nLab
nuclear C*-algebra

Contents

Context

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

Contents

Idea

A type of C*-algebra.

Many equivalent definitions, not all of which were known to be equivalent when they were made. Usually credited to Takesaki (1964). An important characterization in terms of an appropriate “approximation property” was given by Lance (1972).

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Example

Every commutative C*-algebra is nuclear.

If HH is an infinite-dimensional Hilbert space then B(H)B(H) is not nuclear.

Properties

Relation to KK-theory

On K-nuclear C *C^\ast-algebras CC, the KK-theory functor KK(C,)KK(C,-) preserves short exact sequences in the middle (satisfies excision). (Kasparov 80, Skandalis 88).

Hence restricted to nuclear C*C*-algebras the canonical functor KKEKK \to E from KK-theory to E-theory (see there) is a full and faithful functor.

References

The notion was studied as a condition for excision in KK-theory in section 7 of

The generalization to K-nuclearity was introduced and discussed in

  • Georges Skandalis, Une notion de nuclearité en K-theorie, K-Theory 1 (1988) 549-574.

Last revised on September 26, 2016 at 18:48:11. See the history of this page for a list of all contributions to it.