Amenability and paradoxicality in semigroups and C*-algebras

We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable,unital) semigroups and corresponding semigroup rings. We consider also Følner’s type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Føln...

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Autores: Lledó, Fernando, Martínez, Diego
Formato: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2019
País:España
Recursos:Consejo Superior de Investigaciones Científicas (CSIC)
Repositorio:DIGITAL.CSIC. Repositorio Institucional del CSIC
OAI Identifier:oai:digital.csic.es:10261/197106
Acesso em linha:http://hdl.handle.net/10261/197106
Access Level:acceso abierto
Palavra-chave:Amenability
Paradoxical decompositions
Følner condition
Semigroups
Semigroup rings
Inverse semigroup C*-algebra
Proper infiniteness
Amenable traces
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spelling Amenability and paradoxicality in semigroups and C*-algebrasLledó, FernandoMartínez, DiegoAmenabilityParadoxical decompositionsFølner conditionSemigroupsSemigroup ringsInverse semigroup C*-algebraProper infinitenessAmenable tracesWe analyze the dichotomy amenable/paradoxical in the context of (discrete, countable,unital) semigroups and corresponding semigroup rings. We consider also Følner’s type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups S we give a characterization of invariant measures on S (in the sense of Day) in terms of two notions: domain measurability and localization. Given a unital representation of S in terms of partial bijections on some set X we define a natural generalization of the uniform Roe algebra of a group, which we denote by RX. We show that the following notions are then equivalent: (1) X is domain measurable; (2) X is not paradoxical; (3) X satisfies the domain Følner condition; (4) there is an algebraically amenable dense *-subalgebra of RX; (5) RX has an amenable trace; (6) RX is not properly infinite and (7) [0] 6= [1] in the K0-group of RX . We also show that any tracial state on RX is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of X. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of C ∗ (X) implies the amenability of X. The converse implication is false.Partially supported by MINECO and European Regional Development Fund, jointly, through the grant MTM2017-83487-P, by MINECO through the Mar´ıa de Maeztu Programme for Units of Excellence in R&D (MDM2014-0445) and by the Generalitat de Catalunya through the grant 2017-SGR-1725. 2 Supported by research projects MTM2017-84098-P and Severo Ochoa SEV-2015-0554 of the Spanish Ministry of Economy and Competition (MINECO), Spain. 3 Supported by research projects MTM2017-84098-P, Severo Ochoa SEV-2015-0554 and BES-2016-077968 of the Spanish Ministry of Economy and Competition (MINECO), Spain.Peer ReviewedMinisterio de Economía y Competitividad (España)Ministerio de Ciencia, Innovación y Universidades (España)Generalitat de Catalunya2019201920192019info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Preprintinfo:eu-repo/semantics/submittedVersionhttp://hdl.handle.net/10261/197106reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Inglés#PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE#info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MDM-2014-0445info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/SEV-2015info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/MTM2017-84098-Pinfo:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/MTM2017-83487-PMTM2017-84098-P/AEI/10.13039/501100011033info:eu-repo/semantics/openAccessoai:digital.csic.es:10261/1971062026-05-22T06:33:51Z
dc.title.none.fl_str_mv Amenability and paradoxicality in semigroups and C*-algebras
title Amenability and paradoxicality in semigroups and C*-algebras
spellingShingle Amenability and paradoxicality in semigroups and C*-algebras
Lledó, Fernando
Amenability
Paradoxical decompositions
Følner condition
Semigroups
Semigroup rings
Inverse semigroup C*-algebra
Proper infiniteness
Amenable traces
title_short Amenability and paradoxicality in semigroups and C*-algebras
title_full Amenability and paradoxicality in semigroups and C*-algebras
title_fullStr Amenability and paradoxicality in semigroups and C*-algebras
title_full_unstemmed Amenability and paradoxicality in semigroups and C*-algebras
title_sort Amenability and paradoxicality in semigroups and C*-algebras
dc.creator.none.fl_str_mv Lledó, Fernando
Martínez, Diego
author Lledó, Fernando
author_facet Lledó, Fernando
Martínez, Diego
author_role author
author2 Martínez, Diego
author2_role author
dc.contributor.none.fl_str_mv Ministerio de Economía y Competitividad (España)
Ministerio de Ciencia, Innovación y Universidades (España)
Generalitat de Catalunya
dc.subject.none.fl_str_mv Amenability
Paradoxical decompositions
Følner condition
Semigroups
Semigroup rings
Inverse semigroup C*-algebra
Proper infiniteness
Amenable traces
topic Amenability
Paradoxical decompositions
Følner condition
Semigroups
Semigroup rings
Inverse semigroup C*-algebra
Proper infiniteness
Amenable traces
description We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable,unital) semigroups and corresponding semigroup rings. We consider also Følner’s type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups S we give a characterization of invariant measures on S (in the sense of Day) in terms of two notions: domain measurability and localization. Given a unital representation of S in terms of partial bijections on some set X we define a natural generalization of the uniform Roe algebra of a group, which we denote by RX. We show that the following notions are then equivalent: (1) X is domain measurable; (2) X is not paradoxical; (3) X satisfies the domain Følner condition; (4) there is an algebraically amenable dense *-subalgebra of RX; (5) RX has an amenable trace; (6) RX is not properly infinite and (7) [0] 6= [1] in the K0-group of RX . We also show that any tracial state on RX is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of X. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of C ∗ (X) implies the amenability of X. The converse implication is false.
publishDate 2019
dc.date.none.fl_str_mv 2019
2019
2019
2019
dc.type.none.fl_str_mv info:eu-repo/semantics/article
http://purl.org/coar/resource_type/c_6501
Preprint
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10261/197106
url http://hdl.handle.net/10261/197106
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
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#PLACEHOLDER_PARENT_METADATA_VALUE#
#PLACEHOLDER_PARENT_METADATA_VALUE#
#PLACEHOLDER_PARENT_METADATA_VALUE#
#PLACEHOLDER_PARENT_METADATA_VALUE#
info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MDM-2014-0445
info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/SEV-2015
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/MTM2017-84098-P
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/MTM2017-83487-P
MTM2017-84098-P/AEI/10.13039/501100011033
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.source.none.fl_str_mv reponame:DIGITAL.CSIC. Repositorio Institucional del CSIC
instname:Consejo Superior de Investigaciones Científicas (CSIC)
instname_str Consejo Superior de Investigaciones Científicas (CSIC)
reponame_str DIGITAL.CSIC. Repositorio Institucional del CSIC
collection DIGITAL.CSIC. Repositorio Institucional del CSIC
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