II •

Since M (S) c M (S), we have poh £ LUC(S). a - n

Noting that

pohov = (peh)ov = po(hev) and peh, hev £ M(S)*, it follows that pohov £ UC(S). 73

The argument involving MB(S) in our Lemma is taken from Paterson 2.3 LEMMA.

Let m be a LIM on . UC(S).

Lao].

Then

(i) m is a TLIM on UC(S); (ii) m(pohov) = m(nohov) Proof.

For any

~· E

fo~

P(M(S)) and f

m(~of) = J m(xof)d~(x) •

aLL v,p,n

m(f)~(S)

E

E

P(Ma(S)) and h

E

M(S)*.

UC(S) we have

by Remark 3.4.7

• m(f);

and so item (i) follows. We now prove item (ii). By item 2.2.11 we may take (pa) to be a bounded (norm) approximate identity forM (S). Then a

lml

-+ 0 and our result follows.

Let A1 and A2 be any of the spaces: UC(S), LUC(S), RUC(S), C(S), MB(S), Ma(S)* and M(S)*. Then the foLLowing items ~e eq~ivaLent: 2.4 THEOREM.

(i)

The~e ~sts

a P(Ma(S)) - LIM on A1 ;

(ii)

the~e

exists a LIM on A2 ;

(iii)

the~e

exists a TLIM on A2 •

Proof. We note that UC(S) ~ A1 £ M(S)* for all choices of A1 • So the existence of any of the three types of left invariant mean on M(S)* implies (by restriction) the existence of the same type of left invariant mean on A1 • Now let m be a P(Ma(S)) -LIM on UC(S). 74

Then, with the above remarks in

mind and recalling that a LIM on UC(S) is a TLIM, our Theorem will follow if we can show that there exists a TLIM on M(S)*. To this end we fix measures n and v in P(Ma(S)) and define the function M on M(S)* by M(h) := m(nohov) Since nohov for any

~ E

E

for all

h

M(S)*.

UC(S), we evidently have that M is a mean on M(S)*.

P(M(S)) we have that

M(~oh)

E

~*n E

Further,

P(Ma(S)) and so

:= m(no(~oh)ov) = m(~*nohov)

= m(nohov) = M(h).

by Lemma 2.3(ii)

This completes our proof. 2.5 SOME OPEN PROBLEMS.

Of course the success of Theorem 2.4 is pivoted

around the object Ma(S).

It is natural to ask what the situation would be

when existence of Ma(S) is not assumed. For instance in Chapter 3 we studied an important class of semigroups S which behave as if they were foundation semigroups though the latter is still unknown.

The following

problem is therefore reasonable. PROBLEM A.

Let S be a stip.

We conjecture that

(i) there is a LIM on C(S) if UC(S) has a LIM; (ii) there is a TLIM on C(S) if there exists a LIM on C(S). Should the question whether every stip is a foundation semigroup turn out to be answered affirmatively then Problem A would follow.

However it would

still be interesting if Problem A could be settled without the use of Ma(S). The preceeding remarks invoke another question.

As stated before, our

Theorem 2.4 is a generalization of a result well known for groups - see e.g. Greenleaf [53, Theorem 2.2.1]. Now for a topological group G we know that G is locally compact if and only if Ma(G) is non-zero.

Hence considering

the situation where Ma(G) is zero, we have the following problem. PROBLEM B. Let G be any (non-locally compact) topological group. We do not know whether the existence of a LIM on C(G) implies the existence of a TLIM on C(G).

Further, we do not know whether there is a LIM on C(G) as 75

soon as there is a LIM on UC(G). First we warn the reader that, for the remainder of this

2. 6 TERMINOLOGY.

section, S is not necessarily as stated at the beginning of the section. Let S be any locally compact topological semigroup and assume the notations introduced in item 2.1 for this S. A net or sequence (pa) c P(Ma(S)) is said to be conve~ent

to topotogicat inuaPiance if v*pa - Pa

~eakty

+

(or

st~ngty)

0 and Pa*v - Pa + 0

weakly (or strongly, respectively) in Ma(S) for all v

€

Ma(S).

Let IM(WUC(S)) denote the set of (left and right) invariant means on WUC(S). We now commence an argument aimed at estimating the size of IM(WUC(S)).

Let S be a tocatty compact topotogicat then

2. 7 LEMMA.

Petativety

semi~up.

If S is not

neo-compact~

m(f) • 0 foP att f

€

C (S) and 0

m

€

IM(WUC(S)).

Let f € C (S) be positive and put K := supp(f). Proof. Since S is not oo relatively neo-compact, we can choose a sequence {xn} in S such that X

n+

So if n

~

l

! -1 -1 -1 't K(x 1· Ku ••• ux K)

for all n

n

-1

k we have xn K n

sup.p(x f) n supp( n

~

-1

·~

K=

+ or,

€

1N •

equivalently,

+·

f) •

Consequently nm(f) • m(

xl

f + ••• +

xn

f)


0, ~e can find n > n 0 such that lln ((T1u ••• uTn ) n Tn ) 0 'Proof.


e: 0

>

0 such that

for all n > n 0 •

77

Let f E C0 {S) be a positive function with f

1 on T1u ••• uTn

and note that 0

~

(f)

n

> £

-

for all

n

>

n • o

Let m be any weak*-cluster point of

{~

n

invariant mean on WUC(S) such that m(f) m(f) = 0.

} in WUC(S)* and note that m is an ~ £,

By Lemma 2.7, we must

hav~

This contradiction implies our result.

2.10 LEMMA. Let S be a a-compact locally compa~t topological semigroup (with Ma(S) non-zero) and let there be an invariant mean on l~C(S). Then there exists a sequence {pn) in P(Ma(S)) converging strongly to topological invariance and such that Kn := supp{p n ) is compact~ for aU n E ~.if any

one of the foUowing conditions holds: (a) S

is a foundation semigroup

~ith

an identity element.

(b) The centre of Fa(S) is not Ma(S)-negligible. Proof.

First we show that there exists a net in P(Ma(S)) weakly convergent

to topological invariance, if (a) or (b) holds. holds.

To this end, suppose (a)

Then there exists a topological (left and right) invariant mean m

on Ma{S)*, by Theorem 2.4. Now m E weak*-closure (P(Ma{S))) in Ma(S)**· Consequently, there exists a net (~ a ) in P(Ma (S)) such that ~ a (h) + m(h) for all h E Ma(S)*. In particular, for each v E P(Ma(S)) we have

for all h E Ma(S)*, Similarly ~ a *v - ~ a + 0 weakly. is weakly convergent to topological invariance,

Thus

(~

a ) C P(Ma (S))

Next suppose condition (b) holds. Then we can choose T E P(Ma{S)) such that supp(T) c Z(F (S)) • where Z(F (S)) denotes the centre of F (S). l-Ie then a a a have T*v =.v*T for all v E P(M (S)). Now if m is any invariant mean on a o WUC(S) we have that m0 is topologically invariant, by arguing as in Lemma 2.3(i).

Let (n) be a net in P(Ma {S)) such that na (f)+ mo {f), for all a f E WUC(S). Then, for any v E P(Ma(S)) and hE Ma(S)*, we have that

T*vohoT, TohoT E WUC(S) by Lemma 3.2;

78

hence, if

~a

:= T*na*T,

• na(T*vohoT) - na(TohoT) m0 (vo(TOhOT)) - m0 (TOhOT)

+

a

0.

Similarly h(pa*v - pa) + 0. Thus (pa) is weakly convergent to topological invariance. Now suppose either (a) or (b) holds. Then there exists a net (n 8) S P(Ma(S)) strongly convergent to topological invariance, by Lemma 2.8. Fix any A E P(Ma(S)) and set Ps :• A*ns*A. Note that (p 8) is also strongly convergent to topological invariance. As S is a-compact, we can choose an

.

increasing sequence of compact neighbourhoos

n1 S n2 S ...• such that

Noting that the maps X+ x*ps and y + Ps*Y of s into Ma(S) are u Dn. n=l norm continuous (see e.g. Corollary 2.2.4) we can choose a sequence S =

{ps ,P 8 •••• } trom the ps's such that 1 2 llx*ps *Y - Ps n

n

II

{cf» E

(t"")* :

cf»

~ 0,

11+11 = 1 and +(g) = 0 for all g

such that g(k)

~

0 as k

~

E

t""

""} and c denote the

cardinal of continuum.

Let S be a a-compact 'locaZZ.y compact topo'logica'l semigroup such that S is not l'e'lative'ly neo-compact~ Ma (S) is non-zero and thel'e e:cists an inual'iant mean on WUC(S). Suppose eithel' 2 .11 THEOREM.

is a foundation semigroup IIlith identity eLement~ Ol' (b) the centl'e of Fa(S) is not Ma(S)-neg'ligib'le.

(a) S

Then thel'e exists a lineal' isomet'!'y T(F)

Proof.

T :

(t"") ~

S IM(WUC(S)) and so card(IM(WUC(S)))

WUC(S)* such that ~ 2c.

Note that we have the hypothesis of Lemma 2.10 met, and let {pn}

and {Kn} be as in Lemma 2.10.

Choose v

P(Ma(S)) with C := supp(v)

E

compact and note that the sequence {v*pn*v} also converges strongly to topological invariance. Observi11g that Tn := supp(v*pn*v) = CKnC is compact (n E :m) and recalling Lemma 2.9, there exists a subsequence v*p

80

~

{~}

*v (T \ (T u ••• uT )) > 1-2-k ~ nl ~-1

of {n} such that

for k

=

2,3, ••••

Let

'If

:

.

WUC(S) ..... R.

'lr(f)(k) : = p

~

To see that follows:

1r

be the linear mapping defined by

(f).

for all f e WUC{S) and k e

is onto, let g

..

E R.

be fixed.

1N.

We now define "' g

E R.

..

as

Let

~(1) := g(l) and g(k) := g(k)-

k-1 I:

i=l

g(i)JJk(F.)/lJ.(F.), fork= 2,3, ••• 1 1 1

(Indeed~ e .t"" since, by our definition of the Fk's we have 180>1 ~ 2llgll .. and on assuming ls(i>l ~ 2llgll .. fori= 1,2, ••• ,k-l we are led to k-1 k-1 . lg(k) I~ 11811 .. + 2 11811 .. i:l ~(Fi)/}.li (Fi) ~ 11811 .. (1+2 i:l 2-k/(1-2~ 1 ))

~2llgil ... >

Now the pairwise disjointness of members of the sequence {Fk}, that "' g e R... and 'IJk(Fk) h

. l,

>

:=

imply that the function

is a linear functional in M(S)*. Consequently vohov e WUC(S), by Lemma 3.2. n(vohov)

Now, for all k e

~.

we have

= p (vohov) r'k

v*p

*v(h)

..

~

'IJk(.E (g(i)/'IJi(Fi))xF.) 1=1 1 k-1 = g(k) + I: g(i)'IJk(F.)/lJ.(F.) = g(k). i=l 1 1 1 Thus

1r

maps the function vohov onto g and

1r

is onto.

Further, we clearly have 11811 ..

=

lln 11 .. = llvohovll 5 •

It follows that the dual map n* :

{R.00 )* -+

WUC(S)* is a linear isometry.

81

~•F

To see that ~*+

> 0

S IM(WUC(S)), let+£ F be fixed. ~*+(1)

and

Then, clearly

= +(1) = 1.

Now for any n £ P(Ma(S)) and f £ WUC(S) we have

:c p

~(nof-f)(k)

~

(nof-f)

Recalling the definition of ~*+(nof-f)

Similarly

=

= (n*P

-p )(f)+ ~~

o

ask+~.

F we have

o.

~*+(fon-f)

= 0

and so

~*+

£ IM(WUC(S)).

Taking SlN to be the Stone-Cech compactification of lN, we have SlN\lN SF.

Since card(SlN\:fi) = 2c and~· is an isometry, we thus get

card(IM(WUC(S))) ~ card(f) > 2c. Theorem is proved.

SoT:= ~* is the required map and our

Let S be a non-FeZativeZy neo-compact~ a-compact~ ZocaZZy compact topoZogicaZ semigroup ~th Fespect to which the hypothesis of TheoFem 2.11 hoZds and Zet E(S) := {f £ WUC(S) : m(f) = a constant as m runs through IM(WUC(S))}. Then the quotient space WUC(S)fE(S) is nonsepaFabZe. 2.12 COROLLARY.

Proof.

Suppose, on the contrary, WUC(S)fE(S) is separable.

Then there

exists a countable set F := {fn : n £ lN} in WUC(S) such that E(S) + L(F) is dense in WUC(S), where L(F) denotes the linear span of F.

So each

p £ IM(WUC(S)) is determined by the sequence {p (f ) } and so n

card(IM(WUC(S))) 2.13 NOTES.

~c.

This contradicts Theorem 2.11.

Various results on the sizes of sets of invariant means can

be found in the literature : for discrete semigroups see e.g. Chou ([15] and [16]), -Granirer [51] and Klawe [68], and for locally compact topological groups see e.g. Chou ([16] and [17]) and Granirer [49]. Our Theorem 2.11 generalizes some of these results and our techniques follow closely those given in [17].

[49].

The proof of our Corollary 2.12 is similar to that mentioned for

groups in [18] •

82

For gorups, Corollary 2.12 was first proved by Granirer

3. THE SIZE OF THE DIFFERENCE WUC(S)\WAP(S) In this section, for a large class of topological semigroups

s,

we indicate

how one can use absolutely continuous measures to explicitly construct many functions in WUC(S)\WAP(S).

FoP convenience ~e assume that aZZ the spaces of measupes~ jUnctions (and jUnctionaZs) mentioned in this section aPe FeaZ-vaZued and PemaPk that the genePaUzation to the compZex-vaZued case can be achieved tPivici.Uy. First we prove a lemma which plays a pivotal role in the construction of functions in WUC(S)\WAP(S).

Let s be any topoZogicaZ semigPOup

~hich

is not PeZativeZy neoLet C := C0 u {1} D := D u{l} ~hePe 1 is an identity eZement of S (if thePe is one) OP an 0 adjoined isoZated identity eZement of s. Then theFe exist infinite sequences {xl'x2 , ••• }and {y1 ,y 2 , ••• } in S such that 3.1 LEMMA.

compact~

C0 and D0 any fixed compact subsets of S.

C-l(Cx y D)D-l n C-l(Cx.y.D)D-l n m

if any one of the (a) n < m

~

foZZo~ing

and i

=+

thPee conditions hoZds:

> j;

(b) n > m, i > j

and n

(c) n ~ m, i ~ j

and

Proof.

J

~

i;

m ~ j.

Our proof is by induction.

Suppose, by the inductive hypothesis,

we have finite sequences Xp := {x 1 ,x 2 , ••• ,xp} and Yp := {y 1 ,y 2 , ••• ,yp} inS such that the lemma holds for n,m,i,j in {1,2, ••• ,p}. For convenience, let p-1

and

Rp :=

u

m=l In terms of the latter notation, the conclusion of our lemma under item (a) for the finite sequences X and Y , is equivalent to p

p

(1)

83

We now establish the inductive step, that is choose xp+l and yp+l such that the lemma is valid for n,m,i,j in {1,2, ••• ,p+l}. Since S is not relatively neo-compact while both T :• C-l(C(C-lL D-l)D)(Y D)-l p

and

p

T' :~ C-l(C(C-l(CX Y D)D-l)D)(Y D)-l p p p are relatively neo-compact, we can choose xp+l in S\(TuT'). Now, that xp+ 1 ~ T is equivalent to C-l(Cx 1Y D)D-l n C-lL D-l • p+ p p

(2)

+

while that xp+l ~ T' is equivalent to (3)

Also the subsets

Q :• (CXp+l)-l (C(C-l(CXPYPD)D-l)D)D-l and Q' :• (CXp+l)-l (C(C-l(RpuCxp+lYpD)D-l)D)D-l

are relatively neo-compact, and so we can choose yp+l in S such that yp+l~ Q and yp+l ~ Q'. Equivalently, this is such that

+

(4)

n C-l(CX Y D)D-l •

(5)

n C-l(R u Cx 1Y D)D-l • p p+ p

p p

and

+

(respectively). Now for the finite sequences Xp+l and Yp+l" item (3) and the inductive hypothesis show that the lemma holds under condition (b), item (4) and the inductive hypothesis show that the lemma holds under condition (c), and to verify the lemma under condition (a) it is sufficient to establish item (1) with p+l in place of p. To the latter end, we note that the inductive hypothesis, items (2) and (5) imply that

84

C-lL

p+l

D-l n C-lR D-l • C-l(L u CX 1y 1D)D-l n C-l(R U Cx Y D)D-l p+l p p+ p+ p p+l p (C-lL D-l n C-l(R u Cx 1Y D)D-l) p p p+ p (C

-1

(CXp+lyp+lD)D

-1

-1

n C

(Rpu Cxp+lYpD)D

-1

)

= ••

Repeating the argument countably many times we get our lemma.

This

completes the proof. Remark.

We warn the reader that in general conditions (a), (b) and (c)

together, are weaker and not equivalent to the condition (d): (n,m)

~

(i,j).

It remains an open problem whether one can establish the conclusion of our lemma under (the stronger) condition (d). group S, this is not known to us.

In fact even for a topological

We strongly believe that if one can

supply an affirmative answer to this problem, for a locally compact topological group S, then property (E) mentioned by Ching Chou [18] may be dropped in most of the results of [18]. Next we introduce the role of absolutely continuous measures in the following lemma.

Let s be a c-distinguished topological semigPOup~ v and ~ measures in M(S) such that the maps X ..... v*x and X ... x*~ of s into M(S) a1'e weakly continuous. Then 3.2 LEMMA.

voho~ £

WUC(S) for all

h

£

M(S)*.

We first note that for all A £ M(S) and h £ M(S)*, vohoA £ C(S), Proof. since vohoA(x) = heA(v*x) and the map x + v*x is weakly continuous. To show that voho~ is in RWUC(S), for example, we take a t £ C(S)* and must show that the function x

+

t((voho~)x)

is continuous.

But this follows from the

fact that (voho~)x = vohox*~ and the functional A + t(vohoA) is in M(S)*. Similarly voho~ £ LWUC(S) and we are done. We now prove the main result of this section.

85

Let 5 be a C-distinguished topological semigPoup admitting a non-aePO absolutely continuous measUPe (i.e. Ma(S) is non-zero). Then if 5 is not Nlatively neo-compaot, we have that the quotient space WUC(S)fwAP(S) contains a lineaP isometPic copy of 1~ and so is nonsepaPable. 3.3 THEOREM.

~

As noted in item 1.5, we trivially have WAP(S)

Proof.

WUC(S).

Since Ma(S) is solid (by Theorem 2.1.7), we can choose a positive measure n

Ma(S) such that

£

X ~

v*x and

X ~ x*~

measures, v and

~.

and K : .. supp(n) is compact.

By Corollary

such that if v := n*u and ~ := v*n then the maps

£ 5

of 5 into M(S) are weakly continuous.

We keep these

fixed for the remainder of our proof and note that both

C := supp(v) and D := 0

=1

llnll

2.1.13 we can find u,v

0

supp(~)

are compact, (in fact C .

0

= Ku

and D = vK), 0

and II v II ... II ~ II ... 1. Let C :• C0 u{l} and D :• D0 U{l}, where 1 is the identity of 5 (if there is one) or an adjoined isolated identity of S. Let sequences A :• {x1 ,x2 , ••. } and B := {y1 ,y 2 , ••• } be chosen as in Lemma 3.1 with respect to the compact sets C and D. We can choose infinite subsequences~

:•

{~ ·~

1 ~

, ••• } of A and Bk :- {yk ,yk , ••• } of B such that 2

1

2

~

(a)

and

u Bk k=l

then

An n Am

C

B;

(B)

if n I< m

(y)

Lemma 3.1 remains valid with k ,k ,k. and k. in place of n

n,m,i and j, respectively (i.e.

m

J

1

when~

and Bk replace A

and B, respectively). Let ~

~

:•

~

and

We define the functions fk on 5 by

Let {ck} be any element in 1~.

86

Fk ·= •

U

U

j=l i>j

C~.Yk.D. 1

J

.

We now show that

I

ckfk is in WUC(S).

From (B) and (y) we note that

k=l all the

~·s

and Fk's are pairwise disjoint and so can define the functional

h e: M(S)* by

.

Now a simple exercise on our definitions shows that

..

and so

.

I ckfk e: WUC(S) by Lemma 3.2. k=l

It remains to show the (clearly linear) map

.

..

is an isometry of 1 demonstrating that

where f :=

.. I

~fk

into WUC(S)/WAP(S)"

and g e: WAP(S).

To achieve this we start by

Suppose on the contrary there exists

k•l a g in WAP(S) and e: > 0 such that

We can find a positive integer k' such that ~·

may assume that (2)

llf+gll S


j

(3)

From (2) and (3) we obtain

{~ ~ ~

(4)

1

> J

implies

g(~! Yk!) ~ -ck' + lck,+g(~! Yk!>l < J

1

implies

g(~! Yk!) ~ ck' - 1-ck,+g(~! yk!>l > J

1

£

J

1

1

£.

J

From (4) and Proposition 1.2. we have that g 4 WAP(S), which contradicts our original choice of gin WAP(S). Hence (1) holds. Noting that ao

and recalling (1), we have that ao

II k..I l

ao

~fk + WAP(S) II WUC(S) I

g

E:

WAP(S)}

WAP(S)

ao

I ~fk + WAP(S) is a linear isometry of k•l into WUC(S)fwAP(S) and our proof is complete.

Consequently the mapping {ck} Lao

+

LetS be a non reZativeZy neo-compact stip with Ma(S) nonaero. (In partiouZar S may be any ZocaZZy compact group which is not compact.) Then the quotient space UC(S)fwAP(S) contains an isometric Zinear co'PiJ of tao. 3.4 COROLLARY.

Proof. We have UC(S) • WUC(S), by Theorem l.S(i), and so our result follows from Theorem 3.3. Our next result and Theorem 3.7 are versions of Theorem 3.3 with proofs essentially contained in that of Theorem 3.3.

88

Let S be a C-distinguished topological semigroup, p a positive measure in Ma(S) with compact support, IIPII = 1 and suppose that S is not relatively neo-compact. Then WUC(S)op\ WAP(S) is non-separable (in C(S)). 3.5 PROPOSITION.

Proof.

F :=

Let

A:= F n

norm-cl(WUC(S)op),

WAP(S) and E

:=

supp(p).

In

the proof of Theorem 3.3 take D := (Eu{l})(D 0 u{l}) and let the functions fk be as constructed there.

is an isometry of 1~ into

Then the (clearly linear) map

F;A

and hence

F \ A is

non-separable.

This implies

our result. For the purpose of our next lemma, we recall that if A is any set of functionals then A+ := {a E A : a~ 0} and A- := {a E A : a~ 0}. L

For any n E M(S), let n := {hE M(S)* h(v*n) = 0 for all Then if p and n are any positive measures in M(S) with p S 1s w1t

c-1o

and Dc- 1

compact for all compact subsets C and D of S; (ii) S is pseudocompact but not compact (see e.g. [44]); (iii) Ma(S). is zero; (iv) C(S) • WAP(S),

(by (ii) and the above Proposition).

We have already noted that a stip S behaved as if Ma(S) were non-zero. So the following conjecture seems reasonable. CONJECTURE B [37].

If Sis a stip that is not relatively neo-compact, the

quotient space UC(S)fwAP(S) contains an isometric linear copy of 1~.

92

Even if it turns out that every stip S is such that M (S) is non-zero, it a would be interesting to solve this conjecture without the use of M (S). We a suspect that the notion of functions with separable orbits may be useful in finding a function in UC(S)\WAP(S). 4. THE STONE-~ECH COMPACTIFICATION OF A SEMIGROUP

In this section semigroup.

~e

take s to be any compLeteLy reguL2r semitopoLogicaL

We now define the Stone-~ech compactification BS of S.

For each

f E C(S), let Df :={A E C: IAI ~ llfll S} and note that Df is a compact disc. Hence, by Tychonoff's theorem, E := x{Df : f E C(S)} is compact. In a natural manner we embed S into E via the mapping 6 : S + E given by 6(x)(f) := f(x) Then BS := cl(6(S)).

for all

xES

and

f E C(S).

For our convenience we shall write x in place of 6(x)

(and hence S in place of 6(S)), for all x inS. Suppose that S is locally compact.

Then C(S)* is the space of bounded

finitely additive regular Borel measures on S.

It is interesting to study

C(S)* as a Banach algebra with an Arens or convolution product.

This idea

is suggested in Hewitt and Ross [59, page 275] and has been studied to some extend by Butcher [14], Olubummo ( [75J and [76]) and Pym and Vasudema m}. have f E WAP(S) and our result follows.

Recalling Proposition 1.2 we thus

4.3 THEOREM. Let s be (topoZogicaZZy) no~z~ ZocaZZy compact~ right canceZZative and such that c-1n is reZativeZy compact for aZZ compact subsets C and D of s. Then the foZZo~ing items are equivaZent: (i) (SS,o) is a Zeft semitopoZogicaZ semigroup; (ii) Proof.

s is discrete or countabZy compact. From our definitions and lemma 4.2, it is trivial to note that

item (ii) implies that LMC(S)

= C(S)

and hence item (i) follows by

Proposition 4.1. Now suppose that (i) holds with S neither discrete nor countably compact. We can find a relatively compact infinite set

{sn : n

E

m} in S and a

sequence {Un } of subsets of S such that Un is a neighbourhood of s n and sm ~ Un for m n. Let C := cl({sn : n E m}) and note that C is compact. We can choose a sequence {tn } in S without a cluster point and such that

+

for all

n

E

m.

(1)

Next we consider the closure X of the set X : = {smtn : m, n (sm(a)tn(a)) be a set in X converging to x

E

S.

E

m}.

Let

Let D be a compact neigh-

bourhood of x and choose a 0 such that a~ a 0 implies that sm(a)tn(a) ED. 94

-1 Since c- 10 is relatively compact and {tn} Thus tn(a) E C D for a ~ a 0 • has no cluster point, we must have {t ( ) : a > a } finite. Hence x = ctn n a - o for some n E 1N and c E C. The representation x = ctn is unique, for if ctn

c 1 tm for some c 1

E

C and m

E

1N , then n

= m by

(1) and so c • c 1 by

right cancellation. It is now clear that if we define f on X by setting f(smtn) := Jl

l_o

f(ctn) :=

0

n

if

m

if

m> n




C(S)* is such that +(f .g)

E

= cj>(f)+(g)

LetS be a k 0 -space and f 4.4 THEOREM. Ol" equivaZ.ent: (i) f

E

E

we say cj> is muUiplicative.

C(S).

Then the

foZ.Z.o~ing

items

LMC(S).

Fol" aZ.Z. sequences {xn} and {yn} in S ~th {xn n E 1N} l"eZ.ativeZ.y compact~ ~e have the cZ.osul"es of the sets {f(xnym) : n < m} and {f(xnm y ) n > m} not disjoint. (ii)

(iii) { f X

(iv) f

E

x

E

K} is

~eakZ.y

compact fol"

eve~

compact K c s.

LWUC(S).

95

: n Em}) and let n P be the set of all multiplicative means on C(S) endowed with the weak*Proof.

To show that (i) implies (ii), let K := cl({x

topology.

Thus both K and P are compact.

F : K x P

~

For f

E

LMC(S), we have the map

C, given by

F(x,n) := n(xf)

for all

(x,n)

E

K x P,

separately continuous. Assuming the notation of (ii), let x' be a cluster point of {xn} inK and v a cluster point of

{ym} in

P.

Consequently every neighbourhood of v( ,f) x meets infinitely many columns and infinitely many rows of the double So the closures of the sets {x f(ym) : n < m} and n {x f(ym) : n > m} in C are not disjoint. Thus (i) implies (ii). n

The equivalence of (ii) and (iii) follows from Grothendieck's Theorem (-see Appendix B.7). That (iii) and (iv) are equivalent is left as a simple exercise for the reader. Now from Theorem 4.4 and the proof of Theorem 4.3 we note that if a semigroup S satisfies the hypothesis of Theorem 4.3, then C(S) only if S is either discrete or countably compact.

LWUC(S) if and

We now show that an

easy adjustment of the proof of Theorem 4.3 gives a more general result of independent interest.

4.5 THEOREM. Let s be no~az~ ZocaZZy compact and right JanceZZative. Suppose S is neither countabZy compact nor discrete and c- 1n is compact for aZZ compact subsets C and D of s. Then~ for some cZosed subset Xof s~ ~e have that (C(S)\LWUC(S)) IX contains a Zinear isometric copy oft~ and so the quotient space C(S)fLWUC(S) is non-separabLe.

Proof.

We urge the reader to first study the proof of Theorem 4.3 as we

will omit closely related details in this proof. Let thesequences {s } and {t } and the set C be as constructed in the n

proof of Theorem 4.3.

n

Choose infinite subsequences Tk :s {tk ,tk , ••• } of 1 2 T := {t 1 ,t 2 , •••• } such that 96

.

u Tk '"' T, and k=l Tk n Tk' =; Let

~

:= {smtk n

k I k'.

if and only if

: m,n E 1N}, X :• {smn t : m,n

E

:tl}

and note that our

construction of the Tk's and T imply (a) ~ ~

(b)

..

(c)

c

C and

E

n

E

lN} •

n ~· ~;if and only if k I k', ~

u

k=l

= X.

(To verify these items, see proof of Theorem 4.3 for relevant techniques.)

Next we define the functions fk :

fk(sm.tk ) m

·-r . -1

fk(ctk ) := -1 n

if

if

m< n

if

m> n

~ E

lR by

-

C E

C \ {sm

m

E lN} •

Then (as similarly shown in the proof of Theorem 4.3) fk is continuous, for all k

E

':tl.

Corresponding to each element {~,} in t."", let F(~,) be the

function defined on X by for some I. By items (b) and (c) we thus have

F(~,)

lN.

E

well defined as a function.

To see that F(~,) is continuous, suppose (catn(a)) is a net in converging to some point

c~

1

, for some c a • c

by the definition of T, we have n(a) = k (c t ) c ana -

X•

E

C and t na ( )' tk 1

E

X T.

Then,

eventually and so, eventually

Consequently, eventually

97

Thus F(dk,) is continuous. Now noting that if

m < n

if

m > n

(*)

Theorem 4.4 and Tietze's Extension Theorem imply the existence of a function F(dk,) £ C(S)\LWUC(S) such that F(dk,>lx=F(dk,)

and

IIF(dk,)lls=

IIF(dk,)llx=ll{dk,liiQ).

Thus the (clearly) linear map {dk,} ~ F(dk,) li is isometric.

of tQ) into (C(S)\LWUC(S))Ii

Since tQ) is non-separable it follows that C(S)\LWUC(S) and hence C(S)fLWUC(S) is non-separable. Theorems 4.3 and 4.4 are taken from Baker and Butcher [7].

The argument

used in showing that (i) implies (ii) in the proof of Theorem 4.4 is inspired by the proof of [106, Theorem 1]. Also in the paper [7], Baker and Butcher showed that SS is a left semitopological semigroup under a certain operation extending that of S while C(S) # LMC(S), for many locally compact semitopological semigroups S.

Such

arbitrary products unfortunately do not seem to reflect much on the structure of SS or LMC(S). of interest.

On the other hand the Arens type of products seem to be

In what follows, we make use of Arens products defined wi·th

respect to various subspaces of C(S). 4.6 NOTATION.

For the remainder of this section let AS£ {LMC(S), LWUC(S),

WUC(S), C(S)} and A(S) be the compact space constructed with respect to AS as we constructed SS with respect to C(S).

(Thus A(S) is the spectrum of

the algebra AS.) For all f £As and a£ A(S), let aof be the function on S 4.7 EXERCISES. given by aof(x) := a(xf) for all x £ S. Then if As £ LMC(S) we have (i) aof £ As

98

(ii) (A(S),o) is a left semitopological semigroup with the Arens operation 'o' given by aoS(f) := a(Sof), for all a,S 4.8 PROPOSITION.

The

foZZo~ing

E

A(S).

items are equivalent:

(i) A(S) is a semitopoZogicaZ semigPoup

~th

subsemigroup

s;

(ii) AS= WAP(S). Proof.

That (i) implies (ii) follows from our definition of A(S) and

Lemma 4.2.

Now suppose (ii) holds and consider the semigroup (A(S),o)

defined in Exercise 4.7(ii).

To verify item (i) it is sufficient to show

that (A(S),o) is a right semitopological semigroup.

To this end, let f

be fixed and suppose (a) is a net converging to a in A(S). y

E

AS

We have

Hence (ay of) S} compact and (a of) c Cf. y has a weak cluster point g (say) in As. In particular for all x in S we Cf := weak-closure of { f : x X

E

have that a y of(x) := a y ( X f)

~

a( X f)

= aof(x)

and g(x) is a cluster point of (a y of(x)). Hence g = aof. It follows that a~of ~ aof in the a(AS,A~)-topology. So for all S E A(S), we have

Thus (A(S),o) is a right semitopological semigroup.

Let S be Locally compact and such that c-1D and DC-l ape If S is eitheP a-compact compact foP aU compact subsets C and D of S. and As = C(S) oP S is a topological semigroup suppopting a non-aero absolutely continuous measUPe~ ~e have the foZZo~ing items equivalent: 4.9 THEOREM.

(i) A(S) is a semitopoZogicaZ semigroup

~th

subsemigroup S;

(ii) S is compact.

Proof.

This is an immediate consequence of Corollary 3.9, the Theorem

mentioned in item 3.10 and Proposition 4.8. The advantage of the compactification A(S) of S is that already, if AS E {LMC(S), LWUC(S), WUC(S)}, A(S) is a left semitopological semigroup. Our preceding Theorem now says that, for a large class of non-compact S, 99

A(S) is not a semitopological semigroup.

A natural question to ask is:

What is the largest semitopological semigroup BA(S) contained in (A(S),o) and such that S is a subsemigroup of BA(S)? first introduce the following notation.

To formalize this question we

Let

BA (S) : = {S E A(S) : the map a + Boa of A(S) into A(S) is continuous}. BA(S) is called the set of bicontinuous points in A(S) in [88] (where S is a group).

4.10 EXERCISE.

We always

haveS~

BA(S).

The following conjecture seems reasonable. Let S be locally compact and such that c- 1D and DC-l are 4.11 CONJECTURE. compact for all compact subsets C and D of S. Then if S is either a-compact or such that S is a topological semigroup admitting a non-zero absolutely continuous measure, we have BA(S)

= S.

For the case of a discrete topological abelian group, the above conjecture is due to Baker and Milnes

[a].

4.12 THEOREM (Ruppert [88]). foz:Lowing cases:

We quote the following cases now known.

We have that BA (S)

S for each of the

s is an abeZian ZocaZZy compact topoZogicaZ group; s is a connected ZocaZZy compact topoZogicaZ group; s is a discrete topoZogicaZ group with the set of a~Z eZements in s

(i)

(ii) (iii)

of order tw countabZe. We conclude this section by giving the reader an exercise promised before.

4.13 EXERCISE.. In general every k-space is a k 0 -space. We required S to be a k -space in Theorem 4.4. We now mention the following semitopological 0 semigroup S which is a k -space but not a k-space. 0 Let

T

0\P

with 0 open in the usual topology of 1R and P a countable subset of 1R.

Let 1R

be the topology on the real numbers with generic open sets of the type

T

denote the resultant space.

Then with the additive operation S := 1RT

is a semitopological semigroup that is a k 0 -space but not a k-space, e.g. [9] for details.)

100

(See

5. REGULARITY OF MULTIPLICATION IN SEMIGROUP ALGEBRAS Let A be a Banach algebra with first and second dual spaces A* and A**• respectively. hf in A* by voh(~)

For f

:=

€

h(v.~)

A**• h

€

A* and v

hov(~)

and

:=

€

A we define voh, hov, hf and

h(~.v)

(~ € A) (~ €

A).

We then define the Arens products ( [1] and [2]) e and e' on A** by

for all

f,~ €

A** and h

€

A*.

If the two Arens products coincide (i.e. we say multiplication in A is regular.

irregular if it is not regular.

f•~

=

f•'~

for all

f.~

in A**)

Multiplication in A is said to be

The products were first introduced by Arens

and have been used in various publications.

It is unusual to find a

property of Banach algebras that is invariant under passage both to subalgebras and to quotient algebras. of regularity of multiplication.

One such invariant property is that In this section we are interested in

finding when we have this property for algebras like Ma(S).

With respect to either product • ore', A**

is a Banach algebra into

which A is embedded isometrically by the canonical homomorphism v (v

€

~

n(v)

A) where n(v) (h) : = h(v)

(h

€

A*).

It is immediate from the definitions that either Arens product on A** is Any pair {n(v )}, {n(~ )} of separately continuous for a(A**• A*). n m bounded sequences in n(A) will have cluster points f,~ (respectively) in A** for a(A**,A*). h

€

If multiplication in A is regular, it follows that if

A* is such that both repeated limits of the double sequence {h(v

exist, then these two limits are equal (being in fact

n

·~

)} m

f•~(h)).

Recalling Proposition 1.2, we summarise the above remarks (for measure algebras) in the following lemma. 101

5.1 LEMMA. Let s be a c-distinguished topoLogicaL semig~up and A a cLosed subaLgebra of M(S). Then the foLLowing items are equivaLent: (i)

A has reguLar muLtipLication;

(ii) if {vn }, {~m} are any bounded sequences in A and h £ A* such that b := lim lim h(vn *~m) and c :~ lim lim h(vn *~m) e~st, then b = c; m n n m (iii) evePy h

£

semigroup (P(A),

A* is a weakLy aLmost periodic fUnction on the topoLogicaL

II II).

(Recall the definition of P(A) from 2.1.)

For any A S M(S) we recall that dA denotes the density function as defined in item 1.3.2.

be a c-distinguished topoLogicaL semig~up with identity eLement 1 and A a convoLution measure aLgebra with foundation equaL to s. Let W be a subset of s such that 1 is not isoZated in W and WS dA(S). Then there ~st sequences {Cn}• {Dm} of non-A-negLigibLe compact subsets of W such that, for aZZ n,m,i,j £ 1N with n < m and i > j we have 5.2 LEMMA.

Let

S

Cn Dn n C.D. = 1 J

Proof.

+•

Suppose, by the induction hypothesis, for some positive integer p

we have finite sequences {c1 ,c 2 , ••• ,Cp}• {D1 ,n 2 , ••• ,Dp} of non-A-negligible compact subsets of W such that if p-·1 p p X := u cn• y :• u Dn• L := u u cn Dm p p p n=l n=l n<m_3) n""l and p-1

:= u p m=l

u

R

p~n>m

cn Dm

then (a) 1 • X

p U Yp u Lp u Rp

(B) xp n

YP

(y 1 ) LP n YP

~

(y 2) RP n xP

+ +

(15) Lp n Rp •

102

+

To prove the inductive step, we now choose Cp+l and Dp+l and verify items (a) to (~) with p+l in place of p. (Note that all sets mentioned in items (a) to (6) are compact.)

We can choose a non-A-negligible compact set Cp+l c W such that (1)

1 ~ cp+l u cp+l YP

(2)

cp+l n (YP u RP) =

(J)

cp+lyp n (LP u XP u cp+l) •

by (a)

+

by (a)

+

by (B), (y 1) and since 1 ~ YP.

Next we can choose a non-A-negligible compact set D 1 c W such that p+

(4)

1 1 D

by (a)

(5)

Dp+l n (XpDp+ l u Lp u Xp+ 1)

=•

..

p+l uXD p p+l

by (a) and since 1 • Cp+ 1 •

We are now in a position to establish items (a) to (6) with p+l in place of p. Noting that Lp+l ~ Lp u XpDp+l and Rp+l = Rp u Cp+ 1Yp , it follows from (a), (1) and (4) that 1 ~ Xp+l u Yp+l u Lp+l u Rp+l.

From (8), (2) and (5) we get

From (y 1 ), (5) and (6) we get (Lp U Xp Dp+ 1) n Yp+l

From

(~),

(J) and (6) we have

p+l n Rp+l

L

=

(L

p u XpDp+ 1 ) n (Rp u Cp +ly p )

.. (Lp n Rp) u (Lp n Cp+lYp) u (XpDp+l n (Rp

U

Cp+l Yp))

- cf>. 103

This completes our verification of the inductive step.

By repeating the

argument countably many times we note, in particular, that item (6) leads to the conclusion of our lemma.

Let S denote a c-distinguished topoZogicaZ semig~up with an identity eZement 1 and A a convoLution measupe aZgebra with foundation equaZ to s. Then 5.3 THEOREM.

(i)

If

1

is not isoZated in s we have muZtipZication in A iX"l'eguZar.

If the foundation of Ma(S) coincides with S and the sets x -l{y} and 1 {y}x- are finite (x,y e S), then A has reguZar muZtipZication if and onZy (ii)

if

S

is finite.

Proof.

To prove (i) suppose that 1 is not isolated and choose sequences

{Cn }, {Dn } as in Lemma 5.2 (with W := S). positive measures vn .~ n in

II vnll = vn(Cn) = 1

A such and

Since

A is

solid, we can find

that

n~nll = ~ n (Dn ) = 1

(n e 1N).

00

Since L := u u CD nm is a-compact, we can define h n""l n<m h(n) := n(H) (n e A).

E

A* by

We then get h(vn *~m) := vn *~m(L)

=

if

n < m

if

n

>

m.

Recalling Lemma 5.1, our proof for item (i) is complete. To prove item (ii) we first remark that if S is finite then trivially multiplication in

A is

regular.

case where S is discrete. Lemma 3.1(a)· (with C

=D=

For the converse we first consider the

Now if S is discrete and infinite, recalling {1}), an argument similar to that used in the

proof of item (i) (which we omit) easily shows that multiplication.

Thus S must be finite if

A has

A has

irregular

regular multiplication.

Finally we consider the case where S is not discrete.

In view of item (i)

our proof will be complete if we can show that 1 is not isolated.

Since S

is the foundation of Ma(S), if 1 is isolated than I e Ma(S) and Sis discrete -see e.g. Exercise 2.3.10(c)(ii). By this conflict if Sis not discrete, 1 104

is not isolated and our result follows.

Let A be a convolution measuPe algebPa whose foundation (semigPOup) A is a topological subsemigPOup of a topological gpoup G. the following items aPe equivalent: 5 • 4 THEOREM.

(i) Multiplication in

Then

A is PegulaP;

(ii) S is finite. Proof.

That (ii) implies (i) is trivial.

Suppose (i) holds.

assume that S is closed in G for otherwise we may take

S in

We may

place of S and

note that S is a closed subsemigroup which we may also take to be the foundation of A.

If S is discrete, that (i) implies (ii) can be verified as

done in Theorem 5.3(ii). Suppose S is not discrete.

If S is not compact then by arguing in a

manner simpler to the proof of Lemma 3.l(a) we can find sequences {C }, {D } n m of non-A-negligible compact subsets of S such that n < m and i > j imply that C D n C. D. = 4» n m 1. J

(n,m,i,j

£

1N).

Recalling our proof of Theorem 5.3(i), this similarly leads us to a contradiction.

Next we suppose that S is compact.

Then S is a compact group,

by Theorem 1.2.3, and Theorem 5.3(ii) gives us the result. 5.5 ~·

(i) Civin and Yood [19] proved that if G is an infinite abelian locally compact topological group, multiplication in L1 (G) is irregular.

Then Young [105] improved the result to include the case where G is not necessarily abelian.

Pym [81] studied the regularity of multiplication

for certain convolution measure algebras supported on semigroups and so did Young [106].

Motivated by these results Dzinotyiweyi [35] proved the

preceding two Theorems.

In particular Theorem 5.4 is a generalization of

Theorem 5 of Pym [81] • (ii) At this stage it should be clear to the reader that: of continuity of multiplication in

as.

the phenomenon

the weakly almost periodicity of

functions on S and the regularity of multiplication of convolution measure algebras "living" on S, are somewhat related. Indeed if Sd denotes the semigroup S with discrete topology, then the reader may easily extract a proof for the following Theorem of Young [106] from Sections 3,4 and 5.

105

The

THEOREM.

foll~ng aPe

equivalent for any c-distinguished topological

semigroup S: (a) M(S) has regular multiplication; (b) t 1 (s)

has regular multiplication;

(c) BSd admits the structure of a semitopotogical semigroup

~ith

S as a

subsemigroup; there is no pair of sequences {xn }, {ym} in {x y y : n > m} are disjoint; nm : n < m} and {xnm (d)

(e) m(S)

S

such that the sets

= WAP(Sd).

(iii) A major difference between Lemma 3.l(a) and Lemma 5.2 is that;

in

the former sequences {x }, {y } are chosen in a "scattered manner" where as n m in Lemma 5.2 the sequences {C }, {D } are chosen within a given "vicinity" n m of the identity element of S. Now letS := [0,1] with maximum operation. Then S = 1-1 {1} and Sd is relatively neo-compact.

However, from Lemma 5.2 and the above Theorem, we

have that BSd is not a semitopological semigroup with subsemigroup S.

Thus

the Stone-Cech compactification of a locally compact relatively neo-compact topological semigroup is not necessarily a "nice" semitopological semigroup. (iv) THE RADICAL OF Ma(S)**·

Throughout this item S denotes a

foundation semigroup with identity element 1 (unless otherwise stated) and Ra (S) the radical of the Banach algebra Ma (S)** with Arens multiplicaton •'· For convenience we assume all spaces to be real. We are interested in showing that, for many cases, the space Ra(S) is very large.

Towards this

end we first prove the following result, also of independent interest.

Let S be non-discrete and right cancellative. Then the quotient spaces Ma(S)*fc(S) and Ma(S)*fLWUC(S) contain isometric linear copies of t"". PROPOSITION.

Proof.

Let W be a compact neighbourhood of 1 and corresponding to each

function g in C(S) let G be the function in G(x,y) := g(xy)

106

for all

x,y

£

W.

C(W~W)

given by

Then a simple compactness argument shows that the set {G(x,.) : x

£

W}

is

relatively (norm and hence) weakly compact in C(W). We can find a sequence {Vk} of disjoint open neighbourhoods contained in w. Choose vk £ Vk n s1 and recall that Vkv;1 is a neighbourhood of 1. So there is a sequence {Uk} of open neighbourhoods of 1 such that U~ ~ Vkv~ 1 • for all k £ B. By Lemma 5.2 we can choose sequences {~ }, {Dk} of nonn

n

Ma(S)-negligible compact subsets of Uk such that, for all n,m,i,j

£

~.

we

have ~

n

Dk n m

~.

Dk.

1

J

=+

whenever n < m and i > j.

By right cancellation we have Ck

n

:a

Dk vk n ck. Dk. vk m 1 J ~a(S)

and choose sequences of points {ck }, {ek } such that n

ck n

£

(1)

+whenever n < m and i > j.

da(~)

and

n

ek n

£

da(Dk

n

(2)

vk).

n

...

...

u u ck. Dk. vk, and define the Let Ek := u Dk. vk and Fk := u i>j i .. l i<j ~.1 1 j-1 J J

function

~

Noting that

by ~ £

~

:= X~ - XFk

.

Ma(S)* we now show that (in the norm of Ma(S)*)

II~+ gil

> 1

If not, for some g

£

for all g

£

(3)

C(S).

C(S) we can find

£

> 0 such that

II~ + gil ~ 1-e

In particular for

va,~B £

P(Ma(S)) with supp(va) C

~n

and

supp(~B) C

Dkm vk,

we have

and so (recalling our definition of

~)

107

Jif n < m then 11 + g(va.*lJs) I ~ 1-£

lif n > m then 1-1 + g(va.*JJs>l ~ 1-£. Letting (va.) shrink to ck

and (JJB) to ~ we thus get (by continuity of g)

n

{

m

if n < m then 11 + g(ck ek ) I ~ 1-£

n m if n > m then 1-1 + g(~ ek >I < 1-£. n m I t follows that < -£

if n

>

if n > m


.

Thus A is a right ideal of Ma (S)** such that Ao'Ma (S)** is zero.

Hence

A c R (S), (see e.g. Rickart, "General theory of Banach algebras", Van -

a

Nostrand (1960);

Theorem 2.3.5(ii)).

By the preceding Proposition, there exists an isometric linear map w of t~ into Ma(S)*ILWUC(S)•

So for some closed subsapce P of Ma(S)*, we have -1

~

w(1) dense in PfLWUC(S)•

Hence the inverse map w

~

(defined on w(t )) ~

extends to a unique isometric linear map T of P/LWUC(S) onto t . The dual map T* : (1~)* ~ (P/LWUC(S))* is isometric, linear and onto. But then A= LWUC(S)

.L

can be identified with the dual of Ma(S)*ILWUC(S)•

So each

element of (P/LWUC(S))* can be identified with the restriction of some element of A to P. This completes our proof (on noting that (1~)* is nonseparable). Let S be any topological semigroup satisfying the hypothesis of Theorem 2.11 and note that WUC(S)* is a Banach algebra under an Arens operation given by ae'a(f) := a(aof) for all a,a

£

WUC(S)*;

and f

£

aof(x) := a(xf), WUC(S) and x

£

S.

Now, by Theorem 2.11,

card(IM(WUC(S))) ~ 2c. Let B := {~ £ WUC(S)* : ~(1) = 0 and w(xf) = w(f), for all x £ S and f £ WUC(S)}. Note that B is a right ideal of WUC(S)* such that B•'B

= {0}. By the result referred to Rickart's book in the proof

of the preceding Theorem, we have that the radical of the algebra WUC(S)* contains B. Now fix ~ 0 £ IM(WUC(S)) and note that IM(WUC(S)) - ~ 0 is c contained in B, and so card(B) ~ 2 • Consequently the radical of WUC(S)* is nonseparable.

109

PROBLEM. Let S be a cancellative infinite foundation semigroup. We conjecture that the radical of the algebra WUC(S)* is nonseparable. (The preceding remarks say that, to a large extent, the conjecture is true if there exists a left invariant mean on WUC(S) and S is not compact. Even for groups, this conjecture is still open [52j.) REFERENCES. of L~(G)*.

Let G be a locally compact topological group and R the radical Civin and Yood [19] showed that R is infinite dimensional for

G nondiscrete and abelian or G • z. E.E. Granirer [52] proved that R is nonseparable if G is amenable or nondiscrete. Our theorem above is inspired by the latter paper of Granirer. In [56] Gulick showed that the quotient space L~(G)fc(G) is nonseparable if G is not extremely disconnected. results on the radical of L~(G) or UC(G)* can be found in [16].

110

Other

5 Characterizations of absolutely continuous measures This chapter forms a natural continuation of Chapter 2 bearing in mind our experiences in Chapter 3.

The message of Chapter 2 is that one can study

various spaces of measures, on a topological semigroup, that can be different in general but all coinciding with the group algebra in the case of a locally compact topological group;

and further one such space, namely the

space of measures which are absolutely continuous, seems to have many interesting properties.

Chapter 3 then presents a very large class of

topological semigroups, the so called stips, which have attractive topological "homogeneous" properties and include the class of all locally compact topological groups as a "very special" case. In this chapter we shall study various characterizations of the absolutely continuous measures on a stip and also include some results very close to characterizing such measures.

Throughout this chapteP S denotes a stip unless othePwise explicitly stated. Blanket Assumption.

1. CONTINUITY OF MEASURES UNDER TRANSLATION ON A STIP Noting that the map

(x,~) ~ x*~

of S x M(S) into M(S) defines a left action

of Son M(S), the following result follows as a special case of Theorem 3.4.6.

(If S is also the foundation of M (S), then one may also deduce our a next Theorem from Corollary 2.2.4.) 1.1 THEOREM.

If

~ ~

M(S), the following items aPe equivalent:

the map 1 is weakly continuous; ~ (ii) the map 1 is noPm continuous. (i)

~

Let p E Ma (S) and V be a neighboUPhood of 1. -isolated idempotent e in V such that

1.2 COROLLARY.

exists a

;

(ii) \.1

€

M1 (S);

(iii) 1\.11

n

€

M!<s>;

(iv) R.\.1 is

~akty

continuous at the point 1.

1.7. Theorem 1.1 was first proved by Sleijpen [94] in a way relying heavily on measure theoretic techniques. The proof given here is due to The other results of this section are taken from Dzinotyiweyi [34). Sleijpen ( [94] and [92]).

114

2. MEASURES WITH SEPARABLE ORBITS Recalling Definitions 3.4.1 we consider weakly and norm separable left orbits of measures in M(S) in terms of the left action of S on M(S) given by (x,~) ~ i•~ for all x £Sand~£ M(S). First we have the following consequence of Theorem 3.4.4 and item 1.6. 2 .1 THEOREM.

£

M(S). M1 (s) for a

£

s1 ;

(ii) If~ Proof.

and~£

If~ has a weakLy separabLe Left orbit over U, then x*~

(i)

aU x

Let U be a compact neighbourhood of 1

Let

£

M!(s), then~ has a norm separabLe Left orbit over E >

O(y) := {s

u.

0 be given, y £ S fixed and consider the set £

S

To prove item (i) it is sufficient to show that O(y) is closed and Theorem 3.4.4 will imply the result.

To the latter end we first note that the

continuity of the function s ~ s*~(f) defined on S (f £ C(S)) implies that

is closed in S.

Since M(S) is the first continuous dual of C (S), it is 0

trivial to note that, from

we have that

Thus O(y) is closed and item (i) follows. Now item 1.6 says that ~ £ M!(s) implies that {i*~ compact and so item (ii) follows. 2.2 Notes.

x £ U}

is'norm

The subject of absolute continuity of measures with separable

orbits particularly on locally compact groups, has been studied quite extensively. This seems to have started with a paper of R. Larsen [7oJ, and subsequently various other papers appeared- see e.g. Tam [100]; T.S. Liu, A. van Rooij and J.K. Wang [73], G.L.G. Sleijpen [93] 115

and H.A.M. Dzinotyiweyi ( [33] and [34]). In this chapter, Theorem 2.l(i)", (or rather its general form: Theorem 3.4.4) has enormous applications.

In fact almost every Theorem we prove in

this chapter can be traced back to have its roots in Theorem 2.l(i). Theorem 2.l(i) was first proved for stips admitting a certain measure theoretic condition in [93] and its proof in the general form given above first appeared in [33] • 2.3 Remark.

In general there are stips (even foundation semigroups with

identity element) admitting measures having norm separable left orbits but are not absolutely continuous.

See for instance Example 4.4.

In preparation for our next Theorem we give a characterization of ~(S) in the following Lemma. 2.4 LEMMA.

The fottowing items

a~

equivatent foP

any~ E

M(S):

(i) ~ E M!(S); (ii)

foP each 6-isotated idempotent e of s, we have e*~*e EM (eSe). a

Proof.

Evidently (i) implies (ii).

We now assume (ii) and prove (i).

Let

U be an open relatively compact neighbourhood in S and A any countable sub-

U n s 1 • By (the right handed form of) Lemma 3.3.4 we can find a 6-isolated idempotent e E S such that

set of

A(suppl~l u {1}) SeSe

and hence

So item (ii) implies that {a*e*~*e : a E A} is rela:ively norm compact, by the compactness of

U n eSe.

for all countable subsets A of {x*~ : x E

U n s1 }

Thus {a*~ : a E A} is relatively norm compact

U n s1 •

Hence, we have that

is relatively norm compact.

Since each norm-compact subset of M(S) is weak*-closed and each weak*closed subset of M(S) is norm-closed, the fact that the weak*-closure of {x*~ x E U n s 1 } is {x*p : x E U} implies that the latter set is norm 116

compact. Hence i U- is norm continuous, by Lemma 2.2.1 and item (i) llt follows, by item 1.6. 2.5 DEFINITIONS.

Recalling the definition of equi-absolute continuity just

before Lemma 2.1.6, we now formally define spaces of such measures. M!q-a(S) := {p E M(S) : x*IPI


0 and any finite number of eZements s 1 ,s 2 , ••• ,sn ins, there exists a e: t 1 (s) such that 2.3 PROPOSITION.

i

= 1,2, ••• ,n.

Towards our main result, (Theorem 2.7), we prove some lemmas. 139

LetS be an inve~se semigroup and e 1 , ••• ,ek Then the~e emsts ~ € t 1 (E 5 ) such that 2.4 LEMMA.

11~11 ~ Proof.

2k-l - 1

and

~·ej

= ej

fo~

€

E5 be given.

j "' 1, ••• ,k.

Recalling that E5 is a commutative semigroup an easy induction

argument on k with

yields the result. 2.5 DEFINITION.

Let S be an inverse semigroup and k a positive integer.

say E5 satisfies condition (Dk) if given u 1 , ••• ,~+l and integers i,j such that 1 < i < j

~

k+l,

eu.1

and

€

E5 we can find e

€

We E5

u •• J

If s is an inve~se semigroup~ then t 1 (E 5 ) admits a bounded approximate identity if and onLy if E5 satisfies condition (Dk) fo~ some positive intege~ k. 2.6 LEMMA.

Proof.

Suppose E5 satifies condition (Dk) for some positive integer k and

let {u1 , ••• ,un} be any finite subset of E5 •

Then condition (Dk) implies

the existence of e 1 , ••• ,~ € E5 such that for each i € {l, ••• ,n} there exists j € {l, ••• ,k} with e.u. u 1•• Then choosing ~ € t 1 (E 5 ) as in J 1 Lemma 2.4 we get e.u ... e.*~. J

1

J

l, •.. ,n.

ak*e:ll: J 1

1

Now recalling Proposition 2.3 (and the commutativity of E5 ) we conclude that t 1 (E 5 ) has a· bounded approximate identity with bound 2k-1. Conversely suppose that t 1 (E 5 ) admits a bounded approximate identity with bound M (say). Let e 1 , ••• ,ek+l € E5 be given, where k is an integer such that k > M. Proposition 2.3 says that we can find a € t 1 (E 5 ) such that

llall

140

< M

and

lle.-a*e·ll 1 1


h(~)

Then, as in-the proof of Theorem 2.5.3, one can easily show that yh E S~ and that each y E SAn arises from a unique h E M (S) with yh = y; thus the n map h ~ yh of Mn(S)A into SA is both one-to-one and onto. n We now show that the map h ~ yh is a homeomorphism in a similar but simpler way to the proof of Theorem 2.5.6. Suppose ha yh

~

a

166

~

h in Mn(S)A and let T c S be compact.

yh uniformly on T.

We can choose

~

E Mn (S) with

We wish to show that h(~) =

1 and may

assume that, for all a, we have jha(~)- h(~)j


0 we can choose n n p E M(S) such that T := supp(p) is compact and llv-p II < £. For large enough a, we have Ya • y on T, so f yadp = f y dp and (with ya = yh and a y .. yh)

Hence h

a

~

h in M (S)~. n

4.3 THEOREM. Let s be to~sional and compatible with K. Then the canonical map of Mn (S) into t•(SA) is an isomet~ic algeb~ homomoFphism. Proof.

First we note that the map ~ ~ ~ of M (S) into t~(S~) has the n

properties (v,~ E

M (S)). n

Since S is compatible with K, it is p-free and so M(S) has power multiplicative norm, by Theorem 3.2.

So for every ~ E M (S) a result of n Springer.(see Theorem 1.3) implies the existence of a multiplicative linear

map

~

: M(S)

~

IH~> I ..

L, where L is some complete field containing K, such that II~ II

and

II~ II

..

1.

167

Setting y(x) := ~(x) we have a bounded homomorphism y : S ~ L. y(x)~(~)

Now

= ~(x)~(~) = ~(x*~)

(x £ S), so y is continuous. Since S is compatible with K, it follows that y(x) £ K for all x £ Sandy£ SA. Recalling the remarks preceding Theorem 4.1, for any v £ M(S), we have

~(v)~(~) = ~(v*~) = ~I = I~I ~ and so

11~11

= lliill.

=

= ~(y).

v(y)~(~). Hence

IIOII

This completes our proof.

A comparison of Theorems 4.3 and 3.5 motivates one to seek for the image of M (S) under the Fourier-Stieltjes transform. Towards this end we first n give a definition. 4.4 DEFINITION. A continuous function f : SA ~ K is said to be a:l'bitrur>i'ty sma'tt outside equicontinuous sets if given £ > 0 we can find an equicontinuous set A C SA such that If I ~ £ on SA \ A; the set of all such f is denoted by EQ (SA). 0

The following lemma is also of independent interest.

4.4 LEMMA.

We have the following inclusion relations.

(ii)

Proof. equivalence relation x ~ y

if

Given ~

£

>

0, we consider an

on S defined by

y(x) = y(y)

for all

y £ SA with lf(y) I ~

£.

The equivalence classes are evidently clopen and hence the quotient semigroup R :• Sf~ is discrete. Let w : S ~ R be the quotient map and w: RA ~ SA the dual map of n. Since RA is compact (as R is discrete) and

n is 168

continuous, we have that n(RA) is a compact subset of SA.

By our

definition of ~we have lfl < f E C (SA).

E

outside the compact set n(RA) and so

0

(ii) It is sufficient to show that a characteristic function of a compact and open subset of SA is in UC(SA). Let V c SA be compact and open. Recalling the definition of compact open topology on SA, for each a E V we can find a compact Ka c S such that

(and o(a) is a neighbourhood of a).

So for some n E lN, we have

a1 , ••• ,an E V such that V = O(S1 )u ••• uo(an).

Setting T := Ka u ••• uKB 1

we

n

see that V is a union of elements of the clopen partition UT (defined in item 3.3) and so the characteristic function of V lies in UC(SA).

(i) The M (S) onto

s be

Let

4. 5 THEOREM.

to~sionat

and SA

Fourie~-Stiettjes t~nsfo~

is a Banach

atgeb~

s.

Then

isomoPphism of

EQ (SA).

n

0

(ii) If every compact subset of SA is

Stiettjes

points of

sep~te

t~sfo~

Proof. (i) Given A:= {y E SA :

~

is a Banach atgebra

E M (S) and

n IPI ~

E}.

E

then the Fou~e~­ of Mn (S) onto C0 (SA).

equicontinuous~

isomo~phism

> 0, we consider the set

For all x,y E S

llx*~- Y*~ll >sup l<x*~- Y,..~)A(Y)I =sup jy(x)- y(y>II~I -yEA

YEA

>

E

sup ly(x) - y(y)l; yEA

hence A is equicontinuous. By Lemma 4.4(i) and Theorem 3.5 we have that the map ~ + ~ is an isometri algebra homomorphism of M (S) into EQ (SA). To show that this map is onto, n 0 fix f E EQ (SA) and choose v E M(S) such that f = (which can be done, by

v

0

Theorem 3.5 and Lemma 4.4). We now show that v E Mn(S). Fix x £ S and E > 0. By our choice of f the set A:= {y E SA

lv(y)l ~ E} is equicontinuous, so

O(x) := {y E S : ly(x) - y(y) I
ll~(y) I

(*)

{

£

~ llvll -l

O(x) we have llvll =

£

£

if y

A