Abstract
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 \(C_k(X)\) for several natural classes of non-compact Tychonoff spaces X. The concept of \(\Delta _1\)-spaces introduced recently in Ka̧kol et al. (Some classes of topological spaces extending the class of \(\Delta \)-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 \(C(X)^{\prime }\) and the class of compact \(\Delta \)-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 \(C(X)^{\prime }\) possesses a stronger property called effective \(w^{*}\)-binormality. We provide several illustrating examples and pose open questions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Independently and simultaneously an analogous description of distinguished \(C_p\)-spaces (but formulated in different terms) appeared in [5].
References
Arkhangel’skii, A.V.: Topological Function Spaces. Kluwer, Dordrecht (1992)
Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)
Fabian, M.: Gâteaux Differentiability of Convex Functions and topology Weak Asplund Spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)
Ferrando, J. C., Ka̧kol, J., Leiderman, A., Saxon, S. A.: Distinguished\(C_p(X)\)spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 115(1): 27 (2021)
Ferrando, J.C., Saxon, S.A.: If not distinguished, is \(C_{p}(X)\) even close? Proc. Am. Math. Soc. 149, 2583–2596 (2021)
Gabriyelyan, S., Ka̧kol, J., Kubiś, W., Marciszewski, W.: Networks for the weak topology of Banach and Fréchet spaces. J. Math. Anal. Appl. 432, 1183–1199 (2015)
Haydon, R.: Three examples in the theory of spaces of continuous functions. C. R. Acad. Sci. Sér. A 276, 685–687 (1973)
Hrušák, M., Tamariz-Mascarúa, A., Tkachenko, M. (eds.): Pseudocompact Topological Spaces. A Survey of Classic and New Results with Open Problems. Springer, Berin (2018)
Juhasz, I., van Mill, J.: Countably compact spaces all countable subsets of which are scattered. Comment. Math. Univ. Carol. 22, 851–855 (1981)
Ka̧kol, J., Kubiś, W., Lopez-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics. Springer, Berlin (2011)
Ka̧kol, J., Kurka, O., Leiderman, A.: Some classes of topological spaces extending the class of \(\Delta \)-spaces (submitted for publication)
Ka̧kol, J., Leiderman, A.: A characterization of \(X\) for which spaces \(C_p(X)\) are distinguished and its applications. Proc. Am. Math. Soc. Ser. B 8, 86–99 (2021)
Ka̧kol, J., Leiderman, A.: Basic properties of \(X\) for which the space \(C_p(X)\) is distinguished. Proc. Am. Math. Soc. Ser. B 8, 267–280 (2021)
Ka̧kol, J., Leiderman, A.: When is a locally convex space Eberlein–Grothendieck? Res. Math. 77, 36 (2022)
Ka̧kol, J., Saxon, S., Todd, A.T.: The analysis of Warner boundedness. Proc. Edinb. Math. Soc. 47, 625–631 (2004)
Komisarchik, M., Megrelishvili, M.: Tameness and Rosenthal type locally convex spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM. 117, 113 (2023)
Kunen, K.: Set Theory, Studies in Logic (London), vol. 34. College Publications, London (2011)
Kuratowski, K.: Topology, vol. I. Academic Press, New York (1966)
Kurka, O.: On binormality in non-separable Banach spaces. J. Math. Anal. Appl. 371, 425–435 (2010)
Leiderman, A., Tkachenko, V.: Metrizable Quotients of Free Topological Groups. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114, 124 (2020)
Marciszewski, W.: private conversation (2023)
Mazon, J.M.: On the df-spaces of continuous functions. Bull. Soc. Roy. Sci. Liege 53, 225–232 (1984)
Namioka, I., Phelps, R.R.: Banach spaces which are Asplund spaces. Duke Math. J. 42, 735–750 (1975)
Przymusiński, T. C.: Normality and separability of Moore spaces. In: Set-Theoretic Topology. Academic Press, New York, 325–337 (1977)
Reed, G.M.: On normality and countable paracompactness. Fund. Math. 110, 145–152 (1980)
Semadeni, Z.: Banach Spaces of Continuous Functions, vol. I. PWN - Polish Scientific Publishers, Warszawa (1971)
Schmets, J.: Espaces de fonctions continues. (French) Lecture Notes in Mathematics, Vol. 519. Springer, Berlin (1976)
Tkachuk, V.V.: A \(C_p\)-Theory Problem Book. Special Features of Function Spaces. Springer, New York (2014)
Tkachuk, V.V.: A \(C_p\)-Theory Problem Book. Compactness in Function Spaces. Springer, New York (2015)
Warner, S.: The topology of compact convergence on continuous function spaces. Duke Math. J. 25, 265–282 (1958)
Wheeler, R.: The Mackey problem for the compact-open topology. Trans. Am. Math. Soc. 222, 255–265 (1976)
Yost, D.: Asplund spaces for beginners. Acta Univ. Carolin. Math. Phys. 34, 159–177 (1993)
Funding
The research of the first named author Jerzy Ka̧kol is supported by the GAČR project 20-22230 L and RVO: 67985840. The research of the second named author Ondřej Kurka is supported by the GAČR project 22-07833K and RVO: 67985840.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that no other support was received during the preparation of this manuscript. The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the first named author is supported by the GAČR Project 20-22230L and RVO: 67985840. The research of the second named author is supported by the GAČR Project 22-07833K and RVO: 67985840.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ka̧kol, J., Kurka, O. & Leiderman, A. On Asplund Spaces \(C_k(X)\) and \(w^{*}\)-Binormality. Results Math 78, 203 (2023). https://doi.org/10.1007/s00025-023-01979-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-01979-3