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Eugene Bilokopytov

Publications and source records attributed to Eugene Bilokopytov.

At least 19 recordsLinked to original sources
arXiv2026

Variants of order semicontinuity in Banach lattices

In this article we consider various properties of a normed lattice, which are similar to semicontinuity (also known as the Fatou property), establish relations between these properties, and discuss stability of these properties under renorming. In particular, we show that a normed lattice $F$ is order continuous iff every renorming of $F$ is semicontinuous. We also prove that the weakly Fatou (we propose the term ``demicontinuous'') normed lattices are precisely the ones isomorphic to regular sublattices of monotonically complete Banach lattices. In order to do so we introduce the concept of the Lorentz completion of a demicontinuous normed lattice, which is somewhat analogous to the universal completion from the vector lattice theory. Furthermore, we unify and simplify the proofs of the characterizations of monotone completeness from \cite{aw} and \cite{taylor} and provide their quantitative versions. The semicontinuity-related properties in AM-spaces have some additional features. While the classical Kakutani theorem states that AM-spaces are precisely the closed sublattices of $\Co\left(K\right)$-spaces, we show, using two different methods, that the semicontinuous AM-spaces are precisely the closed regular sublattices of $\Co\left(K\right)$-spaces. Finally, we prove that for a normed space $E$, the AM-space of positively homogeneous weak* continuous functions on $\Ba_{E^{*}}$ is semicontinuous iff it is a regular sublattice of $\Co\left(\Ba_{E^{*}},\mathrm{w}^{*}\right)$ iff $\dim E<\8$.

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arXiv2026

Relative uniform completion of a vector lattice

In the paper, we revisit several approaches to the concept of uniform completion $X^{\mathrm{ru}}$ of a vector lattice $X$. We show that many of these approaches yield the same result. In particular, if $X$ is a sublattice of a uniformly complete vector lattice $Z$ then $X^{\mathrm{ru}}$ may be viewed as the intersection of all uniformly complete sublattices of $Z$ containing $X$. $X^{\mathrm{ru}}$ may also be constructed via a transfinite process of taking uniform adherences in $Z$ with regulators coming from the previous adherences. If, in addition, $X$ is majorizing in $Z$ then $X^{\mathrm{ru}}$ may be viewed as the uniform closure of $X$ in $Z$. We show that $X^{\mathrm{ru}}$ may also be characterized via a universal property: every positive operator from $X$ to a uniformly complete vector lattice extends uniquely to $X^{\mathrm{ru}}$. Moreover, the class of positive operators here may be replaced with several other important classes of operators (e.g., lattice homomorphisms). We also discuss conditions when the uniform adherence of a sublattice equals its uniform closure, and present an example (based on a construction by R.N. Ball and A.W. Hager) where this fails.

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arXiv2026

The Dedekind completion of an Archimedean ordered vector space as a reflector

We consider the category $\mathbf{AOVS}$ of Archimedean ordered vector spaces with linear maps which preserve all existing suprema, and its full subcategories $\mathbf{DAOVS}$, $\mathbf{DVL}$ and $\mathbf{UVL}$, consisting of directed spaces, Dedekind complete vector lattices and universally complete vector lattices, respectively. We deduce from some results in the literature that $\mathbf{DVL}$ and $\mathbf{UVL}$ are reflective subcategories of $\mathbf{DAOVS}$, with the usual Dedekind completion being the reflector in $\mathbf{DVL}$. In contrast to these facts, we show that a non-directed Archimedean ordered vector space of dimension greater than $1$ has no reflector in either $\mathbf{DVL}$ or $\mathbf{UVL}$. In particular, there are no free Dedekind complete vector lattices over a set with more than one element. We also use the occasion to show that a free vector lattice with $α$ generators embeds into a free vector lattice with $β$ generators if and only if $α\leβ$, and explore the concept of the free completion of an Archimedean vector lattice with a strong unit.

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arXiv2026

Normed lattices majorizing in their norm completions

This note is a follow-up to \cite{bt}. We focus on conditions under which a normed lattice $X$ is majorizing in its norm completion. We show that \cite[Question 8.17]{bt} -- namely, whether this holds whenever every norm-null sequence in $X$ has an order-bounded subsequence -- is equivalent to the question whether every P-ideal on $\N$ is meager. This is a longstanding open problem in Set Theory, and it has a negative answer under various set-theoretical assumptions, in particular under the Continuum Hypothesis. We also present several equivalent conditions to both of the two aforementioned properties, and give a simple proof of a well-known Riesz-Fischer-style characterization of completeness of a normed lattice.

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arXiv2026

A universal approximation theorem and its applications to vector lattice theory

A classical result in approximation theory states that for any continuous function \( φ: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{φ\circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if \( φ\) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improve the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}.

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arXiv2025

Countability conditions in locally solid convergence spaces

We study (strong) first countability of locally solid convergence structures on Archimedean vector lattices. Among other results, we characterise those vector lattices for which relatively unform-, order-, and $σ$-order convergence, respectively, is (strongly) first countable. The implications for the validity of sequential arguments in the contexts of these convergences are pointed out.

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arXiv2025

Norm-attaining lattice homomorphisms and renormings of Banach lattices

A well-known theorem due to R. C. James states that a Banach space is reflexive if and only if every bounded linear functional attains its norm. In this note we study Banach lattices on which every (real-valued) lattice homomorphism attains its norm. Contrary to what happens in the Banach space setting, we show that this property is not invariant under lattice isomorphisms. Namely, we show that in an AM-space every lattice homomorphism attains its norm, whereas every infinite-dimensional $C(K)$ space admits an equivalent lattice norm with a lattice homomorphism which does not attain its norm. Furthermore, we characterize coordinate functionals of atoms and show that whenever a Banach lattice $X$ supports a strictly positive functional, there exists a renorming with the property that the only (non-trivial) lattice homomorphisms attaining their norm are precisely these coordinate functionals. As a consequence, one can exhibit examples of Dedekind complete Banach lattices admitting a renorming with a non-norm-attaining lattice homomorphism, answering negatively questions posed by Dantas, Rodríguez Abellán, Rueda Zoca and the fourth author.

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arXiv2025

Disjointly non-singular operators and various topologies on Banach lattices

We continue the study of dispersed subspaces and disjointly non-singular (DNS) operators on Banach lattices using topological methods. In particular, we provide a simple proof of the fact that in an order continuous Banach lattice an operator is DNS if and only if it is $n$-DNS, for some $n\in\mathbb{N}$. We characterize Banach lattices with order continuous dual in terms of dispersed subspaces and absolute weak topology. We also connect these topics with the recently launched study of phase retrieval in Banach lattices.

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arXiv2024

Locally hulled topologies

We present a general result about generating group topologies by pseudo-norms. Namely, we show that if a topology has a base of sets which are closed in a certain sense, then it can be generated by a collection of pseudo-norms such that the balls in these pseudo-norms are also closed in the same sense. The examples include linear and locally convex topologies on vector spaces, locally solid and Fatou topologies on vector lattices and Fréchet-Nikodým topologies on Boolean algebras.

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arXiv2024

Composition of locally solid convergences

We carry on a more detailed investigation of the composition of locally solid convergences as introduced in [BCTvdW24], as well as the corresponding notion of idempotency considered in [Bil23]. In particular, we study the interactions between these two concepts and various operations with convergences. We prove associativity of the composition and show that the adherence of an ideal with respect to an idempotent convergence is equal to its closure. Some results from [KT18] about unbounded modification of locally solid topologies are generalized to the level of locally solid idempotent convergences. A simple application of the composition allows us to answer a question from [BCTvdW24] about minimal Hausdorff locally solid convergences. We also show that the weakest Hausdorff locally solid convergence exists on an Archimedean vector lattice if and only if it is atomic.

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arXiv2024

Characterizations of the projection bands and some order properties of the lattices of continuous functions

We show that for an ideal $H$ in an Archimedean vector lattice $F$ the following conditions are equivalent: $\bullet$ $H$ is a projection band; $\bullet$ Any collection of mutually disjoint vectors in $H$, which is order bounded in $F$, is order bounded in $H$; $\bullet$ $H$ is an infinite meet-distributive element of the lattice $\mathcal{I}_{F}$ of all ideals in $F$ in the sense that $\bigcap\limits_{J\in \mathcal{J}}\left(H+ J\right)=H+ \bigcap \mathcal{J}$, for any $\mathcal{J}\subset \mathcal{I}_{F}$. Additionally, we show that if $F$ is uniformly complete and $H$ is a uniformly closed principal ideal, then $H$ is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces.

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arXiv2024

Order continuity and regularity on vector lattices and on lattices of continuous functions

We give several characterizations of order continuous vector lattice homomorphisms between Archimedean vector lattices. We reduce the proofs of some of the equivalences to the case of composition operators between vector lattices of continuous functions, and so we obtain a characterization of order continuity of such operators. Motivated by this, we investigate various properties of the sublattices of the space $C\left(X\right)$, where $X$ is a Tychonoff topological space. We also obtain several characterizations of a regular sublattice of a vector lattice, and show that the closure of a regular sublattice of a Banach lattice is also regular.

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arXiv2024

Locally solid convergences and order continuity of positive operators

We consider vector lattices endowed with locally solid convergence structures, which are not necessarily topological. We show that such a convergence is defined by the convergence to $0$ on the positive cone. Some results on unbounded modification which were only available in partial cases are generalized. Order convergence is characterized as the strongest locally solid convergence in which monotone nets converge to their extremums (if they exist). We partially characterize sublattices on which the order convergence is the restriction of the order convergence of the ambient lattice. We show that homomorphism is order continuous iff it is uo-continuous. Uo convergence is characterized independently of order convergence. We show that on the space of continuous function uo convergence is weaker than the compact open convergence iff the underlying topological space contains a dense locally compact subspace. For a large class of convergences we prove that a positive operator is order continuous if and only if its restriction to a dense regular sublattice is order continuous, and that the closure of a regular sublattice is regular with the original sublattice being order dense in the closure. We also present an example of a regular sublattice of a locally solid topological vector lattice whose closure is not regular.

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arXiv2024

Uniformly closed sublattices of finite codimension

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension $n$ contains a uniformly closed ideal of codimension at most $2n$. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of [AL90a,AL90b] and , as well as Kakutani's characterization of closed sublattices of $C(K)$ spaces.

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arXiv2024

Atomicity of Boolean algebras and vector lattices in terms of order convergence

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.

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arXiv2021

Order and uo-convergence in spaces of continuous functions

We present several characterizations of uo-convergent nets or sequences in spaces of continuous functions $C(Ω)$, $C_b(Ω)$, $C_0(Ω)$, and $C^\infty(Ω)$, extending results of [vdW18]. In particular, it is shown that a sequence uo-converges iff it converges pointwise on a co-meagre set. We also characterize order bounded sets in spaces of continuous functions. This leads to characterizations of order convergence.

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arXiv2021

Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology

In this article we investigate the disjointly non-singular (DNS) operators. Following [8] we say that an operator $T$ from a Banach lattice $F$ into a Banach space $E$ is DNS, if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We partially answer a question from [8] by showing that this class of operators forms an open subset of $L\left(F,E\right)$ as soon as $F$ is order continuous. Moreover, we show that in this case $T$ is DNS if and only if the norm topology is the minimal topology which is simultaneously stronger than the unbounded norm topology and the topology generated by $T$ as a map (we say that $T$ "complements" the unbounded norm topology in $F$). Since the class of DNS operators plays a similar role in the category of Banach lattices as the upper semi-Fredholm operators play in the category of Banach spaces, we investigate and indeed uncover a similar characterization of the latter class of operators, but this time they have to complement the weak topology.

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