Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaces

Abstract

In this paper, we study and establish a positive answer to a long-standing open problem raised by A.T.-M. Lau in 1976. It is about whether the left amenability property of the Banach algebra WAP($S$), of all weakly almost periodic functions, on a given semitopological semigroup $S$ is equivalent to the existence of a common fixed point of any separately weakly continuous and weakly quasi-equicontinuous nonexpansive action of $S$ on a nonempty weakly compact convex subset of a separated locally convex space. We establish here an affirmative answer; and among other things, we show that the affine counterpart of this question holds and also the AP($S$) formulation of this problem is true.