Compact semitopological semigroups: An intrinsic theory
Compact semigroups arise in three different types: The first has a jointly continuous multiplication (x,y)\(\mapsto xy:\) \(S\times S\to S\); the second has continuous translation \(x\mapsto sx, xs:\) \(S\to S\) for all \(x\in S\), and the third has continuous translation on ONE side, say, \(x\mapsto sx:\) \(S\to S\) for all \(s\in S\). Semigroups of the first type are called topological, those of the second type semitopological, and with the third type, the terminological trouble starts. Some authors call these semigroups left topological when left translations are continuous; the present author would call these semigroup right topological, because the continuous argument is on the right side. Let us accede to the author and stick with his terminology throughout the review. Topological compact semigroups occur naturally as compact subsemigroups of the semigroup of all real matrices of degree n and as a great variety of compact semilattices such as they may arise from continuous lattices, and they occur less naturally as constructs in compact semigroup theory as a branch of compact topological algebra. While it would be exaggerated to call their theory classical, after its liftetime of over thirty years it is very well developed, and a number of books and monographs is devoted to it [\textit{A. B. Paalman-de Miranda}, Topological semigroups (1964; Zbl 0136.269); the reviewer and \textit{P. S. Mostert}, Elements of compact semigroups (1966; Zbl 0161.019); the reviewer, The duality of compact semigroups and \(C^*\)-bigebras (Lect. Notes Math. 129) (1970; Zbl 0211.433); \textit{J. H. Carruth, J. A. Hildebrant}, and \textit{R. J. Koch}, The theory of topological semigroups, Volumes I and II (1983; Zbl 0515.22003) and (1986; Zbl 0581.22001)]. This does not even count books in which a substantial portion is devoted to special compact topological semigroups such as compact semilattices [\textit{G. Gierz}, the reviewer, \textit{K. Keimel, J. D. Lawson, M. Mislove}, and \textit{D. S. Scott}, A compendium of continuous lattices (1980; Zbl 0452.06001)]. The theory of compact topological semigroups is a structure theory; very little harmonic analysis is on record in this area, the reviewer's monograph on the duality of compact semigroups being somewhat of an exception. In contrast with this situation, semitopological semigroups arise, as a rule, in the context of harmonic analysis. [\textit{J. F. Berglund} and the reviewer, Compact semitopological semigroups and weakly almost periodic functions (Lect. Notes Math. 42) (1967; Zbl 0155.187); \textit{R. B. Burckel}, Weakly almost periodic functions on semigroups (1970; Zbl 0192.486); \textit{C. F. Dunkl} and \textit{D. E. Ramirez}, Representations of commutative semitopological semigroups (Lect. Notes Math. 435) (1975; Zbl 0302.22001).] In more recent years, right topological semigroups began to attract attention, partly once more through harmonic analysis, where they arise as universal compactifications, partly through number theory in as much as the Stone- Čech compactification \(\beta\) \({\mathbb{N}}\) of the natural numbers supports the structure of a compact right topological semigroup. First texts have appeared on this aspect of compact semigroup theory. [\textit{J. F. Berglund, H. D. Junghenn}, and \textit{P. Milnes}, Compact right topological semigroups and generalizations of almost periodicity (Lect. Notes Math. 668) (1978; Zbl 0406.22005); a new project by these same authors along similar lines is under way.] However, until the appearance of the excellent monograph under review, there was no coherent treatment of a structure theory of compact semitopological semigroups (let alone right topological compact semigroups). The author paints a panorama of results on the general structure of compact semitopological and, as far as they reach, also of compact right topological semigroups. He demonstrates a variety and depth of mathematical tools which is remarkable; they include cohomology, homotopy, transformation groups, and a substantial measure of Lie group and manifold theory. The presentation is careful; the documentation of the background and history complete and extensive, the apparatus, including table of contents, a problem section, bibliographical references, and an alphabetical index is meticulously assembled. The user of the text will find a welcome touch here and there such as, for instance, at the end of each chapter, a compact summary of its essential results. The figures attest to the high standards of constructive geometry which the author, among other things, is in charge of instructing at his institution. Some typographical slips here and there remain but appear to stay on the level of typing errors in a photo reproduced typescript of otherwise very high standard. The book is a very welcome complement to the book literature on the subject of compact semigroups. It is certainly suitable as a text in graduate level seminars. The first chapter deals with what is traditionally known as the First Fundamental Theorem, namely the existence of idempotents and a completely simple minimal ideal in a compact semigroup under the most general circumstances, namely, even in the right topological case. The proofs have now received a final streamlined form which reveals the gist of the matter best if they have to get by with the weakest hypotheses. In this fashion it is precisely illustrated, how the successivley stronger hypotheses of separate continuity in BOTH variables and finally, joint continuity increase the information on the structure and embedding of the minimal ideals. A concerted effort to exhibit and construct as many examples as are necessary to make the point is noticeable here as well as in the whole book. The second chapter deals with a central theme of compact semitopological algebra, namely, the question: To which degree does separate continuity imply joint continuity? It provides a comprehensive overview of this topic ranging from the Bourbaki Lemma (the core Baire category argument) to the gradual elimination of metrizability and separability conditions through Lawson's transport arguments arriving, in the end, at Lawson's theorem that joint continuity is a consequence at all pairs (x,y) for which one of x and y at least is a unit. This subsumes the result of Ellis', important in transformation group theory, saying that a group on a locally compact space is already a topological group if its multiplication is continuous in each variable separately. Additional work also yields the remarkable fact (due to Lawson) that a semitopological compact semilattice is always topological. If the underlying space of a semitopological compact semigroup is very special, say is a one manifold or is a totally ordered compact space, then particular results become available on the theme ''separate continuity implies joint continuity'', as Berglund and the author have observed previously. The book is now an encyclopedia on those matters, too. The third chapter turns to one of the most pressing applications of the theory of compact semitopological semigroups which is the theory of compactifications of topological groups and semigroups via weakly almost periodic and almost periodic functions. In particular, every locally compact group and so notably every Lie group has a universal semitopological compactification, into which it is algebraically and topologically embedded. By the continuity results of the preceding chapter, this group acts as a transformation group by multiplication on either side on the compactification. The isotropy groups at points in the boundary play a crucial role. If G is a connected Lie group and H is a normal subgroup of G, and if S is a semitopological compactification of G, then \(sH=Hs\) for all s in S. Every left ideal of S is also a right ideal, and the idempotents of S are central and form a complete lattice with respect to their natural order. The left and right isotropy groups coincide. The weak almost periodic compactification of \({\mathbb{R}}\) is a prototype whose complex structure is still largely a mystery. One has known for some time through results of Brown and Moran that the lattice of idempotents in this semigroup is formidable. The author's representation tells us that these complications are an abelian phenomenon very much attached to such groups as \({\mathbb{R}}\) and \({\mathbb{Z}}\). Indeed, if G is a simple Lie group with finite center, then its weak almost periodic compactification is simply the one point compactification (Theorem of Chou, Veech). If, however, its center is infinite, then the very complicated weak almost periodic compactification of this center will show up in the weak almost periodic compactification of the whole group. The reader will find many worthwhile pieces of information on the 67 pages of this chapter alone. Of no lesser interest is the fourth and last chapter of the book which treats the topic of semitopological semigroups on compact manifolds. A compact topological semigroup is a Lie group as one has known for some thirty years after a result of Mostert and Shields. In view of this fact, the very rich theory of compact semigroups on manifolds is a proper domain of semitopological semigroups. In the case of topological semigroups one has learned to understand the deeper reasons behind the theorem of Mostert and Shields which are to be found in the idea that all units in a compact topological semigroup containing nonunits are located on the periphery. What peripherality is to mean is not a trivial matter, and several definitions have been tried. The most satisfying from an aesthetical point of view and the most successful in the case of compact topological semigroups is the cohomological one. However, the author, in analyzing the various proofs shows that the peripherality methods used by Mostert and the reviewer in their book of 1966 are the ones that are suitable for generalizations into the domain of semitopological semigroups and even, to some degree, of right topological semigroups. He shows that a semitopological semigroup has an open group of units if it has an identity with a Euclidean neighborhood and it can be embedded into a compact semitopological semigroup, or else if it is locally connected at the identity which is also assumed to be a local weak cut point. The one-dimensional prototypes of compact semitopological semigroups on manifolds are the circle group and the one-point compactification of \({\mathbb{R}}\). In the two-dimensional realm, the author has an ingeniuous construction showing abelian compact semigroups to exist on all compact connected orientable surfaces of genus k. These examples alone show the complexity which compact semitopological semigroups on manifolds are capable of exhibiting. The book contains a virtually complete structure theory of such semigroups, and the formulation of the summary theorem alone occupies more than a full page. The manifolds supporting these semigroups have to have even Euler-Poincaré characteristic (we assume that the semigroup is not itself a group), and the semigroup is a semidirect product of the closure of a subgroup homeomorphic to some \({\mathbb{R}}^ n\) and a compact subgroup; as a semilattice of groups it is stratified by the subgroups in a fashion that is rigorously controlled by the lattice of idempotents. The book concludes with an instructive Appendix on unsolved problems which gives a very good impression of the current status of the ''intrinsic'' theory of compact semitopological semigroups.
semitopological semigroups on compact manifolds
scientific article; zbMATH DE number 3979388