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

HTML articles powered by AMS MathViewer

- by Beatriz Abadie, Søren Eilers and Ruy Exel PDF
- Trans. Amer. Math. Soc.
**350**(1998), 3043-3054 Request permission

## 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

- Beatriz Abadie,
*Generalized fixed-point algebras of certain actions on crossed products*, Pacific J. Math.**171**(1995), no. 1, 1–21. MR**1362977** - —,
*The range of traces on the $K_0$-group of quantum Heisenberg manifolds*, Preprint, Universidade de São Paulo, February 1994. - Bruce Blackadar,
*Shape theory for $C^\ast$-algebras*, Math. Scand.**56**(1985), no. 2, 249–275. MR**813640**, DOI 10.7146/math.scand.a-12100 - Bruce Blackadar,
*$K$-theory for operator algebras*, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR**859867**, DOI 10.1007/978-1-4613-9572-0 - Lawrence G. Brown, Philip Green, and Marc A. Rieffel,
*Stable isomorphism and strong Morita equivalence of $C^*$-algebras*, Pacific J. Math.**71**(1977), no. 2, 349–363. MR**463928** - Lawrence G. Brown, James A. Mingo, and Nien-Tsu Shen,
*Quasi-multipliers and embeddings of Hilbert $C^\ast$-bimodules*, Canad. J. Math.**46**(1994), no. 6, 1150–1174. MR**1304338**, DOI 10.4153/CJM-1994-065-5 - Huu Hung Bui,
*Morita equivalence of twisted crossed products*, Proc. Amer. Math. Soc.**123**(1995), no. 9, 2771–2776. MR**1260162**, DOI 10.1090/S0002-9939-1995-1260162-1 - F. Combes,
*Crossed products and Morita equivalence*, Proc. London Math. Soc. (3)**49**(1984), no. 2, 289–306. MR**748991**, DOI 10.1112/plms/s3-49.2.289 - Raúl E. Curto, Paul S. Muhly, and Dana P. Williams,
*Cross products of strongly Morita equivalent $C^{\ast }$-algebras*, Proc. Amer. Math. Soc.**90**(1984), no. 4, 528–530. MR**733400**, DOI 10.1090/S0002-9939-1984-0733400-1 - Siegfried Echterhoff,
*Morita equivalent twisted actions and a new version of the Packer-Raeburn stabilization trick*, J. London Math. Soc. (2)**50**(1994), no. 1, 170–186. MR**1277761**, DOI 10.1112/jlms/50.1.170 - Ruy Exel,
*A Fredholm operator approach to Morita equivalence*, $K$-Theory**7**(1993), no. 3, 285–308. MR**1244004**, DOI 10.1007/BF00961067 - Ruy Exel,
*Circle actions on $C^*$-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence*, J. Funct. Anal.**122**(1994), no. 2, 361–401. MR**1276163**, DOI 10.1006/jfan.1994.1073 - —,
*Twisted partial actions: a classification of regular $C^*$-algebraic bundles*, Proc. London Math. Soc. (3)**74**(1997), no. 2, 417–443. - J. M. G. Fell and R. S. Doran,
*Representations of $^*$-algebras, locally compact groups, and Banach $^*$-algebraic bundles. Vol. 1*, Pure and Applied Mathematics, vol. 125, Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups and algebras. MR**936628** - S. Kaliszewski,
*Morita equivalence methods for twisted $C^*$-dynamical systems*, Ph.D. thesis, Dartmouth College, 1994. - Terry A. Loring,
*$C^*$-algebras generated by stable relations*, J. Funct. Anal.**112**(1993), no. 1, 159–203. MR**1207940**, DOI 10.1006/jfan.1993.1029 - Gert K. Pedersen,
*$C^{\ast }$-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006** - M.V. Pimsner,
*A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\mathbb {Z}$*, Free Probability Theory (Waterloo, Ont., 1995; D.-V. Voiculescu, editor), Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189–212. - Marc A. Rieffel,
*Induced representations of $C^{\ast }$-algebras*, Advances in Math.**13**(1974), 176–257. MR**353003**, DOI 10.1016/0001-8708(74)90068-1 - Marc A. Rieffel,
*Deformation quantization of Heisenberg manifolds*, Comm. Math. Phys.**122**(1989), no. 4, 531–562. MR**1002830**

## 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
- 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