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Antonio Avilés

Publications and source records attributed to Antonio Avilés.

At least 19 recordsLinked to original sources
arXiv2026

Tight $σ$-filtrations and the BFPP of $C(ω^*)$

Assuming that $\mathcal{P}(ω)$ is tightly $σ$-filtered, we prove that $C(ω^*)$ fails the ball fixed point property. This axiom holds under the continuum hypothesis and in the $\aleph_2$-Cohen model. The proof shows that the Stone space of the countable--cocountable algebra $\operatorname{Cc}(ω_1)$ is a continuous retract of $ω^*$ and uses the equivalence between tight $σ$-filteredness of $\operatorname{Cc}(ω_1)$ and of $\mathcal{P}(ω)$. We also prove that the quotient of a tightly $σ$-filtered Boolean algebra $B$ with the countable separation property by an $\aleph_1$-generated $P$-ideal is a Boolean retract of $B$.

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arXiv2026

On almost disjoint families and Johnson-Lindenstrauss spaces

Every almost disjoint family $A$ of infinite subsets of $ω$ gives rise to a scattered compact space $K_A$ and a family of Johnson-Lindenstrauss spaces JL$_p(A)$. We consider almost disjoint families arising from finitely branching trees and investigate whether they define homeomorphic compacta and isomorphic Banach spaces.

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arXiv2026

Homeomorphism between close relatives of Hilbertian balls

We present a solution to some problems posed by the author and Kalenda. We show that the closed ball of nonseparable Hilbert space in its weak topology is homeomorphic to its positive part, as well as to its product with the Hilbert cube. In the separable setting we obtain that there is a weak homeomorphism of the closed unit ball of $\ell_2$ onto its positive part that preserves the norm, and via a result of Dijkstra and van Mill, the same is true for the ball of $\ell_\infty=\ell_1^*$ in the weak$^*$ topology. All spaces $B(κ,a,b)$ considered by the author and Kalenda are shown to be homeomorphic. The solution has been found by AI (Chatgpt 5.5), the role of the author has been to ask the right questions, check and understand the answers, and adapt the writing to his personal human taste.

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arXiv2026

Transfinite Daugavet property

We extend the Daugavet property and a perfect version of it to transfinite cardinals in order to distinguish between spaces with the ordinary Daugavet property by some kind of complexity (topological, density\ldots), providing a number of examples and results. First, we characterise the transfinite Daugavet $C(K)$ spaces in terms of a cardinal index $\mathfrak r(K)$, which generalises the notion of the reaping number of a Boolean algebra. Besides, the perfect Daugavet property characterizes the absence of $G_δ$-points in $K$. We also study several inheritance results of the transfinite Daugavet properties by almost isometric ideals, absolute sums, and tensor product spaces, with a number of applications. We classify these properties for $L_1(μ)$ and $L_\infty(μ)$ spaces in terms of the Maharam's decomposition of the measure. We also show that the space of Lipschitz functions $\Lip(M)$ on a complete length metric space has the $ω$-perfect Daugavet property, improving the previous knowledge.

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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

Coordinate systems in Banach spaces and lattices

Using methods of descriptive set theory, in particular, the determinacy of infinite games of perfect information, we answer several questions from the literature regarding different notions of bases in Banach spaces and lattices. For the case of Banach lattices, our results follow from a general theorem stating that (under the assumption of analytic determinacy), every $σ$-order basis $(e_n)$ for a Banach lattice $X=[e_n]$ is a uniform basis, and every uniform basis is Schauder. Moreover, the notions of order and $σ$-order bases coincide when $X=[e_n].$ Regarding Banach spaces, we address two problems concerning filter Schauder bases for Banach spaces, i.e., in which the norm convergence of partial sums is replaced by norm convergence along some appropriate filter on $\mathbb N$. We first provide an example of a Banach space admitting such a filter Schauder basis, but no ordinary Schauder basis. Secondly, we show that every filter Schauder basis with respect to an analytic filter is also a filter Schauder basis with respect to a Borel filter.

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arXiv2026

A Classification of Order Convergence via a Transfinite Fatou Hierarchy

We investigate the descriptive complexity of order convergence in separable Banach lattices. While uniform convergence is Borel and $σ$-order convergence is known to be ${\bf Δ}^1_2$, it is unclear in general when $σ$-order convergence is analytic. We introduce a transfinite hierarchy of weakenings of the classical Fatou property, indexed by countable ordinals, and show that it provides a complete structural classification of this definability problem. For a separable Banach lattice $X$, we prove that the following are equivalent: (i) the set of decreasing positive sequences with infimum zero is Borel; (ii) $σ$-order convergence is analytic; and (iii) $X$ satisfies the $α$-Fatou property for some countable ordinal $α$. We further establish that the hierarchy is proper: for every countable ordinal $α$ there exists a separable Banach lattice with a countable $π$-basis that fails to be $α$-Fatou, but is $β$-Fatou for some $β>α$. Thus the Borel definability of order convergence is governed by a canonical ordinal invariant intrinsic to the lattice, and the descriptive complexity can be arbitrarily high below $ω_1$. These results identify projective complexity as a genuine structural invariant in Banach lattice theory.

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arXiv2025

The ball fixed point property in spaces of continuous functions

A Banach space $X$ has the ball fixed point property (BFPP) if for every closed ball $B$ and for every nonexpansive mapping $T\colon B\to B$, there is a fixed point. We study the BFPP for $C(K)$-spaces. Our goal is to determine topological properties over $K$ that may determine the failure or fulfillment of the BFPP for the space of continuous functions $C(K)$. We prove that the class of compact spaces $K$ for which the BFPP holds lies between the class of extremally disconnected compact spaces and the class of compact $F$-spaces. We give a family of examples of $F$-spaces $K$ for which the BFPP fails. As a result, we prove that for every cardinal $κ$, $κ$-order completeness or $κ$-hyperconvexity of $C(K)$ are not enough for the BFPP and we obtain that $\ell_\infty/c_0 = C(\mathbb{N}^*)$ fails BFPP under the Continuum Hypothesis. The space $C([0,+\infty)^*)$ is also analyzed. It is left as an open problem whether all compact spaces for which the BFPP holds are in fact the extremally disconnected compact sets.

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arXiv2025

Basis problem for analytic multiple gaps

A k-gap is a finite k-sequence of pairwise disjoint monotone families of infinite subsets of N mixed in such a way that we cannot find a partition of N such that each family is trival on one piece of the partition. We prove that, relative to the comparison given by restriction to infinite subsets of N, for every positive integer k there is a finite basis for the class of all analytic k-gaps . We also build the fine structure theory of analytic k-gaps and give some applications. The content of Chapter 1 of this manuscript have been published as: A. Avilés, S. Todorcevic, Finite basis for analytic multiple gaps, Publ. Math. IHES. 121 (2015), 57-79. The content of Chapter 2 (except some technical results from 2.5 and 2.6) and Section 3.1, largely revised and improved, has ben published as: A. Avilés, S. Todorcevic, Types in the n-adic tree and minimal analytic gaps, Adv. Math. 292 (2016), 558-600. The content of Sections 3.4, 4.1 and 4.3 have been published as: A. Avilés, S. Todorcevic, Isolating subgaps of a multiple gap, Monatsh. Math. 186 (2018), 373--392. The rest of contents may appear elsewhere.

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arXiv2025

On the class of NY compact spaces of finitely supported elements and related classes

We prove that a compact space $K$ embeds into a $σ$-product of compact metrizable spaces ($σ$-product of intervals) if and only if $K$ is (strongly countable-dimensional) hereditarily metalindelöf and every subspace of $K$ has a nonempty relative open second-countable subset. This provides novel characterizations of $ω$-Corson and $NY$ compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the $σ$-product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space $K$ the property of being $NY$ compact is determined by the topological structure of the space $C_p(K)$ of continuous real-valued functions of $K$ equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.

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arXiv2025

A Banach space with $L$-orthogonal sequences but without $L$-orthogonal elements

We prove that the existence of Banach spaces with $L$-orthogonal sequences but without $L$-orthogonal elements is independent of the standard foundation of Mathematics, ZFC. This provides a definitive answer to \cite[Question~1.1]{AvilesMartinezRueda}. Generalizing classical $Q$-point ultrafilters, we introduce the notion of $Q$-measures and provide several results generalizing former theorems by Miller \cite{Miller} and Bartoszynski \cite{Bartoszynski} for $Q$-point ultrafilters.

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arXiv2024

Local complementation in Banach spaces and its preservation under free constructions

In this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation of Banach spaces. As an application, we show that local complementation is naturally preserved under certain free constructions in Functional Analysis, including Lipschitz-free spaces and free Banach lattices.

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arXiv2024

Barely alternating real almost chains and extension operators for compact lines

Assume $\text{MA}(κ)$. We show that for every real chain of size $κ$ in the quotient Boolean algebra $P(ω)/fin$ we can find an almost chain of representatives such that every $n\inω$ oscillates at most three times along the almost chain. This is used to show that for every countable discrete extension of a separable compact line $K$ of weight $κ$ there exists an extension operator $E:C(K)\longrightarrow C(L)$ of norm at most three.

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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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arXiv2023

Some examples concerning $LΣ(\leqω)$ and metrizably fibered compacta

The class of $LΣ(\leqω)$-spaces was introduced in 2006 by Kubiś, Okunev and Szeptycki as a natural refinement of the classical and important notion of Lindelöf $Σ$-spaces. Compact $LΣ(\leqω)$-spaces were considered earlier, under different names, in the works of Tkachuk and Tkachenko in relation to metrizably fibered compacta. In this paper we give counterexamples to several open questions about compact $LΣ(\leqω)$-spaces that are scattered in the literature. Among other things, we refute a conjecture of Kubiś, Okunev and Szeptycki by constructing a separable Rosenthal compactum which is not an $LΣ(\leqω)$-space. We also give insight to the structure of first-countable $(K)LΣ(\leqω)$-compacta.

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arXiv2023

Compact spaces associated to separable Banach lattices

We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what $C(K)$ spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum $K$ admits a strictly positive regular Borel measure of countable type that is analytic, and in the nonmetrizable case these compacta are saturated with copies of $β\mathbb N$. Some natural questions about this class are left open.

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arXiv2022

Transfinite almost square Banach spaces

It is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate them to the containment of isomorphic copies of $c_0(κ)$, where $κ$ is some uncountable cardinal. We also provide several examples and stability results of the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter two property and octahedral norms, we obtain a solution to an open question from [8].

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