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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA23-04776S" target="_blank" >GA23-04776S: Interakce algebraických, metrických, geometrických a topologických struktur na Banachových prostorech</a><br>

Návaznosti

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

Rok uplatnění

2024

Kód důvěrnosti údajů

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

Název periodika

JOURNAL OF FUNCTIONAL ANALYSIS

ISSN

0022-1236

e-ISSN

1096-0783

Svazek periodika

286

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

20

Strana od-do

—

Kód UT WoS článku

001133583000001

EID výsledku v databázi Scopus

2-s2.0-85178593772