nLab model structure on operator algebras

Overview

So far there are only few works on homotopy theory for operator algebras. One of the basic checks for good homotopy theory of operator algebras is that the Kasparov KK-groups should be obtained from Hom-s in the appropriate stable category of operator algebras. This subject is important in order to introduce more systematic homotopic methods in noncommutative geometry à la Alain Connes.

Related concepts

• homotopical structure on C*-algebras

References

Quillen model category structures on various categories of operator algebras have been introduced and studied in

• Michael Joachim, Mark W. Johnson, Realizing Kasparov’s KK-theory groups as the homotopy classes of maps of a Quillen model category, An alpine anthology of homotopy theory, Contemp. Math. 399, Amer. Math. Soc. 2006, pp. 163–197, MR2007c:46070

• Paul Arne Østvær, Homotopy theory of $C^*$-algebras, Frontiers in Mathematics, Springer Basel, 2010, (arxiv/0812.0154, pdf)

and a category of fibrant objects approach in

• Otgonbayar Uuye, Homotopy theory for $C^*$-algebras, arxiv/1011.2926

In fact, Baues fibration category structure has been constructed in 1997, not only on the category of $C^\ast$-algebras but on a wider class of similar categories:

• Kasper K. S. Andersen, Jesper Grodal, A Baues fibration category structure on Banach and $C^\ast$-algebras, pdf

See at homotopical structure on C*-algebras.

The derived category/triangulated category approach to operator algebras is introduced in

• Ralf Meyer, Ryszard Nest, Homological algebra in bivariant K-theory and other triangulated categories, math.KT/0702146

• Ralf Meyer, Ryszard Nest, The Baum-Connes conjecture via localisation of categories), Topology 45 (2006), no. 2, 209–259, math.KT/0312292, MR2006k:19013

where a noncommutative analogue of the stable category of spectra is introduced.

There is also a related survey

• Ralf Meyer, Categorical aspects of bivariant K-theory, K-theory and noncommutative geometry, European Math. Soc. Ser. Congr. Rep., math.KT/0702145, MR2010h:19008