Module amenability of restricted semigroup algebras under module actions

Abstract

In this article, we show that module amenability with the canonical action of restricted semigroup algebra $l^_r(S)$ and semigroup algebra $l^1(S_r)$ are equivalent, where $S_r$ is the restricted semigroup of associated  to the inverse semigroup $S$. We use this to give a
characterization of module amenability of restricted semigroup
algebra $l^1_r(S)$  with the canonical action, where $S$ is a Clifford semigroup.