Tracing projective modules over noncommutative orbifolds

TL;DR

This paper studies how fundamental projective modules over noncommutative tori extend to crossed product algebras under finite cyclic group actions, clarifying the trace range and Morita equivalence classes.

Contribution

It provides sufficient conditions for extending projective modules over noncommutative tori to crossed products and determines the trace range and Morita classes for specific cases.

Findings

Extended projective modules under group actions are characterized.

The trace range on crossed products is explicitly determined.

Morita equivalence classes are classified for certain crossed products.

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*-algebra $A_\theta \rtimes F.$ Our results allow us to understand the range of the canonical trace on $A_\theta \rtimes F$, 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 $A_\theta \rtimes \mathbb{Z}_2,$ in terms of the Morita equivalence class of $A_\theta.$