Efimov spaces and the separable quotient problem for spaces C(K)

Ka̧kol, Jerzy; Śliwa, Wiesław

doi:10.1016/j.jmaa.2017.08.010

Cited by 17 publications

(12 citation statements)

References 18 publications

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“…Hence C p (βN) lacks infinite-dimensional separable quotient algebras. Nevertheless, as proved in [19,Theorem 4], the space C p (K) has infinite-dimensional separable quotient for any compact space K containing a copy of βN.…”

Section: Introduction and The Main Problemmentioning

confidence: 99%

“…K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). Although, it is unknown if Efimov spaces exist in ZFC (see [6], [7], [8], [9], [11], [12], [13], [15]) we showed in [19] that under ♦ for some Efimov spaces K the function space C p (K) has an infinite dimensional metrizable quotient.…”

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: Introduction and The Main Problemmentioning

confidence: 99%

Section: Introduction and The Main Problemmentioning

confidence: 99%

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

Банах

Ka̧kol

Śliwa

2019

RACSAM

Self Cite

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“…As the space K from Example 1 does not contain βN, the assumption of Theorem 1 is not satisfied. Note that in [18,Example 17] we provided (again under ♦) an example of an Efimov space K for which Theorem 2 cannot be applied.…”

Section: Theorem 1 and Proposition 1 Yield Immediatelymentioning

confidence: 99%

“…K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). Although, it is unknown if Efimov spaces exist in ZFC (see [5], [6], [7], [8], [10], [11], [12], [14]) we showed in [18] that under ♦ for some Efimov spaces K the function space C p (K) has SQ.…”

Section: Introductionmentioning

confidence: 99%

Metrizable quotients of C-spaces

Банах

Ka̧kol

Śliwa

2018

Topology and its Applications

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The famous Rosenthal-Lacey theorem asserts that for each infinite compact set K the Banach space C(K) admits a quotient which is either a copy of c or ℓ 2 . What is the case when the uniform topology of C(K) is replaced by the pointwise topology? Is it true that C p (X) always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space X the function space C p (X) has an infinitedimensional 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 . Applying the latter result, we show that under ♦ there exists a zero-dimensional Efimov space K whose function space C p (K) has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Kakol and Sliwa on infinite-dimensional separable quotients of C p -spaces.1991 Mathematics Subject Classification. 54C35, 54E35.

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“…The famous Rosenthal-Lacey theorem [40], [32], see also [24,Corollary 1], asserts that for each infinite compact space K the Banach space C(K) admits a quotient map onto c 0 or ℓ 2 ; we refer the reader to a survey paper [18,Theorem 18] for a detailed discussion on the theorem. The case of C p -spaces remains however open, namely, it is still unknown whether for every infinite compact space K the space C p (K) admits a quotient map onto an infinite-dimensional metrizable space, see [29]. Nevertheless, Theorem 1.4 yields the following corollary.…”

Section: Introductionmentioning

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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Efimov spaces and the separable quotient problem for spaces C(K)

Cited by 17 publications

References 18 publications

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

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

Metrizable quotients of C-spaces

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

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