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On quasi-multipliers. (English) Zbl 0824.46053

Summary: A quasi-multiplier 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 \(M_ \ell(A)\) (resp. \(M_ r(A)\)) denotes the algebra of left (right) multipliers on \(A\), then \(M_ \ell(A)+ M_ r(A)\) is strictly dense in \(QM(A)\), thereby generalizing a theorem due to Lin.

MSC:

46H05 General theory of topological algebras
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