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BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

  • Johann Langemets (a1) and Ginés López-Pérez (a2)

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.

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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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