A bivariant theory for the Cuntz semigroup

Bosa, J., Tornetta, G. and Zacharias, J. (2019) A bivariant theory for the Cuntz semigroup. Journal of Functional Analysis, 277(4), pp. 1061-1111. (doi: 10.1016/j.jfa.2019.05.002)

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We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many features formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C⁎-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C⁎-algebras. Finally, we show how to use the bivariant Cuntz semigroup of the present work to classify unital and stably finite C⁎-algebras.

Item Type:Articles
Additional Information:This research was also supported by Juan de la Cierva - Incorporacio ́n (IJCI2015-25237) and is partially supported by the project MTM2014-53644-P
Glasgow Author(s) Enlighten ID:Tornetta, Mr Gabriele and Bosa, Dr Joan and Zacharias, Professor Joachim
Authors: Bosa, J., Tornetta, G., and Zacharias, J.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Functional Analysis
ISSN (Online):1096-0783
Published Online:11 May 2019
Copyright Holders:Copyright © 2019 The Authors
First Published:First published in Journal of Functional Analysis 277(4):1061-1111
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
626331The Cuntz Semigroup and the Fine Stucture of Nuclear C*-AlgebrasJoachim ZachariasEngineering and Physical Sciences Research Council (EPSRC)EP/I019227/2M&S - MATHEMATICS