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.

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

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

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

Revised on January 10, 2014 07:43:22
by Zoran Škoda
(161.53.130.104)