Equivalence bimodule between non-commutative tori

Oh, Sei-Qwon ; Park, Chun-Gil

Czechoslovak Mathematical Journal, Tome 53 (2003) no. 2, pp. 289-294.

Résumé

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.

Détail

Classification : 46L05, 46L87, 46L89, 55R15
Keywords: Morita equivalent; twisted group $C^*$-algebra; crossed product