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Banach spaces with weak*-sequential dual ball


Author: Gonzalo Martínez-Cervantes
Journal: Proc. Amer. Math. Soc. 146 (2018), 1825-1832
MSC (2010): Primary 57N17, 54D55, 46A50; Secondary 46B20, 46B50
DOI: https://doi.org/10.1090/proc/13843
Published electronically: November 10, 2017
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Abstract: A topological space is said to be sequential if every subspace closed under taking limits of convergent sequences is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $ X$ is a Banach space with weak*-sequentially compact dual ball and $ Y \subset X$ is a subspace such that $ Y$ and $ X/Y$ have weak*-sequential dual ball, then $ X$ has weak*-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space $ JL_2$ and $ C(K)$ for $ K$ a scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball. These results provide a negative solution to a question of A. Plichko, who asked whether the dual ball of a Banach space is weak*-angelic whenever it is weak*-sequential.


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Additional Information

Gonzalo Martínez-Cervantes
Affiliation: Departamento de Matemáticas Facultad de Matemáticas Universidad de Murcia 30100 Espinardo, Murcia Spain
Email: gonzalo.martinez2@um.es

DOI: https://doi.org/10.1090/proc/13843
Keywords: Sequential, sequentially compact, countable tightness, Fr\'echet-Urysohn, angelic, Efremov property, Banach space
Received by editor(s): December 19, 2016
Received by editor(s) in revised form: May 17, 2017
Published electronically: November 10, 2017
Additional Notes: The author was partially supported by the research project 19275/PI/14 funded by Fundación Séneca - Agencia de Ciencia y Tecnología de la Región de Murcia within the framework of PCTIRM 2011-2014 and by Ministerio de Economía y Competitividad and FEDER (project MTM2014-54182-P)
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society

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