The first Cohomology Group of Semidirect Products of Banach Algebras

摘要

Let A and U be Banach algebras such that U is a Banach A-bimodule with compatible algebra multiplication, module actions and norm. By defining an appropriate action, we turn '1-direct product A x U into a Banach algebra such that A is a closed subalgebra and U is a closed ideal of A x U. This algebra is, in fact, the semidirect product of A and U, AnU, and as well as, every semidirect product of Banach algebras can be represented as this form. Our aim in this paper is to study the first cohomology group of AnU and investigate the relation between the first cohomology group of AnU and those of A and U. As an application of our results, we show that the earlier results obtained for the first cohomology group of various classes of Banach algebras such as direct products of Banach algebras, module extension Banach algebras and hLau products of Banach algebras can be obtained directly by applying our main results and techniques of proofs. Some examples are also given.