DERIVATIONS ON GENERALIZED SEMIDIRECT PRODUCTS OF BANACH ALGEBRAS

Let A and B be Banach algebras, let theta : A -> 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 infinity(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 infinity(theta) I and then we compute the first cohomology group of A infinity(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 infinity(id) A. Furthermore, we calculate the first cohomology group of some concrete Banach algebras related to locally compact groups.