Cyclic Cohomology and Index Theory for Crossed Product C*-Algebras of Twisted Groupoid Actions

This monograph investigates the cyclic cohomology of crossed product C*-algebras arising from twisted actions of locally compact groupoids. We extend the spectral sequence machinery of Feigin-Tsygan-Nistor to the setting of twisted groupoid dynamical systems, providing a computable homological invariant for non-commutative spaces associated with foliations and orbifolds. Furthermore, we construct an explicit Chern-Connes character mapping the K-theory of the twisted crossed product to its periodic cyclic cohomology. By invoking the tangent groupoid formalism, we formulate a generalized index theorem that relates the analytic index of longitudinally elliptic operators to a topological index defined via the twisted cohomology of the classifying space. Our results generalize the Connes-Skandalis index theorem to the context of twisted groupoid actions, offering new insights into the non-commutative geometry of singular spaces.