Almost periodic functions on semidirect products of transformation semigroups

Abstract

The notion of semidirect product of two transformation semigroups is introduced, and its space of almost periodic functions is expressed as a tensor product. The general techniques developed are applied to the special case of a semidirect product S © T of two semigroups. As a consequence new results are obtained on the characterization of the almost periodic compactification of S © Γ as a semidirect product of compact semigroups. A related result is the splitting of the enveloping semigroup of a semidirect product of certain flows into a semidirect product of enveloping semigroups. 0. Introduction. Let S and T be semitopological semigroups and S (?) Γ a semidirect product of S and T. In an earlier paper [10] we showed that, under certain conditions, the almost periodic (a.p.) compactification (S © T) r of S © T is a semidirect product of the a.p. compactification of T and a certain compact topological semigroup containing a dense homomorphic image of S. A simple corollary of this result is that the space of a.p. functions on S © T is a tensor product of the space of a.p. functions on T and a subspace of a.p. functions on S. In this paper we introduce the notion of semidirect product of transformation semigroups and determine exactly when its space of a.p. functions may be expressed as a tensor product in analogy with the semigroup case described above. Cast in this general setting the problem of characterizin g the space of a.p. functions on a semidirect product of semigroups becomes clear, and the techniques developed lead to elegant necessary and sufficient conditions for (S © TY to be a semidirect product. As a consequence we are able to show that (S © T)' is a semidirect product for all semitopological semigroups S with identity and all semitopological groups T, thus generalizing results of [10, 11, 12]. The same conclusion holds if T merely contains a dense subgroup. In a similar vein, but using different techniques, we show that in a wide variety of cases the enveloping semigroup of the semidirect product of two equicontinuous flows is (canonically isomorphic to) a semi- direct product of the original enveloping semigroups.