Grothendieck C(K)-spaces and the Josefson-Nissenzweig theorem

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F23%3A00579686" target="_blank" >RIV/67985840:_____/23:00579686 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.4064/fm218-6-2023" target="_blank" >https://doi.org/10.4064/fm218-6-2023</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4064/fm218-6-2023" target="_blank" >10.4064/fm218-6-2023</a>

Result language

angličtina

Original language name

Grothendieck C(K)-spaces and the Josefson-Nissenzweig theorem

Original language description

For a compact space K, the Banach space C(K) is said to have the l(1)-Grothendieck property if every weak* convergent sequence (mu(n) : n is an element of omega) of functionals on C(K) such that mu(n) is an element of l(1)(K) for every n is an element of omega is weakly convergent. Thus, the l(1)- Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that C(K) has the l(1)-Grothendieck property if and only if there does not exist any sequence of functionals (mu(n) : n is an element of omega) on C(K), with mu(n) is an element of l(1)(K) for every n is an element of omega, satisfying the conclusion of the classical Josefson-Nissenzweig theorem. We construct an example of a separable compact space K such that C(K) has the l(1)-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces K their Banach spaces C(K) do not have the l(1)-Grothendieck property.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF20-22230L" target="_blank" >GF20-22230L: Banach spaces of continuous and Lipschitz functions</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2023

Confidentiality

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

Name of the periodical

Fundamenta Mathematicae

ISSN

0016-2736

e-ISSN

1730-6329

Volume of the periodical

263

Issue of the periodical within the volume

2

Country of publishing house

PL - POLAND

Number of pages

27

Pages from-to

105-131

UT code for WoS article

001108631400001

EID of the result in the Scopus database

2-s2.0-85179096168