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On linear continuous operators between distinguished spaces \(C_p(X)\)

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space \(C_{p}(X)\) is distinguished if and only if X is a \(\Delta \)-space. If there exists a linear continuous surjective mapping \(T:C_p(X) \rightarrow C_p(Y)\) and \(C_p(X)\) is distinguished, then \(C_p(Y)\) also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator \(T:C_p(X) \rightarrow C_p(Y)\) above is open? Secondly, we devote a special attention to concrete distinguished spaces \(C_p([1,\alpha ])\), where \(\alpha \) is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping \(T:C_p([1,\alpha ]) \rightarrow C_p(Y)\) is given. We also observe that for every countable ordinal \(\alpha \) all closed linear subspaces of \(C_p([1,\alpha ])\) are distinguished, thereby answering an open question posed in Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of \(\Delta \)-spaces we prove that a linear continuous surjection \(T:C_p(X) \rightarrow C_k(X)_w\), where \(C_k(X)_w\) denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact \(X \subset {\mathbb {R}}^n\)).

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Notes

  1. We should mention that independently and simultaneously an analogous description of distinguished \(C_p\)-spaces (but formulated in different terms) appeared in [10].

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614, Poznań, Poland

    Jerzy Ka̧kol

  2. Institute of Mathematics Czech Academy of Sciences, Prague, Czech Republic

    Jerzy Ka̧kol

  3. Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva, Israel

    Arkady Leiderman

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  1. Jerzy Ka̧kol
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  2. Arkady Leiderman
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Correspondence to Arkady Leiderman.

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Jerzy Ka̧kol: supported by the GAČR project 20-22230L and RVO: 67985840. The authors thank J. C. Ferrando for providing a short argument in Remark 2.5 (iii) and W. B. Johnson and T. Kania for a very helpful advice and discussion about Theorem 3.2.

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Ka̧kol, J., Leiderman, A. On linear continuous operators between distinguished spaces \(C_p(X)\). Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 199 (2021). https://doi.org/10.1007/s13398-021-01121-4

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