Tracing projective modules over noncommutative orbifolds

Abstract

For an action of a finite cyclic group $F$ on an $n$-dimensional noncommutative torus $A_{\theta }$, we give sufficient conditions when the fundamental projective modules over $A_{\theta }$, which determine the range of the canonical trace on $A_{\theta }$, extend to projective modules over the crossed product $C^{\ast }$-algebra $A_{\theta }⋊F$. Our results allow us to understand the range of the canonical trace on $A_{\theta }⋊F$, and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of ${\mathbb{Z}}_2$ on any $n$-dimensional noncommutative torus. As an application, for the flip action of ${\mathbb{Z}}_2$ on a simple $n$-dimensional torus $A_{\theta }$, we determine the Morita equivalence class of $A_{\theta }⋊{\mathbb{Z}}_2,$ in terms of the Morita equivalence class of $A_\theta.