Automatic Continuity of Derivations on Semidirect Products of Banach Algebras

Let A and U be Banach algebras such that U is also a Banach A-bimodule with compatible algebra operations, module actions and compatible norm. By defining an appropriate multiplication, we turn ℓ1-direct product A×U into a Banach algebra so that A is a closed subalgebra and U is a closed ideal of it. This algebra is, in fact, the semidirect product of A and U which we denote by A imes U. In this paper, we study automatic continuity of derivations on A imes U in a general setting. As an application of our results, we present various results about the automatic continuity of derivations of module extension Banach algebras and θ-Lau products of Banach algebras. Some examples are also given.