Josefson–Nissenzweig property for Cp-spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F19%3A00508352" target="_blank" >RIV/67985840:_____/19:00508352 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1007/s13398-019-00667-8" target="_blank" >http://dx.doi.org/10.1007/s13398-019-00667-8</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s13398-019-00667-8" target="_blank" >10.1007/s13398-019-00667-8</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Josefson–Nissenzweig property for Cp-spaces

Popis výsledku v původním jazyce

The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K) admits a quotient isomorphic to Banach spaces c or ℓ 2 . The aim of the paper is to study a natural variant of this result for the space C p (X) of continuous real-valued maps on a Tychonoff space X with the pointwise topology. Following Josefson–Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for C p (X) -spaces. We prove: for a Tychonoff space X the space C p (X) satisfies the JNP if and only if C p (X) has a quotient isomorphic to c0:={(xn)n∈N∈RN:xn→0} (with the product topology of R N ) if and only if C p (X) contains a complemented subspace isomorphic to c. The last statement provides a C p -version of the Cembranos theorem stating that the Banach space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space c with the sup-norm topology. For a pseudocompact space X the space C p (X) has the JNP if and only if C p (X) has a complemented metrizable infinite-dimensional subspace. An example of a compact space K without infinite convergent sequences with C p (K) containing a complemented subspace isomorphic to c is given.

Název v anglickém jazyce

Josefson–Nissenzweig property for Cp-spaces

Popis výsledku anglicky

The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K) admits a quotient isomorphic to Banach spaces c or ℓ 2 . The aim of the paper is to study a natural variant of this result for the space C p (X) of continuous real-valued maps on a Tychonoff space X with the pointwise topology. Following Josefson–Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for C p (X) -spaces. We prove: for a Tychonoff space X the space C p (X) satisfies the JNP if and only if C p (X) has a quotient isomorphic to c0:={(xn)n∈N∈RN:xn→0} (with the product topology of R N ) if and only if C p (X) contains a complemented subspace isomorphic to c. The last statement provides a C p -version of the Cembranos theorem stating that the Banach space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space c with the sup-norm topology. For a pseudocompact space X the space C p (X) has the JNP if and only if C p (X) has a complemented metrizable infinite-dimensional subspace. An example of a compact space K without infinite convergent sequences with C p (K) containing a complemented subspace isomorphic to c is given.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF16-34860L" target="_blank" >GF16-34860L: Logika a topologie v Banachových prostorech</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

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

Název periodika

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales

ISSN

1578-7303

e-ISSN

—

Svazek periodika

113

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

ES - Španělské království

Počet stran výsledku

16

Strana od-do

3015-3030

Kód UT WoS článku

000483725900005

EID výsledku v databázi Scopus

2-s2.0-85064475223