Derivations on semidirect products of Banach algebras

TL;DR

This paper investigates derivations on semidirect product Banach algebras, focusing on automatic continuity and cohomology, and relates these properties to the component algebras.

Contribution

It provides a unified framework for semidirect products of Banach algebras and analyzes derivations and cohomology within this context, extending known results.

Findings

Derivations on semidirect product algebras can be characterized and studied via component algebras.

Automatic continuity of derivations on the product relates to that on the individual algebras.

First cohomology groups of the product are connected to those of the component algebras.

Abstract

Let A and U be Banach algebras such that U is also a Banach A- bimodule with compatible algebra operations, module actions and norm. By defining an approprite action, we turn l1-direct product A item U into a Banach algebra such that A is closed subalgebra and U is a closed ideal of it. This algebra, is in fact semidirect product of A and U which we denote it by A litem U and every semidirect products of Banach algebras can be represented as this form. In this paper we consider the Banach algebra A litem U as mentioned and study the derivations on it. In fact we consider the automatic continuity of the derivations on A litem U and obtain some results in this context and study its relation with the automatic continuity of the derivations on A and U. Also we calculate the first cohomology group of A litem U in some different cases and establish relations among the first cohomology group of A litem U and those of A and U. As applications of these contents, we present various results about the automatic continuity of derivations and the first cohomology group of direct products of Banach algebras, module extension Banach algebras and theta Lau products of Banach algebras.