Josefson-Nissenzweig property for $C_p$-spaces

TL;DR

This paper introduces the Josefson-Nissenzweig property for $C_p$-spaces, characterizes when these spaces contain complemented $c_0$ subspaces or quotients, and explores their structure in relation to pseudocompactness and specific compact spaces.

Contribution

It establishes a characterization of the JNP for $C_p$-spaces and links it to the existence of complemented $c_0$ subspaces and quotients, extending classical Banach space results.

Findings

$C_p(X)$ satisfies JNP iff it has a $c_0$ quotient or complemented subspace.

For pseudocompact $X$, $C_p(X)$ has JNP iff it contains a complemented metrizable infinite-dimensional subspace.

$C_p(etabN)$ contains $c_0$ but lacks a $c_0$ quotient.

Abstract

The famous Rosenthal-Lacey theorem asserts that for each infinite compact space $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c_{0}$ or $\ell_{2}$. The aim of the paper is to study a natural variant of this result for the space $C_{p}(K)$ of continuous real-valued maps on $K$ with the pointwise topology. Following famous Josefson-Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson-Nissenzweig property, briefly, the JNP) for $C_{p}$-spaces. We prove: For a Tychonoff space $X$ the space $C_p(X)$ satisfies the JNP if and only if $C_p(X)$ has a quotient isomorphic to $c_{0}$ (with the product topology of $\mathbb R^\mathbb{N}$) if and only if $C_{p}(X)$ contains a complemented subspace, isomorphic to $c_0$. For a pseudocompact space $X$ the space $C_p(X)$ has the JNP if and only if $C_p(X)$ has a complemented metrizable infinite-dimensional subspace. This applies to show that for a Tychonoff space $X$ the space $C_p(X)$ has a complemented subspace isomorphic to $\mathbb R^{\mathbb N}$ or $c_0$ if and only if $X$ is not pseudocompact or $C_p(X)$ has the JNP. The space $C_{p}(\beta\mathbb{N})$ contains a subspace isomorphic to $c_0$ and admits a quotient isomorphic to $\ell_{\infty}$ but fails to have a quotient isomorphic to $c_{0}$. An example of a compact space $K$ without infinite convergent sequences with $C_{p}(K)$ containing a complemented subspace isomorphic to $c_{0}$ is constructed.