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Quasi-Multipliers and Embeddings of Hilbert C*-Bimodules

Published online by Cambridge University Press:  20 November 2018

Lawrence G. Brown ,
James A. Mingo  and
Nien-Tsu Shen
Lawrence G. Brown
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U.S.A., e-mail: lgb@math.purdue.edu
James A. Mingo
Affiliation:
Department of Mathematics, Queen's University Kingston, Ontario K7L 3N6, e-mail: mingoj@qucdn.queensu.ca
Nien-Tsu Shen
Affiliation:
MicroModule Systems 10500-A Ridgeview Court Cupertino, California 95014-0736, U.S.A., e-mail: Shen@MMS.com
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Abstract

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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1150 - 1174
Copyright
Copyright © Canadian Mathematical Society 1994

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