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

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