Assume that $A$ is a Banach algebra. We define the $beta-$topology and the $gamma-$topology on the space $QM_{el}(A^{*})$ of all bounded extended left quasi-multipliers of $A^{*}.$ We establish further properties of $(QM_{el}(A^{*}),gamma)$ when $A$ is a $C^{*}-$algebra. In particular, we characterize the $gamma-$dual of $QM_{el}(A^{*})$ and prove that $(QM_{el}(A^{*}),gamma)^{*},$...