ON QUASI-MULTIPLIERS

A quasi-multipliers is a generalization of the notion of a left (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 strict topology on the quasi-multiplier space QM(A) of a Banach algebra A with a bounded approximate identity. We prove that, if M(l) (A) (resp. M, (A)) denotes the algebra of left (right) multipliers on A, then M(l)(A) + M(r)(A) is strictly dense in QM(A), thereby generalizing a theorem due to Lin.