The Structure of φ -Module Amenable Banach Algebras

Abstract

We study the concept of $\phi$-module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions of $\phi$-amenability and $\phi$-module amenability of Banach algebras. As a consequence, we show that, if $S$ is an inverse semigroup with finite set $E$ of idempotents and $l^1\left(S\right)$ is a commutative Banach $l^1\left(E\right)$-module, then $l^1{\left(S\right)}^{**}$ is ${\phi }^{**}$-module amenable if and only if $S$ is finite, when $\phi \in {\text{H}\text{o}\text{m}}_{l^1\left(E\right)}\left(l^1\left(S\right)\right)$ is an epimorphism. Indeed, we have generalized a well-known result due to Ghahramani et al. (1996).