The equivalence between CPCP and strong regularity under Krein-Milman property

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F24%3A00379046" target="_blank" >RIV/68407700:21230/24:00379046 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1016/j.jfa.2023.110273" target="_blank" >https://doi.org/10.1016/j.jfa.2023.110273</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jfa.2023.110273" target="_blank" >10.1016/j.jfa.2023.110273</a>

Result language

angličtina

Original language name

The equivalence between CPCP and strong regularity under Krein-Milman property

Original language description

We obtain a result in the spirit of the well-known W. Schachermayer and H. P. Rosenthal research about the equivalence between Radon-Nikodym and Krein-Milman properties, by showing that, for closed, bounded and convex subsets C of a separable Banach space, under Krein-Milman property for C, one has the equivalence between convex point of continuity property and strong regularity both defined for every locally convex topology on C, containing the weak topology on C. Then, under Krein-Milman property, not only the classical convex point of continuity property and strong regularity are equivalent, but also when they are defined for an arbitrary locally convex topology containing the weak topology. We also show that while the unit ball B of c0 fails convex point of continuity property and strong regularity (both defined for the weak topology), there is a locally convex topology tau on B, containing the weak topology on B, such that B still fails convex point of continuity property for tau, but B surprisingly enjoys strong regularity for tau-open sets. Moreover, B satisfies the diameter two property for the topology tau, that is, every nonempty tau-open subset of B has diameter two even though every tau-open subset of B contains convex combinations of relative tau-open subsets with arbitrarily small diameter, that is, B fails the strong diameter two property for the topology tau. This stresses the known extreme differences up to now between those diameter two properties from a topological point of view.(c) 2023 Elsevier Inc. All rights reserved.

Czech name

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Czech description

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Type

J_imp - 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/GA23-04776S" target="_blank" >GA23-04776S: Interplay of algebraic, metric, geometric and topological structures on Banach spaces</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

JOURNAL OF FUNCTIONAL ANALYSIS

ISSN

0022-1236

e-ISSN

1096-0783

Volume of the periodical

286

Issue of the periodical within the volume

4

Country of publishing house

US - UNITED STATES

Number of pages

20

Pages from-to

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UT code for WoS article

001133583000001

EID of the result in the Scopus database

2-s2.0-85178593772