Banach spaces of continuous functions without norming Markushevich bases

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F23%3A00390254" target="_blank" >RIV/68407700:21230/23:00390254 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1112/mtk.12217" target="_blank" >https://doi.org/10.1112/mtk.12217</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1112/mtk.12217" target="_blank" >10.1112/mtk.12217</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Banach spaces of continuous functions without norming Markushevich bases

Popis výsledku v původním jazyce

We investigate the question whether a scattered compact topological space K such that C(K)$C(K)$ has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hajek, Todorcevic and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that C([0,& omega;1])$C([0,omega _1])$ does not embed in a Banach space with a norming M-basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact K for C(K)$C(K)$ not to embed in a Banach space with a norming M-basis. Examples of such conditions are that K is a zero-dimensional compact space with a P-point, or a compact tree of height at least & omega;1+1$omega _1 +1$. In particular, this allows us to answer the said question in the case when K is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than & omega;(2) are Valdivia.

Název v anglickém jazyce

Banach spaces of continuous functions without norming Markushevich bases

Popis výsledku anglicky

We investigate the question whether a scattered compact topological space K such that C(K)$C(K)$ has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hajek, Todorcevic and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that C([0,& omega;1])$C([0,omega _1])$ does not embed in a Banach space with a norming M-basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact K for C(K)$C(K)$ not to embed in a Banach space with a norming M-basis. Examples of such conditions are that K is a zero-dimensional compact space with a P-point, or a compact tree of height at least & omega;1+1$omega _1 +1$. In particular, this allows us to answer the said question in the case when K is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than & omega;(2) are Valdivia.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

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

Název periodika

Mathematika

ISSN

0025-5793

e-ISSN

2041-7942

Svazek periodika

69

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

19

Strana od-do

992-1010

Kód UT WoS článku

001034796500001

EID výsledku v databázi Scopus

2-s2.0-85165983886