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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Morita equivalence for crossed products
by Hilbert $C^*$-bimodules


Authors: Beatriz Abadie, Søren Eilers and Ruy Exel
Journal: Trans. Amer. Math. Soc. 350 (1998), 3043-3054
MSC (1991): Primary 46L55, 46L05, 46C50; Secondary 46L45, 46L80
MathSciNet review: 1467459
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Abstract: We introduce the notion of the crossed product $A \rtimes _X{\Bbb{Z}}$ of a $C^*$-algebra $A$ by a Hilbert $C^*$-bimodule $X$. It is shown that given a $C^*$-algebra $B$ which carries a semi-saturated action of the circle group (in the sense that $B$ is generated by the spectral subspaces $B_0$ and $B_1$), then $B$ is isomorphic to the crossed product $B_0 \rtimes _{B_1}{\Bbb{Z}}$. We then present our main result, in which we show that the crossed products $A \rtimes _X{\Bbb{Z}}$ and $B \rtimes _Y{\Bbb{Z}}$ are strongly Morita equivalent to each other, provided that $A$ and $B$ are strongly Morita equivalent under an imprimitivity bimodule $M$ satisfying $X\otimes _A M \simeq M\otimes _B Y$ as $A-B$ Hilbert $C^*$-bimodules. We also present a six-term exact sequence for $K$-groups of crossed products by Hilbert $C^*$-bimodules.


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Additional Information

Beatriz Abadie
Affiliation: Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, 05508-900 São Paulo, Brazil
Address at time of publication: Centro de Mathemáticas, Facultad de Ciencias, Universidad de la República, Eduardo Acevedo 1139, CP 11200 Montevideo, Uruguay
Email: abadie@cmat.edu.uy

Søren Eilers
Affiliation: Matematisk Institut, Københavns Universitet, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
Email: eilers@math.ku.dk

Ruy Exel
Affiliation: Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, 05508-900 São Paulo, Brazil
Email: exel@ime.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02133-3
PII: S 0002-9947(98)02133-3
Keywords: Crossed products, Morita equivalence, \cstar-algebras, Hilbert \cstar-bimodules, spectral subspaces, Pimsner-Voiculescu sequence
Received by editor(s): April 6, 1995
Additional Notes: The first author was supported by FAPESP, Brazil, on leave from Facultad de Ciencias, Montevideo, Uruguay. The second author was supported by Rejselegat for matematikere, Denmark, on leave from Københavns Universitet. The third author was partially supported by CNPq, Brazil.
Article copyright: © Copyright 1998 American Mathematical Society