QUASI-MULTIPLIERS AND EMBEDDINGS OF HILBERT C-ASTERISK-BIMODULES

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 C*(A(.)), the enveloping C*-algebra of the C*-algebra A with its product perturbed by a positive quasi-multiplier s:a.b = asb, is isomorphic to the closure of s(1)/As-2(1)/(2) (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.