nLab
equivariant KK-theory

Contents

Context

Index theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Group Theory

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Noncommutative geometry

Contents

Idea

The equivariant cohomology version of KK-theory.

Definition

Let GG be a locally compact topological group.

Definition

KK GKK_G is the category

(Blackadar, section 20.2) The composition is in (Blackadar, theorem 20.3.1) .

Definition

The KK-theoretic representation ring of GG is the ring

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

(Blackadar def. 20.4.2)

Properties

The Green-Julg theorem:

Theorem

(Green-Julg theorem)

Let GG be a topological group acting on a C*-algebra AA.

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

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

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

  2. if GG is a discrete group then the descent map

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

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

(Blackadar, 20.2.7).

Proposition

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

(Blackadar, prop. 20.4.4)

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.