The abstract machinery of Homotopical Algebra introduced by D.
Quillen, coupled with Gel’fand-Naimark’s theorem (which asserts
nothing but an (anti-)equivalence of categories C*-Alg ∼ LCHaus)
suggests how natural the existence of homotopical methods in
C*-algebra theory should be.
The main problem faced in Uuye's paper is that the category of
C*-algebras admits an homotopical calculus which can’t be extended to
a full model structure in the sense of Quillen, a theorem Uuye takes
from an unpublished argument by Andersen and Grodal; the plan to
overcome this difficulty is to seek for a weaker form of Homotopical
Calculus, still fitting our needs.
REFERENCES:
[Brown] Kenneth S. Brown, Abstract Homotopy Theory and Generalized
Sheaf Cohomology, Transactions of the American Mathematical Society,
Vol. 186 (Dec. 1973), 419-458.
[Quillen] Daniel G. Quillen, Homotopical algebra, Lecture Notes in
Mathematics, No. 43, Springer-Verlag, Berlin, 1967. MR 0223432.
[Uuye] Otgonbayar Uuye, Homotopy Theory for C*-algebras,
http://arxiv.org/abs/1011.2926 , to appear in Journal of
Noncommutative Geometry.