Metrizable quotients of C-spaces

Банах, Тарас; Ka̧kol, Jerzy; Śliwa, Wiesław

doi:10.1016/j.topol.2018.09.012

Cited by 17 publications

(8 citation statements)

References 8 publications

(13 reference statements)

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“…A topological space X is pseudocompact if it is Tychonoff and each continuous real-valued function on X is bounded. It is known (see [3]) that a Tychonoff space X is not pseudocompact if and only if C p (X) contains a complemented copy of R N . Combining this characterization with Theorem 1, we obtain another characterization related to Problem 1.…”

Section: The Main Resultsmentioning

confidence: 99%

“…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 (K n ) n∈N of compact subsets such that for every n the space K n contains two disjoint topological copies of K n+1 . If fact, 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 ℓ ∞ = {(x n ) ∈ R N : sup n |x n | < ∞} of R N or to the product R N .…”

Section: Introduction and The Main Problemmentioning

confidence: 99%

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Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Банах

Ka̧kol

Śliwa

2019

RACSAM

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We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E ′ , σ(E ′ , E)) → (E ′ , β * (E ′ , E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP.We show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak * null-sequence {µn}n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1(X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP. We also define two modifications of the JNP, called the universal JNP and the JNP everywhere (briefly, the uJNP and eJNP), and thoroughly study them in the classes of locally convex spaces, Banach spaces and function spaces. We provide a characterization of the JNP in terms of operators into locally convex spaces with the uJNP or eJNP and give numerous examples clarifying relationships between the considered notions.2010 Mathematics Subject Classification. Primary 46A03; Secondary 46E10, 46E15.

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Section: The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Problemmentioning

confidence: 99%

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Банах

Ka̧kol

Śliwa

2019

RACSAM

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“…Proof of (4): Since the space X is pseudocompact and contains discrete ω which is C * -embedded into X, we apply Theorem 1 from [4] to deduce that C p (X) has a quotient C p (X)/W isomorphic to the subspace (ℓ ∞ ) p of R ω endowed with the product topology, where…”

Section: Proof Of Theorem 14 and Its Consequencesmentioning

confidence: 99%

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

Ka̧kol¹,

Marciszewski²,

Sobota³

et al. 2020

Preprint

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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 .

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“…Both of the contexts, despite originating in functional analysis, have deep connections with set theory (as demonstrated e.g. in [25], [2], [22], [9], [37], and [36]).…”

Section: Introductionmentioning

confidence: 99%

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

Marciszewski¹,

Sobota²

2022

Preprint

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For a free filter F on ω, endow the space NF = ω ∪ {pF }, where pF ∈ ω, with the topology in which every element of ω is isolated whereas all open neighborhoods of pF are of the form A ∪ {pF } for A ∈ F . Spaces of the form NF constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson-Nissenzweig theorem from Banach space theory. We prove, e.g., that a space NF carries a sequence µn : n ∈ ω of normalized finitely supported signed measures such that µn(f ) → 0 for every bounded continuous real-valued function f on NF if and only if there is a density submeasure ϕ on ω such that the dual ideal F * is contained in the exhaustive ideal Exh(ϕ). Consequently, we get that if F is a free filter on ω contained in the filter dual to a density ideal, then: (1) if X is a Tychonoff space and NF is homeomorphic to a subspace of X, then the space C * p (X) of bounded continuous real-valued functions on X contains a complemented copy of the space c0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and NF is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. We also prove that the set of filters which can be covered by filters dual to density ideals contains a family of 2 ω many mutually non-isomorphic Fσ P-filters and a family of 2 2 ω many mutually non-isomorphic non-Borel filters.

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Metrizable quotients of C-spaces

Cited by 17 publications

References 8 publications

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

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

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

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