Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category

TL;DR

This paper constructs a Quillen model category structure on certain C*-algebras, enabling the homotopy classes of maps to directly correspond to Kasparov KK-theory groups, thus providing a new categorical perspective.

Contribution

It introduces a Quillen model category framework for KK-theory of C*-algebras, answering an open question and linking homotopy theory with operator algebra K-theory.

Findings

Homotopy classes of maps coincide with KK-groups for separable C*-algebras.

Provides a categorical model for KK-theory using Quillen model categories.

Answers an open question about categorical descriptions of KK-theory.

Abstract

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.