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Félix Cabello Sánchez

Publications and source records attributed to Félix Cabello Sánchez.

29 records · Page 2Linked to original sources
arXiv2019

Splitting chains, tunnels and twisted sums

We study splitting chains in $\mathscr{P}(ω)$, that is, families of subsets of $ω$ which are linearly ordered by $\subseteq^*$ and which are splitting. We prove that their existence is independent of axioms of $\mathsf{ZFC}$. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of $C(ω^*)=\ell_\infty/c_0$ and $c_0(\mathfrak c)$. Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.

math.LO↗
arXiv2019

On Banach spaces whose group of isometries acts micro-transitively on the unit sphere

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly convex and uniformly smooth, and that they form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known results on it. Besides, we show that if there is a non-Hilbertian non-separable Banach space with uniform micro-semitransitive (or micro-transitive) norm, then there is a non-Hilbertian separable one. Finally, we show that an $L_p(μ)$ space is micro-transitive or uniformly micro-semitransitive only when $p=2$.

math.FA↗
arXiv2019

Quiz your maths: do the uniformly continuous functions on the line form a ring?

The paper deals with the interplay between boundedness, order and ring structures in function lattices on the line and related metric spaces. It is shown that the lattice of all Lipschitz functions on a normed space $E$ is isomorphic to its sublattice of bounded functions if and only if $E$ has dimension one. The lattice of Lipschitz functions on $E$ carries a "hidden" $f$-ring structure with a unit, and the same happens to the (larger) lattice of all uniformly continuous functions for a wide variety of metric spaces. An example of a metric space whose lattice of uniformly continuous functions supports no unital $f$-ring structure is provided.

math.FA↗
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.

math.FA↗
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.

math.OA↗
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.

math.FA↗
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.

math.FA↗
arXiv2014

Quasi-Banach spaces of almost universal disposition

We show that for each $p\in(0,1]$ there exists a separable $p$-Banach space $\mathbb G_p$ of almost universal disposition, that is, having the following extension property: for each $ε>0$ and each isometric embedding $g:X\to Y$, where $Y$ is a finite dimensional $p$-Banach space and $X$ is a subspace of $\mathbb G_p$, there is an $ε$-isometry $f:Y\to \mathbb G_p$ such that $x=f(g(x))$ for all $x\in X$. Such a space is unique, up to isometries, does contain an isometric copy of each separable $p$-Banach space and has the remarkable property of being "locally injective" amongst $p$-Banach spaces. We also present a nonseparable generalization which is of universal disposition for separable spaces and "separably injective". No separably injective $p$-Banach space was previously known for $p<1$.

math.FA↗
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.

math.FA↗
arXiv2013

On ultrapowers of Banach spaces of type $\mathscr L_\infty$

We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain $c_0$ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs had been infected by that statement. In particular we provide proofs for the following statements: (i) All $M$-spaces, in particular all $C(K)$-spaces, have ultrapowers isomorphic to ultrapowers of $c_0$, as well as all their complemented subspaces isomorphic to their square. (ii) No ultrapower of the Gurari\u ı space can be complemented in any $M$-space. (iii) There exist Banach spaces not complemented in any $C(K)$-space having ultrapowers isomorphic to a $C(K)$-space.

math.FA↗