arXiv subjects

Valentin Ferenczi

Publications and source records attributed to Valentin Ferenczi.

At least 19 recordsLinked to original sources

arXiv2025

Descriptive set theory of separable Fr\'echet spaces

In the past few decades, much has been done regarding the descriptive set theory of separable Banach spaces. However, the descriptive properties of separable Fr\'echet spaces have not yet been investigated. In these notes, we look at this problem, its relation with the (now standard) theory for separable Banach spaces, and we compute/estimate the descriptive complexity of some classical classes of separable Fr\'echet spaces such as Fr\'echet-Hilbert, Schwartz, nuclear, and Montel spaces. Our main result shows that the class of Montel spaces is complete coanalytic. Noticeably, this applies outside the realm of descriptive set theory and solves an old problem regarding Fr\'echet spaces satisfying the Heine--Borel property (i.e., Montel spaces). Precisely, we show that there is no separable Montel space containing isomorphic copies of all separable Montel spaces.

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arXiv2025

Equivariant liftings in Lipschitz-free spaces

We consider Banach spaces $X$ that can be linearly lifted into their Lipschitz-free spaces $\mathcal{F}(X)$ and, for a group $G$ acting on $X$ by linear isometries, we study the possible existence of $G$-equivariant linear liftings. In particular, we prove that such lifting exists when $G$ is compact in the strong operator topology, or an increasing union of such groups and $\mathcal{F}(X)$ is complemented in its bidual by an equivariant projection. As an example of application, we define and study a complex version of the Lipschitz-free space $\mathcal{F}(X)$ when $X$ is a subset of a complex Banach space stable under the action of the circle group.

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arXiv2024

Extremes of interpolation scales of Banach spaces

M. Daher gave conditions so that the spheres of the spaces in the interior of a complex interpolation scale are uniformly homeomorphic. We look for sufficient conditions for the validity of this result and related ones on the extremes of the scale, with applications to uniform homeomorphism between spheres of Banach spaces and the sphere of $\ell_2$.

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arXiv2023

Tight-minimal dichotomies in Banach spaces

We extend the methods used by V. Ferenczi and Ch. Rosendal to obtain the `third dichotomy' in the program of classification of Banach spaces up to subspaces, in order to prove that a Banach space E with an admissible system of blocks with admissible set A, contains an infinite dimensional subspace with a basis which is either A-tight or A-minimal. In this setting we obtain, in particular, dichotomies regarding subsequences of a basis, and as a corollary, we show that every normalized basic sequence has a subsequence which either satisfies a tightness property or is spreading. Other dichotomies between notions of minimality and tightness are demonstrated, and the Ferenczi-Godefroy interpretation of tightness in terms of Baire category is extended to this new context.

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arXiv2023

Almost Fra\"iss\'e Banach spaces

Continuing with the study of Approximately ultrahomogeneous and Fra\"iss\'e Banach spaces introduced by V. Ferenczi, J. L\'opez-Abad, B. Mbombo and S. Todorcevic, we define formally weaker and in some aspects more natural properties of Banach spaces which we call Almost ultrahomogeneity and the Almost Fra\"iss\'e Property. We obtain relations between these different homogeneity properties of a space $E$ and relate them to certain pseudometrics on the class $\mathrm{Age}(E)$ of finite dimensional subspaces of $E$. We prove that ultrapowers of an almost Fra\"iss\'e Banach space are ultrahomogeneous. We also study two properties called finitely isometrically extensible and almost finitely isometrically extensible, respectively, and prove that approximately ultrahomogeneous Banach spaces are finitely isometrically extensible.

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arXiv2023

Interpolator symmetries and new Kalton-Peck spaces

Diagrams generated by three interpolators in an abstract Kalton-Montgomery complex like interpolation scheme. We will consider in detail the case of the first three Schechter interpolators associated to the usual Calderón complex interpolation method in two especially interesting cases: weighted $\ell_2$ spaces, i.e., interpolation pairs $(\ell_2(w^{-1}), \ell_2(w))_θ$, and $\ell_p$ spaces, i.e., the interpolation pair $(\ell_\infty, \ell_1)_θ$, both at $θ=1/2$.

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arXiv2021

Envelopes in Banach spaces

We define the notion of isometric envelope of a subspace in a Banach space, and relate it to a) the mean ergodic projection on the space of fixed points of a semigroup of contractions, b) results on Korovkin sets from the 70's, and c) extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces $L_p$, $p \notin 2\mathbb N+4$ are the only separable Approximately Ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strenghtening of the Approximately Homogeneous Property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive $L_p$-spaces are the only reflexive rearrangement invariant spaces on $[0,1]$ for which all $1$-complemented subspaces are envelopes. We also identify the isometrically unique "full" quotient space of $L_p$ by a Hilbertian subspace, for appropriate values of $p$, as well as the associated topological group embedding of the unitary group into the isometry group of $L_p$.

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arXiv2021

Group actions on twisted sums of Banach spaces

We study bounded actions of groups and semigroups $G$ on exact sequences of Banach spaces from the point of view of quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of $G$-centralizer and $G$-equivariant map. We will show that when (A) $G$ is an amenable group and (U) the target space is complemented in its bidual by a $G$-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of $G$-centralizers; and that, under (A) and (U), $G$-centralizers are bounded perturbations of $G$-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of $L_p(0,1), p\neq 2$ on the Kalton-Peck space $Z_p$, certain non-unitarizable triangular representations of the free group $F_\infty$ on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of $L_p$-spaces. In the last section we consider the category of $G$-Banach spaces and study its exact sequences, showing that, under (A) and (U), $G$-splitting and usual splitting coincide.

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arXiv2021

On Mazur rotations problem and its multidimensional versions

The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach's famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraïssé theory and Ramsey theory, and led to development of concepts of interest independent of Mazur problem. This survey focuses on results that have been published after 2000, stressing two lines of research which were developed in the last ten years. The first one is the study of approximate versions of Mazur rotations problem in its various aspects, most specifically in the case of the Lebesgue spaces Lp. The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results. Some new results are also included.

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arXiv2021

There is no largest proper operator ideal

An operator ideal is proper if the only operators of the form $Id_X$ it contains have finite rank. We answer a question posed by Pietsch (1979) by proving that there is no largest proper operator ideal. Our proof is based on an extension of the construction by Aiena-González (2000), of an improjective but essential operator on Gowers-Maurey's shift space $X_S$ (1997), through a new analysis of the algebra of operators on powers of $X_S$. We also prove that certain properties hold for general $\mathbb{C}$-linear operators if and only if they hold for these operators seen as real: for example this holds for the ideals of strictly singular, strictly cosingular, or inessential operators, answering a question of González-Herrera (2007). This gives us a frame to extend the negative answer to the question of Pietsch to the real setting.

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arXiv2020

On the ${\Ext}^2$-problem for Hilbert spaces

We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves the four-space problem for local convexity.

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arXiv2019

On the stability of the differential process generated by complex interpolation

We study the stability of the differential process of Rochberg and Weiss associated to an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit sphere while there is no (bounded) stability for families of four or more Köthe spaces. In the context or arbitrary pairs of Banach spaces we present local stability results and global isometric stability results.

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arXiv2019

On Disjointly singular centralizers

We study ``disjoint" versions of the notions of trivial, locally trivial, strictly singular and super-strictly singular quasi-linear maps in the context of Köthe function spaces. Among other results, we show: i) (locally) trivial and (locally) disjointly trivial notions coincide on reflexive spaces; ii) On non-atomic superreflexive Köthe spaces, no centralizer is singular, although most are disjointly singular. iii) No super singular quasi-linear maps exist between superreflexive spaces although Kalton-Peck centralizers are super disjointly singular; iv) Disjoint singularity does not imply super disjoint singularity.

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arXiv2018

Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

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arXiv2017

On the classification of positions and of complex structures in Banach spaces

A topological setting is defined to study the complexities of the relation of equivalence of embeddings (or "position") of a Banach space into another and of the relation of isomorphism of complex structures on a real Banach space. The following results are obtained: a) if $X$ is not uniformly finitely extensible, then there exists a space $Y$ for which the relation of position of $Y$ inside $X$ reduces the relation $E_0$ and therefore is not smooth; b) the relation of position of $\ell_p$ inside $\ell_p$, or inside $L_p$, $p \neq 2$, reduces the relation $E_1$ and therefore is not reducible to an orbit relation induced by the action of a Polish group; c) the relation of position of a space inside another can attain the maximum complexity $E_{\rm max}$; d) there exists a subspace of $L_p, 1 \leq p <2$, on which isomorphism between complex structures reduces $E_1$ and therefore is not reducible to an orbit relation induced by the action of a Polish group.

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arXiv2015

Complex structures on twisted Hilbert spaces

We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck $Z_2$ space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of Köthe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant Köthe function spaces then there are complex structures on it that can be extended to a complex structure of the twisted Hilbert space. Regarding the hyperplane problem we show that no complex structure on $\ell_2$ can be extended to a complex structure on an hyperplane of $Z_2$ containing it.

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arXiv2014

Singular twisted sums generated by complex interpolation

We present new methods to obtain singular twisted sums $X\oplus_ΩX$ (i.e., exact sequences $0\to X\to X\oplus_ΩX \to X\to 0$ in which the quotient map is strictly singular), in which $X$ is the interpolation space arising from a complex interpolation scheme and $Ω$ is the induced centralizer. Although our methods are quite general, in our applications we are mainly concerned with the choice of $X$ as either a Hilbert space, or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces, including the only known example so far: the Kalton-Peck space $Z_2$. In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. We then use Rochberg's description of iterated twisted sums to show that there is a sequence $\mathcal F_n$ of H.I. spaces so that $\mathcal F_{m+n}$ is a singular twisted sum of $\mathcal F_m$ and $\mathcal F_n$, while for $l>n$ the direct sum $\mathcal F_n \oplus \mathcal F_{l+m}$ is a nontrivial twisted sum of $\mathcal F_l$ and $\mathcal F_{m+n}$. We also introduce and study the notion of disjoint singular twisted sum of Köthe function spaces and construct several examples involving reflexive $p$-convex Köthe function spaces, which include the function version of the Kalton-Peck space $Z_2$.

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arXiv2014

Non-unitarisable representations and maximal symmetry

We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of ${\rm Aut}(T)$ on $\ell_2(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of ${\rm GL}(\mathcal H)$ extending a well-known non-unitarisable representation of $\mathbb F_\infty$. As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex.

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