Tracing projective modules over noncommutative orbifolds

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) (sic) F. Our results allow us to understand the range of the canonical trace on A(theta) (sic) 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 Z(2) on any n-dimensional non -commutative torus. As an application, for the flip action of Z(2) on a simple n-dimensional torus A(theta), we determine the Morita equivalence class of A(theta) (sic) Z(2), in terms of the Morita equivalence class of A(theta).