arXiv subjects

Pedro Tradacete

Publications and source records attributed to Pedro Tradacete.

At least 19 recordsLinked to original sources

arXiv2026

Banach lattices and phase retrieval: A case study for the use of AI in mathematics

The ability of large language models to assist professional mathematicians has been progressing rapidly. Earlier this year, a group of researchers in Banach lattice theory and phase retrieval began incorporating this technology into their research workflows. Facing challenges about the reliability of these models, they also decided to couple the discovery process with Lean verification. Here, we present a case study of how this has led to a more united community and a deeper understanding of our field.

math.FA↗

arXiv2026

Measure-valued valuations on star bodies

A complete classification of weak$^*$~continuous, measure-valued valuations is established on star bodies in $\R^n$. Consequences are an integral representation of rotation equivariant, measure-valued valuations and a characterization of dual area measures.

math.MG↗

arXiv2026

Free Products of Banach Lattices

We study free products, that is, coproducts, in the category of Banach lattices and contractive lattice homomorphisms. We give a concrete construction of the free product of an arbitrary family of Banach lattices as a quotient of a free Banach lattice, and prove its basic structural properties. We also establish stability results for sublattice embeddings and projective Banach lattices, and also analyze the behavior of quotient maps. For compact Hausdorff spaces $K_1$ and $K_2$ we identify $C(K_1)\ast C(K_2)$ lattice isomorphically with $C(K_1\ast K_2)$, where $K_1\ast K_2$ denotes the topological join, and we derive an explicit formula for the free product norm in this representation. We further discuss free factors of free Banach lattices, and exploit the existence of non-trivial homological spheres to show that a free Banach lattice can have free factors which are not isomorphic to free Banach lattices.

math.FA↗

arXiv2026

On the free Banach lattice generated by a lattice

We study structural properties of the free Banach lattice $FBL\langle L\rangle$ generated by a distributive lattice $L$. We characterize when $FBL\langle L\rangle$ has a strong unit, compute its density character, analyze the density character of order intervals and study when is $FVL\langle L\rangle$ order dense in $FBL\langle L\rangle$. We also study projection bands, quasi-interior points, and Banach lattice homomorphisms induced by lattice homomorphisms. Finally, we show that $FBL\langle L\rangle$ is lattice isometric to $FBL\langle L^{\mathrm{op}}\rangle$, where $L^{\mathrm{op}}$ denotes the opposite lattice.

math.FA↗

arXiv2026

Free Banach $f$-algebras

We construct and analyze the free Banach $f\!$-algebra $\operatorname{FB{\it f}A}[E]$ generated by a Banach space $E$, extending recent developments on free Banach lattices to the setting of Banach $f\!$-algebras, where multiplication interacts with the lattice structure. Starting from the explicit realization of the free Archimedean $f\!$-algebra as a sublattice-algebra of $\mathbb{R}^{E^*}$, we develop a new structure theorem for normed $f\!$-algebras that allows us to identify the kernel of the maximal submultiplicative lattice seminorm as precisely those functions vanishing on the unit ball $B_{E^*}$. This yields a representation of the free normed $f\!$-algebra inside $C(B_{E^*})$. We prove that this representation extends to an injective map on the completion $\operatorname{FB{\it f}A}[E]$ if and only if $\operatorname{FB{\it f}A}[E]$ is semiprime, and we establish that $\operatorname{FB{\it f}A}[E]$ is indeed semiprime whenever $E$ is finite-dimensional or $E = L_1(μ)$. This is closely related to approximating operators into a Banach $f\!$-algebra by operators into finite-dimensional Banach $f\!$-algebras. We also use the newly constructed free objects to provide an example of a semiprime normed $f\!$-algebra whose norm completion is not semiprime. Using the tools developed for the study of free objects, we show the following extension property: if $A$ is a closed sublattice-algebra of a Banach $f\!$-algebra $B$, then every real-valued lattice-algebra homomorphism on $A$ extends to a real-valued lattice-algebra homomorphism on $B$.

math.FA↗

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.

math.FA↗

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.

math.FA↗

arXiv2026

Non-continuous valuations on convex bodies and a new characterization of volume

This paper investigates the use of automatic continuity techniques in the context of valuations on convex bodies. We first provide an automatic continuity theorem for valuations restricted to parallelotopes with respect to a fixed basis. This result in combination with a counting argument provides a strengthened version of a classical characterization of volume due to Hadwiger. As a byproduct of the proof it is shown that $[0,n-1]\cup\{n\}$ are precisely the possible degrees of homogeneity of bounded translation invariant valuations on $n$-dimensional convex bodies.

math.MG↗

arXiv2026

Banach lattices with upper $p$-estimates: Renorming and factorization

The notions of $p$-convexity and concavity are fundamental tools for studying Banach lattices, as they partition the class of Banach lattices into a scale of spaces with $L_p$-like properties. Upper and lower $p$-estimates provide a refinement of this scale, modeled by the Lorentz spaces $L_{p,\infty}$ and $L_{p,1}$, respectively. In this article, we provide a comprehensive treatment of Banach lattices with upper $p$-estimates. In particular, we show that many well-known theorems about $p$-convex Banach lattices have analogues in the upper $p$-estimate setting, including the ability to represent all such spaces inside of infinity sums of model spaces, to canonically factor the convex operators and identify their associated operator ideals, as well as to give a precise description of the free objects and push-outs. Proving these results is far from straightforward and will require the development of a variety of new tools that avoid convexification and concavification procedures. In fact, we will identify many fundamental differences between the theories of $p$-convexity and upper $p$-estimates, particularly with regards to isometric problems and renormings.

math.FA↗

arXiv2025

Free quasi-Banach lattices

We study different versions of \emph{free objects} in the setting of quasi-Banach spaces and quasi-Banach lattices. Special attention is devoted to the free $p$-convex $p$-Banach lattice $\operatorname{FpBL}^{(p)}[E]$ generated by a $p$-natural quasi-Banach space $E$, for which we provide a functional representation by means of operators into $L_p[0,1]$. This representation yields, among other consequences: (1) Operators from a Banach space $E$ to any $p$-convex $(0<p<1)$ quasi-Banach lattice $X$ can be extended to lattice homomorphisms $\operatorname{FBL}[E] \to X$ with control of the norm. (2) The space $\ell_p(Γ)$ $(0<p<1)$ is a projective $p$-Banach lattice precisely when $Γ$ is countable. (3) The free vector lattice generated by $E$ sits inside $\operatorname{FpBL}^{(p)}[E]$ as a dense sublattice.

math.FA↗

arXiv2025

Minimizing measures for the doubling condition

We study those measures whose doubling constant is the least possible among doubling measures on a given metric space. It is shown that such measures exist on every metric space supporting at least one doubling measure. In addition, a connection between minimizers for the doubling constant and superharmonic functions is exhibited. This allows us to show that for the particular case of the euclidean space $\mathbb R^d$, Lebesgue measure is the only minimizer for the doubling constant (up to constant multiples) precisely when $d=1$ or $d=2$, while for $d\geq3$ there are infinitely many independent minimizers. Analogously, in the discrete setting, we can show uniqueness of the counting measure as a minimizer for regular graphs where the standard random walk is a recurrent Markov chain. The counting measure is also shown to be a minimizer in every infinite graph where the cardinality of balls depends solely on their radii.

math.CA↗

arXiv2025

Complemented subspaces of Banach lattices

We survey recent developments on the structure of complemented subspaces of Banach lattices, including in particular the construction of a complemented subspace of a $C(K)$-space which is not linearly isomorphic to any Banach lattice. Motivated by this, several natural questions and directions of future research are presented. We provide an approach to some of these problems using tools from the theory of free Banach lattices.

math.FA↗

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.

math.FA↗

arXiv2024

Banach lattice AM-algebras

An analogue of Kakutani's representation theorem for Banach lattice algebras is provided. We characterize Banach lattice algebras that embed as a closed sublattice-algebra of $C(K)$ precisely as those with a positive approximate identity $(e_γ)$ such that $x^{*}(e_γ)\to \|x^{*}\|$ for every positive functional $x^{*}$. We also show that every Banach lattice algebra with identity other than $C(K)$ admits different product operations which are compatible with the order and the algebraic identity. This complements the classical result, due to Martignon, that on $C(K)$ spaces pointwise multiplication is the unique compatible product.

math.FA↗

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 $\ell_2$.

math.FA↗

arXiv2024

Banach lattices of positively homogeneous functions induced by a Banach space

Motivated by the construction of the free Banach lattice generated by a Banach space, we introduce and study several vector and Banach lattices of positively homogeneous functions defined on the dual of a Banach space $E$. The relations between these lattices allow us to give multiple characterizations of when the underlying Banach space $E$ is finite-dimensional and when it is reflexive. Furthermore, we show that lattice homomorphisms between free Banach lattices are always composition operators, and study how these operators behave on the scale of lattices of positively homogeneous functions.

math.FA↗

arXiv2024

The strong Nakano property in Banach lattices

We study the strong Nakano property in Banach lattices with a special focus on free Banach lattices. We show that for every finite dimensional Banach space $E$, the free Banach lattice $FBL[E]$ has the strong Nakano property with a constant independent of the dimension. It is also shown that if $FBL[E]$ has the strong Nakano property, then $E$ cannot contain subspaces isomorphic to neither $c_0$ nor $L_1$.

math.FA↗

arXiv2023

Band projections in spaces of regular operators

We introduce inner band projections in the space of regular operators on a Dedekind complete Banach lattice and study some structural properties of this class. In particular, we provide a new characterization of atomic order continuous Banach lattices as those for which all band projections in the corresponding space of regular operators are inner. We also characterize the multiplication operators $L_AR_B$ which are band projections precisely as those with $A,B$ being band projections up to a scalar multiple.

math.FA↗