On structural methods and results in the theory of compact semitopological semigroups

Abstract

Since its very beginning, the theory of compact semitopological semigroups has always relied heavily on the use of functional analytic methods; many important results have been found and proved solely (or at least almost solely) with the help of auxiliary theorems from functional analysis This fact is not surprizing; after all, the main applications of semitopological semigroups lie in the theory of certain function spaces; functional analysis is a longestablished and well-developed theory which provides powerful tools for many branches of topological algebra. However it seems to be desirable also to have a structural approach to the theory of compact semitopological semigroups, an approach which emphasizes algebraico-topological methods and general results about the algebraic and/or topological structure of compact semitopological semigroups without immediate reference to functional analysis. Such methods and results could in turn be most useful for applications iX functional analysis. Furthermore, it is to be hoped that in this way some of the traditional proofs can be simplified or extended and that new properties of semigroups or topological groups can be exhibited.