The Bundle Structure of Noncommutative Tori over $UHF$-Algebras

Abstract

The noncommutative torus $C^*(\mathbb Z^n,\,\omega)$ of rank $n$ is realized as the $C^*$-algebra of sections of a locally trivial continuous $C^*$-algebra bundle over $\widehat{S_{\omega}}$ with fibres $C^*(\mathbb Z^n/S_{\omega},\, \omega_1)$ for some totally skew multiplier $\omega_1$ on $\mathbb Z^n/S_{\omega}$. It is shown that $C^*(\mathbb Z^n/S_{\omega},\,\omega_1)$ is isomorphic to $A_{\varphi}\otimes M_k(\mathbb C)$ for some completely irrational noncommutative torus $A_{\varphi}$ and some positive integer $k$, and that $A_{\omega} \otimes M_{l^{\infty}}$ has the trivial bundle structure if and only if the set of prime factors of $k$ is a subset of the set of prime factors of $l$. This is applied to understand the bundle structure of the tensor products of Cuntz algebras with noncommutative tori.