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

0575131 - MÚ 2024 RIV DE eng J - Journal Article
Kąkol, Jerzy - Kurka, Ondřej - Leiderman, A. G.
On Asplund spaces Ck(X) and w∗-binormality.
Results in Mathematics. Roč. 78, č. 5 (2023), č. článku 203. ISSN 1422-6383. E-ISSN 1420-9012
R&D Projects: GA ČR(CZ) GF20-22230L; GA ČR(CZ) GF22-07833K
Institutional support: RVO:67985840
Keywords : Asplund property * compact space * compact-open topology * scattered space
OECD category: Pure mathematics
Impact factor: 2.2, year: 2022
Method of publishing: Limited access
https://doi.org/10.1007/s00025-023-01979-3

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.
Permanent Link: https://hdl.handle.net/11104/0344990