On the continuity of derivations over locally regular Banach algebras

We study the problem of continuity of derivations over Banach algebras. More specifically, we consider a class of Banach algebras that contain a dense ‘$C^{\ast }$-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 $X$.