Classification of simple $C^{*}$-algebras and higher dimensional noncommutative tori

Abstract

We show that unital simple $C^*$-algebras with tracial topological rank zero which are locally approximated by subhomogeneous $C^*$-algebras can be classified by their ordered $K$-theory. We apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered $K$-theory. In particular, irrational higher dimensional noncommutative tori of the form $C({\Bbb T}^k)\times_{\theta}{\Bbb Z}$ are in fact inductive limits of circle algebras.