##
**Analysis on semigroups: function spaces, compactifications, representations.**
*(English)*
Zbl 0727.22001

Canadian Mathematical Society Series of Monographs and Advanced Texts; A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. xiv, 334 p. £51.75 (1989).

Let me say straight away that, in my opinion, this book will become the standard text on the basic theory of semigroups with compact topologies. That claim is likely to evoke the question of why anyone should want such a text. I reckon that the theory satisfies both the criteria which make a subject of value: it is intrinsically interesting and it is useful. The first of these qualities will become apparent when the content of the present book is discussed. Let me briefly mention three examples of its utility. The reader who demands yet more evidence is referred to a recent collection of articles which survey the whole field, The analytical and topological theory of semigroups, edited by K. H. Hofmann, J. D. Lawson and the reviewer (de Gruyter, Berlin, 1990; Zbl 0702.00013).

(a) A fundamental situation in topological dynamics is of a group \(G\) acting on a compact space \(X\). The closure of \(G\) in the product space \(X^X\) (identified with all selfmaps of \(X\)) is a compact semigroup with multiplication generally continuous in one variable only. The algebraic structure of this semigroup – and especially of its minimal ideals – is one key to the analysis of the dynamics (see, for example, R. Ellis’s book “Lectures on topological dynamics.” New York: Benjamin (1969; Zbl 0193.51502)).

(b) If the additive group \(\mathbb{N}\) of positive integers is given the discrete topology, the Stone-Čech compactification \(\beta\mathbb{N}\) has a natural noncommutative semigroup operation extending \(+\). This compact semigroup has yielded results in combinatorial number theory which can at present be obtained in no other way, besides providing a very elegant proof of van der Waerden’s theorem (for the latter see “An algebraic proof of van der Waerden’s theorem”, V. Bergelson, H. Furstenberg, N. Hindman and Y. Katznelson [Enseign. Math., II. Sér. 35, 209–215 (1989; Zbl 0704.22004)]).

(c) Let \({\mathcal F}\) be a translation invariant space of bounded functions on a locally compact group \(G\). The closure in the weak operator topology of the image of \(G\) under its left regular representation on \({\mathcal F}\) is frequently a semigroup compactification of \(G\). By changing the space \({\mathcal F}\), a range of representations can be studied, and the structure of each compactification reflects properties of the corresponding representation.

The present book emphasizes the abstract theory rather than applications, though it contains a wealth of examples. In principle, it could be read by any student with a little knowledge of the theory of Banach spaces; there is a good appendix devoted to weak compactness, a crucial tool in the subject. The book is elegantly and economically written and altogether a pleasure to use. This contrasts with the earlier book on the topic by the same triple of authors [Compact right topological semigroups... Lect. Notes Math. 663 (1978; Zbl 0406.22005)] which was, till now, indispensible to experts but not exactly user-friendly. The present work is considerably more substantial and comprehensive than that volume whilst maintaining the general perspective appropriate for a text. This contrasts with W. Ruppert’s impressive “Compact semitopological semigroups” [Lect. Notes Math. 1078 (1984; Zbl 0606.22001)] which covers some of the same basic material but is essentially a research monograph.

The first chapter is devoted to the basic structure theory of compact semigroups S in which all the maps \(x\mapsto xy\) (y\(\in S)\) are continuous. These semigroups are called right topological here; the reviewer hopes that the appearance of this book will encourage mathematicians to follow this terminology (some currently use the description “left topological” instead!) and to use right topological semigroups where possible. Such an S has minimal left ideals L which are algebraically isomorphic with \(E\times G\), where E has multiplication \(e_ 1e_ 2=e_ 1\) and G is a group. It also has minimal right ideals R with a dual structure and each product LR is the unique minimal ideal, while RL is a group isomorphic with G. This algebraic structure, the reader will observe, comes from few hypotheses on S. The next step should be to describe the topology of L. Here, however, the general theory has loose ends: basic questions remain unanswered. Further assumptions, for example that S be also left topological (and so separately continuous), allow the conclusion that L is the direct product of compact topological semigroups E and G, so that L is in fact jointly continuous. The highly nontrivial fact used here is that sometimes separate continuity implies joint continuity. In a very welcome second appendix, this text provides a complete account of Christensen’s approach to such questions. Also in their first chapter the authors describe the connection between semigroups and topological dynamics. This allows them to include Namioka’s very elegant proof of the Ryll-Nardzewski Fixed-point theorem.

The next two chapters are devoted to seeing how compact right topological semigroups can arise. First comes a study of means on a Banach space \({\mathcal F}\) of bounded functions on a semigroup S. If \({\mathcal F}\) is closed and contains the constants, a mean \(\mu\) is a positive linear functional of norm 1. The set of means often forms a compact convex right topological semigroup in the weak topology. Invariant means on function spaces on groups, and in particular locally compact groups, are considered, and the algebraic properties of invariant and idempotent means are described. Of course, this subject is immense, and only a small part of it can be dealt with in one chapter.

A short third chapter discusses another, very general, way in which compact semigroups can be produced, namely by a universal mapping procedure. The approach adopted is by means of subdirect products, a most elegant tactic in this context, enabling a whole string of universal compactifications to be obtained at one stroke. Each such compactification of a semigroup S determines, and is determined by, a space of bounded functions on S. In Chapter 4, these function spaces are the subject of study. This part of the book serves as a dictionary of spaces important in the field, from the extensively studied almost periodic and weakly almost periodic functions to the less familiar almost automorphic and minimal functions, though the student will have to work through some tough examples to extract all the information the book offers. Although this chapter is very substantial, taking up almost a third of the work, it too cannot be exhaustive in such a massive field. The authors appropriately concentrate on properties relating to general semigroups but do of course mention important special cases such as groups.

The remaining two chapters relate to more specific questions. Chapter 5 is concerned with how various compactifications of semigroups relate to one another, for example when one is a subsemigroup of another, or when a semigroup is a product of two others. These problems are remarkably difficult and answers are given only for very special cases. The final chapter looks at weakly almost periodic semigroups of operators. These have natural separately continuous compactifications, and the detailed structure of the compactification gives properties of the action (for example, decomposability). These results are applied to give ergodic properties of group actions and to study Markov operators.

Other features of the book should be noticed. A third appendix gives a complete proof (following Namioka) of the existence of “left invariant” measures on certain compact right topological groups; the quotation marks are to warn that the invariance is in fact less than total. Also, throughout the book there are historical notes, showing how the subject developed and giving some problems to allow the reader to find a place in its future.

A reviewer should find some point of criticism to make the authors fell less than perfect. These gentlemen should have found place to say that their left norm continuous functions are exactly the left uniformly continuous functions familiar from the literature. Of course the definition makes this plain, but if they are attempting to change a terminology and notation now fairly standard (and I hope they succeed) they should at least argue the case. But even if they are not perfect, I reckon these authors may feel pretty pleased with themselves.

(a) A fundamental situation in topological dynamics is of a group \(G\) acting on a compact space \(X\). The closure of \(G\) in the product space \(X^X\) (identified with all selfmaps of \(X\)) is a compact semigroup with multiplication generally continuous in one variable only. The algebraic structure of this semigroup – and especially of its minimal ideals – is one key to the analysis of the dynamics (see, for example, R. Ellis’s book “Lectures on topological dynamics.” New York: Benjamin (1969; Zbl 0193.51502)).

(b) If the additive group \(\mathbb{N}\) of positive integers is given the discrete topology, the Stone-Čech compactification \(\beta\mathbb{N}\) has a natural noncommutative semigroup operation extending \(+\). This compact semigroup has yielded results in combinatorial number theory which can at present be obtained in no other way, besides providing a very elegant proof of van der Waerden’s theorem (for the latter see “An algebraic proof of van der Waerden’s theorem”, V. Bergelson, H. Furstenberg, N. Hindman and Y. Katznelson [Enseign. Math., II. Sér. 35, 209–215 (1989; Zbl 0704.22004)]).

(c) Let \({\mathcal F}\) be a translation invariant space of bounded functions on a locally compact group \(G\). The closure in the weak operator topology of the image of \(G\) under its left regular representation on \({\mathcal F}\) is frequently a semigroup compactification of \(G\). By changing the space \({\mathcal F}\), a range of representations can be studied, and the structure of each compactification reflects properties of the corresponding representation.

The present book emphasizes the abstract theory rather than applications, though it contains a wealth of examples. In principle, it could be read by any student with a little knowledge of the theory of Banach spaces; there is a good appendix devoted to weak compactness, a crucial tool in the subject. The book is elegantly and economically written and altogether a pleasure to use. This contrasts with the earlier book on the topic by the same triple of authors [Compact right topological semigroups... Lect. Notes Math. 663 (1978; Zbl 0406.22005)] which was, till now, indispensible to experts but not exactly user-friendly. The present work is considerably more substantial and comprehensive than that volume whilst maintaining the general perspective appropriate for a text. This contrasts with W. Ruppert’s impressive “Compact semitopological semigroups” [Lect. Notes Math. 1078 (1984; Zbl 0606.22001)] which covers some of the same basic material but is essentially a research monograph.

The first chapter is devoted to the basic structure theory of compact semigroups S in which all the maps \(x\mapsto xy\) (y\(\in S)\) are continuous. These semigroups are called right topological here; the reviewer hopes that the appearance of this book will encourage mathematicians to follow this terminology (some currently use the description “left topological” instead!) and to use right topological semigroups where possible. Such an S has minimal left ideals L which are algebraically isomorphic with \(E\times G\), where E has multiplication \(e_ 1e_ 2=e_ 1\) and G is a group. It also has minimal right ideals R with a dual structure and each product LR is the unique minimal ideal, while RL is a group isomorphic with G. This algebraic structure, the reader will observe, comes from few hypotheses on S. The next step should be to describe the topology of L. Here, however, the general theory has loose ends: basic questions remain unanswered. Further assumptions, for example that S be also left topological (and so separately continuous), allow the conclusion that L is the direct product of compact topological semigroups E and G, so that L is in fact jointly continuous. The highly nontrivial fact used here is that sometimes separate continuity implies joint continuity. In a very welcome second appendix, this text provides a complete account of Christensen’s approach to such questions. Also in their first chapter the authors describe the connection between semigroups and topological dynamics. This allows them to include Namioka’s very elegant proof of the Ryll-Nardzewski Fixed-point theorem.

The next two chapters are devoted to seeing how compact right topological semigroups can arise. First comes a study of means on a Banach space \({\mathcal F}\) of bounded functions on a semigroup S. If \({\mathcal F}\) is closed and contains the constants, a mean \(\mu\) is a positive linear functional of norm 1. The set of means often forms a compact convex right topological semigroup in the weak topology. Invariant means on function spaces on groups, and in particular locally compact groups, are considered, and the algebraic properties of invariant and idempotent means are described. Of course, this subject is immense, and only a small part of it can be dealt with in one chapter.

A short third chapter discusses another, very general, way in which compact semigroups can be produced, namely by a universal mapping procedure. The approach adopted is by means of subdirect products, a most elegant tactic in this context, enabling a whole string of universal compactifications to be obtained at one stroke. Each such compactification of a semigroup S determines, and is determined by, a space of bounded functions on S. In Chapter 4, these function spaces are the subject of study. This part of the book serves as a dictionary of spaces important in the field, from the extensively studied almost periodic and weakly almost periodic functions to the less familiar almost automorphic and minimal functions, though the student will have to work through some tough examples to extract all the information the book offers. Although this chapter is very substantial, taking up almost a third of the work, it too cannot be exhaustive in such a massive field. The authors appropriately concentrate on properties relating to general semigroups but do of course mention important special cases such as groups.

The remaining two chapters relate to more specific questions. Chapter 5 is concerned with how various compactifications of semigroups relate to one another, for example when one is a subsemigroup of another, or when a semigroup is a product of two others. These problems are remarkably difficult and answers are given only for very special cases. The final chapter looks at weakly almost periodic semigroups of operators. These have natural separately continuous compactifications, and the detailed structure of the compactification gives properties of the action (for example, decomposability). These results are applied to give ergodic properties of group actions and to study Markov operators.

Other features of the book should be noticed. A third appendix gives a complete proof (following Namioka) of the existence of “left invariant” measures on certain compact right topological groups; the quotation marks are to warn that the invariance is in fact less than total. Also, throughout the book there are historical notes, showing how the subject developed and giving some problems to allow the reader to find a place in its future.

A reviewer should find some point of criticism to make the authors fell less than perfect. These gentlemen should have found place to say that their left norm continuous functions are exactly the left uniformly continuous functions familiar from the literature. Of course the definition makes this plain, but if they are attempting to change a terminology and notation now fairly standard (and I hope they succeed) they should at least argue the case. But even if they are not perfect, I reckon these authors may feel pretty pleased with themselves.

Reviewer: John S. Pym (Sheffield)

### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

43A07 | Means on groups, semigroups, etc.; amenable groups |

22A15 | Structure of topological semigroups |

22A20 | Analysis on topological semigroups |

43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |

43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |

54H20 | Topological dynamics (MSC2010) |

54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |

47D03 | Groups and semigroups of linear operators |