The second cohomology group of module extensions banach algebras

Abstract

We study on the second cohomology of the module extension Banach algebra $A{\oplus }_{1}X$, where $A$ is a Banach algebra and $X$ is a Banach $A$-bimodule. We characterize the defined 2-cocycles and 2-coboundaries on $A{\oplus }_{1}X$ and as an interesting result we show that when $H^2\left(A,A\right)=0$, then $H^2\left(A{\oplus }_{1}A,A{\oplus }_{1}A\right)=QM\left(A\right)$, where $QM\left(A\right)$ is the set of quasi-multipliers on $A$. Also, we give some examples related to vanishing and nonvanishing of the second cohomology of various module extensions of Banach algebras.