Derivations on generalized semidirect products of Banach algebras

Abstract

Let $A$ and $B$ be Banach algebras, let $\theta :A\to B$ be a continuous Banach algebra homomorphism, and let $I$ be a closed ideal in $B$. Then the $l^1$-direct sum of $A$ and $I$ with a special product becomes a Banach algebra, denoted by $A{\bowtie }^{\theta }I$, which we call the generalized semidirect product of $A$ and $I$. In this article, among other things, we first characterize derivations on $A{\bowtie }^{\theta }I$ and then we compute the first cohomology group of $A{\bowtie }^{\theta }I$ when the first cohomology groups of $A$ with coefficients in $A$ and $I$ are trivial. As an application we characterize the first cohomology group of second duals of dual Banach algebras. Then we provide a nice characterization of the first cohomology group of $A{\bowtie }^{\mathrm{id}}A$. Furthermore, we calculate the first cohomology group of some concrete Banach algebras related to locally compact groups.