Quasi-Multipliers and Embeddings of Hilbert C*-Bimodules

Abstract This paper considers Hilbert C *-bimodules, a slight generalization of imprimitivity bimodules which were introduced by Rieffel [20]. Brown, Green, and Rieffel [7] showed that every imprimitivity bimodule X can be embedded into a certain C *-algebra L, called the linking algebra of X. We consider arbitrary embeddings of Hilbert C *-bimodules into C *-algebras; i.e. we describe the relative position of two arbitrary hereditary C *-algebras of a C *-algebra, in an analogy with Dixmier's description [10] of the relative position of two subspaces of a Hilbert space. The main result of this paper (Theorem 4.3) is taken from the doctoral dissertation of the third author [22], although the proof here follows a different approach. In Section 1 we set out the definitions and basic properties (mostly folklore) of Hilbert C *-bimodules. In Section 2 we show how every quasi-multiplier gives rise to an embedding of a bimodule. In Section 3 we show that , the enveloping C *-algebra of the C *-algebraA with its product perturbed by a positive quasi-multiplier , is isomorphic to the closure (Proposition 3.1). Section 4 contains the main theorem (4.3), and in Section 5 we explain the analogy with the relative position of two subspaces of a Hilbert spaces and present some complements.