Completely continuous and weakly completely continuous abstract Segal algebras

Abstract

Let \(\mathcal{A}\) be a Banach algebra. It is obtained a necessary and sufficient condition for the complete continuity and also weak complete continuity of symmetric abstract Segal algebras with respect to \(\mathcal{A}\), under the condition of the existence of an approximate identity for \(\mathcal{B}\), bounded in \(\mathcal{A}\). In addition, a necessary condition for the weak complete continuity of \(\mathcal{A}\) is given. Moreover, the applications of these results about some group algebras on locally compact groups are obtained.