EQUIVARIANT PERIODIC CYCLIC HOMOLOGY

Abstract

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