On character amenability of semigroup algebras

Abstract

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a nite number of idempotents, inverse semigroups with uniformly locally nite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra ℓ¹(S) in relation to the structures of the semigroup S: In particular, we show that for any semigroup S; if ℓ ¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ℓ ¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being nite. Finally, we show that for a Rees semigroup S with a zero over the group G; the left character amenability of ℓ ¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.

Keywords: Semigroup, semigroup algebra, Banach algebra, character, left character amenable, character amenable