On the continuity of some linear maps on certain Banach algebras

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

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