Generalized non-commutative tori

Chun-Gil Park¹

¹ Department of Mathematics Chungnam National University Taejon 305-764, South Korea

Studia Mathematica, Tome 149 (2002) no. 2, pp. 101-108

Résumé

The generalized non-commutative torus $T_{\varrho }^{k}$ of rank $n$ is defined by the crossed product $A_{{m}/{k}} \times _{\alpha _3} {\mathbb Z} \times _{\alpha _4}\dots\times _{\alpha _n} {\mathbb Z}$, where the actions $\alpha _i$ of ${\mathbb Z}$ on the fibre $M_k({\mathbb C})$ of a rational rotation algebra $A_{{m}/{k}}$ are trivial, and $C^*(k{\mathbb Z} \times k{\mathbb Z}) \times _{\alpha _3} {\mathbb Z} \times _{\alpha _4}\dots\times _{\alpha _n} {\mathbb Z}$ is a non-commutative torus $A_{\varrho }$. It is shown that $T^k_{\varrho }$ is strongly Morita equivalent to $A_{\varrho }$, and that $T_{\varrho }^{k} \otimes M_{p^{\infty }}$ is isomorphic to $A_{\varrho } \otimes M_{k}({\mathbb C}) \otimes M_{p^{\infty }}$ if and only if the set of prime factors of $k$ is a subset of the set of prime factors of $p$.

Détail

DOI : 10.4064/sm149-2-1

Keywords: generalized non commutative torus varrho rank defined crossed product times alpha mathbb times alpha dots times alpha mathbb where actions alpha mathbb fibre mathbb rational rotation algebra trivial * mathbb times mathbb times alpha mathbb times alpha dots times alpha mathbb non commutative torus varrho shown varrho strongly morita equivalent varrho varrho otimes infty isomorphic varrho otimes mathbb otimes infty only set prime factors subset set prime factors