On dual of group algebras under a locally convex topology

For a locally compact group $G$, $L^1(G)$ is its group algebra and $L^\infty(G)$ is the dual of $L^1(G)$. We consider on $L^\infty(G)$ the $\tau$-topology, i.e. the weak topology under all right multipliers induced by measures in $L^1(G)$. For such an arbitrary $G$ the $\tau$-topology is not weaker than the weak$^*$-topology and not stronger than the norm topology on $L^\infty(G)$. Among the other results we mention that except for discrete $G$ the $\tau$-topology is always different from the norm-topology. The properties of $\tau$ are then studied further and we pay attention to the $\tau$-almost periodic elements of $L^\infty(G)$.