Automatic continuity of Some linear mappings from certain products of Banach algebras

Let $\mathcal{A}$ and $\mathcal{U}$ be Banach algebras and $\theta$ be a nonzero character on $\mathcal{A}$. Then the \textit{Lau product Banach algebra} $\mathcal{A}{\times }_{\theta }\mathcal{U}$ associated with the Banach algebras $\mathcal{A}$ and $\mathcal{U}$ is the $l^1$-direct sum $\mathcal{A}\oplus \mathcal{U}$ equipped with the algebra multiplication $(a,u)(a’,u’)=(ab,\theta(a)u’+\theta(a’)u+uu’), (a,a’\in$\mathcal{A}$,u,u’\in$\mathcal{U}$)$and$l^1$-norm. In this paper we shall investigate the derivations and multipliers from this Banach algebras and study the automatic continuity of these mappings. We also study continuity of the derivations for some special cases of Banach algebra $\mathcal{U}$ and Banach $\mathcal{A}{\times }_{\theta }\mathcal{U}$-bimodule $\mathcal{X}$ and establish various results on the continuity of derivations and give some examples.