Abstract

The spectrum of an admissible subalgebra \(\mathscr{A}(G)\) of \(LUC(G)(G)\), 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 \(G^{LUC(G)}\) 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 \(G^⁎G^⁎\), where \(G^⁎ = G^{\mathscr{A}}\setminus G\). 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 \(G^⁎G^⁎\).