Characterization of $2$-cocycles and $2$-coboundaries on Direct Sum of Banach Algebras

Abstract

‎Let $A$ and $B$ be Banach algebras‎. ‎In this paper‎, ‎we investigate the structure of $2$-cocycles and $2$-coboundaries on $A\oplus B$‎, ‎when $A$ and $B$ are unital‎. ‎Actually‎, ‎we provide a specific criterion for each $2$-cocycle maps and establish a connection between $2$-cocycles and $2$-coboundaries on $A\oplus B$ and $2$-cocycles and $2$-coboundaries on $A\oplus B$ on $A$ and $B$‎. ‎Finally‎, ‎our results lead to a connection between $\mathcal{H}^2(A‎, ‎A^{*})$,$ \mathcal{H}^2(B‎, ‎B^{*})$ and $\mathcal{H}^2(A\oplus B‎, ‎A^{*}\oplus B^{*})$‎.