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Jesús M. F. Castillo

Publications and source records attributed to Jesús M. F. Castillo.

At least 37 records · Page 2Linked to original sources
arXiv2020

The isomorphic Kottman constant of a Banach space

We show that the Kottman constant $K(\cdot)$, together with its symmetric and finite variations, is continuous with respect to the Kadets metric, and they are log-convex, hence continuous, with respect to the interpolation parameter in a complex interpolation schema. Moreover, we show that $K(X)\cdot K(X^*)\geqslant 2$ for every infinite-dimensional Banach space $X$. We also consider the isomorphic Kottman constant (defined as the infimum of the Kottman constants taken over all renormings of the space) and solve the main problem left open in [CaGoPa17], namely that the isomorphic Kottman constant of a twisted-sum space is the maximum of the constants of the respective summands. Consequently, the Kalton--Peck space may be renormed to have Kottman's constant arbitrarily close to $\sqrt{2}$. For other classical parameters, such as the Whitley and the James constants, we prove the continuity with respect to the Kadets metric.

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arXiv2020

On the ${\Ext}^2$-problem for Hilbert spaces

We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves the four-space problem for local convexity.

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arXiv2019

Banach spaces of almost universal complemented disposition

We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-FDD are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces.

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arXiv2019

On the stability of the differential process generated by complex interpolation

We study the stability of the differential process of Rochberg and Weiss associated to an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit sphere while there is no (bounded) stability for families of four or more Köthe spaces. In the context or arbitrary pairs of Banach spaces we present local stability results and global isometric stability results.

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arXiv2019

On Disjointly singular centralizers

We study ``disjoint" versions of the notions of trivial, locally trivial, strictly singular and super-strictly singular quasi-linear maps in the context of Köthe function spaces. Among other results, we show: i) (locally) trivial and (locally) disjointly trivial notions coincide on reflexive spaces; ii) On non-atomic superreflexive Köthe spaces, no centralizer is singular, although most are disjointly singular. iii) No super singular quasi-linear maps exist between superreflexive spaces although Kalton-Peck centralizers are super disjointly singular; iv) Disjoint singularity does not imply super disjoint singularity.

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arXiv2018

Derivation of vector-valued complex interpolation scales

We study complex interpolation scales obtained by vector valued amalgamation and the derivations they generate. We study their trivial and singular character and obtain examples showing that the hypotheses in the main theorems of [J.M.F. Castillo, V. Ferenczi and M. González, \emph{Singular exact sequences generated by complex interpolation}, Trans. Amer. Math. Soc. 369 (2017) 4671--4708] are not necessary.

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arXiv2018

On the bounded approximation property on subspaces of $\ell_p$ when $0<p<1$ and related issues

This paper studies the bounded approximation property (BAP) in quasi Banach spaces. In the first part of the paper we show that the kernel of any surjective operator $\ell_p\to X$ has the BAP when $X$ has it and $0<p\leq 1$, which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec-Pełczyński-Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

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arXiv2018

Bilinear forms and the $\Ext^2$-problem in Banach spaces

Let $X$ be a Banach space and let $κ(X)$ denote the kernel of a quotient map $\ell_1(Γ)\to X$. We show that $\Ext^2(X,X^*)=0$ if and only if bilinear forms on $κ(X)$ extend to $\ell_1(Γ)$. From that we obtain i) If $κ(X)$ is a $\mathcal L_1$-space then $\Ext^2(X,X^*)=0$; ii) If $X$ is separable, $κ(X)$ is not an $\mathcal L_1$ space and $\Ext^2(X,X^*)=0$ then $κ(X)$ has an unconditional basis. This provides new insight into a question of Palamodov in the category of Banach spaces.

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arXiv2017

1-complemented subspaces of Banach spaces of universal disposition

We first unify all notions of partial injectivity appearing in the literature ---(universal) separable injectivity, (universal) $\aleph$-injectivity --- in the notion of $(α, β)$-injectivity ($(α, β)_λ$-injectivity if the parameter $λ$ has to be specified). Then, extend the notion of space of universal disposition to space of universal $(α, β)$-disposition. Finally, we characterize the $1$-complemented subspaces of spaces of universal $(α, β)$-disposition as precisely the spaces $(α, β)_1$-injective.

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arXiv2016

Twisting non-commutative $L_p$ spaces

The paper makes the first steps into the study of extensions ("twisted sums") of noncommutative $L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra $\mathcal M$. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss saying that whenever one finds a given Banach space $Y$ as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension $ 0\longrightarrow Y\longrightarrow X\longrightarrow Y \longrightarrow 0. $ For semifinite algebras, considering $L^p=L^p(\mathcal M,τ)$ as an interpolation space between $\mathcal M$ and its predual $\mathcal M_*$ one arrives at a certain self-extension of $L^p$ that is a kind of noncommutative Kalton-Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented. For general algebras, including those of type III, the interpolation mechanism produces two (rather than one) extensions of one sided modules, one of left-modules and the other of right-modules. Whether or not one can find (nontrivial) self-extensions of bimodules in all cases is left open.

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arXiv2015

Complex structures on twisted Hilbert spaces

We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck $Z_2$ space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of Köthe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant Köthe function spaces then there are complex structures on it that can be extended to a complex structure of the twisted Hilbert space. Regarding the hyperplane problem we show that no complex structure on $\ell_2$ can be extended to a complex structure on an hyperplane of $Z_2$ containing it.

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arXiv2014

On nested sequences of convex sets in a Banach space

In this paper we study different aspects of the representation of weak*-compact convex sets of the bidual $X^{**}$ of a separable Banach space $X$ via a nested sequence of closed convex bounded sets of $X$.

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arXiv2014

$\aleph$-injective Banach spaces and $\aleph$-projective compacta

A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to E$. We say that $E$ is $\aleph$-injective (respectively, universally $\aleph$-injective) if the preceding condition holds for Banach spaces $X$ (respectively $Y$) with density less than a given uncountable cardinal $\aleph$. We perform a study of $\aleph$-injective and universally $\aleph$-injective Banach spaces which extends the basic case where $\aleph=\aleph_1$ is the first uncountable cardinal. When dealing with the corresponding "isometric" properties we arrive to our main examples: ultraproducts and spaces of type $C(K)$. We prove that ultraproducts built on countably incomplete $\aleph$-good ultrafilters are $(1,\aleph)$-injective as long as they are Lindenstrauss spaces. We characterize $(1,\aleph)$-injective $C(K)$ spaces as those in which the compact $K$ is an $F_\aleph$-space (disjoint open subsets which are the union of less than $\aleph$ many closed sets have disjoint closures) and we uncover some projectiveness properties of $F_\aleph$-spaces.

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arXiv2014

Complex interpolation and twisted twisted Hilbert spaces

We show that Rochberg's generalizared interpolation spaces $\mathscr Z^{(n)}$ arising from analytic families of Banach spaces form exact sequences $0\to \mathscr Z^{(n)} \to \mathscr Z^{(n+k)} \to \mathscr Z^{(k)} \to 0$. We study some structural properties of those sequences; in particular, we show that nontriviality, having strictly singular quotient map, or having strictly cosingular embedding depend only on the basic case $n=k=1$. If we focus on the case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces, then $\mathscr Z^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space; we then show that $\mathscr Z^{(n)}$ is (or embeds in, or is a quotient of) a twisted Hilbert space only if $n=1,2$, which solves a problem posed by David Yost; and that it does not contain $\ell_2$ complemented unless $n=1$. We construct another nontrivial twisted sum of $Z_2$ with itself that contains $\ell_2$ complemented.

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arXiv2013

Stability constants and the homology of quasi-Banach spaces

We affirmatively solve the main problems posed by Laczkovich and Paulin in \emph{Stability constants in linear spaces}, Constructive Approximation 34 (2011) 89--106 (do there exist cases in which the second Whitney constant is finite while the approximation constant is infinite?) and by Cabello and Castillo in \emph{The long homology sequence for quasi-Banach spaces, with applications}, Positivity 8 (2004) 379--394 (do there exist Banach spaces $X,Y$ for which $\Ext(X,Y)$ is Hausdorff and non-zero?). In fact, we show that these two problems are the same.

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