On character amenability of semigroup algebras

Let S be a foundation locally compact topological semigroup, and let M _a (S) be the space of all measures μ∈M(S) for which the maps x?|μ|?δ _x and x?|μ|?δ _x from S into M(S) are weakly continuous. The purpose of this article is to develop a notion of character amenability for semigroup algebras. The main results concern the χ-amenability of M _a (S). We give necessary and sufficient conditions for the existence of a left χ-mean on M _a (S)^?.