An inductive system of noncommutative tori

초록

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.