Recasting the Elliott Conjecture

Let A be a simple, unital, finite, and exact C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this em...

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Authors: Perera Domènech, Francesc|||0000-0002-4669-4736, Toms, Andrew S.
Format: article
Publication Date:2006
Country:España
Institution:Universitat Autònoma de Barcelona
Repository:Dipòsit Digital de Documents de la UAB
Language:English
OAI Identifier:oai:ddd.uab.cat:44150
Online Access:https://ddd.uab.cat/record/44150
Access Level:Open access
Keyword:C*-àlgebres
Invariants
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spelling Recasting the Elliott ConjecturePerera Domènech, Francesc|||0000-0002-4669-4736Toms, Andrew S.C*-àlgebresInvariantsLet A be a simple, unital, finite, and exact C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomor phism, and prove the conjecture in several cases. In these same cases - Z-stable algebras all - we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of Z-stable C*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott's classification conjecture -that K-theoretic invariants will classify separable and nuclear C*-algebras- with the recent appearance of counterexamples to its strongest concrete form.Centre de Recerca MatemàticaCentre de Recerca Matemàtica 22006-01-0120062006-01-01Articlehttp://purl.org/coar/resource_type/c_6501AOhttp://purl.org/coar/version/c_b1a7d7d4d402bcceinfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/44150reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.https://creativecommons.org/licenses/by-nc-nd/2.5/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:441502026-06-06T12:50:31Z
dc.title.none.fl_str_mv Recasting the Elliott Conjecture
title Recasting the Elliott Conjecture
spellingShingle Recasting the Elliott Conjecture
Perera Domènech, Francesc|||0000-0002-4669-4736
C*-àlgebres
Invariants
title_short Recasting the Elliott Conjecture
title_full Recasting the Elliott Conjecture
title_fullStr Recasting the Elliott Conjecture
title_full_unstemmed Recasting the Elliott Conjecture
title_sort Recasting the Elliott Conjecture
dc.creator.none.fl_str_mv Perera Domènech, Francesc|||0000-0002-4669-4736
Toms, Andrew S.
author Perera Domènech, Francesc|||0000-0002-4669-4736
author_facet Perera Domènech, Francesc|||0000-0002-4669-4736
Toms, Andrew S.
author_role author
author2 Toms, Andrew S.
author2_role author
dc.contributor.none.fl_str_mv Centre de Recerca Matemàtica
dc.subject.none.fl_str_mv C*-àlgebres
Invariants
topic C*-àlgebres
Invariants
description Let A be a simple, unital, finite, and exact C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomor phism, and prove the conjecture in several cases. In these same cases - Z-stable algebras all - we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of Z-stable C*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott's classification conjecture -that K-theoretic invariants will classify separable and nuclear C*-algebras- with the recent appearance of counterexamples to its strongest concrete form.
publishDate 2006
dc.date.none.fl_str_mv 2
2006-01-01
2006
2006-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AO
http://purl.org/coar/version/c_b1a7d7d4d402bcce
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/44150
url https://ddd.uab.cat/record/44150
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/2.5/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/2.5/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Centre de Recerca Matemàtica
publisher.none.fl_str_mv Centre de Recerca Matemàtica
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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