The First Module Cohomology Group of Banach Algebras on Different Modules

Abstract

‎Let $X$ be Banach $\mathfrak{A}$-$A$-module and Banach $\mathfrak{B}$-$A$-module‎. ‎In this paper we convert $A$ and $X$ to Banach $\mathfrak A\oplus\mathfrak B$-$A$-module and show that‎ ‎\[‎ ‎\HH^1_{\mathfrak A\oplus\mathfrak B}(A,X^*)\simeq \HH^1_{\mathfrak A}(A,X^*)\oplus \HH^1_{\mathfrak B}(A,X^*)‎ ‎\]‎ ‎and in particular‎, ‎$A$ is $\mathfrak A\oplus\mathfrak B$-module amenable (weak $\mathfrak A\oplus\mathfrak B$-module amenable) if and only if $A$ is $\mathfrak A$-module amenable and $\mathfrak B$-module amenable (weak $\mathfrak A$-module amenable and weak $\mathfrak B$-module amenable)‎.