A generalization of the $n$-weak module amenability of banach algebras

Abstract

In this paper, we generalize the notion of $n$-weak module amenability of a Banach algebra $A$ which is a Banach module over another Banach algebra $U$ with compatible actions to that of $(\sigma )-n$-weak module amenability for $n\in \mathbb {N}$ and $\sigma \in Hom_{U}(A)$. We also investigate the relation between this new concept of amenability of $A$ and the quotient Banach algebra $A/J$ where $J$ is the closed ideal of $A$ generated by elements of the form $(a\cdot \alpha )b-a(\alpha \cdot b)$ for $a,b\in A$ and $\alpha \in U$. As a consequence, we show that the semigroup algebra $l^{1}(S)$ is $(\sigma )$-$(2n+1)$-weakly module amenable as an $l^{1}(E)$-module for each $n\in \mathbb {N}$ and $\sigma \in Hom_{l^{1}(E)}(l^{1}(S))$, where $S$ is an inverse semigroup with the set of idempotents $E$.