Partial C*-dynamics and Rokhlin dimension

Abadie, Fernando - Gardella, Eusebio - Geffen, Shirly

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.


Detalles Bibliográficos
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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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
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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
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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