Cyclic homology, tight crossed products, and small stabilizations

Autores
Cortiñas, Guillermo Horacio
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In [1] we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases. Mathematics In [1] (arXiv:1212.5901) we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases.
Fil: Cortiñas, Guillermo Horacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Materia
Cyclic Homology
Relative K-Theory
Homotopy Invariance
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:11336/18899

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network_name_str CONICET Digital (CONICET)
spelling Cyclic homology, tight crossed products, and small stabilizationsCortiñas, Guillermo HoracioCyclic HomologyRelative K-TheoryHomotopy Invariancehttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In [1] we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases. Mathematics In [1] (arXiv:1212.5901) we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases.Fil: Cortiñas, Guillermo Horacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaEuropean Mathematical Society2014-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/18899Cortiñas, Guillermo Horacio; Cyclic homology, tight crossed products, and small stabilizations; European Mathematical Society; Journal of Noncommutative Geometry; 8; 4; 12-2014; 1191-12231661-6952CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1304.3508info:eu-repo/semantics/altIdentifier/doi/10.4171/JNCG/184info:eu-repo/semantics/altIdentifier/url/http://www.ems-ph.org/journals/show_abstract.php?issn=1661-6952&vol=8&iss=4&rank=11info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-15T14:23:02Zoai:ri.conicet.gov.ar:11336/18899instacron: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:34982025-10-15 14:23:03.209CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Cyclic homology, tight crossed products, and small stabilizations
title Cyclic homology, tight crossed products, and small stabilizations
spellingShingle Cyclic homology, tight crossed products, and small stabilizations
Cortiñas, Guillermo Horacio
Cyclic Homology
Relative K-Theory
Homotopy Invariance
title_short Cyclic homology, tight crossed products, and small stabilizations
title_full Cyclic homology, tight crossed products, and small stabilizations
title_fullStr Cyclic homology, tight crossed products, and small stabilizations
title_full_unstemmed Cyclic homology, tight crossed products, and small stabilizations
title_sort Cyclic homology, tight crossed products, and small stabilizations
dc.creator.none.fl_str_mv Cortiñas, Guillermo Horacio
author Cortiñas, Guillermo Horacio
author_facet Cortiñas, Guillermo Horacio
author_role author
dc.subject.none.fl_str_mv Cyclic Homology
Relative K-Theory
Homotopy Invariance
topic Cyclic Homology
Relative K-Theory
Homotopy Invariance
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In [1] we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases. Mathematics In [1] (arXiv:1212.5901) we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases.
Fil: Cortiñas, Guillermo Horacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
description In [1] we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases. Mathematics In [1] (arXiv:1212.5901) we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin’s correspondence. In the current article we compute the relative cyclic homology HC. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ K/ D K top .A/. Similarly, we prove that if A is a unital Banach algebra and `1 D S q<1 ` q , then K.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 it contains K top .A/ as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC. 1.A/ W IS.A// in terms of HC.`1.A/ W S.A// for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases.
publishDate 2014
dc.date.none.fl_str_mv 2014-12
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/11336/18899
Cortiñas, Guillermo Horacio; Cyclic homology, tight crossed products, and small stabilizations; European Mathematical Society; Journal of Noncommutative Geometry; 8; 4; 12-2014; 1191-1223
1661-6952
CONICET Digital
CONICET
url http://hdl.handle.net/11336/18899
identifier_str_mv Cortiñas, Guillermo Horacio; Cyclic homology, tight crossed products, and small stabilizations; European Mathematical Society; Journal of Noncommutative Geometry; 8; 4; 12-2014; 1191-1223
1661-6952
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1304.3508
info:eu-repo/semantics/altIdentifier/doi/10.4171/JNCG/184
info:eu-repo/semantics/altIdentifier/url/http://www.ems-ph.org/journals/show_abstract.php?issn=1661-6952&vol=8&iss=4&rank=11
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 European Mathematical Society
publisher.none.fl_str_mv European 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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