arXiv subjects

Zdeněk Silber

Publications and source records attributed to Zdeněk Silber.

9 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↗

arXiv2025

Banach spaces with arbitrary finite Baire order

We investigate intrinsic Baire classes of Banach spaces defined by Argyros, Godefroy and Rosenthal (2003). We introduce a construction, for any Banach space $X$ with a basis, of an $\ell_1$-saturated separable Banach space $Y$ such that for any $α\leqslant ω_1$ we have $Y^{**}_{1+α} \cong Y \oplus X^{**}_α$, where $X^{**}_α$ denotes the $α$-th intrinsic Baire class of $X$. We apply this construction to answer two open problems by Argyros, Godefroy and Rosenthal (2003), namely we build separable Banach spaces of any Baire order less or equal to $ω$, and a non-universal separable Banach space of order $ω_1$. Finally, we apply the construction to show an analogue of a result of Lindenstrauss (1971) by constructing, for any Banach space $X$ with a basis and any $n \in \mathbb{N}$, a Banach space $Y$ such that $Y^{**}_n \cong Y^{**}_{n-1} \oplus X$, showing that any such $X$ can appear as the space of functionals in a bidual Banach space $Y^{**}$ that are of $n$-th intrinsic Baire class but not of $(n-1)$-th intrinsic Baire class.

math.FA↗

arXiv2025

On Subspaces of Indecomposable Banach Spaces

We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit $\ell_\infty$ as a quotient (equivalently do not admit a subspace isomorphic to $\ell_1(\cc)$). This includes all Asplund spaces and all weakly Lindelöf determined Banach spaces of density not bigger than the continuum. However, we also show that this class includes some Banach spaces admitting $\ell_\infty$ as a quotient. This sheds some light on the question asked in [S. Argyros, R. Haydon, \emph{Bourgain-Delbaen $L^\infty$-spaces, the scalar-plus-compact property and related problems}, Proceedings of the International Congress of Mathematicians (ICM 2018), Vol. III, 1477--1510. Page 1502] whether all Banach spaces not containing $\ell_\infty$ embed in some indecomposable Banach spaces. Our method of constructing indecomposable Banach spaces above a given Banach space is a considerable modification of the method of constructing Banach spaces of continuous functions with few$^*$ operators developed before by the first-named author.

math.FA↗

arXiv2025

On weak sequential completeness of spaces where weakly compact sets are super weakly compact

We show that every Banach space in which weakly compact sets are super weakly compact in automatically weakly sequentially complete answering a question by Silber (2024). In the proof we show how to build a weakly compact set which is not super weakly compact from an arbitrary nontrivial weakly Cauchy sequence using the notion of a summing subsequence of Rosenthal or Singer.

math.FA↗

arXiv2024

Weak compactness in Lipschitz-free spaces over superreflexive spaces

We show that the Lipschitz-free space $\mathcal{F}(X)$ over a superreflexive Banach space $X$ has the property that every weakly precompact subset of $\mathcal{F}(X)$ is relatively super weakly compact, showing that this space "behaves like $L_1$" in this context. As consequences we show that $\mathcal{F}(X)$ enjoys the weak Banach-Saks property and that every subspace of $\mathcal{F}(X)$ with nontrivial type is superreflexive. Further, weakly compact subsets of $\mathcal{F}(X)$ are super weakly compact and hence have many strong properties. To prove the result, we use a modification of the proof of weak sequential completeness of $\mathcal{F}(X)$ by Kochanek and Pernecká and an appropriate version of compact reduction in the spirit of Aliaga, Noûs, Petitjean and Procházka.

math.FA↗

arXiv2023

Countably tight dual ball with a nonseparable measure

We construct a compact Hausdorff space $K$ such that the space $P(K)$ of Radon probabiblity measures on $K$ considered with the weak$^*$ topology (induced from the space of continuous functions $C(K)$) is countably tight which is a generalization of sequentiality (i.e., if a measure $μ$ is in the closure of a set $M$, there is a countable $M'\subseteq M$ such that $μ$ is in the closure of $M'$) but $K$ carries a Radon probability measure which has uncountable Maharam type (i.e., $L_1(μ)$ is nonseparable). The construction uses (necessarily) an additional set-theoretic assumption (the $\diamondsuit$ principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of $P(K\times K)$ implies that all Radon measures on $K$ have countable type. So, our example shows that the tightness of $P(K\times K)$ and of $P(K)\times P(K)$ can be different as well as $P(K)$ may have Corson property (C) while $P(K\times K)$ fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avilés, Martínez-Cervantes, Rodríguez and Rueda Zoca.

math.FA↗

arXiv2022

On subspaces whose weak* derived sets are proper and norm dense

We study long chains of iterated weak* derived sets, that is sets of all weak* limits of bounded nets, of subspaces with the additional property that the penultimate weak* derived set is a proper norm dense subspace of the dual. We extend the result of Ostrovskii and show, that in the dual of any non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual, we can find for any countable successor ordinal α a subspace, whose weak* derived set of order α is proper and norm dense.

math.FA↗

arXiv2021

Quantification of Banach-Saks properties of higher orders

We investigate possible quantifications of Banach-Saks sets and weak Banach-Saks sets of higher orders and their relations to other quantities. We prove a quantitative version of the characterization of weak $ξ$-Banach-Saks sets using $\ell_1^{ξ+1}$-spreading models and a quantitative version of the relation of $ξ$-Banach-Saks sets, weak $ξ$-Banach-Saks sets, norm compactness and weak compactness. We further introduce a new measure of weak compactness. Finally, we provide some examples showing the limitations of these quantifications.

math.FA↗

arXiv2021

Weak$^*$ derived sets of convex sets in duals of non-reflexive spaces

We investigate weak$^*$ derived sets, that is the sets of weak$^*$ limits of bounded nets, of convex subsets of duals of non-reflexive Banach spaces and their possible iterations. We prove that a dual space of any non-reflexive Banach space contains convex subsets of any finite order and a convex subset of order $ω+ 1$.

math.FA↗