Banach spaces of continuous functions without norming Markushevich bases
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Banach spaces of continuous functions without norming Markushevich bases
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - 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
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Mathematika
ISSN
0025-5793
e-ISSN
2041-7942
Volume of the periodical
69
Issue of the periodical within the volume
4
Country of publishing house
GB - UNITED KINGDOM
Number of pages
19
Pages from-to
992-1010
UT code for WoS article
001034796500001
EID of the result in the Scopus database
2-s2.0-85165983886