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

Abstract

Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces $X$ and $Y$ the Banach space $C(X\times Y)$ of continuous real-valued functions on $X\times Y$ 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 $C_p(X\times Y)$ of continuous functions on $X\times Y$ endowed with the pointwise topology contains either a complemented copy of ${\mathbb{R}}^{\omega }$ or a complemented copy of the space $(c_0{)}_p=\{(x_n{)}_{n\in \omega }\in {\mathbb{R}}^{\omega }:x_n\to 0\}$, both endowed with the product topology. We show that the latter case holds always when $X\times Y$ 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 $C_p(X\times X)$ 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 $C_p(X\times Y)$ is linearly homeomorphic to the space $C_p(X\times Y)\times \mathbb{R}$, although, as proved earlier by Marciszewski, there exists an infinite compact space $X$ such that $C_p(X)$ cannot be mapped onto $C_p(X)\times \mathbb{R}$ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form $C_p(X\times Y)$. Another corollary asserts that for every infinite Tychonoff spaces $X$ and $Y$ the space $C_k(X\times Y)$ of continuous functions on $X\times Y$ 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 $c_0$.

30 pages, comments are welcome

References in corpus (3)

• Metrizable quotients of $C_p$-spaces
• On the weak and pointwise topologies in function spaces II
• The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

Cited by in corpus (1)

• The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces