arXiv subjects

Abraham Rueda Zoca

Publications and source records attributed to Abraham Rueda Zoca.

At least 55 records · Page 3Linked to original sources

arXiv2022

Residuality in the set of norm attaining operators between Banach spaces

We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if $C$ is a bounded subset of a Banach space $X$ which admit an LUR renorming satisfying that, for every Banach space $Y$, the operators $T$ from $X$ to $Y$ for which the supremum of $\|Tx\|$ with $x\in C$ is attained are dense, then the $G_δ$ set of those functionals which strongly exposes $C$ is dense in $X^*$. This extends previous results by J.\ Bourgain and K.-S.\ Lau. The particular case in which $C$ is the unit ball of $X$, in which we get that the norm of $X^*$ is Fréchet differentiable at a dense subset, improves a result by J.\ Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the $G_δ$ set of absolutely strongly exposing operators from $X$ to $Y$ by requiring that the set of strongly exposing functionals on $X$ is dense and conditions on $Y$ or $Y^*$ involving RNP and discreteness on the set of strongly exposed points of $Y$ or $Y^*$. These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications, among which we point out that we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A.

math.FA↗

arXiv2022

The Daugavet property in spaces of vector-valued Lipschitz functions

We prove that if a metric space $M$ has the finite CEP then $\mathcal F(M)\widehat{\otimes}_π X$ has the Daugavet property for every non-zero Banach space $X$. This applies, for instance, if $M$ is a Banach space whose dual is isometrically an $L_1(μ)$ space. If $M$ has the CEP then $L(\mathcal F(M),X)=\Lip(M,X)$ has the Daugavet property for every non-zero Banach space $X$, showing that this is the case when $M$ is an injective Banach space or a convex subset of a Hilbert space.

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 characterization of a local vector valued Bollobás theorem

In this paper, we are interested in giving two characterizations for the so-called property {\bf L}$_{o,o}$, a local vector valued Bollobás type theorem. We say that $(X, Y)$ has this property whenever given $\eps > 0$ and an operador $T: X \rightarrow Y$, there is $η= η(\eps, T)$ such that if $x$ satisfies $\|T(x)\| > 1 - η$, then there exists $x_0 \in S_X$ such that $x_0 \approx x$ and $T$ itself attains its norm at $x_0$. This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair $(X, Y)$ has the {\bf L}$_{o,o}$ for compact operators if and only if so does $(X, \K)$ for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $(X \pten Y, \K)$ satisfies the {\bf L}$_{o,o}$ for linear functionals under strict convexity or Kadec-Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $(L_p(μ) \times L_q(ν); \K)$ cannot satisfy the {\bf L}$_{o,o}$ for bilinear forms.

math.FA↗

arXiv2021

Norm-attaining tensors and nuclear operators

Given two Banach spaces $X$ and $Y$, we introduce and study a concept of norm-attainment in the space of nuclear operators $\mathcal{N}(X,Y)$ and in the projective tensor product space $X \widehat{\otimes}_πY$. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in $\mathcal{N}(X,Y)$ and in $X\widehat{\otimes}_πY$ is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces $X$ and $Y$ which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces $X$ and $Y$ failing the approximation property in such a way that the class of elements in $X\widehat{\otimes}_πY$ which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.

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

A characterisation of $L_1$-preduals in terms of extending Lipschitz maps

We characterise the Banach spaces $X$ which are $L_1$-predual as those for which every Lipschitz compact mapping $f:N\longrightarrow X$ admits, for every $\varepsilon>0$ and every $M$ containing $N$, a Lipschitz (compact) extension $F:M\longrightarrow X$ so that $\Vert F\Vert\leq (1+\varepsilon)\Vert f\Vert$. Some consequences are derived about $L_1$-preduals and about Lipschitz-free spaces.

math.FA↗

arXiv2021

On norm-attainment in (symmetric) tensor products

In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product $\widehat{\otimes}_{π,s,N} X$ of a Banach space $X$, which turns out to be naturally related to the classical norm-attainment of $N$-homogeneous polynomials on $X$. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in $\widehat{\otimes}_{π,s,N} X$ is dense. We show that the set of all norm-attaining symmetric tensors is dense in $\widehat{\otimes}_{π,s,N} X$ for a large set of Banach spaces as $L_p$-spaces, isometric $L_1$-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if $X^*$ satisfies the Radon-Nikodým and the approximation property, then the set of all norm-attaining symmetric tensors in $\widehat{\otimes}_{π,s,N} X^*$ is dense. From these techniques, we can present new examples of Banach spaces $X$ and $Y$ such that the set of all norm-attaining tensors in the projective tensor product $X \widehat{\otimes}_πY$ is dense, answering positively an open question from the paper by S. Dantas, M. Jung, Ó. Roldán and A. Rueda Zoca.

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

Daugavet points in projective tensor products

In this paper, we are interested in studying when an element $z$ in the projective tensor product $X \widehat{\otimes}_πY$ turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of $X \widehat{\otimes}_πY$ having the Daugavet property implies the existence of a great amount of isometries from $Y$ into $X^*$. Having this in mind, we provide methods for constructing non-trivial Daugavet points in $X \widehat{\otimes}_πY$. We show that $C(K)$-spaces are examples of Banach spaces such that the set of the Daugavet points in $C(K) \widehat{\otimes}_πY$ is weakly dense when $Y$ is a subspace of $C(K)^*$. Finally, we present some natural results on when an elementary tensor $x \otimes y$ is a Daugavet point.

math.FA↗

arXiv2021

Daugavet points and $Δ$-points in Lipschitz-free spaces

We study Daugavet points and $Δ$-points in Lipschitz-free Banach spaces. We prove that, if $M$ is a compact metric space, then $μ\in S_{\mathcal F(M)}$ is a Daugavet point if, and only if, there is no denting point of $B_{\mathcal F(M)}$ at distance strictly smaller than two from $μ$. Moreover, we prove that if $x$ and $y$ are connectable by rectifiable curves of lenght as close to $d(x,y)$ as we wish, then the molecule $m_{x,y}$ is a $Δ$-point. Some conditions on $M$ which guarantee that the previous implication reverses are also obtained. As a consequence of our work, we show that Lipschitz-free spaces are natural examples of Banach spaces where we can guarantee the existence of $Δ$-points which are not Daugavet points.

math.FA↗

arXiv2020

$L$-orthogonality in Daugavet centers and narrow operators

We study the presence of $L$-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that, if $\dens(Y)\leq ω_1$ and $G:X\longrightarrow Y$ is a Daugavet center, then $G(W)$ contains some $L$-orthogonal for every non-empty $w^*$-open subset of $B_{X^{**}}$. In the context of narrow operators, we show that if $X$ is separable and $T:X\longrightarrow Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^*$-open subset $W$ of $B_{X^{**}}$ then $W$ contains some $L$-orthogonal $u$ so that $T^{**}(u)=T(y)$. In the particular case that $T^*(Y^*)$ is separable, we extend the previous result to $\dens(X)=ω_1$. Finally, we prove that none of the previous results holds in larger density characters (in particular, a counterexample is shown for $ω_2$ under continuum hypothesis).

math.FA↗

arXiv2020

Daugavet property in projective symmetric tensor products of Banach spaces

We show that all the symmetric projective tensor products of a Banach space $X$ have the Daugavet property provided $X$ has the Daugavet property and either $X$ is an $L_1$-predual (i.e.\ $X^*$ is isometric to an $L_1$-space) or $X$ is a vector-valued $L_1$-space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise "localised" versions of the Daugavet property (i.e.\ Daugavet points and $Δ$-points recently introduced) for $L_1$-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real $L_1$-preduals in terms of the absence of $Δ$-points and also to provide new examples of $L_1$-preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for $L_1$-preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results about the Daugavet property for projective tensor products is also got.

math.FA↗

arXiv2020

$L$-orthogonality, octahedrality and Daugavet property in Banach spaces

In contrast with the separable case, we prove that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the existence of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$, which shows that the answer to an environment question in [9] is negative. Furthermore, we prove that the abundance of almost $L$-orthogonal vectors in a Banach space $X$ (almost Daugavet property) whose density character is $ω_1$ implies the abundance of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$. In fact, we get that a Banach space $X$ whose density character is $ω_1$ verifies the Daugavet property if, and only if, the set of vectors in $X^{**}$ being $L$-orthogonal to $X$ is weak-star dense in $X^{**}$. We also prove that, under CH, the previous characterisation is false for Banach spaces with larger density character. Finally, some consequences on Daugavet property in the setting of $L$-embedded spaces are obtained.

math.FA↗