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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
โข 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 โข 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
/ โขunltCpoint 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 โขi' 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 ~(v)/~w)
=
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 ยง (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,
โขW/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 ~ ยง
~ ~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 .
.
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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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Trans.
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Pac. J. Math.
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: 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.
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15.
: The point spectrum of weak almost periodic functions. Michigan Math. J. ~ (1955-56), 137-139.
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16. R. Ellis: Locally compact transformation groups. Math. J. 24 (1957), 119-126.
Duke
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