# Article

 Title: Equivalence bimodule between non-commutative tori  (English) Author: Oh, Sei-Qwon Author: Park, Chun-Gil Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 Volume: 53 Issue: 2 Year: 2003 Pages: 289-294 Summary lang: English . Category: math . Summary: The non-commutative torus $C^*(\mathbb{Z}^n,\omega )$ is realized as the $C^*$-algebra of sections of a locally trivial $C^*$-algebra bundle over $\widehat{S_{\omega }}$ with fibres isomorphic to $C^*(\mathbb{Z}^n/S_{\omega }, \omega _1)$ for a totally skew multiplier $\omega _1$ on $\mathbb{Z}^n/S_{\omega }$. D. Poguntke [9] proved that $A_{\omega }$ is stably isomorphic to $C(\widehat{S_{\omega }}) \otimes C^*(\mathbb{Z}^n/S_{\omega }, \omega _1) \cong C(\widehat{S_{\omega }}) \otimes A_{\varphi } \otimes M_{kl}(\mathbb{C})$ for a simple non-commutative torus $A_{\varphi }$ and an integer $kl$. It is well-known that a stable isomorphism of two separable $C^*$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_{\omega }$-$C(\widehat{S_{\omega }}) \otimes A_{\varphi }$-equivalence bimodule. Keyword: Morita equivalent Keyword: twisted group $C^*$-algebra Keyword: crossed product MSC: 46L05 MSC: 46L87 MSC: 46L89 MSC: 55R15 idZBL: Zbl 1028.46102 idMR: MR1983452 . Date available: 2009-09-24T11:01:35Z Last updated: 2012-05-31 Stable URL: http://hdl.handle.net/10338.dmlcz/127800 . Reference: [1] L.  Baggett and A.  Kleppner: Multiplier representations of abelian groups.J.  Funct. Anal. 14 (1973), 299–324. MR 0364537 Reference: [2] M.  Brabanter: The classification of rational rotation $C^*$-algebras.Arch. Math. 43 (1984), 79–83. MR 0758343 Reference: [3] L.  Brown, P.  Green and M.  Rieffel: Stable isomorphism and strong Morita equivalence of $C^*$-algebras.Pacific J.  Math. 71 (1977), 349–363. MR 0463928 Reference: [4] S.  Disney and I.  Raeburn: Homogeneous $C^*$-algebras whose spectra are tori.J.  Austral. Math. Soc. (Series A) 38 (1985), 9–39. MR 0765447 Reference: [5] R. S.  Doran and J. M. G.  Fell: Representations of $*$-Algebras, Locally Compact Groups, and Banach $*$-Algebraic Bundles.Academic Press, San Diego, 1988. Reference: [6] G. A.  Elliott: On the $K$-theory of the $C^*$-algebra generated by a projective representation of a torsion-free discrete abelian group.In: Operator Algebras and Group Representations, Vol.  1, Pitman, London, 1984, pp. 157–184. Zbl 0542.46030, MR 0731772 Reference: [7] P.  Green: The local structure of twisted covariance algebras.Acta Math. 140 (1978), 191–250. Zbl 0407.46053, MR 0493349 Reference: [8] D. Poguntke: Simple quotients of group $C^*$-algebras for two step nilpotent groups and connected Lie groups.Ann. Scient. Ec. Norm. Sup. 16 (1983), 151–172. Zbl 0523.22007, MR 0719767 Reference: [9] D.  Poguntke: The structure of twisted convolution $C^*$-algebras on abelian groups.J.  Operator Theory 38 (1997), 3–18. Zbl 0924.46046, MR 1462012 Reference: [10] M. Rieffel: Morita equivalence for operator algebras.Operator Algebras and Applications. Proc. Symp. Pure Math. Vol. 38, R. V.  Kadison (ed.), Amer. Math. Soc., Providence, R. I., 1982, pp. 285–298. Zbl 0541.46044, MR 0679708 .

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