On the Schauder fixed point property II

The Schauder fixed point property (F) was introduced and studied by Lau and Zhang as a semigroup formulation in the general setting of convex spaces of the well-known Schauder fixed point theorem in Banach spaces. What amenability property should possess a semigroup or a topological group to satisfy the Schauder fixed point property. Recently, the author provided a partial answer to that question and as a sequel, it is the purpose of this paper to study in more deep this problem. Our main result establishes that for a compact semitopological semigroup S we have: LUC(S) is left amenable if, and only if, S has the fixed point property (F). Furthermore, we also prove that totally bounded topological groups, semitopological groups S with the property that LUC(S) subset of aa( S), and strongly left amenable semitopological semigroups, possess all the Schauder fixed point property.