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.
Highlights
The classic Rosenthal–Lacey theorem, see [19,23,27], asserts that the Banach space C(K ) of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to Banach spaces c or 2, or equivalently, there exists a continuous linear map from C(K ) onto c or 2, see a survey paper [14]
Problem 1 has been already partially studied in [3], where we proved that for a Tychonoff space X the space C p(X ) has an infinite-dimensional metrizable quotient if X either contains an infinite discrete C∗-embedded subspace or else X has a sequence (Kn)n∈N of compact subsets such that for every n the space Kn contains two disjoint topological copies of Kn+1
The first case asserts that C p(X ) has a quotient isomorphic to the subspace ∞ = {(xn) ∈ RN : supn |xn| < ∞} of RN or to the product RN. This theorem reduces Problem 1 to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). It is unknown if Efimov spaces exist in ZFC we showed in [22] that under ♦ for some Efimov spaces K the function space C p(K ) has an infinite dimensional metrizable quotient
Summary
Problem 1 has been already partially studied in [3], where we proved that for a Tychonoff space X the space C p(X ) has an infinite-dimensional metrizable quotient if X either contains an infinite discrete C∗-embedded subspace or else X has a sequence (Kn)n∈N of compact subsets such that for every n the space Kn contains two disjoint topological copies of Kn+1. The first case (for example if compact X contains a copy of βN) asserts that C p(X ) has a quotient isomorphic to the subspace ∞ = {(xn) ∈ RN : supn |xn| < ∞} of RN or to the product RN This theorem reduces Problem 1 to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). Problem 2 Characterize those spaces C p(K ) which contain a complemented copy of c0 with the product topology of RN
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