Related papers: Some equivalence relations which are Borel reducib…

We give a simple example of a countable metric space $M$ that does not embed bi-Lipschitz with distortion strictly less than 2 into any Asplund space. Actually, if $M$ embeds with distortion strictly less than 2 to a Banach space $X$, then…

A Banach space contains asymptotically isometric copies of $\ell_1$ if and only if its dual space contains an isometric copy of $L_1$.

A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $\mc A$, we can find a reflexive Banach space $E$, and an…

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism),…

Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also…

We calculate ordinal $L_p$ index defined in "An ordinal L_p index for Banach spaces with an application to complemented subspaces of L_p" authored by J. Bourgain, H. P. Rosenthal and G. Schechtman, for Rosenthal's space $X_p$, $\ell_p$ and…

For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the…

This paper initiates the study of the structure of a new class of $p$-Banach spaces, $0<p<1$, namely the Lipschitz free $p$-spaces (alternatively called Arens-Eells $p$-spaces) $\mathcal{F}_{p}(\mathcal{M})$ over $p$-metric spaces. We…

See title. (A Banach space is said to be L-embedded if it is complemented in its bidual such that the norm between the two complementary subspaces is additive.)

The aim of this paper is twofold. On the one hand, we compute, in terms of $r$ and $s$, the indices $p$ for which $\ell_p$ isomorphically embeds into the mixed-norm separable spaces $L_s(L_r)$, $\ell_s(L_r)$, $L_s(\ell_r)$ and…

Given a set $B$ of operators between subspaces of $L_p$ spaces, we characterize the operators between subspaces of $L_p$ spaces that remain bounded on the $X$-valued $L_p$ space for every Banach space on which elements of the original class…

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\ell_p^n$, $p > 0$, into various type of Banach or…

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of $\ell_1$. This result has many consequences for the structure…

We prove that non-Hilbertian separable Orlicz sequence spaces are ergodic, i.e., the equivalence relation $\mathbb{E}_0$ Borel reduces to the isomorphism relation between subspaces of every such space. This is done by exhibiting…

We prove that the outer $L^p_\mu(\ell^r)$ spaces, introduced by Do and Thiele, are isomorphic to Banach spaces, and we show the expected duality properties between them for $1 < p \leq \infty, 1 \leq r < \infty$ or $p=r \in \{ 1, \infty \}$…

For $n\geq 2, p<2$ and $q>2,$ does there exist an $n$-dimensional Banach space different from Hilbert spaces which is isometric to subspaces of both $L_{p}$ and $L_{q}$? Generalizing the construction from the paper "Zonoids whose polars are…

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver and Harrington-Kechris-Louveau show that with respect to Borel reducibility, any…

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most…

Let $X$ be a Banach space with a separable dual. We prove that $X$ embeds isomorphically into a $\cL_\infty$ space $Z$ whose dual is isomorphic to $\ell_1$. If, moreover, $U$ is a space so that $U$ and $X$ are totally incomparable, then we…

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…