An inductive system of noncommutative tori

Abstract

We define the $UHF$-noncommutative torus $A_{[\frac{1}{p}]\omega}$ of type $p^{\infty}$ by the inductive limit of an inductive system of noncommutative tori $A_{\frac{1}{p^{2s}}\omega}$ with natural inclusions. We prove that $A_{[\frac{1}{p}]\omega}$ is isomorphic to $A_{\omega} \otimes M_{p^{\infty}}$. It is moreover shown that if $A_{[\frac{1}{p}]\omega}$ is an inductive limit of circle algebras then $A_{\omega`}$ is an inductive limit of circle algebras for $\omega`$ a suitable multiplier related to $\omega$, and that $A_{[\frac{1}{p}]\omega}$ has real rank $0$ if $A_{\omega}$ is completely irrational.