## Morita equivalence for crossed products by Hilbert $C^\ast$-bimodules

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- by Beatriz Abadie, Søren Eilers and Ruy Exel
- Trans. Amer. Math. Soc.
**350**(1998), 3043-3054 - DOI: https://doi.org/10.1090/S0002-9947-98-02133-3
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## Abstract:

We introduce the notion of the crossed product $A \rtimes _X\mathbb {Z}$ 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 $B_0 \rtimes _{B_1}\mathbb {Z}$. We then present our main result, in which we show that the crossed products $A \rtimes _X\mathbb {Z}$ and $B \rtimes _Y\mathbb {Z}$ 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.## References

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## Bibliographic 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
- MR Author ID: 609837
- Email: eilers@math.ku.dk
**Ruy Exel**- Affiliation: Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, 05508-900 São Paulo, Brazil
- MR Author ID: 239607
- Email: exel@ime.usp.br
- 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.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 3043-3054 - MSC (1991): Primary 46L55, 46L05, 46C50; Secondary 46L45, 46L80
- DOI: https://doi.org/10.1090/S0002-9947-98-02133-3
- MathSciNet review: 1467459