Operators on (C[0,1]) preserving copies of asymptotic (\ell_1) spaces

Complementability of isometric copies of $\ell_1$ in transportation cost spaces

This work aims to establish new results pertaining to the structure of transportation cost spaces. Due to the fact that those spaces were studied and applied in various contexts, they have also become known under different names such as Arens-Eells spaces, Lipschitz-free spaces, and Wasserstein spaces. The main outcome of this paper states that if a metric space $X$ is such that the transportation cost space on $X$ contains an isometric copy of $\ell_1$, then it contains a $1$-complemented isometric copy of $\ell_1$.

Embedding Banach spaces into the space of bounded functions with countable support

We prove that a WLD subspace of the space $\ell_\infty^c(Γ)$ consisting of all bounded, countably supported functions on a set $Γ$ embeds isomorphically into $\ell_\infty$ if and only if it does not contain isometric copies of $c_0(ω_1)$. Moreover, a subspace of $\ell_\infty^c(ω_1)$ is constructed that has an unconditional basis, does not embed into $\ell_\infty$, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of $c_0(ω_1)$).

$(1+)$-complemented, $(1+)$-isomorphic copies of $L_{1}$ in dual Banach spaces

The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on Pełczyński's classical work on dual Banach spaces containing $L_{1}$ ($=L_{1}[0,1]$) and the Hagler--Stegall characterisation of dual spaces containing complemented copies of $L_{1}$. We prove the following quantitative version of the Hagler--Stegall theorem asserting that for a Banach space $X$ the following statements are equivalent: $\bullet$ $X$ contains almost isometric copies of $(\bigoplus_{n=1}^{\infty} \ell_{\infty}^{n})_{\ell_1}$, $\bullet$ for all $\varepsilon>0$, $…

Slicely Countably Determined Banach spaces

We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternati…

Polylog dimensional subspaces of $\ell_\infty^N$

We show that a subspace of of $\ell_\infty^N$ of dimension $n>(\log N\log \log N)^2$ contains $2$-isomorphic copies of $\ell_\infty^k$ where $k$ tends to infinity with $n/(\log N\log \log N)^2$. More precisely, for every $η>0$, we show that any subspace of $\ell_\infty^N$ of dimension $n$ contains a subspace of dimension $m=c(η)\sqrt{n}/(\log N\log \log N)$ of distance at most $1+η$ from $\ell_\infty^m$.

Subspaces of separable $L_1$-preduals: $W_α$ everywhere

The spaces $W_α$ are the Banach spaces whose duals are isometric to $\ell_1$ and such that the standard basis of $\ell_1$ is $w^*$-convergent to $α\in \ell_1$. The core result of our paper proves that an $\ell_1$-predual $X$ contains isometric copies of all $W_α$, where the norm of $α$ is controlled by the supremum of the norms of the $w^*$-cluster points of the extreme points of the closed unit ball in $\ell_1$. More precisely, for every $\ell_1$-predual $X$ we have $$ r^*(X)=\sup\left\lbrace \left\|g^*\right\|: g^*\in \left(\mathrm{ext}\, B_{\ell_1}\right)'\right\rbrace =\sup \left\lbrace \l…

Dual Banach spaces which contain an isometric copy of $L_1$

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

由零均值函数的独立副本张成的Orlicz空间子空间

On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

本文研究由零均值函数的独立副本张成的Orlicz空间子空间，刻画其强嵌入性及单位球范数等度连续性，并给出当且仅当条件（指数不等式）成立的一类广泛空间。

The Kalton-Peck space is the complexification of the real Kalton-Peck space

The Kalton-Peck $Z_2$ space is the derived space obtained from the scale of $\ell_p$ spaces by complex interpolation at $1/2$. If we denote by by $Z_2^{real}$ the derived space obtained from the scale of $\ell_p$ spaces by real interpolation at $(1/2, 1/2)$, we show that $Z_2$ is the complexification of $Z_2^{real}$. We also show that $Z_2^{real}$ shares the most important properties of $Z_2$: it is isomorphic to its dual, it is singular and contains no complemented copies of $\ell_2$.

Coarse embeddings into superstable spaces

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 $X$ must contain an isomorphic copy of $\ell_p$, for some $p\in[1,\infty)$. In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then $X$ has a spreading model isomorphic to $\ell_p$, for some $p\in[1,\infty)$. In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any sup…

$\mathcal{L}(\ell_p^n,\ell_p^m)$, $(p\in \{1,\infty\})$ 上保持平行对的线性映射

Linear maps on $\mathcal{L}(\ell_p^n,\ell_p^m)$, $(p\in \{1,\infty\})$ preserving parallel pairs

研究 $\mathcal{L}(\ell_p^n,\ell_p^m)$ ($p\in\{1,\infty\}$) 上保持平行对和 TEA 对的线性映射，证明非零映射保持 TEA 对、保持平行对且秩大于1或可逆、为等距标量倍数四者等价。

Classifications of linear operators preserving elliptic, positive and non-negative polynomials

We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [9] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.

将ℓ₂和J嵌入到JT和JT*的子空间中

Embedding $\ell_2$ and $J$ into subspaces of $JT$ and $JT^*$

论文研究了JT和JT*子空间中ℓ₂和J的嵌入性质，证明了闭子空间包含补集的ℓ₂，并展示了非反射子空间包含J的等距复制。

On the structure of tensor products of l_p spaces

We examine some structural properties of (injective and projective) tensor products of $\ell_p$-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals. Our main results are: \medbreak \item{(1)} If $1<p<\infty$, then $B(\ell_p)\approx B(L_p)$ ($B(X)$ consists of the bounded linear operators on $X$). \medbreak \item{(2)} If ${1\over p_i}+{1\over p_j}\leq1$ for every $i\neq j$, or if all of the $p_i$'s are equal, then $\ell_{p_1}\hat{\otimes}\cdots \…

$\mathcal{L}(\ell_{\MakeLowercase{p}},\mathcal{Y})$ 中保持平行对和 TEA 对的线性映射

Linear maps in $\mathcal{L}(\ell_{\MakeLowercase{p}},\mathcal{Y}) $ preserving parallel and TEA pairs

研究从 $\ell_p$ 到 Banach 空间 $\mathcal{Y}$ 的有界线性映射保持平行对和三角形等式达到（TEA）对的条件，给出了不同 $p$ 和 $\mathcal{Y}$ 下的刻画。

Generalized transportation cost spaces

The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete infinite metric spaces transportation cost spaces on which do not contain isometric copies of $\ell_1$, this result answers a question raised by Cuth and Johanis (2017); (2) locally finite metric spaces which admit isometric embeddings only into…

保持 $\ell_p$-范数平行向量的线性映射

Linear maps preserving $\ell_p$-norm parallel vectors

本文刻画了赋范空间中保持平行对与三角形等式达到（TEA）对的线性映射，给出了 $\ell_p$-范数下各类情形的具体刻画，并将其推广至无限维空间。

A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals

We show that every Lorentz sequence space $d(\textbf{w},p)$ admits a 1-complemented subspace $Y$ distinct from $\ell_p$ and containing no isomorph of $d(\textbf{w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\textbf{w},p)$ yet known. We also give an explicit representation of $Y$ in the special case $\textbf{w}=(n^{-θ})_{n=1}^\infty$ ($0<θ<1$) as the $\ell_p$-sum of finite-dimensional copies of $d(\textbf{w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal{L}(d(\textbf{w},p))$, of which only five …

Banach actions preserving unconditional convergence

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence $(a_n)_{n\inω}$ in $A$ and unconditionally convergent series $\sum_{n\inω}x_n$ in $X$ the series $\sum_{n\inω}a_nx_n$ is unconditionally convergent. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^*\in Y^*$ the operator $D_{y^*}:X\to A^*$, $D_{y^*}(x)(a)=y^*(ax)$, is absolutely summing. Combining …

Banach Spaces from Barriers in High Dimensional Ellentuck Spaces

A new hierarchy of Banach spaces $T_k(d,θ)$, $k$ any positive integer, is constructed using barriers in high dimensional Ellentuck spaces \cite{DobrinenJSL15} following the classical framework under which a Tsirelson type norm is defined from a barrier in the Ellentuck space \cite{Argyros/TodorcevicBK}. The following structural properties of these spaces are proved. Each of these spaces contains arbitrarily large copies of $\ell_\infty^n$, with the bound constant for all $n$. For each fixed pair $d$ and $θ$, the spaces $T_k(d,θ)$, $k\ge 1$, are $\ell_p$-saturated, forming natural extensions of…

Preserving positive polynomials and beyond

Following the classical approach of Pólya-Schur theory we initiate in this paper the study of linear operators acting on $\mathbb{R}[x]$ and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.