On the continuity of some linear maps on certain Banach algebras

Subject Areas : Functional analysis

H.  Farhadi ¹ (Department of Mathematics, University of Kurdistan, Sanandaj, Iran)
E.  Ghaderi ² (Department of Mathematics, University of Kurdistan, Sanandaj, Iran)

Received: 2022-02-23 Accepted : 2022-06-12 Published : 2022-06-13

Keywords: derivation, Multiplier, Automatic continuity, Lau product,

Abstract :

‎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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