TY - JOUR
T1 - C*-ALGEBRAS OF STABLE RANK ONE AND THEIR CUNTZ SEMIGROUPS
AU - Antoine, Ramon
AU - Perera, Francesc
AU - Robert, Leonel
AU - Thiel, Hannes
N1 - Funding Information:
de Catalunya grant 2017-SGR-1725. Thiel’s work was partially supported by the German Research Foundation (DFG) under the project SFB 878 (Groups, Geometry and Actions) and under Germany’s Excellence Strategy EXC 2044-390685587 (Mathematics Münster: Dynamics-Geometry-Structure).
Funding Information:
Antoine and Perera’s work was partially supported by Ministry of Economic Affairs and Digital Transformation (MINECO) grants MTM2014-53644-P and MTM2017-83487-P, and by Comissionat per Universitats i Recerca de la Generalitat Since B A ˝ K, the displayed formula also evaluates to 0 in B. That is, for every " > 0 there exists w 2 B such that kb w wk < " and kbww k < ". Since B is separable, [32, Theorem 2.1 and Proposition 2.2] implies that B is stable.
Funding Information:
Acknowledgments. This work was initiated during the intensive research program “Operator Algebras: Dynamics and Interactions” at the Centre de Recerca Matemàtica (CRM) in Barcelona, in the Spring of 2017. The authors would like to thank the CRM for financial support and for providing inspiring working conditions. Part of this research was conducted while the authors attended the workshop “Future Targets in the Classification Program for Amenable C*-Algebras” at the Banff International Research Station (BIRS), September 2017, and while they attended the mini-workshop on the Cuntz semigroup at the University of Houston, June 2018. The authors would like to thank the involved institutions for their kind hospitality. It is also a pleasure to thank the anonymous referees for their thorough reading of a former version of this work and for providing many detailed comments and suggestions which have greatly improved the paper.
Publisher Copyright:
© 2022 Duke University Press. All rights reserved.
PY - 2022/1/15
Y1 - 2022/1/15
N2 - The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: we answer affirmatively, for the class of stable rank-one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the global Glimm halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.
AB - The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: we answer affirmatively, for the class of stable rank-one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the global Glimm halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.
UR - http://www.scopus.com/inward/record.url?scp=85124151656&partnerID=8YFLogxK
U2 - 10.1215/00127094-2021-0009
DO - 10.1215/00127094-2021-0009
M3 - Article
AN - SCOPUS:85124151656
SN - 0012-7094
VL - 171
SP - 33
EP - 99
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 1
ER -