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Timur Oikhberg

Publications and source records attributed to Timur Oikhberg.

At least 19 recordsLinked to original sources
arXiv2026

Upper bound properties of free and related Banach lattices via operators

It is known that the free $p$-convex Banach lattice on a Banach space $X$ can be represented as a space of functions on the unit ball of $X^*$. In this way, it gives rise to certain related (larger) lattices. To investigate such lattices, we introduce a new tool, related to operators into $X$. This tool is then used to (i) determine whether the lattices in question possess, or fail, properties involving upper bounds of upward directed sets -- namely, the Fatou property, and the related property of monotonic boundedness; (ii) investigate the regularity of embeddings between spaces in question. We find connections between the Fatou-like properties of ${\textrm{FBL}}^{(p)}[X]$ and the Radon-Nikodym property of $X$. In addition, we give an example of $X \subset Y$ such that ${\textrm{FBL}}^{(p)}[X]$ is not a regular sublattice of ${\textrm{FBL}}^{(p)}[Y]$.

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arXiv2026

On Matricial Order Operator Spaces

We investigate the category of ``matricial order operator spaces,'' which generalize operator systems, being equipped with both matricial norms and matricial order. For these objects, we develop duality theory. Taking a cue from the theory of ordered normed spaces, we introduce two important properties describing the interplay between order and norm -- ``normality'' and ``generation,'' and show that they are dual to each other. As examples, we consider operator systems (in particular, C*-algebras), and Schatten spaces. We also describe the minimal and maximal matricial order structures (which, again, turn out to be in duality), and show how Banach lattices can be equipped with such structures.

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arXiv2026

Maximal inequalities, frames and greedy algorithms

The aim of this article is to use Banach lattice techniques to study coordinate systems in function spaces. We begin by proving that the greedy algorithm of a basis is order convergent if and only if a certain maximal inequality is satisfied. We then show that absolute frames need not admit a reconstruction algorithm with respect to the usual order convergence, but do allow for reconstruction with respect to the order convergence inherited from the double dual. After this, we investigate the extent to which such coordinate systems affect the geometry of the underlying function space. Most notably, we prove that a Banach lattice $X$ is lattice isomorphic to a closed sublattice of a $C(K)$-space if and only if every unconditional sequence in $X$ is absolute.

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arXiv2025

The lattice Sch\"affer constant

For a Banach lattice $X$, its lattice Sch\"affer constant is defined by: \begin{gather*} \lambda^+(X)=\inf\{\max\{\|x+y\|,\|x-y\|\}\,\colon\,\|x\|=\|y\|=1,x,y\geq{\bf0}\}. \end{gather*} In this paper, we investigate this constant, as well as the companion parameter \begin{gather*} \beta(X)=\inf\{\|x\vee y\|\,\colon\,\mbox{$\|x\|=\|y\|=1$, $x,y\geq{\bf0}$ and $x\wedge y={\bf0}$}\}. \end{gather*} Our main results fall into two groups. (1) We link the behavior of the parameters $\lambda^+$ and $\beta$ to the global properties of the lattice $X$. For instance, we prove that (i) if $\lambda^+(X)>1$, then the Banach lattice $X$ is a KB-space, and moreover, it satisfies a lower $q$-estimate for some $q\in(1,\infty)$; (ii) $\lambda^+(X)=1$ if and only if $X$ contains lattice-almost isometric copies of $\ell_\infty^2$; and (iii) that $\lambda^+(X)=2$ if and only if $X$ is an abstract $L$-space. (2) We establish inequalities relating $\lambda^+(X)$ to the characteristics of monotonicity, $\varepsilon_{0,m}(X)$ and $\tilde\varepsilon_{0,m}(X)$. Along the way, we compute $\lambda^+(X)$ and $\beta(X)$ for various Banach lattices $X$.

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arXiv2024

Lattice renormings of $C_0(X)$ spaces

Suppose $X$ is a locally compact Polish space, and $G$ is a group of lattice isometries of $C_0(X)$ which satisfies certain conditions. Then we can equip $C_0(X)$ with an equivalent lattice norm $| \! | \! | \cdot | \! | \! |$ so that $G$ is the group of lattice isometries of $(C_0(X), | \! | \! | \cdot | \! | \! |)$. As an application, we show that for any locally compact Polish group $G$ there exists a locally compact Polish space $X$, and an lattice norm $| \! | \! | \cdot | \! | \! |$ on $C_0(X)$, so that $G$ is the group of lattice isometries of $(C_0(X), | \! | \! | \cdot | \! | \! |)$.

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arXiv2023

Counterexamples in isometric theory of symmetric and greedy bases

We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to $1$. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be $1$-suppression unconditional, thus settling Problem 4.4 from [F. Albiac and J.L. Ansorena, Characterization of $1$-almost greedy bases, Rev. Mat. Complut. 30 (2017), no. 1, 13-24]. In particular, our construction demonstrates that bases with Property (A) need not be $1$-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

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arXiv2023

Geometry of unit balls of free Banach lattices, and its applications

We begin by describing the unit ball of the free $p$-convex Banach lattice over a Banach space $E$ (denoted by ${\mathrm{FBL}}^{(p)}[E]$) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space $E$ has the $\lambda$-Approximation Property, then ${\mathrm{FBL}}^{(p)}[E]$ has the $\lambda$-Positive Approximation Property. Further, we show that operators $u \in B(E,F)$ (where $E$ and $F$ are Banach spaces) which extend to lattice homomorphisms from ${\mathrm{FBL}}^{(q)}[E]$ to ${\mathrm{FBL}}^{(p)}[F]$ are precisely those whose adjoints are $(q,p)$-mixing. Related results are also obtained for free lattices with an upper $p$-estimate.

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arXiv2022

Coarse geometry of operator spaces and complete isomorphic embeddings into $\ell_1$ and $c_0$-sums of operator spaces

The nonlinear geometry of operator spaces has recently started to be investigated. Many notions of nonlinear embeddability have been introduced so far, but, as noticed before by other authors, it was not clear whether they could be considered ``correct notions''. The main goal of these notes is to provide the missing evidence to support that \emph{almost complete coarse embeddability} is ``a correct notion''. This is done by proving results about the complete isomorphic theory of $\ell_1$-sums of certain operators spaces. Several results on the complete isomorphic theory of $c_0$-sums of operator spaces are also obtained.

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arXiv2020

Renorming AM-spaces

We prove that any separable AM-space $X$ has an equivalent lattice norm for which no non-trivial surjective lattice isometries exist. Moreover, if $X$ has no more than one atom, then this new norm may be an AM-norm. As our main tool, we introduce and investigate the class of so called Benyamini spaces, which ``approximate'' general AM-spaces.

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arXiv2020

Order extreme points and solid convex hulls

We consider the "order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn-Banach type separation result is obtained, which allows us to establish an "order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining functionals. Finally,we introduce the "solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikodym Property.

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arXiv2018

Lebesgue inequalities for Chebyshev Thresholding Greedy Algorithms

We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper.

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arXiv2017

Injectivity and projectivity in $p$-multinormed spaces

We find large classes of injective and projective $p$-multinormed spaces. In fact, these classes are universal, in the sense that every $p$-multinormed space embeds into (is a quotient of) an injective (resp. projective) $p$-multinormed space. As a consequence, we show that any $p$-multinormed space has a canonical representation as a subspace of a quotient of a Banach lattice.

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arXiv2016

Almost band preservers

We study the stability of band preserving operators on Banach lattices. To this end the notion of $\varepsilon$-band preserving mapping is introduced. It is shown that, under quite general assumptions, a $\varepsilon$-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of $\varepsilon$-band preserving maps are also obtained.

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arXiv2016

Reducing the number of inputs in nonlocal games

In this work we show how a vector-valued version of Schechtman's empirical method can be used to reduce the number of inputs in a nonlocal game $G$ while preserving the quotient $β^*(G)/β(G)$ of the quantum over the classical bias. We apply our method to the Khot-Vishnoi game, with exponentially many questions per player, to produce another game with polynomially many ($N\approx n^8$) questions so that the quantum over the classical bias is $Ω(n/\log^2 n)$.

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arXiv2015

Almost disjointness preservers

We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an "almost disjointness preserving" operator is well approximable by a disjointess preserving one. However, this approximation is not always possible, as our examples show.

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arXiv2014

2-local triple derivations on von Neumann algebras

We prove that every {\rm(}not necessarily linear nor continuous{\rm)} 2-local triple derivation on a von Neumann algebra $M$ is a triple derivation, equivalently, the set Der$_{t} (M)$, of all triple derivations on $M,$ is algebraically 2-reflexive in the set $\mathcal{M}(M)= M^M$ of all mappings from $M$ into $M$.

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arXiv2014

Subprojective Banach spaces

A Banach space $X$ is called subprojective if any of its infinite dimensional subspaces $Y$ contains a further infinite dimensional subspace complemented in $X$. This paper is devoted to systematic study of subprojectivity. We examine the stability of subprojectivity of Banach spaces under various operations, such us direct or twisted sums, tensor products, and forming spaces of operators. Along the way, we obtain new classes of subprojective spaces.

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