Related papers: Interpolating hereditarily indecomposable Banach s…

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for $p\in [1,\infty]$, every proper subset of $L_p$ is almost Lipschitzly embeddable into a Banach space $X$ if and only if $X$…

We provide two examples of complex homogeneous quadratic polynomials P on Banach spaces of the form l_1(I). The first polynomial P has both separable and nonseparable maximal zero subspaces. The second polynomial P has the property that…

In this paper, we describe an example of a hyperkaehler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach…

This paper deals with the following types of problems: Assume a Banach space $X$ has some property (P). Can it be embedded into some Banach space $Z$ with a finite dimensional decomposition having property (P), or more generally, having a…

We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ ($2<p<\infty$) is ergodic. This reinforces the conjecture that $\ell_2$ is the only non ergodic Banach space. As an application of our criterion for…

Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of $\ell_p$, for some $p\in[1,\infty)$. In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then…

Let $0<p<q\leq\infty$ and $\alpha \in (0,\infty]$. We give a characterization of quasi-Banach interpolation spaces for the couple $(L_p(0,\alpha),L_q(0,\alpha))$ in terms of two monotonicity properties, extending known results which mainly…

The objective of this paper is to construct separable Banach spaces $S{D^p}[\mathbb{R}^\infty]$ for $1\leq p \leq \infty$, each of which contains the $L^p[\mathbb{R}^\infty] $ spaces, as well as finitely additive measures, as compact dense…

The purpose of this paper is to construct a new class of separable Banach spaces $\K^p[\mathbb{B}], \; 1\leq p \leq \infty$. Each of these spaces contain the $ \mcL^p[\mathbb{B}] $ spaces, as well as the space $\mfM[\R^\iy]$, of finitely…

We present a reflexive Banach space $\mathfrak{X}_{_{^\text{usm}}}$ which is Hereditarily Indecomposable and satisfies the following properties. In every subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a weakly null normalized…

For every $n\geq 3,$ we construct an $n$-dimensional Banach space which is isometric to a subspace of $L_{1/2}$ but is not isometric to a subspace of $L_1.$ The isomorphic version of this problem (posed by S. Kwapien in 1969) is still open.…

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

Suppose $X$ is a real or complexified Banach space containing a complemented copy of $\ell_p$, $p\in(1,2)$, and a copy (not necessarily complemented) of either $\ell_q$, $q\in(p,\infty)$, or $c_0$. Then $\mathcal{L}(X)$ and…

It is shown that any Banach space X of sufficiently large density contains an (infinite) unconditional sequence and a separable quotient. If a density of X is a weakly compact cardinal, then X contains an unconditional sequence of…

In this note we show that every Banach space $X$ not containing $\ell_1^n$ uniformly and with unconditional basis contains an arbitrarily distortable subspace.

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal T$ of a certain type on a space X is…

A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l_p. Employing methods from interpolation theory,…

We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\Bbb N}/\ell_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called H\"older$(\alpha)$ embeddability between…

A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic…

The sequence of entropy numbers quantifies the degree of compactness of a linear operator acting between quasi-Banach spaces. We determine the asymptotic behavior of entropy numbers in the case of natural embeddings between…