Realization of a simple higher dimensional noncommutative torus as a transformation group C*-algebra

Abstract

Let $θ$ be a nondegenerate skew symmetric real $d$ by $d$ matrix, and let $A_θ$ be the corresponding simple higher dimensional noncommutative torus. Suppose that $d$ is odd, or that $d$ is greater or equal to 4 and the entries of $θ$ are not contained in a quadratic extension of $\mathbb{Q}$. Then $A_θ$ is isomorphic to the transformation group C*-algebra obtained from a minimal homeomorphism of a compact connected one dimensional space locally homeomorphic to the product of the interval and the Cantor set. The proof uses classification theory of C*-algebras.