Chern character for twisted K-theory of orbifolds

For an orbifold X and α ∈ H³ (X, Z), we introduce the twisted cohomology H_c^* (X, α) and prove that the non-commutative Chern character of Connes-Karoubi establishes an isomorphism between the twisted K-groups K_α^* (X) ⊗ C and the twisted cohomology H_c^* (X, α). This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem-Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai-Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.