Character semigroups of locally compact inverse semigroups

Abstract

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