Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Josefson-Nissenzweig property for $C_p$-spaces   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.

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The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: nlin