Related papers: Subspaces of separable $L_1$-preduals: $W_\alpha$ …

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to…

Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has…

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing…

In the first part of the paper we show that every closed subspace of $JT$ or $JT^*$ contains $\ell_2$ complemented in $JT$ or $JT^*$ respectively, and $JT$ contains uncomplemented copies of $\ell_2$. As a result, the predual $\B$ of $JT$,…

We study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space $X$.…

For every $\alpha<\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\omega^{\alpha\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed…

Let $X$ be a Banach space and $(\Omega,\Sigma)$ be a measure space. We provide a characterization of sequences in the space of $X$-valued countably additive measures on $\Omega,\Sigma)$ of bounded variation that generate complemented copies…

We study the space $c_{0,\mathcal{I}}$ of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the sup norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that two such spaces,…

If a separable Banach space $X$ is such that for some nonquasireflexive Banach space $Y$ there exists a surjective strictly singular operator $T:X\to Y$ then for every countable ordinal $\alpha $ the dual of $X$ contains a subspace whose…

We study those Banach spaces $X$ for which $S_X$ does not admit a finite $\eps$-net consisting of elements of $S_X$ for any $\eps < 2$. We give characterisations of this class of spaces in terms of $\ell_1$-type sequences and in terms of…

We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation…

We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded…

A recent result of Freeman, Odell, Sari, and Zheng states that whenever a separable Banach space not containing $\ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis…

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…

Let $X$ be a Banach space with the unit ball $B(X)$ and $A\subset X$ be a convex origin-symmetric compact in $X$. Let $\mathrm{j}:X\rightarrow \widetilde{X}$ be an isometric extension of $X$. It is well-known that linear widths $\lambda…

The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if…

We show that any Banach space contains a continuum of non isomorphic subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of $X$, and show…

In this article, we show that the predual $\mathcal{G}_w(U)$ of the weighted space of holomorphic functions has the $\mathcal{I}$- approximation property if and only if $E$ has the $\mathcal{I}$- approximation property, where $\mathcal{I}$…

Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\in\Gamma}\ell_{q}(X)$, where $\Gamma$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain…

For an asymptotic $\ell_1$ space $X$ with a basis $(x_i)$ certain asymptotic $\ell_1$ constants, $\delta_\alpha (X)$ are defined for $\alpha <\omega_1$. $\delta_\alpha (X)$ measures the equivalence between all normalized block bases…