Wavelet and Linear Algebra

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 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 $\mathcal{H}^2(A\oplus B, A^{*}\oplus B^{*})$.