arXiv subjects

Jerzy Kąkol

Publications and source records attributed to Jerzy Kąkol.

15 recordsLinked to original sources

arXiv2026

A consistent failure of separable quotients for pointwise function spaces

Assuming Jensen's diamond principle, we construct an infinite compact zero-dimensional space $K$ such that $C_p(K)$ has no infinite-dimensional Hausdorff separable linear quotient. The space $K$ is separable and crowded, has weight $\aleph_1$ and cardinality $2^{\aleph_1}$, and is an Efimov space. We construct $K$ as an inverse limit of compact metrisable spaces indexed by the countable ordinals. At each nontrivial successor step, the projection has two-point fibres over a chosen closed set and singleton fibres elsewhere; this changes the weak-star limit of a selected sequence of finitely supported measures. We also prove that, for compact $X$, the existence of an infinite-dimensional separable quotient of $C_p(X)$ is equivalent to the existence of an infinite-dimensional metrisable quotient.

math.FA↗

arXiv2026

Normal sequences without strongly normal subsequences

A sequence $(y_n^*)$ in the unit sphere of $E^*$ is strongly normal if the vectors $x\in E$ satisfying $\sum_n|y_n^*(x)|<\infty$ form a dense subspace. For every set $Γ$ of cardinality at least the continuum, we construct a normalized weakly null sequence in $\ell_1(Γ)^*$ with no strongly normal subsequence. This answers a question of Śliwa negatively in ZFC. Together with the classical selection argument below the bounding number $\mathfrak b$, the construction gives positive and negative bounds for this subsequence property on $\ell_1(κ)$. Its validity on $\ell_1(ω_1)$ is independent of ZFC. The intermediate range $\mathfrak b\leqslantκ<\mathfrak c$ remains undecided by these results. We also give a counterexample in a Banach sequence space that is not isomorphic to any $\ell_1(Γ)$.

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arXiv2026

Pointwise function spaces over compacta are not weak Banach spaces

Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h\colon C_p(K)\to E_w$ whose inverse is continuous at $h(0)$. Consequently, $C_p(K)$ and $C_w(L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.

math.FA↗

arXiv2026

Continuous Linear Surjections from $C_p(X)$ onto Symmetric Sequence Ideals in $c_0$

Rosenthal's classical theorem says that, for every infinite compact space $X$, the Banach space $C(X)$ admits a quotient isomorphic to either $c_0$ or $\ell_2$. The corresponding question for $C_p(X)$, the space $C(X)$ endowed with the topology of pointwise convergence, is much subtler and still open; the only compact spaces for which the existence of an infinite-dimensional metrizable quotient is not presently settled in ZFC are Efimov compacta. We prove that the Banach space case cannot be reproduced with the usual sequence spaces carrying their pointwise topologies: Let $X$ be a Tychonoff space and let $E\subseteq c_0$ be a non-trivial symmetric sequence ideal endowed with the subspace topology inherited from $\mathbb{R}^{\mathbb{N}}$. Then the existence of a continuous linear surjection $T:C_p(X)\rightarrow E_p$ implies $E=c_0$, where $E_p$ means $E$ with the topology inherited from $\mathbb{R}^{\mathbb{N}}$. Hence, no proper non-zero symmetric sequence ideal of $c_0$ can be realized as a continuous linear image of a $C_p$-space. Combining this result with the characterization of the Josefson--Nissenzweig property for $C_p(X)$ obtained by Banakh, Kąkol, and Śliwa, we derive a complete characterization of all pairs $(X,E)$ for which such a surjection exists. In particular, for every $0<q<\infty$, there is no continuous linear surjection $C_p(X)\rightarrow(\ell_q)_p$.

math.FA↗

arXiv2026

Talagrand compacta, 2DCP, and pointwise quotients

We revisit Talagrand's CH compactum as a test object for the two-disjoint-copies property and for pointwise quotient questions. The two-disjoint-copies property, or 2DCP, is a topological sufficient condition for the existence of infinite-dimensional metrisable quotients of spaces $C_{\operatorname{p}}(X)$; recent work asks whether Talagrand's compactum has this property. Assuming $\diamondsuit(S)$ for a stationary co-stationary $S\subseteqω_1$, we carry out Talagrand's inverse-limit construction with additional diagonalisation. The resulting compactum $T$ keeps Talagrand's conclusions: $C(T)$ is Grothendieck, the weak-star compact ball $M_1(T)$ contains no copy of $βω$, and $T$ has no non-trivial convergent sequences. At the same time, no two disjoint non-metrisable closed subspaces of $T$ are homeomorphic; hence $T$ has no 2DCP and is not locally homogeneous. We also give a ZFC example of a perfect compact space with 2DCP which is not locally homogeneous and contains neither $βω$ nor $2^ω$. Finally, we isolate a general locally convex observation, in the spirit of the Banakh--Gabriyelyan theory of the Josefson--Nissenzweig property, showing that pointwise quotients onto $(\ell_p)_{\operatorname{p}}$, $1\leqslant p<\infty$, force the Josefson--Nissenzweig property. Consequently Talagrand compacta have no classical pointwise sequence quotients $(c_0)_{\operatorname{p}}$, $(\ell_p)_{\operatorname{p}}$, or $(\ell_\infty)_{\operatorname{p}}$. The full metrisable quotient problem for these $C_{\operatorname{p}}$-spaces remains open. Several open problems are included.

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arXiv2026

Amenability constants for unconditional sums of Banach algebras

We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family $(A_i)_{i\in I}$ of Banach algebras and a Banach sequence lattice $E$ on~$I$, the $E$-sum $\bigl(\bigoplus_{i\in I} A_i\bigr)_{\!E}$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that $C_E := \sup\{\|χ_F\|_E: F\subseteq I\text{ finite}\}<\infty$, we prove that this $E$-sum is amenable if and only if the amenability constants of the summands are uniformly bounded, and we establish the two-sided estimate \[ \sup_{i\in I}\text{AM}(A_i) \;\le\; \text{AM}\Bigl(\bigl(\textstyle\bigoplus_{i\in I} A_i\bigr)_{\!E}\Bigr) \;\le\; C_E^2\,\sup_{i\in I}\text{AM}(A_i). \] We show that the factor $C_E^2$ is sharp by exhibiting finite-dimensional examples where equality holds. We further prove that finiteness of $C_E$ is necessary whenever infinitely many summands are non-zero and the sum admits a bounded approximate identity. As applications, we recover the classical formula $\text{AM}\bigl(c_0\text{-}\bigoplus_{i\in I} A_i\bigr) = \sup_{i\in I}\text{AM}(A_i)$ for arbitrary (possibly uncountable) index sets, extend it to weighted $c_0$-spaces, and characterise amenability for Orlicz sequence algebra sums. We also record how these unconditional criteria give obstructions within the conditional framework of James-type $J$-sums. Finally, we investigate weak amenability of $E$-sums. We prove that weak amenability passes to summands, that $E$-sums of commutative weakly amenable algebras are weakly amenable, and--contrasting sharply with the Johnson amenability picture--that for $1 < p < \infty$, the $\ell_p$-sum of infinitely many copies of a non-commutative weakly amenable algebra fails to be weakly amenable. In the $c_0$-type regime ($C_E < \infty$), we establish two-sided estimates for weak amenability constants with constants depending only on $C_E$.

math.FA↗

arXiv2026

The two-disjoint-copies property for compact spaces, homogeneity and connection with $C_p$-theory

A Tychonoff space $X$ has the two-disjoint-copies property (2DCP) if there exists a sequence $(K_n)_{n\inω}$ of non-empty compact subsets of $X$ such that each $K_n$ contains two disjoint subsets homeomorphic to $K_{n+1}$. Banakh, Kąkol and Śliwa showed that 2DCP yields an infinite-dimensional metrizable quotient of $C_p(X)$, while it is still a long-standing open question whether $C_p(X)$ has such a quotient for any infinite compact space $X$. The above concept as well as the last problem are closely related to Efimov's problem that has remained open for 40 years. We will discuss a number of conditions that imply 2DCP. For example, every locally homogeneous compact space, every space containing a copy of $βω$ or $2^ω$ has 2DCP although compact $h$-homogeneous spaces with 2DCP without such copies exist in ZFC. We prove that no scattered compact space has 2DCP and there exist in ZFC compact perfect spaces without 2DCP. This implies that for compact metric spaces $X$ the 2DCP is equivalent to uncountability of $X$. There exist explicit uncountable separable compact spaces failing 2DCP, for example the Isbell-Mrówka compacta. We give positive classes among zero-dimensional compact spaces; for example, the Brech, as well as the Sobota-Zdomskyy compact spaces of Efimov type have 2DCP. Open questions are included.

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arXiv2025

Lotz-Peck-Porta and Rosenthal's theorems for spaces $C_p(X)$

For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. Applications will be provided. We prove also a $C_p$-version of Rosenthal's theorem by showing that for an infinite compact $X$ the space $C_p(X)$ contains a closed copy of $c_{0}(Γ)$ (with the pointwise topology) for some uncountable set $Γ$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$. Illustrating examples, additional supplementing $C_p$-theorems and comments are included.

math.FA↗

arXiv2024

Continuous operators from spaces of Lipschitz functions

We study the existence of continuous (linear) operators from the Banach spaces $\mbox{Lip}_0(M)$ of Lipschitz functions on infinite metric spaces $M$ vanishing at a distinguished point and from their predual spaces $\mathcal{F}(M)$ onto certain Banach spaces, including $C(K)$-spaces and the spaces $c_0$ and $\ell_1$. For pairs of spaces $\mbox{Lip}_0(M)$ and $C(K)$ we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. It is also showed that if a metric space $M$ contains a bilipschitz copy of the unit sphere $S_{c_0}$ of the space $c_0$, then $\mbox{Lip}_0(M)$ admits a continuous operator onto $\ell_1$ and hence onto $c_0$. Using this, we provide several conditions for a space $M$ implying that $\mbox{Lip}_0(M)$ is not a Grothendieck space. Finally, we obtain a new characterization of the Schur property for Lipschitz-free spaces: a space $\mathcal{F}(M)$ has the Schur property if and only if for every complete discrete metric space $N$ with cardinality $d(M)$ the spaces $\mathcal{F}(M)$ and $\mathcal{F}(N)$ are weakly sequentially homeomorphic.

math.FA↗

arXiv2023

On complemented copies of the space $c_0$ in spaces $C_p(X,E)$

We study the question for which Tychonoff spaces $X$ and locally convex spaces $E$ the space $C_p(X,E)$ of continuous $E$-valued functions on $X$ contains a complemented copy of the space $(c_0)_p=\{x\in\mathbb{R}^ω\colon x(n)\to0\}$, both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Domański and Drewnowski, proved for the case of Banach and Fréchet spaces $C_k(X,E)$. Also, for given infinite Tychonoff spaces $X$ and $Y$, we show that $C_p(X,C_p(Y))$ contains a complemented copy of $(c_0)_p$ if and only if any of the spaces $C_p(X)$ and $C_p(Y)$ contains such a subspace.

math.FA↗

arXiv2023

Grothendieck $C(K)$-spaces and the Josefson--Nissenzweig theorem

For a compact space $K$, the Banach space $C(K)$ is said to have the $\ell_1$-Grothendieck property if every weak* convergent sequence $\big\langleμ_n\colon\ n\inω\big\rangle$ of functionals on $C(K)$ such that $μ_n\in\ell_1(K)$ for every $n\inω$, is weakly convergent. Thus, the $\ell_1$-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that $C(K)$ has the $\ell_1$-Grothendieck property if and only if there does not exist any sequence of functionals $\big\langleμ_n\colon\ n\inω\big\rangle$ on $C(K)$, with $μ_n\in\ell_1(K)$ for every $n\inω$, satisfying the conclusion of the classical Josefson--Nissenzweig theorem. We construct an example of a separable compact space $K$ such that $C(K)$ has the $\ell_1$-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces $K$ their Banach spaces $C(K)$ do not have the $\ell_1$-Grothendieck property.

math.FA↗

arXiv2022

On complementability of $c_0$ in spaces $C(K\times L)$

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces $K$ and $L$ the product $K\times L$ admits a sequence $\langleμ_n\colon n\in\mathbb{N}\rangle$ of normalized signed measures with finite supports which converges to $0$ with respect to the weak* topology of the dual Banach space $C(K\times L)^*$. Our approach is completely constructive -- the measures $μ_n$ are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos and Freniche which states that for every infinite compact spaces $K$ and $L$ the Banach space $C(K\times L)$ contains a complemented copy of the space $c_0$.

math.FA↗

arXiv2022

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

Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces $X$ and $Y$ the Banach space $C(X\times Y)$ of continuous real-valued functions on $X\times Y$ 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 $C_{p}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the pointwise topology contains either a complemented copy of $\mathbb{R}^ω$ or a complemented copy of the space $(c_{0})_{p}=\{(x_n)_{n\inω}\in \mathbb{R}^ω\colon x_n\to 0\}$, both endowed with the product topology. We show that the latter case holds always when $X\times Y$ 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 $C_{p}(X\times X)$ 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 $C_{p}(X\times Y)$ is linearly homeomorphic to the space $C_{p}(X\times Y)\times\mathbb{R}$, although, as proved earlier by Marciszewski, there exists an infinite compact space $X$ such that $C_{p}(X)$ cannot be mapped onto $C_{p}(X)\times\mathbb{R}$ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form $C_p(X\times Y)$. Another corollary asserts that for every infinite Tychonoff spaces $X$ and $Y$ the space $C_{k}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: $\mathbb{R}^ω$, $(c_{0})_{p}$ or $c_{0}$.

math.GN↗

arXiv2021

The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

The celebrated Josefson-Nissenzweig theorem implies that for a Banach space $C(K)$ of continuous real-valued functions on an infinite compact space $K$ there exists a sequence of Radon measures $\langleμ_n\colon\ n\inω\rangle$ on $K$ which is weakly* convergent to the zero measure on $K$ and such that $\big\|μ_n\big\|=1$ for every $n\inω$. We call such a sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we study the situation when the space $K$ admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others that $K$ admits such a Josefson-Nissenzweig sequence if and only if $C(K)$ does not have the Grothendieck property restricted to functionals from the space $\ell_1(K)$. We also investigate miscellaneous analytic and topological properties of finitely supported Josefson-Nissenzweig sequences on general Tychonoff spaces. We prove that various properties of compact spaces guarantee the existence of finitely supported Josefson-Nissenzweig sequences. One such property is, e.g., that a compact space can be represented as the limit of an inverse system of compact spaces based on simple extensions. An immediate consequence of this result is that many classical consistent examples of Efimov spaces, i.e. spaces being counterexamples to the famous Efimov problem, admit such sequences of measures. Similarly, we show that if $K$ and $L$ are infinite compact spaces, then their product $K\times L$ always admits a finitely supported Josefson--Nissenzweig sequence. As a corollary we obtain a constructive proof that the space $C_p(K\times L)$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology--this generalizes results of Cembranos and Freniche. Finally, we provide a direct proof of the Josefson-Nissenzweig theorem for the case of Banach spaces $C(K)$.

math.FA↗

arXiv2020

$ω^ω$-Base and infinite-dimensional compact sets in locally convex spaces

A locally convex space (lcs) $E$ is said to have an $ω^ω$-base if $E$ has a neighborhood base $\{U_α:α\inω^ω\}$ at zero such that $U_β\subseteq U_α$ for all $α\leqβ$. The class of lcs with an $ω^ω$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $D'(Ω)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $ω^ω$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $ω^ω$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $φ$ endowed with the finest locally convex topology has an $ω^ω$-base but contains no infinite-dimensional compact subsets. It turns out that $φ$ is a unique infinite-dimensional locally convex space which is a $k_{\mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.

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