Hermann Pfitzner
Nonreflexive Banach spaces that are complemented in their bidual by an L-projection - like preduals of von Neumann algebras or the Hardy space $H^1$ - contain, roughly speaking, many copies of $l^1$ which are very close to isometric copies. Such $l^1$-copies are known to fail the fixed point property. Similar dual results hold for $c_0$.
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We characterize Cesàro-Orlicz function spaces $Ces_φ$ containing order isomorphically isometric copy of $l^\infty$. We discuss also some useful applicable conditions sufficient for the existence of such a copy.
In this article we prove that every isometric copy of C(L) in C(K) is complemented if L is compact Hausdorff of finite height and K is a compact Hausdorff space satisfying the extension property, i.e., every closed subset of K admits an extension operator. The space C(L) can be replaced by its subspace C(L|F) consisting of functions that vanish on a closed subset F of L. We also study the class of spaces having the extension property, establishing some closure results for this class and relating it to other classes of compact spaces.
Isometric copies of $\ell_\infty^n$ in Lipschitz-free spaces over finite metric spaces
针对每个自然数n，构造有限度量空间M_n，使其Lipschitz-free空间包含ℓ_∞^n的等距拷贝，从而解决Khan等人提出的问题，并刻画可嵌入的有限维空间为多面体空间。
A Banach space contains asymptotically isometric copies of $\ell_1$ if and only if its dual space contains an isometric copy of $L_1$.
Main results: (a) If a metric space contains $2n$ elements, the transportation cost space on it contains a $1$-complemented isometric copy of $\ell_1^n$. (b) An example of a finite metric space whose transportation cost space contains an isometric copy of $\ell_\infty^4$. Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ 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 $\ell_1$.
Isometric rigidity and Fraïssé properties of Orlicz sequence spaces
论文研究了Orlicz序列空间的近等距刚性性质，证明了某些空间guarded Fraïssé但非ω-范畴，且不含ℓ₂的拷贝，回答了Cúth-de Rancourt-Doucha的问题。
Two normal functionals on a JBW$^*$-triple are known to be orthogonal if and only if they are $L$-orthogonal (meaning that they span an isometric copy of $\ell_1(2)$). This is shown to be stable under small norm perturbations in the following sense: if the linear span of the two functionals is isometric up to $δ>0$ to $\ell_1(2)$, then the functionals are less far (in norm) than $\eps>0$ from two orthogonal functionals, where $\eps\to0$ as $δ\to0$. Analogous statements for finitely and even infinitely many functionals hold as well. And so does a corresponding statement for non-normal functiona…
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…
We show that Müntz spaces, as subspaces of $C[0,1]$, contain asymptotically isometric copies of $c_0$ and that their dual spaces are octahedral.
We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-FDD are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separa…
A metric vector space is asymptotically metrically normable (AMN) if there exists a norm asymptotically isometric to the distance. We prove that AMN vector spaces are rigid in the class of metric vector spaces under asymptotically isometric perturbations. This result follows from a general metric normability criterium. If the distance is translation invariant and satisfies an approximate multiplicative condition then there exists a lipschitz equivalent norm. Furthermore, we give necessary and sufficient conditions for the distance to be asymptotically isometric to the norm.
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 $\varepsilon>0$, $…
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…
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 $c_0(ω_1)$).
Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0(L)$ have very strong diameter $2$ properties and, for every real number $\varepsilon$ with $0<\varepsilon<1$, contain an $\varepsilon$-isometric copy of $c_0$. If $L$ does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of $\ell_1$.
For each ordinal $ξ$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $ξ$-$p$-asymptotically uniformly smooth norm. We also introduce the notion of $ξ$-asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $ξ$-asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $ξ$-asymptotic smoothness of $C(K)$ in terms of the Cantor-Bendixson index of $K$. We also prove that for …
We investigate structure of the optimal domains for the Hardy-type operators including, for example, the classical Cesàro, Copson and Volterra operators as well as for some of their generalizations. We prove that, in some sense, the abstract Cesàro and Copson function spaces are closely related to the space $L^1$, namely, they contain "in the middle" a complemented copy of $L^1[0,1]$, asymptotically isometric copy of $\ell^1$ and also can be renormed to contain an isometric copy of $L^1[0,1]$. Moreover, the generalized Tandori function spaces are quite similar to $L^\infty$ because they contai…
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…
Structure properties of Banach spaces of I-null sequences
本文研究理想零序列空间c_{0,I}及其相关商空间的结构性质，通过Dini原理和Sobczyk定理等工具，刻画了对偶空间、紧性、补余拷贝等经典问题，并构造了统计理想以展示非补余现象。
Projection schemes and the fixed point property in nonreflexive Banach spaces
该研究引入Banach空间投影格式的抽象框架，得到非自反Banach空间满足不动点性质的通用判据，还刻画了变指数Lebesgue空间的不动点性质，给出首个具有该性质的非自反Banach函数格例子。
On embedding separable spaces $\mathcal{C}(L)$ in arbitrary spaces $\mathcal{C}(K)$
针对紧空间$K$和度量紧空间$L$的连续函数Banach空间，刻画$\mathcal{C}(L)$到$\mathcal{C}(K)$等距/同构嵌入的存在性，证明可分$\mathcal{C}(L)$时经典定理等价，并给出关于相对胞腔度的新结果。
Fixed point results for mappings of asymptotically Hölder nonexpansive type
将Goebel-Kirk不动点定理推广到渐近Hölder非扩张型映射，通过例子说明新框架严格包含经典情形，并证明若Banach空间包含c₀或ℓ₁的等距同构副本则此类映射的不动点性质失效，最后构造无限维Banach空间中存在无不动点的紧凸集K及其上的渐近Hölder非扩张型仿射自映射T。
We give a criterion ensuring that the elementary class of a modular Banach space E (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of E) consists of all direct sums E\oplus_m H, where H is an arbitrary Hilbert space and \oplus_m denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.
We investigate the structure of the free $p$-convex Banach lattice $FBL^{(p)}[E]$ over a Banach space $E$. After recalling why such a free lattice exists, and giving a convenient functional representation of it, we focus our study on how properties of an operator $T:E\rightarrow F$ between Banach spaces transfer to the associated lattice homomorphism $\overline{T}:FBL^{(p)}[E]\rightarrow FBL^{(p)}[F]$. Particular consideration is devoted to the case when the operator $T$ is an isomorphic embedding, which leads us to examine extension properties of operators into $\ell_p$, and several classical…
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