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.

Mathematics Subject Classification (2010):

• Primary: 46H20
• Secondary: 46H10
• 46H25

Key words:

• Semigroup
• semigroup algebra
• Banach algebra
• character
• left character amenable
• character amenable