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

Nigel Kalton's work in isometrical Banach space theory

This paper surveys some of the late Nigel Kalton's contributions to Banach space theory. The paper is written for the Nigel Kalton Memorial Website http://mathematics.missouri.edu/kalton/, which is scheduled to go online in summer 2011.

Almost isometric ideals in Banach spaces

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.

The dual of a non-reflexive L-embedded Banach space contains $\ell^\infty$ isometrically.

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.)

关于几乎等距理想和局部再退化在Banach和度量空间中的注记

Note on almost isometric ideals and local retracts in Banach and metric spaces

本文提出了一种新方法，证明了非分离Banach空间中存在大量几乎等距理想和度量空间中大量几乎等距局部再退化，同时展示了在非分离Banach空间的每个对偶空间上存在大量非平凡投影。

Isometric stability property of certain Banach spaces

Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach space, $p>1,$ and $(X,σ)$ be a space with a finite measure. We prove that $E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if $E$ is isometric to a subspace of $F.$ Moreover, every isometry $T$ from $E$ into $L_p(X;F)$ has the form $Te(x)=h(x)U(x)e, e\in E,$ where $h:X\rightarrow R$ is a measurable function and, for every $x\in X,$ $U(x)$ is an isometry from $E$ to $F.$

Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces

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 operators if (and only if) $X$ does not contain a copy of $\ell_{1}$, has the metric compact approximation property, and has property~$(M)$. The investigation of special versions of property~$(M)$ leads to results on almost isometric structure of some classes of Banach spaces. For instance, we give a simple necessary and sufficient condition for a Banach spac…

Banach 空间中的超限（几乎等距）理想

Transfinite (almost isometric) ideals in Banach spaces

本文提出并研究 Banach 空间中（几乎等距）理想的超限版本，借此刻画超限内射空间与（几乎）万有配置空间，并构造反例、修正经典结果。

Ideal boundedness of subseries and rearrangements in Banach spaces vs Banach spaces possessing a copy of $c_0$

Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are meager. We use Bessaga-Pełczy{ń}ski $c_0-$Theorem and concept of uniformly unconditionally bounded series. Moreover we prove similar result for the ideal boundedness for a class of Baire ideals using Talagrand's characterisation.

Isometric embeddings of Banach bundles

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an \emph{into} isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.

等距 universally 的 Banach 空间与单调 Schauder 基

An isometrically universal Banach space with a monotone Schauder basis

本文提出一个等距互补 universal Banach 空间 B，具有单调 Schauder 基，并与 Pełczyński 的 universal 基空间及 Kadec 的互补 universal 空间相关联。

Isometric embeddings of Banach spaces under optimal projection constants

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 further an FDD $(E_n)$ whose strong bimonotonicity projection constant is not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if $(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 + \varepsilon)$-unconditional. The proof uses renorming and skipped blo…

On isometric reflexions in Banach spaces

We obtain the following characterization of Hilbert spaces. Let $E$ be a Banach space whose unit sphere $S$ has a hyperplane of symmetry. Then $E$ is a Hilbert space iff any of the following two conditions is fulfilled: a) the isometry group ${\rm Iso}\, E$ of $E$ has a dense orbit in S; b) the identity component $G_0$ of the group ${\rm Iso}\, E$ endowed with the strong operator topology acts topologically irreducible on $E$. Some related results on infinite dimentional Coxeter groups generated by isometric reflexions are given which allow to analyse the structure of isometry groups containin…

Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals

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 $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.

可分Banach空间到$(\ell^\infty \setminus c)\cup\{0\}$的等距嵌入

Isometric embeddings of separable Banach spaces into $(\ell^\infty \setminus c)\cup\{0\}$

该研究证明可分Banach空间可等距嵌入到与收敛序列空间$c$仅交于原点的$\ell^\infty$子集中，且相关可分子空间可保避开$c$的性质扩展为包含任意可分Banach空间等距副本的子空间，推广了经典嵌入结论。

Isometric copies of $l^\infty$ in Cesàro-Orlicz function spaces

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.

双对偶巴拿赫空间的等距直接极限

Isometric direct limits of bidual Banach spaces

本文引入Dir和Inv两个函子，研究双对偶巴拿赫空间的塔系统及其极限，证明Dir保持直和分解及算子谱、弗雷德霍姆性和紧性，并应用于函数代数表示测度支集定位。

Banach空间算子的等距、对称与等对称交换$d$元组

Isometric, symmetric and isosymmetric commuting $d$-tuples of Banach space operators

该研究将Banach空间算子相关性质推广到交换$d$元组，阐明其诸多结构性质可通过单算子情形的常用论证推导，厘清该推广的内在逻辑。

Remarks on quasi-isometric non-embeddability into uniformly convex Banach spaces

We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed.

在Banach空间中为贪婪基构造等距重范数，及其在$ L_p[0,1] $中Haar系统的应用

Isometric renormings for greedy bases in Banach spaces, with applications to the Haar System in $L_p[0,1]$, $1<p<\infty$

研究通过等距重范数改进基的贪婪常数，证明若Banach空间具有无条件和双民主基且基本函数满足特定正则性，则可重范使其成为等距贪婪基，并应用于Haar系统等。

Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces

We study the horofunction boundaries of Hilbert and Thompson geometries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space.

Banach spaces of almost universal complemented disposition

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…

On the second dual space of Banach space of vector-valued little Lipschitz functions

Let \(X\) be a compact metric space and \(E\) be a Banach space. \(\lip (X, E)\) denotes the Banach space of all \(E\)-valued little Lipschitz functions on \(X\). We show that \(\lip (X, E)^{**}\) is isometrically isomorphic to Banach space of \(E^{**}\)-valued Lipschitz functions \(\Lip(X, E^{**})\) under several conditions. Moreover, we describe the isometric isomorphism from \(\lip (X, E)^{**}\) to \(\Lip (X, E^{**})\).

Non-universal families of separable Banach spaces

We prove that if $ C $ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $ C $ is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for $ C $ but still not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.

Special symmetries of Banach spaces isomorphic to Hilbert spaces

In this paper Hilbert spaces are characterized among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical here: If X is a real Banach space isomorphic to a Hilbert space and convex-transitive with respect to the isometric finite-dimensional perturbations of the identity, then X is already isometric to a Hilbert space.

Fibred cofinitely-coarse embeddability of box families and proper isometric affine actions on uniformly convex Banach spaces

In this paper we show that a countable, residually amenable group admits a proper isometric affine action on some uniformly convex Banach space if and only if one (or equivalently, all) of its box families admits a fibred cofinitely-coarse embedding into some uniformly convex Banach space.

复域上的巴拿赫等距猜想

Banach's Isometric Conjecture over the Complex Field

该研究完成复域巴拿赫等距猜想证明，还证四元数域对应结论，用Lu和Yang的丛次数机制完成核心几何论证，且将结果推广到多类空间与度量。

Isomorphic and isometric structure of the optimal domains for Hardy-type operators

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…

Isometric embeddings of a class of separable metric spaces into Banach spaces

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $\ell_\infty$.

Genericity of Fréchet smooth spaces

If a separable Banach space contains an isometric copy of every separable reflexive Fréchet smooth Banach space, then it contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach spaces with Fréchet smooth dual space. This improves a result of G. Godefroy and N. J. Kalton.

Almost Fraïssé Banach 空间

Almost Fraïssé Banach spaces

本文定义了更弱的几乎超齐性与几乎 Fraïssé 性质，建立其与伪度量的关系，证明几乎 Fraïssé 空间超幂的超齐性及近似超齐性空间的有穷等距可扩展性。

An isometrically universal Banach space induced by a non-universal Boolean algebra

Given a Boolean algebra $A$, we construct another Boolean algebra $B$ with no uncountable well-ordered chains such that the Banach space of real valued continuous functions $C(K_A)$ embeds isometrically into $C(K_B)$, where $K_A$ and $K_B$ are the Stone spaces of $A$ and $B$ respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density $κ$, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality $κ$. Such a phe…

Operators which factor through Banach lattices not containing c_0

In this supplement to [GJ1], [GJ3], we give an intrinsic characterization of (bounded, linear) operators on Banach lattices which factor through Banach lattices not containing a copy of $c_0$ which complements the characterization of [GJ1], [GJ3] that an operator admits such a factorization if and only if it can be written as the product of two operators neither of which preserves a copy of $c_0$. The intrinsic characterization is that the restriction of the second adjoint of the operator to the ideal generated by the lattice in its bidual does not preserve a copy of $c_0$. This property of an…

巴拿赫等距猜想的一个解

A solution to Banach's isometric conjecture

该研究解决了实巴拿赫空间的巴拿赫等距猜想，通过结合纤维丛拓扑与布劳威尔度理论，证明了所有奇数n的情形，与Gromov的偶数维结果共同完成了该猜想的实情形证明。

Amalgamations of classes of Banach spaces with a monotone basis

It was proved by Argyros and Dodos that, for many classes $ C $ of separable Banach spaces which share some property $ P $, there exists an isomorphically universal space that satisfies $ P $ as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that $ C $ consists of spaces with a monotone Schauder basis. For example, we prove that if $ C $ is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every $ X \in C $ is reflexive and has a monotone Schauder basis, then there exists a…

The spectra of Banach algebras of holomorphic functions on polydisk type domains

R.M. Aron et al. proved that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra $\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelin showed that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$ in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets $U$ of a Banach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk type domains $U$…

Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space $E$ there is a complex Banach lattice $FBL_{\mathbb C}[E]$ containing a linear isometric copy of $E$ and satisfying the following universal property: for every complex Banach lattice $X_{\mathbb C}$, every operator $T:E\rightarrow X_{\mathbb C}$ admits a unique lattice homomorphic extension $\hat{T}:FBL_{\mathbb C}[E]\rightarrow X_{\mathbb C}$ with $\|\hat{T}\|=\|T\|$. The free complex Banach lattice $FBL_{\mathbb C}[E]$ is shown to ha…

Distances between Banach spaces

The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces $X$ and $Y$, the Kadets distance is defined to be the infimum of the Hausdorff distance $d(B_X,B_Y)$ between the respective closed unit balls over all isometric linear embeddings of $X$ and $Y$ into a common Banach space $Z.$ This is compared with the Gromov-Hausdorff distance which is defined to be the infimum of $d(B_X,B_Y)$ over all isometric embeddings into a common metric space $Z$. We prove continuity type results for the Kadets distance including a result that shows that t…

Equivariant geometry of Banach spaces and topological groups

We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, i.e., continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.

A note on universal operators between separable Banach spaces

We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator $Ω$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $Ω$ is generic in the sense of a natural infinite game.

On a characterization of dual Banach spaces through determinant subspaces of norm-attaining linear forms

Necessary and sufficient conditions for Banach space to be(isometrically isomorphic to) a dual space will be given.

关于Banach空间条件基对称类型的注记

Remarks about symmetry-type conditions of conditional bases of Banach spaces

该研究探讨了准Banach空间中条件基的对称性和扩散性，通过重范数构造使排列和递增映射成为等距映射，扩展了经典等距重范数定理的应用范围。

非自反Banach空间中的投影格式与不动点性质

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。

New axiomatizable classes of Banach spaces via disjointness-preserving isometries

Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class C^B of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its "script class" SC, so does C^B with SC^B. This allows us to give new examples of axiomatizable classes of Banach spaces, …

Envelopes in Banach spaces

We define the notion of isometric envelope of a subspace in a Banach space, and relate it to a) the mean ergodic projection on the space of fixed points of a semigroup of contractions, b) results on Korovkin sets from the 70's, and c) extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces $L_p$, $p \notin 2\mathbb N+4$ are the only separable Approximately Ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé B…

关于Banach空间算子的严格等距与严格对称交换$d$元组

On strict isometric and strict symmetric commuting $d$-tuples of Banach space operators

该研究刻画了Banach空间上交换算子$d$元组张量积对的严格$m$-等距、严格$m$-对称性质与原算子对对应性质的关系，给出了张量积严格性质成立时原算子对的分解形式及阶数关系，证明反向仅能推出非严格性质。

The normed and Banach envelopes of Weak L^1

The space Weak L^1 consists of all measurable functions on [0,1] such that q(f) = sup_{c>0} c λ{t : |f(t)| > c} is finite, where λdenotes Lebesgue measure. Let ρbe the gauge functional of the unit ball {f : q(f) \leq 1} of the quasi- norm q, and let N be the null space of ρ. The normed envelope of Weak L^1, which we denote by W, is the space (Weak L^1/N, ρ). The Banach envelope of Weak L^1, \overline{W}, is the completion of W. We show that \overline{W} is isometrically lattice isomorphic to a sublattice of W. It is also shown that all rearrangement invariant Banach function spaces are isometr…

Linear properties of Banach spaces and low distortion embeddings of metric graphs

We characterize non-reflexive Banach spaces by a low-distortion (resp. isometric) embeddability of a certain metric graph up to a renorming. Also we study non-linear sufficient conditions for $\ell_1^n$ being $(1+\varepsilon)$-isomorphic to a subspace of a Banach space $X$.

Zippin's embedding theorem and amalgamations of classes of Banach spaces

It was proved by Dodos and Ferenczi that the classes of Banach spaces with a separable dual and of separable reflexive Banach spaces are strongly bounded. In this note, we provide an isometric version of this result.

作为非交换拟-Banach空间的 $S_q^m$ 到 $S_p^n$ 的等距嵌入可能性

On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach space

本文研究拟-Banach范围内交换与非交换Schatten空间的等距嵌入，利用非交换Clarkson不等式等工具证明多类 $S_q^m$ 到 $S_p^n$ 的嵌入不存在。

On Asymptotic Transitivity in Banach Spaces

We introduce a flexible almost isometric version of the almost transitivity property of Banach spaces. With the help of this new notion we generalize to several directions a strong recent rotational characterization of Hilbert spaces due to Randrianantoanina. This characterization is a partial answer to the classical Banach-Mazur rotation problem.

Kurzweil-Henstock 理论的 Banach-Lamperti 定理

Banach-Lamperti for Kurzweil-Henstock

本文刻画 Kurzweil-Henstock 可积函数空间上的等距同构，证明其对应于双绝对连续的变量替换，建立相应 Banach-Lamperti 型结果。

Isometric dilations and $H^\infty$ calculus for bounded analytic semigroups and Ritt operators

We show that any bounded analytic semigroup on $L^p$ (with $1<p<\infty$) whose negative generator admits a bounded $H^{\infty}$ functional calculus with respect to some angle $< π/2$ can be dilated into a bounded analytic semigroup $(R_t)_{t\geq 0}$ on a bigger $L^p$-space in such a way that $R_t$ is a positive contraction for any $t$. We also establish a discrete analogue for Ritt operators and consider the case when $L^p$-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations o…

Compensated Compactness in Banach Spaces and Weak Rigidity of Isometric Immersions of Manifolds

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div-curl lemma can be employed to establish the global weak rigidity of the Gauss-Codazzi-Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.

A bicommutant theorem for dual Banach algebras

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 isometric, weak$^*$-weak$^*$-continuous homomorphism $π:\mc A\to\mc B(E)$ such that $π(\mc A)$ equals its own bicommutant.