Some topologies on the space of quasimultipliers

Assume that A is a Banach algebra. We define the β−topology and the γ−topology on the space QMel(A^∗) of all bounded extended left quasi-multipliers of A^∗. We establish further properties of (QMel(A^∗), γ) when A is a C^∗−algebra. In particular, we characterize the γ−dual of QMel(A^∗) and prove that (QMel(A^∗), γ)^∗, under the topology of bounded convergence, is isomorphic to A^∗∗∗.