On character amenability of semigroup algebras

Abstract

Let S be a foundation locally compact topological semigroup, and let Ma(S) be the space of all measures μ ∈ M (S) for which the maps x → |μ| * δx and x → |μ| * δx 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 Ma(S). We give necessary and sufficient conditions for the existence of a left χ-mean on Ma(S) * .

Some Characterizations of Character Amenable Banach Algebras

“…A foundation semigroup S is said to have the Hahn-Banach theorem property if for every left action of S on a Banach space X, if ρ is a continuous invariant seminorm on X, and φ is a χ-invariant linear functional on an invariant subspace Y of X such that |φ| ≤ ρ, then there exists a χ-invariant extension ψ of φ to X such that |ψ| ≤ ρ. Ghaffari in [5] studied character amenability of semigroup algebras. He proved that if the unital foundation locally compact semigroup S has the Hahn-Banach theorem property, then it is LCA and converse is true when m = 1, where m is the linear functional defined on M a (S) * , [5,Theorem 6] (m is the linear functional defined in Section 1).…”

“…He proved that if the unital foundation locally compact semigroup S has the Hahn-Banach theorem property, then it is LCA and converse is true when m = 1, where m is the linear functional defined on M a (S) * , [5,Theorem 6] (m is the linear functional defined in Section 1). By B(X) we mean the space of all linear bounded operators from Banach space X into itself.…”

“…Recently character amenability of various classes of Banach algebras considered, for instance Fourier-Stieltjes algebras and Fourier algebras studied in [11], abstract Segal algebras and Segal algebras studied in [2], and semigroup algebras in [5]. Similar to the amenability of Banach algebras, versions of CA such as character contractibility and essential character amenability, have been defined and studied (see [2,11,19]).…”