Primitive idempotent measures on compact semigroups

Abstract

Let S be a compact semigroup (with jointly continuous multiplication) and let P(S) denote the probability measures on S, i.e. the positive regular Borel measures on S with total mass one. Then P(S) is a compact semigroup with convolution multiplication and the weak* topology. Let II(P(S)) denote the set of primitive (or minimal) idempotents in P(S). Collins (2) and Pym (5) respectively have given complete descriptions of II(P(S)) when S is a group and when K(S), the kernel of S, is not a group. Choy (1) has given some characterizations of II(P(S)) for the general case. In this paper we present some detailed and intrinsic characterizations of II((P(S)) for various classes of compact semigroups that are not covered by the results of Collins and Pym.