Uniform semigroups and fixed point properties

Abstract

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group.