On representations of certain semigroups

Abstract

A theory of representations for compact semigroups has been lacking due in large part to the absence of a translationinvariant carrying measure that exists for compact groups. The object in this paper is to show that for a compact, groupextremal affine semigroup there is a sufficient system of representations by linear operators on finite-dimensional complex linear spaces; in the abelian case, a sufficient system of affine semicharacters is obtained. As a result, a compact groupextremal affine semigroup is the inverse limit of compact, finite-dimensional, group-extremal affine semigroups.