Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

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 $p$.