A survey on operator K-theory via homotopical algebra

2023/11/28 by Bunke, Ulrich, Land, Markus, Pennig, Ulrich
#FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2311.17191

Abstract

This is a survey article with the goal to advertise spectrum valued versions of K- and KK- theory for C^*-algebras via a (stable and symmetric monoidal) ∞-categorical enhancement of Kasparov's classical KK-theory. The main purpose is to present, in the simplest case, homotopy theoretic arguments for classical results on operator K-theory, including Swan's theorems, Künneth and universal coefficient formulas, the bootstrap class, variations of Karoubi's conjecture, and spectra of units for strongly self-absorbing C^*-algebras, as well as some new aspects on twisted K-theory and coherent multiplicative structures on C^*-algebras, viewed as objects in the previously mentioned ∞-category.

Related

• The generators and relations picture of KK-theory
• KK-theory spectra for C^∗-categories and discrete groupoid C^∗-algebras
• The spectral model of (Real) K-theory
• Operator K-theory algebra spectra of C^*-algebras
• Real K-Theory for C^*-Algebras: Just the Facts