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Damian Sobota

Publications and source records attributed to Damian Sobota.

At least 19 recordsLinked to original sources
arXiv2026

Symmetric compactifications of the integers and separable quotients of spaces $C_p(X)$

A compactification of the discrete space $ω$ of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of $ω$. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of $ω$, and the Arkhangel'ski\uı--Bereznitski\uı--Schachermayer space studied in $C_p$-theory, which is in a sense the maximal symmetric compactification of $ω$. We investigate symmetric compactifications of $ω$ and their relations to the Separable Quotient Problem for spaces $C_p(X)$ and to the existence of Josefson--Nissenzweig sequences of finitely supported Borel measures on spaces $X$, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces $C_p(K)$, $K$ compact, to the case when $K$ is a totally asymmetric compactification of $ω$. Our results shed some new light on the Grothendieck property of Banach spaces $C(K)$.

math.FA↗
arXiv2026

A small Banach space $C(K)$ without nice renormings

We prove that consistently $ω_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $ω_1$, and admits no renorming which is strictly convex or sequentially Kadets--Klee.

math.FA↗
arXiv2026

A small remark on small-dimensional normed barrelled spaces

Combining the methods of Brian and Stuart with the classical Dvoretzky theorem, we show that no infinite-dimensional Banach space contains a barrelled subspace of (algebraic) dimension $<\mbox{cov}(\mathcal{N})$, the covering number of the Lebesgue null ideal $\mathcal{N}$. Consequently, every infinite-dimensional normed barrelled space has dimension $\ge\mbox{cov}(\mathcal{N})$ and so it is consistent with \textsf{ZFC} that no normed barrelled space has dimension equal to the bounding number $\mathfrak{b}$.

math.FA↗
arXiv2026

The Nikodym and Grothendieck properties of Boolean algebras and rings related to ideals

For an ideal $\mathcal{I}$ in a $σ$-complete Boolean algebra $\mathcal{A}$, we show that if the Boolean algebra $\mathcal{A}\langle\mathcal{I}\rangle$ generated by $\mathcal{I}$ does not have the Nikodym property, then it does not have the Grothendieck property either. The converse however does not hold -- we construct a family of $\mathfrak{c}$ many pairwise non-isomorphic Boolean subalgebras of the power set $\wp(ω)$ of the form $\wp(ω)\langle\mathcal{I}\rangle$ which, when thought of as subsets of the Cantor space $2^ω$, belong to the Borel class $\mathbb{F}_{σδ}$ and have the Nikodym property but not the Grothendieck property, and a family of $2^\mathfrak{c}$ many pairwise non-isomorphic non-analytic Boolean algebras of the form $\wp(ω)\langle\mathcal{I}\rangle$ with the Nikodym property but without the Grothendieck property. Extending a result of Hernández-Hernández and Hrušák, we show that for an analytic P-ideal $\mathcal{I}$ on $ω$ the following are equivalent: 1) $\mathcal{I}$ is totally bounded, 2) $\mathcal{I}$ has the Local-to-Global Boundedness Property for submeasures, 3) $\wp(ω)/\mathcal{I}$ contains a countable splitting family, 4) $\mbox{conv}\le_K\mathcal{I}$. Moreover, proving a conjecture of Drewnowski, Florencio, and Paúl, we present examples of analytic P-ideals on $ω$ with the Nikodym property but without the Local-to-Global Boundedness Property for submeasures (and so not totally bounded). Exploiting a construction of Alon, Drewnowski, and Łuczak, we also describe a family of $\mathfrak{c}$ many pairwise non-isomorphic ideals on $ω$, induced by sequences of Kneser hypergraphs, which all have the Nikodym property but not the Nested Partition Property -- this answers a question of Stuart. Finally, Tukey reducibility of a class of ideals without the Nikodym property is studied.

math.LO↗
arXiv2026

Complementability of separable spaces $\mathcal{C}(K)$ in Banach spaces

For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the product $E \times E^*$, based on new descriptions of the Banach spaces $\mathcal{C}([1, ω^α])$ for countable ordinal numbers $α$ and $\mathcal{C}(2^ω)$. Applying this general result in the case where $E=\mathcal{C}(K)$ for some compact space $K$, we further obtain a characterization of the existence of a positively $1$-complemented positively isometric copy of $\mathcal{C}(L)$ inside $\mathcal{C}(K)$ in terms of the topology of $K$ and the space of probability Radon measures on $K$. In the process, we also prove a variant of the classical Holsztyński theorem for isometric embeddings onto complemented subspaces.

math.FA↗
arXiv2024

On embedding separable spaces $\mathcal{C}(L)$ in arbitrary spaces $\mathcal{C}(K)$

Supplementing and expanding classical results, for compact spaces $K$ and $L$, $L$ metric, and their Banach spaces $\mathcal{C}(L)$ and $\mathcal{C}(K)$ of continuous real-valued functions, we provide several characterizations of the existence of isometric, resp. isomorphic, embeddings of $\mathcal{C}(L)$ into $\mathcal{C}(K)$. In particular, we show that if the embedded space $\mathcal{C}(L)$ is separable, then the classical theorems of Holsztyński and Gordon become equivalences. We also obtain new results describing the relative cellularities of the perfect kernel of a given compact space $K$ and of the Cantor--Bendixson derived sets of $K$ of countable order in terms of the presence of isometric copies of specific spaces $\mathcal{C}(L)$ inside $\mathcal{C}(K)$.

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↗
arXiv2024

Complemented subspaces of Banach spaces $C(K\times L)$

We prove that, for every compact spaces $K_1,K_2$ and compact group $G$, if both $K_1$ and $K_2$ map continuously onto $G$, then the Banach space $C(K_1 \times K_2)$ contains a complemented subspace isometric to the Banach space $C(G)$. Consequently, $C(K_1\times K_2)$ contains a complemented copy of $C([0,1])$ for every non-scattered $K_1,K_2$. Also, answering a question of Alspach and Galego, we get that $C(βω\timesβω)$ contains a complemented copy of $C([0,1]^κ)$ for every cardinal number $1\leκ\le{\mathfrak c}$ and hence a complemented copy of $C(K)$ for every metric compact space $K$. On the other hand, for the pointwise topology, we show that $C_p(βω\timesβω)$ contains no complemented copy of $C_p(2^ω)$.

math.FA↗
arXiv2024

Construction under Martin's axiom of a Boolean algebra with the Grothendieck property but without the Nikodym property

Improving a result of M. Talagrand, under the assumption of a weak form of Martin's axiom, we construct a totally disconnected compact Hausdorff space $K$ such that the Banach space $C(K)$ of continuous real-valued functions on $K$ is a Grothendieck space but there exists a sequence $(μ_n)$ of Radon measures on $K$ such that $μ_n(A)\to0$ for every clopen set $A\subseteq K$ and $\int_Kfdμ_n\not\to0$ for some $f\in C(K)$. Consequently, we get that Martin's axiom implies the existence of a Boolean algebra with the Grothendieck property but without the Nikodym property.

math.FA↗
arXiv2024

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

For a free filter $F$ on $ω$, endow the space $N_F=ω\cup\{p_F\}$, where $p_F\not\inω$, with the topology in which every element of $ω$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. Spaces of the form $N_F$ 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, for a filter $F$, the space $N_F$ carries a sequence $\langleμ_n\colon n\inω\rangle$ of normalized finitely supported signed measures such that $μ_n(f)\to 0$ for every bounded continuous real-valued function $f$ on $N_F$ if and only if $F^*\le_K\mathcal{Z}$, that is, the dual ideal $F^*$ is Katětov below the asymptotic density ideal $\mathcal{Z}$. Consequently, we get that if $F^*\le_K\mathcal{Z}$, then: (1) if $X$ is a Tychonoff space and $N_F$ 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 $c_0$ endowed with the pointwise topology, (2) if $K$ is a compact Hausdorff space and $N_F$ 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.

math.FA↗
arXiv2023

On sequences of finitely supported measures related to the Josefson--Nissenzweig theorem

Given a Tychonoff space $X$, we call a sequence $\langleμ_n\colon n\inω\rangle$ of signed Borel measures on $X$ a finitely supported Josefson--Nissenzweig sequence (in short a JN-sequence) if: 1) for every $n\inω$ the measure $μ_n$ is a finite combination of one-point measures and $\|μ_n\|=1$, and 2) $\int_Xf\,\mathrm{d}μ_n\to0$ for every continuous function $f\in C(X)$. Our main result asserts that if a Tychonoff space $X$ admits a JN-sequence, then there exists a JN-sequence $\langleμ_n\colon n\inω\rangle$ such that: i) $\mbox{supp}(μ_n)\cap\mbox{supp}(μ_k)=\emptyset$ for every $n\neq k\inω$, and ii) the union $\bigcup_{n\inω}\mbox{supp}(μ_n)$ is a discrete subset of $X$. We also prove that if a Tychonoff space $X$ carries a JN-sequence, then either there is a JN-sequence $\langleμ_n\colon n\inω\rangle$ on $X$ such that $|\mbox{supp}(μ_n)|=2$ for every $n\inω$, or for every JN-sequence $\langleμ_n\colon n\inω\rangle$ on $X$ we have $\lim_{n\to\infty}|\mbox{supp}(μ_n)|=\infty$.

math.GN↗
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

Convergence of measures after adding a real

We prove that if $\mathcal{A}$ is an infinite Boolean algebra in the ground model $V$ and $\mathbb{P}$ is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any $\mathbb{P}$-generic extension $V[G]$, $\mathcal{A}$ has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.

math.LO↗
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↗
arXiv2022

There is a P-measure in the random model

We say that a finitely additive probability measure $μ$ on $ω$ is \emph{a P-measure} if it vanishes on points and for each decreasing sequence $(E_n)$ of infinite subsets of $ω$ there is $E\subseteqω$ such that $E\subseteq^* E_n$ for each $n\inω$ and $μ(E) = \lim_{n\to\infty}μ(E_n)$. Thus, P-measures generalize in a natural way P-points and it is known that, similarly as in the case of P-points, their existence is independent of $\mathsf{ZFC}$. In this paper we show that there is a P-measure in the model obtained by adding any number of random reals to a model of $\mathsf{CH}$. As a corollary, we obtain that in the classical random model $ω^*$ contains a nowhere dense ccc closed P-set.

math.LO↗
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↗