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We provide a homotopy theorist's point of view on KK -and E-theory for C *algebras.We construct stable ∞-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 constructions from C *algebra theory, in particular of Cuntz' q-construction.We will also discuss operator algebra K-theory in this framework.
Ulrich Bunke (2024) studied this question.