Mahanta, Snigdhayan (2015) Colocalizations of noncommutative spectra and bootstrap categories. ADVANCES IN MATHEMATICS, 285. pp. 72-100. ISSN 0001-8708, 1090-2082
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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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | K-THEORY; THEOREM; HOMOTOPY; ALGEBRA; ORDER; Noncommutative spectra; Stable infinity-categories; Triangulated categories; Bootstrap categories; (Co)localizations; C-*-algebras; Bivariant K-theory |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 07 May 2019 09:57 |
| Last Modified: | 07 May 2019 09:57 |
| URI: | https://pred.uni-regensburg.de/id/eprint/4468 |
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