Table Of ContentLecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
663
J. F. Berglund
H. D. Junghenn
p. Milnes
Compact Right Topological
Semigroups and Generalizations
of Almost Periodicity
Springer-Verlag
Berlin Heidelberg New York 1978
Authors
John F. Berglund
Virginia Commonwealth University
Richmond, Virginia 23284/USA
Hugo D. Junghenn
George Washington University
Washington, D.C. 20052/USA
Paul Milnes
The University of Western Ontario
London, Ontario
Canada N6A 5B9
AMS Subject Classifications (1970): 22A15, 22A20, 43A07, 43A60
ISBN 3-540-08919-5 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-08919-5 Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1978
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
2141/3140-543210
INTRODUCTION
The primary objective of this monograph is to present a
reasonably self-contained treatment of the theory of compact
right topological semigroups and, in particular, of semigroup
compactifications. By semi group compactification we mean a
compact right topological semigroup which contains a dense
continuous homomorphic image of a given semi topological semi
group. The classical example is the Bohr (or almost periodic)
compactification (a,AR) of the usual additive ~eal numbers R.
Here AR is a compact topological group and a: R + AR is a con
tinuous homomorphism with dense image. An important feature of
the Bohr compactification is the following universal mapping
property which it enjoys: given any compact topological group
G and any continuous homomorphism ~: R + G there exists a con
tinuous homomorphism ¢: AR + G such that ~ = ¢ 0 a. Such
universal mapping properties are central to the theory of
semigroup compactifications.
Compactifications of semigroups can be produced in a
variety of ways. One way is by the use of operator theory,
a technique employed by deLeeuw and Glicksberg in their now
classic 1961 paper on applications of almost periodic compacti
fications. In this setting, AR appears as the strong operator
closure of the group of all translation operators on the C*
algebra AP{R) of almost periodic functions on R. More
IV
generally, but using essentially the same ideas, deLeeuw and
Glicksberg were able to construct the almost periodic and
weakly almost periodic compactifications of any semi topological
semigroup with identity.
Another method of obtaining compactifications is based on
the Adjoint Functor Theorem of category theory. The first
systematic use of this technique appeared in the 19.67 monograph
of Berglund and Hofmann, where it was shown that any semitopo
logical semigroup, with or without identity, possesses both
almost periodic and weakly almost periodic compactifications.
One important advantage of the category theory approach is
that it provides a vantage point from which the fundamental
unity of the subject may be viewed. In addition, category
theory suggests other semigroup compactifications. An
appendix here shows how the Adjoint Functor Theorem can be
applied to produce a variety of semigroup compactifications.
A third method, and the one primarily used in this mono
graph, is based on the Gelfand-Naimark theory of commutative
C*-algebras. Compactifications of a semitopological semigroup
S now appear as the spectra of certain C*-algebras of functions
on S. (For example, AR is taken as the spectrum of AP(R).)
This method yields (perhaps somewhat more elegantly) compacti
fications which could also be produced using the operator
theoretic approach, and still allows the use of functional
analytic tools to facilitate their study. Furthermore it
suggests a parallel theory of affine compactifications and
v
provides a natural setting in which to study the interplay
between the two theories via measure theory.
The main part of Chapter I is devoted to constructing
compactifications (section 4). The necessary preliminary
information about means on function spaces, from which the
compactifications are constructed, is assembled in section 3.
Sections 1 and 2 contain the basic facts and definitions con
cerning semigroups, flows, and probability measures on compact
semi groups needed in later sections.
Chapter II is devoted primarily to structure theory. In
section 1 the relevant algebraic structure theory is developed.
The main result is the Rees-Suschkewitsch Theorem (Theorem 1.16) •
In the latter part of the section applications are made to
transformation semigroups. Section 2 contains the structure
theory of compact right topological semigroups. As might be
expected, the theory is more complicated than the corresponding
theory for compact semi topological (let alone topological)
semigroups. One complication is the fact that, in contrast
to the semi topological case, minimal right ideals and maximal
subgroups of the minimal ideal need not be closed. The struc
tures of compact right topological groups and of compact
affine right topological semi groups are treated in sections 3
and 4 respectively. The last section of Chapter II examines
the topologico-algebraic structure of the support of a mean
on an algebra of functions defined on a semigroup.
Chapter III is the heart of the monograph. Much of the
material presented in this chapter is new, beginning with the
VI
general theory of affine compactifications, the subject of
section 1. The parallel theory of non-affine compactifications
is treated in section 2. The emphasis of both of these sections
is on the universal mapping property that a compactification
enjoys (relative to the function space which defines the
compactification). In sections 3-13, eleven different kinds
of semigroup compactifications are constructed, including the
familiar almost periodic and weakly almost periodic compacti
fications. The relevant functional analytic properties of the
underlying function spaces are also examined. The universal
mapping property that distinguishes each compactification is
readily derived from the general theory developed in sections
1 and 2. General and specific inclusion relationships among
the function spaces are presented in section 14; they suggest
a dual theory of homomorphic image relationships among the
corresponding compactifications. Section 15 treats the follow
ing interesting question: when can a function with certain
properties on a subsemigroup S of a semigroup S' be extended
to a function with the same properties on S'? This problem
is essentially the same as the problem of determining when a
compactification of S is canonically contained as a closed
subsemigroup of the corresponding compactification of S'.
The final section of Chapter III uses the structure theory
developed in Chapter II to determine when a given C*-algebra
of functions on a semigroup is a direct sum of an ideal of
"flight functions" and a subalgebra of "reversible functions".
VII
Chapter IV characterizes the existence of left invariant
means on the function spaces of Chapter III in terms of the
existence of fixed points for various types of flows. The
presentation is in the spirit of the fixed point theorems of
Day (1961) and Mitchell (1970).
Chapter V is a collection of examples which illuminate
and test the sharpness of many of the results of previous
chapters. It is by no means complete, a fact which we hope
will inspire further research in the field.
The authors were influenced by many mathematicians before
and during the preparation of this monograph. We would like
to acknowledge our indebtedness, spiritual and otherwise,
particularly to M. M. Day, I. Glicksberg, K. deLeeuw, J. S.
pym, K. H. Hofmann, T. Mitchell, and J. W. Baker.
Thanks go to Mrs. wendy Waldie and Mrs. Barbara Smith
for their skilful preparation of the typescript.
The research of the last-named author was partially
supported by National Research Council of Canada grant A7857.
J. F. Berglund
H. D. Junghenn
P. Milnes
TABLE OF CONTENTS
CHAPTER I. PRELIMINARIES 1
l. Semigroups 1
2. Actions 8
3. Means 12
4. Semigroups of means 17
CHAPTER II. THE STRUCTURE OF COMPACT SEMI GROUPS 28
l. Algebra 28
2. Compact right topological semi groups 50
3. Compact right topological groups 61
4. Compact affine right topological semi groups 68
5. Support of means 79
CHAPTER III. SUBSPACES OF C(S) AND COMPACTIFICATIONS
OF S 91
l. General theory of affine compactifications 92
2. General theory of non-affine compactifications 98
3. The WLUC-affine compactification 101
4. The LMC-compactification 103
5. The LUC-compactification 104
6. The K-compactification 106
7. The CK-affine compactification 107
8. The WAP-compactification 107
9. The AP-compactification 114
10. The SAP-compactification 117
ll. The LWP-compactification 121
12. The KWP-compactification 121
l3. The CKWP-affine compactification 122
14. Inclusion relationships among the subspaces 123
15. Extension of functions 132
16. Direct sums of subspaces of C(S) 141
x
CHAPTER IV. FIXED POINTS AND LEFT INVARIANT MEANS
ON SUBSPACES OF C{S) 150
1. Fixed points of affine flows and left invariant
means 150
2. Fixed points of flows and multiplicative left
invariant means 161
CHAPTER V. EXAMPLES 166
1. Structure examples 166
2. Extension examples and examples to show the
subspaces can be different 175
APPENDIX A. AN APPROACH THROUGH CATEGORY THEORY 180
APPENDIX B. SYNOPSIS 222
NOTATION 229
INDEX 233
REFERENCES 239
CHAPTER I
PRELIMINARIES
1. SEMIGROUPS
1.1. Definition: A semigroup is a non-empty set S together
with an associative binary operation (s,t) + st: S x S + S,
called multiplication. S is commutative if st = ts, s, t E S.
If S is a semigroup, then for each t E S the maps
Pt : S + S, Pt(s) st
At: S S, At (s) ts
-?
are called, respectively, right and left mUltiplication
maps (by t). We define
L (S) = {At I t E s}, R(S) = {pt I t E S}.
As a consequence of the identities
the sets L(S) and R(S) are semigroups under composition of
mappings.
If s E S and A, B are subsets of S, we shall write
sA = A (A), As = P (A), AB = usB.
s s sEA
A non-empty subset T of a semigroup S is called a
subsemigroup of S if TT c T, a right ideal of S if TS c T,
or a left ideal of S if ST c T. A right (left) ideal of S
which properly contains no right (left) ideal is called a
minimal right (left) ideal.
An element s E S is called an idempotent if s2 = s, a
left (right) identity if st t (ts = t) for all t E S, or
a left (right) zero if st = s (ts = s) for all t E S. If
