Amenability and Hahn-Banach extension property for set-valued mappings

Abstract

Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterization of a semitopological semigroup $S$ on the existence of a left invariant mean on $\text{\rm LUC}(S)$, $\text{\rm AP}(S)$ and $\text{\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of Banach algebras related to amenability of groups and semigroups.