Completely continuous and weakly completely continuous abstract Segal algebras

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.