Equivariant periodic cyclic homology

TL;DR

This paper develops a new form of equivariant periodic cyclic homology for locally compact groups, extending noncommutative geometry tools to a broader setting with novel features and key invariance properties.

Contribution

It introduces a noncommutative equivariant cyclic homology theory that generalizes previous work and proves fundamental properties like homotopy invariance, stability, excision, and a Green-Julg theorem.

Findings

Defines equivariant periodic cyclic homology for locally compact groups.

Proves homotopy invariance, stability, and excision of the theory.

Establishes a Green-Julg theorem for compact and discrete groups.

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.