A note on Haar-like measure for group-extremal semi-groups

Abstract

If 5 is a compact group-extremal affine semigroup, it is natural to ask how much of the group structure of the extreme points carries over to S. In [4], the author shows it is possible to extend sufficiently many unitary representations to separate points of S; in the abelian case, this leads to sufficiently many affine semicharacters to separate points of S. We investigate in this note the possibility of extending Haar measure to S. Using results in [l], [2], and [3], it can be seen that a compact affine semigroup which supports an idempotent measure must be of the form XX Y where X and Y are compact convex sets and multiplication is given by (x, y) •(*'> y')~ix, y')-Thus, one cannot hope in general to extend Haar measure and retain all of its properties. However, we will show that if 5 is a compact group-extremal affine semigroup, then there is a probability measure p. supported on 5 (i.e. /^-measure of each nonvoid open set is positive) satisfying