On the continuity of some linear maps on certain Banach algebras

Let A and U be Banach algebras, θ be a nonzero character on A and let A ×θ U be the corresponding Lau product Banach algebra. In this paper we investigate derivations and multipliers of A ×θ U and study the automatic continuity of these maps. We also study continuity of the derivations for some special cases of U and the Banach (A ×θ U)-bimodule X and establish various results in this respect. Some of the results are devoted to find conditions under which one can represent a derivation on A ×θ U as sum of two derivations in such a way that one of them is continuous. Some examples are also given.