Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

Abstract

Quasi-multipliers for a Hilbert $C^*$-bimodule $V$ were introduced by L. G. Brown, J. A. Mingo, and N.-T. Shen [3] as a certain subset of the Banach bidual module $V^{**}$. We give another (equivalent) definition of quasi-multipliers for Hilbert $C^*$-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over $C^*$-algebras, provided these $C^*$-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule $l_2(A)$ and for bimodules of sections of Hilbert $C^*$-bimodule bundles over locally compact spaces.