KK- and E-theory via homotopy theory

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