nLab crossed product C*-algebra

Idea

A crossed product C*-algebra.

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.

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

Related concepts

• Baum-Connes conjecture

References

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

• Bruce Blackadar, K-Theory for Operator Algebras

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

• Alain Connes, A walk in the non-commutative garden (arXiv:math/0601054)

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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