Josefson-Nissenzweig property for $C_p$-spaces

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.

14 pages