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

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.