Josefson\u2013Nissenzweig 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 isomorphic to Banach spaces c or ell _{2}. The aim of the paper is to study a natural variant of this result for the space C_{p}(X) of continuous real-valued maps on a Tychonoff space X with the pointwise topology. Following Josefson–Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for C_{p}(X)-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}:={(x_n)_{nin mathbb N}in mathbb {R}^mathbb {N}:x_nrightarrow 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. The last statement provides a C_{p}-version of the Cembranos theorem stating that the Banach space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space c_{0} with the sup-norm topology. 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. An example of a compact space K without infinite convergent sequences with C_{p}(K) containing a complemented subspace isomorphic to c_{0} is given.