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 derivations over a class of Banach algebras that contain dense ' * -like' subalgebras.The results we prove are then applied to -crossed products and symmetrized -crossed products.For example, it follows that every derivation over the -crossed product (, , ) is continuous, provided that is infinite, finitely generated, has polynomial growth, and acts freely on the compact Hausdorff space .