Compact Semitopological Semigroups and Weakly Almost Periodic Functions

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z~Jrich

42

9

J. F. Berglund K. H. Hofmann Tulane University, New Orleans

Compact Semitopological Semigroups and Weakly Almost Periodic Functions

1967

Springer-Verlag. Berlin. Heidelberg-New York

I

This work was supported in part by NSF Grant GP 6219. The second author is a Fellow of the Alfred P. Sloan Foundation

All rights, especially that oftranslatlon into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanlca] means (photostat, mlcrofdm and/or microcard)or by other procedure without written permission from Springer Verlag. @ by Springer-Verlag Berlin" Heidelberg 1967. Library of Congress Catalog Card Numbet 67-29251. Printed in Germany. Title No. 7362

TABLE OF C O N T E N T S

INTRODUCTION

CHAPTER

I.

............................................

1

PRELIMINARIES

. Compactness

Criteria

..............................

12

T h e o r e m 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E q u i v a l e n t conditions for c o m p a c t n e s s in function spaces.

16

. E q u l c o n t l n u o u s S e m i g r o u p s of L i n e a r Operators and Afflne T r a n s f o r m a t i o n s . Affine S e m l g r o u p s

.... 21

T h e o r e m 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The almost periodic subspace.

26

Theorem 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The w e a k l y almost periodic subspace.

27

P r o p o s i t i o n 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kakutanl fixed point theorem.

30

T h e o r e m 2.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R y l l - N a r d z e w s k l fixed point theorem.

34

3. Ellis'

Theorem

....................................

36

. Actions of Compact Groups on T o p o l o g i c a l V e c t o r Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

P r o p o s i t i o n 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A B a n a c h weak G-module is a s t r o n g G-module if G is a l o c a l l y compact group.

CHAPTER

If.

COMPACT

1. A l g e b r a i c

SEMITOPOLOGICAL

Background

Material

41

S~4IGROUPS

.....................

44

P r o p o s i t i o n 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rees Theorem.

47

P r o p o s i t i o n 1.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The group s u p p o r t i n g subspace.

57

P r o p o s i t i o n 1.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The s e m i g r o u p w i t h zero s u p p o r t i n g subspace.

59

2. Locally

........................

60

P r o p o s i t i o n 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The structure of a minimal ideal in a locally compact s e m i t o p o l o g l c a l s@migroup.

61

3. C o m p a c t

Compact

Paragroups

Semitopological

Semi~roups

................

Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The first fundamental theorem of compact s e m i t o p o l o g i c a l semlgroups.

67

P r o p o s i t i o n 3.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The strongly almost periodic subspace.

71

Theorem 3.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The main theorem on semigroups of operators on a B a n a c h space.

80

. Invariant

Measures

on Locally

Compact

Semlgroups

.. 88

Theorem 4.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N e c e s s a r y and sufficient conditions for invariance of a measure. CHAPTER

65

III.

I. Various

AI/~OST PERIODIC AND W E A K L Y AI~OST PERIODIC FUNCTIONS ON S E M I T O P O L O G I C A L S E M I G R O U P S Universal

. The D e f i n i t i o n

Functors

of Almost

.......................

Periodic

Functions

3. Invariant Means Theorem

3.2

112

...... 120

P r o p o s i t i o n 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D e c o m p o s i t i o n of weakly almost periodic functions.

4. Locally

97

126

..................................

127

....................................

127

Compact

Semitopological

Semigroups

.......

130

P r o p o s i t i o n 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Necessary and sufficient conditions for the e m b e d d i n g into the weakly almost periodic c o m p a c t l f i c a t l o n to be topological.

CHAPTER

Proposition 4.6 ................................ Equivalent conditions on the embedding of S into GAS.

133

Proposition 4.11 ............................... Analogue of 4.5 for the almost periodic compactlflcation.

136

Proposition 4.12 ............................... Analogue of 4.6 for A S .

137

Proposition 4.14 ............................... The almost periodic compactlfication of a topologically left simple semigroup.

138

Proposition 4.16 ............................... Partial analogue of 4.14 for the weakly almost periodic compactlficatlon.

140

IV.

BIBLIOGRAPHY

EXAMPLES

.................................

.........................................

146

158

-

1

-

INTRODUCTION After a lifetime of about fifteen years the theory of compact topological definition a compact Hausdorff

semigroups

topological

space with a continuous

multiplication considerable

has come of age.

semigrou p S is a compact associative

(x,y) r~--~ xy : S x S ---> S.

body of information

general compact topological from special subclasses

semigroups

P. S. Mostert,

Elements

Topological

Mathematical 1964).

yet been fully exploited perhaps.

questions

of functional

which do arise

Centre Tracts ll,

semigroups

One might expect that

And indeed, many

(for instance,

functions

have not

analysis would be especially

to such applications.

almost periodic

Charles E.

The possible applications

of the theory of compact topological

susceptible

(K. ~ H. Hofmann and

of Compact Semlgroups,

Amsterdam

certain branches

and is assembled

1966 and A. B. Paalman - de Miranda,

Semigroups,

Math. Centrum,

and of semigroups

is now available,

in books and monographs

Columbus

A

about the structure of

and accessible

Merrill,

By

in the study of

or in the characterization

of

the measure algebra of a locally compact abelian group) lead automatically semigroups.

to considerations

However,

these applications

class of compact semlgroups, are

(compact)

Hausdorff

in S being continuous semigroups

involving compact

namely,

call for a wider

semigroups

S which

spaces with the multiplication

in each variable

separately.

will be called semitopolo~ical

Such

semigroups,

-

2

-

even though some authors have called them topological semigroups.

The distinction is necessary

are discussed

if both types

concurrently.

No coherent structure theory of compact a~mitopological

semigroups

is a~ailable yet.

The first

authors to emphasize that compact semitopological

semi-

groups need to be considered were I. Glicksberg and K. de Leeuw, who, in their studies of weakly almost periodic functions

in the early sixties, already provided sufficient

motivation for investigating this topic.

J. L. Taylor in

his characterization of the measure algebra of a locally compact abelian group was also led to consider certain compact semitopological State University,

1964).

semigroups

(Dissertatlon, Loulslana

Beyond the general observations

in the work of Glicksberg and de Leeuw, we know of no attempt to begin laying the foundations of a structure theory of compact semitopological right.

semigroups

in their own

It is an unfortunate but indisputable fact that

most of the features of compact topological

semigroups

(familiar to anyone who spends only a l~ttle time contemplating them) do not carry over to compact semitopological semigroups.

Even very simple constructions

produce counterexamples

to most assertions about compact

semitopological

semigroups which are modeled after the

analogous assertions for compact topological One of the most elementary constructions compactification

~ ~ o f

semigroups.

is the one-point

the additive group of real

numbers with the operation extended by x + ~ = ~ + x = ~.

-3-

In fact, after some experience with compact semi~ topological

semigroups,

it begins to seem surprising

that there should be any remnants topological

theory which remain valid under the

weaker hypotheses.

At the present time there seems

to be only one substantial salvaged namely,

at all of the

fact which can be partially

from the theory of compact topological

semigroups,

the result that has been called the first

fundamental

theorem of compact topological

semigroups.

It states the existence of a unique compact minimal ideal M(S) and describes

its structure and imbedding in

a compact

topological

instance,

there are serious deficiencies

topological

case.

semigroup S.

Comparatively

M(S) need not be closed

~ech cohomology

true.

in the presence of an identity, ideal).

Practically

The primitive

by any maximal counterexample

nothing is left of the special

of Green's relations

topological

ideal M(S)

this may fall to remain

and often very useful structural compact

More

semigroup S that all of the

shows how completely

properties

show that

need not be compact.

of S is carried by the minimal

group in the minimal ~ U ~

easy examples

nothing remains of the familiar phenomenon

in a compact topological

(or, even,

in the semi-

in a compact semitopological

semlgroup S and, therefore, drastically,

But even in this

semigroup.

of the five relations

considered

which yielded an early decomposition

of a

A coset relative to any by Green need not be

closed in a compact semitopologlcal

semigroup;

in

-4-

particular, maximal subgroups need not be closed, as th~ example ~

v ~

shows.

Also, the two relations ~

no longer agree in general.

and

Furthermore, the so-

called swelling l~mma (see, for example, Hofmann and Mostert, lo 9. cit., p. 15) is not ~rue for compact semitopological semigroups; in addition, a compact semitopologlcal semigroup may very well contain a bicyclic semigroup (as the one-point compactification of the discrete bicyclic semigroup shows), this fact being unheard of in compact topological semigroups (ibid., p. 77).

The more subtle facets of the theory

of compact topological semigroups, like the theory of one parameter semigroups or of irreducible semigroups, remain completely obscure in the case of compact semitopological semigroups, and no basic research has yet been undertaken in this direction. This set of notes has two major objectives: Firstly, it presents the major motivations for the consideration of compact semltopological semigroups; notably, the foundations of the theory of almost periodic and weakly almost periodic functions based on a reasonably general theory of semigroups of operators on topological vector spaces, where the semigroups in question are compact in the strong operator topology or in the weak operator topology.

The tools used in this study are

general methods concerning topological vector spaces and compactness criteria for function spaces.

-5-

Secondly,

it displays the rudiments of a general

structure theory of compact semitopological

semigroups

with particular emphasis on the study of the minimal ideal.

Apart from standard devices borrowed from

algebraic and topological

semigroup theory, we use Ellis

results on locally compact transformation groups, integration theory, and fixed point theorems for semigroups of affine transformations

on compact convex

sets as tools in this effort. In combining the two main trends

in the notes, we

apply the semigroup theory to operator semigroups and thus develop the theory of almost periodic the spirit of Glicksberg and de Leeuw.

functions

in

In the general

existence theorem we give a proof w~ich is based on the adJoint functor theorem of category theory. There is some indication that the theory of compact semitopologlcal

semigroups,

on the one hand, and the

theory of compact topological

semigroups,

on the other,

(as well as the emphases of the methods used in one or the other) takes on a distinctly different flavor. the topological theory,

the semitopological

theory seems

to lean strongly towards functional analysis. the best results concerning Green's relations compact semitopological

One of in a

semlgroup is the observation

that the minimal ideal M(H-)

of the closure of sny

~aximal group H~i~ a compact topological group. theorem is a consequence of Ryll-Nardzewski's theorem.

Unlike

This

fixed point

A purely semigroup theoretical proof, although

-6-

likely to exist, has not yet been found. These notes are not entirely self-contained.

We use

the theory of locally convex topological vector spaces freely, and we take the theory of integration of vectorvalued functions

for granted.

We do, however,

the pertinent portions of Grothendieck's compactness Ellis'

in function spaces.

present

theory of

On the other hand, again,

results about the Joint continuity of actions of

operations

in a locally compact transformation group on a

locally compact space are quoted without proofs.

The

interested reader must consult the original literature; nevertheless,

the results themselves are easily

understood, but the methods in their proofs would have led us too far astray.

We assume a certain familiarity

with the theory of compact topological and of algebraic semigroups,

although such a familiarity

is not an

absolutely necessary prerequisite for any prospective reader.

In one place the adJoint functor theorem is

used, but the existence theorem resulting from this categorical argument is also proved by somewhat more constructive methods

in the most interesting special

cases. The bibllography

is given for each chapter separately

We do not claim completeness

in our historical references,

nor do we bother to trace all results to their origins. The content of the notes Chapter

I. Preliminaries:

is roughly as follows: Section 1 is devoted to

-

compactness

7-

criteria ~n function spaces.

This material

is necessary for any discussion of weakly almost periodic functions

including such standard,

superficial,

but not a~ all

facts as the equivalence of right and left

weak almost periodicity.

Section 2 discusses equl-

continuous semigroups of linear operators on locally convex vector spaces and certain natural generalizations. The guiding theme is to single out the set of vectors which have relatively compact orbits under the semlgroup in various topologies

in order to show that these elements

form a closed subvector space,

to restrict the semlgroup

to this subvector space, and, finally, this restriction

to observe that

is a compact topological semi~roup or

a compact semitopological

semi~roup,

as the case may be.

The section concludes wlbh the discussion of some fixed point theorems for semigroups of continuous affine transformations

of compact convex sets.

In particular,

we give Namioka and Asplund's very elegant proof of Ryll-Nardzewski's formulate Ellis' transformation

fixed point theorem.

In section 3 we

results about locally compact

~roups.

Section four gives a brief

account of compact topological groups acting linearly on topological vector spaces;

this section complements

section 1 and generally known facts of the representation of compact groups on Hilbert spaces. Chapter

II.

Compact semitopologlcal

semigroups:

Section 1 handles some background material from the algebraic theory of semigroups.

The theory of completely

-8-

simple minimal ideals is developed is needed for semitopological

in exactly the way it

semigroups.

The second part

of this section is concerned with elementary facts about semigroups of linear transformations

on vector spaces.

We pay particular attention to the presence of a completely simple minimal

ideal in the semigroup and to

the consequences that result for the action.

Section 2

is devoted to a discussion of locally compact completely simple semigroups and gives their complete structure theory.

Section 3 describes the first fundamental theorem

of compact semitopological

semigroups.

(As mentioned

before, at the present stage of the theory of compact semitopological theorem.)

semigroups,

there is no second fundamental

The latter part o f the section discusses

compact semitopological

semigroups of linear operators

on Banach spaces.

The section concludes with a very

important theorem,

in which we collect all the pertinent

facts.

Since all the important partial results were

proved previously, its formulation,

its proof is considerably shorter than

which extends over six pages.

Section 4

contains a discussion of various types of invariant measures and integrals on a locally compact semitopological semigroup.

In this particular respect,

little d ~ e r e n c e semigroups.

there is very

between topological and semitopological

A series of papers have been writen relating

to this topic.

Some facets of our presentation are new

as far as we can tell, however. Chapter III. Almost periodic and weakly almost

-9-

periodic functions on semitopoloEical

semigroups:

In

section 1 we point out the functorial aspects of certain universal

constructions.

Just as any topological

carries its Stone-Cech compactification along, any semitopological

space

functorially

semigroup S determines

functorially a compact semitopological a continuous semigroup morphism

~S

semigroup

: S ---> ~ S

~S

and

such

that for any continuous semigroup morphism ~ : S ---> T into a compact semitopologlcal morphism $'

T there is a unique

: ~-~S ---> T such that ~ = $ ' ~ S "

In fact,

we give a series of similar functors and discuss their mutual relationship.

Section 2 at last presents the

definitions of weakly almost periodic and almost periodic functions from various aspects.

All of these algebras

of functions are functorially related to the objects discussed

in section 1.

biJective

correspondence between the weakly almost

periodic functions on O S

For example,

there Is a natural

f on S and the continuous

which is given by F ~ S = f.

functions F

Particular attention

is paid to the delicate question of how the presence or absence of an identity in the semlgroup S simplifies or complicates matters. avoided

This issu~ had been carefully

in earlier treatments.

The constructive proof

of the weakly almost periodic compactification ~).S of S given by Glicksberg and de LeeuW section)

(and repreduced

in this

in the case that S has an identity is not

adequate to cover the general case.

Section 3 discusses

invariant means for all of the classes of almost periodic

-

functions introduced

lO-

in section 2.

We note that all the

various classes of almost periodic functions admit invarlant means,

if the semigroup in question is

algebraically a group.

(For the case of the weakly

almost periodic functions, Nardzewski

this requires the Ryll-

fixed point theorem.)

Section 4 is devoted

to the question of how a locally compact semitopological semlgroup S (and thus, in particular,

a locally compact

topological group) is mapped into its weakly almost periodic compactification

QS.

given to a characterization which the morphism

~S

Specific emphasis is

of those semigroups S for

is a homeomorphism of S onto an

open subsemigroup of ~ S

(which turns out always to be

the case for locally compact topological groups). of the significant mentioned earlier: semitopological

One

results of this section has been If G is any subgroup of a compact

semigroup,

then the minimal ideal M(G--)

of its closure is a compact topological group. Chapter IV.

Examples:

A short catalogue of examples

illustrating some of the properties of compact semitopological semlgroups concludes the discussion. These notes originate from three different, but not independent occasions.

K. H. Hofmann gave a course about

weakly almost periodic functions and compact semitopological semigroups at the University of TGbingen during the Summer Semester 1966, and held a seminar about the same topic at Tulane University in the academic year

-

1966-67.

ll

-

J. F. Berglund took notes of this seminar,

and

at the same time he worked on his dissertation which is closely related to the present topic.

A 6ood part of his

results found their way into these notes. of some of his results

An announcement

is to appear in the Czechoslovak

Journal of Mathematics. While the curiosity about compact topological

semi-

groups has been awake in one of the authors for quite some time,

it was

Frank

Birtel who evoked interest in

weakly almost periodic functions

in both of us, and the

slant towards a more general attempt at the theory of compact semltopologlcal

semlgroups

then came quite

naturally.

John F. Berglund Karl Heinrich Hofmann Tulane University

-

12

CHAPT~

-

I

PRELIMINARIES 1. Compactness 1.1

Notation: denote

Criteria

For a topological

vector space E, let E w

the vector 8pace E with the weak ~(E,E')

topology.

If S is a topological

will denote the commutative

C*-algebra

functions

of all bounded

complex-valued

continuous

uniform norm.

If D is a dense subset of S, denote

by C(S) D the set C(S) endowed pointwise 1.2

space, 1 then C(S)

Observe

convergence

that

identity

with the topology

of

on D.

if DC_ D' ~

function

on S in the

S and ~ = S, then the

defines

continuous

maps

c(s) w ---> c(s) o, ---> c(s) D.

1.3

Suppose

K ~ C(S) w is pre-compact.

compact

in C(S) D for any dense subset

K is bounded Proof:

that bounded

same for any locally

O]

, Chap.

sets

1.4

IV,

pre-compactness,

Let S be compact, is relatively

Also,

in E and E w are the

convex topological

vector

of topological

By

groups

we have the first assertion

and let K ~ C(S) be bounded.

sequentially

space

2, N ~ 4 , Th . 3 , Cot.).

1.2 and the fact that morphisms preserve

D of S.

in C(S).

Observe

(Bourbaki

Then K is pre-

compact

lln these notes, unless otherwise cal spaces are Hausdorff spaces.

If K

in C(S) S, then K

noted,

all topologi-

-

is relatively Proof:

compact

13

-

in C(S) w.

Take a sequence n ~

fn in K.

Pick a sub-

sequence n n--> gn such that g = lim gn in C(S) S. By the ~ebesgue

Let /~ be a Radon measure on S. convergence

theorem,

I I

lira s gn dr

= t g d#.

Hence g = lim gn in C(S) w. sequentially

compact

Therefore

in C(S) w.

K is relatively

Now we use the

V

Eberlein-Smulian relatively

theorem to conclude

compact

in C(S) w.

that K is

(For the Eberleln-

~mulian theorem see, for example,

Dunford,

N., and

i

Schwartz,

J. T., Linear Operators

I, Inteirsc. Publ.,

New York 1958, p. 430. ) A topological

space

is countably compact

if every countable

open cover has a finite subcover.

1.5 Let S be compact, statements

and let K c C(S)s.

are equlvalent:

(a) K is relatively

countably

(b) K is relatively

sequentially

Proof:

(b) ==>

(a) ==>

(b):

(a):

: fn(X)

: h = 1,2,...~

in K, and

relation on S given by = fn(y), n = 1,2,...~.

Then all of the functions

exists an f'

compact.

Let n n---~ fn be a sequence

R = ~(x,y)

equivalence

compact.

Trivial.

let R be the equivalence

~fn

The following

in the closure F of

in C(S) S are constant on the

classes of [~; and to every f & F there : S/R ---> ([~ with f = f ' ~ ,

where

-

l~

-

: S ---> S/R denotes the quotient map. f ^

The mapping

~ f' is a homeomorphism of F into C(S/R).

F' be the image of F in C(S/R).

~ach countable subset

of F' is relatively compact in C(S/R), holds for F in C(S).

In particular,

has a cluster point in C(S/R). notation

Let

since this !

then, n r~-~ fn

We simplify the

in that we now assume that the fn separate

the points of S.

In which case the smallest conJugat

closed subalgebra of C(S) containing all of the fn is dense in C(S) by the Stone-Welerstrass Therefore, metrizsble.

C(S)

is separable,

theorem.

and so, S is separable

Hence, S has a countable dense subset M.

With the diagonal method we may obtain a subsequence of the fn which converges

in C(S) M.

On the other

hand, after the preceding,

this subsequence has a

cluster point g in C(S) S.

Furthermore,

the sub-

sequence converges to g on a dense subset; whence the convergence

is everywhere,

and g is actually a limit

function. 1.6

Definition: s : IN x t ~

Let X be a topological space. ---> x be a double sequence.

Let Then s ~ X

is a double cluster point if for every neighborhood V of x, there exists

(N,M) s /~ x i~ such that if

n ~ N and m ~ M, then the sets and

~

~/L~: ~(/~,n) t V

: ~(m, ~ ) ~ V~ are infinite.

-

1.7

15

-

Let D be compact, T dense in S. such that K(s) = ~f(s)

Suppose K ~ C(S) S is

: f~ K~ Is compact for all s ~ T .

Then the following are equivalent: (a) K is relatively compact in C(S)s; (b) K is relatively countably compact In C(S)s; (c) If n ~ ( f n , X n )

Is a sequence in K x T, and If

Z = ~x n : n=1,2,...~,

then

~f &C(Z)such

f Is a cluster point of n ~ f n l Z (d) If n ~ ( f n , X n )

that

In C(Z) Z.

is a sequence In K x T, then

( m , n ) ~ f m ( X n) has a double cluster point. Proof:

(a) ===~

(b) : Trivial.

(b) ===~

(c):

Let f be a cluster point of {fn :n=l,~,...

(c) ==-~

(d):

For the given sequence choose an f as In

(c).

Let x be a cluster point of the set ~x n : n = l ~ , . . . ~

In the compact space S.

Then c = f(x) is a double clust~

point of ( m , n ) ~ f m ( X n ) -

for if V were a neighborhood of

c such that for at most a finite number of indices J, we had f m ( X j ) ~ V

for m>im o(j), then f ( x j ) ~ V

for almost all

J; therefore,

c = f(x) ~V, contradicting the choice of V

On the other hand, if for at most a finite number of m, fm(Xj)~V

for J ~ J o ( m ) ,

Hence c = f ( x ) ~ V ,

then fro(x) ~V for almost all m.

which again Is not possible.

(d) ===~ (a):

Now we have K(x) relatively compact In

for all x ~ S .

Therefore there is a function f : S - - ~

and a net f on K with lira f(x) = f(x) for each x C S and each net y In T with x = llm y. In that case there Is a point x & S

Suppose not, and a net

-

16-

y in T with x = lim y such that there is an a > O ,~Ith lfCz)

fCx)t

-

holdln~ throughout.

The sequence n~-~*Cfn,Xn),

in K x T is constructed x O = x.

Suppose

Let fo = f,

(fi, xi) 6, K x T have been chosen

for i = I,...,n-I, induction

as follows:

n=1,...

n52.

(It is obvious

step how (f1'xl) is chosen.)

in the On the basis

of the fact that f(z) = lim f(z) for all z ~ S, we determine an fn so that (I) I f n ( x i ) -

f(xi) I < I/n

for i = O,I,...,n-I,

With the aid of the net y, Xn will now be chosen so that, first of all,

fi(x)i~

(2) I f i ( X n ) -

I/n

f o r i = 0,1,...,n,

and, secondly, (3) IfCxn) - fr Because

1~

of (I) we have f(xl) = liram fm(Xi) and

f(x) = lim m fm(X).

By (2), fi(x) = llm m fi(Xm)-

Now let c be chosen as in (d) for the sequence n%(f

n, Xn).

Then c is necessarily

of the sequences

i%f(xi)

c must equal f(x). 1.8

Theorem:

and i ~ f i ( x ) ;

But this contradicts

Let S be a topological

be the commutative valued continuous

f~f

a cluster point

C*-algebra functions

therefore, (3).

space, and let C(S)

of all bounded

on S.

complex

Let

:CCS)--r

be t h e Gelfand i s o m o r p h i s m .

Let K , C(S) be bounded.

Then the following are equivalent:

17(a) ~ is relatively compact in C(S)w; -

(a') k is relatively compact in C(~S)w; (b) K is relatively compact in C(@S)~S; (c) K is relatively countably compact in C(r (d) K is relatively sequentiall~ c o m p a ~ t ~ n i C ( ~ S (e) For each sequence n~--~(fn, x n) on K x S, (m,n)--~fm(X n) has a double cluster point. Note that (a) and (a') imply K bounded. Proof: With the observation that because of the isomorphism

(a)-i==--~(a'), everything else follows

from 1.4, 1.5, and 1.7. 1.9

Corollary: Let C:S--$~S be the natural map.

Suppose

K has the property that f

f

) = o.

LIn particular, if s is locally compact and K~Co(S). If K is bounded in C(S), then the following are equivalent : (a) K is relatively compact in C (S)w; (b) K is relatively compact in C(S)S; (c) K is relatively countably compact in C(S)s; (d) K is relatively sequentially compact in C(S) Proof: Observe that 1.8(b)<~==>1.9(b) under the present hypotheses. I .I0 Corollary: Let f:S x S--~ ~i be bounded and separately continuous. Define K I = ~f(-,y) : y6S~. and K r = Lf(x, 9 ) : x ~Si.

Then the following~re~qui~lent

(a) K 1 is relatively compact in C(S)w; (b) K r is relatively compact in C(S)w;

-

18

-

Since K~ and K i are bounded,

Proof:

by theorem.l.8

we have K r is relatively

compact

in C(S) w

<===> for each sequence n ~--~ (f (xn, 9 ),Yn ) in K r x S,

(m,n) ru_, (f(Xm,Yn))

has

a double cluster point <===> for each sequence n ~ in K I x S,

(m,n)~

(f(.,yn),Xn) (f(Xm,Yn))

has

a double cluster point <===> K 1 is relatively Note: Corollary

compact

in C(S) w.

1.10 is false if C(S) w is replaced by

C(S) throughout:

Comslder the continuous

function

J

f : S x S ---> (~ defined by f(x,y) = x l/y, where S is the locally

compact space S = ~0,1].

situation we have that ~ f(x,-) compact

: x 6 S~ is relatively

in C(S), but that ~f(.,y)

relatively I.II Corollary:

compact

In this

: Y & S~ is not

in C(S).

Let A be a C*-algebra with identity i.

Define A r to be the vector space A with the weakest topology making all irreducible

representations

continuous.

the statements

For K C__A bounded,

(a) K is relatively

compact

in A w

(b) K is relatively

compact in A r

and

are equivalent provided Proof:

that A is commutative.

Since A is a commutativelC*-algebra,

identify it with C(S), where S is the compact

-

structure

19-

space of A given by the Gelfand theory.

Then A r is Just C(S) S and the result follows 1.12 Corollary:

from I.~

Let A be a C*-algebra with identity

Suppose K,L c~ A are bounded. relatively

I.

If K and L are

compact In A w, then so is KL provided

that A is commutative. Proof: Let a~-,a^ :A--~C(S) be the Gelfand isomorphism where S is the compact structure by 1.8, E

Is compact In A w Iff

space of A.

Now

(KL)A is compact in

C(S) S (closures being taken in the appropriate topolo61es).

Let

(ab) ^ - aAb ~ be a net in (EL) ~.

Since K ~ is compact In C(S) S, a ^ has a convergent subnet a'_A; and since L ~ is compact, net b r^ has a convergent a"_A--~f and b"_% - - ~ . (a"^bl~)(t)

subnet b ''^.

the correspondi1~ Suppose

Then for each t ~ S, = a'--~(t)b"^(t)--@f(t)g(t)

since multiplication

in ~ ~ r ~ ] is continuous.

every net in (KL) ~ has a C(S)s-convergent whence Question:

= fg(t) Thus

subnet;

(KLi ~ is compact In C (S)s. Are corollaries

the proviso 1.13 Corollary:

1.11 and 1.12 true without

that A be commutative? Let S be compact.

is separately

continuous

Suppose f:S x S - - ~ C&

and bounded.

Then

(1) K r = ~f(.,y)

: y~S~

Is compact in C(S)w;

(ll) K 1 = ~f(x,.)

: x~S~

Is compact in C(S) w.

Proof: Let Fr:S--~C (S )s be defined by Fr(x ) = f(-,x). Since f Is separately

continuous,

F r is continuous.

-

20-

im F r = K r is compact

So

is relatively

compact

in C(S)w;

of polntwise

convergence

K r is closed

in C(S)w.

1.14 Corollary:

in C(S) S.

Thus, by 1.8, K r

but since the topology

is weaker than the weak topolog~ The proof of (ii) ls slmllar.

Under the hypotheses

of 1.13,

if

Fr, F 1 : ~ ---~ C(S)w are defined by

Fr(X) = f(-.x), Fl(X) = f(x,.), then F r and F I are continuous. Proof: Fr(S)

We consider Fr, the proof for F I b e i n g similar. = Kr~

C(S) w is compact by 1.13.

C(S) S is weaker than that of C(S)w; of K r in C(S)w,

so by the compactness

C(S) S induces the same topology on K r.

But Fr: S ---> C(S)s ly continuous.

The topology of

is continuous

because

f is separate-

Since F r factors through Kr,

F r : S ---> C(S)w is continuous. 1.15 Let S be compact.

Suppose

f ~ C(S x S).

Then

(i) K r = ~f(.,y)

: y & S~ is compact

in C(S);

(ii) K 1 = ~f(x,.)

: x 5. S~ is compact

In C(S).

Proof:

We prove

(i); the proof of

(ii) Is similar.

Let F r : B ---~ C(S) be defined by Fr(x) show F r is continuous:

Since

= f(.,x).

f is continuous,

We

for a

fixed So ~ S and for each t & S, there is a neighborhood Vt x W t of

(t,So) In S x S such that

If(r,s) - f(t,So) ~ < g, for all

(r,s) s V t x W t.

Cover S by finitely many such V t, say VI,

..., V n.

Let

n

W =

f'] Wi. Then ~ f ( t , s ) - f(t,So) i < ~-, for all (t,s) i=l S x W; i.e., liFt(s) - F r ( S o ) ~ < ~, for all s ~ W. Thus

K r is compact as the continuous

image of the compact S.

-

2. Equicontinuous Affine 2.1

Semigroups

Trsnsformstions,

91

-

of Linear Operators ~"

Notation and Conventions: following hypotheses notation

and notation,

Let E and F be topological (We will a m b i g u o u s l y Then F E =
as well as the

this section:

vector spaces over ~

: f is a function~

in the product

topology

I~.)

(topology of

Also & linear}

vector subspace of F E.

write L(E) rather than L(E,E). is an equicontlnuous

or

is a topologlce

= If & F E : f is continuous

is a topological

the

denote the scalar field by

convergence).

L(E,F)

Affine Semigroups. We will maintain

of 1.1, throughout

and

set.

If F = E,

Suppose S % L(E,F)

Let S denote the closure

of S in F E. 2.2

(I) ~ c_ L(E,F);

and

(ii) ~ is equicontinuous. Proof:

Let U be a closed neighborhood

Let V be a neighborhood for all s 6 S.

of 0 in E such that sV ~ U

Take t ~ 5.

Then there exists a net

s on S with tx = llm sx for all x 6 E. for v s V ,

tv = llm s_v s ~ = U.

Furthermore,

of 0 in F.

Therefore,

So ~ is e q u l c o n t l n u o u

if x,y ~ E, ~ ~ ( [ , we have t(kx + y) = llm s(~x

+ y)

= ~[lim s_x + lira sy = ~tx Thus, ~ c_~ L(E,F).

+ ty.

-

2.3

The

followin~

statements

(a) ~ is compact (b) ~ Proof:

(a)==~(b):

is a

we have S - ~

: x~ E~ ~ F ~.

~x.

Since each S-~

: x & E l is compact by the Tychonof~

Thus ~ is a closed subset of a compact

space and is therefore Lemma:

in F for all x ~ E.

~ince fr~*f(x):FE--*F

S~-~{~

is compact, ~ { S x

2.4

are equivalent:

vector space morphism,

(b)==~(a):

theorem.

-

in L(E,F);

is compact

continuous

22

compact.

Let E be a locally convex quasi-complete I

topological completion

vector space. of E.

Suppose E denotes

Let C ~ E be bounded.

the

Then the

closure of C in Ew is the same as the closure of C in (~)w; whence the following

statements

are

equivalent : (a) C is relatively

compact in Ew;

(b) C is relatively

compact

Proof:

Let ~ denote the closure of C in ( ~ w "

Consider

c--o(C), the closed

Since ~ is quasi-complete 6o(C)

in (E) w.

is complete.

(in E) convex hull of C. and c0(C) is bounded,

Hence c0(C) is closed

But since c-~(C) is convex,

in E.

it is also closed

in (E)w"

But then C c co(C) ~ E, and we have the assertion. Remark:

In the sequel we will need the following fact:

Let E,F be locally convex topological

vector spaces;

let u be a linear function from E into F. metrizable,

then the following statements

I quasi-complete

is complete

If E is are

means a closed bounded

set

-

23

-

equivalent : (a) u 5 L(E,F); (b) u 6 L(Ew,Fw). In particular, L(E,F) = L(Ew,F w) = L(E,F~). (See Schaefer [7], P. 158 and p. 132.) 2.5

Suppose that one of the following conditions is satisfied: (I) F is locally convex and quasi-complete. (2) E is metrizable and F = (Fo) w for some locally convex quasi-complete topological vector space F o . Let G = ~x ~ E : S-~ is compact in F~.

Then

(i) G is a vector subspace of E. (li) If F = E and S is a semigroup, then G is an invariant subspace. (iii) SJG, closure taken in F G, is compact and contained in L(G,F). (iv) G is closed in E. Proof:

(i) Let x,y 6 G, A 6 ~K; then S(Ax +y) C A ~

(ii) Let x ~. G, s ~. S. (iii) Let S' = SIG.

Then SsxCl Sx, so Sex c_ Sx.

By 2.3, since S ' ~

c~osure of S' in F G, -S', is compact. contained

in L(G,F).

+ S-~.

L(G,F), the By 2.2, S' is

(Indeed, in case (2), S' ~_ L.rG ~ ' .~O~. ~ ~

(iv) Let "~" denote the completion of a topological vector space.

Let ~

in case (1)

FI = [ (Fo)w in case (2). Now, each s ~ ~T extends uniquely to ~

: ~ ---> F 1.

(For case (2), use the preceding remark in the following

-

fashion:

24

-

s~ L(G,F o) extends

to "s ~ L ( ~ , F o) = L ( ~

Let T= ~.~ _ L ( ~ , F I) : stS~'~. T is equlcontinuous.

F I)

Then

Take x a ~, y 8 G, s , t & ~ ,

and

consider =

By equicontinuity,

y) + ( t -

the first summand

small independently The second summand

in L(~,F I).

is small if s is sufficiently

Therefore

of the pointwise

is continuous.

iqow take x 5 ~ , sx

can be made

of s and t if y is close to x.

close to t by definition So s~*~x:S-r--*F1

s)y.

=

topology.

Hence T is compact

s ~S;

we have

( s l G ) "~x ~ T x

Sx is relatively

compact in F I 9

In case

(I

then, S-~, closure in F, is complete since F is quasicomplete and Sx is bounded; whence Sx c F is compact. have Sx relatively

compact

in either case, x a G ; 2.6

The following

In case

(2), we

in F by lemma 2.4.

Thus,

so G is closed.

functions

(i) ( s , x ) - ~ s x : S

are continuous: x E--~F;

(ii) (s,t) ~-*st:S x L(E)--~L(E,F). Proof:

We prove

(ii); the proof of (i) is similar.

Let X ~_ E be finite;

U a neighborhood

Using the equlcontinuity

of 0 in F.

of S, take V a neig~hborhood

of 0 in E such that sV ~ U for all s ~ S.

Fix So& S,

to~ L(E).

U~.

Let W(K,U) m {tJ~ L(E',F') : tK c

let t ~ (t o + W(X,V)) and s ~ (s o + W ( t o X , U ) ) ~ S . for x g X ,

we have s(t - t o ) x g s V

9_ U, and

Now Then

-

25

-

(s - s o)tox ~_ U. Therefore st-

Sot o = s(t - to) + (s - So)t o

~ w(x,u) + w(x,u), which yields 2.7

the desired

The ring multiplication space L(Ew) Proof:

F,G:I,~ E

on the topological

is continuous

Let x cE,

s o' :~'--~E'

continuity.

y ~E',

in each variable so~L(Ew);

be the adJolnt

of s o .

x ~ --2 I~E x E' defined

= ~(SoX,Y),

G($)(x,y)

= ~(X,So, Y),

and let The functions

With f(s) = ss o and g(s) = SoS, and

with the function i(s)(x,y)

separately

by

F(~)(x,y)

are continuous.

vector

i:L(Ew)--,#~ E

= <sx,y>,

x E'

defined by

we have a commutative

L(~w ) __i ~

diagram:

x E' %

h(~ w) - ! - - ~ t k ~:x ~ ' , and a corresponding

diagram

f and F, respectively. pointwise

topology,

homeomorphism. 2.8

Definition:

such that

Remark:

the corestriction

Therefore

S together

(x,y)r~,xy:S

[separately

By the definition

A topological

is a semigroup

with g and G Instead of the

of i is a

f and g are continuous. [semitopolodica~ with a Hausdorff

x S--~S

semigrou~ topology

is continuous

continuous ~.

By 2.7, L(E w) is a semltopological

relative

to multiplication.

of

semigroup

-

2.9 Theorem:

26-

Let E be s locally convex quasl-complete

topological vector space over ~ or ~. S c: L(E) is an equicontlnuous A = Ix ~ E

Suppose

semigroup.

Define

: S-'x is compact in Ei;

and define T = SiA, closure being taken in A A. Then (1) A is a closed invariant vectorsubspace of (li) T is a compact topological semlgroup in L(E ~ (iii)

~ :S--sT, defined by

~(s)

= slA, is a

morphism of topological semigroups with ~-~

= T.

If, in addition,

E is an algebra such that x ~ a x

x~,xa, Vx,a~E,

are continuous,

s 6 S are endomorphisms

and

and if all of the

of the algebra,

then A is a

subalgebra of E and all t 5 T are algebra endomorphism Proof:

By 2.5, we have

(i).

Also by 2.5, T is a

compact subset of L(E), and by 2.6, (s,t)~st:T

So we have

x L(E)--~L(E)

(ll).

is continuous.

Statement

But then

(iil) is trivial.

Now suppose E is an algebra satisfying the given conditions.

The function

~ : E - - ~ E E with

~g(a)x = ax

is in L(E) by definition of the pointwlse topology. Let a ~ A; then

~(Sa)

is equicontlnuous. in E A is compact, L(A,E). K ~E.

is equicontinuous because S

So by 2.5, the closure H of T(Sa) equicontlnuous,

By 2.6(i), Now let b EA.

an8 a subset of

HK is compact in E for all compac Then we have

-

27-

SabC_ (Sa)(Sb)

Therefore,

Sab is compact and a b 6 A.

Now let t &T,

and let s be a net on S with t = lira s~A. tab = llm s_ab = lim

Th~nby 2.6,

(sa)(s_b)

= llm

= (lira ~(s_a)) (lira sb) = ( ~ (ta)) (tb) = (ta) (tb). So t is an algebra Remarks :

endomorphism.

With the hypotheses (i)

(t,x)~tx:T

of Theorem 2.9,

x A--~A is continupus;

(li) the topology

of pointwise

convergence

on T

is the same as the topoloEy of compact convergence.

2.10 Theorem:

Let E be a Frechet

Suppose S ~ L(E)

space over ~Cor

is an equicontinuous

2.

semiEroup.

Define W = and define

~xwE

T = S!W,

: Sx is compact

in EweS; W

closure being taken in

Then (i) W is a closed

invariant

vector subspace

in E and F~; (ii) T is a compact

semitopological

semigroup

in L (~,w) ; (iii)

~:S--~T, morphism with

defined

by

~(s)

of semitopological

~(S)

= T.

= sIW,

is a

semigroups

If,

in addition,

28

-

E is a commutative

C*-algebra,

if all of the s ~ S are C*-endomorphisms, a C*-subalgebra Proof:By vector

2.5,

we have

that W is a closed

of E.

Since

W is eonvex and @losed in E w.

of L(Ew).

invariant

E is locally

in E, by Mazur's

Also by 2.5,

Let s 6 S .

c@nvex and

theorem,

T is a compact

it

subset

Then

since W is invariant. S such that

then W is

and all t & T are C*-endomorphisms.

subspace

is closed

and

So if t ~T and s is a net on

t = llm slW , then

for all t' -~ T, by 2 .6 ,

t't = t' (llm s) = lira t's i.T, since

T is closed.

it is a s u b s e m i g r o u p L(E w)

(2.7),

Thus

and

of the semitepolegical

it is a semitopological

Therefore is again

T is a semigroup;

we have

since

semigreup

semigroup.

(il).

Statement

trivial.

If E is a commutative s g S are algebra

C*-algebra,

endomorphisms,

and

if all

then we have

gab c_ (Sa)CSb) and

(S'~)(~-~) is compact

a,b in E w by corollary

That each t s T is a C * - e n d o m o r p h l s m manner Remarks:

(iii)

similar

to the proof

With the hypotheses

is proved

of corollary

of theorem

I .12. in a

I .12.

2. I0,

(i)

(t,x)~tx:T

x W--~W w is continuous;

(li)

(s,t)~st:T

x (TfIL(W))--~T

if T ~*L(W) has the topology

is continuous induced

from W '~'

-

Definition: convex

29-

Let K be 8 convex subset of s locsll. topological

vector space E.

: K ---> F is called affine (i) ~

is continuous,

(ii) for all r,s

A function

if

and

& [0,~

with r + s = i,

o~(rx + sy) = r ~ ( x )

+ s ~.(y),

x,y & K. Let A(K) = ~ o L : K ---> K : ~ 2.11 Let K be as in 2.10.

Then A(K)

semigroup under composition Furthermore,

is affine~. is a semitopological

in the topology

A(K) has a minimal

of K K.

ideal M(A(K));

in

particular, M(A(K))

= ~ o L ~ A(K)

: there is a k ~ K w i t h ~

or is a minimal

= ~k~

left ideal.

2.12 Let K be a compact convex subset of a locally topological

vector space E.

subsemigroup

Suppose

T' ~

convex

A(K)

such that T' is equicontinuous

is a

on K.

Let T be the closure of T' in K K. (i) T is compact

in K K and equicontinuous

(ii) The uniform and polntwise

topologies

on K on T

are the same. (iii) T ~ A(K). (iv) T is a topological Proof:

semigroup.

(i) Similar to the proofs of 2.2(ii)

(ii) and

(iv) follow from the general

function spaces

(see, for example,

and 2.3.

theory of

Kelley,

Jr L.,

-

Topology,

General

30

-

D. Van Nostrand

Co.,

N. Y. 1955,

p. 232), (iil) A(K)

is closed

in C(K,K),

the space

of all

continuous

functions

f : K ---> K in the u n i f o r m

topology. In the proof few results

of the f o l l o w i n g w h i c h we will

of the results 2.13

locally

establish

of the present

Proposition:

proposition

topological

semigroup

tions

that,

on K, T is topologically dense

in T for all

point

in K.

Proof:

By 2.12,

generality in which

right

that T is a compact

Let T

transforma-

simple

topology

(i.e.,

without

topological

semigroup, But

that

G is a compact

x G,

topological

group and E(T),

the set of idempotents

in T, is a right-zero

(see II.3.6 and

infra).

prove

tT is

loss of ~en

tT = T for all t a T.

T = E(T) where

of affine

E.

of a

Then T has a fixed

we may assome

then we may assume

space

subset

in the pointwlse

t s T).

case we have

independently

convex

vector

be an eguicontinuous Suppose

to a

chapter.

Let K be a compact

convex

of K.

later

we refer

II.1.3

that G has a fixed

and e is the identity

point

semigrou

Now it suffices

in K, for if Gk = k

of G, by II.l.18

infra,

fk = ek = k for all f a E(T). So we may assume group.

that

to

T is a compact

topological

-

Fix k o & K.

-

Let/%be

the compact group T. on C(T,K)

31

normed Haar measure on

We may define I an integral

in such a fashion

and that f f ~ vector-valued

coincides integral

that

with the usual weak

(Bourbaki

[2], p. 12).

Let f : T ---> K be defined by fCs) = sk o, s 6 T. Note that t f(s) = f(ts).

Now there is a k S K

such that

k = ; f(s) d/~(s), and since t S T is an affine tk = f tof(s) d~(s)

transformation

of K,

= f f(ts) d/~(s) = I f(s) d/a(s) = k.

That is, k is a fixed point of T.

iAn indication

of the definition

of ! f d~:

Suppose X is any compact set, and F_ is a regular probability measure on X. Let ~ be the uniformity on X. For each entourage U ~ I/~, subdivide X into a disjoint union X I ~ X & ~ . . . ~ X n of ~ - m e a s u r a b l e sets such that X i x X i c U, i = 1,...,n. Let E be a locally convex space with fundamental system of convex neighborhoods of zero #~. Since f ~ C(X,E) is uniformly continuous, for each V ~ ~ , there is a U ~'ij~ such that

(x,y) ~ u

==>

f(x) - f(y) ~ v.

Let (Xi, i = l,...,n) be a subdivision corresponding to U ~%/{. Pick (xl,...,x n) s X I x ...x X n. Define n

~( (~)I (xi)) = ~ fCxi)/~Cxi )" Then g((Xi)~(x~ )) forms a Cauehy net which is independent 'of- the choice of (xj,...,Xn). Define

; f dr

= lira ~((X i)ICxi)).

That this integral has the properties easily verified.

stated

is

-

2.14 Lemma: subset

32

-

Let K be s non-empty, of a locally

convex

and let p be a continuous

separable,

topological pseudo-norm

convex

vector on E.

space E, For a

subset X of E, define

p-dlam(xl

= sup

p(x- y)

: x,y

If K is compact

in Ew,

then for each

a closed

subset

Q ~ K such that

convex p-diam

Proof:

(KkQ)

Let B be the B

=

Jx

:~ E

of K.

countable

Since

: p(x)

closed

~

convex body

.

in E w of the set of extreme K is separable,

there

is a

B + k i : i = 1,2,...~.

Since D is compact

in Ew,

in itself with respect (Choquet's

theorem:

Therefore,

set W which

is open

Let K 1 = c 0 ( D ~ W ) ,

to the relative

weak

topoloEy

J., Les C*-alg~bres

Gauthar-Villars:

Paris

and let K 2 = c o ( D ~ W ) .

in E w.

and K is compact

Since

in Ew, K 1 and

~o, by the theorem

of Kreln

K = co(K 1 U K 2)

Furthermore,

and Schwartz, K 1%

1964,

in E w such that

and Milman

Dunford

category

we can find a point k & K and a

KI,K 2 ~ K are closed K 2 are compact

it is of the second

see Dixmier,

representation,

P. 355).

(cf.,

is

set ~k i : i = 1 , 2 , . . . ~ ~ K such that

D C K q

et leurs

> O, there

~.

(weakly)

Let D be the closure points

<

~

op. ci___~t., p. 415).

K: for otherwise,

D \W

would

-

33

-

contain all the extreme points of K (Kelley, and Namioka,

I., Linear Topologicsl

Nostrand Co., New York 1963, the fact that W ~ D since

(B + k ) ~ D

% ~.

D. Van

p. 132) contradicting E p-diam(K 2) _< 2

Obviously,

~WAD.

Spaces,

J. L.,

Now let r ~ ~

O < r ~ l, and let fr : K 1 x K 2 ~ [r,l~

with ---> K be

defined by fr(X I , x2,t) = tx I

+

(1

t

-

)x 2

.

Then the image Qr of fr is closed in E w and convex. Moreover,

Qr % K: For otherwise,

each extreme point

z

of K would be of the form z = tx I + ( 1 where x i ~ Ki, would and,

t)x 2,

i = 1,2, and t ~ [ r , ~ .

But that

imply that each extreme point of K was in K 1 therefore,

Finally,

thst K = K1, a contradiction.

if y ~ K \ Q r ,

then

y = tx I + (1 - t)x2, where x i ~ Ki, i = 1,2, and t ~ LO,r).

It follows

that p(y-

x 2) = tp(x I - x 2) _< rd,

where d = p-diam(K)

G Now since p-diam(K 2) _< 2 '

< ~.

we have p-diam

(K kQr)

< ~ -

Choose r =

and let Q = Qr" p-diam

(K\Q)

<-~ --

2.15 Definition:

+ 2rd.

2

2

Then + 2(-~-~)d = ~ . 4d

Let K be a convex subset of a locally

convex topological

vector space E, and let S c A(K).

Suppose D is an S-invariant

subset of K.

S is said t,

-

act n o n - c o n t r a c t i y e l y x ~ y,

zero is not

34

-

on D if for all x,y & D with

in the closure

(in E) of

sCx- yl : s Clearly, only

S acts n o n - c o n t r a c t l v e l y

on D ~ K if and

if, for x,y ~ D with x ~ y, there

continuous

pseudo-norm

p = Px,y on E such that

0 < Inf ~p(s(x 2.16 Theorem

- y))

(Ryll-Nardzewskl):

afflne

transformations

subset

K of a locally

space E.

Suppose

point

Proof:

Assume

formation

of a weakly convex

1K.

case,

By a familiar to prove

standard

then

closed

x ~ K.

convex

of S.

subset < 2" --

point

In K.

Let x o ~ K be If this Is not the pseudo-norm

p on E,

~ > 0 with for all t ~ S.

K = co(SXo) , K Is a weakly subset

argument

we may also assume

is a continuous

s o ~ S, and an

trans-

that each finitely

of S has a fixed

for every

there

p-diam(K\Q)

Then S has

compactness

reduction,

(*) P(tSoX o - tx o) > ~

separable

vector

the identity

We show that Sx o = ~Xo~.

an ~lement

Since

x ~ K.

that S contains

that K = ~ ( S x ) flxed.

topological

convex

that S acts n o n - c o n t r a c t l v e l y

subsemlgroup

By a n o t h e r

compact

of

in K.

it Is sufficient generated

: s ~ S~.

Let S be a semlgroup

on each orbit slx = S x U ~ x ~ , a fixed

is a

Hence,

compact,

by 2.14,

convex,

there

Is a

Q of K such that K ~ Q and Let Yo =

Xo

+

SoXo

.

Now by

(*)

2

p(ty o - tx o) >--~-~ and p(ty o - tSoXo) 2

>!

2

for every

-

t ~ S.

35-

But then, ty o ~ Q for every t 6 S.

Whence

~ ( S y o) ~ Q since Q is closed and convex; and chat is a contradiction since Q is a proper subset of K and K = c-~(Syo). Remark:

This elegant proof of Ryll-Nardzewski's fixed

point theorem and the proof of lemma 2.14 are due to I. Namioka and E. Asplund: A geometric proof of Ryll-Nardzewski's fixed point theorem (to appear).

-

3. Ellis'

3.1

36

-

Theorem

We state the following proof,

theorem

which may be found

Theorem: Suppose

due to Ellis without

in Ellis

Let G and X be locally that G is a l g e b r a i c a l l y (~,x)~--~;g.x

[4~:

compact

a group and that

: G x X --, X

is a function such that fer all g , h ~ G , (gh).x

Furthermore, semigroup

suppose

that G is a s e m i t o p o l o g i c a l

: X --~ X,

g----~g-x : G - - 2 are continuous.

X,

Remark:

g ~ G, x ~ X

Then

(i) G is a topological (il)

(g,x)~g-x

We do not require

x 8X

= g. ( h . x ) .

and that the functions x-"~g.x

spaces.

group;

is continuous. that

I "x = x.

-

4. Actions

4.1

of Compact

Definition :

37-

~roups

on T o p o l o g i c a l

Let ~ be a topological

over

=~_

or

group.

Suppose

I~ = ~ 9

Vector Speces

vector

space

Let G be a t o p o l o g i c a l

we have a function

(g,x)~"~g-x

: G x E --9 E

such that for all g,h aG,

x,y 6 E

(gh)'x = g'(h'x), g-(x

+ y) = g.x + g-y,

1"X = X . If the functions

are

x'~'~g-x

: E - - @ E,

g a G,

gA-~g-x

: G--@

x ~ E

continuous,

call E a s_.eparate G-module;

(~,x)~-~.x is continuous,

4.2

E,

if

: G x V~ --@ E

then call E a G-module.

If E is a separate

G-module--G

topological

E a Baire

group,

a locally

space--,

compact

then E is a

G-module. Proof:

Let U be a closed n e i g h b o r h o o d

and C a compact n e i g h b o r h o o d T = TU = ~{g-1-U Then T is a closed g~-~g-x

continuous,

of 0 in E;

of I in G.

Define

: E 6 C~.

subset of E.

Fix x & E .

so C.x is compact;

Therefore,

there

is a p o s i t i v e

C.x _c nU.

But then x 6 (q{n g-1.U

We have

hence,

boundeg

inteser n such that : ggC

= nT.

Thus

m@

E = ~ / nT. n=1 closed sets,

Since E is a Balre

space,

say, nT has n o n - e m p t y

one of these

interior.

But

-

38

-

then T itself must have non-empty

interior.

be the interior of T, and let xo~ V. x~V, Thus

Let V

Then for all

g.(x - x o) ~ U - U; so g-(V - Xo) ~ U - U. (g,x)~-~g-x

is continuous

at

(1,O).

It follows

easily that the function must be continuous everywhere

4.3

Let G be a locally compact

Notation : group.

Define Coo(G)

=

C(G)

Let E be a topological subset,

topological

- supp

is compact ~.

f

IfMC_Eis

vector space.

a

we again denote the closed convex hull of M

by ~ ( M ) .

We say that E satisfies

E is locally

condition

convex and

C C] E is compact Suppose that E is a locally

==--> B'~(C) is compsct i n E w convex topological

space and that E is a separate G-module. denote a fixed left invariant (normalized continuous

(K) if

If

~:G--,E

function with compact support, ~(g)

element of the algebraic

Let dg

Haar measure on G

if G is compact).

define the integral

vecto~

is a then we

dg to be the unique

dual of the topological

dual of E (denoted E'*) satisfying G

-G

If E satisfies

condition

member of E (identified

(K), then

with its canonical

in E'*). [See Bourbakl C21, p If f g C ( G )

and s & G ,

I~(~)

12.]

Define s f a C ( G )

sf(t) = f(s'It),

by

t&S.

dg is a image

39-

4.4

Let G be a locally compact topological group.

Suppos

E is a separate G-module which satisfies condition For f ~ C o o ( G ) ,

x~E,

(K

define

f.x =~G f(s) s.x ds, where ds is a fixed left-invarlant Haar measure on G, Then (i) E is a Coo(G)-module, in the sense of algebra, when Coo(G) is considered as a subalgebra of L I(G); if G is compact, then fr~f'x:C(G)--~

~s continuous

(with

respect to the topology on C(G) induced

by L I (G)); (il) if f & C o o ( G )

and s ~ G ,

then s-(f-x) = sf-X

(iii) the set Coo(G).E = ~f-x : f & C o o ( G ) ,

x ~E~

is dense and G-Invariant in E. proof:

(i) a. The algebraic properties of (1) are

straightforward (f*g).x

=

except, perhaps, /GCf*~)(s)

for convolutlon:

s.x ds

= JG(JG f(t)g(t "Is) dt) s'x ds = JG f(t) fG g(t-ls) s'x ds dt = JG f(t) t ' %

g(t-ls)

(t-ls)-x ds dt

= JG f(t) t-JG g(s) s-x ds dt = f- (~-xl

b. If G Is compact,

for each x a E, we have G-x compac

since E is a separate G-module.

Fix x & E, and let

B be the closed convex circled hull of G.x. for all u g E ' ,

we have

Now

- ~r

-

l! = =

IJo
u> as i

~. ,G Ifr u>l ds l!fll~sup~,l<s.x, u>l. s.~.o~. so f-x ~ llflliB. And thls ylelds the dt~Ired ~ u . ' l . ~ (II) Let f & C o o ( G ) , s ~;G. s.(f.x)

Then

= s- JG f ( t )

t-x

= ~G f(s-lst) c s f(st) = JG

dt

(st)-x dt

(st).x dt

= j% sf(t) t-x dt _- sf.X (Iii) Invariance of Coo(G).E follows from (ll). ]J4 be the nelg~hborhood fllter of I In G. U~I~. plck F u & C o o ( G ) ~G FU(S) ds = I .

For each

such that supp F U c_ U and

Suppose V Is a glven convex symmetrl

closed nelg~hborhood of 0 In E. U~ ~;L so that s-x - x s u~_V~

Let

Let x GE.

whenever s &U.

E* (i.e., l~v, u ~

Choose Now If

I for all v 6 V ) ,

then

l~ = I<~ Fuc,~ c,.x- x~ d,, u>I - Is

- x),

1

.< j~ L F~(. ~I I<. ' ~ - x, u>l a. : Yu I F u ( . ) 1 % < . . x - ~, u > [ d . <

"%,

So FU-X - x s 'Coo (G) .'E = E.

1.

by the bipolar theorem.

Thus we have

-

4.5

41

-

Let E be a Banach space, and let G be a locally compact topological group. G-module.

Suppose E w is a separate

Then E is a G-module with the same action.

~Compare deLeeuw and Glicksberg's Theorem 2.8 in "The Decomposition of Certain Group Representations, " Journ. d'Analyse Math., 15(1965), pp. 135-192. 7 Proof:

Define H = ~x ~ E

: g~'x

is continuous

.

We show that E = H. (a) H is a closed vector subspace of E: that 0 ~H,

so it is non-empty.

Since addition and

scalar multiplication are continuous that H is a vector subspace of E. closed,

compact neighborhood ~(g)x = g.x.

is bounded

To show that H is

Steinhaus

is continuou

Let V be a fixed symmetric

of I in G.

Define ~:G--@L(E)

Since for each y ~ E,

!~(g)y : g & V ~

in E w and therefore in E, we have that

IT(g) : g ~V~

if g g V ,

in E, we have

it suffices to show that g ~ E ' x

at I in G for each x ~ .

by

We note first

is bounded

theorem,

in L(E) by the Banach-

say !11~(g)ll ~ M for all g &V.

x'~ H, and x g S ,

Thus

we have

(M + 1)t/x-

x'it

By the continuity of g - ~ g . x ' ,

+ Itg'x'

- x'l,1.

we can make this as

small as we please by choosing x' sufficiently close to x and g sufficiently close to I. (b) Coo(G)-E % H:

In order to talk about Coo(G).E,

we note that E satisfies Banach space.

condition

(K) since it is a

Now let V be as in (a); let f ~ Coo(G),

and K = supp f; fix x sE.

For s ~V,

we have,

i~s.(f-x)- f-xli : ~sZ'X- z.xll <

f JVK isf(t)-

f(t)l

l!S(t) xl~ dt

t

since sr'~sflvK can

be

marie

continuous

as

: v--, small

as

ii

c(vK) is continuous, I l s f -

fl

we w i s h .

i,,

Thus

at I, hence everywhere.

s~'--~s 9 (f.x)

So f-x ~H.

(c) H = E because Coo(G)'E is dense in E and H is closed.

4.6

Let G be a compact

topological

separate G-module E satisfies

group.

Suppose

condition

(K).

the Let

F be the space generated by all finite dimensional G-invariant Proof:

subspaces

of E.

Then F = E.

Let R(G) ~ C(G) be the ring of representative

functions

(i.e~

fsR(G)

finite dimensional Peter-Weyl

<=-=--~sf : s ~ G ~

subspace of C(G)).

theorem, R - ~

= C(G).

spans a

Then by the

But if f ~ R ( G ) ,

x ~

then f.xs Isf.X : s a G , which spans a finite dimensional R(G).E ~ F.

G-Invariant

subspace of E.

Now since f ~ f - x : C ( G ) - - ~ E

when C(G) has the uniform topology, C(G).E = R(--'~V~.E a ~. But C(G)'E is dense in E, so F = E.

Hence, is continuous

we have

-

~3

-

REFERENCES

I .

Bourbakl, N.: El~ments de Math~matlque. XVIII. Premiere partie. Les structures'fondam~ntales de l'analyse. Livre V. Espaces vectorlelles topologique. Chap. III-V. Actualit~s Sci. et Ind. 1229, Paris: Hermann & Cie. 1955. f

/

. .

/

.

Bourbaki, N.: Elements de ~athematique. XXV. s partle. Les structures fondamentales de l'analyseo Livre VI. Int~gratlon. Chap. V~ Actuallt~s Sci. et Ind. 1281. Paris: Hermann & Cie. 1 959.

.

deLeeuw K., and Glicksberg, I.: Applications of almost periodic compactifications. Acta Math. 105 (1961), 63-97.

.

Ellis, R.: Locally compact transformation ~roups. Duke Math. J. 24 (1957), 119-125.

.

Grothendleck, A.: Crlt~res de compacite dans les espaces fonctlonelles generaux, Amer. J. Math. 7__4 (1952), 168-186. d

f

.

Nalmark, M . A . : Normed Rings. N. V. 1964.

.

Schaefer, H. H.: Topological Vector Spaces. York: Macmillan Co. 1966.

Groningen : p. ~oordhof New

CHAPTER COMPACT I. Algebraic Notation: 1.1

II

SEMITOPOLOGICAL

Background

SEMIGROUPS

Material

Let S denote a semi~roup

Definition:

Let I ~ S.

throughout

this s e c t l o

Then

(I) I is a subsemi6roup (2) I is a ri6ht ~ e a l

if and only if II ~ I; if and o n l y

if IS ~

I;

(3) I is a left ideal if and only if SI ~ I; (4) I is an ideal if and only if SI u I S ~ Let E(S) denote

the set of Idempotents

S; i.e., E(S) = ~ s ~ S partial

order ~

: s 2 = s~.

I.

in a semigroup

There is a natural

on E(S) given by

e ~ f if and only if ef = fe = e. An idempotent (I) a ~

e aE(S)

is

identity

(2) an identity

if ~ s ~ S

if Is a S

(3) a right zero if ~s ~ S (4) a zero if ~s a S

In general,

statement

: se = e~ = S, and

"right",

about

there is a

"left"

p

and vice versa

we will not make both statements,

will use both statements 1.1 cont.

: es = se = s~ = S,

: es = se = e~ = S.

NB.: For every statement about corresponding

: se = s~ = S,

If Y is any set,

freely.

it can be made into a semlgro~

by giving it the trivial m u l t i p l i c a t l o n xy = y,

but we

all x , y ~ Y .

-

Such

~ote

a semigroup that

a right

in a right

H(e)

H(1) 1.2

: se

idempotent is called

every

element

maximal

e.

is

define

= es = s snd there is an s' a such that s's = ss' = e~. group

containing

If e = I is the identity

the group of units

S

the of S,

of S.

Definition: (I) S is called proper

right

right

(2) S is called a,x,y~S,

The following

ax = ay implies a right

(a) S is right

defined

by

if these

and Y is a

and left ca ncellstive; and

for all x ~ S ,

conditions

xS = S;

hold,

if e ~ E ( S ) ,

is a rlg~t

zero

_Ce(g, Y) = gY is an isomorphism

with

and

Y = E(S)

the function

: G x Y --, S

inverse

%e'1(s)

inverse

of se in the group

Proof:

if s is isomorphic

group.

then G = Se is a group,

~e

x = y.

are equivalent:

simple

(c) S is a right

semigroup,

group

if for all

G is a group,

is non-empty,

Specifically,

no

zero semigroup.

statements

(b) E(S)

if S contains

ideals.

to G x Y, where right

simple

left csncellative

(3) S is called

1.3

zero semigroup.

identity~

e 5 E(S),

is a group--the

given

a right

zero semigroup,

zero and a left

= (s ~ S

-

is called

..'or an idempotent

H(e)

45

= (se,

See Clifford

(se)-Is),

where

(se) -I is the

G = Se.

& Preston

[I], p. 38.

-

1.4

46

-

If I ~ S is a minimal right ideal, and if E(I) is not empty, Proof:

then I is a right group.

Let x ~ I.

Since

(xI)S = x(IS) G xl,

xI is a right ideal in S. minimal,

so xl = I.

But x l ~

II ~ I and I is

Whence by 1.3(b),

I is a right

group.

1.5

Let X and Y be any non-empty sets whatever; any group.

let G be

Take a function ~ : Y x X --2 G, denoted

or(y, x) = [Y'X~"

The set X x G x Y becomes a

semigroup when endowed with the multiplication

(x, g, yl(x' This semigroup, ideals.

,

denoted

g' , y' 1

=

(x, g[y,x']

g' ,y).'

(X,G,Y)e- , has no proper

Its minimal right ideals are of the form

~x~ x G x Y; its minimal left ideals, X x G x ~y~; its Idempotents, x

G

Proof: 1.6

(x, [y,x] -I, y); its maximal groups,

Straightforward.

Definition: (I) S is simple if it contains no proper I d e a s ; (2) e s

is primltlve if it is minimal with

respect to the partial order ~ on E(S)

(1.1

(3) S is completely simple if it is simple and contains a primitive idempotent; (4) S is a

P,

aragrou~ if there are sets X and Y, , ,[ ,

a group G, a sandwich functlon ~: Y x X - - ~ @ such that S is isomorphic to (X,G,Y)~ Note that in a paragroup, hence, a paragroup

(I .5

every Idempotent is primitive;

is completely simple.

-

1.T

If e s

and eS is rIsht

group

(1.3)),

Proof:

simple

Let I & Se be a left

is a group,

so x

-I

exists

Then

I = Ie.

Let x ~eI.

in eSe.

By 1.3,

Now

& x - l e I @ SeI c I.

So Se = (Se)e _~ Sel c_ I ~_ Se.

Let J be a right

a right

(a left group).

ideal.

eI = ele ~- eSe ~ I .

e = x'Ix

1.8

(hence,

then Se is left simple

Consequently, eSe

4T-

ideal

Thus

in S and

I = Se.

I a right

ideal

in J.

Then (i) For every and

i s I, iJ is a right

right

then

I = J; i.e.,

J is

simple;

If I is minimal ideal

Proof:

in S

IJ ~_ I;

(ii) If J is minimal,

(iii)

ideal

in J, then

I is a right

in S.

(I) (IJ)S = l(JS) c ij.

(li) ~

(I) and mlnlmality,

(111) For each I E l , IJ = I.

Suppose

(I) and mlnlmallty.

that

e s

and

Then G = eSe is a group;

zero semigroup; Further,

IJ = I by

I. S,

Now IS = (IJ)S = I(JS) c__ IJ = I.

1.9 Proposition: simple.

I c_ j = IJ _~ I for i ~

with

Y = E(eS),

that eS is righ

X = E(Se),

a right zero

a left

semigroup.

o- : Y x X --@ G given by o-(y, x) = yx,

the function ~e

: (X,G,Y)~--*

is an inJective

S with

morphism : s

Re(X,

g, y) = xgy

of semizroups.

--9

The function

-

48

-

defined by ~' ~e(S)

= (s(ese) "I , ese,

(ese) -I s)

(inversion in G = eSe~ is a left inverse of ~e.

In

particular, ~e is an isomorphism of semigroups when its codomain is corestrlcted in S.

Moreover,

~

to SeS, the image of ~e

is a surmorphlsm of semigroups

if and only if (I) se and sx are in the same maximal 6roup for every x t X and every s ES, and (2) es and ys are in the same maximal ~roup for every y ~ Y

and every s s

In any event, ~e Y(e' is a retraction of S onto SeS in the category of sets. Proof:

By 1.3 and 1.7, X, G, and Y are as stated.

That ~e is a morphism is straightforward. g &G, y ~Y.

Let x ~ X,

We have

ex = e = ye, xe = x, ey = y, and exgye = g. Therefore " ~ ~e(X,

g, y) = ( x g y ( e x ~ y e ) I ,exgye,(exgyo)-1x~y ) = (x, ~, y).

so )~ is a left inverse for ~e, and ~e is an isomorphism of semi8roups when its codomain is corestrlcted

Since (~e ~ )

to SeS.

(~e ~ ) 2 = ~ e ~ e ~' e Q e is Idempotent;

the category of sets.

and ~ e'~ e

and therefore,

is the identity a retraction

in

We

49-

have he(St)

= (st(este) -1, este,

(este)-Ist)

and (II)

~e(S)~e(t)

Now suppose idempotent

that

= (s(ese) -I, este,

(I) and

(2) hold.

(ete)'It)

There exists an

x E A such that xte = te. Fur t h e r m o ~ ,

~y (1

sx is in the same group as se--let f be the identity of that group.

Since s(ese) -I is also in the same

group as se, and s(ese) -I is idempotent, f = s(ese) -I.

we must have

Now we compute-.

st(este) "I = s(xte)(este) -I = (sx)(te)(este) "I = fsxte(este) -I = feste(este) -I = fe = s(ese) -I. The proof that

(este) -I st = (ete)- I t is similar using

(2); whence ~e' is a morphism. ~

is a morphism.

by

(I) and

Conversely,

Let t a x .

Then for any s ~S,

(II) we have st(este) -I = s(ese) -I.

this is an idempotent;

vlz.,

But

the identity of the

maximal group containing st and se. get

suppose

Similarly,

we

(2).

1.10 Corollary:

The following statements

(a) S is completely (b) S is simple,

are equivalent:

simple;

and there is an e ~ E ( S )

such

that eS is a right group; (c) S is a paragroup. Proof:

(a) <===~ (c): Clifford

and Preston [I],

(c) ===~ (b):

Straightforward.

(b) ===~

B y 1.9, ~e is InJectlve.

(c):

SeS is an ideal, and S is simple;

p. 92

But im ~e =

so ~e is surjectlve

-

50-

1.11 The following statements (a)

are equivalent:

(I) S contains a minimal right ideal, and (2) Every right i deal in S contains

an

idempotent; (b)

S contains

a unique minimal

ideal M(S),

which is a paraEroup. Proof:

(b) ===>

(a): Every minimal right ideal in M(S"

is a minimal right ideal in S by 1.8(iii). if I is a right ideal in M(S);

ideal in S, then I ~ M ( S ) whence

Moreover,

is a rlg~ht

there is an idempotent

e ~ I f'~M(S) ~ I. (a) ---~

(b):

Since E(I)

Let I be a minimal right

is not empty,

ideal in S

I is a right group

Now SeS is simple since eS is right simple, e(SeS)

= (eS)eS = eS is a right group;

p a r a g r o u p by 1.10. minimal,

(the uniqueness

the fact that the intersection always

contains

~(s)

the category

following from

of two ideals

I and J

ideal M(S) w h i c h is a

Let e ~ M ( S ) .

= ese.

and it is

IJ).

1.12 Suppose S has a minimal paragroup.

and

so SeS is a

But SeS is an ideal,

so SeS = M(S)

(1.4).

Define

Then eSe = H(e),

~ : S - - ~ eSe by

• is a retraction

(in

of sets), and the following statements

are equivalent: (a) 9~ is a morphism; (b) ~ I M ( S ) (c) M(S)

is a morphism;

is isomorphic

(d) E(M(S))

= E(S)~M(S)

with E(Se) x H(e) x E(eS); is a s u b s e m i g r o u p

of S;

-

51

-

There exists a surmcrphlsm ~'

(e)

such that ~IH(e) Moreover,

is inJective.

the above conditions

equivalent

: S --2 H(e)

imply the following

conditions :

(f) If X, G, Y, o, ~ ~e

are as in 1.9, then

: S --~ (X,G,Y) o- is a morphism of

semlgroups ; (5) s~'-~s(ese)-Is

: S --~M(S)

is a morphism

of semigroups. Proof:

(a) ~==--~ (b): Trivial.

(c) ~===~ (d): Trivial in view of 1.11 and 1.10. (a) ===> (e) : Trivial. (b) < = = = 7 (c): Straightforward

computation

in a

paragroup. (e) ==='~ (d): If f , g , h ~ E ( S ) , and efghe

then ~'(efghe)

= e 5 = e=

H(e); so efghe = e by the inJectlvlty of

~'IH(e).

It follows that

So f e f g E ( S ) ~ H ( f ) ,

(Eef) 2 = ~efgef = g e f c E ( S )

whence fef = f.

Then

(fg)2 = fgfg = (fef)g(fef)g = f(efgfe)f~ = fefg = fg. Thus f g 6 E ( S ) . (f) ~==--~ (S): Clear by 1.9. (a) ===>

(f):

~e(S)=

(sSO(s)-I, ~(s), ~ ( s ) - I s ) a n d

~le' is a morphism if s~--~(Se(s),~(s)-Is):S.->eSe~CE(eS) and s~-~(s~(s) "1,9~(s)) morphisms,

." S - - ~ E ( S e )

x eSe are

which by I .3 is the case if and only if

~: S --, eSe is a morphism. Remark:

By considerinE any paragroup which is not a direct

product,

one sees that (f) does not imply

(c).

-

It will be n e c e s s a r y groups

acting

in 1.2.

Notation:

insight

into

Throughout

denote

some

a vector

information

vector

involving

the remainder over

tC&a

maps

Most

linear

but

of the semigroup.

!7~ and K a convex

(under composition) affine

let V

subset

of

of afflne

if and only

if

~,~).

Note that

the set of all

of K into K is a semigroup.

of the time we will

consider

semigroups

of

maps.

1.13 Suppose

T is a left-zero

semigroup.

P = ULtK is convex, Proof:

then

If

: t ~ T~

T is convex.

Take s,t s T, k ~ K, k ~ [ o , i ] . sk

since

no topology

+ (1 - k)b) = ~ta + (1 - ~)tb

for all a,b ~ K, ~ h affine

we are

of this section

of K into K (t is called

semi-

Some results

section

the structure

space

about

spaces.

In this

Let T be a semigroup

maps

Remark:

-

w i t h some statements

giving more

V.

to have

on topological

have been presented concerned

52

+

P is convex.

k' ~ K with

Ask

T is a left-zero ~sk

(i

-

~)tk

Therefore

+ (i -~K)tk

there

is a u ~ T and a

= uk'

= u2k ' +

= ~usk =

+

(i

(I

- A)tk)

+ (1 - ~ ) u t k +

and since k was arbitrary,

But then,

we have

= u(ksk

Is

~ P

+ (I - )~)tk = uk'. semigroup,

Then

(I

-

l)uk

we must have

- )~)t

=

u

~_ T.

=

uk;

since

-

1.14 Suppose and

53

that T is a l e f t - z e r o

that

all the e l e m e n t s

following

statements

(a) P = ~ t V

(c) card

since

P is convex,

The

space;

Take

E a c h tV is a v e c t o r it is a v e c t o r

s,t

6 T, k 6 V.

P is a v e c t o r

left-zero

of T are l i n e a r .

T = 1.

(a)==>(b):

and a k' ~

that K = V,

: t 6 T~ is convex;

Proof:

(b)==>(c):

semlgroup,

are equivalent:

(b) P is a v e c t o r

because

-

space.

V s u c h that semigroup,

space;

thus,

space. Then

Thus

sk - tk ~ P,

there

sk - tk = uk'.

is a u & T

Since

T is a

we h a v e

sk - tk = uk'

= u2k '

= u(sk

- tk)

= usk - u t k = uk - uk = O. Hence

1.15

s = t and

card T = 1.

If K = V and T is a r i g h t - ( r e a p ,

left-)

zero

semigrou;

then T* = ~i - t is a l e f t - ( r e a p , maps

zero semlgroup

of a f f i n e

of V into V.

Proof:

1.16 Suppose and

right-)

: t ~ T~

Straightforward.

that

T is a r i g h t - z e r o

t h a t all the e l e m e n t s

following

statements

are

semigroup,

of T are l i n e a r . equivalent:

t h a t K = V, The

-

Q = ~ker

(a)

t

54

: t ~. T ~ is convex;

(b) Q is a v e c t o r (c) card Proof:

1.17

-

space;

T = i.

Clear

from 1 . 1 5

If T is a l e f t - z e r o

and

1.14.

semigroup,

then,

for s,t

~. T, we

have sK = tK if and o n l y Proof:

The

k ~ K.

"if"

Then

sk = tk'.

is o b v i o u s .

sk & tK,

Since

1.18

S

=

so t h e r e

= t2k'

"only

if",

let

is a k' ~. K w i t h

= t(sk)

semigroup,

we h a v e

= tk;

t.

If T is a r i g h t - z e r o s,t

F o r the

T is a l e f t - z e r o

sk = tk' SO

if s = t.

s emigroup,

then sK = tK for all

~. T.

Proof:

sK = tsK c. tK = stK ~ s K .

1 . 1 9 S u p p o s e S has a c o m p l e t e l y s i m p l e m i n i m a l ideal M(S). If e ~ E ( M ( S ) ) and s ?~ S, t h e n the f o l l o w i n g are e q u i v a l e n t : (a) se ~. eS; (b) se = ese; (c) sR c R, w h e r e

R is the m i n i m a l

(d) sf & fS for all f 6 E(R) If,

in a d d i t i o n ,

equivalent

S = T, then

right

ideal

eS

= E(M(S))F~R.

the a b o v e

conditions

are

to

(e) s e K <~eK. Proof:

(a)==>(b):

so ese = e(et) (b)==>(c):

By

(a), t h e r e

is a t & S w i t h se=et;

= e2t = et = se.

Since

se = ese,

sR = s(eS)

= seS

we have = eseS ~ eS = R.

-

(c)==>(d): so by

55

If f g E(R),

-

then R = fS

(1.3 and 1.4);

(c), sf s sR ~ R = fS.

(d)==>(a) :

Trivial.

(d)==>(e):

Since

E(R)

is a r i g h t - z e r o

semigroup

by 1.18 we have fK = eK for every f s E(R); f &E(R)

(1.9)

so, lettirJ

be such that fse = se, we have seK = fseK (..-fK = eK.

(e)==>(b):

By

(e), for each k ~ K, there

with sek = ek';

so esek = e(sek)

is a k' g K

= e(ek')

= ek'

= sek.

Thus se = ese.

1.20 Suppose

S has a c o m p l e t e l y

For a minimal

right

simple m i n i m a l

ideal M(S).

ideal R, define

S R = ?is ~L S

: sR ~R~._

Then S R is a s u b s e m i g r o u p

of S w i t h minimal

ideal

M(S R) = R. Proof:

R is an ideal

in S R since srR ~,.sR ~ _ R

rsR C_rR C R for all s 6 Sp~, r ~ R. simple

since R is

(1.8), R = M(S R).

1.21 S u p p o s e linear, M(T),

So,

and

that K = V, that all the elements that T has a c o m p l e t e l y

and that e E E(M(T))

minimal

left and minimal

of T are

simple minimal

ideal

so that L = Se, R = eS are

right

ideals

of S, resp.

Define V R = eV, (by 1.18,

1.15,

on R, resp. Then

V L = ker e = (1 - e)V

and 1.9 V R, resp.

VL, depends

L, and not on e 6 E(R),

resp.

only

e % E(L)).

-

56

-

V = VL @ VR

(i)

(ii) V R is i n v a r l a n t M(T)

Proof:

(i) That

(ii) By 1.19,

T if and

only if

under

T if and

only

= R;

(lii) V L is i n v a r i a n t M(T)

under

if

= L; V = V L @ V R follows

seV ~ e V

for all

from the definition.

s &

T

iff ese = se for all s ~

T

iff T R = T Iff R = M(T) (iii)

If M(T) (1

-

: L,

e)s(1

-

(1.20).

then e)

es = ese for all s ~ T; so

=

s

=

s(l

-

se

-

-

es

+

ese

=

s

-

se

e).

Whence, s(l - e ) V =

(I - e)s(l

- e)V Q

(i - e ) V

m

for all s ~. T. for all

s ~ T, then

s(1 - e)v = s(1

- e) =

implying

that

so M(T)

1.22 S u p p o s e linear, ideal

(1 - e)v';

-

se

=

s

so,

- e)V ~. (i - e ) V

-

se

is a v' &

as b e f o r e ,

- e); -

that

is,

es

ese,

+

ese = es for all s ~ T.

V with

we h a v e

Thus

e2 = eTe,

= L.

that K = V, that and

if s(l

for v ~. V, t h e r e

(i - e)s(l s

and

Conversely,

that

T has

all

the e l e m e n t s

a completely

of T are

simple

minimal

M(T).

(1)

~QeV

: e ~ E(M(T))}

maximal

contains

a unique

invarlant

subspace

Vg;

(ii) Vg is the m a x i m a l

subspace

invariant

T s u c h that

T iVg is a g r o u p

of a u t o -

under

-

morphisms

57-

of Vg; : T ---> TIVg is a

(iii) The function t~

morphism of semigroups onto a group. Proof:

(i) If W, W' are invariant subspaces of '

V : e i E(M(T)),~, so is W + W'.

Let Vg be the

sum of all the invariant subspaces contained in

O[ev

" e ~ E(M(T))~.

(ii) For a fixed e h E(M(T)) and v ~ Vg, we have tv = (ete)v for all t ~ T. But etelVg has the inverse

(ete)'l~vg with (ete)-l

being the inverse of ete in the group eTe.

Thus

T IVg is a group of automorphisms of Vg. On the other hand, let W be an invariant subspac~ of V such that T iW is a group of automorphisms of W. Then for each e ~ E(T), elW must be the identity on W; hence W is contained in f ~ e V

: e s E(M(T))~.

But

then W ~ VS. (iii) Trivial. 1.23 Proposition: Let S be a semigroup of endomorphisms of a real vector space V.

Suppose S has a completely

simple semigroup M(S) as a minimal ideal.

The

following statements are equivalent: (a) M(S) is a minimal right ideal; (b) eV = e'V for all e,e' ~ E(M(S)); (c) eV = Vg for some e & E(M(S)); (d) eV = Vg for all e a E(M(S)); (e) ~.leV : e • E(M(S))~

is convex;

(f) ~.1{eV : e ~ E(M(S))}

is a vector space;

(g) SIeV is a group for some e ~. E(M(S)).

-

Proof:

(a) is equivalent

(a') csrd E(Se) (see 1.9, I.IO). Clearly, 1.22,

(b) ==> (f).

linear,

to

(a) <==> And

(e) <==> (f) by 1.14.

(a) ==> (b) by 1.18.

Obviously,

(c) ==> (a) by 1.21. 1.24 Suppose

-

= I for e & E(M(S))

Hence

(b) ==> (d).

M(T),

58

(d) ==> (c).

Also by 1.21,

that T has a completely

and that x ~ V.

And

(c) <==>

that K = V, that all the elements

(g).

of T are

simple minimal

The followin~

By

statements

ideal are

equivalent : (a) There

is a t & T with tx = O;

(b) O ~ Tx (c) It ~ T : tx = O i is a left ideal; (d) There

is a minimal

left ideal L % T with

Lx = ~0~; (e) There is an e ~ E(M(T)) Proof:

Trivially

(a),

(b), and

Since every left ideal contains in a semigroup Also by 1.11, 1.25 Suppose

with minimal (d) ==>

(e).

with ex = O. (c) are equivalent. a minimal

ideal

(1.11),

Obviously,

that K = V, that all elements

and that T has a completely (i) ~ { ( 1

- e)V

maximal

left ideal (c) ==>

(e) ==>

invariant

(ii) V z is the maximal

contains

subspace subspace

morphism

t~tlV

ideal M(T). a unique

Vz; invariant

T such that T IVz is a semi~roup (iii) The function

(a).

of T are linear,

simple minimal

: e & M(T)~

under

with zero;

z : T ---> T!V z is

of semig~oups.

(d).

a

-

Proof:

(i) is clear

(li ): Certainly t Z M(T)

59-

(see also 1.22)}

(iii) is trivial.

T IVz is a semigroup with zero slnce fo~

and v [~ V z we have tv = t(1 - e)v = tv - tv = 0

when we take a right

identity e for t in M(T).

On the other hand,

if W is an invarlant

subspace

such that T~W is a semlgroup with zero, then, since the morphism t ~ t onto zero, W <~

IW must map the minimal

ideal M(T)

(I - e)W = W for all e 6 E(M(T)).

I - e)V

1.26 Proposition:

: e 6 E(M(T))

.

Thus,

Hence

W C_ V z.

Let S be a semigroup of endomorphisms

a real vector space V.

Suppose S has a completely

simple semigroup M(S) as a minimal following statements (a) M(S)

of

ideal.

The

are equivalent:

is a minimal left ideal;

(b) ker e = ker e' for all e,e' E E(M(S)); (c) (i - e)V = V z for some e ~ E(M(S)); (d) (I - e)V = V z for all e & E(M(S)); (e) \J~(l - e)V

: e ~ E(M(S))~

is convex;

(f)

: e i E(M(S))~

is a vector space~

O~(1

- e)V

(g) Sl(1 - e)V is a semigroup with zero for some e #~ ~([ii(b)); (h) ~ { ( 1 Proof:

We have

(see 1.9, Clearly, 1.25,

(b) ==>

Hence (f).

(d).

(a) by 1.21.

Trivislly

: e s Z(M(S))~

is invsriant.

(s) iff card ~(eS) = 1 for e 6 E(M(o))

1.10).

(b) ==>

(c) = = >

e)V

(a) <==> And

(e) <==>

(a) ==> (b) by 1.17.

Obviously,

(d) --=> (c).

Also by 1.21,

(d) ==> (h), and

(f) by 1.14

(c) <==>

And (g).

(h) ==> (f) is easy.

By

-

60-

2. Locally Compact Paragroups Notation:

Let ~ denote a locally compact semitopologlcal

semigroup throughout 2.1

Let G C

this section.

S be a subgroup,

G itself locally compact.

(1) G is a topological group. (ii) The following functions are continuous:

Proof: (i),

I. ( g , s ) ~ E s

: G x S --~S;

2.

(s,g)~gs

: S x G --, S;

3.

(g,s,h)e~gsh

: G x S x G - - ~ S;

4.

(g,s) "1~Ssg -I

: G x S --~ S (inversion i n G

By Ellis'

Theorem

(ii) (I), and

is continuous,

Lemma: e&E(T),

(ii) (2).

Since g ~ g - 1 : G

( ( g , h ) , s ) ~ , g s h -I

is a group action. 2.2

(1.3.1) we immediately get --, G

: (G x G) x S - - 9 S

From this we get

Let T be a semitopological

(li)~)an8 (ii)(4).

semigroup.

If

then Te, eT, eTe are all retracts of T,

and are therefore closed. Proof: Clear. 2.3

If S is a right group,

then S is a topological

group, and S is isomorphic topological semigroups)

semi-

(in the category of

to G x Y, where G is a

locally compact topological group, and Y is a right zero is

(topological) semlgroup.

In particular,

E(S)

closed.

Proof:

Let e ~ E ( S ) .

By 1.3, G = Se is a group.

By

2.2, G Is closed in S; hence G is a locally compact

o

topological

group

61

-

(2.1). The functions

R e : S --, G with Re(S) : G --, G with and

<(~) = g-1 (inversion in G)

o~: G x S --~ S with

are continuous

by definition

semi~roup and 2.1.

= se,

O<(6,s) = gs

of semltopolo5ical

Now, as in 1.3, Y = E(S)

right zero semi~roup,

is a

and we have the algebraic

isomorphism ~e

: G x Y --, S wlth

~e(g,Y)

Since ~e = 3t I G x Y, it is continuous.

= gY.

Furthermore,

since

~-'~-1(8) e ~ells

= (se ' (se)-ls)

continuous

= (ReS' ,~L(~'(ReS)' s))

the composition

as

'

of continuous

functions. 2.4 Proposition: semigroup.

Let S be a locally Suppose

that S has a minimal

w h i c h is a paragroup. Proposition

compact semltopologlcal

Let e s

ideal M(S) As in

1.9, let G = eSe, X = E(Se),

Y = E(eS),

O- : Y x X - - , G with 6(y,x) qe

= LY,X

= yx,

: S - - , (X,G,Y)g with ~(s)

and

fe

= (s(ese) -I , ese,

(ese)

Is),

: (X,G,Y) O- - - ~ S with

~ e(X,

6, Y) = xgy.

Then (I)(I) G is a locally compact and is closed

topological ~rou

in S;

(2) ~, Y are locally

compact left-, resp.

-

62

-

zero semigroups and are closed in S;

right-,

(3) (X,G,Y)cr is a locally compact semitopological

semlgroup relative to the

product topology. (ii) ~e is continuous, ~elM-T~ is a morphlsm of semitopologlcal

semlgroups,

blJective;

(M-T~) 2 ~ M(S).

also

and ~e IM(S ) is

(Ill) The following statements are equivalent:

(a) ~elM(S) is an isomorphism; (b) ~e is continuous; (c) ~ e ~ e

!

: S ---> S is a continuous retraction;

(d) M(S) is closed in S; (e) M(S) is locally compact. (iv) If ~ is continuous,

then E(M(S))

(X,G,Y)~ is a topological

is closed, and

semlgroup.

On the

other hand, if G is compact and E((X,G,Y)~) closed, Proof:

then @

is

is continuous.

For the algebraic properties listed, see 1.9.

The function o is separately continuous by definition of semitopologlcal

semigroup.

By 2.2, eSe, Se, and eS are

closed, so locally compact.

Therefore,

eS = G x Y and

Se = X x G by 2.3 and its dual, and X and Y are closed in S.

By an argument analogous

continuous

in 2.3, we get ~e continuous.

' I M-T~ is a morphism. sm = ~e~e'(Sm)t = lira t. lira

Then

and _t a net on M(S) with

Then st = llm st = lira

Thus we have

~e

By 2.5 Infra,

Let s K M-T~, m ~ M(S).

Let t ~ ~

(~e~e(S)~e~e(t))

M(S).

to the one yielding

e

e

= llm ( ~ e ~ ( s ) ) t (i) and (ii).

---

=~e~e(S)t

-

(iii):

(s) ===~

(b): Clear

63

in view of the fsct that

the corestrlctlon

of ~e to M(S)

(b) ===>

(c) ===>

(d) ===>

(e) ===~

(a):

= SeS is (~e~M(S))'l.

(e): Trivial.

Without loss of generality,

that S = M(S).

we assume

We must show that the biJective

continuous morphlsm open in S, Uo~. U. neighborhood

-

~'e is an open map.

Let U be

In order to show that

~,ie '(U)

is

!

of ~:[e(Uo), it will be sufficient

a

to

show that for each open set U in S, the sets Ue and eU are open in Se and eS, respectively, function

~ e I : eS - - , eSe x E(eS)

because

is a homeomorphism

by 2.3 and a similar statement holds for Se. with the eS case:

We deal

We must show that, for each UotU,

the set eU is a neighborhood fs

the

of eu o.

There is an

such that UoS = fS; whence fe = f and

ef = e.

The functions

defined by

: eS --~ fS and

):o (x) = fx and

/ : fS --~>e[

~y(y) = ey thus are inverse

to each other,

and so are homeomorphlsms.

a neighborhood

of fu o in fS, since U ~ f S

is open in

fS and f U o t U o f S

c fU.

neiEhborhood

(fUo) = eu o in eS since ~ is a

of

But then

Now fU is

)~(fU) = eU is a

homeomorphism. (iv):

If o-is continuous,

topological

semigroup.

If and only if ~ = E((X,G,Y)~r) closed.

as the inverse

~e

(X,G,Y)~ is a

By 1.5,

V,x , ~

is practically

Since

then

.

(x,g,y) 6 E((X,G,Y)o-)

So if o-is continuous,

the graph of c7, so it is

is continuous,

E(M(S))

image of E((X,G,Y)o-).

Is closed

Conversely,

if

-

-

G is compact and ~((X,G,Y)~)

is closed, then the

graph of ~ is closed and its range is compact; whence ~ is continuous.

2.5

Let TI, T2, T 3 be semltopological semigroups with T1C

T 2 8nd ~l = T2"

Suppose that $ : T 2 ---> T 3

is a continuous function with $1Tl: T 1 ---> T 3 a morphlsm of semitopological semigroups.

Then

is a morphism of semltopologlcal semigroups. Proof:

(1) Fix s 6 T 1.

t ~

Then the function

(%(s)#(t),~(st)): T 2 ---> T 3 x T 3

is continuous and maps T 1 into the diagonal of T 3 x T 3.

Since the diagonal is closed and T 1 is

dense in T2, it maps T 2 into the diagonal. %(st) = #(s)#(t),

s ~ TI, t 9 T 2.

(2) Similarly, @(ts) = #(t)#(s),

s a TI, t & T 2.

(3) Thus, by taking nets, we get that ~ is a morphlsm of semltopologlcal semlgroups.

That is,

-

3. Compact Semitopological Notation:

Semigroups

Let S denote a compact

throughout

3.1

65-

this section

semitopological

semigroup

(excluding the final theorem).

Let S' be any semitopologlcal semi~roup. Lemma: / L e t X, Y, T ~ S' with Y ~ l ~ e d . (i) If TX C_ y, then TX c y; (ii) If T is a subsemigroup subsemlgroup (iii)

of S', then T is a

of S.

If T is a right ideal,

then T is a right

ideal. (iv) If T is a commutative commutative Proof:

set,

then T is a

set.

(i) Clear.

(ii) With X = T, Y = T in (i), we have TT ~ T; so by the dual of (i), TT~. T. (lii) With X = T, Y = ~ in (i), we have T--S'q 3. (iv) For a set X q S', define

the centralizer of X to

be Z X = ~s ~ S'': sx = xs for all x s X~. Z a = ~s ~ S' : sa = as~ is a closed S'.

If a g S"

subsemigroup

of

Therefore zT

is a closed commutative,

-

a

: a

T

subsemigroup of S'.

Now if T is

this says that T C ZT, w h i c h implies

that ~ C Z T since Z T is closed.

But then T ~ Z_,

-

w h i c h again is closed;

T

so T ~_ Z_, as desired. T

-

3.2

Let ~

66-

= ~J ~ ~ : J is a right ideal in S_ ; and let

c = [J~[~

: ~ = J~"

(i) If J s

Ii

Then

, then there is an I s

such that

J;

(ll) I is minimal in J

if and only if I is

minimal in j/c ; (iii) Minimal right ideals exist in S and are closed. Proof:

(i) Let J~ ~/, and ~& J.

Then I = sS % J, and

sS is compact; so I ~ ~/c" (ii) Let J be mlnlmal~in ~/; s tl.

Then I = s S q

and I 5~c, and trivially J is minimal in /c"

J

The

converse follows trivially from (1). (lii) The system

~c,~i

is inductive by compactness.

Hence there is a J 6 ~c which is minimal.

The

statement then follows from (ii).

3.3

If S is abellan, then M(S) exists and is a compact topological group. Proof:

If S is abelian, then

I is a ri6ht ideal iff I is a left ideal iff I is an ideal. Thus by 3.2, there is a minimal is compact.

(rlg~ht) ideal which

It must be unique since I n J ~ IJ for

ideals I and J.

But by 1.8, M(S) is right simple,

and dually, it is left slmple--thus a group. it is a topological group.

By 2.1,

- 67-

3.4

E(S) is non-empty. Proof:

Let T = ~s,sP,s 3 pooch@ ~ Then T is a commutative

subsemlgroup of S.

By 3.1, ~ Is a compact esmmut~tive

subsemlgroup of S.

Whence by 3.3, M(~) exists and

is a group.

3.5

Theorem:

Thus there ls an e = e P ~ M ( ~ ) ~ S.

Every compact semltopologlcal

semigroup S

has a minimal ideal M(S) which is a paragroup. e ~E(M(S)).

As in Propositions

Let

1.9 and 2.4, let

G = eSe, X = E(Se), Y = E(eS), o" : Y x X ---.S with Cy,x)

:

[y,xj

: yx,

,lie : S --@ (X,G,Y) O-with l'(s) = (s(ese) -I ,~e

,

ese,

(ese)

-I

s)

and tie : (X,G,Y) o --a S with )~e(X, g, y ) =

xgy.

Then (i)(I) G is a compact topological group; (2) X, Y are compact left-, resp. right-, zero semigroups; (3) (X,G,Y)~ is a compact semitopologlcal semlgroup relative to the product topology; (ii) ~e is continuous and ~!eIM--~ is a morphism of semitopological semigroups; ~$~M(S) is bl Jectlve ; (ill) The followin~ statements are equivalent: (a))~e IM(S) is an isomorphism; (b) ~e is continuous; (c) ~e)l e' ".

S

--3 S is a continuous

retraction;

-

M(S)

(d)

68

-

is compact;

(e) M(S)

is locally

(iv) g is continuous closed;

and

compact;

if and only

if these conditions

( X , G , Y ) ~ Is a topological (v) Minimal

(vl) Proof:

3.6

Suppose

right

The existence Everything

of M(~)

T is t o p o l o g i c a l l y of T is dense S is a right

in S and are closed;

in T).

follows

from 3.4,

topological

simple

3.2,

semigroup

Further

(i.e.,

Then ~ = M(S);

and

suppose

every right

that

ideal

in particular,

group. simple:

If t ~ T, then tS is a closed

T ~tS.

semigroup;

ei2st

subsemigroup.

right

(1) ~ is right

is a closed

hold,

else from 2.4 an~ 3.2.

that S is a compact

that T C_ S is a dense

Proof:

idesls

is

M(s).

2

and I.ii.

if E(M(S))

right

ideal

right

ideal

in S.

So tSC%T

in T; but then T = tSg%T,

But T is dense

in S and tS is closed.

i.e.,

Thus

S = tS for every t & T. Let s ~ S, and suppose For each x ~ S, there S x S is compact, necessary, topological

t is a net on T w i t h s = llm t. is a net r on S w i t h x = tr.

by p a s s i n g

we may assume

to a subnet of

lim r = r exists.

(t,r),

Since if

Since S is a

s e m l g r o u p I $ we have

x = limtr

= (llm t ) ( l i m r) = st.

So S = sS for every

s ~

(2) That S is a right 1At this point semi groups.

S.

group n o w follows

the a r g u m e n t

breaks

from 1.3 and 3.4.

down for s e m i t o p o l o g l c a

-

In a m a n n e r semigroups

analogous

topological

about

the action

semigroups

1.2.9 and

semitopological

3.7

to the c o n s i d e r a t i o n s

a c t i n g on vector spaces

observetions

theorems

69-

Suppose

of linear

in II.1 we make some

of compact

on t o p o l o g i c a l

linear

occur

From

vector spaces.

1.2.10 we k n o w that compact

semigroups

semi-

in a n a t u r a l

linear context.

E in L(E w) ~. E w .

that T is compact

(i) t~'-~tx : T ---> E w is continuous

for each

x ~ E; (ii) The closure

of an invariant

subspace

is

invariant ; (iii)

If e g~E(M(T)),

then

E = EL 9 ER, with E L = ker e = (i - e)E and E R = eE, the sum b e i n g a direct topological

vector

EL ) depending

sum of closed

spaces

and E R

only on R = eT

(resp.,

L = Te) and not on the p a r t i c u l a r of e in E(R)

(resp.,

E R is invariant the s u b s p a c e if M(T) (iv) There ~[eE

E(L)).

(resp.,

choice

The s u b s p a c e

if and only

if M(T)

E L is invariant

= R;

if and only

= L;

is a maximal

invariant

: e ~ E(M(T))} which

subspace is closed

Eg ii and

is

70

-

the largest subspace such that TIEg is

a compact topological group. Proof:

(1) follows from the definition of the topologl

(ii):Let F be an invsriant subspace.

Then

tF ~ F ~ K for all t ~ T. Since t is continuous,

t~

~.

(iii) follows from 1.21. (iv):

Immediate from

(ii), 1.22, and 1.3.1. E

3.8

Suppose that T is compact in L(E w) ~ E w

and that x ~

The following statements are equivalent: (a) Tx is compact in E; (b) T-~ is compact in E; (c) Tx ~ K, for some compact K ~ E; (d) s~-~sx (a)==>(b):

: T ---> E is continuous.

Trivial.

(b)<==>(c): (b)==>(d):

Trivial. If Tx is compact,

then E and E w induce the

same topology on T-~. But s ~ s x

: T ---> E w is

continuous. (d)==>(a):

3.9

Trivial since T is compact.

The set A of all x ~ E satisfying the conditions of 3.8 is an invariant subspace, Proof:

Trivially,

plication;

where T is as in 3.8.

A is invariant under scalar multi-

and if x ~ A, then for t & T, T(tx) ~ TTx ~ Tx,

so tx & A by 3.8(c).

With x,y & A, we have

T(x + y) ~ Tx + Ty, and therefore x + y ~ A by 3.8(c) since Tx + Ty is compact.

-

71

-

3.10 If E is a Bansch space and T ~ L(E w) ~ Ew E is compact (or even if E is Just quasi-complete to be equicontinuous), Proof:

then A is closed.

If E is a Banach space,

Tx is weakly bounded, bounded

and T is known

then for every x ~ E,

hence bounded;

but then, T is

in norm, hence equicontinuous.

equicontinuous

and E Is quasi-complete,

If T is then theorem

1.2.9 proves the assertion. 3.11 Let E be a Banach space. compact.

Suppose T ~ L(E w) ~ E w

is

Then

(i) ~

~ A;

(II) The sum of all flnlte dimensional subspaces Proof:

E

(I):

invarlant

of Eg is dense in Eg.

By 1.4.5,

s~-~ese~-~esev

= SV

:

T

---> Eg

is continuous for each v 6 E~. By 3.8, then, E~ ~oA. (ii) then follows from 1.4.6. Thus, using the results

of section

1.2, we have the

following propositlon: 3.12 Proposition: subsemlgroup invariant

Let E be a Banach space and S a compact of L(Ew).

subspace

Then there exists a closed

Eg ~ A such that

(i) SIEg is a compact

topological

group, Eg is

maximal with respect to this property,

and

s ~ - ~ s IEg : S ---> SIEg is a morphlsm of semigroups;

-

72

-

(ii) x a A if and only if and only

if Sx is compact if s ~

sx

in E

:S - - - > E

is continuous; (iii)

For each idempotent

e E M(S),

(iv) Eg = eE for each idempotent if and only

if M(S)

Eg c eE;

e in M(S)

is a minimal

right

ideal; (v) S iA is a topological Proof:

Only

(v) needs

from the fact 3.10)

(excluding locally

the remainder

the final

convex

theorem),

topological

Let T be a convex compact

in L(E)

subset

semitopological

multiplication. T on a convex

follows

(see proof

is strongly

on every norm bounded

Throughout

in L(A).

The assertion

that S is norm bounded

and m u l t i p l i c a t i o n

continuous

Notation:

proof:

semigroup

of

Jointly

set.

of this section let E denote

vector

space over

a ~.

of E such that T is a affine

(An affine set such that

semigroup

semigroup

under some

is a semigroup

the maps

t ~--~ st

: T ---> T

t r~-* ts

: T ---> T

and

are affine

3.13

for every

s a T.)

If T has an identity element)

i, then every unit

in T is an extreme

Proof:

Suppose

0 < ~<

1.

1 =~x

+ ~y,

point where

If q is an extreme q = q(~x

+ ~y)

=%qx

(invertible

of T. x,y

point

& T, @L + ~ = l,

of T, then

+ ~qy;

-

73

-

n

so q = qx = qy. ~I

Now

~ O, qi e x t r e m e

n

If z = i=IZ ~ l q l , w i t h

points

of T,

I=IE ~ i

I = I, 2,...,

= i,

n,

then n

n

zx =

E ~ l q l x = Z odlqiy = zy. i=l i=l N o w by the t h e o r e m of K r e l n and M i l m a n , 1 is the limit

of a net ~ w i t h

z x = ~y.

x = (lim ~ ) x = lim Thus

(~x)

(~y) = (lim ~)y

1 is an e x t r e m e Finally,

= lim

Hence

point

= y.

of T.

if t ~ T is a unit

t =~x

+ ~y,

and

T, "#,~

x,y ~

> O, ~ + ~ = l,

then i = t-lt So i = t-Ix

3.14 S u p p o s e

= t-ly,

S q

implying

T is a compact

T = c-~ S and

such

is c o n t i n u o u s . s o ~ S such

=@~t-lx

that

Then

+ ~t-ly.

that

t = x = y.

subsemigroup

(t,s)

there

"~--~ ts

exist

s u c h that

: T x S ---> T

points

t o & T,

that toTs o = StoSo~.

Proof:

Let A=

be finite F

: E --->

{aI ,..., %

subsets

[ ~ S, B =

of S and E',

{l l , . . .

respectively.

, An .

E'

Define

~.,n by F(x)

= (<x, l l > , . . . , <x, A n > ) , x a E.

Since m

t ~

1 r ta i 9 T - - - > T m i=l continuous, there is a t'

is a f f l n e and 1 m t' = -- r t'a i. m

Therefore,

i=l

(*)

= t'(A,B)

m

F(t's)

= -i m

E F(t'ais) i=l

for all s ~ S.

with

-

74

-

Let s' = s'(A,B) ~ S be such that

-- sup{liFct's

[]

tl

:

I=I ~(t's')il _> l!F(t'als')ll, i = 1,..., m (slnce als' s --

From this and

m

(*) we get

F(t's')

= F(t'als') , i = I,..., m;

hence
,

A j > =
By a compactness

t

Aj>,

i=l,...,m,

argument on the directed

J=l,...,n. system

given by the finite subsets of S and the finite subsets of E', using the continuity (t,s) r - ~

of

ts : T x S ---> T, we get points t o a T ,

s o ~ S such that

=

for all s s S, A~E'.

Whence toSS 0 = ~toSoQ which implies that toTs o = ~toSo~. 3.15 Suppose G ~ T is a compact group such that T = c-~ G. Then T has a zero. Proof:

By 1.3.1,

(t,g)'-'-~ tg : T x G ---> T and

(g,t) f\-~ gt : G x T ---> T are continuous. 3.14,

By

there are points s I ~ T, gl 6 G such that

~Slgl~

= SlTgl;

left zero for T.

so ~Sl~

=SlT;

that is, s I is a

By the dual of 3.14, we get a

right zero s 2 for T.

But then s2s I is a zero for T.

The proofs of 3.13, 3.14, and 3.15 are due to Cohen, and Collins, Soc. ~

H. S.: Affine semigroups,

(1959), pp. 97-113.

H.

Trans. Amer. Math.

-

3.16

(i)

T is s group

(ii) M(T) Proof:

-

if and only

if card

T = 1.

= E(M(T)).

Clear

3.17 Suppose

?5

S c

T = c--~ S.

from 3.15.

T is a compact If M(S)

then M(T)

is a minimal

is a minimal

particular,

M(T)

subsemigroup

left

is a left

left

ideal

such that

ideal of S,

of T;

in

zero semigroup.

Proof:

Let e g E(M($)).

Then eS is a compact

group.

Let S o = c-o(eS).

The function !~" : T - - - > T

defined

by 'Nzlt) = et,

is continuous restriction ~co(S)

since

and affine.

and

% T,

Let

corestriction

is a morphism,

a surmorphism.

t

$ : T - - - > S o be the of~.

By 2.5,

~ is a morphism;

By 3.15,

S o has a zero,

since

in fact, say f.

$ is Now

f = fe, we have fT = feT = if~-

Then,

by 3.5, M(T)

by 3.16,

is a minimal

it is a left

ideal;

and then,

zero semigroup.

A special

case of an affine

semigroup

of endomorphisms

vector

left

semigroup

is a convex

of a locally

convex

topological

space E.

3.18 Suppose compact

that E is a B a n a c h and

convex.

Eg = ~x g A that

space and T ~

With Eg as in 3.12,

L(E w) is we have

: tx = x for all t ~ T~;

is, Eg is the fixed

point

set of T.

-

Proof:

By 3.12,

of automorphisms transformation,

76

-

T IEg is a compact topological of r

Since restriction

T Eg is convex.

T ~g is frivlal,

group

is a linear

Whence, by 3.16(i),

i.e., Tx = ~x~ for all x ~ E g .

3.19 Let T be a compact convex semigroup of endomorphisms of a locally convex topological vector space E. the followin~ statements are equivalent: (a) M(T) is a minimal right ideal; (b) M(T)

is a rlght-zero

Then

semigroup

(c) TIeE is a group for all e 6 M(T); (d) Tx contains Proof:

(a) (==~

a fixed point of T for all x a E.

(b): 3.16(ii).

(a) K==~ (c): 1.23. (b) ==~ (d):

Let x & E, e a M(T).

Then for every

t 6T, tex = (te)ex = ex w

since te ~ M(T). (d) ==~ (c):

Thus ex ~ Tx is a fixed point of T.

Let e c M(T),

fixed point of Tex.

x a E.

Suppose

tex is a

Then

tex = etex = ex. So ex is a fixed point in Tex. 3.20 Definition:

Whence TIeE = leE.

Let S be a topological

space,

and let F

be a subspace of C(S) containing the constants. A mean on F is a continuous

linear functional

m ~ F'

such that (1)

~l,

m~

=

(il) Kf, m~ 2 0 Suppose S is a semigroup.

i,

and for f _~ O, f & F. For s ~ S define the right

-

77-

operator r s ~ L(C(S)) by

translation

rsf(x)

= f(xs),

x ~ s.

The subspace F is right invariant all s ~ S whenever f ~ F.

if rsf & F for

A mean m on a right

invarlant subspace F is called right Invariant

if

(lii) = for all s ~ S , f a F. 3.21 Let TI, T2, T 3 be semitopological T 2 and T 3 affine.

Suppose T l C

semlgroups

with

T 2 and co(T l) = T 2.

Further suppose that ~ : T 2 ---> T 3 is a continuous affine function with $~TI:T1--->T 3 a morphism. is a morphism of semitopological

Then

affine semlgroups.

Proof: Since ~iTl is a morphism and ~ is affine, ico(T1) is a morphism.

Then % is a morphism by 2.5.

3.22 Suppose S ~ T is a compact subsemlgroup

with identity

1 such that T = co S. The following statements

are

equivalent : (a) M(S)

is a minimal left

(resp., right)

ideal

of S. (b) C(S) has a right

(resp.,

left) invariant

mean. (c) M(T) is a minimal left

(resp., right)

ideal

of T. Proof:

(a) ==> (c): 3.17.

(b) ==> (a):

Suppose that Se' and Se are distinct

for e, e' & E(M(S)). and disjoint. flSe'

Then Se' and Se are closed

So there is an f ~ C(S) such that

= O and fiSe = 1.

But then re,f = O and

ref = l; so a right invariant mean cannot exist

-

78

-

for C(S). The proof for the other csse is the same. (c) ==> (b): Let r s 6 L(C(S) w) be right translation by s ~. S.

Let ~ = ~r s : s s S~ and ~ = c'~(S~) in

L(C(S)w).

Since s ~

continuous

(see I.l.14), S is compact in L(C(S)w);

r s : S ---> L(C(S) w) is

and since C(S) is a Banach space, T is compact in L(C(S) w) (Dunford, N. and Schwartz, J. T., Linear Operators

I, Intersc. Publ., N. Y. 1958, p. 511,

exercise 3, noting that L(C(S) w) is L(C(S)) in the weak operator topology). Let u be an element of L(C(S)) and

~C(S)'.

Then k

~

f u <s, A> d~(s)

: E' --->

is a continuous linear functional on E', which is denoted

~(u).

Clearly, ~

: L(C(S)) ---> E" is

linear;

it is in fact continuous if L(C(S)) is

given the weak operator topology and E" has the weak* topology g(E", E' 1. convex subset of E"

Then 4~(T~) is a compact

namely the closed convex

W* '

hull of a~(~).

Thus, if ~ ( ~ )

is in E (where we

make the canonical identification of E with a subspace of E"), then

~(~)

is in E by the bipolar

theorem. Now we specialize ~ to the point evaluation ~l at 1 %

S.

Then, for s a S, < @ ( r s), >,> = f r s dfl(t) = f dfl(t) = <S,

k>;

hence

~ ( r s) = s and

@(T)

~ E and T =

~(~)

~(T~).

~:

79-

= S c E; therefore, Let

T ---> T

be the restriction

and corestrlction

of

ap.

If s,t ~ S, then ~ ( r s r t) = ~(rst) So ~IS is a morphlsm. affine,

= st = ~ ( r s) ~(rt).

Since ~ is continuous

and

by 5.21, ~ is a morphism of semitopologlcal

afflne semlgroups. Now if M(T) then eT = e. !

m = eo~I

is a minimal left ideal and e ~ M(T)~

Let e o a ~ ' l ( e ) ,

& 3(S)

!

.

Suppose

and let

f ~. C(S)

is of the

.k rE'.

Let t s S.

form f(s) = <s, ~> for some

Then

= Y eort <s, A > dR ( s ) = < ~ ( e o r t ) , )~> = < ~(e O) ~ ( r t), A > = <et, A > -

<e,

~>.

In particular, = = <e, A > = . But functions of the form s r ~ points of S.

Thus,

<s, A >

separate

since the set of all f s C(S)

with = for all t ~ S also

includes

the constants and is therefore dense in C(S) Weierstrass), If M(T) then Te = e.

the

m is a right-lnvarlant is a minimal

(Stone-

mean on C(S).

right ideal and e & M(T),

Let eo, m, f, and t be as before.

i t a L(C(S)) denote left translation to show that = .

by t.

Let

We have

By the preceding,

-

it suffices

80

-

to prove

= f rte o <s, X > d~l(S). But ltr s = rsl t for all s ~ S, hence I t is in the c e n t r a l i z e r of the closed in L(C(S)).

convex hull T of all r s

Hence eol t = lteo.

But then

= I eol t <s, X > d~l(S ) = f lte o ~s, ~ > d~l(S) = (eof) (t) = f rte o <s, x> d~l(S). We now collect following

the accumulated

information

in the

theorem:

3.23 Theorem:

Let E be a Banach space and S @ L(E) an

equicontinuous

semigroup.

convex hull of S.

Let S' = co(S) be the

Define

W = ix ~ E : Sx is relatively

compact

W' = ~x a E : S'x is relatively A = ~x ~ E : Sx is relatively

compact

compact

A' = ~x a E : S'x is relatively Then the following statements

in Ewe, in Ewe,

in El, and

compact

in E~.

are true:

(i) (1) W = W' and A = A'. (2) x ~ A if and only if s ~

sx : S---> E

is continuous. (il)

(1) W and A are closed

invariant

subspaces

of E. (2) There are semigroups TW_ 'c L(W),

T A (-_ L(A),

T W ~ L(W), and T A' c L ( a )

such that T W and T'W are compact semitopological

semlgroups

in the

-

81

-

of Ww W and T A and Ti are

topology

compact topological topology of A A.

semlgroups

Moreover,

are affine semlgroups with

in the

y

!

T W and T A TW!

= c-o(TW)

and T A' = B-~(TA) in the appropriate topologies. (3) The maps qW

: S ---> TW, ~A : S ---> TA,

~

: S'

--->

T'W'

~ A'

: S'

---> TA'

: T W ---> TA, and %' : T'W "--> TA' defined by

~w(s) = slw,

~(s)

= slA,

~(t) : tIA , and ~'r

:

t'iA

are morphisms of semltopologlcal

semi-

groups such that

~

= T W, ~A--~ = T A, !

w

(TW) = T A, and ~'(T W)' = T A' and such that the following diagram commutes:

'

3'

TW

>

T' A

>

TA

r

! S

TW

;r

(All unmarked maps are inclusions.)

-

(lii)

82

(1) There are invariant G~A,

G' c

-

vector subspaces

A such that

T G = TwIG = TAIG and

T~ : % t o '

: T~IG'

a r e compact t o p o l o g i c a l

of G G, resp. , G,G' .

topology more,

groups

G and G' are maximal

respect

the

Further-

with !

to this property.

Since T G

is also an affine semigroup, =

in

card T~

I.

(2) The maps

---> %. ~

s

and ~G : T G - - - > defined

T'G

by -~!

%(s)

---> %.

s,

: s !a, % ( s

and t G ( t )

!

!

) : s Is',

= tlG'

a.~e morphisms

of semltopologlc&l

~emi~

groups and the diagram B

TQ - - - - * ~ commutes.

(3) F o r e a c h resp.

....

>

S'

,

T;

In particular,

idempotent

e ~ E(M(Tw)) ,

e ~ E(M(TA}) , o n e h a s

G c eW, r e s p . ,

G q eA.

(4) The weak and strong operator topologies ' and T G are the same. on T A, T A,

-

83

-

(5) The following statements are equivalent : (a) M(T W) is a minimal right ideal of T W. !

(a') M(T~)

is a minimal right ideal of T W.

(b) eW = e'W for all e,e' a E(M(Tw)). (b') M(T~)

is a right zero semigroup.

(c) G = eW for some e ~ E(M(Tw)). (c') G' = eW for some e ~ M(T~). (d) G = eW for all e 6 E(M(Tw)). (d') G' = eW for all e ~ M(T&). (e)~eW

: ea E(M(Tw))~

(e') ~{eW

: edM(T~)~

(f) u~eW

: e ~ E(M(T W))~

is convex.

is convex. is a vector

space. (f') u{eW

: e a M(T~)~ is a vector

space. (g) TwIeW is a group for some e~E(M(Tw)). (g') T~,IeW is a group for some e ~ M(T~). !

(g") T W' few = lew for some e ~ M(Tw). !

(h) T~x contains a fixed point of T W for every x s W. (h') T~x contains a fixed point of S for every x g W. (6) The following statements are equivalent : (a) T A is a group. (b) T A = T G(c)

A=G.

-

(iv)

84

-

(i) There are invarlant vector subspaces N ~ W, N' ~ W such that !

!

T N = T W N, resp., T~ = T W N , is a compact semitopological semlgroup with zero in the topology of Nw N, resp., N' W N'

Furthermore N and N'

"

are maximal with respect to this property. (2) The maps

~N

: S

---> T N , ~

---> T s'

: S'

and 9N, : TN, __.> T N defined by

~N(s) = sl~, @'~(s') = s' iN', !

!

and ~N ( t )

= t'IN

are morphisms of semitopologlcal semlgroups and t~e diagram $

___>

Sf

t

", ~N T N <. . . . . .

TN

commutes.

(3) For each Idempotent e t E(M(Tw)) , resp., e ~ M(T~), N ~

(I

-

e)W,

reap.,

(4) NAQ = ~O~ and N'OG'

N' % ( i

-

e)W.

= {04.

(5) The following statements are equivalent: (a) M(T W) is s minimal left ideal of T W(a') M(T~) is s minimal left i ~ s l

of T~.

-

85

-

(b) ker e = ker e' for ~11 e,e' t E(M(T W) ). (b') M(T~)

is a left-zero

semigroup.

(c) N = (1 - e)W for some e & E(M(Tw)). (c') N' = (1 - e)W for some e a M(Tw). (d) N = (i - e)W for all e ~ E(M(T W)). (d') N' = (i - e)W for all e ~ M(Tw). (e) U{(I-e)W

: e ,E(M(Tw)) } is convex.

(e') U{(l-e)W

: e~M(T~)%

(f) U~(I-e)W

: e s E(M(Tw)) ~ is a

is convex.

vector space. (f') U~(l-e)W

: e~ M(TCv) ~ is a vector

space. (g) U~(I-e)W

: etE(M(Tw)) ~ is ~-invarlsnt.

~') U~(I-e)W

: e % M(T' )I is ~invariant.

(~'9 U~(I-e)W

: es M(T' )} is S'-Invarisnt.

(h) TWI(I - e)W is a semisroup

with

zero for some e % E(M(Tw)). ' (h') TWI(1 - e )W is a semigroup

with

zero for some e ~ M(T~.). (v) (1) M(T W) is a sroup

if and only if

T W' has a zero. (~) If M(T W) is a group,

then

W = N | G and W = N' @ G'; in addition, identity

if e, resp.,

of M(Tw),

resp.,

e', is the M(T~),

N = (1 - e)W and G = eW, resp., N' = (1 - e')W and G' = e'W.

-

(vl)

The following

86

-

diagram

is commutative:

!

TN

< ...........

~N

T f

N

/ / TW

..... ~-

___.>

S'

~'

....

TG

TA

T !

....

G

(All unmarked restrictions

morphisms

are either

or inclusions.)

PrOof:

Assertions

(i) and

(ii) are from 1.2.9,

1.2.10,

3.15, and the fact that in a Banach space

th~ convex hull of a relatively relatively compact,

weakly compact,

resp.,

~elatively

compact,

resp.,

set is again relatively weakly

compact

(see, for

-

87

-

example, Dunford and Schwartz, op. cit., pages 416 and 434).

Assertions

(ili)(1) - (iii)(3) are contained in

3.12 (except for the commutative diagram which is obvious).

For (iii)(4) observe that since TA, TA, ' and

T G are compact in the strong operator topology and since the weak operator topology is coarser, the assertion follows.

The equivalent statements in

(iii)(5) are a combination of 3.22, 1.23, and 3.19 (except (h) ~==> (h') which is obvious).

In (iii)(6)

we clearly have (a) ==> (b) ==> (c) ==> (a). proof of (iv) Is similar to that of (iii). (Iv)(5) the implication

The (In

(g") ==> (g') follows from the

fact that for each x ~ W, T~x is the strong closure of S'x.)

Then (v) follows from (ill) and (iv).

And

(vi) is Just a collection of the various diagrams.

- 88 -

4. Invariant Measures on Locally Compact SemIEroups Notation and conventions: 8emitopological

Let S be a locally compact

semisroup;

Coo(S),

vector space of all continuous

the topological

functions f : S --->'~[

with compact support and the topolosy defined by the s eminorms

f"*tt f IIc C s S compact;

= sup

f(x)

: xs C

~,~(S), the vector space of all

regular Borel measures on S (i.e., the dual of Coo(S)) Let Coo(S) + , resp. j~(S) +, be the cones of nonnegative elements.

For~

~(S)

+ define

~I(s,~)

to be the completion of Coo(S) with respect to II'~I For sL S let R s : S --9 S be defined by _qs(X) = xs. Then ~s IX = { y 6 S m

If f : S - - ~ -~ 4

: yss

for X ~ S.

let Sf(x) = f(xs). 4.1

Let X ~ S. (I)

Then -I

(%

-I

x)s = x c_~ s (xs);

(ll) Rs iz~ __CRsI~; (ill) If I is a left ideal and s4, I, then aslI = S and R s I ( S \ I ) (iv)

If f

:

S --9~,

Proof: Straishtforward (li) implies 4.2

If ~

supp sf ~ Rs1(Supp

since R s is continuous and

(iv).

~ ( S ) +, then (a),

of equivalent

then

= ~;

statements,

(b)and where

(A), (B) are pairs

- 89-

(a) F o r a l l

s ~ S and 0 ~- f ~ ~1 ( ~ , / ~ ) ,

: s f at~

,~ alp;

..

(b) For all s 5 S and all /g-measurable sets X

(A) f o r a l l

s~.S and 0 $ f 5,LI(s,/J.),

/ s f d;,-~- ,ff d ~ ; (B) For all s~ S and all f-measurable

sets X C~

/J(.~S 1 X) ~ / . ; . ( X ) . Proof : Let ~ X denote the characteristic function of X.

Then

s( k )=

x

for a l l

(*)

X c_ s.

>-~:'x

Hence, if X is ~(-measurable,

/S(kx)d/Q

= /)(.a;Ix d f

=

p(R;'X).

That (a)==>(b) ~nd (A)==>(B) is now clear from (*). Let 0 g f ~

(S,/r

Then f can be ~l-approximated

from below by an increasing sequence of finite measurable non-negative step functions fin' n=1,2,...

By ( * ) , /sffn d/r

:~n d~

in case (b)

and / s o n d/~ :':f o"n d/a in case (B). The sequence n-~,s~ n is non-decreasing; so lim /s~ n d

exists in [0,~].

But l i m / S @ n d]~ =

Jsf d/Q since lim Sqn(X ) = Sf(x) for /~-almost all x (cf. HalmosC~], p 112).

But

lira /orn cI/~ = /f ~,.

by the definition of on . So by (*), sf ~

(S, ,~) and

:sf d~ ~ Yf d~ in case (b

and in case (B).

-

4.3

For

90

-

/K t J~I(S)+, we have

is equivalent to

(

)

.<

Proof: Straightforward 4.4

from 4.1 (1).

Definition: /x& ~ ( S ) + is called (I) right sub-invariant, (2) right contra-invarlant, (3) rig~ht infra-invariant if (I) ~

satisfies

(a) of 4.2,

(2) /~ satisfies

(A) of 4.2,

(3) for all s ~ S and all ~-measurable

sets X

(xs respectively. 4.5

Examples : (I) Let S be a discrete right cancellatlve semigroup. ~(X)

Define ~ 6 , ~ ( S ) =

(ca~

(I .2)

+ by

X if card X is finite if card X is infinite

Then ~

is a right sub- and infra- invarlant

measure. (2) Let S be ~ +

= LO,C@Eunder addition, and let / ~

be Lebesgue measure.

Then ~ - i s

sub-and Infra-

Invariant, but not contra-lnvarlant. (3) Let S = LO,I] under multiplication, be Lebesgue measure.

and let ~ -

Then ~. is Infra-invariant

but neither contra-invariant nor sub-invarlant.

(4)

91

-

Let S be a locally compact group. Haar measure

~a~(S)

Then any right

+ satisfies all right

invarlance properties. (5) Let S be a locally compact left-zero semigroup, and let ~

~(S)+

be arbltrary.

Then /~

satisfies all right invariance properties. (6) Let S be a locally compact right-zero semlgroup, and let ~ J ~ ( S )

+ be arbitrary.

Then

is neither right contra-lnvariant, nor rig~ht sub-lnvariant. (7) Let S be a locally compact left group, say S = X x G, where X is any locally compact space under left-zero multiplication and G is any locally compact group.

Let ~ ~ ( X )

a rlght Haar measure on G.

+, and

Then ~

=

#

~ |

has all right invariance properties. (8) Let S be a locally compact semltopological semlgroup with a minimal ideal M(S) which is a left group.

Suppose ~ ' g J ~ ( M ( S ) ) + Is a rlght

invarlant measure such as in (7).

Define

~ ' ~ S ) + by

is rlght sub-lnvariant and right contrainvariant. (9) Let S be the space of all ordinals less than or equal to the first uncountable ordlnal ~'~. Let

/ •unltCpoint bmasszl eatif ~Sl, ~i.e. x /~X)

=
otherwise

-

then ~

92

is sub-invariant

not Infra-invariant cation

-

~na contra-invariant

if ~ is given the max multipli.

(st = max s,tL Cn the other hand, ~

infra-invariant

but

is

but not sub- or contra- invarlant

if S is given the min multiplication.

The

space S cannot carry any measure with dense support. 4.6

Let __,h.~g,~(S)+ be right sub-invariant. (i) s u p p ~

~ t~se

: e ~ E(S)~

Then

so that all

idempotents are right identities for supp ~ ; (ii) supp1~

is a closed right ideal;

(iil) for e ~ E ( S ) ,

Proof:

(i) Let e = e 2.

compact, and U O S e Ve = ReV = Ue.

Hence (ii)

~(U)

= ~.

jf ~

=

Suppose U is open, relativel~ Let V = U ~ Ue.

= O.

So supp LA ~_Se. J x , s & S , and x s S s u p p

there is an f ~ C o o ( S ) + such that Let g = S f & c ( s ) +.

xs g s u p p ~

~.

Then

Consequently,

for all s ~,S.

(ill) Immediate from (1).

~

/f d~

.

Then

= 0 but

Then g(x) ~ O, and :

-

SO x ~ s u p p

for all

Now

Suppose t h a t

f(xs) ~ O.

fef d~

x &suppe

o

implies

-

4.T

Let

~

93

-

6 ,~(S)+ be right contra-invariant.

be the set of all closed left ideals in S.

Let ~ c Then

(~

supp~ ~

~j~C C_ ~ S e

In particular, if ~ closed left ideal.

: eSE(S)j.

~ O, there is a unique minimal Note that ~%[Se : e a E ( S ) ~ =

C~/

if each closed left ideal contains an idempotent. Proof:

Let I s ~c, and let s & I.

Suppose U is open,

relatively compact, and U t~I = ~. by 4. I (iii).

Hence ~(U) ~

So s u p p ~ 4.8

Then RsIu =

~ I.

~(RsIU)

= O.

The rest is clear.

Let S be a paragroup and ~ s

+, /U~ % O, right

sub-invariant or right contra-invariant.

Then S is

a left group, say S = X x G, where X is a locally compact left-zero semi5roup, and G is a locally compact topological group, and ~(X)

+ and

// = ,~ @ t) where

~ is a (non-zero) right Hear measure

on G. Proof:

If e& E(S), then if ~

is right subinvarlant

/

or right contra-invariant, resp.).

supp~

C_ Se (4.6 or 4.7,

If S ~ Se, then there is an e'g E(S) with

Se % Se' such that ReSe' = Se, R e,Se = Se', and R e~se = ~.

Now if ~

is right contra-lnvariant,

~(Se) ~ ~(Re]Se) contradicting ~

~ O.

= O,

On the other hand, i f ~

is

right sub- invariant, 0 = ~(Se') = ~(See') ~ ~(Se), again a contradiction.

Whence S is a left ~roup (I.

The

4.9

94

-

remainder of the proof will follow from 4.9.

Let S = X x G, where X is a locally compact left-zero semigroup and G is a locally compact group.

Let

f~.~ J ~ S ) +, and suppose that y_ is right sub-invarian + rig~ht contra-invarlant, Then ~

or right infra-lnvarlant.

= ~ ~ ~ , where )~ & ~ ( X )

+, and # is a rig~ht

Haar measure on G. Proof:

For each h ~ C o o ( X ) , f~/h(x)

f a Coo(G) ,

f(t) d/~(x,t), |

defines a Radon measure

~ h on G.

If /Z satisfies

any of the right invarlance properties

of 4.4, then

A

~h

is a right Haar measure.

right Haar measure

~

Hence, for a fixed

on G, for each h ~ C o o ( X ) ,

there is a real number l(h) such that /h(x)

f(t) d/Y.(x,t)=

Now I is a positive continuous

l(h)/f(t)

d ~(t).

linear form on Coo(X)

So I(h) = / h ( x ) for some Borel measure /hCx)

d~k(x)

~. on X.

f(t) d ~ ( x , t )

But then the identlt~

: /h(x)

d~(x)/f(t)

d ,)(t

#

shows ~

: A | ~.

4.10 Let L(~ ,~(S) + be right sub-invariant. E(S)~ supp//

~ ~.

semigroup by 4.6).

Denote supp j~ Let e ~ E ( S o ) ,

Then X is a closed left-zero

morphism of semigroups.

by So (which is a T = eS o, X = E(So).

subsemigroup,

locally compact semitopological : X x T --3 So with

Suppose

~(x,t)

semigroup,

X x T is a and

= xt is a surJective

Furthermore,

we have

-

~(x,t)

=

~(x' j

'

95

t') if and only if xt = xt'

S o is left cancellative, group,

then ~

-

~

is blJectlve.

Thus if If T is a

is an isomorphism and S o is a left

group. Proof:

The elements of E(S o) are all right

of S o by 4.6. closed

But the space of right identities

in S o and is a left zero semigroup.

T is closed by 2.2. directly checked. 4.11 L e t ~

The properties

of ~

The last assertion

surmorphism of semigroups such that

~(s)

x ~S, then supp ~&

Proof:

= G; whence Since

of ~ - I ( I ) ~ S

So ~I~(So) is a

~(So)

= G since a

1 6 ~ ( S o ) , by hypothesis

o are right identities

is a closed left-zero

~T(1) = G.

Then

semi~roup.

Let I be

~T[(1) is a left ideal of G;

Thus there is an e 6 ~ T "I(I) O I

But then S o = Soe ~ I ~ S o.

Hence S o is left simple.

Thus So is a left group by I .3, and the assertion follows

of S(

every right identity of S o is in ~ - I ( i ) .

a left edeal of So. therefore,

semlgroup of

of S o = supp I~.

group is right simple.

So ~ - I ( I )

If there is a

is a locally compact left group,

right ideal in ~ S )

Conversely,

may be

= I implies that xs = x for all

By 4.6, S o is a right ideal.

all elements

Clearly,

?K: S --9 G, where G is a

and 'TT-I (I) is the closed left-zero right identities

is

follows from 2.3.

& ~ ( S ) + be right sub-lnvarlant.

sroup,

Identltle

from 2.3.

-

96-

Let ~ g ~j~(S)+, /~ ~ O, be right sub-invariant.

Notation:

Denote by ~g.2(S,/a) the set of all square Integrable functions.

Let 76 denote the set of/~-null functione

in ~2(S,/~).

Take L 2 ( S , ~ )

= ~Z~2(S,/~)/TZ.

.< / f we have t h a t f 6 ~

Since

o,

implies sf~

for all

saS.

Thus for the class fa L2(S,/~) of a function f ~ ~2(~;!,/~), we can define ~ v

s 7

=

(sf).

Therefore we define 7F : S --J, L(L 2(S,/~c)) by ~(s)f

= (sf).

(The range of ~T is L(L2(S,/~))

since

llf So,

in

fact,

,.)

4.12 Let/~ ~ ~ ( S ) +, /~ / O, be right sub-lnvarlant.

Then

(i) 7[(s) = ]/(t) if and only if xs = xt for all x a supp/s (il)

;

---- I if and only if xs = x for all

7[(S)

x a supp/~ . Proof:

(l)

"E(s) =

f 6 ~L-,2 (S,/~ ). 0

= /~4([x

7[(t) i f f

sf.

t f a*~6 f o r a l l

For any f s ~;~2 (S ,/~c) then, :

Sf(x) / tf(x)~)

= /~(-i~ : r(~s) / f(xt)~ ). If f Is continuous,

then ~x : f(xs) ~ f(xt)

SO

supp/Q. ~

{x:: f(xs) ~ f(xt)} = ~'.

Is open;

-

Thus for continuous x asupp~A

.

97

-

f, f(xs) = f(xt) for all

Consequently,

since Coo(S) separates

xs = xt for all x a s u p p ~

points of S.

(ii) Similar. 4.13 L e t ~

& U ~ ( S ) +, }tO ~ O, be right sub-lnvarlant.

Suppose

~(S)

is a group.

a closed left group.

Then M(S) exists and is

Furthermore,

So = s u p p ~

~

S o is a left group,

and S o is a union of maximal

subgroups

Moreover,

of M(S).

M(S

for e a E ( M ( S ) ) , for all f a ~ I ( s , / ~ ) .

Proof:

Let L be a left ideal. SoL ~

L ~ So

since S O is a right ideal

(4.6(ii)).

left ideal of So; however, S o is a left group Lo = ~L Thus a minimal So contains

(4.11).

since

So So ~ Lo, where

viz. L o.

Hence Lo is closed by 2.2.

is a paragroup,

and equals Lo since

left ideal in S.

a union of maximal groups follows from 4.9.

4.14 Theorem: semigroup, on S.

That S o is The last

is 4.6(iii).

Let S be a locally compact semitopological and l e t ~

~ 0 be a regular Borel measure

The following statements (G) ~

Since

(4.11) and Lo is left simple,

(I .3).

Lo is the unique minimal

assertion

is a

U~(S) is a group,

left ideal exists,

Lo is a left group

Also L ~ S o

: L is a left ideal of S~.

idempotents

So M(S) exists,

Then

are equivalent:

is right sub-lnvariant

and

-

98

-

the image of S under the natural representat of S on L 2 ( S , ~ )

is a group;

(L) Each left ideal In S contains an Idempotent and ~

is right sub-lnvarlant or right

contra-lnvarlant; (M) S has a closed left group as mlnlmal ideal; if X = E(M(S)), G = eSe for some e a X ,

then

there is a positlve regular Borel measure on X and a right Haar measure ~

on G such

that

If d#

dCX

=

)Cx,g)

for all f 6 C o o ( S ) . Proof:

(G) ===> (L): 4.13 and 4.8.

(L) ===>

(M): Since E(L) { ~ for each left 1deal L,

Lo = ~ { S e

: eEE(S)~

by 4.6 and 4.7. ideal.

= ~{L

Thus L 0 is the unique minimal left

By 1.8 and 1.3, L is a left group; by 2.2, It

Is closed.

The rest follows from 4.6(111) and 4.8.

(M) ===> (G): Let e6 E(M(S)). Hence

: L is a left Ideal~ ~

Then e f _

~(e)

= I (the identity operator).

~(S) =

"K(e) qy(S)w[(E) = 9y(eSe),

fa~

(4.6(i

So

which Is a ~roup since eSe Is a group. 4.15 Corollary:

Let S be a locally compact semltopologlc~

semigroup, and l e t ~ on S which is rlg~ht Invariant.

>

0 be a regular Borel measure

sub-invarlant or right contra-

The following condltlons are sufflclent l

that the conditions of the theorem hold:

-

99-

(I) S is compact; (2) Every element of S is in some compact subsemigroup of ~; (3) S has a unique minimal left ideal L and there is an 1 5 L

such that ~ s s S : sl = ~

is compact. Moreover,

if~

is both right sub-invariant and right

contra-lnvarlant,

the following conditions are also

sufficient that the conditions of the theorem hold. (4) S has a minimal left ideal or a minimal right ideal; (5) S has a minimal ideal M(S) and E(M(S)) ~ ~. Proof:

(I) ==-~ (2): Trivial.

(2) ==--~ (L) of 4.14: (3) = = 3

3.4

(M) of 4.14: Since S~ is a left ideal,

L ~ Sl ~ L, therefore l a S1 = L. C = {sis

: sl = I~.

C is a subsemigroup.

Let

Since it Sl, C ~ ~. Now C ~ L

Trivially,

is a left ideal of C.

Every left ideal in a compact semitopological semigroup contains a minimal left ideal; let L' ~ be a minimal left ideal of C. and so E(L') ~ ~ (3.4). left group (1.4). Note that if ~

L' is compact

C~L (3.2)

Whence M(S) = L is a closed

The rest then follows from 4.8.

is both right sub-invariant and right

contra-invariant,

then llqT(s)~l = I.

So the range of

T[ is the group of unitary operators on the Hilbert Bpace Lr(S, ix,).

-

(4) ===~ (G): S.

l O 0 -

Suppose R is a minimal right ideal of

Then 9f(R) is a minimal

particular,

right ideal of ~(S);

T<(-q) is right simple.

As a subsemigroup

of a group, 7[(R) is left cancellative. I .3, "K(R) is a right group. idempotent

In a group,

particular,

Similarly

so ~ R )

(5) ===~" (G):

In

Then

rs) c

left ideals.

Since M(S)

simple,

and since E(M(S))

7~M(S))

ls a group,

is simple, 7[(M(S)) ~,

I&7[(M(S)).

is Whence

since for each a~9[(M(S)) , the

eq~Jatlo~ zsy = I is solvable therefore,

by

must be a group. with 1~(r) = I.

=

for minimal

Whence,

But there Is only one

there is an r g R =

In

a -I = YX&T[(~(S

).

for x,ygq~(M(S))} Now as in (4), one

shows that ?C(S) = 7~M(S)). 4.16 Corollary: semlgroup.

Let S be a compact semltopologlcal The following conditions

are equivalent:

(a) M(S) Is a group; (b) M(S)

Is a compact group;

(c) S has a positive

(non-zero)

sub-invarlant

(non-zero)

contra-lnvarlant

measure; (d) S has a positlve measure.

-

101

-

4.17 Definition: Let S be compact.

Let P(S) be the set of

all regular probability measures on S; i.e., / ~ if and only if /~ > 0 and ll#~i = f d/~ = 1.

P(S)

Let P(S)

have the topology induced on it as a subset of C(S)', where C(S)' has the weak*

(~(C(S)',C(S)))

4.18 Let S be compact, P = P(S). S =

%) C P

topology.

Define

: ~ is unit point mass at some t z S~.

Then S and ~ are hcmeomorphic,

P = c'~ ~, and P is

compact. Proof: Dunford and Schwartz, op. __cit-, p. 441. 4.19 Let S be compact.

Suppose f : S x S --->

and separately continuous.

For all ~ , P

(

is bounded

~ P, f is

integrable with respect to the product measure ~ ) . Proof:

(1) Suppose

)) is unit point mass at p a S.

Then f f(x,y) d ~(y) = f(x,p) = Fr(P)(X). Since Fr(p) E C(S), f d/~(x) f f(x,y) d ~)(y) and equals

! f(x,p) d/~(x).

exists

On the other hand, if

= s f(x.y)

then # Z C(S) since it is the composition of F r and a continuous linear functional on C (S)w.

Z d ~(y) f f(x,y)

Thus,

dj~.(x) = f ~'(y) d ~ (y)

exists and equals ~/(p) = f f(x,p) d/~(x). n

(2) Suppose where each

~

is a convex combination

@i ~ ~"

f dt, L(x) f

f(x,y)

(3) Now suppose

~ ~ i •i' i=l Then, by a similar argument to (1),

d ~'(y) = f d ~ ( y ) f f(x,y) ~) is arbitrary.

By 4.18,

d2 (x).

there is

-

a net W

on P with

9

102

-

---> ~) and

t d~Cx) t ~Cx,y) ds

= t d~Cy) t t(x,y) d~.(x).

Now since f f(x,y) d/~(x) is continuous

t d ~ (y) t ~ ( x , y ) d ~ ( x )

--->

in y,

t d~ (y) t ~ ( x , y ) d/~(x).

On the other hand,

t f(x,y) d~(y)

--->

t f(x,y) d~(y)

--->

t t(x,y) d~(y)

polntwise; so, by 1 . 1 . 8 ,

t Z(x,y) d_~(y)

weakly since {f f(x,y) d A(y)

(II,

: A & P~ is bounded

(y)il _llFz(xllllll_ llflll.

s d/~(x) t Z(x,y) d _~(y)

--->

t ~ / ~ ( x ) t Z(x,y) d ~ ( y ) .

In view of Theorem 4.14, s positive regular Borel measure ~c on a compact semltopological

semigroup will be called

right ijnvarlant in the sequel if it is right sub-invariant or right contra-invariant.

-

Notation:

105

-

Throughout the remainder of this section, let

S be a compact semltopologlcal semlgroup, and let P = P(S) be the set of all re6ular probability measures on S.

For /~,~ e P we define

the convolution

k = /~.9

t f(8) dX(s) f Z C(S) 4.20

=

H

by

f(xy) d/~(x) d~(y),

(well-deflned by 4.19).

The convolution semigroup P is a compact semitopological affine semigroup. Proof:

In view of

4.18,

we need only show that the

semigroup P is semitopologlcal: on P and ~_ - - - > ~

in the weak* topology.

Since the unit ball of ~ ( S )

Let

~ z P.

is weak* compact, we need

only show that /~_* ~) ___>/a,~) pointwlse on C(S).

Suppose /~_ is a net

and ~ * ~ - - - >

Let f g C(S).

~)~/~

The function

defined by

~O(y) : s f(xy) d ~ (x) is in C(S), so

s f(s) d ~ . ~ ( s )

=

ff f(xy) d ~ ( x )

=

s ~(y) d#(y)

--->

d/~(y)

f ~(y) d/~(y) =

; f(s) d~./~(s),

On the other hand, the net of functions ~ defined by ~(Y) = I f(xy) d/~(x) is in C (S), and since/~

--->/~

weak*ly, ~ --->

weakly, where ~(y) = f f(xy) d/~(x). But then,

10#

-

~' fCs) a~.>~Cs) /

-

=

.r.r

=

f

--->

f(xy) d/~_Cx) d~)Cy) #(y)

d))(y)

f %(y) d~.'(y)

=

! f(s) d ~ . # /

(s)

4.21 Iff~,t) Z P, then supp/L~*P Proof:

C

((supp /~)(supp

Let A = supp //x, B = 8upp

~I.

~)))--. By the

regularity of /x.~) , for

6 9 0, there is an open set

U

U and

C S such that (AB)-- C

~*~(u)

_< 2 ~ * ~ ( C A B ) - )

By the Urysohn lemma,

where

~X

there is a function f & C(S) with

~(A~)-

9(AB <

+ ~.

-~ f

-< ~ u ,

denotes the characteristic

Now ~Ar162

_< ~(~r

1 = /x(A) ))(B)

=

If

function of X.

so ")(A(X) ~(B(y) d/a.(x) d~' (y)

_< fl X A B C X y )

d~(x)

d ~ (y)

<

.ff f(xy) d2~.(x) d ~ ( y )

=

! fCs) d ~ . ~

(s)

_< j" XuCS) d~.~, (s) = ~ . ~ (u)

_< 2..~ ((~B)-) + <

I +

E.

Since

was arbitrary,

CAB)-- O

supp /~. ~), as desired.

4.22 N(s) c__. ~supp /~.:

~Mr

/~.~) ((AB)--) = I; so we have

c_ Mr

Specifically,

if 2~ is Haar measure on a maximal group in MCS), then

E M(P). Proof:

If s & M(S), then there is a maximal group G

- 105 such that s & G.

N o w G is a compact

topological group;

let 2~ be normalized Haar measure on G. supp /u. = G, we have s a supp /~.

Since

Let

~ P, f ~ C ( S ) .

Then

d ~ . ~./,- (s)

~" f(s)

=

fff f(xyz) d ~ ( x )

d~(y)

d/~-(z)

- tt[s yzf(x) e~(x)] d~(y)d2~(z)

Since ~ . ) ~ . ~

=

=

,t [t fc )

=

s f ( x ) e/.,.(x).

for every

d

~ ~. P, we must have that

/~ ~ ~(P). Now suppose tl~at ~ let ~

& M(P).

Let e E E(M(S)), and

be unit point mass at e.

*)~.~

= ~.

By 3.16(ii),

By 4.21 we have

supp ~x = supp ~ . ~.#~

c ((supp ~ ) (supp

~) (supp ~ ) ) -

= ((supp ~ ) e ( s u p p

/w.))--

c_ (SeS)= MCs )-. 4.93 The following statements are equivalent: (a) S has a positive right Invariant measure; (b) M(S) is a minimal left ideal of S; (c) M(P) is a minimal left ideal of P. Proof:

(a)==>(b): 4.1/~.

(b)==>(c): 4.20 and 3.17. (c)==>(a): Let ~,P

/ ~ & M(P) and

)3 6 P.

Then

=/~., so if f ~ C(S) and s ~ S, then f f(t) d/~(t)

=

fI f(ts) d/~(t) d~)(s)

:

ts SfCt) d ~ ( t )

d~Cs).

But f fCt) d/~.(t) = ff f(t) d/~.(t) d~'Cs), so since

P

106

-

separates the points of C(S), the functions

s ~

f sf(t) d/~(t) and s ~

f f(t) d/%(t)

must be equal; that is, I sf(t) d//~(t) 4.24 If / ~ 6 E(P) and supp2~

:

f f(t) d/~(t).

= G v H, where G and H are

compact subgroups of S, then s u p p ~

is a subseml-

group of S. Proof:

If G /% H ~ ~, then G t; H = G or G Lt H = H, so

supp~

is a subsemigroup of S.

G /% H = ~.

Therefore, assume

By 4.21,

B = supp/~

C_ ((supp/~)2)-- = B 2-.

Let U be an open set in S such that U ~

B 2 ~ ~.

Suppose st e U (% B 2 and, say, s & G, t a B. By 1.3.],there is a set V open in G and a set W open in S such that s 6 V, t s W, and

(VW)--~ U.

Since H is closed, there

is an open set V' in S with V' (A B = V. V' and W are open and V ' ~

Now, since

B ~ ~ and W C% B % ~, we

have 0 < /~(W) and 0 < /~(V') = /~(V' /% B) = /4&(V). By Urysohn's lemma, there is an f ~ C(S) with

~(~)where

~X

-< f

-< Xu,

is the characteristic

0
=

ff

_< f2

function of i.

~v(X)~w(y)

d/~(x) d2~(y)

3[(V~)_(xy) d ~ (x) d/~(y)

_< s~ f(xy) d/~(x) d/~(y) : s f(s) d~(s) _< t ~U(s) d/~(s) : 2~ (u). So B2 ~ s u p D ~

= B.

Now,

107

-

4.25

If M(P)

is convex,

of S or M(S) Proof: ideals S.

then M(S)

is a m i n i m a l

Suppose in S.

is a m i n i m a l

left

respectively,

groups and

Let e and

G = R l~

let

~

and

Haar m e a s u r e s

only are

if e S f = G and disjoint. L2 ~

4.26

But

H = Sf ~

fSe = H s i n c e

ideals

in

identities L2,

Since

M(P)

supp~

is

= G V H groups

is p o s s i b l e

R 1 = eS and R 2 = fS

then, fSf = fSf = fSe

T is an a f f l n e

semigroup,

segment

in T.

If there

exist

~

on L, then L c o n s i s t s

idempotents

and xLx = ~x~

Se = L1;

and L is a llne

three

idempotents

e between

right

H = R2 ~

in G U H, w h i c h

Suppose

with

left

then the m a x i m a l

L 1 = L 2.

Let e,

H.

By 4 . 2 4

so L l g ~ L 2 ~ ~, w h e n c e

Proof:

ideal

~' be the r e s p e c t i v e

convex,/~A

fSe are c o n t a i n e d

minimal

f be the

L 1 and

on G and

= ~--~ M(P). 2 is a s u b s e m l g r o u p of S. But

right

of S.

Let L 1 and L 2 be m i n i m a l

of the c o m p a c t

e S f and

ideal

R 1 and R 2 are ~ i s t i n c t

We s h o w that L 1 = L 2.

normalized

-

distinct

entirely

of

for all x & L.

f, and g be d i s t i n c t

idempotents

f and

exists

g.

Then

s u c h that e = af + (1 - a)~;

there

on L,

0 < a < 1

so

af + (1 - a ) g = e = e2 = a 2 f + a(l - a ) f g Multiplication

+ a(l - a ) g f

+ (1 - a)2~.

on the left by f y i e l d s

af + (1 - a ) f g = a2f Rewriting

this

+ a(l - a ) f g

+ a(l - a ) f g f

as a(l - a)f = a(l - a ) f g f

+ (i - a) Ag.

and u s i n g

the

-

I08

that a is n e i t h e r

fact

By s i m i l a r that b o t h fact

arguments

-

zero n o r one,

we o b t a i n

f = fgf.

one can s h o w g f g = g, and

fg and gf are i d e m p o t e n t s .

Again

it f o l l o w s

using

the

that e is an i d e m p o t e n t ,

af + (1 - a ) g = e = a2f This

+ a(1 - a)fg

can be r e w r i t t e n a(1

-

a)f

x = b f § (1 - b)g,

= b2f

+ (I - a)2g.

as

+ a(1 - a ) 8 = a(1 - a ) f g

so f + g = fg + gf.

x 2 = b2f

+ a(1 - a ) g f

If, now,

+ a(l - a)gf,

x is a n y p o i n t

on L, s a y

then

+ b(l - b ) f g

+ b(l - b ) g f

+ b(l

+ g)

- b)(f

+ (I - b ) 2 g

+ (I - b)2

= bf + (i - b ) g = x;

4.27

so e a c h

x s L is an i d e m p o t e n t .

it t h e n

follows

Every and

element

only

if x T x = {x]

requirement

for e v e r y

semitopological

t h a t T be s i m p l e

T is I d e m p o t e n t

x ~ T. affine

if

In a d d i t i o n , semigroup,

is e q u i v a l e n t

if

the

to e a c h of the

If xTx = ~x~ for all x in T, t h e n x 3 = x; so

x 2 = x3x = x x 2 x Idempotents,

= x.

and x,y

Conversely, & T, x ~ y,

J o i n i n g x and y c o n t a i n s by 4.26,

4.26

semlgroup

conditions.

Proof:

Note:

computation

that x L x = {x~.

of an a f f i n e

T is a c o m p a c t

above

readily

By d i r e c t

xyx

s xLx = ~x~;

and 4 . 2 7

semigroups,

more

Pac.

are due

if T c o n s i s t s then

than two

i.e.,

12

segment

Idempotents;

so,

xTx = {xl.

to Collins,

J. Math.

the l i n e

of

(1962),

Remarks

on a f f l n e

449-455.

- 1o9

4.28 Theorem:

-

Let S be a compact semltopological semlgroup,

and let P be the set of regular probability measures on S.

Then the following statements hold: (1) In the weak'* topology P is a compact semitopological afflne semigroup under the operation of convolution defined by f f(s) d / a . P (s) : If f(xy) d/x(x) d ~ ( y ) , f ~ c (s),

~ , ~ ,.

P.

(il) S is isomorphic to : {#~s g P : f f(t) d/~s(t) : f(s), s z S~, and P = co S. (Ill) The following statements are equivalent: (a) S has 8 positive right

(resp., left)

invarlant measure. (b) M(P) = { ~ z P :/~ is right (resp.,left)invsriantl. (c) M(P) is a minimal left (resp., right) ideal of P. (d) M(P) is a left

(resp., right) zero

semlgroup. (e) M(S) is a minimal left (resp., right) ideal of S. (f) M(S) is a left group X x G (resp., a right group G x Y), and M(P)

=

{k|

:

(respectively,

M(P) = ~ ~

zP, supp A c X and %) is normalized Haar measure on G~ i

:

is supp ~ C y and ~ is normalized Haar measure on G~).

110

-

-

(iv) The following statements

are equivalent:

(a) M(P) is convex. (b) M(P) is midpoint convex. (c) S has a positive right invariant measure or a positive left Invariant measure. (Compare

(iv) with (lii)(a).)

(v) The following statements are equivalent: (a) P is simple. (b) S is a left zero or a right zero semigroup. (c) P is a left zero or a right zero semigroup.

(vi) M(~) c_ U { s u p p / ~

9

~ M(P)~ c_ M(s)-.

(1) and (il) are from 4.20 and 4.18.

Proof:

(iii) Suppose

/~g

P is right invariant,

and

9EP.

Let f z C(S), then f f(t) d / ~ , ~ (t)

So ~

=

ff f(xy) d/~(x) d ~ ( y )

=

ts Yf(x) ~ ( x )

=

;t f(x) d/~(x) dW (y)

=

t f(x) d/~(x).

is a left zero for P.

is a minimal left ideal of P.

dW(y)

Hence /z~ ~ M(P), and M(P) Now the proof of 4.23

shows that M(P)c_ {~) c P : )2 is right invarlant~. Thus,

(a)==>(b).

follows

Since

(b)==>(a)

from 4.23 and 4.14.

(iv) (a)<==>(b) 4.27.

(a)==>(c) 4.25. (c)==>(a) Trivial by (iil). (v) Trivial by (ii) and

(vi) 4.22.

(iv).

is trivial,

the rest

-

~11

-

REFERENCES

.

Ber~lund, J. F.: Various topics in the theory of compact semitopological semigroups and weakly almost periodic functions. Dissertation. New Orleans: Tulane University.

.

Bourbakl, N.: Elgments de Mathematique. XXV. Premiere partie. Les structures fondamentales de l'analyse. Livre VI. Intggration. Chap. VI Actualit4s Scl. et Ind. 1281. Paris: Hermann & Cie. 1959.

.

Clifford, A. H., and Preston, G. B.: The Algebraic Theory of Semigroups, Vol. 1. Mathematical Surveys, No. 7. Providence: A m e r i c a n ~ a t h e m ~ t i ~ l Society 1961.

.

deLeeuw K., and Glicksberg, I.: Applications of almost periodic compactlficatlons. Acta Math. lO~ (1961), 63-97.

.

Hofmann, K. H., and Mostert, P. S.: Elements of Compact Semlgroups. Columbus: Charles E. Merrill 1966.

.

Paalman de-Miranda, A. B.: Topological Semlgroups. Amsterdam: Mathematisch Centrum 1964.

-

112-

CHAPTE

III

ALMOST PERIODIC AND WEAKLY ALMOST PERIODIC FUNCTIONS ON SEMITOPOLOGICAL 1. Various Universal 1.1

Lemma:

SEMIGROUPS

Functors

Let S be a semitopological

is a set ~ o f semigroups

morphisms

semigroup.

There

~/: S ---> T of semltopologics

such that

(1) ~-T~I = T, and (2) every morphism *~': S ---> S' of semitopolo~ic~ semigroups

decomposes S -~>

~

as follows:

S' ~ inclusion

T

with some % & ~ Proof:

and an isomorphism *~i~~-

(I) There is only a set of images of S in the

category of sets since each such image is given (within isomorphy) by a quotient map of S. (II) For any set X, there is only a set of topologies on X (a subset of the power set of the power set of X (III) For any topological of topological homeomorphy)

space X, there is only a se

spaces Y with X ~

Y and ~ = Y (up to

since the cardlnallty

the cardinallty

of Y is bounded by

of the set of filters on X.

Thus from each class of morphisms with ~

~/ : S ---> T

= T such that for any two morphlsms

the class there is an isomorphism such that the diagram

~/,~/' of

~ : T ---> T'

-

11.5

-

S T .... > T' commutes,

we pick one representative.

Thus we obtain

the desired set ~ . Notation:

Let

be the category of semitopological semigroups; ~, the category of semitopological semi~roups wlth zero ~, the category of topological semigroups; C, the category of compact semitopological

~,

the category of topological

groups;

semigroups;

and,

61., the category of commutative C*-~l~ebrss with identity.

1.2 There is

s,

commutative diagram of covarlant functors L

c

all of which are coatiJoints of the respective inclusion functors;

except that for N we make the special

definition

that it is the coadJoint for the inclusio~

functor of the full subcategory of morphisms mapping the minimal

ideal into the minimal ideal

wise for the functor directly beneath

~

(and likein the

diagram ). Proof: With inclusion functors diagram

in all places,

the

114

-

commutes.

-

Since all inclusion functors are limit

preserving and by I.I the solution set condition for the coatiJoint existence

theorem holds in all places,

we have the result by the coadJoint existence (see Mitchell

1.3

tg], p.

The front adJunctions are denoted by

~,

124). of fl,

%,

Thus, for each $ 6 ~

theorem

~,

~, ~,

[", N , ~, and

E,

and J ~

~, resp.

, there is a commuting diagram of

morphisms ~S

fkCs ) >

Moreover,

if S & ~

> N(s)

ACs)

, then

~(S)

= S ~

and 9S is Just

the quotient map. Remark: Recall that the properties rive us information

of an adJoint situation

like the following,

for example:

If ~ : S ---> T is any morphism of semltopologlca semlgroups

and T is compact,

there is a unique

morphi~

4' :i~S ---> T such that $ = $'~S" Similar statements functors.

are true for all the other

-

1 1 5 -

Only the "Moreover"

Proof of 1.3:

The Rees quotient map : S ---> S / M ~

assertion

(Hofmann and Mostert

fsctors

a semigroup with zero.

through

~S since S ~

semigroups

group with zero mapping the minimal M(T) = {0~, then by the continuity So ~/ factors uniquely

N(s)

1.4

is

property

into a semi-

ideal M(S)

into

of 4, ~ ( ~ )

through S/M-T~V.

=

O

Thus S / ~

characterizing

~(~);

henc~

:

Let S ~ ~.

Then any continuous

unitary representation Conversely,

dimensional with IS" between

through ~S"

finite dimensional

w of [~(S) defines a continuous

unitary representation

representations

unitary finite

on S by composing

Thus there is a bijective

continuous

finite dimensional

correspondence unitary

of S and r(S).

Proof:

For any n, U(n)

group.

So any w 9 S ---> U(n)

factors

through ~S"

Definition:

finite dimensional

of S factors

any continuous

representation

1.5

[1], P. 25)

But if ~/ : S ---> T is a

morphism of semitopological

has the universal

requires proof:

(resp.,

O(n)) (resp.,

is a compact w : h---~O(n))

The other way is clear.

Define C : ~ - - - > 0 [

to be the contra-

variant functor given by C(S) : ~f : S - - - > ~

: f is bounded and~ , S ~ continuous J

and C(9)(f)

: fog, where r

and f ~ C(T).

(Note that the image of a C*-algebra under s *-morphisn

is

closed.)

C*-algebras

116-

Thus, we define the following commutative with identity for S c $:

W(S) = C(~s)(C(ftS))

c CCS),

ACs) = C ( ~ s ) ( C ( A S ) )

c_ C(S),

aCs) = C ( x s ) ( C ( ~ S ) ) ~ _ And,

C(S)-

somewhat differently, we define,

for S ~ ~,

N(S) = C(<~Sa)~)(C(N~IS) o) ~ C(S), where fa C(/q~IS)o if and only if f~ C(~QLS) and f(O) = O. Note:

Unlike the other C*-algebras,

N(S) does not have an

identity. 1.6

Definition:

Let S ~ %.

by RsX = xs, x a S. ~s(S) = C(Rs);

For s ~ S, define R s : S - - - >

Define ~S : S ---> L(C(S)) by

that is,

~sCs)Cf) = foRs = sf for f ~ c(s). Note that w S is a morphlsm of semigroups.

!.7

Let S & ~, and let s s S.

Then

(i) ~s(s)c(~s) : c(,~s)~ns(~s(S)); (II) ~s(s)c(~ s) = c ( ~ s ) ~ s ( % ( s ) ) ; (III) ~s(S)C(Is)

(iv) ~s(s)C(gs%) In particular,

=

C(~s)~rS(|S(S)); =

cCgs~s)~.nsC~s~s(s)).

W(S), A(S), G(S), and N(S) are

invsrlent under ~s(S). Proof:

We have the commutative

s __~s___> ~ls

s

....... > fLs.

diagram of morphisms

- 117

-

The adJoint diagram

c(s) <. c (~s) c(~) c(~s) ; ;c(~ cCs)

yields

< .......

(s))

c(~s)

(i) since C(R s) = TTS(B !) and C(RcoS(s)) =

~GS(~s(s)).

The proofs of (ii),

(ill), and

(iv) are

similar. 1.8

Let ~ : S ---> S' be a morphism in'~.

Then there are

i

unique morphlsms

i

~,'(,I,)

: w(s')

A(~)

:

--->

w s),

A(s') --->A~s),

G(%) : G(S') ---> G(S), and N(~)

:

NCs') --->N~Cs). i

Furthermore,

w:

~

A:

- - - > ~,

~--->,~,

G

: t

--->r

and N : $ - - - > i O t are contravarlant Proof:

functors.

We consider

are similar.

the first case only.

The others

To the commutln 6 diagram S - - - ~ - - > ClS

Sf

uS

>KIS'

we apply the functor C and obtain the commuting diagra

c(s) < c(~ s) c(ns) cC,)[ It(n(,)) cCs' )< c(~ s, ) c(ns' ). But thls means that

-

118

-

c(~)',#(s' ) = c(~)c(~) s, )c(a~') = c (~s)c (a(~))c (~s')

~_ c (~s)C ~ s , ) : w(s). Hence we define W(#) = C(9)iW(S' ). this definition 1.9

(i) ~ C

It is clear that

is functorial.

: CIC; i.e., w(s) : C(S) for compact semitopological semigroups ;

(ii) AI~a~: = CIC~%;

i.e., A(S) = C(S) for compact topological semigroups ;

(iii) G I ~ . ~ =

ClOne;

i.e., G(S) = C(S) for compact topological groups;

(Iv) For s~e,,~t, N(~) : C(~) o. Proof:

(i) If S r

then~S

on S; hence the assertion.

= S,and ~S is the identity The proofs of the others

are similar. I.I0 Lemma:

Let ~/ be the category of semitopological

semigroups with identity, and ~l, ~ l

the correspondin;

categories associated with ~ and g. (1) The inclusion functor

~--->~l

which associates with each S s ~

has a coatiJoint,

the semigroup

obtained by adjoining a discrete identity

and with

each morphism S ---> T in ~ the obvious extension S1 ___> T 1. Similar assertions are true for ~ and ~. adJunction

The front

kS : S ---> S 1 is Just the embedding.

- 1~9

(2) (i) cCs I) = C(S) @ (

-

and C ( ~ )

:

C(S I) ---> cCs) is

the projection; (il) W(S l) = W(S) @ (

and W(~ S) : W(S l) ---> W(S) i~

the projection; (iii) A(S l) = A(S) @ ~ and A(L S) : A(S l) ---> A(S) i the projection; (iv)

a(s I) = G(S);

(v) NCs I) = NCs) @ f a n d N ( ~ s )

:

N(S I) ---> NCs)

the projection. Proof:

(i) Straightforward.

(2) (i) Clear.

(il) w(s I) = C(~sl)C(~(sl)).

But by the following

diagram, il(ZI) = (D~)l:

>i1(s I)

~o, l ~ <

- ~? ( s l ) .

Now s ince S

S

>~S

LS! I a ....

commutes,

tr~S S l_>(~)l

W(~ S) = W~s)C(~Ds)W(~S1)

-1

.

But by (1),

C(~9~S) : C(~ISl) ---> C(~S) is the projection

in

C(~)Sl) = C(AS) @ E and W(m S), W(a~l) are isomorphisms. (iii) and (v) are proved similarly. (iv) G(S l) = G(~s1)C(['(S1)).

But C(S l) = ~(S), and a

similar computation as in (ii) finishes the proof.

2.

120

-

The Definition of Almost Periodic Functions

Notation:

Throughout

semigroup,

this section,

let S be a semitopological

and for s ~ S, let R s : S ---> S be the right

translation defined by Rsx = xs, x s S.

Denote

C(R s) : C(S) ---> C(S) by r s so that for f ~ C(S), s a S, (rsf)(x) = f(xs), x a S; we also use the notation sf = rsf. If S has an identity, as in i.i0.

let S I = S; otherwise,

SI = S ~I~,

Each f s C($) is canonically extended to

fl ~ C(S l) by fl(1) = 0 if 1 $ S.

Let ~ ~ L(C(S1)) denote

the semigroup ~r s : s ~ sll. I 2.1

S is an equicontinuous Proof:

~Irsll = sup

sfli :

= sup~sup 2.2

semigroup.

If(xs)!

fll~ I~ : x~S~

(i) There are closed S-invariant C*-algebras)

: sup If(y)l ~ I Y

subspaces

~L

(in fact,

W, A, G, and N of C(SI); and the closures

TW, TA, TG, and T N of SiW in Ww W, of SIA in A A, of WIG in G G, and of SIN in Nw N, resp., are compact semitopological

semigroups

(and, in fact, ~A and T G are

topological

semigroups).

The maps, defined by restrictions,

•W/I

TW

,.% TG

are morphisms of semitopological

semigroups.

iThe S I defined above is not the same as in i.i0.

-

(li)

1 2 1

-

W = {f 6 c(sl) 9 Sf is relatively compact in C(SI~ A = {f & C($ l) 9 Sf is relatively compact in C(S1)~ N is the maximal S-invsriant subspace in W(-L1)o = {f

~ C(S l) : 0 is a weak cluster point of Sf~.

Proof: Theorem II.3.23. 2.3

(i)

w(s 1)

c_w;

( i i ) A(S 1) c_ A; (iii) G(S I) c G; (iv) N(S I) c N. Proof:

(I)" Let f s W(S I).

Then there is a g 6 C(&ISI)

such that f = C(~aS1)g. i~ow slf = ~rsf : s s S l ~.

=

c(~s)C(~l)g

~

C(~s1)~C(Rt)g

: 8 ~. sl~ : t &~sl~,

(1.7).

But the last term is relatively compact in C(SI)w, since C(~SI) is an isomorphism and I.I.13 applies. Hence f s W. (ii): Similar with I.I.15 in place of I.i.13. (ill): G(S I) = C(~S I)C(PS), and (with closures taken in

e(sl) w) TwIG(S1 ) C_ (g G(SI)) - ~ (C(FsI)C(RrS))- by 1.7. But C(RrS) is a compact subgroup of L(C([~S)) in the strong and weak operator topologies weak case).

(see 1.14 for the

Hence, since C(~S I) is an isomorphism,

TWIG(S1 ) C C(~sI)C(qf, S) and this is a group.

Thus

TwIG(S I) is a compact semltopological subsemigroup of a group and is therefore a group.

By the definition

of G, then, G(S I) C G. (iv) N(S I) = C(~SI~sI)C(NSI)o,

with

c(~sl)o : ~f ~ c(gs I) : f(o) : o~.

For

122

-

all f ~ i~(S1), f = C(951~1__ )5 for some g &

But C(RO)g = 0 snd ~C([~sl~sl)g dense in C ( ~ B l ) g ; isomorphism, S of f.

: s ~. sl~ is weakly

hence, since C(gSI~s1)

is an S1

0 is a weak cluster point of

2.4 (i) We define morphlsms

C(NS I)o

of semitopological

f, whence

semigroups

~x : sl ---> Tx by 9X(S) :~x(rs), s ~ Sl, where X is W, A, G, or N. (ii) There exist unique

isomorphisms

r

:i~l

---> TW'

r

: AS1 ---> TA'

CG : CS1 --'> TG' and CN : ~S1 "--> TN such that the following TN

is a commutative

<-

diagram: ~SI

%

TA

TG

%A

.........

< . . . . . . . . . . . .CG .........

~sl

,S 1

where the maps of the left hand vertical sequence are restriction

maps.

D

"123

-

Proof:

-

(i) The continuity of sr~--~r s : S I ---> L(C(S1) w)

follows from separate continuity of multiplication

in

S 1 and from i.l.14. The continuity of sr~--~rslA : S 1 ---> L(A)follow~ from 3.12. (ii) The existence and uniqueness

of ~W, etc. are

obtained from the definition of ~S1 a n d e s l, etc. and 1.3). compact,

(1.2

Since the image of ~W is dense and ~ S 1 is the surJectivity of %, follows.

holds for all the other SX" inJectivity:

The same

Thus it remains to show

For that it is sufficient

to show the

existence of a right inverse for ~W' etc.,

i.e. a

morphism %~ : T W ---> ~'~51 such that ~W 0j' W = IT W' etc. For this purpose

it is sufficient

a morphism of C*-algebras that

~ w C ( ~ w ) = 1C(Tw ).

C(~W) is monic;

XW

to show that there is

: C (~S I) ---> C (T W) such

Since ~W has a dense image,

thus it suffices to show that

C ( ~ W ) k w C ( $ w ) = C((~W).

For f e, W, define ~W f ~. C(T W)

by (~wf)(t) = (tf)(1) for t ~ T W = SiW.

(Here is the

only place where we use the identity of T W.)

Now

~W : W ---> C(T W) is a morphism of C*-algebras. t = ~W(S), (C(Rs)f)(1)

s & S l, then

((~wf)(t) = (CPw(S)f)(1) =

= f(s); thus C(~W)~W = inclusion:W___>C(S ])

By 2.3, W(S I) = i m be the inclusion,

C(~SI) C W; let i : W(S I) ---> W then =

let

If

~ W = ~wiW(a~S1)"

C(%l);

Thus the requirement

The other cases are similar.

is satisfie~

-

2.5

12Zl-

-

(I) w = ',,(s1); (il) ~ = A(sl); (ili) G = G(S I);

(iv) ~ = N(S 1). Proof:

(i) By 2.3, W(SI)c_- W.

By the proof of 2.4,

W = C(~W)~w(W ) ~_ Im C(qW) = im C(#W~S1) (_ im C(~S1)

= w(sl). The other proofs are the same. 2.6

Let f ~ C(S I).

Then f ~ G if and only if f is the

uniform limit of linear combinations diminsional

representations

(f ~ C(S l) is a coefficient

of unitary flnit~

of S 1. of a unitary reprementation

of S 1 on a Hilbert space E if f(s) = <x, ~(s)y> for some x,y ~ E.) Proof: G = G(S l) = C(Is1)C(r'S). representative

functions

The rlng R(CS) of

(coefficients of finite

dimensional unitary representations)

of FS is dense

in C (rS) by a theorem of Peter and Weyl. dimensional unitary representations

The finite

of S 1 are exactly

the W ONsI with finite dimensional unitary representations ~ of r'S. coefficients

of the finite dimensional unitary

representations 2.7

Thus C(~sI)R(f'S) is the ring of

of S 1.

(i) f ~ W(S) if and only if fl s W(SI); (ii) f ~ A(S) if and only if fl a A(S 1); (ill) f ~ G(S) if and only if fl & G(SI); (Iv) f 6 N(S) if and only If fl ~ N(SI).

-

Proof: defined

(i):

125

The function

-

L : C(bl) ---> C(S) @ C

L(f) = (flS, f(1)) is an isomorphism by 1.10.

If f s C(S I), then

(slf) = ($1f IS, f(S 1)).

f is contlnuous, f(S l) is bounded. implies flS & W ( S )

Since

Thus f ~ W(S l)

and f ~ W(S) implies f l a

W(S1).

The other cases are similar.

2.8

Definition:

Let S be a semltopolo61cal semlgroup.

A

bounded continuous complex-valued function f on S is called (1) weakly almost periodic, (2) almost periodic, (3) strongly almost periodic, (4) alssipative if

2.9

If

(I)

f

a

w(s),

(2)

f

a

~(s),

(3)

f

~

GCs),

S is a compact semitopological

semigroup,

then all

continuous functions on S are weakly almost perlodic. If, in addition, S is topological, functions on S are almost periodic.

then all continuou~ If S is a compact

group, then all continuous functions on S are strongl) almost periodic.

If S is a compact semitopologlcal

semlgroup with zero, then all continuous functions on S which vanish at the zero are dissipative.

-

2.10 Proposition:

126

-

Let S be a semitopological

such that M(CIS) (resp., M(AS))

semigroup

is a group.

Then

~(s) = G(S) 9 N(S) (resp.,

A(s) = G(S) e ( N ( S ) n A ( S ) ) ) and the idempotent projection

e of M(~IS) (resp.,of M(AS))

onto G(S) with kernel N(S)

with kernel

is the

(resp., onto CK[

N(S)C%A(S)).

Proof: Theorem II.3.23.

2.11 Definition:

In view of the preceding results,

we cal~

the morphisms (i) (0S : S --->~qS, (2) ~'S

: S

--->AS,

(3) ~s : s - - - > r s , (4) 9 s ~ s : s - - - > ~ s the

(I) weak almost periodic

compactifioation _

(2) slmost

periodic

of S,

,

c.om~sctificatlon

(3) strong almost periodic (4) dissipatiye

..,.

of S,

compactification

compagtification

of S~

of S.

These terms are also used fer~IS, AS, rS, and NS,re~. Note:

If S has no identity, "~S --- TW\II~, AS

then T

, PS =

, NS ~- T

1 .

Also note that TW, etc. are obtained by first adding an identity to S then proceedin~ as indicated. If T~W is the closure in Ww W of ~rsiW : s & S~, then ~W may be a proper homomorphic image of Tw\~l~ as, for example,in the case when S Is any compact space endowed with the multiplication xy = 0 for all x,y where @ is some fixed element of S.

-

127

-

3. Invariant Means

3.1

Definition:

Let S be a s e m i t o p o l o g i c a l

A mes_~n for W(5)

is an element m

~W(~)'

= I and > 0 w h e n e v e r ~, L(~v(~)) let $' transformation.

: W(S)'

semigroup. such thst

f ~ O.

- - - > W(S)'

For any

be the sdJoint

We sey that a mean m is

(1) right subinvariant

if

<sf, m> = < for all s ~ S, f ~ W(S),

i.e.,

if

C(R s)'m < m for all s & S, (2) ri6ht

infrainvariant

if

~sf, m> = > for all s E S, f & W(S),

i.e.,

if

C(R s)'m > m for all s & S. Similar definitions

hold for A(S) and G(S).

is called right invariant invariant and right

3.2

Theorem:

if it is both right sub-

infrainvariant.

Let S be a s e m i t o p o l o g i c a l

a right subinvariant

A mean m

semigroup.

Then

mean or a right infrainvariant

mean exists (i) for W(S)

if and only if M(~S)

is a

left group (ii) for A(S)

if and only if M(AS)

is a

left group (iii) for G(S), Furthermore,

always.

all right subinvariant

means are right

invariant.

or infrainvariant

An invariant mean exists

-

128

-

(and is unique) (i) for W(S) if and only if M(~S) is a group (ii) A(S) if and only if M(~S) is a group (ill) for G(S), always. Proof:

(1) W(S) has a right subinvariant mean m a W(S)

if and only if C(GS) has one since

cC

)'

: w(s)'

--->c(ms)'

is an isomorphism and C(Rs)C(~)

= C(~sIC(a~(s))

implies

= and since

=CLS.

assertions (ii) and

By II.4.14 and 11.4.16, the

then follow.

(Ill) are similar to (i); in (ill) normed Hma~

integral on C(rS) always exists and is unique. Note:

It should be observed

m on W(S), for exsmple,

that a right invariant mean and a corresponding right

invarlant normed m e a s u r e ) a o n i q S f f d~=

.

are related by

-

3.3

Definition: let

129

-

Let S be a semitopologlcal

~ E ~(S).

semigroup,

and

For f,g E C(S), the convolution f*g of

f and g (with respect to

~) is defined to be a function

h ~ C(S) such that h d~ where ~

= f d X,

= d~.P,

9 = g d~,

if the convolutlon/~*J)

exists snd if such a function h exists.

3.4

Suppose W(S) admits a right Invarlant or a left invariant mean m.

Let f,g a W(S).

Then the convolution

f*g of f

and g with respect to m exists and is in W(S). if m is right invarlant,

Moreover,

then f.g is strongly almost

periodic. Proof: m.

Let ~

be the measure on M ( ~ S )

Let f = W ( ~ S )-If , g~ = W ( ~ S )-Is.

invariant,

corresponding

to

If m is right

then for F ~ C(~-~S), and for some e L E(M(~S)),

If FCxy) fCx) g(y) d/~(x) d2~(y) =

ff F(xeye) f(xe) g(ye) d/~(xe) d/~.(ye)

=

ff eyeF(xe) eyef(x(eye)-l)

=

ft F(xe) ~(x(eye) "I) g(ye) d/~(xe) d/~(ye)

=

f; F(x) ~(x(eye) -I) g(y) ~

:

; F(x)

g(ye) d ~ ( x e )

d~(ye)

(x) d/~(y)

h(x)

where h(x) = f ~(x(eye) -I) g(y) d/~(y).

Let

A

f.g = h = W(Cgs)h. The calculation

is similar if m is left-lnvariant.

If m is right invariant,

then eh = h, so h is strongly

almost periodic by 3.23(ili)(5).

-

130

-

4. Locally Compact Semitopological Semi~roups

Notation:

Throughout this section let S denote a locally

compact, non-compact,

semitopological semigroup.

Let Coo(S) and Co(S) be the functions in C(S) with compact support and the ones vanishing at infinity, respectively.

Let J~

: S --->~S(S)

corestriction of ~3 : S - - - > ~ S . and F = {f ~ C(~S)

: flJ = 0~.

be the

Define J = ~ S \ ~ s ( S ) Let

K : S ---> S u ~

be the embedding in the one-point compactification. 4.1

Lemma:

The following are equivalent statements:

(a) ~S is a homeomorphism; (b) ~S is injective and ~S(S) Proof:

is open i n O S .

(a)==>(b): Since ~ S is a homeomorphism, we h a w

that ~ S

is inJective and ~S(S) is locally compact.

But a locally compact subspace of a locally compact space is the intersection of an open set and a closed set.

Since ~s(S) is dense i n ~ ,

it follows that

~s(S) is open inl~S. (b)==>(a)~ Let(IS/J be the space obtained by collapsin~ the closed set J to a point.

Consider the diagram

of naturally defined functions:

inclu~~sion

(s) / quotient map

!

~nclusion

> s/J

-

Clearly~(S)

151

-

---> ( ~ I S / J ~ J ~

is a homeomorphlsm.

Since ~ is a continuous injection of a compact space onto a compact space, it is a homeomorphlsm.

But the~

~@~is a homeomorphism onto its image, viz.,

~ S / J )\~J!~

whence ~S is a homeomorphism. 4.2

Lemma:

if for every pair Sl, s 2 a S ,

s1%

s2, there i~

an f E. W(S) such that f(s l) ~ f(s2) , then 9S is inJective. Proof:

Suppose ~S(Sl) = ~s(S2) and f ~ W(S).

g = C(~S )'If"

Let

Then

fCs 1) = g(~S(Sl))

= g(~S(s2))

= f(s2);

hence s I = s 2 by hypothesis.

4.3

Lemma:

(i) c(~ s

)-l

(Co(S) 6w(s)) a F.

(ii) If ~S is a homeomorphism, = C(%)IF

: F --->

then Co(S)nw(s)

is an isomorphism of C*-algebras. Proof:

(i) Let f ~ Co(S)fBW(S).

that f : C ~ ) g . that

Let g a C(~IS) be suc

For t ~ J, take a net ~ on S such

lim ~S(~) = t.

Now s cannot have a cluster

point in S, whence 0 = lim f(s) = lira g(a~s(S)) = g(t). (ii) Since W(~ S) : C((IS) ---> W(S) is an isomorphism, ~is

inJective.

We must show that W(~s)FC_

Take g ~ F, and let f = C(~ S)g. with no cluster point. = O.

Let s be a net on S

We must show that

Let ~' be a subnet of ~ such that

exists.

Co(S).

lim f(s) llm f(~')

Let s" be a subnet of s' such that

lira ~oS(S'') = t exists.

Then t ~ ~s(S) since s' has

- 132 -

no cluster point in S and since (~S is inJective and ~s(S)

is open (4.1).

Hence

lim f(s') = lira f(s") = lim g(~S(S")) = g(t) = O. 4.4

Lemma: The following are equivalent statements: (a) ~S is a homeomorphism;

(b) Co(S)c Proof: C(~s)IF

w(s).

(a)==>(b): By ~.i, J is closed. : F--->

Co(S)~W(S)

By 4.3(ii),

is an isomorphism.

F separates the points of ~s(S)

Henc~

(i.e., if tl,t2602S(S )

t I ~ t2, then there is an f & F with f(t l) = 1 and f(t 2) = 0).

So Co(S)(%W(S ) separates the points of S;

hence Co(S)~%W(S)

= Co(S).

(b)==>(a): By 4.3(i), W(~S)-Ilco(s) maps Co(S) isomorphioally

onto a C*-subalgebra F' of F.

But

Co(S) separates the points of ~; hence F' sepsrstes the points of 62S(S). that F' = F.

This yields that aJs(S) is open and

Since Co(S) separates the points of S,

W(S) certainly satisfies the hypothesis of 4.2; hence a~S is inJective and the conclusion follows from 4.1.

4.5

Proposition:

If S is a locally compact, non-compact,

semitopological

semigroup,

then the following

statements are equivalent: (a) ~S is a homeomorphism the corestriction

(~S : S ---> ~ ( S )

of G)S : S ---> ~S);

(b) ~S is open and inJective; (c) a~ is inJective and ~s(S) is open in mS; (d) Co(S) C_ w(S).

is

-

Proof:

133

-

(s) <==> (c): 4.1

(b) :=> (c): Trivial. (a) and (c) ==> (b): Trivial.

(a) ~::> (d): 4.3. 4.6

Proposition:

If S is a locally compact, non-compact,

semitopological semigroup,

then the following

statements are equivalent: (a) Co(S) is invariant ~ d e r

right and left

translstions; (b) The one-point compactification S u ~ o f

S

is a semitopological semigroup; (b') If s ~ S and K ~ S is compact, then there is a compact set K' ~ S such that if x $ K', then sx, xs ~ K; (c) ~

: S---~

~(S),

: S --->~S,

the corestriction of is an isomorphism of

semitopological semigroups a n d ~ S \ ~ s ( S )

is

a closed ideal o f ~ S . In particular,

any of these equivalent statements

implies the equivalent statements in Proposition 4.5. Proof:

(b) <==> (b'): Clear.

(b) ==> (c): If S U ~

is a semitopological semigroup,

then the commutativity of the d i a ~ a m s .... ~ ..... > O s

yields that the diagram of the corestrictions

-

commutes.

But from

~-S is inJective (e) ==>

(b):

is a closed

-

the latter

diagram

and open and that

If ~

then S ~

we get that

K'-I(~)

is an Isomorohlsm

ideal,

a semlgroup,

154

= QS\~s(S).

and J = ~ S ~ ( S )

is isomorphio

and homeomorphic

toQS/J

a~

to it as a topological

space. (a) <==>

(b):

1.5), where

Co(S)

f s H iff f ( ~ )

topological, suppose

Identify

then

with H = C ( S v { ~ ) o (see = O.

If ~ V ~

H is S u { ~ - i n v a r i a n t .

H is invariant

(under right

Let s be a net on S such that llm ss = t s S u [ ~ .

is semiConversely,

translations).

lira s = ~ and

If f s H, by invariance,

C(R s)f ~ H, hence

o : lira C ( R s ) f ( s_) = lira f(s_s) : f(t). So all functions Hence

lim ~s = ~

Similarly,

using

of H vanish at t implying

that t =

for every net ~ with llm ~ = ~. invariance

under

left

translation,

lim ss = ~ for all such nets. (c) :=> 4.5(b) :

4.7

If S is a locally

Obvious. compact

group,

then

the conditions

of 4.5 and 4.6 are satisfied. Proof: group, Suppose

If S is a locally then

compact

it is a topological

group

(b') of 4.6 were violated.

be a compact

semitopological (I.3.1).

Then there would

set K g S and an element

s ~ S such that

- 155 for each compact

set E' ~ S, there would be an

element ~(K') ~ K with ~(K')s 6 K, say.

Then for

s subnet ~ of ~ we would have llm Zs = k & K; hence llm ~ = ks "I s Ks -I. But for sufficiently

large indices

i,

Z(i) r Ks-IV for any fixed compact neighborhood

4.8

If S satisfies

the equivalent

Co(S) Proof:

= Co(S)nw(s)

V of i.

conditions

of 4.6, then

~ N(S).

By 4.3(i) and 4.6,

(~u)'l(co(~))

c F

o~

- ~f ~ c(Qs)

: flJ =

c ~f ~ c(~s)

: flM(Qs)

= o~

= c (~s) (~s). 4.9

If S satisfies

the equivalent

if e is sn idempotent

conditions

in M(Tw),

of 4.6,

then

we have

ef = O, f ~. Co(S 1,. Proof: CIearly, Now

~:I satisfies

the conditions

of 4 6.

C(~oS1)-lf vanishes o~tside ~sl(S1), hence, in

particular,

on M(K~S1).

(e f) (x) but XCw-l(e) 6 M(s

4.10 Examples:

By 2.4, = (C(~sl)-lf)(X%w-l(e)__),

).

(I) Let S be any locally compact space

endowed with the multiplication distinguished

element

0 ~ S.

xy = 0 for some Then W(S) = C(S) and

- 136 -

W(S I) = C(SI). conditions

Hence Co(S) ~ W(S); so we have the

of 4.5 satisfied.

Co(S) $ so the conditions

However,

N(S) = c(s) o, of 4.6 cannot be fulfilled

in

view of 4.8. (2) Let S be the non-negative multiplication

reals with the "max"

(i.e., xy = max {x,y~).

as

=

Then

s~.

So

Co(S) ~ c(su~.~) o = c(nS)o = N(S). Now we investigate periodic the

the analogous

functions.

corestriction

4.11 Propo~itlo~ ~

for almost

Let ~ S : S ---> ~B(S) be of ~S

: S ---> AS.

if S,-~e s local ~

semitopological statements

situation

semigroup,

come,aCt, non-compact,

then the following

are equival@nt:

(a) JS is a homeomorphism; (b) ~S is open and inJective; (c) ~B is inJective

and ks(S)

is open in ~S;

(d) Co(S) c_ a(S). Moreover,

a necessary

condition

hold is that S be a topological Proof:

The equivalence

proof of 4.5.

of

that these statements semigroup.

(a) - (d) is similar to th~

The last assertion

and the fact that ~ S

follows

is topological.

from

(a)

-

4.12 Proposition:

(a)

-

If S is a locally compact,

semitopological statements

137

semigroup,

then the following

are equivalent: The one-point

compactification

is a topological (a')

non-compact,

S~of

S

semigroup with ~ as zero;

(i) For each compact set K ~ S there is a compact set K' ~ S such that x,y ~ K' implies xy,yx

~K;

and,

(ii) For each compact set K ~ S and each compact neighborhood

V of a point s,

there is a compact set K' ~

S such

that x ~ V, y ~ K ~ implies xy,yx ~ K; (b)

~S ~S

: S ---~ ~ ( S ) ,

: S ---~ #~S, is an isomorphism

topological a closed In particular,

4.13

semigroups

of

and I~S\~s(S) is

any of these equivalent

Again analogous

statements

statements

in Proposition

4.11

to that of 4.6.

If S is a locally compact group, conditions

of

ideal of RS.

implies the equivalent Proof:

~h6 corestriction

then none of the

of 4.11 or 4.12 is satisfied.

Proof: We need only show that the conditions

of 4.11

are not satisfied.

But that will follow immediately

from the subsequent

statement:

138

-

4.14 Proposition: non-compact,

Suppose

-

that S is a locally compact,

topologically

left simple

ideal is dense) semltopologlcal the following statements

(every left

semigroup.

Then

hold:

(i) For f 6 A(S), 0 ~ TAf only if f = O. (li) N ( S ) ~ A ( S )

= ~0~.

Co(S)6A(S) equivalent

(So, in particular,

= ~O% if S satisfies conditions

the

of 4.6.)

(iii) 8S is a left group. (iv) If S has an identity,

then BS ~ T A is a

group and G(S) = A(S). Proof:

(1) Let f & A(S) with 0 ~ TAf.

Then there is

a net s on S such that llm f(x~) = O uniformly

in x.

For each open set U q S, there is a net t on S with ts since,

~U,

for each index i, the left ideal Ss_(1) is

dense in S and therefore f ~ O.

intersects

U.

Now suppose

Then for some (. > O, the open set u :

is not empty.

t s

But then

cannot held uniformly

:

{f(x)l

If~s)l

in x.

>

c

> %, so llm f(xs_) = 0

This contradiction

shows

that f = O. (il) If f & N(S)/~A(S), (II.3.231(ii) from (i). (ill) II.3.6.

and

then 0 s Twf = TAf

(iv))). The assertion

then follows

- 139

-

(iv) If S has an identity, then T A ~ ~S (2.4) is also a left group; but a left group having an identity is a group.

By II.3.23,

G(S) = A(S).

(Note that if

S has no identity, then T A ~ ~S I ~ (AS) I = ASu~l~, so T A is the union of a left group as minimal ideal and an isolated identity.) 4.15 Suppose that for any compact subset K of S there is a closed ideal I such that (1) K ~ I

= ~, and

(2) S/I (the Rees quotient,

~l~, p. 25) is

a compact topological semigroup. Then Co(S) ~ A(S). Proof:

Let F I = ~f 6 C(S)

: fll = 0~.

: S ---> S/l is the quotient map. C(~)

If

The morphism

: C(S/I) ---> C(S) maps ~f ~ C(S/I)

Isomorphically onto F I.

Now A(S/I) = C(S/I) because

S/I is a compact topological semigroup FI C_ A(~)A(S/I) Coo(S) C ZI F I.

: f(1) = O~

(2.9); hence

CA(S).

Thus r FI c A(S). By (I) I Since A(S) is closed, then,

C o ( S ) c A(s).

Remark:

Proposition 4.14 did not need the hypothesis

thst S be locally compact, but its principal applications are in the realm of locally compact semigroups.

The following series of assertions,

however, are of interest outside the context of local compactness.

Note thmt 4.16 is an analogue

of 4.14(iv) and its corollary

(4.17) is an

-

140

-

Also, 4.19 is a generalization

analogue of II.3.6. of II.3.15. 4.16 Proposition:

Let T be a topologically

right) simple semltopological identity 1. Proof:

Then M(~.T)

left (resp.,

semigroup with

is a group.

We prove the left simple case, noting that

the other case follows from this one by considering the dual multiplication

x'y = yx.

(1) Every element in ~T(T)

is a left unit

(has a

left Inverse) : If s ~ T, then ideal of A T .

(~iT)~.~T(S) is a closed left

So

".~T(T) ,~ ( (~'~T) OaT(S 1) is a closed left ideal of OT(T);

hence

&'T(T) =~T(T)C%((~'IT)~T(S); i.e., ~ T ( T ) q so "O

T

(~'2T) O~T(S).

( ~ T ) ~JT(S)

=

a u ~ ~T

But ~aT(T) is dense in(IT;

Thus for each s ~ T, there is

such that u o.~(s) = i

(2) If f t W, then ~ ( T f ) Suppose n ~

s n is a sequence in T with

I!rsnfli --->

o

(where r s is as in III.2). such that Un(rsnlW)

condition f s W.

Then, with u n s T w "--'"lIT

= i,

!~fi~ = llUnrsn~l

whence f = O.

contains a constant.

ilUn ii iirSn fll < ilrsnf!l---> o;

Thus ~ = ~r tlW : t i T~ satisfies

(D) of 1.2.15, where K = co(Tf) for any

By 1.2.16, K cantains a fixed point of T;

141

-

-

that is, co(Tf) contains a constant function for every f ~- W. (3) M(~'~.T) is a right group. This follows immediately from II.3.23(iii)(5). (4) M(C2T)

is a group.

By II.3.5, there is a compact semitopological paragroup

(X,G,Y) O and a continuous function

f~ : C ~ T - - - >

(X,G,Y) O such thst

biJective morphlsm. so is (X,G,Y)o-.

~IM(~'IT)

is a

Since M(CIT) is a right group,

So by II.l.12, ~e' is a morphism.

Since T is topologically left simple, so is ~e' &~T(T) But then, since

(X,G,Y)~ is a compact topological

semlgroup, by II.3.6, Whence

(X,G,Y)~ is a left group.

(X,G,Y) o- is a group and so is M ( Q T ) .

4.17 Corollary:

Suppose that S is a compact semi-

topological semigroup and that T ~ o is a dense subsemigroup with identity.

Further suppose that

T is topologically left (resp.,right) simple.

Then

M(S) is a group. Proof: There is a morphism ~ :<.~T ---> S such that

'~/

T ~Inclusion

(IT ........ > S commutes.

Since T is dense in S, # is a surmorphism.

Thus %(M(C~T)) = M(S).

By 4.16, M(~'IT) is a group,

so its homomorphlc image M(S) is also a group.

4.18

142

Corollary:

If G 18 a subgroup of a compact semi-

topological

semlgroup S, then M(G--) Is a compact

topological

group.

Proof:

Immediate

4.19 Corollary:

from 4.17.

If S is a compact semltopologlcal

semlgroup wlth identity, consists Proof: 4.20

-

afflne

and If S = c--o T, where T

of left units of S, then S has a zero. 4.17 and II.3.17.

(i) If S is a semltopologlcal

semlgroup,

then the

action S x W(S) w ---> W(S) w defined by (s, f) ~--~ sf, where Sf(x) = f(xs),

Is separately

(II) If, In addition, then W(S) Proof:

S Is a locally compact group,

ls an S-module

(I.4.1).

(i) The action ~IS x C ( O S ) w ---> C(~-~S) w

Is separately ~S

continuous.

continuous

: S ---> O S

by I.l.14.

Is continuous

under W ( ~ S ) , the continuity

Since

and W(S) w ~ C(~-2S) w

of s ~--> sf follows.

The continuity of f '~-~ sf Is clear since the polntwlse

topology on W(S)

Is coarser than the weak

topology. Now (11) follows

from

(I) and 1.4.5.

4.21 Every weakly almost periodic

function on a locally

compact group is unlformly.continuous. Proof:

Immediate

from 4.20(II).

4.22 Let E be a Banach space, and let S be a bounded semlgroup of operators

on E.

Suppose

that Sxo is

-

relatively every

~

compact

143

-

in E w for some x o E E.

6 E', the function f(s)

Then for

f defined by

= <SXo,A

>

is in W(S). Proof:

Define a continuous

T : E--->

linear operator

C(S) by Tx(s) = <sx, )~>, s a S, x ~ E.

By the remark on pase 22, T 6 L(Ew,C(S)w). T : E w ---> C(S)w is continuous, compact

in C(S) w.

So, sinc~

T(Sx o) is relatively

But T(Sx o) = Sf since

TSXo(t) = = sf(t). 4.23 Corollary: continuous

Let ~ : S ---> L(Ew) be a bounded weakly representation

of a semitopological

semigroup S on a reflexive Banach space E. function

f defined by f(s)

A >,

=

for some x s E, Proof:

~ a E', is weakly almost periodic.

Since the unit ball in a reflexive Banach

space is relatively weakly compact, follows

the assertion

from 4.22.

4.24 Corollary:

Let S be a semltopological

and let f ~ W(S),

)~ d W(S)'.

defined by h(s) = <Sf, A > is weakly almost periodic. Proof:

Then the

4.20(i)

and 4.23.

semigroup,

Then the function h

-

4.25 Corollary:

144

-

Every positive definite function on a

topological group Is weakly almost perlodlc. Proof:

It is well-known that a positive definite

function Is a coefficient of a unitary representation of the group on some Hllbert space. example, Nalmark, M. A., Normed Rings translation), P. 393. )

(See, for (English

Gronlngen: P. Noordhoff N. V. 196~, )

The assertion then follows from 4.23.

-

145

-

REFERENCES i.

Hofmann, K. H., and Mostert, P. ~.: Elements of Compact Semlgroups, Columbus: Charles E. Merrill 1966.

2. Mitchell, B. 9 Theory of Categorles. Academlc Press 1965.

New York"

_

1LI-6

-

CHAPTER IV EXAMPLES 1. Locally Compact Groups 1.1

Let H be a locally compact, non-compact, group.

topological

Then the following hold:

(a) H O, the one-point compactiflcation

of H

with the point at infinity as a zero, is a compact semitopologlcal a topological

semlgroup,

but not

semigroup.

(b) H is isomorphic to corestriction of

~H(H) ~ ~'~H, under the

c~H : H - - - >

~H;

~H(H)

is the group of units of -(~H. (c) ~'iH\~aH(H) Is a closed maximal ideal in ~IH. (d) M(~iH)

is a compact topological

M(~-~H) is isomorphic to is isomorphic to

group, and

~H, which in turn

~H.

(e) A(H) = G(H). (f) W(H) and A(H) have unique invariant means. If m denotes the invariant mean on W(H), resp., A(H), and if f 5 W(H), resp., f & A(H), then there is a unique constant function in ~-~(Hf) whose value is ; if ~ = W ( ~ s ) - l f , resp., ~ = A(OLs)-lf, then I,

= f f(t) dt, where the integration normalized

is with respect to

Haar measure on the compact

-

~$7

-

topological group M(~XH), resp., M ( ~ H ) = ~ H . (g) Let ~

be the family of compact subsets of

H, and let m denote the invariant mean on W(H).

If f ~ W(H), then the following are

equivalent : (a) f ~ N(H);

(b) <~f~2, m> = O; (c) for every

s

> O and every K & ~ ,

there is an h ~ H such that :k~

K~.

(See ~lJ, II,4.9.) (h) NCH)-~ Co(H). (1) W(H) = A(H) 9 N(H). (J) WCH) D P(H) = {f & CCH)

: f is positive definite~.

(k) If f & W(H), then f is uniformly continuous. I.i.i Let H be a locally compact, non-compact, abelian topological group. mean of W(H) a n d / ~

~-compact,

If m is the invariant

is a left Haar measure on H,

then there is an increasing sequence U1C

U 2 C ... C U n ~

...

of relatively compact open sets such that = nlim - - > ~ / t~~ ( ~ ~ for

1.1.1.1

every

Let ~

f

~ wCH)

(cf.,

~4~,~p.

:~(~) d ~ , ( x ) 256).

be the additive group of real numbers.

Then we have the following: (a) the one-point compactlflcation ~ *

of 72

- 18-8 -

with the operation extended by X

+~

:

~ +

X

=

~

is a compact connected semltopological semigroup.

In particular,

it is a compact

semltopologlcal semigroup with a nonisolated zero (cf., [63, p. 69). 4

(b) The Cech cohomologj of ~ * by the minimal ideal M ~ * )

is not carried = ~(of.,

[5],

p. 52). (c) The unique invariant mean m on W(~) is given by =

lim I/T T-->~ _T f(x) dx,

where the integration is ordinary Ri.~mann integration. (d) In addition to the maximal ideal ~ ~ ( ~ ) and the minimal ideal M ( C I ~ ) , ~

contains

closed ideals~ ~

and

Furthermo re,

M(C2 ~) ~ IR ~ I~, and, of course, I R ~ in ~ D ~ .

IL is a closed ideal iJ;

(For the fact that M ~ " I ~ )

IR6~ IL, see 47].) I.i .I .i.I

Let ~ _ denote the additive group of integers.

Then we have the following:

- 449 (a)~

is a topological

Zwhlch

compactlflcation

of

is not totally disconnected.

(Note that

~ =

((~/~/~)~

totally dlsconne~ted torsionrgroup,

~

not

since~/~/is

and M ( i - / ~

not a

~ ~;~.)

(b) If e is the identity of M(~//__7) and : O~--->

M(~Z/)

Clifford-Miller

is the so-called

endomorphism

given by

~(t) = te = et = ere, then ker ~

where Z

= {t

: te = e~

=

(n'.

." n=1,2,3,...

ls identified with

~22.(~)

and the closures are taken in l-2~_~ ~. is the congruence

If R

relation

(called the kernel congruence of f ~ A(Z),

,

f = W(6o~)'if,

constant on all R-classes;

~), and

then ? is in particular,

lim Rn,.f = f. n-->6~ (See [I], II.4.7.) 2.

A Semitopologlcal

Group

2.1 Let R be algebraically

the additive group of real

numbers and give R the t o p o l o g y # follows: U 6 ~

defined as

if and only if there is a countable

set C:'(which may be empty) such that U ~ C is open in ~ ,

the group of real numbers with its usual

topology.

R has the following properties:

-

-

(a) R is a semltopologlcal a topological

semlgroup.

(b) R is not completely topological (c) Inversion

group which is not

regular

(whereas any

group is completely

regular).

in R is continuous.

(d) C(R) = C ( ~ ) ,

W(R) = W ~ ) ,

A(R) = A m ) .

(e) The only compact sets In R are finite sets. (f) R*, the one-polnt

compactlflcatlon

of R

with the operation extended by X

+~

=

L~ +

X

Is a quasi-compact

:

<~j

semltopologlcal

semigroup. (g) q i R = ~ , 3.

3.1

~R

= ~

= r~

=

rR.

Double Modules Let S be an addltively written necessarily

commutative)

multlpllcatlvely module

(though not

semigroup,

written group.

and H a

Call S an H-double

if there are functions (h,s) r'-'-* h.s

: H x S ---> S

and (s,h)n--~ s h : S x H ---> S such that h.(s + t) = h.s + h.t, (s + t) h = s h + t h,

(gh).s = g.(h.s), sg h = ( s g ) h

(g.s)h

= g" (sh).

-

If S is an H-double relative

151

-

module,

then S x H is a s e m i g r o u p

to the m u l t i p l i c a t i o n (s,g)(t,h)

If S has an identity

= (s h + g.t, 0 and

gh).

if

h.O = o h = o for all h & H and 1.s = s l

=s

for all s

S, then e = (0,i)

S x H.

in addition,

If,

H a semitopological sr--~

is an identity

S is a topological

semlgroup,

h.s

: S

--->

for semlgroup

and the functions

S

and sr~--> s h : S ---> S are continuous

for every

semitopologlcal

h s H, then S x H is a

semigroup.

3.1.1 Let H be the s e m l t o p o l o g i c a l and let S be the additive real numbers. s h = se -h.

Let

S = (L,v),

h.s =

semigroup

module,

so S x H

with identity

h) = (se -h + egt,

be a l a t t l c e - o r d e r e d

S ~ = (L~+o~,V),

be any automorphlsms o~(h)

H-double

of n o n - n e g a t i v e

to the m u l t i p l i c a t i o n

(L,+,v,~)

o~, ~

2.1,

S, let h.s = ehs and

Then S is an H-double

(s, g)(t,

3.2

of example

semigroup

For h s H, s %

is a s e m i t o p o l o g i c a l relative

group

+ s

modules.

and

g + h).

group.

and H = (L,+). of H.

Let Let

Under the a~tlons

sh = s + ~(h),

S and S ~ are

So S x H and S ~ x H are semigroups

-

152

-

relative to the multiplication (s,g)(t,h) = ((s + ~ ( h ) ) V In particular, and ~

( ~ (g) + t), g + h).

if oL is the identity automorph!sm

is the automorphlsm of period 2 given by

h ~

-h, then L x L and (L U ~ + ~ )

x L are semigroups

relative to the multiplication (s,

g)(t,

Moreover,

h) = ( ( s

- h)~(8

+ t),

8 + h).

there are subsemlgroups defined by

lhl _< and by -~(s,h) Finally, ordered

: S + h = 2q f o r

if

(L,+,V,A)

group,

topological

then

is

some q ~ .

a topological

L x L and

semigroups,

~here

lattice-

(LU~+~) the

+c~

x L are is

taken

to

be the point at infinity of the one-point compactlfication of the positive cone of (L,+,~/,A). 3.2.1 Let (L,+,V,A) be a locally compact lattlce-ordered topological group.

Let

T = t(s'h) 8 (L u ~ §

~ ~hi < s and x L : 48 + ~ = 2q f o r (some q 5 L u ~ + a o ~ w

~.

Under the multiplication (s,g)(t,h) = ((s - h)k/(g + t), g + h), T is a locally compact semltopological semlgroup with identity, and the following hold: (a) The one-polnt compactiflcatlon T O of T is a compact semltopologlcal semlgroup. (b) T has a minimal ideal M(T) = ~+oo~x L.

-

155

-

(c) W(L) ~ W(M(T))

= W(T)IM(T),

A(L) ~ A(M(T))

= A(T)IM(T),

G(L) ~ G(M(T)) = G(T)iM(T). (d)

~ T Is a compact topological fa~t:,: .u

=

group and, In

/~M(T) ~ A L .

(e) A(T) = G(T).

(see

II.2.9.)

3.2.1.1 Let (L,+,V,A) be either and let T be as In 3.2.1.

(~,+,kf, A) or (~,+,~/,A); Define S = T \ M ( T ) .

Then

we also have the following: (a) W(S) = W(T)IS and A(S) = A(T)~S;

hence,

W(S) "--"W(T) and A(S) ~ A(T). (b) A(S) Is canonically

isomorphic

(c) G(S) = A(S) and

=

~S

to A(L).

rs.

(d) [~,S = /4T. (e) W(S) has an lnvarlant mean, so w(s)

(f) S O

= Q(s)

e ~(s)

= A(s)

e N(S).

the one-polnt compactlfloatlon

of S wlth

the point at Inflnlty as a zero, Is a compact semltopologlcal

semlgroup whlch ls

O-slmple, but not completely O-simple. (g) S O contains a blcycllc semlgroup

(ln fact,

If L = Z/_, then S Is the blcycllc semlgroup)

(cf.,

p. 77).

(h) S O contains mag~nlfyln~ elements P. 77).

(cf., ~5~,

-

(i) S O violates

154

-

the swelling lemma of compact

topological semlgroup theory

([5~, P. 15).

(See [13, 1.3.10 and II.2.10.) .

4.1

Modules Let S be an additively written semlgroup and H a multipllcatively

written group.

If S is an H-module

under the action (s,h) ~ - *

s h .- S x H ---> S,

then H x S is a semigroup relative to the multiplication (g,s)(h,t) = (gh, sh + t). If S has an identity 0 and O h = 0 for every h / H and s I = s for every s ~ S, then e = (I,0) is an identity for H x S.

If S is a topological semigroup,

H is a semltopologlcal

semigroup,

and

s,~--> s h : S - - - > S is continuous,

then H x S is a semitopological

semigroup. 4.1.1 Let S be the additive semigroup of positive real numbers,

and let H be the multiplicative

positive real numbers.

group of

Under the usual topology,

or the discrete topology,

T = H x S is a locally

compact topological semigroup relative to the multiplication (g,s)(h,t) Furthermore,

= (gh, s h + t).

Green's relations ~

and

~

do not

agree in T (since T is simple and thus consists of

-

a single ~ - c l a s s , point

155

-

while every O~-class

is a single

(see, ~2~, pp. 50-51)).

4 1.2 Let T be the discrete semlgroup of 4.1.1 the one-point compactiflcation

Then T O

of T with the point

at infinity as a zero, is a compact semitopological semigroup for which Green's relations ~ not a~ree 5-

5.1

(cf.,

do

and

~5], P. 30),

Null. Semlgroxzps Let S be any locally compact space with the multiplication 0 ~ S.

xy = 0 for some distinguished

element

Then

(a) W(S) = C(S) and W(S l) = c(sl),

(b) Co( ) C._ w ( s ) , (c) coS : S ---> ~ S ~s(S)

is open and InJectlve,

so

is open in ~/S,

(d) Co(S) % N ( S ) , a n d

I~S\~)s(S)

is not an ideal

in i--2S. 6.

6.1

Semilattices Let S be the semigroup of non-negatlve

real numbers

(resp., InteEers) under the lattice operation V. Then the following statements

hold"

(a) S O , the one-point compactificatlon

of S with

the point at infinity as a zero, is a compact topological semigroup. (b) sO = A S

= KS

(c) G(S) = ~ ,

=

NS.

NCS) = Co(S), and

- 156

w(s)

.

The Minimal

-

= a(s)

= Co(S)

~ ~:.

Ideal in a Compact S e m i t o p o l o g i c a l

S e m i g r o u p Need Not Be Closed. 3.1

Let I = [O,lJ.

Take a function f : I x I - - - > I

w h i c h is separately

continuous,

on a dense subset of I x I. f as follows:

Let

Pi,

For each pi==

define

be a countable (qi,ri),

: I x I ---> I by

!(x- q l ) ( Y - r i ) l

gpl (x,y) =

(x-

if

(For instance,

i=1,2,3,..,

dense subset of I x I. define gPl

but discontinuous

qi )2 + ( y - rl)2

(x,y) ~ ~I, and gpi(Pi ) = O, f ( x , y ) = o~ z

Define f by

2_Igpl ( x , y ) . )

i=l Define a m u l t i p l i c a t i o n

(x, y, z ) ( x ' ,

on S = I x I x I by

y ' , z ' ) = (x, f ( x , z ' ) ,

With this m u l t i p l i c a t i o n a compact semitopological

~.(s) = E ( s ) = Thus, M(S)

z')

and the usual topology, semigroup.

But

~(x, f(x,z), z) : x,z ~ I~.

is not closed and the equivalent

of II.3.5(Iii)

S is

do not hold.

conditions

-

157

-

REFERENCES .

.

Berglund, J. F.: Various topics in the theory of compact semitopologlcal semigroups and weakly almost periodic functions. Dissertation. New Orleans: Tulane University 1967. Clifford, A. H., and Preston, G. B.: The Algebraic Theory of Semlgroups, Vol. 1. Mathematical Surveys, No. 7. Providence: Amer. Math. Soc. 1961.

3. d~eLeeuw K., and Glicksberg, I.: Applications of almost periodic compactifications. Acta Math. 105 (1961), 63-97. 4. Hewitt, E., and Ross, K. A.: Abstract Harmonic Analysis I. New York: Academic Press 1963. .

.

.

Hofmann, K. H., and Mostert, P. S.: Elements of Compact Semigroups. Columbus: Charles E. Merrill 1966. Paalman de - Miranda, A. B.: Topological Semlgroups. Amsterdam: Mathematisch Centrum 1964. Ramlrez, D. E.: Uniform approximation by FourierStielJes coefficients. Proc. Cambridge Philos. Soc. (to appear).

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158

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BIBLIOGRAPHY

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: ~l~ments de Math~matique. Livre Vi. integration. Chap. VI. Actualit~s Sci. et Ind. 1281. Paris: Hermann & Cie. 1959.

5. A. H. Clifford and G. B. Preston: The Algebraic Theory of Semigroups I. Mathemstical Surveys, No. 7. Providence: Amer. Math. Soc. 1961. 6. H. Cohen and H. S. Collins: Afflne semlgroups, Amer. Math. Soc. 93 (1959), 97-113.

Trans.

7. H. S. Collins: The kernel of a semigroup of measures. Duke Math. J. 28 (1961), 387-392. .

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Pac. J. Math.

9. K. de Leeuw and I. Glicksberg: Almost periodic functions on semigroups. Acta Math. l ~ (1961), 99-140. lO.

: Applications of almost periodic compactifications. Acta Math. 105 (1961), 63-98.

II.

: The decomposition of certain group representations. Journ. d'Analyse Math. ~5 (1965), 135-192.

12. J. Dixmier: Les C*-algebres et leurs representation. Paris: Gauthar-Villars, 1964. 13. N. Dunford and J. T. Schwartz: Linear Operators I. New York: Interscience, 1958. 14. W. F. Eberlein: Abstract ergodic theorems and weak almost periodic functions. Trans. Amer. Math. Soc. 6~ (1949), 217-240.

15.

: The point spectrum of weak almost periodic functions. Michigan Math. J. ~ (1955-56), 137-139.

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159

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16. R. Ellis: Locally compact transformation groups. Math. J. 24 (1957), 119-126.

Duke

17. I. Gllcksberg: Convolution semigroups of measures. Pac. J. Math. ~ (1959), 51-67. 18. U. Grenander: Probabilities on Algebraic Structures. New York: John Wiley & Sons, 1963. 19. A. Grothendieck: Crit~res de compacit~ dans les espaces fonctlonnels ggn~raux. Amer. J. Math. 74 (1952), 168-186. 20. P. R. Halmos: Measure Theory. Nostrand Co., 1950.

Princeton: D. Van

21. E. Hewitt and K. A. Ross: Abstract Harmonic Analysis I. New York: Academic Press, 1963. 22. K. H. Hofmann and P. S. Mostert: Elements of Compact Semigroups. Columbus: Charles E. Merrill, 1966. 23. J. L. Kelley: General Topology. Nostrand Co., 1955.

Princeton: D. Van

24. J. L. Kelley, I. Namioka, et al.: Linear Topological Spaces. Princeton: D'U.Van Nostrand Co., 1963. 25. E. S. LJapin: Semigroups. Providence: Amer. Math. Soc. Translations ~ (1963) (Russian ed., 1960).

26. B. Mitchell:

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New York: Academic

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33. C. Ryll-Nardzeweki: Generalized random ergodic theorems and weakly almost periodic functions. Bull. L'Acad. Pol. des Sci. I0 (1962), 271-275. 34. H. H. Schaefer: Topological Vector Spaces. Macmillan Co., 1966.

New York:

35. J. L. Taylor: The structure of convolution measure algebras. Dissertation, Louisiana State University, 1964. 36. A. Well: L'integration dane les Groupes Topologlques et see Applications. Paris: Hermann & Cie., 1940. 37. J. G. Wendel: Haar measure and the semigroup of measures on a compact group. Proc. Amer. Math. Soc. (1954), 923-928.

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