nLab equivariant KK-theory

group theory

Cohomology and Extensions

Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

Contents

Idea

The equivariant cohomology version of KK-theory.

Definition

Let $G$ be a locally compact topological group.

Definition

$KK_G$ is the category

Definition

The KK-theoretic representation ring of $G$ is the ring

$R(G) \simeq KK_G(\mathbb{C}, \mathbb{C}) \,.$

Properties

Theorem

(Green-Julg theorem)

Let $G$ be a topological group acting on a C*-algebra $A$.

1. If $G$ is a compact topological group then the descent map

$KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)$

is an isomorphism, identifying the equivariant operator K-theory of $A$ with the ordinary operator K-theory of the crossed product C*-algebra $G \ltimes A$.

2. if $G$ is a discrete group then the descent map

$KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})$

is an isomorphism, identifying the equivariant K-homology of $A$ with the ordinary K-homology of the crossed product C*-algebra $G \ltimes A$.

Proposition

If $G$ is a compact topological group, then the KK-theoretic representation ring, def. coincides with the ordinary representation ring of $G$.

Section 20 of