Equivariant periodic cyclic homology

Abstract

We define and study equivariant periodic cyclic homology for locally compact groups. This can be viewed as a noncommutative generalization of equivariant de Rham cohomology. Although the construction resembles the Cuntz-Quillen approach to ordinary cyclic homology, a completely new feature in the equivariant setting is the fact that the basic ingredient in the theory is not a complex in the usual sense. As a consequence, in the equivariant context only the periodic cyclic theory can be defined in complete generality. Our definition recovers particular cases studied previously by various authors. We prove that bivariant equivariant periodic cyclic homology is homotopy invariant, stable and satisfies excision in both variables. Moreover we construct the exterior product which generalizes the obvious composition product. Finally we prove a Green-Julg theorem in cyclic homology for compact groups and the dual result for discrete groups.