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Abraham Rueda Zoca

Publications and source records attributed to Abraham Rueda Zoca.

At least 37 records · Page 2Linked to original sources
arXiv2024

Stability results of octahedrality in tensor product spaces

We prove that there exists a finite-dimensional Banach space $X$ such that $L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter two property and $L_\infty^\mathbb C([0,1])\widehat{\otimes}_πX^*$ fails to have octahedral norm. This proves that the octahedrality of the norm (respectively the strong diameter two property) is not automatically inherited from one factor by taking projective tensor product (respectively injective tensor product), which answers [16,Question 4.4].

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arXiv2024

Computing Borel complexity of some geometrical properties in Banach spaces

We compute the Borel complexity of some classes of Banach spaces such as different versions of diameter two properties, spaces satisfying the Daugavet equation or spaces with an octahedral norm. In most of the above cases our computation is even optimal, which completes the research done during the last years around this topic for isomorphism classes of Banach spaces.

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arXiv2024

Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $Δ$-points

We prove that there exists an equivalent norm $\Vert\vert\cdot\vert\Vert$ on $L_\infty[0,1]$ with the following properties: (1) The unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is weakly dense; (3) The set of ccw $Δ$-points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is norming. We also show that there are points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ which are not $Δ$-points, meaning that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ fails the diametral local diameter 2 property. Finally, we observe that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.

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arXiv2024

Slicely countably determined points in Banach spaces

We introduce slicely countably determined points (SCD points) of a bounded and convex subset of a Banach space which extends the notions of denting points, strongly regular points and much more. We completely characterize SCD points in the unit balls of $L_1$-preduals. We study SCD points in direct sums of Banach spaces and obtain that an infinite sum of Banach spaces may have an SCD point despite the fact that none of its components have it. We then prove sufficient conditions to get that an elementary tensor $x\otimes y$ is an SCD point in the unit ball of the projective tensor product $X \widehat{\otimes}_πY$. Regarding Lipschitz-free spaces on compact metric spaces, we show that norm one SCD points of their unit balls are exactly the ones that can be approximated by convex combinations of strongly exposed points of the unit ball. Finally, as applications, we prove a new inheritance result for the Daugavet property to its subspaces, we show that separable Banach spaces for which every convex series of slices intersects the unit sphere must contain an isomorphic copy of $\ell_1$, and we get pointwise conditions on an operator on a Banach space with the Daugavet property to satisfy the Daugavet equation.

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arXiv2023

(Almost isometric) local retracts in metric spaces

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of locally complemented and almost isometric ideals from Banach spaces. We prove that given two metric spaces $N\subseteq M$ there always exists an almost isometric local retract $S\subseteq M$ with $N\subseteq S$ and $dens(N)=dens(S)$. We also prove that metric spaces which are local retracts (respectively almost isometric local retracts) can be characterised in terms of a condition of extendability of Lipschitz functions (respectively almost isometries) between finite metric spaces. Different examples and counterexamples are exhibited.

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arXiv2023

A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions

Let $M$ be a metric space and $X$ be a Banach space. In this paper we address several questions about the structure of $\mathcal F(M)\widehat{\otimes}_πX$ and $\mathop{Lip}(M,X)$. Our results are the following: (1) We prove that if $M$ is a length metric space then $\mathop{Lip}(M,X)$ has the Daugavet property. As a consequence, if $M$ is length we obtain that $\mathcal F(M)\widehat{\otimes}_πX$ has the Daugavet property. This gives an affirmative answer to [13,Question 1] (also asked in [24,Remark 3.8]). (2) We prove that if $M$ is a non-uniformly discrete metric space or an unbounded metric space then the norm of $\mathcal F(M)\widehat{\otimes}_πX$ is octahedral, which solves [6, Question 3.2 (1)]. (3) We characterise all the Banach spaces $X$ such that $L(X,Y)$ is octahedral for every Banach space $Y$, which solves a question by Johann Langemets.

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arXiv2023

Diameter, radius and Daugavet index of thickness of slices in Banach spaces

We construct a Banach space $X$ with the r-BSP such that the infimum of the diameter of the slices of the unit ball is $1$, which gives negative answer to a 2006 question by Y. Ivakhno in an extreme way. This example is performed by considering modifications of the classical James-tree space $JT_\infty$ constructed on a tree with infinitely many branching points $T_\infty$. Moreover we prove that every Banach space with the Daugavet property admits, for every $\varepsilon>0$, an equivalent renorming for which its Daugavet index of thickness is bigger than $2-\varepsilon$ and there are slices of the unit ball of diameter strictly smaller than $2$, which solves an open question from [7].

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arXiv2023

Weak precompactness in projective tensor products

We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have the following property: whenever $W_1 \subseteq X$ and $W_2 \subseteq Y$ are sets such that $\{x\otimes y: \, x\in W_1, \, y\in W_2\}$ is weakly precompact in the projective tensor product $X \widehat{\otimes}_πY$, then either $W_1$ or $W_2$ is relatively norm compact. For instance, such a property holds for the pair $(\ell_p,\ell_q)$ if $1<p,q<\infty$ satisfy $1/p+1/q\geq 1$. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of $\ell_1$ into $X \widehat{\otimes}_πY$ for arbitrary Banach spaces $X$ and $Y$, in connection with the compactness of all operators from $X$ to $Y^*$.

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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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arXiv2023

On weakly almost square Banach spaces

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(Ω,Σ)$ be a measurable space, let $E$ be a Banach lattice and let $ν:Σ\to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(ν)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue-Bochner space $L_1(μ,Y)$ is weakly almost square for any Banach space~$Y$ and for any non-atomic finite measure~$μ$. On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space which fails the diameter two property. In this line we prove that if $X$ is any Banach space containing a complemented isomorphic copy of~$c_0$, then for every $0<\varepsilon<1$ there exists an equivalent norm $|\cdot|$ on~$X$ satisfying: (i)~every slice of the unit ball $B_{(X,|\cdot|)}$ has diameter~$2$; (ii) $B_{(X,|\cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; and (iii)~$(X,|\cdot|)$ is $(r,s)$-SQ for all $0<r,s < \frac{1-\varepsilon}{1+\varepsilon}$.

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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.

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arXiv2023

Diametral notions for elements of the unit ball of a Banach space

We introduce extensions of $Δ$-points and Daugavet points in which slices are replaced by relative weakly open subsets (super $Δ$-points and super Daugavet points) or by convex combinations of slices (ccs $Δ$-points and ccs Daugavet points). We first give a general overview on these new concepts and provide some isometric consequences on the spaces. As examples: if a Banach space contains a super $Δ$-point, then it does not admit an unconditional FDD with suppression constant smaller than two; if a real Banach space contains a ccs $Δ$-point, then it does not admit a one-unconditional basis; if a Banach space contains a ccs Daugavet point, then every convex combination of slices of its unit ball has diameter two. We next characterize the notions in some classes of Banach spaces showing, for instance, that all the notions coincide in $L_1$-predual spaces and that all the notions but ccs Daugavet points coincide in $L_1$-spaces. We next remark on some examples which have previously appeared in the literature and provide some new intriguing examples: examples of super $Δ$-points which are as closed as desired to strongly exposed points (hence failing to be Daugavet points in an extreme way); an example of a super $Δ$-point which is strongly regular (hence failing to be a ccs $Δ$-point in the strongest way); a super Daugavet point which fails to be a ccs $Δ$-point. The extensions of the diametral notions to point in the open unit ball and the consequences on the spaces are also studied. Last, we investigate the Kuratowski measure of relative weakly open subsets and of convex combinations of slices in the presence of super $Δ$-points or ccs $Δ$-points, as well as for spaces enjoying diameter 2 properties. We conclude the paper with a section on open problems.

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arXiv2022

Banach spaces which always produce octahedral spaces of operators

We characterise those Banach spaces $X$ which satisfy that $L(Y,X)$ is octahedral for every non-zero Banach space $Y$. They are those satisfying that, for every finite dimensional subspace $Z$, $\ell_\infty$ can be finitely-representable in a part of $X$ kind of $\ell_1$-orthogonal to $Z$. We also prove that $L(Y,X)$ is octahedral for every $Y$ if, and only if, $L(\ell_p^n,X)$ is octahedral for every $n\in\mathbb N$ and $1<p<\infty$. Finally, we find examples of Banach spaces satisfying the above conditions like $\Lip(M)$ spaces with octahedral norms or $L_1$-preduals with the Daugavet property.

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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 $X\widehat{\otimes}_πX$.

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arXiv2022

Several remarks on norm attaining in tensor product spaces

The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if $X\hat{\otimes}_πY$ is strongly subdifferentiable and either $X$ or $Y$ has the metric approximation property then every bounded operator from $X$ to $Y^*$ is compact. We also prove that $(\ell_p(I)\hat{\otimes}_π\ell_q(J))^*$ has the $w^*$-Kadec-Klee property for every non-empty sets $I,J$ and every $2<p,q<\infty$, obtaining in particular that the norm of the space $\ell_p(I)\hat{\otimes}_π\ell_q(J)$ is strongly subdifferentiable. This extends several results of Dantas, Kim, Lee and Mazzitelli. We also find examples of spaces $X$ and $Y$ for which the set of norm-attaining tensors in $X\pten Y$ is dense but whose complement is dense too.

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arXiv2022

Extremal structure in ultrapowers of Banach spaces

Given a bounded convex subset $C$ of a Banach space $X$ and a free ultrafilter $\mathcal U$, we study which points $(x_i)_\mathcal U$ are extreme points of the ultrapower $C_\mathcal U$ in $X_\mathcal U$. In general, we obtain that when $\{x_i\}$ is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then $(x_i)_\mathcal U$ is an extreme point (respectively denting point, strongly exposed point) of $C_\mathcal U$. We also show that every extreme point of $C_{\mathcal U}$ is strongly extreme, and that every point exposed by a functional in $(X^*)_{\mathcal U}$ is strongly exposed, provided that $\mathcal U$ is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of $C_\mathcal U$ in the case that $C$ is a super weakly compact or uniformly convex set.

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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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