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

Abstract

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

34 pages