# nLab crossed product C*-algebra

Contents

### Context

#### Noncommutative geometry

noncommutative geometry

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

# Contents

## Properties

### Equivalence to convolution algebra

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

## 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 $C^*$-algebras (arXiv:0907.5570)

Detailed lecture notes are in

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

A survey of noncommutative geometry with a bunch of examples of crossed product $C^\ast$-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.