Bidual octahedral renormings and strong regularity in Banach spaces
Functional Analysis
2021-07-01 v2
Abstract
We prove that every separable Banach space containing can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its unit ball has diameter two).
Keywords
Cite
@article{arxiv.1902.04170,
title = {Bidual octahedral renormings and strong regularity in Banach spaces},
author = {Johann Langemets and Ginés López-Pérez},
journal= {arXiv preprint arXiv:1902.04170},
year = {2021}
}
Comments
Compared to the previous version, we have now fixed a mistake that appeared in Proposition 2.5 and added a new Corollary 4.3