Lecture Notes in Mathematics

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1 Lecture 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

2 Authors John F. Berglund Virginia Commonwealth University Richmond, Virginia 23284/USA Hugo D. Junghenn George Washington University Washington, D.C /USA Paul Milnes The University of Western Ontario London, Ontario Canada N6A 5B9 AMS Subject Classifications (1970): 22A15, 22A20, 43A07, 43A60 ISBN Springer-Verlag Berlin Heidelberg New York ISBN Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer-Verlag Berlin Heidelberg 1978 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/

3 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 semigroup. The classical example is the Bohr (or almost periodic) compactification (a,ar) of the usual additive ~ e anumbers l R. Here AR is a compact topological group and a: R + AR is a continuous 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 continuous 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 compactifications. 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

4 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 monograph of Berglund and Hofmann, where it was shown that any semitopological 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 monograph, 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) compactifications which could also be produced using the operatortheoretic approach, and still allows the use of functional analytic tools to facilitate their study. Furthermore it suggests a parallel theory of affine compactifications and

5 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 concerning 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 structures 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

6 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 compactifications. 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 following 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".

7 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

8 TABLE OF CONTENTS CHAPTER I. PRELIMINARIES l. Semigroups 2. Actions 3. Means 4. Semigroups of means CHAPTER II. THE STRUCTURE OF COMPACT SEMI GROUPS l. Algebra 2. Compact right topological semi groups 3. Compact right topological groups 4. Compact affine right topological semi groups 5. Support of means CHAPTER III. SUBSPACES OF C(S) AND COMPACTIFICATIONS OF S 91 l. General theory of affine compactifications General theory of non-affine compactifications The WLUC-affine compactification The LMC-compactification The LUC-compactification The K-compactification The CK-affine compactification The WAP-compactification The AP-compactification The SAP-compactification 117 ll. The LWP-compactification The KWP-compactification 121 l3. The CKWP-affine compactification Inclusion relationships among the subspaces Extension of functions Direct sums of subspaces of C(S) 141

9 x CHAPTER IV. FIXED POINTS AND LEFT INVARIANT MEANS ON SUBSPACES OF C{S) 1. Fixed points of affine flows and left invariant means 2. Fixed points of flows and multiplicative left invariant means CHAPTER V. EXAMPLES 1. Structure examples 2. Extension examples and examples to show the subspaces can be different APPENDIX A. APPENDIX B. NOTATION INDEX REFERENCES AN APPROACH THROUGH CATEGORY THEORY SYNOPSIS