Right topological groups, distal flows, and a fixed-point theorem

Abstract

Introduction. It is well known that the study of minimal equicontinuous transformation groups on compact Hausdorff spaces reduces to the study of compact Hausdorff topological groups and their quotients. In [3], Ellis has shown that the study of minimal distal transformation groups on compact Hausdorff spaces, which include the equicontinuous ones, reduces to the study of certain compact "right topological" groups and their quotients. A right topological group is a group provided with a topology such that x ~ x y is continuous for each y. This result of Ellis was exploited by Furstenberg in his striking and profound analysis of distal transformation groups or distal flows [7]. In the present paper, we make right topological groups the primary objects of study. This study is facilitated by the observation that each right topological group can be given a second topology (called the o-topology here) with pleasant properties. This topology is stronger than Furstenberg's "F-topology" when they can be compared, and they have many desirable features in common. Yet the definition of the o-topology is simpler. (At this point the reader may wish to peek at the definition of the g-topology in Section 1.) The treatment of the o-topology in Section 1 is more general than needed in the subsequent sections, but, at any rate, the entire Section 1 is quite elementary. The central issue of Section 2 is whether a compact Hausdorff r igh t topological group admits a quotient which is a non-trivial compact Hausdorff topological group. The positive answer requires some sort of metrizability condition (cf. Theorem 2.2). This is not surprising because each non-trivial compact Hausdorff topological group admits a non-trivial finite-dimensional representation. The proof of the main theorem (Theorem 2.2) uses a category argument which goes directly back to Furstenberg [7]. As an application, it is proved that a compact Hausdorff right topological group such that x ~ z x is continuous for each z belonging to a countable dense subset is a transfinite extension of a point by compact Hausdorff topological groups (Corollary 2.1).