K-homology and K-theory for the lamplighter groups of finite groups

Let F be a finite group. We consider the lamplighter group L = F ≀ Z over F. We prove that L has a classifying space for proper actions EL which is a complex of dimension 2.We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that theassembly map µLi: K...

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Autores: Flores Díaz, Ramón Jesús, Pooya, Sanaz, Valette, Alain
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2017
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/64166
Acceso en línea:http://hdl.handle.net/11441/64166
https://doi.org/10.1112/plms.12061
Access Level:acceso abierto
Palabra clave:K-theory
C ∗ -algebra
Proper actions
Baum-Connes conjecture
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spelling K-homology and K-theory for the lamplighter groups of finite groupsFlores Díaz, Ramón JesúsPooya, SanazValette, AlainK-theoryC ∗ -algebraProper actionsBaum-Connes conjectureLet F be a finite group. We consider the lamplighter group L = F ≀ Z over F. We prove that L has a classifying space for proper actions EL which is a complex of dimension 2.We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that theassembly map µLi: KLi(E L) → Ki(C∗L)(i =0, 1) is an isomorphism. Actually, K0(C∗L) is free abelian of countable rank, with an explicit basis consisting of projections in C∗L, while K1(C∗L) is infinite cyclic, generated by the unitary of C∗L implementing t he shift. Finally we show that,for F abelian, the C∗-algebra C∗L is completely characterized by |F | up to isomorphism.Ministerio de Ciencia e InnovaciónNational Science FoundationLondon Mathematical SocietyGeometría y TopologíaFQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y HomotopíaMinisterio de Ciencia e Innovación (MICIN). EspañaNational Science Foundation (NSF). United States2017info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/64166https://doi.org/10.1112/plms.12061reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésProceedings of the London Mathematical SocietyMTM2010-20692DMS-1440140http://onlinelibrary.wiley.com/doi/10.1112/plms.12061/epdfinfo:eu-repo/semantics/openAccessoai:idus.us.es:11441/641662026-06-17T12:51:07Z
dc.title.none.fl_str_mv K-homology and K-theory for the lamplighter groups of finite groups
title K-homology and K-theory for the lamplighter groups of finite groups
spellingShingle K-homology and K-theory for the lamplighter groups of finite groups
Flores Díaz, Ramón Jesús
K-theory
C ∗ -algebra
Proper actions
Baum-Connes conjecture
title_short K-homology and K-theory for the lamplighter groups of finite groups
title_full K-homology and K-theory for the lamplighter groups of finite groups
title_fullStr K-homology and K-theory for the lamplighter groups of finite groups
title_full_unstemmed K-homology and K-theory for the lamplighter groups of finite groups
title_sort K-homology and K-theory for the lamplighter groups of finite groups
dc.creator.none.fl_str_mv Flores Díaz, Ramón Jesús
Pooya, Sanaz
Valette, Alain
author Flores Díaz, Ramón Jesús
author_facet Flores Díaz, Ramón Jesús
Pooya, Sanaz
Valette, Alain
author_role author
author2 Pooya, Sanaz
Valette, Alain
author2_role author
author
dc.contributor.none.fl_str_mv Geometría y Topología
FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
Ministerio de Ciencia e Innovación (MICIN). España
National Science Foundation (NSF). United States
dc.subject.none.fl_str_mv K-theory
C ∗ -algebra
Proper actions
Baum-Connes conjecture
topic K-theory
C ∗ -algebra
Proper actions
Baum-Connes conjecture
description Let F be a finite group. We consider the lamplighter group L = F ≀ Z over F. We prove that L has a classifying space for proper actions EL which is a complex of dimension 2.We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that theassembly map µLi: KLi(E L) → Ki(C∗L)(i =0, 1) is an isomorphism. Actually, K0(C∗L) is free abelian of countable rank, with an explicit basis consisting of projections in C∗L, while K1(C∗L) is infinite cyclic, generated by the unitary of C∗L implementing t he shift. Finally we show that,for F abelian, the C∗-algebra C∗L is completely characterized by |F | up to isomorphism.
publishDate 2017
dc.date.none.fl_str_mv 2017
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/64166
https://doi.org/10.1112/plms.12061
url http://hdl.handle.net/11441/64166
https://doi.org/10.1112/plms.12061
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Proceedings of the London Mathematical Society
MTM2010-20692
DMS-1440140
http://onlinelibrary.wiley.com/doi/10.1112/plms.12061/epdf
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv London Mathematical Society
publisher.none.fl_str_mv London Mathematical Society
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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