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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Nondentable solid subsets in Banach lattices failing RNP. Applications to renormings


Author: Elisabeth Werner
Journal: Proc. Amer. Math. Soc. 107 (1989), 611-620
MSC: Primary 46B30; Secondary 46B22
DOI: https://doi.org/10.1090/S0002-9939-1989-1017225-2
MathSciNet review: 1017225
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Abstract: We show that for every Banach lattice $ E$ failing RNP and not containing $ {c_0}$ (resp. containing $ {c_0}$) and for every $ \varepsilon > 0$ there exists a solid convex closed subset $ D$ of the unit ball of $ E$, such that

$\displaystyle \operatorname{dist} [\operatorname{ext} ({\overline D ^{\sigma ({... ...)}}),E] > 1 - \varepsilon (\operatorname{resp} . > \tfrac{1}{2} - \varepsilon )$

and such that every slice of $ D$ has diameter bigger than $ 2 - \varepsilon $.

We also prove that these results are optimal. We apply them to construct rough lattice norms with almost optimal constant on non-Asplund Banach lattices.


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DOI: https://doi.org/10.1090/S0002-9939-1989-1017225-2
Article copyright: © Copyright 1989 American Mathematical Society