Module Amenability for Semigroup Algebras

Abstract

We extend the concept of amenability of a Banach algebra $A$ to the case that there is an extra $\mathfrak{A}$-module structure on $A$, and show that when $S$ is an inverse semigroup with subsemigroup $E$ of idempotents, then $A={\ell }^1(S)$ as a Banach module over $\mathfrak{A}={\ell }^1(E)$ is module amenable iff $S$ is amenable. When $S$ is a discrete group, ${\ell }^1(E)=\mathbb{C}$ and this is just the celebrated Johnson's theorem.

12 pages