On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$

DOI10.1007/S11856-022-2334-2zbMath1518.46013arXiv2007.14723MaRDI QIDQ6346089

Damian Sobota, Witold Marciszewski, Lyubomyr Zdomskyy, Jerzy Kąkol

Publication date: 29 July 2020

Abstract: Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces $X$ and $Y$ the Banach space $C(X imes Y)$ of continuous real-valued functions on $X imes Y$ endowed with the supremum norm contains a complemented copy of the Banach space $c_{0}$. We extend this theorem to the class of $C_p$-spaces, that is, we prove that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X imes Y)$ of continuous functions on $X imes Y$ endowed with the pointwise topology contains either a complemented copy of $mathbb{R}^{omega}$ or a complemented copy of the space $(c_{0})_{p}={(x_n)_{ninomega}in mathbb{R}^omegacolon x_n o 0}$, both endowed with the product topology. We show that the latter case holds always when $X imes Y$ 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 $C_{p}(X imes X)$ does not contain a complemented copy of $(c_{0})_{p}$. As a corollary to the first result, we show that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X imes Y)$ is linearly homeomorphic to the space $C_{p}(X imes Y) imesmathbb{R}$, although, as proved earlier by Marciszewski, there exists an infinite compact space $X$ such that $C_{p}(X)$ cannot be mapped onto $C_{p}(X) imesmathbb{R}$ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form $C_p(X imes Y)$. Another corollary asserts that for every infinite Tychonoff spaces $X$ and $Y$ the space $C_{k}(X imes Y)$ of continuous functions on $X imes Y$ endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: $mathbb{R}^omega$, $(c_{0})_{p}$ or $c_{0}$.

Mathematics Subject Classification ID

Function spaces in general topology (54C35) Topological linear spaces of continuous, differentiable or analytic functions (46E10)

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