Compact Semitopological Semigroups and Weakly Almost Periodic Functions

165 Pages·1967·4.76 MB·English

by  J. F. Berglund

Lecture Notes ni Mathematics
A collection of informal reports dna seminars
Edited yb .A Dold, Heidelberg dna .B Eckmann, hcirJ~Z
24
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J. .F Berglund
.K .H nnamfoH
Tulane University, New Orleans
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.galreV-regnirpS Berlin. Heidelberg-New kroY
This work was supported in part by NSF Grant GP .9126 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 CONTENTS
INTRODUCTION ............................................ 1
CHAPTER I. PRELIMINARIES
. Compactness Criteria .............................. 12
Theorem 1.8 ..................................... 16
Equivalent conditions for compactness
in function spaces.
. Equlcontlnuous Semigroups of Linear Operators
and Afflne Transformations. Affine Semlgroups .... 21
Theorem 2.9 ..................................... 26
The almost periodic subspace.
Theorem 2.10 .................................... 27
The weakly almost periodic subspace.
Proposition 2.13 ................................ 30
Kakutanl fixed point theorem.
Theorem 2.16 .................................... 34
Ryll-Nardzewskl fixed point theorem.
3. Ellis' Theorem .................................... 36
. Actions of Compact Groups on Topological
Vector Spaces ..................................... 37
Proposition 4.5 ................................. 41
A Banach weak G-module is a strong G-module
if G is a locally compact group.
CHAPTER If. COMPACT SEMITOPOLOGICAL S~4IGROUPS
1. Algebraic Background Material ..................... 44
Proposition 1.9 ................................. 47
The Rees Theorem.
Proposition 1.23 ................................ 57
The group supporting subspace.
Proposition 1.26 ................................ 59
The semigroup with zero supporting subspace.
2. Locally Compact Paragroups ........................ 60
Proposition 2.4 ................................. 61
The structure of a minimal ideal in a
locally compact semitopologlcal s@migroup.
3. Compact Semitopological Semi~roups ................ 65
Theorem 3.5 .................................... 67
The first fundamental theorem of compact
semitopological semlgroups.
Proposition 3.12 ................................ 71
The strongly almost periodic subspace.
Theorem 3.23 .................................... 80
The main theorem on semigroups of operators
on a Banach space.
. Invariant Measures on Locally Compact Semlgroups .. 88
Theorem 4.14 .................................... 97
Necessary and sufficient conditions for
invariance of a measure.
CHAPTER III. AI/~OST PERIODIC AND WEAKLY AI~OST PERIODIC
FUNCTIONS ON SEMITOPOLOGICAL SEMIGROUPS
I. Various Universal Functors ....................... 112
. The Definition of Almost Periodic Functions ...... 120
Proposition 2.10 ............................... 126
Decomposition of weakly almost periodic
functions.
3. Invariant Means .................................. 127
Theorem 3.2 .................................... 127
4. Locally Compact Semitopological Semigroups ....... 130
Proposition 4.5 ................................ 132
Necessary and sufficient conditions for the
embedding into the weakly almost periodic
compactlficatlon to be topological.
Proposition 4.6 ................................ 133
Equivalent conditions on the embedding of
S into GAS.
Proposition 4.11 ............................... 136
Analogue of 4.5 for the almost periodic
compactlflcation.
Proposition 4.12 ............................... 137
Analogue of 4.6 for AS.
Proposition 4.14 ............................... 138
The almost periodic compactlfication of
a topologically left simple semigroup.
Proposition 4.16 ............................... 140
Partial analogue of 4.14 for the weakly
almost periodic compactlficatlon.
CHAPTER IV. EXAMPLES ................................. 146
BIBLIOGRAPHY ......................................... 158
- 1 -
INTRODUCTION
After a lifetime of about fifteen years the theory
of compact topological semigroups has come of age. By
definition a compact topological semigrou p S is a compact
Hausdorff space with a continuous associative
multiplication (x,y) ~--~r xy : S x S ---> S. A
considerable body of information about the structure of
general compact topological semigroups and of semigroups
from special subclasses is now available, and is assembled
and accessible in books and monographs .K( ~ H. Hofmann and
P. S. Mostert, Elements of Compact Semlgroups, Charles E.
Merrill, Columbus 1966 and A. B. Paalman - de Miranda,
Topological Semigroups, Mathematical Centre Tracts ll,
Math. Centrum, Amsterdam 1964). The possible applications
of the theory of compact topological semigroups have not
yet been fully exploited perhaps. One might expect that
certain branches of functional analysis would be especially
susceptible to such applications. And indeed, many
questions which do arise (for instance, in the study of
almost periodic functions or in the characterization of
the measure algebra of a locally compact abelian group)
lead automatically to considerations involving compact
semigroups. However, these applications call for a wider
class of compact semlgroups, namely, semigroups S which
are (compact) Hausdorff spaces with the multiplication
in S being continuous in each variable separately. Such
semigroups will be called semitopolo~ical semigroups,
- 2 -
even though some authors have called them topological
semigroups. The distinction is necessary if both types
are discussed concurrently.
No coherent structure theory of compact a~mi-
topological 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 semigroups (Dissertatlon, Loulslana
State University, 1964). 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 semigroups in their own
right. 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 semi-
topological semigroups. Even very simple constructions
produce counterexamples to most assertions about compact
semitopological semigroups which are modeled after the
analogous assertions for compact topological semigroups.
One of the most elementary constructions is the one-point
compactification ~ ~ o f 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 at all of the
topological theory which remain valid under the
weaker hypotheses. At the present time there seems
to be only one substantial fact which can be partially
salvaged from the theory of compact topological semigroups,
namely, 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 semigroup S. But even in this
instance, there are serious deficiencies in the semi-
topological case. Comparatively easy examples show that
M(S) need not be closed in a compact semitopological
semlgroup S and, therefore, need not be compact. More
drastically, nothing remains of the familiar phenomenon
in a compact topological semigroup S that all of the
~ech cohomology of S is carried by the minimal ideal M(S)
(or, even, in the presence of an identity, by any maximal
group in the minimal ideal). The primitive counterexample
~ U ~ shows how completely this may fall to remain
true. Practically nothing is left of the special
properties of Green's relations which yielded an early
and often very useful structural decomposition of a
compact topological semigroup. A coset relative to any
of the five relations considered 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 ~ and
no longer agree in general. 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 sa( 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 semi-
topological 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 functions in
the spirit of Glicksberg and de Leeuw. 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. Unlike
the topological theory, the semitopological theory seems
to lean strongly towards functional analysis. One of
the best results concerning Green's relations in a
compact semitopological semlgroup is the observation
that the minimal ideal M(H-) of the closure of sny
~aximal group H~i~ a compact topological group. This
theorem is a consequence of Ryll-Nardzewski's fixed point
theorem. A purely semigroup theoretical proof, although