Abstract
We construct a compactly generated and closed symmetric monoidal stable infinity-category NSp' and show that hNSp'(op) contains the suspension stable homotopy category of separable C*-algebras Sigma Ho-C* constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of NSp', namely, NSp'[K-1] and NSp'[Z(-1)], both of which are shown to be ...
Abstract
We construct a compactly generated and closed symmetric monoidal stable infinity-category NSp' and show that hNSp'(op) contains the suspension stable homotopy category of separable C*-algebras Sigma Ho-C* constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of NSp', namely, NSp'[K-1] and NSp'[Z(-1)], both of which are shown to be compactly generated and closed symmetric monoidal. We prove that Kasparov KK-category of separable C*-algebras sits inside the homotopy category of KK infinity := NSp'[K-1](op) as a fully faithful triangulated subcategory. Hence KK infinity should be viewed as the stable infinity-categorical incarnation of Kasparov KK-category for arbitrary pointed noncommutative spaces (including non separable C*-algebras). As an application we find that the bootstrap category in hNSp'[K-1] admits a completely algebraic description. We also construct a K-theoretic bootstrap category in hKK(infinity) that extends the construction of the UCT class by Rosenberg-Schochet. Motivated by the algebraization problem we finally analyze a couple of equivalence relations on separable C*-algebras that are introduced via the bootstrap categories in various colocalizations of NSp'. (C) 2015 Elsevier Inc. All rights reserved.
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