Module Amenability and Weak Module Amenability of Banach algebras as a Direct Sum of Modules

Abstract

‎Let $َ\mathfrak{A}‎, ‎\mathfrak{B}$ and $A$ be Banach algebras and $X$ be Banach $\mathfrak{A}$-$A$-module and Banach $\mathfrak{B}$-$A$-module‎. ‎In this paper show that $A$ is $\mathfrak A\oplus\mathfrak B$-module amenable (weak $\mathfrak A\oplus\mathfrak B$-module amenable) if and only if $A$ is both $\mathfrak A$-module amenable and $\mathfrak B$-module amenable (weak $\mathfrak A$-module amenable and weak $\mathfrak B$-module amenable)‎.