Realizing Kasparovs KK-theory groups as the homotopy classes of maps of a Quillen model category

In this article we build a Quillen model category structure on the category of sequentially complete l.m.c.-C*-algebras such that the corresponding homotopy classes of maps Ho(A,B) for separable C*-algebras A and B coincide with the Kasparov groups KK(A,B). This answers an open question posed by Mark Hovey about the possibility of describing KK-theory for C*-algebras using the language of Quillen model categories.

💡 Research Summary

The paper establishes a full Quillen model category structure on the category 𝓒 of sequentially complete locally multiplicatively convex C*-algebras (l.m.c.-C*-algebras) and shows that for separable C*-algebras A and B the set of homotopy classes of maps Ho(A,B) in this model category coincides with Kasparov’s bivariant K‑theory group KK(A,B). This result directly answers an open problem posed by Mark Hovey, who asked whether KK‑theory could be expressed within the language of model categories.

The authors begin by defining the ambient category 𝓒. An l.m.c.-C*-algebra is a topological ‑algebra whose topology is generated by a family of submultiplicative seminorms; “sequentially complete’’ means that every Cauchy sequence converges with respect to this family. The authors prove that 𝓒 is complete and cocomplete: all small limits and colimits exist, and they are constructed using tensor products with the commutative C-algebras C(