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

TL;DR

This paper investigates the automatic continuity of derivations and multipliers on Lau product Banach algebras, providing conditions and examples for their continuity in various algebraic settings.

Contribution

It establishes new results on the automatic continuity of derivations and multipliers on Lau product Banach algebras, including special cases and examples.

Findings

Derivations and multipliers are automatically continuous under certain conditions.

Continuity results are extended to specific Banach algebra and bimodule cases.

Examples illustrating the theoretical results are provided.

Abstract

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.