Homotopy Theory for C^{*}-algebras

TL;DR

This paper applies the category of fibrant objects framework to C^{*}-algebras, unifying homotopy theory, KK-theory, and E-theory within a single homotopical approach.

Contribution

It introduces a unified homotopy-theoretic framework for C^{*}-algebras, encompassing various existing theories through fibrant categories.

Findings

Unified treatment of homotopy, KK-theory, and E-theory for C^{*}-algebras

Expresses these theories as homotopy categories of fibrant objects

Provides a categorical framework for analyzing C^{*}-algebras

Abstract

Category of fibrant objects is a convenient framework to do homotopy theory, introduced and developed by Ken Brown. In this paper, we apply it to the category of C^{*}-algebras. In particular, we get a unified treatment of (ordinary) homotopy theory for C^{*}-algebras, KK-theory and E-theory, as all of these can be expressed as the homotopy category of a category of fibrant objects.