A survey on operator $K$-theory via homotopical algebra

TL;DR

This survey explores spectrum-based operator K-theory and KK-theory for C*-algebras using homotopical algebra and ∞-categories, providing new insights and simplified proofs of classical results.

Contribution

It introduces spectrum-valued versions of K- and KK-theory via ∞-categorical frameworks, offering a homotopy theoretic perspective on classical operator algebra results.

Findings

Homotopy theoretic proofs of Swan's theorems

Künneth and universal coefficient formulas in spectrum context

New aspects of twisted K-theory and multiplicative structures

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) $\infty$-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, K\"unneth 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 $\infty$-category.