(Weakly) Almost Periodic Functions and Fixed Point Properties on Norm Separable \(*\)-Weak Compact Convex Sets in Dual Banach Spaces

Abstract

Given a semitopological semigroup \(S\), let \(\operatorname{WAP}(S)\) and \(\operatorname{AP}(S)\) be the algebras of weakly and strongly almost periodic functions on \(S\), respectively. This paper centers around the study of the fixed point property (\(\mathbf{F}_{*,s}\)): whenever \(\pi\colon S\times K \to K\) is a jointly \(*\)-weak continuous nonexpansive action on a non-empty norm separable \(*\)-weak compact convex set \(K\) in the dual \(E^*\) of a Banach space \(E\), then there is a common fixed point for \(S\) in \(K\). We are primarily interested in answering the following problems posed by Lau and Zhang. (1) Let \(S\) be a discrete semigroup. If the fixed point property (\(\mathbf{F}_{*,s}\)) holds, does \(\operatorname{WAP}(S)\) have a left invariant mean? (2) Is the existence of a left invariant mean on \(\operatorname{WAP}(S)\) a sufficient condition to ensure the fixed point property (\(\mathbf{F}_{*,s}\))? (3) Do the bicyclic semigroups \(S_2=\langle e,a,b,c \colon ab=ac=e\rangle\) and \(S_3=\langle e,a,b,c,d \colon ac=bd=e\rangle\) have the fixed point property (\(\mathbf{F}_{*,s}\))? Among other things, characterization theorems of the amenability property of the algebras \(\operatorname{WAP}(S)\) and \(\operatorname{AP}(S)\) are also given.