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Gonzalo Martínez-Cervantes

Publications and source records attributed to Gonzalo Martínez-Cervantes.

At least 19 recordsLinked to original sources
arXiv2026

A positive version of Miljutin's Theorem

In this paper we provide a positive version of Miljutin's theorem. Namely, we show that for any uncountable compact metric spaces $K$ and $L$ there exists a positive isomorphism $T:C(K)\longrightarrow C(L)$. Moreover, the inverse of $T$ is a regular operator, and $T$ may be chosen so that $\|T\|_\mathrm{r} \cdot \|T^{-1}\|_\mathrm{r} \leq 9+6\sqrt3$. The method also gives positive versions of classical nonmetrizable results for Miljutin and Dugundji compacta. In particular, if $G$ and $H$ are infinite compact groups of the same weight, then $C(G)$ and $C(H)$ are positively and regularly isomorphic with the same uniform quantitative estimate.

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↗
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.

math.LO↗
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.

math.FA↗
arXiv2023

On the property (C) of Corson and other sequential properties of Banach Spaces

A well-known result of R. Pol states that a Banach space $X$ has property ($\mathcal{C}$) of Corson if and only if every point in the weak*-closure of any convex set $C \subseteq B_{X^*}$ is actually in the weak*-closure of a countable subset of $C$. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of $B_{X^*}$ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under $\mathrm{MA}+\neg \mathrm{CH}$ for the class of $\mathcal{C}(K)$-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom ($\mathrm{PFA}$) the following conditions are equivalent for an arbitrary Banach space $X$: 1) $X$ has property $\mathcal{E}'$; 2) $X$ has weak*-sequential dual ball; 3) $X$ has property ($\mathcal{C}$) of Corson; 4) $(B_{X^*},w^\ast)$ has countable tightness. This provides a partial extension of a former result of Arhangel'skii. In addition, we show that every Banach space with property $\mathcal{E}'$ has weak*-convex block compact dual ball.

math.FA↗
arXiv2023

Rank-one perturbations and norm-attaining operators

The main goal of this article is to show that for every (reflexive) infinite-dimensional Banach space $X$ there exists a reflexive Banach space $Y$ and $T, R \in \mathcal{L}(X,Y)$ such that $R$ is a rank-one operator, $\|T+R\|>\|T\|$ but $T+R$ does not attain its norm. This answers a question posed by S. Dantas and the first two authors. Furthermore, motivated by the parallelism exhibited in the literature between the $V$-property introduced by V.A. Khatskevich, M.I. Ostrovskii and V.S. Shulman and the weak maximizing property introduced by R.M. Aron, D. García, D. Pellegrino and E.V. Teixeira, we also study the relationship between these two properties and norm-attaining perturbations of operators.

math.FA↗
arXiv2022

Extremal structure of projective tensor products

We prove that, given two Banach spaces $X$ and $Y$ and bounded, closed convex sets $C\subseteq X$ and $D\subseteq Y$, if a nonzero element $z\in \overline{\mathrm{co}}(C\otimes D)\subseteq X\widehat{\otimes}_πY$ is a preserved extreme point then $z=x_0\otimes y_0$ for some preserved extreme points $x_0\in C$ and $y_0\in D$, whenever $K(X,Y^*)$ separates points of $X \widehat{\otimes}_πY$ (in particular, whenever $X$ or $Y$ has the compact approximation property). Moreover, we prove that if $x_0\in C$ and $y_0\in D$ are weak-strongly exposed points then $x_0\otimes y_0$ is weak-strongly exposed in $\overline{\mathrm{co}}(C\otimes D)$ whenever $x_0\otimes y_0$ has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space $X$ isomorphic to $\ell_2$ with a weak-strongly exposed point $x_0\in B_X$ such that $x_0\otimes x_0$ is not a weak-strongly exposed point of the unit ball of $X\widehat{\otimes}_πX$.

math.FA↗
arXiv2022

Topological properties in tensor products of Banach spaces

Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor product $X\widehat{\otimes}_πY$ has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) $X\widehat{\otimes}_πY$ is WLD if and only if both $X$ and $Y$ are WLD and all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range; (ii) $X\widehat{\otimes}_πY$ is subspace of a WCG space if the same holds for both $X$ and $Y$ under the assumption that every operator from $X$ to $Y^*$ is compact; (iii) $\ell_p(Γ)\widehat{\otimes}_π\ell_q(Γ)$ is subspace of a Hilbert generated space for any $1< p,q<\infty$ such that $1/p+1/q<1$ and for any infinite set $Γ$. We also pay attention to the injective tensor product $X\widehat{\otimes}_\varepsilon Y$. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if $K$ is a compact space such that $C(K\times K)$ has property (C), then all regular Borel probability measures on $K$ have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the $w^*$-angelicity of the dual unit ball under injective tensor products.

math.FA↗
arXiv2021

Banach spaces containing $c_0$ and elements in the fourth dual

A recent result of T.~Abrahamsen, P.~Hájek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof passes through a sequential version of being almost square which we call being \textit{sequentially almost square}. In this article we study these conditions in the nonseparable setting. On one hand, we show that a Banach space $X$ contains a copy of $c_0$ if and only if there exists an equivalent renorming $| \cdot |$ on $X$ for which there exists $h\in S_{X^{****}}$ such that $|x+h|=\max\{|x|,1\}$ for every $x\in X$. On the other hand, although it is unclear whether the aforementioned result of T.~Abrahamsen et al. holds in the nonseparable setting, we show that, under the existence of selective ultrafilters, if $X$ is a sequentially almost square Banach space then there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$.

math.FA↗
arXiv2021

Linear versus lattice embeddings between Banach lattices

A well-known classical result states that $c_0$ is linearly embeddable in a Banach lattice if and only if it is lattice embeddable. Improving results of H.P.~Lotz, H.P.~Rosenthal and N.~Ghoussoub, we prove that $C[0,1]$ shares this property with $c_0$. Furthermore, we show that any infinite-dimensional sublattice of $C[0,1]$ is either lattice isomorphic to $c_0$ or contains a sublattice isomorphic to $C[0,1]$. As a consequence, it is proved that for a separable Banach lattice $X$ the following conditions are equivalent: (1) $X$ is linearly embeddable in a Banach lattice if and only if it is lattice embeddable; (2) $X$ is lattice embeddable into $C[0,1]$.

math.FA↗
arXiv2021

Lattice embeddings in free Banach lattices over lattices

In this article we deal with the free Banach lattice generated by a lattice and its behavior with respect to subspaces. In general, any lattice embedding $i\colon \mathbb{L} \longrightarrow \mathbb{M}$ between two lattices $\mathbb{L} \subseteq \mathbb{M}$ induces a Banach lattice homomorphism $\hat \imath\colon FBL \langle \mathbb{L} \rangle \longrightarrow FBL \langle \mathbb{M}\rangle$ between the corresponding free Banach lattices. We show that this mapping $\hat \imath$ might not be an isometric embedding neither an isomorphic embedding. In order to provide sufficient conditions for $\hat \imath$ to be an isometric embedding we define the notion of locally complemented lattices and prove that, if $\mathbb L$ is locally complemented in $\mathbb M$, then $\hat \imath$ yields an isometric lattice embedding from $FBL\langle\mathbb L\rangle$ into $FBL\langle\mathbb M\rangle$. We provide a wide number of examples of locally complemented sublattices and, as an application, we obtain that every free Banach lattice generated by a lattice is lattice isomorphic to an AM-space or, equivalently, to a sublattice of a $C(K)$-space. Furthermore, we prove that $\hat \imath$ is an isomorphic embedding if and only if it is injective, which in turn is equivalent to the fact that every lattice homomorphism $x^*\colon \mathbb{L} \longrightarrow [-1,1]$ extends to a lattice homomorphism $\hat x^*\colon \mathbb{M} \longrightarrow [-1,1]$. Using this characterization we provide an example of lattices $\mathbb{L} \subseteq \mathbb{M}$ for which $\hat \imath$ is an isomorphic not isometric embedding.

math.FA↗
arXiv2021

A renorming characterization of Banach spaces containing $\ell_1(κ)$

A result of G. Godefroy asserts that a Banach space $X$ contains an isomorphic copy of $\ell_1$ if and only if there is an equivalent norm $|||\cdot|||$ such that, for every finite-dimensional subspace $Y$ of $X$ and every $\varepsilon>0$, there exists $x\in S_X$ so that $|||y+r x|||\geq (1-\varepsilon)(|||y|||+\vert r\vert)$ for every $y\in Y$ and every $r\in\mathbb R$. In this paper we generalize this result to larger cardinals, showing that if $κ$ is an uncountable cardinal then a Banach space $X$ contains a copy of $\ell_1(κ)$ if and only if there is an equivalent norm $|||\cdot|||$ on $X$ such that for every subspace $Y$ of $X$ with $dens(Y)<κ$ there exists a norm-one vector $x$ so that $||| y+r x|||=|||y|||+\vert r\vert$ whenever $y\in Y$ and $r\in\mathbb{R}$. This result answers a question posed by S. Ciaci, J. Langemets and A. Lissitsin, where the authors wonder whether the previous statement holds for infinite succesor cardinals. We also show that, in the countable case, the result of Godefroy cannot be improved to take $\varepsilon=0$.

math.FA↗
arXiv2021

Some remarks on the weak maximizing property

A pair of Banach spaces $(E, F)$ is said to have the weak maximizing property (WMP, for short) if for every bounded linear operator $T$ from $E$ into $F$, the existence of a non-weakly null maximizing sequence for $T$ implies that $T$ attains its norm. This property was recently introduced in an article by R. Aron, D. García, D. Pelegrino and E. Teixeira, raising several open questions. The aim of the present paper is to contribute to the better knowledge of the WMP and its limitations. Namely, we provide sufficient conditions for a pair of Banach spaces to fail the WMP and study the behaviour of this property with respect to quotients, subspaces, and direct sums, which open the gate to present several consequences. For instance, we deal with pairs of the form $(L_p[0,1], L_q[0,1])$, proving that these pairs fail the WMP whenever $p>2$ or $q<2$. We also show that, under certain conditions on $E$, the assumption that $(E, F)$ has the WMP for every Banach space $F$ implies that $E$ must be finite dimensional. On the other hand, we show that $(E, F)$ has the WMP for every reflexive space $E$ if and only if $F$ has the Schur property. We also give a complete characterization for the pairs $(\ell_s \oplus_p \ell_p, \ell_s \oplus_q \ell_q)$ to have the WMP by calculating the moduli of asymptotic uniform convexity of $\ell_s \oplus_p \ell_p$ and of asymptotic uniform smoothness of $\ell_s \oplus_q \ell_q$ when $1 < p \leq s \leq q < \infty$. We conclude the paper by discussing some variants of the WMP and presenting a list of open problems on the topic of the paper.

math.FA↗
arXiv2021

Abundance of independent sequences in compact spaces and Boolean algebras

It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal $\mathcal{C}^\perp$ as the class of all compact spaces for which every continuous image in $\mathcal{C}$ is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and $ccc$ compacta), and also classes where independence is scarce (most notably weakly Radon-Nikodým compacta) with classes where it is abundant. We study these problems for zero-dimensional compact spaces with the aid of Boolean algebras and show the main difficulties arising when passing to the general setting. Our main results are the constructions of several relevant examples.

math.FA↗
arXiv2021

$L$-orthogonal elements and $L$-orthogonal sequences

Given a Banach space $X$, we say that a sequence $\{x_n\}$ in the unit ball of $X$ is $L$-orthogonal if $\Vert x+x_n\Vert\rightarrow 1+\Vert x\Vert$ for every $x\in X$. On the other hand, an element $x^{**}$ in the bidual sphere is said to be $L$-orthogonal (to $X$) if $\|x+x^{**}\|= 1+\Vert x\Vert$ for every $x\in X$. A result of V. Kadets, V. Shepelska and D. Werner asserts that a Banach space contains an isomorphic copy of $\ell_1$ if and only if there exists an equivalent renorming with an $L$-orthogonal sequence, whereas a result of G. Godefroy claims that containing an isomorphic copy of $\ell_1$ is equivalent to the existence of an equivalent renorming with $L$-orthogonals in the bidual. The aim of this paper is to clarify the relation between $L$-orthogonal sequences and $L$-orthogonal elements. Namely, we study whether every $L$-orthogonal sequence contains $L$-orthogonal elements in its weak*-closure. We provide an affirmative answer whenever the ambient space has small density character. Nevertheless, we show that, surprisingly, the general answer is independent of the usual axioms of set theory. We also prove that, even though the set of $L$-orthogonals is not a vector space, this set contains infinite-dimensional Banach spaces when the surrounding space is separable.

math.FA↗
arXiv2021

Free Banach lattices generated by a lattice and projectivity

In this article we deal with the free Banach lattice $FBL\langle \mathbb{L} \rangle$ generated by a lattice $\mathbb{L}$. We prove that if $FBL\langle \mathbb{L} \rangle$ is projective then $\mathbb{L}$ has a maximum and a minimum. On the other hand, we show that if $\mathbb{L}$ has maximum and minimum then $FBL\langle \mathbb{L} \rangle$ is $2$-lattice isomorphic to a $C(K)$-space. As a consequence, $FBL\langle \mathbb{L} \rangle$ is projective if and only if it is lattice isomorphic to a $C(K)$-space with $K$ being an absolute neighborhood retract. As an application, we characterize those linearly ordered sets and Boolean algebras for which the corresponding free Banach lattice is projective.

math.FA↗
arXiv2021

On the existence of non-norm-attaining operators

In this paper we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal{L}(E, F)$. By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set $K$ of $\mathcal{L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm closed convex hull, then we can guarantee the existence of an operator which does not attain its norm. This allows us to provide the following generalization of results due to Holub and Mujica. If $E$ is a reflexive space, $F$ is an arbitrary Banach space, and the pair $(E, F)$ has the bounded compact approximation property, then the following are equivalent: (i) $\mathcal{K}(E, F) = \mathcal{L}(E, F)$; (ii) Every operator from $E$ into $F$ attains its norm; (iii) $(\mathcal{L}(E,F), τ_c)^* = (\mathcal{L}(E, F), \| \cdot \|)^*$; where $τ_c$ denotes the topology of compact convergence. We conclude the paper presenting a characterization of the Schur property in terms of norm-attaining operators.

math.FA↗