# nLab equivariant KK-theory

Contents

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

## Higher genera

cohomology

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 $G$.

## References

Section 20 of

Last revised on July 18, 2013 at 13:13:22. See the history of this page for a list of all contributions to it.