Module Contractibility of Banach Algebras as $\ell^1$-direct sum of Modules

Abstract

‎Let $\mathfrak{A}$‎, ‎$\mathfrak{B}$ and $A$ be Banach algebras such that $A$ is both Banach $\mathfrak{A}$-bimodule and Banach $\mathfrak{B}$-bimodule‎. ‎Let $X$ be both Banach $A$-$\mathfrak{A}$-module and Banach $A$-$\mathfrak{B}$-module‎. ‎In this paper, ‎we show that ‎\[‎ ‎\HH^1_{\mathfrak A\oplus\mathfrak B}(A,X)‎\subseteq‎ \HH^1_{\mathfrak A}(A,X)\oplus \HH^1_{\mathfrak B}(A,X)‎. ‎\]‎ ‎In ‎particular,‎ $A$ is $\mathfrak A\oplus\mathfrak B$-module contractible (resp‎. $\mathfrak A\oplus\mathfrak B$-module amenable and ‎weak $\mathfrak A\oplus\mathfrak B$-module amenable) if $A$ is both $\mathfrak A$-module contractible and $\mathfrak B$-module contractible (resp‎. $\mathfrak A\oplus\mathfrak B$-module amenable and ‎weak $\mathfrak A\oplus\mathfrak B$-module ‎amenable)‎.