Invariant means and semigroups of nonexpansive mappings on uniformly convex Banach spaces

Abstract

Let S be a semitopological semigroup, i.e., S is a semigroup with Hausdorff topology such that for each a~ S, the mappings s + sa and s-+ as from S into S are continuous. Let RUC(S) denote the space of bounded right uniformly continuous real-valued functions on S. In [ 141, T. Mitchell, generalizing a well-known fixed point theorem of Day [S], showed that RUC(S) has a right invariant mean if and only if whenever S x C 4 C is an affine jointly continuous anti-action of S on a compact convex subset of a separated locally convex space, then C contains a common fixed point for S. Furthermore, an argument similar to that for the proof of Theorem 1 in [14] shows that if S has the following fixed point property for dual Banach space, then RUC(S) has a right invariant mean: