Morita equivalence and spectral triples on noncommutative orbifolds

Let GG be a finite group. Noncommutative geometry of unital GG-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical orbifold groupoids, a Morita equivalence for the crossed product spectral triples is developed. Noncommutative orbifolds are Morita equivalence classes of the crossed product spectral triples. As a special case of this Morita theory one can study freeness of the GG-action on the noncommutative level. In the case of a free action, the crossed product formalism reduced to the usual spectral triple formalism on the algebra of GG-invariant functions.