On the continuity of derivations over locally regular Banach algebras

TL;DR

This paper investigates the conditions under which derivations over certain Banach algebras, especially those with a dense $C^*$-like subalgebra, are continuous, with applications to $L^p$-crossed products.

Contribution

It establishes the continuity of derivations over a class of Banach algebras containing dense $C^*$-like subalgebras, including $L^p$-crossed products.

Findings

Derivations over $L^p$-crossed products are continuous under specified conditions.

Results apply to Banach algebras with dense $C^*$-like subalgebras.

Provides new insights into the structure of derivations in Banach algebra theory.

Abstract

We study the problem of continuity of derivations over Banach algebras. More specifically, we consider a class of Banach algebras that contain a dense '$C^*$-like' subalgebra. We discuss applications to $L^p$-crossed products and symmetrized $L^p$-crossed products. As an example, our results imply that every derivation over the $L^p$-crossed product $F^p(G,X,\alpha)$ is continuous, provided that $G$ is infinite, finitely generated, has polynomial growth, and acts freely on the compact Hausdorff space $X$.