Idempotent measures on locally compact semigroups

Abstract

The structure of the support F of an idempotent probability measure Μ on a locally compact semigroup S is considered. It is shown that if S satisfies the condition (L): AB −1 is compact whenever A and B are compact subsets of S, then F is a completely simple semigroup and has the canonical representation X ×G×Y of which G and Y are compact. Moreover, Μ is a product measure Μ X ×Μ G ×Μ Y where Μ X and Μ Y are probability measures and Μ G is the Haar measure on the group G. We conjecture that a similar result remains true even without the condition (L). We give also a relation between our conjecture and a conjecture of Argabright on the support of an r *-invariant measure.