On relationships between reversibility and amenability of $\mathrm{WAP}(S)$

It is well-known that whenever $S$ is a discrete left reversible semigroup or a topological group, then the Banach algebra $\mathrm {WAP}(S)$ of weakly almost periodic functions on $S$ has a left invariant mean. Whether this is true for semitopological semigroups is still unknown. By introducing a certain fixed point property, we are able to provide a positive answer for separable semitopological semigroups (or more generally, for strongly left reversible semitopological semigroups) and for all semitopological left groups. Moreover, we prove that whenever $S$ is a semitopological left group, for any right ideal of type $sS$ $(s\in S)$ the space $\mathrm {WAP}(sS)$ has a left invariant mean, extending a well-known property of topological groups.