Partial C*-dynamics and Rokhlin dimension
Resumen:
We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product A G by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing C∗-algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.
2022 | |
Dynamical systems and the theory of C∗-algebras | |
Inglés | |
Universidad de la República | |
COLIBRI | |
https://hdl.handle.net/20.500.12008/38122 | |
Acceso abierto | |
Licencia Creative Commons Atribución (CC - By 4.0) |
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---|---|
author | Abadie, Fernando |
author2 | Gardella, Eusebio Geffen, Shirly |
author2_role | author author |
author_facet | Abadie, Fernando Gardella, Eusebio Geffen, Shirly |
author_role | author |
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collection | COLIBRI |
dc.contributor.filiacion.none.fl_str_mv | Abadie Fernando, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática. Gardella Eusebio, Universidad de Münster Geffen Shirly, Universidad de Negev |
dc.creator.none.fl_str_mv | Abadie, Fernando Gardella, Eusebio Geffen, Shirly |
dc.date.accessioned.none.fl_str_mv | 2023-07-13T14:21:17Z |
dc.date.available.none.fl_str_mv | 2023-07-13T14:21:17Z |
dc.date.issued.none.fl_str_mv | 2022 |
dc.description.abstract.none.fl_txt_mv | We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product A G by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing C∗-algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies. |
dc.format.extent.es.fl_str_mv | 34 h |
dc.format.mimetype.es.fl_str_mv | application/pdf |
dc.identifier.citation.es.fl_str_mv | Abadie, F, Gardella, E y Geffen, S. "Partial C*-dynamics and Rokhlin dimension". Ergodic Theory and Dynamical Systems. [en línea] 2022, 42(10): 2991–3024. 34 h. |
dc.identifier.doi.none.fl_str_mv | 10.1017/etds.2021.82 |
dc.identifier.issn.none.fl_str_mv | 1469-4417 |
dc.identifier.uri.none.fl_str_mv | https://hdl.handle.net/20.500.12008/38122 |
dc.language.iso.none.fl_str_mv | en eng |
dc.publisher.es.fl_str_mv | Cambridge University Press |
dc.relation.ispartof.es.fl_str_mv | Ergodic Theory and Dynamical Systems, 2022, 42(10): 2991–3024 |
dc.rights.license.none.fl_str_mv | Licencia Creative Commons Atribución (CC - By 4.0) |
dc.rights.none.fl_str_mv | info:eu-repo/semantics/openAccess |
dc.source.none.fl_str_mv | reponame:COLIBRI instname:Universidad de la República instacron:Universidad de la República |
dc.subject.es.fl_str_mv | Dynamical systems and the theory of C∗-algebras |
dc.title.none.fl_str_mv | Partial C*-dynamics and Rokhlin dimension |
dc.type.es.fl_str_mv | Artículo |
dc.type.none.fl_str_mv | info:eu-repo/semantics/article |
dc.type.version.none.fl_str_mv | info:eu-repo/semantics/publishedVersion |
description | We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product A G by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing C∗-algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies. |
eu_rights_str_mv | openAccess |
format | article |
id | COLIBRI_25e0c8a273b84a2c4f010a55b7cb2f6a |
identifier_str_mv | Abadie, F, Gardella, E y Geffen, S. "Partial C*-dynamics and Rokhlin dimension". Ergodic Theory and Dynamical Systems. [en línea] 2022, 42(10): 2991–3024. 34 h. 1469-4417 10.1017/etds.2021.82 |
instacron_str | Universidad de la República |
institution | Universidad de la República |
instname_str | Universidad de la República |
language | eng |
language_invalid_str_mv | en |
network_acronym_str | COLIBRI |
network_name_str | COLIBRI |
oai_identifier_str | oai:colibri.udelar.edu.uy:20.500.12008/38122 |
publishDate | 2022 |
reponame_str | COLIBRI |
repository.mail.fl_str_mv | mabel.seroubian@seciu.edu.uy |
repository.name.fl_str_mv | COLIBRI - Universidad de la República |
repository_id_str | 4771 |
rights_invalid_str_mv | Licencia Creative Commons Atribución (CC - By 4.0) |
spelling | Abadie Fernando, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.Gardella Eusebio, Universidad de MünsterGeffen Shirly, Universidad de Negev2023-07-13T14:21:17Z2023-07-13T14:21:17Z2022Abadie, F, Gardella, E y Geffen, S. "Partial C*-dynamics and Rokhlin dimension". Ergodic Theory and Dynamical Systems. [en línea] 2022, 42(10): 2991–3024. 34 h.1469-4417https://hdl.handle.net/20.500.12008/3812210.1017/etds.2021.82We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product A G by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing C∗-algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.Submitted by Faget Cecilia (lfaget@fcien.edu.uy) on 2023-07-13T12:34:16Z No. of bitstreams: 2 license_rdf: 19875 bytes, checksum: 9fdbed07f52437945402c4e70fa4773e (MD5) 101017etds202182.pdf: 369989 bytes, checksum: 961d5ad10898d8f19d4eecba72b0670a (MD5)Approved for entry into archive by Faget Cecilia (lfaget@fcien.edu.uy) on 2023-07-13T12:40:34Z (GMT) No. of bitstreams: 2 license_rdf: 19875 bytes, checksum: 9fdbed07f52437945402c4e70fa4773e (MD5) 101017etds202182.pdf: 369989 bytes, checksum: 961d5ad10898d8f19d4eecba72b0670a (MD5)Made available in DSpace by Luna Fabiana (fabiana.luna@seciu.edu.uy) on 2023-07-13T14:21:17Z (GMT). No. of bitstreams: 2 license_rdf: 19875 bytes, checksum: 9fdbed07f52437945402c4e70fa4773e (MD5) 101017etds202182.pdf: 369989 bytes, checksum: 961d5ad10898d8f19d4eecba72b0670a (MD5) Previous issue date: 202234 happlication/pdfenengCambridge University PressErgodic Theory and Dynamical Systems, 2022, 42(10): 2991–3024Las obras depositadas en el Repositorio se rigen por la Ordenanza de los Derechos de la Propiedad Intelectual de la Universidad de la República.(Res. Nº 91 de C.D.C. de 8/III/1994 – D.O. 7/IV/1994) y por la Ordenanza del Repositorio Abierto de la Universidad de la República (Res. Nº 16 de C.D.C. de 07/10/2014)info:eu-repo/semantics/openAccessLicencia Creative Commons Atribución (CC - By 4.0)Dynamical systems and the theory of C∗-algebrasPartial C*-dynamics and Rokhlin dimensionArtículoinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionreponame:COLIBRIinstname:Universidad de la Repúblicainstacron:Universidad de la RepúblicaAbadie, FernandoGardella, EusebioGeffen, ShirlyLICENSElicense.txtlicense.txttext/plain; charset=utf-84267http://localhost:8080/xmlui/bitstream/20.500.12008/38122/5/license.txt6429389a7df7277b72b7924fdc7d47a9MD55CC-LICENSElicense_urllicense_urltext/plain; charset=utf-844http://localhost:8080/xmlui/bitstream/20.500.12008/38122/2/license_urla0ebbeafb9d2ec7cbb19d7137ebc392cMD52license_textlicense_texttext/html; charset=utf-838552http://localhost:8080/xmlui/bitstream/20.500.12008/38122/3/license_text2fc523bba4df4b71d4fa008ef2dea84bMD53license_rdflicense_rdfapplication/rdf+xml; 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- Universidad de la Repúblicafalse |
spellingShingle | Partial C*-dynamics and Rokhlin dimension Abadie, Fernando Dynamical systems and the theory of C∗-algebras |
status_str | publishedVersion |
title | Partial C*-dynamics and Rokhlin dimension |
title_full | Partial C*-dynamics and Rokhlin dimension |
title_fullStr | Partial C*-dynamics and Rokhlin dimension |
title_full_unstemmed | Partial C*-dynamics and Rokhlin dimension |
title_short | Partial C*-dynamics and Rokhlin dimension |
title_sort | Partial C*-dynamics and Rokhlin dimension |
topic | Dynamical systems and the theory of C∗-algebras |
url | https://hdl.handle.net/20.500.12008/38122 |