Uniform embeddings, homeomorphisms and quotient maps between Banach spaces (A short survey)

A Characterization of Subspaces and Quotients of Reflexive Banach Spaces with Unconditional Bases

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property has the unconditional tree property. Then we prove that a separable reflexive Banach space with the unconditional tree property embeds into a reflexive Banach space with an unconditional basis. This solves several long standing open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite dimensional decomposition embeds into a reflexive Banach space with an unconditional basis.

Isolated spectral points and Koliha-Drazin invertible elements in quotient Banach algebras and homomorphism ranges

In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.

Asymptotic properties of Banach spaces and coarse quotient maps

We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space $Y$ is a coarse quotient of a subset of a Banach space $X$, where the coarse quotient map is coarse Lipschitz, then the ($β$)-modulus of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants. In particular, if the coarse quotient map is a coarse homeomorphism, then the modulus of asymptotic uniform convexity of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants.

A Banach space dichotomy for quotients of subspaces

A Banach space $X$ with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable (QHI) property if $X/Y$ is hereditarily indecomposable (HI) for any infinite codimensional subspace $Y$ with a successive finite-dimensional decomposition on the basis of $X$. A reflexive space with the restricted QHI property is in particular HI, has HI dual, and is saturated with subspaces which are HI and have HI dual. The following dichotomy theorem is proved: any infinite dimensional Banach space contains a quotient of subspace which either has an unconditional basis, or has the…

A solution of one Problem of Lindenstrauss and Rosenthal on Subspace Homogeneous and Quotient Homogeneous Banach Spaces with Application to the Approximation Problem and to the Schroeder - Bernstein Problem

In article is constructed a wide couple of pairwice non-isomorphic separable superreflexive Banach spaces E that are subspace homogeneous. Their conjugates are quotient homogeneous. None of this couple neither isomorphic to its Cartesian square nor has the approximation property. At the same time, any such E is isomorphic to E+ E+ W for some Banach space W and, hence, solves the Schroeder - Bernstein problem.

可分商问题与Banach空间中的Lotz性质

The Separable Quotient Problem and the Lotz Property in Banach Spaces

本文为可分商问题建立新条件，并证明缺乏Lotz性质的无限维Banach空间存在可分商，揭示SQP与半群理论的联系。

On the separable quotient problem for Banach spaces

While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of a particular infinitedimensional separable quotient in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the results presented are consequence of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c_0 and l_p, for 1 <= p <= \infty. Also recent results of Argyros - Dodos - Kanellopoulos, and Sliwa are provided. This makes our presentation supplementary to a previo…

Quotients of continuous convex functions on nonreflexive Banach spaces

On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction gives also a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.

一个具有无限维自反商代数 L(X)/SS(X) 的 Banach 空间

A Banach space with an infinite dimensional reflexive quotient algebra L(X)/SS(X)

构造 Gowers-Maurey 型 Banach 空间 X，使有界算子代数 L(X) 为无限维自反空间与严格奇异算子理想的直和，从而商代数 L(X)/SS(X) 为无限维自反空间。

Interpolating hereditarily indecomposable Banach spaces

It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(λ)$, $1<p<\infty $, is a quotient of a H.I Banach space.

Quotients of Banach spaces with the Daugavet property

We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative.

On the Bounded Approximation Property in Banach spaces

We prove that the kernel of a quotient operator from an $\mathcal L_1$-space onto a Banach space $X$ with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky --case $\ell_1$-- and Figiel, Johnson and Pełczyński --case $X^*$ separable. Given a Banach space $X$, we show that if the kernel of a quotient map from some $\mathcal L_1$-space onto $X$ has the BAP then every kernel of every quotient map from any $\mathcal L_1$-space onto $X$ has the BAP. The dual result for $\mathcal L_\infty$-spaces also hold: if for some $\mathcal L_\infty$-space $E$ some quo…

On the category of quotient Banach spaces after Wegner

We study Waelbroeck's category of Banach quotients after Wegner, focusing on its basic homological and functional analytic properties.

Nonlinear quotients

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces $X$, $Y$ for which any Lipschitz function from $X$ to $Y$ can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient mappings between Banach spaces. It is proved, in particular, that any Banach space which is a uniform quotient of $L_p$, $1<p<\infty$, is already isomorphic to a linear quotient of $L_p$.

Total subspaces in dual Banach spaces which are not norming

The main result: the dual of separable Banach space $X$ contains a total subspace which is not norming over any infinite dimensional subspace of $X$ if and only if $X$ has a nonquasireflexive quotient space with the strictly singular quotient mapping.

The cofinal property of the Reflexive Indecomposable Banach spaces

It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably $\ell_p$ saturated space with $1<p<\infty$ and of a $c_0$ saturated space.

n-赋范线性空间中的最佳逼近与 Banach 代数研究

A study on Best Approximation and Banach Algebra in n-normed linear space

本文研究 n-赋范空间中的最佳逼近、唯一性及 n-Hilbert 情形，并建立 n-Banach 代数，讨论可逆元、拓扑零因子与复同态，证明 Gleason–Kahane–Zelazko 型定理。

Ando-Choi-Effros liftings for regular maps between Banach lattices

The Ando-Choi-Effros lifting theorem provides conditions under which a bounded linear mapping taking values in a quotient space can be lifted through the quotient map. We prove two versions of said theorem for regular maps between Banach lattices. Our conditions mirror the classical ones, but additionally taking into account the order structure.

具有逼近性质失败的巴拿赫空间上的近似可算算子商代数

Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

本文研究了具有逼近性质失败的巴拿赫空间上紧-近似可算算子商代数的结构性质，并给出了相关嵌入结果。

可分Banach空间的对偶作为Calkin代数与泛型理想商

Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients

本文通过oracle相对的双类型有限扩张构造，证明每个可分Banach空间的对偶可实现为可分Banach空间的Calkin代数，且存在泛型理想商覆盖所有可分空间。

The Existence of Linear Selection and the Quotient Lifting Property

Lifting properties for Banach spaces are studied. An alternate version of the lifting property due to Lindenstrass and Tzafriri is proposed and a characterization, up to isomorphism, is given. The quotient lifting property for pairs of Banach spaces $(X,J)$, with $J$ proximinal in $X$, is considered and several conditions for the property to hold are given.

A topological group observation on the Banach--Mazur separable quotient problem

The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fréchet space has an infinite-dimensional separable quotient space, namely $\mathbb{R}^ω$. It is proved in this paper that every infinite-dimensional Fréchet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace whic…

Banach空间插值尺度的端点

Extremes of interpolation scales of Banach spaces

本文将 Daher 关于复插值尺度内部球面一致同胚的结果推广到端点情形，并给出 Banach 空间球面与 $\ell_2$ 球面一致同胚的应用。

Quotients of Banach algebras acting on $L^p$-spaces

We show that the class of Banach algebras that can be isometrically represented on an $L^p$-space, for $p\neq 2$, is not closed under quotients. This answers a question asked by Le Merdy 20 years ago. Our methods are heavily reliant on our earlier study of Banach algebras generated by invertible isometries of $L^p$-spaces.

Automatic norm continuity of weak* homeomorphisms

We prove that in a certain class E of nonseparable Banach spaces the norm topology of the dual ball is definable in terms of its weak* topology. Thus, any weak* homeomorphism between duals balls of spaces in E is automatically norm-continuous.

The quotient shapes of normed spaces and application

The quotient shape types of normed vectorial spaces(over the same field) with respect to Banach spaces reduce to those of Banach spaces. The finite quotient shape type of normed spaces is an invariant of the (algebraic) dimension, but not conversely. The converse holds for separable normed spaces as well as for the bidual-like spaces (isomorphic to their second dual spaces). As a consequence, the Hilbert space $l_2$, or even its (countably dimensional, unitary) direct sum subspace may represent the unique quotient shape type of all $2^\aleph_0$-dimensional normed spaces. An application yields …

Banach space properties forcing a reflexive amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If $A$ is a reflexive, amenable Banach algebra such that for each maximal left ideal $L$ of $A$ (i) the quotient $A / L$ has the approximation property and (ii) the canonical map from $A \check{\otimes} L^\perp$ to $(A / L) \wtensor L^\perp$ is open, then $A$ is finite-dimensional. As an application, we show that, if …

New examples of $c_0$-saturated Banach spaces II

For every Banach space $Z$ with a shrinking unconditional basis satisfying upper $p$-estimates for some $p > 1$, an isomorphically polyhedral Banach space is constructed having an unconditional basis and admitting a quotient isomorphic to $Z$.

Asymptotic geometry of Banach spaces and uniform quotient maps

Recently, Lima and Randrianarivony pointed out the role of the property $(β)$ of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of $(β)$ of the domain space. We also provide conditions under which this comparison can be improved.

On $c_0$-saturated Banach spaces

A Banach space E is c_0-saturated if every closed infinite dimensional subspace of E contains an isomorph of c_0. A c_0-saturated Banach space with an unconditional basis which has a quotient space isomorphic to l^2 is constructed.

Quotient algebras of Banach operator ideals related to non-classical approximation properties

We investigate the quotient algebra $\mathfrak{A}_X^{\mathcal I}:=\mathcal I(X)/\overline{\mathcal F(X)}^{||\cdot||_{\mathcal I}}$ for Banach operator ideals $\mathcal I$ contained in the ideal of the compact operators, where $X$ is a Banach space that fails the $\mathcal I$-approximation property. The main results concern the nilpotent quotient algebras $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for the quasi $p$-nuclear operators $\mathcal{QN}_p$ and the Sinha-Karn $p$-compact operators $\mathcal{SK}_p$. The results include the following: (i) if $X$ has cotype 2, t…

Real hereditarily indecomposable Banach spaces and uniqueness of complex structure

There exists a real hereditarily indecomposable Banach space $X$ such that the quotient space $L(X)/S(X)$ by strictly singular operators is isomorphic to the complex field (resp. to the quaternionic division algebra). Up to isomorphism, the example with complex quotient space has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable. So there exist two Banach spaces which are isometric as real spaces but totally incomparable as complex spaces. This extends results of J. Bourgain and S. Szarek from 1986. The quaternionic example, on the …

Convex-transitivity of Banach algebras via ideals

We investigate a method for producing concrete convex-transitive Banach spaces. The gist of the method is in getting rid of dissymmetries of a given space by taking a carefully chosen quotient. The spaces of interest here are typically Banach algebras and their ideals. We also investigate the convex-transitivity of ultraproducts and tensor products of Banach spaces.

On Banach Spaces of Large Density Character and Unconditional Sequences

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 cardinality that is equal to the density of X

New examples of $c_0$-saturated Banach spaces

For every $ 1 < p < \infty $ an isomorphically polyhedral Banach space $E_p$ is constructed having an unconditional basis and admitting a quotient isomorphic to $\ell_p$. It is also shown that $E_p$ is not isomorphic to a subspace of a $C(K)$ space for every countable and compact metric space $K$.

Perturbed Operators on Banach Spaces

Let X be a Banach Space over K=R or C, and let f:=F+C be a weakly coercive operator from X onto X, where F is a C^1-operator, and C a C^1 compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C^1-diffeomorphism. Three corollaries are given. The first one, when F is a linear homeomorphism. The second one, when F is a k-contractive perturbation of the identity. The third one, when X is a Hilbert space and F a particular linear operator. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping…

Banach-Stone-like results for combinatorial Banach spaces

We show that under a certain topological assumption on two compact hereditary families $\F$ and $\G$ on some infinite cardinal $κ$, the corresponding combinatorial spaces $X_\F$ and $X_\G$ are isometric if and only if there is a permutation of $κ$ inducing a homeomorphism between $\F$ and $\G$. We also prove that two different regular families $\F$ and $\G$ on $ω$ cannot be permuted one to the other. Both these results strengthen the main result of \cite{BrechFerencziTcaciuc}.

Actions affines isométriques propres des groupes hyperboliques sur des quotients d'espaces $\ell^p$

We prove that any hyperbolic group admits a proper affine isometric action on a quotient space of a $\ell^p$ Banach space, for all $p>1$ sufficiently close to 1.

酉不变理想球面之间的均匀同胚

Uniform homeomorphisms between spheres of unitarily invariant ideals

本文证明非交换版本的Odell-Schlumprecht定理，通过两类广义Mazur映射（基于p-凸化和量子相对熵最小化）建立酉不变理想球面与Hilbert空间球面间的均匀同胚，提供满足性质(H)的新例子。

Smooth extension of functions on a certain class of non-separable Banach spaces

Let us consider a Banach space $X$ with the property that every real-valued Lipschitz function $f$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (with $C$ depending only on the space $X$). This is the case for a Banach space $X$ bi-Lipschitz homeomorphic to a subset of $c_0(Γ)$, for some set $Γ$, such that the coordinate functions of the homeomorphism are $C^1$-smooth. Then, we prove that for every closed subspace $Y\subset X$ and every $C^1$-smooth (Lipschitz) function $f:Y\to\Real$, there is a $C^1$-smooth (Lipschitz, respectively) extens…

与拓扑动力系统相关的对合巴拿赫代数的非交换谱综合

Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

本文研究与拓扑动力系统关联的交叉积对合巴拿赫代数的谱综合问题，证明在两种模型下谱综合成立当且仅当同胚无周期点。

巴拿赫空间上有界算子代数的商空间的代数范数唯一性

Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

本研究证明三类巴拿赫空间上有界算子代数的商空间对所有闭理想具有唯一代数范数，等价于到巴拿赫代数的同态均连续且有闭值域，核心是恒等算子的可控范数定量分解结论，具独立研究意义。

A Bourgain-Pisier construction for general Banach spaces

We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $\mathcal L_{\infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain $\mathcal L_\infty$ spaces of arbitrary large densities with the Schur and the Radon-Nikodym properties. This extents the a classical result by J. Bourgain and G. Pisier.

Property $(β)$ and uniform quotient maps

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of $\ell_p$, $1 < p \neq 2 < \infty$, must be isomorphic to a linear quotient of $\ell_p$. We apply the geometric property $(β)$ of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case $1<p<2$. We also give a necessary condition for a Banach space to have $c_0$ as a uniform quotient.

Nonlinear Operator Ideals Between Metric Spaces and Banach Spaces (PART I)

In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear versions of operator ideals. We introduce several examples of nonlinear ideals and the relationships between them. For every space ideal $\mathsf{A}$ can be generated by a special nonlinear ideal which consists of those Lipschitz operators admitting a factorization through a Banach space $\mathbf{M}\in\mathsf{A}$. We investigate products and quotients of nonlinea…

Vector-valued Sobolev spaces based on Banach function spaces

It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.

论不可分解Banach空间的子空间

On Subspaces of Indecomposable Banach Spaces

本文刻画了可嵌入不可分解Banach空间的Banach空间类，证明其包含所有密度不超过连续统且不以ℓ∞为商空间的Banach空间，并改进构造方法以涵盖部分以ℓ∞为商空间的情形，为Argyros–Haydon提出的问题提供新见解。

A family of quotient maps of $\ell^\infty$ that do not admit uniformly continuous right inverses

Previously only two examples of Banach space quotient maps which do not admit uniformly continuous right inverses were known: one due to Aharoni and Lindenstrauss and one due to Kalton ($\ell^\infty\to\ell^\infty/c_{0}$). We show through an application of Kalton's Monotone Transfinite Sequence Theorem that a quotient map of a subspace of $\ell^\infty$ of sequences that converge to zero along an ideal in $\mathbb{N}$ toward another such subspace, provided one of the ideals is `much larger' than the other, cannot have a uniformly continuous right inverse. We show in general that pairs of ideals …

Proximity to $\ell_p$ and $c_0$ in Banach spaces

We construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain $\ell_p$ and $c_0$ indices, dualization of the Bourgain $c_0$ index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain $\ell_p$ index of an arbitrary Banach space $X$ in terms of a subspace $Y$ and the quotient $X/Y$.

Banach 格的自由积

Free Products of Banach Lattices

本文在 Banach 格与压缩格同态范畴中研究自由积（即余积），通过自由 Banach 格的商构造给出任意族 Banach 格自由积的具体实现，并证明其基本结构性质，同时建立子格嵌入与投射 Banach 格的稳定性结果，分析商映射行为，对于紧 Hausdorff 空间 K1 和 K2 将 C(K1)∗C(K2) 与 C(K1∗K2) 格同构，其中 K1∗K2 表示拓扑联合，并推导该表示中自由积范数的显式公式，进一步讨论自由 Banach 格的自由因子，利用非平凡同调球的存在性说明自由 Banach 格可以具有不同构于自由 Banach 格的自由因子。

Hyperplane conjecture for quotient spaces of $L_p$

We give a positive solution for the hyperplane conjecture of quotient spaces F of $L_p$, where $1<p\kll\infty$. \[ vol(B_F)^{\frac{n-1}{n}} \kl c_0 \pl p' \pl \sup_{H \p hyperplane} vol(B_F\cap H) \pl.\] This result is extended to Banach lattices which does not contain $\ell_1^n$'s uniformly. Our main tools are tensor products and minimal volume ratio with respect to $L_p$-sections.

A new metric invariant for Banach spaces

We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $ω$ or if the Szlenk index of its dual is larger than $ω$, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.

The transfer of property $(β)$ of Rolewicz by a uniform quotient map

We provide a Laakso construction to prove that the property of having an equivalent norm with the property $(β)$ of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces. On the other hand, we show that the $(β)$-modulus is not quantitatively preserved via such a map by exhibiting two uniformly homeomorphic Banach spaces that do not have $(β)$-moduli of the same power-type even under renorming.

The complexity of classifying separable Banach spaces up to isomorphism

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomo…