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KK- and E-theory via homotopy theory (2304.12607v2)
Published 25 Apr 2023 in math.KT, math.AT, and math.OA
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.
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