Algebraic structure of semigroup compactifications: Pym's and Veech's Theorems and strongly prime points

Abstract

The spectrum of an admissible subalgebra \(\mathscr{A}(G)\) of \(\mathscr{LUC}(G)\), the algebra of right uniformly continuous functions on a locally compact group \(G\), constitutes a semigroup compactification \(G^\mathscr{A}\) of \(G\). In this paper we analyze the algebraic behaviour of those points of \(G^\mathscr{A}\) that lie in the closure of \(\mathscr{A}(G)\)-sets, sets whose characteristic function can be approximated by functions in \(\mathscr{A}(G)\). This analysis provides a common ground for far reaching generalizations of Veech's property (the action of \(G\) on \(G^\mathscr{LUC}\) is free) and Pym's Local Structure Theorem. This approach is linked to the concept of translation-compact set, recently developed by the authors, and leads to characterizations of strongly prime points in \(G^\mathscr{A}\), points that do not belong to the closure of \(G^\ast G^\ast\), where \(G^\ast=G^\mathscr{A}\setminus G.\) All these results will be applied to show that, in many of the most important algebras, left invariant means of \(\mathscr{A}(G)\) (when such means are present) are supported in the closure of \(G^\ast G^\ast\).