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 finite number of idempotents, inverse semigroups with uniformly locally finite 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 ℓ1(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ℓ1(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ℓ1(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.