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

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 $G^\ast G^\ast$.