(Weakly) almost periodic functions and fixed point properties on norm separable $*$-weak compact convex sets in dual Banach spaces

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.