Character Semigroups of Locally Compact Inverse Semigroups

Abstract

We show that if $S$ is a locally compact abelian continuous-inverse semigroup whose idempotent semigroup $E$ satisfies a certain technical condition on its prime ideals, then the maximal subgroups of the character semigroup $S^\wedge$ of $S$ are obtained as inverse limits of the duals of the maximal subgroups of $S$. It is shown that the technical conditions on $E$ are satisfied in each of the following cases: $E$ is compact, $E$ is totally disconnected, or $E$ is a chain. We then obtain necessary and sufficient conditions in order that a given inverse system of compact groups indexed by a totally disconnected semilattice $E$ admit a compatible compact semigroup topology on their disjoint union.