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BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  16 May 2019

Johann Langemets Open the ORCID record for Johann Langemets [Opens in a new window]  and
Ginés López-Pérez Open the ORCID record for Ginés López-Pérez [Opens in a new window]
Johann Langemets
Affiliation:
Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409Tartu, Estonia (johann.langemets@ut.ee)
Ginés López-Pérez
Affiliation:
Universidad de Granada, Facultad de Ciencias, Departamento de Análisis Matemático, 18071-Granada, Spain (glopezp@ugr.es)
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Abstract

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math.95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

Keywords

octahedral normdiameter two propertiesstrong regularityrenorming

MSC classification

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces 46B22: Radon-Nikodým, Kreĭn-Milman and related properties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 2 , March 2021 , pp. 569 - 585
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Copyright
© Cambridge University Press 2019

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Footnotes

The work of J. Langemets was supported by the Estonian Research Council grant (PUTJD702), by institutional research funding IUT (IUT20-57) of the Estonian Ministry of Education and Research, and by a grant of the Institute of Mathematics of the University of Granada (IEMath-GR). The work of G. López-Pérez was supported by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE) and by Junta de Andalucía Grant FQM-0185.

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