Operator ideals and assembly maps in K-theory

Autores
Cortiñas, Guillermo Horacio; Tartaglia, Gisela
Año de publicación
2013
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = S p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map HG ∗ (E(G, Vcyc), K(S)) → K∗(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients S and the use of some results about algebraic K-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu’s result. Our proof uses the usual Chern character to cyclic homology. Like Yu’s, it relies on results on algebraic K-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy K-theory. We prove that the rational assembly map HG ∗ (E(G, Fin), KH(L p )) ⊗ Q → KH∗(L p [G]) ⊗ Q is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.
Fil: Cortiñas, Guillermo Horacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina
Fil: Tartaglia, Gisela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina
Materia
Algebraic K-theory
Schatten ideals
Isomorphism conjecture
Cyclic homology
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/18825
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/18825
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Operator ideals and assembly maps in K-theoryCortiñas, Guillermo HoracioTartaglia, GiselaAlgebraic K-theorySchatten idealsIsomorphism conjectureCyclic homologyhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = S p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map HG ∗ (E(G, Vcyc), K(S)) → K∗(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients S and the use of some results about algebraic K-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu’s result. Our proof uses the usual Chern character to cyclic homology. Like Yu’s, it relies on results on algebraic K-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy K-theory. We prove that the rational assembly map HG ∗ (E(G, Fin), KH(L p )) ⊗ Q → KH∗(L p [G]) ⊗ Q is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.Fil: Cortiñas, Guillermo Horacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; ArgentinaFil: Tartaglia, Gisela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; ArgentinaAmerican Mathematical Society2013-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/18825Cortiñas, Guillermo Horacio; Tartaglia, Gisela; Operator ideals and assembly maps in K-theory; American Mathematical Society; Proceedings of the American Mathematical Society; 142; 12-2013; 1089-10990002-99391088-6826CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1090/S0002-9939-2013-11837-Xinfo:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1202.4999info:eu-repo/semantics/altIdentifier/url/http://www.ams.org/journals/proc/2014-142-04/S0002-9939-2013-11837-X/home.htmlinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:11:25Zoai:ri.conicet.gov.ar:11336/18825instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 10:11:25.275CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Operator ideals and assembly maps in K-theory
title Operator ideals and assembly maps in K-theory
spellingShingle Operator ideals and assembly maps in K-theory
Cortiñas, Guillermo Horacio
Algebraic K-theory
Schatten ideals
Isomorphism conjecture
Cyclic homology
title_short Operator ideals and assembly maps in K-theory
title_full Operator ideals and assembly maps in K-theory
title_fullStr Operator ideals and assembly maps in K-theory
title_full_unstemmed Operator ideals and assembly maps in K-theory
title_sort Operator ideals and assembly maps in K-theory
dc.creator.none.fl_str_mv Cortiñas, Guillermo Horacio
Tartaglia, Gisela
author Cortiñas, Guillermo Horacio
author_facet Cortiñas, Guillermo Horacio
Tartaglia, Gisela
author_role author
author2 Tartaglia, Gisela
author2_role author
dc.subject.none.fl_str_mv Algebraic K-theory
Schatten ideals
Isomorphism conjecture
Cyclic homology
topic Algebraic K-theory
Schatten ideals
Isomorphism conjecture
Cyclic homology
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = S p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map HG ∗ (E(G, Vcyc), K(S)) → K∗(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients S and the use of some results about algebraic K-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu’s result. Our proof uses the usual Chern character to cyclic homology. Like Yu’s, it relies on results on algebraic K-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy K-theory. We prove that the rational assembly map HG ∗ (E(G, Fin), KH(L p )) ⊗ Q → KH∗(L p [G]) ⊗ Q is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.
Fil: Cortiñas, Guillermo Horacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina
Fil: Tartaglia, Gisela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina
description Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = S p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map HG ∗ (E(G, Vcyc), K(S)) → K∗(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients S and the use of some results about algebraic K-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu’s result. Our proof uses the usual Chern character to cyclic homology. Like Yu’s, it relies on results on algebraic K-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy K-theory. We prove that the rational assembly map HG ∗ (E(G, Fin), KH(L p )) ⊗ Q → KH∗(L p [G]) ⊗ Q is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.
publishDate 2013
dc.date.none.fl_str_mv 2013-12
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/18825
Cortiñas, Guillermo Horacio; Tartaglia, Gisela; Operator ideals and assembly maps in K-theory; American Mathematical Society; Proceedings of the American Mathematical Society; 142; 12-2013; 1089-1099
0002-9939
1088-6826
CONICET Digital
CONICET
url http://hdl.handle.net/11336/18825
identifier_str_mv Cortiñas, Guillermo Horacio; Tartaglia, Gisela; Operator ideals and assembly maps in K-theory; American Mathematical Society; Proceedings of the American Mathematical Society; 142; 12-2013; 1089-1099
0002-9939
1088-6826
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1090/S0002-9939-2013-11837-X
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1202.4999
info:eu-repo/semantics/altIdentifier/url/http://www.ams.org/journals/proc/2014-142-04/S0002-9939-2013-11837-X/home.html
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society
publisher.none.fl_str_mv American Mathematical Society
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.13397

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