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On complemented copies of the space c0 in spaces Cp(X × Y)

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Abstract

Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces X and Y the Banach space C(X × Y) of continuous real-valued functions on X × Y endowed with the supremum norm contains a complemented copy of the Banach space c0. We extend this theorem to the class of Cp-spaces, that is, we prove that for all infinite Tychonoff spaces X and Y the space Cp (X × Y) of continuous functions on X × Y endowed with the pointwise topology contains either a complemented copy of ℝω or a complemented copy of the space (c0)p = {(xn)n∈ω ∈ ℝω: xn → 0}, both endowed with the product topology. We show that the latter case holds always when X × Y is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space X such that Cp(X × X) does not contain a complemented copy of (c0)p.

As a corollary to the first result, we show that for all infinite Tychonoff spaces X and Y the space Cp(X × Y) is linearly homeomorphic to the space Cp(X × Y) × ℝ, although, as proved earlier by Marciszewski, there exists an infinite compact space X such that Cp(X) cannot be mapped onto Cp(X) × ℝ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel’ski for spaces of the form Cp(X × Y).

Another corollary—analogous to the classical Rosenthal-Lacey theorem for Banach spaces C(X) with X compact and Hausdorff—asserts that for every infinite Tychonoff spaces X and Y the space Ck(X × Y) of continuous functions on X × Y endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: ℝω, (c0)p or c0.

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Author information

Authors and Affiliations

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

    Jerzy Kąkol

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

    Jerzy Kąkol

  3. Institute of Mathematics and Computer Science, University of Warsaw, 02-097, Warszawa, Poland

    Witold Marciszewski

  4. Kurt Gödel Research Center, Institut für Mathematik, Universität Wien, Augasse 2-6, UZA 1—Building 2, 1090, Wien, Austria

    Damian Sobota & Lyubomyr Zdomskyy

Authors
  1. Jerzy Kąkol
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  2. Witold Marciszewski
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  3. Damian Sobota
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  4. Lyubomyr Zdomskyy
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Corresponding author

Correspondence to Damian Sobota.

Additional information

The research of the first-named author is supported by the GAČR project 2022230L and RVO: 67985840.

The third- and fourth-named authors thank the Austrian Science Fund FWF (Grants I 2374-N35, I 3709-N35, M 2500-N35) for generous support for this research.

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Kąkol, J., Marciszewski, W., Sobota, D. et al. On complemented copies of the space c0 in spaces Cp(X × Y). Isr. J. Math. 250, 139–177 (2022). https://doi.org/10.1007/s11856-022-2334-2

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  • DOI: https://doi.org/10.1007/s11856-022-2334-2

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