A characterization of the weak topology in the unit ball of purely atomic L-1 preduals

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F22%3A00360217" target="_blank" >RIV/68407700:21230/22:00360217 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1016/j.jmaa.2022.126311" target="_blank" >https://doi.org/10.1016/j.jmaa.2022.126311</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jmaa.2022.126311" target="_blank" >10.1016/j.jmaa.2022.126311</a>

Result language
angličtina
Original language name
A characterization of the weak topology in the unit ball of purely atomic L-1 preduals
Original language description
We study Banach spaces with a weak stable unit ball, that is, Banach spaces where every convex combination of relatively weakly open subsets in its unit ball is again a relatively weakly open subset in its unit ball. It is proved that the class of L-1 preduals with a weak stable unit ball agree with those L-1 preduals which are purely atomic, that is preduals of l(1)(Gamma) for some set Gamma, getting in this way a complete geometrical characterization of purely atomic preduals of L-1, which answers a setting problem. As a consequence, we prove the equivalence for L-1 preduals of different properties previously studied by other authors, in terms of slices around weak stability. Also we get the weak stability of the unit ball of C-0 (K, X) whenever K is a Hausdorff and scattered locally compact space and X has a norm stable and weak stable unit ball. This gives a characterization of weak stability of the unit ball in C-0 (K, X) for finite-dimensional X. Finally we prove that Banach spaces with a weak stable unit ball satisfy a very strong new version of diameter two property. (C) 2022 Elsevier Inc. All rights reserved.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/EF16_019%2F0000778" target="_blank" >EF16_019/0000778: Center for advanced applied science</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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