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.