nLab
crossed product 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

Noncommutative geometry

Contents

Idea

A crossed product C*-algebra.

Properties

Equivalence to convolution algebra

If the AA is a C*-algebra of functions on some topological space XX and the action of GG on AA is induced from an action on XX, then in good cases the crossed product algebra is equivalent to the convolution algebra of the action groupoid.

e.g. (Land 13, prop. 4.2, Soltan, section 5.1, Bos 07, example 12.2.8) also (Connes 06, around p. 13)

References

A standard textbook account (with an eye towards KK-theory) is section 10 of

Original articles include

  • M. A. Rieffel, Proper actions of groups on C∗-algebras Mappings of operator algebras (Philadelphia, PA, 1988), 141–182, Progr. Math. 84, Birkhäuser Boston, Boston, MA, 1990. (pdf)

  • Jonathan Henry Brown, Proper actions of groupoids on C *C^*-algebras (arXiv:0907.5570)

Detailed lecture notes are in

  • Piotr Soltan, C *C^\ast-Algebras, group actions and crossed products, lecture notes (pdf)

A survey of noncommutative geometry with a bunch of examples of crossed product C *C^\ast-algebras is

Further discussion in the context of the Baum-Connes conjecture is in

  • Markus Land, The Analytical Assembly Map and Index Theory, (arXiv:1306.5657)

Applications to geometric quantization and specifically geometric quantization by push-forward are indicated in

  • Roger Bos, Groupoids in geometric quantization, 2007 (pdf)

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Last revised on July 12, 2013 at 10:59:10. See the history of this page for a list of all contributions to it.