Quasi-multipliers of Hilbert and Banach C*-bimodules

TL;DR

This paper develops a new, equivalent framework for understanding quasi-multipliers of Hilbert and Banach C*-bimodules, establishing their universal properties and exploring their structure in various contexts.

Contribution

It introduces an alternative definition of quasi-multipliers using the centralizer approach and extends the concept to bimodules in Kasparov's sense and Banach bimodules over C*-algebras.

Findings

Quasi-multipliers are shown to be universal objects in a certain category.

A topological description of quasi-multipliers via the quasi-strict topology is provided.

Explicit descriptions of quasi-multipliers for specific bimodules like l_2(A) and bimodules over bundles are given.

Abstract

Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 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.