A characterization of amenability property of WAP(S) in locally convex spaces

This paper is a sequel of a recent work regarding an open problem posted by Lau and Zhang on a characterization of existence of a left invariant mean on WAP(S), the Banach algebra of weakly almost periodic functions of a semitopological semigroup S, in terms of a fixed point property for nonexpnsive mappings on weakly compact convex sets in locally convex spaces. More precisely, it is about whether (left) amenability of WAP(S) of a given semitopological semigroup S is equivalent to the following fixed point property: (F): Every weakly separately continuous, weakly quasi-equicontinuous and Q-nonexpansive action of S on a weakly compact convex set K in a separated locally convex topological vector space (E, Q) has a common fixed point for S. In this paper, we provide a complete answer to this question for arbitrary semitopological semigroups.