On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$

Abstract

Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces X and Y the Banach space C(X × Y ) of continuous real-valued functions on X × Y endowed with the supremum norm contains a complemented copy of the Banach space c 0 . We extend this theorem to the class of C p -spaces, that is, we prove that for all infinite Tychonoff spaces X and Y the space C p (X ×Y ) of continuous functions on X ×Y endowed with the pointwise topology contains either a complemented copy of R ω or a complemented copy of the space (c 0 ) p = {(x n ) n∈ω ∈ R ω : x n → 0}, both endowed with the product topology. We show that the latter case holds always when X × Y is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space X such that C p (X × X) does not contain a complemented copy of (c 0 ) p .As a corollary to the first result, we show that for all infinite Tychonoff spaces X and Y the space C p (X × Y ) is linearly homeomorphic to the space C p (X × Y ) × R, although, as proved earlier by Marciszewski, there exists an infinite compact space X such that C p (X) cannot be mapped onto C p (X) × R by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form C p (X × Y ).Our another corollary-analogous to the classical Rosenthal-Lacey theorem for Banach spaces C(X) with X compact and Hausdorff-asserts that for every infinite Tychonoff spaces X and Y the space C k (X × Y ) of continuous functions on X × Y endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: R ω , (c 0 ) p or c 0 .

The Josefson--Nissenzweig theorem and filters on $ω$

“…Let us call the dual filter F d = Exh ϕ d * simply the (asymptotic) density filter. The BJNP and JNP of the spaces associated to F d were already studied in [3,Section 4], [26,Section 5.1], and [27,Example 4.2], where it was among others proved that:…”

“…These two properties are closely related to the famous Josefson-Nissenzweig theorem from Banach space theory stating that for every infinite-dimensional Banach space E there exists a sequence x * n : n ∈ ω in the dual space E * such that x * n = 1 for every n ∈ ω and x * n (x) → 0 for every x ∈ E (that is, x * n : n ∈ ω is weak* null). They were introduced and studied in [3], [26], and [27], in the context of the Separable Quotient Problem for spaces C p (X) as well as in order to investigate Grothendieck Banach spaces of the form C(K) (see below for details). Both of the contexts, despite originating in functional analysis, have deep connections with set theory (as demonstrated e.g.…”

“…For many examples of spaces with or without the (B)JNP, we refer the reader to [26] and [27]. It is however immediate that if a space X contains a non-trivial convergent sequence, then X has the JNP (see Fact 4.1), the converse though does not hold (see e.g.…”

“…On the other hand, βω, the Čech-Stone compactification of ω, is an example of a space without the JNP (see Fact 4.2). The JNP and BJNP coincide, of course, in the class of pseudocompact spaces but not in the class of all Tychonoff spaces-in [27,Example 4.2] it is proved that the space N F d , where F d stands for the asymptotic density filter (see Remark 5.12 for the definition), has the BJNP but not the JNP.…”

“…One the most remarkable results here is due to Banakh, Kąkol, and Śliwa [3], who proved that, for an infinite space X, the space C p (X) contains a complemented copy of the space (c 0 ) p = x ∈ R ω : x(n) → 0 endowed with the pointwise topology if and only if X has the JNP. The C * p -variant of this theorem was obtained in [27] and asserts that, for an infinite space X, the space C * p (X) contains a complemented copy of (c 0 ) p if and only if X has the BJNP. Using the latter characterization and the results presented above, we obtain the following sufficient condition for spaces C * p (X) (and hence for spaces C p (X) for X pseudocompact) to contain a complemented copy of (c 0 ) p .…”