On quasi-multipliers

A quasi-multipliers is a generalization of the notion of a lef (right, double) multiplier. The first systematic account of the general theory of quasi-multipliers on a Banach algebra with a bounded approximate identity was given in a paper by McKennon in 1977. Further developments have been made in more recent papers by Vasudevan and Goel, Kassem and Rowlands, and Lin. In this paper we consider the quasi-multipliers of algebras not hitherto considered in the literature. In particular, we study the quasi-multipliers of A*-algebras, of the algebra of compact operators on a Banach space, and of the Pedersen ideal of a C*-algebra. We also consider the stict topology on the quasi-multiplier space QM(A) of a Banach algebra A with a bounded approximate identity. We prove that if ${\displaystyle M_l\left(A\right)}$ (resp. ${\displaystyle M_r\left(A\right)}$) denotes the algebra of left (right) multipliers on A, then ${\displaystyle M_l\left(A\right)+M_r\left(A\right)}$ is strictly dense in QM(A), thereby generalizing a theorem due to Lin.