KK- and E-theory via homotopy theory

Abstract

We provide a homotopy theorist's point of view on $KK$- and $E$-theory for $C^{*}$-algebras. We construct stable $\infty$-categories representing these theories through a sequence of Dwyer-Kan localizations of the category of $C^{*}$-algebras. Thereby we will reveal the homotopic theoretic meaning of various classical construction from $C^{*}$-algebra theory, in particular of Cuntz' $q$-construction. We will also discuss operator algebra $K$-theory in this framework.