Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP

We characterize all separable L1-preduals X for which every weak* closed convex unbounded set in X* lacks the approximate fixed point property for nonexpansive mappings. Our result improves and completes the main theorem in [4], where this property was studied for C(α) and C0(α) spaces with α an inf...

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Autores:
Piasecki, Łukasz
Villada Bedoya, Jeimer Alveiro
Tipo de recurso:
Article of investigation
Fecha de publicación:
2023
Institución:
Tecnológico de Antioquia
Repositorio:
Repositorio Tdea
Idioma:
eng
OAI Identifier:
oai:dspace.tdea.edu.co:tdea/3841
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https://dspace.tdea.edu.co/handle/tdea/3841
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dc.title.none.fl_str_mv Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
title Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
spellingShingle Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
title_short Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
title_full Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
title_fullStr Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
title_full_unstemmed Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
title_sort Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPP
dc.creator.fl_str_mv Piasecki, Łukasz
Villada Bedoya, Jeimer Alveiro
dc.contributor.author.none.fl_str_mv Piasecki, Łukasz
Villada Bedoya, Jeimer Alveiro
description We characterize all separable L1-preduals X for which every weak* closed convex unbounded set in X* lacks the approximate fixed point property for nonexpansive mappings. Our result improves and completes the main theorem in [4], where this property was studied for C(α) and C0(α) spaces with α an infinite countable ordinal as well as for ℓ1-predual hyperplanes in c.
publishDate 2023
dc.date.accessioned.none.fl_str_mv 2023-09-05T17:58:43Z
dc.date.available.none.fl_str_mv 2023-09-05T17:58:43Z
dc.date.issued.none.fl_str_mv 2023
dc.type.spa.fl_str_mv Artículo de revista
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identifier_str_mv 0021-2172
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url https://dspace.tdea.edu.co/handle/tdea/3841
dc.language.iso.spa.fl_str_mv eng
language eng
dc.relation.citationendpage.spa.fl_str_mv 96
dc.relation.citationissue.spa.fl_str_mv 1
dc.relation.citationstartpage.spa.fl_str_mv 87
dc.relation.citationvolume.spa.fl_str_mv 254
dc.relation.ispartofjournal.spa.fl_str_mv Israel Journal of Mathematics
dc.relation.references.spa.fl_str_mv D. Alspach, A ℓ1-predual which is not isometric to a quotient of C(α), in Banach Spaces (Mérida, 1992), Contemporary Mathematics, Vol. 144, American mathematical Society, Providence, RI, 1993, pp. 9–14.
E. Casini, E. Miglierina and Ł. Piasecki, Hyperplanes in the space of convergent sequences and preduals of ℓ1, Canadian Mathematical Bulletin 58 (2015), 459–470.
T. Domínguez, On the failure of the fixed point property for some unbounded sets of ℓ1, Journal of Nonlinear and Convex Analysis 17 (2016), 1249–1257.
T. Domínguez, M. Japón and J. Villada, Linearly and directionally bounded weak-star closed sets and the AFPP, Israel Journal of Mathematics 230 (2019), 509–526.
K. Goebel and T. Kuczumow, A contribution to the theory of nonexpansive mappings, Bulletin of the Calcutta Mathematical Society 70 (1978), 355–357.
H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Die Grundlehren der mathematischen Wissenschaften, Vol. 208, Springer, New York—Heidelberg, 1974.
A. Lazar and J. Lindenstrauss, Banach spaces whose duals are L1spaces and their representing matrices, Acta Mathematica 126 (1971), 165–193.
E. Matoušková and S. Reich, Reflexivity and approximate fixed points, Studia Mathematica 159 (2003), 403–415.
A. Pełczyński, On C(S)-subspaces of separable Banach spaces, Studia Mathematica 31 (1968), 513–522.
W. Ray, Nonexpansive mappings on unbounded convex domains, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 26 (1978), 241–245.
S. Reich, The almost fixed point property for nonexpansive mappings, Proceedings of the American Mathematical Society 88 (1983), 44–46.
I. Shafrir, The approximate fixed point property in Banach and hyperbolic spaces, Israel Journal of Mathematics 71 (1990), 211–223.
R. Whitley, The Krein—Smulian theorem, Proceedings of the American Mathematical Society 97 (1986), 376–377.
M. Zippin, On some subspaces of Banach spaces whose duals are L1spaces, Proceedings of the American Mathematical Society 23 (1969), 378–385.
M. Zippin, Correction to “On some subspaces of Banach spaces whose duals are L1spaces”, Proceedings of the American Mathematical Society 146 (2018), 5257–5262.
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dc.format.extent.spa.fl_str_mv 10 páginas
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dc.publisher.spa.fl_str_mv Springer
dc.publisher.place.spa.fl_str_mv Israel
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spelling Piasecki, Łukasz495bd072-1a5c-4c4a-8414-79b78095db0aVillada Bedoya, Jeimer Alveiro7d95c24b-0d22-4a1a-940d-bc419066b7c12023-09-05T17:58:43Z2023-09-05T17:58:43Z20230021-2172https://dspace.tdea.edu.co/handle/tdea/38411565-8511We characterize all separable L1-preduals X for which every weak* closed convex unbounded set in X* lacks the approximate fixed point property for nonexpansive mappings. Our result improves and completes the main theorem in [4], where this property was studied for C(α) and C0(α) spaces with α an infinite countable ordinal as well as for ℓ1-predual hyperplanes in c.10 páginasimage/jpegengSpringerIsraelhttps://link.springer.com/article/10.1007/s11856-022-2388-1Separable Lindenstrauss spaces whose duals do not contain weak* closed convex unbounded sets having the AFPPArtículo de revistahttp://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/version/c_970fb48d4fbd8a8596187254Israel Journal of MathematicsD. Alspach, A ℓ1-predual which is not isometric to a quotient of C(α), in Banach Spaces (Mérida, 1992), Contemporary Mathematics, Vol. 144, American mathematical Society, Providence, RI, 1993, pp. 9–14.E. Casini, E. Miglierina and Ł. Piasecki, Hyperplanes in the space of convergent sequences and preduals of ℓ1, Canadian Mathematical Bulletin 58 (2015), 459–470.T. Domínguez, On the failure of the fixed point property for some unbounded sets of ℓ1, Journal of Nonlinear and Convex Analysis 17 (2016), 1249–1257.T. Domínguez, M. Japón and J. Villada, Linearly and directionally bounded weak-star closed sets and the AFPP, Israel Journal of Mathematics 230 (2019), 509–526.K. Goebel and T. Kuczumow, A contribution to the theory of nonexpansive mappings, Bulletin of the Calcutta Mathematical Society 70 (1978), 355–357.H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Die Grundlehren der mathematischen Wissenschaften, Vol. 208, Springer, New York—Heidelberg, 1974.A. Lazar and J. Lindenstrauss, Banach spaces whose duals are L1spaces and their representing matrices, Acta Mathematica 126 (1971), 165–193.E. Matoušková and S. Reich, Reflexivity and approximate fixed points, Studia Mathematica 159 (2003), 403–415.A. Pełczyński, On C(S)-subspaces of separable Banach spaces, Studia Mathematica 31 (1968), 513–522.W. Ray, Nonexpansive mappings on unbounded convex domains, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 26 (1978), 241–245.S. Reich, The almost fixed point property for nonexpansive mappings, Proceedings of the American Mathematical Society 88 (1983), 44–46.I. Shafrir, The approximate fixed point property in Banach and hyperbolic spaces, Israel Journal of Mathematics 71 (1990), 211–223.R. Whitley, The Krein—Smulian theorem, Proceedings of the American Mathematical Society 97 (1986), 376–377.M. Zippin, On some subspaces of Banach spaces whose duals are L1spaces, Proceedings of the American Mathematical Society 23 (1969), 378–385.M. Zippin, Correction to “On some subspaces of Banach spaces whose duals are L1spaces”, Proceedings of the American Mathematical Society 146 (2018), 5257–5262.info:eu-repo/semantics/closedAccesshttp://purl.org/coar/access_right/c_14cbTHUMBNAILSeparable Lindenstrauss Spaces whose duals do not contain a weak star closed convex unbounded sets with the AFPP.jpg.jpgSeparable Lindenstrauss Spaces whose duals do not contain a weak star closed convex unbounded sets with the AFPP.jpg.jpgGenerated Thumbnailimage/jpeg8645https://dspace.tdea.edu.co/bitstream/tdea/3841/3/Separable%20Lindenstrauss%20Spaces%20whose%20duals%20do%20not%20contain%20a%20weak%20star%20closed%20convex%20unbounded%20sets%20with%20the%20AFPP.jpg.jpgecc8d7427f3ce491127e5593e42c7029MD53open accessLICENSElicense.txtlicense.txttext/plain; charset=utf-814828https://dspace.tdea.edu.co/bitstream/tdea/3841/2/license.txt2f9959eaf5b71fae44bbf9ec84150c7aMD52open accessORIGINALSeparable Lindenstrauss Spaces whose duals do not contain a weak star closed convex unbounded sets with the AFPP.jpgSeparable Lindenstrauss Spaces whose duals do not contain a weak star closed convex unbounded sets with the AFPP.jpgimage/jpeg98674https://dspace.tdea.edu.co/bitstream/tdea/3841/1/Separable%20Lindenstrauss%20Spaces%20whose%20duals%20do%20not%20contain%20a%20weak%20star%20closed%20convex%20unbounded%20sets%20with%20the%20AFPP.jpge7da93fbbb04842a018d0eb2ec997cf2MD51open accesstdea/3841oai:dspace.tdea.edu.co:tdea/38412023-09-06 03:03:25.835open accessRepositorio Institucional Tecnologico de 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 incorporada en las Obras Colectivas.

b.	Distribuir copias o fonogramas de las Obras, exhibirlas públicamente, ejecutarlas públicamente y/o ponerlas a disposición pública, incluyéndolas como incorporadas en Obras Colectivas, según corresponda.

c.	Distribuir copias de las Obras Derivadas que se generen, exhibirlas públicamente, ejecutarlas públicamente y/o ponerlas a disposición pública.
Los derechos mencionados anteriormente pueden ser ejercidos en todos los medios y formatos, actualmente conocidos o que se inventen en el futuro. Los derechos antes mencionados incluyen el derecho a realizar dichas modificaciones en la medida que sean técnicamente necesarias para ejercer los derechos en otro medio o formatos, pero de otra manera usted no está autorizado para realizar obras derivadas. Todos los derechos no otorgados expresamente por el Licenciante quedan por este medio reservados, incluyendo pero sin limitarse a aquellos que se mencionan en las secciones 4(d) y 4(e).

4. Restricciones.
La licencia otorgada en la anterior Sección 3 está expresamente sujeta y limitada por las siguientes restricciones:

a.	Usted puede distribuir, exhibir públicamente, ejecutar públicamente, o poner a disposición pública la Obra sólo bajo las condiciones de esta Licencia, y Usted debe incluir una copia de esta licencia o del Identificador Universal de Recursos de la misma con cada copia de la Obra que distribuya, exhiba públicamente, ejecute públicamente o ponga a disposición pública. No es posible ofrecer o imponer ninguna condición sobre la Obra que altere o limite las condiciones de esta Licencia o el ejercicio de los derechos de los destinatarios otorgados en este documento. No es posible sublicenciar la Obra. Usted debe mantener intactos todos los avisos que hagan referencia a esta Licencia y a la cláusula de limitación de garantías. Usted no puede distribuir, exhibir públicamente, ejecutar públicamente, o poner a disposición pública la Obra con alguna medida tecnológica que controle el acceso o la utilización de ella de una forma que sea inconsistente con las condiciones de esta Licencia. Lo anterior se aplica a la Obra incorporada a una Obra Colectiva, pero esto no exige que la Obra Colectiva aparte de la obra misma quede sujeta a las condiciones de esta Licencia. Si Usted crea una Obra Colectiva, previo aviso de cualquier Licenciante debe, en la medida de lo posible, eliminar de la Obra Colectiva cualquier referencia a dicho Licenciante o al Autor Original, según lo solicitado por el Licenciante y conforme lo exige la cláusula 4(c).

b.	Usted no puede ejercer ninguno de los derechos que le han sido otorgados en la Sección 3 precedente de modo que estén principalmente destinados o directamente dirigidos a conseguir un provecho comercial o una compensación monetaria privada. El intercambio de la Obra por otras obras protegidas por derechos de autor, ya sea a través de un sistema para compartir archivos digitales (digital file-sharing) o de cualquier otra manera no será considerado como estar destinado principalmente o dirigido directamente a conseguir un provecho comercial o una compensación monetaria privada, siempre que no se realice un pago mediante una compensación monetaria en relación con el intercambio de obras protegidas por el derecho de autor.

c.	Si usted distribuye, exhibe públicamente, ejecuta públicamente o ejecuta públicamente en forma digital la Obra o cualquier Obra Derivada u Obra Colectiva, Usted debe mantener intacta toda la información de derecho de autor de la Obra y proporcionar, de forma razonable según el medio o manera que Usted esté utilizando: (i) el nombre del Autor Original si está provisto (o seudónimo, si fuere aplicable), y/o (ii) el nombre de la parte o las partes que el Autor Original y/o el Licenciante hubieren designado para la atribución (v.g., un instituto patrocinador, editorial, publicación) en la información de los derechos de autor del Licenciante, términos de servicios o de otras formas razonables; el título de la Obra si está provisto; en la medida de lo razonablemente factible y, si está provisto, el Identificador Uniforme de Recursos (Uniform Resource Identifier) que el Licenciante especifica para ser asociado con la Obra, salvo que tal URI no se refiera a la nota sobre los derechos de autor o a la información sobre el licenciamiento de la Obra; y en el caso de una Obra Derivada, atribuir el crédito identificando el uso de la Obra en la Obra Derivada (v.g., "Traducción Francesa de la Obra del Autor Original," o "Guión Cinematográfico basado en la Obra original del Autor Original"). Tal crédito puede ser implementado de cualquier forma razonable; en el caso, sin embargo, de Obras Derivadas u Obras Colectivas, tal crédito aparecerá, como mínimo, donde aparece el crédito de cualquier otro autor comparable y de una manera, al menos, tan destacada como el crédito de otro autor comparable.

d.	Para evitar toda confusión, el Licenciante aclara que, cuando la obra es una composición musical:

i.	Regalías por interpretación y ejecución bajo licencias generales. El Licenciante se reserva el derecho exclusivo de autorizar la ejecución pública o la ejecución pública digital de la obra y de recolectar, sea individualmente o a través de una sociedad de gestión colectiva de derechos de autor y derechos conexos (por ejemplo, SAYCO), las regalías por la ejecución pública o por la ejecución pública digital de la obra (por ejemplo Webcast) licenciada bajo licencias generales, si la interpretación o ejecución de la obra está primordialmente orientada por o dirigida a la obtención de una ventaja comercial o una compensación monetaria privada.

ii.	Regalías por Fonogramas. El Licenciante se reserva el derecho exclusivo de recolectar, individualmente o a través de una sociedad de gestión colectiva de derechos de autor y derechos conexos (por ejemplo, los consagrados por la SAYCO), una agencia de derechos musicales o algún agente designado, las regalías por cualquier fonograma que Usted cree a partir de la obra (“versión cover”) y distribuya, en los términos del régimen de derechos de autor, si la creación o distribución de esa versión cover está primordialmente destinada o dirigida a obtener una ventaja comercial o una compensación monetaria privada.

e.	Gestión de Derechos de Autor sobre Interpretaciones y Ejecuciones Digitales (WebCasting). Para evitar toda confusión, el Licenciante aclara que, cuando la obra sea un fonograma, el Licenciante se reserva el derecho exclusivo de autorizar la ejecución pública digital de la obra (por ejemplo, webcast) y de recolectar, individualmente o a través de una sociedad de gestión colectiva de derechos de autor y derechos conexos (por ejemplo, ACINPRO), las regalías por la ejecución pública digital de la obra (por ejemplo, webcast), sujeta a las disposiciones aplicables del régimen de Derecho de Autor, si esta ejecución pública digital está primordialmente dirigida a obtener una ventaja comercial o una compensación monetaria privada.

5. Representaciones, Garantías y Limitaciones de Responsabilidad.
A MENOS QUE LAS PARTES LO ACORDARAN DE OTRA FORMA POR ESCRITO, EL LICENCIANTE OFRECE LA OBRA (EN EL ESTADO EN EL QUE SE ENCUENTRA) “TAL CUAL”, SIN BRINDAR GARANTÍAS DE CLASE ALGUNA RESPECTO DE LA OBRA, YA SEA EXPRESA, IMPLÍCITA, LEGAL O CUALQUIERA OTRA, INCLUYENDO, SIN LIMITARSE A ELLAS, GARANTÍAS DE TITULARIDAD, COMERCIABILIDAD, ADAPTABILIDAD O ADECUACIÓN A PROPÓSITO DETERMINADO, AUSENCIA DE INFRACCIÓN, DE AUSENCIA DE DEFECTOS LATENTES O DE OTRO TIPO, O LA PRESENCIA O AUSENCIA DE ERRORES, SEAN O NO DESCUBRIBLES (PUEDAN O NO SER ESTOS DESCUBIERTOS). ALGUNAS JURISDICCIONES NO PERMITEN LA EXCLUSIÓN DE GARANTÍAS IMPLÍCITAS, EN CUYO CASO ESTA EXCLUSIÓN PUEDE NO APLICARSE A USTED.

6. Limitación de responsabilidad.
A MENOS QUE LO EXIJA EXPRESAMENTE LA LEY APLICABLE, EL LICENCIANTE NO SERÁ RESPONSABLE ANTE USTED POR DAÑO ALGUNO, SEA POR RESPONSABILIDAD EXTRACONTRACTUAL, PRECONTRACTUAL O CONTRACTUAL, OBJETIVA O SUBJETIVA, SE TRATE DE DAÑOS MORALES O PATRIMONIALES, DIRECTOS O INDIRECTOS, PREVISTOS O IMPREVISTOS PRODUCIDOS POR EL USO DE ESTA LICENCIA O DE LA OBRA, AUN CUANDO EL LICENCIANTE HAYA SIDO ADVERTIDO DE LA POSIBILIDAD DE DICHOS DAÑOS. ALGUNAS LEYES NO PERMITEN LA EXCLUSIÓN DE CIERTA RESPONSABILIDAD, EN CUYO CASO ESTA EXCLUSIÓN PUEDE NO APLICARSE A USTED.

7. Término.

a.	Esta Licencia y los derechos otorgados en virtud de ella terminarán automáticamente si Usted infringe alguna condición establecida en ella. Sin embargo, los individuos o entidades que han recibido Obras Derivadas o Colectivas de Usted de conformidad con esta Licencia, no verán terminadas sus licencias, siempre que estos individuos o entidades sigan cumpliendo íntegramente las condiciones de estas licencias. Las Secciones 1, 2, 5, 6, 7, y 8 subsistirán a cualquier terminación de esta Licencia.

b.	Sujeta a las condiciones y términos anteriores, la licencia otorgada aquí es perpetua (durante el período de vigencia de los derechos de autor de la obra). No obstante lo anterior, el Licenciante se reserva el derecho a publicar y/o estrenar la Obra bajo condiciones de licencia diferentes o a dejar de distribuirla en los términos de esta Licencia en cualquier momento; en el entendido, sin embargo, que esa elección no servirá para revocar esta licencia o que deba ser otorgada , bajo los términos de esta licencia), y esta licencia continuará en pleno vigor y efecto a menos que sea terminada como se expresa atrás. La Licencia revocada continuará siendo plenamente vigente y efectiva si no se le da término en las condiciones indicadas anteriormente.

8. Varios.

a.	Cada vez que Usted distribuya o ponga a disposición pública la Obra o una Obra Colectiva, el Licenciante ofrecerá al destinatario una licencia en los mismos términos y condiciones que la licencia otorgada a Usted bajo esta Licencia.

b.	Si alguna disposición de esta Licencia resulta invalidada o no exigible, según la legislación vigente, esto no afectará ni la validez ni la aplicabilidad del resto de condiciones de esta Licencia y, sin acción adicional por parte de los sujetos de este acuerdo, aquélla se entenderá reformada lo mínimo necesario para hacer que dicha disposición sea válida y exigible.

c.	Ningún término o disposición de esta Licencia se estimará renunciada y ninguna violación de ella será consentida a menos que esa renuncia o consentimiento sea otorgado por escrito y firmado por la parte que renuncie o consienta.

d.	Esta Licencia refleja el acuerdo pleno entre las partes respecto a la Obra aquí licenciada. No hay arreglos, acuerdos o declaraciones respecto a la Obra que no estén especificados en este documento. El Licenciante no se verá limitado por ninguna disposición adicional que pueda surgir en alguna comunicación emanada de Usted. Esta Licencia no puede ser modificada sin el consentimiento mutuo por escrito del Licenciante y Usted.


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