On Asplund spaces Ck(X) and w∗-binormality

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F23%3A00575131" target="_blank" >RIV/67985840:_____/23:00575131 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s00025-023-01979-3" target="_blank" >https://doi.org/10.1007/s00025-023-01979-3</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00025-023-01979-3" target="_blank" >10.1007/s00025-023-01979-3</a>

Result language
angličtina
Original language name
On Asplund spaces Ck(X) and w∗-binormality
Original language description
A celebrated theorem of Namioka and Phelps (Duke Math. J. 42:735-750, 1975) says that for a compact space X, the Banach space C(X) is Asplund iff X is scattered. In our paper we extend this result to the space of continuous real-valued functions endowed with the compact-open topology Ck(X) for several natural classes of non-compact Tychonoff spaces X. The concept of Δ 1-spaces introduced recently in Ka̧kol et al. (Some classes of topological spaces extending the class of Δ -spaces, submitted for publication) has been shown to be applicable for this research. w∗-binormality of the dual of the Banach space C(X) implies that C(X) is Asplund (Kurka in J Math Anal Appl 371:425–435, 2010). In our paper we prove in particular that for a Corson compact space X the converse is true. We establish a tight relationship between the property of w∗-binormality of the dual C(X) ′ and the class of compact Δ-spaces X introduced and explored earlier in Ka̧kol and Leiderman (Proc. Am. Math. Soc. Ser B 8:86-99, 2021, 8:267-280, 2021). We find a complete characterization of a compact space X such that the dual C(X)′ possesses a stronger property called effective w∗ -binormality. We provide several illustrating examples and pose open questions.
Czech name
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Czech description
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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

Project
Result was created during the realization of more than one project. More information in the Projects tab.
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

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