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

Document Type : Research Paper


Department of Mathematics, University of Birjand, Birjand, Iran.



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 map and establish a connection between $2$-cocycles and $2$-coboundaries on $A\oplus B$ and $2$-cocycles and $2$-coboundaries on $A$ and $B$. Finally, our results lead to a connection between  $\mathcal{H}^2(A, A^{*})$,$ \mathcal{H}^2(B, B^{*})$ and $\mathcal{H}^2(A\oplus B, A^{*}\oplus B^{*})$. 


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