On relationships between reversibility and amenability of WAP(S)

It is well-known that whenever S is a discrete left reversible semigroup or a topological group, then the Banach algebra 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∈S) the space WAP(sS) has a left invariant mean, extending a well-known property of topological groups.