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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/18899
Ver los metadatos del registro completo
id |
CONICETDig_c3c3fc1c104a278e52b3e0fcb0bb706f |
---|---|
oai_identifier_str |
oai:ri.conicet.gov.ar:11336/18899 |
network_acronym_str |
CONICETDig |
repository_id_str |
3498 |
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 |
_version_ |
1846082634636066816 |
score |
13.22299 |