Complemented subspaces of products of Banach spaces

$(1+)$-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 $\varepsilon>0$, $…

1-complemented subspaces of Banach spaces of universal disposition

We first unify all notions of partial injectivity appearing in the literature ---(universal) separable injectivity, (universal) $\aleph$-injectivity --- in the notion of $(α, β)$-injectivity ($(α, β)_λ$-injectivity if the parameter $λ$ has to be specified). Then, extend the notion of space of universal disposition to space of universal $(α, β)$-disposition. Finally, we characterize the $1$-complemented subspaces of spaces of universal $(α, β)$-disposition as precisely the spaces $(α, β)_1$-injective.

关于X_p的可补子空间

On the complemented subspaces of X_p

本文证明了与$X_{p}$的补子空间同构分类问题有关的一些结果。本文给出了$X_{p}$中与$X_{p}$同构的可补子空间的特征，证明了这类空间中必有一个与$X_{p}.$同构的标准可补子空间，并给出了与$\ell_{p}\oplus \ell_{2}.$同构的可补子空间的一些特征

When is the sum of complemented subspaces complemented?

We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented. Under this condition a formula for a projection onto the sum is given. We also show that the condition is sharp (in a certain sense). As applications, we get (1) sufficient conditions for the complementability of sums of marginal subspaces in $L^p$ and sums of tensor powers of subspaces in a tensor power of a Banach space and (2) quantitative results on stability of the complementability property of the sum of linearly independent subspaces.

A Survey on the Complemented Subspace Problem

The complemented subspace problem asks, in general, which closed subspaces $M$ of a Banach space $X$ are complemented; i.e. there exists a closed subspace $N$ of $X$ such that $X=M\oplus N$? This problem is in the heart of the theory of Banach spaces and plays a key role in the development of the Banach space theory. Our aim is to investigate some results on complemented subspaces, to give a history of the subject, and to present some open problems.

One-complemented subspaces of real sequence spaces

Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either Luxemburg or Orlicz norm.

Complemented subspaces of spaces obtained by interpolation

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A_0,A_1) such that A_0 and A_1 are isometric to $X\oplus V$, and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces.

Classifying Complemented Subspaces of $L_p$ with Alspach Norm

Understanding the complemented subspaces of $L_p$ has been an interesting topic of research in Banach space theory since 1960. 1999, Alspach proposed a systematic approach to classifying the subspaces of $L_p$ by introducing a norm given by partitions and weights. This paper shows that with Alspach Norm we are able to classify some complemented subspaces of $L_p, 2<p<\infty$.

On the complemented subspaces of the Schreier spaces

It is shown that the Schreier space X admits a set of continuum cardinality whose elements are mutually incomparable complemented subspaces spanned by subsequences of the natural Schauder basis of X.

向量值测度空间中$\ell_1$的可补拷贝及其应用

Complemented copies of $\ell_1$ in spaces of vector valued measures and applications

设$X$是巴拿赫空间，$(Ω,Σ)$是测度空间.给出了序列在$X$ -值可数可加测度空间中的一个刻画，该空间是有界变差的$Ω,Σ)$上生成$\ell_1$的可补拷贝的序列.作为应用，我们证明了：如果对偶巴拿赫空间$E^*$具有Petroczynski性质（V*），则$E^*$ -值有界变差可数可加测度空间也具有Petroczynski性质（V *）.另一个应用，我们证明了对于巴拿赫空间$X$，空间$\ell_\infty(X)$包含$\ell_1$的补拷贝当且仅当$X$包含所有$\ell_1^n$一致补的.

A new isomorphic (\ell_1) predual not isomorphic to a complemented subspace of a (C(K)) space

An isomorphic (\ell_1)-predual space (X) is constructed such that neither (X) is isomorphic to a subspace of (c_0), nor (C(ω^ω)) is isomorphic to a subspace of (X). It follows that (X) is not isomorphic to a complemented subspace of a (C(K)) space.

Banach空间$C(K\times L)$的补子空间

Complemented subspaces of Banach spaces $C(K\times L)$

该研究探讨紧空间上连续函数构成的Banach空间的补子空间问题，证明了紧群相关的补子空间等距结论，回答了Alspach与Galego的相关问题，并给出点态拓扑下的反例结果。

On the complexity of some classes of Banach spaces and non-universality

These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded oper- ators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounde…

The subspace $c_0$ is not complemented in $ac_0$

We prove that the subspace $c_0$ of sequences that converge to zero is not complemented in the space $ac_0$ of sequences that almost converge to zero. We proceed with applying the same approach to inclusion chain $c_0\subset A_0 \subset \ell_\infty$.

On ultrapowers of Banach spaces of type $\mathscr L_\infty$

We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain $c_0$ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs had been infected by that statement. In particular we provide proofs for the following statements: (i) All $M$-spaces, in particular all $C(K)$-spaces, have ultrapowers isomorphic to ultrapowers of $c_0$, as well as all their complemented subspaces isomorphic to their square. (ii) No ultrapower of the Gurari\u ı space can be complemented in any $M$-spac…

由Candido构造的$C(K)$-空间的补子空间

Complemented subspaces of a $C(K)$-space constructed by Candido

本文研究了由Candido构造的非平凡局部紧致Hausdorff空间$L$所对应的$C_0(L \times L)$空间的补子空间分类问题。

$C(K)$空间的补子空间问题：一个反例

The complemented subspace problem for $C(K)$-spaces: A counterexample

本研究针对$C(K)$空间的补子空间问题构造反例，证明存在紧空间$L$使得$C(L)$有不与连续函数空间同构的1-补子空间。

Ordinal indices for complemented subspaces of l_p

We provide complete isomorphic invariance of a class of translation invariant complemented subspaces of L_p constructed by Bourgain, Rosenthal and Schechtman. We compute ordinal L_p-indices for this class. We further show that the isometric index of a tree subspace over a well founded tree is an invariance for the order of the tree. Finally we provide a dichotomy for the subspaces of L_p with small ordinal indices.

无条件基巴拿赫空间的补子空间问题的否定解

A negative solution to the complemented subspace problem for Banach spaces with unconditional bases

本文构造了可分超自反空间上的投影，得到具有 Schauder 基但无无条件基的补子空间，否定相关猜想并给出 Lean 4 形式化证明。

Compactifications of $ω$ and the Banach space $c_0$

We investigate for which compactifications $γω$ of the discrete space of natural numbers $ω$, the natural copy of the Banach space $c_0$ is complemented in $C(γω)$. We show, in particular, that the separability of the remainder of $γω$ is neither sufficient nor necessary for $c_0$ being complemented in $C(γω)$ (for the latter our result is proved under the continuum hypothesis). We analyse, in this context, compactifications of $ω$ related to embeddings of the measure algebra into $P(ω)/fin$. We also prove that a Banach space $C(K)$ contains a rich family of complemented copies of $c_0$ whenev…

Uniqueness of unconditional bases in Banach spaces

We prove a general result on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have the unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis.

On isometric stability of complemented subspaces of ${L_p}$

We show that Rudin-Plotkin isometry extension theorem in $L_p$ implies that when $X$ and $Y$ are isometric subspaces of $L_p$ and $p$ is not an even integer, $1 \leq p < \infty$, then $X$ is complemented in $L_p$ if and only if $Y$ is; moreover the constants of complementation of $X$ and $Y$ are equal. We provide examples demonstrating that this fact fails when $p$ is an even integer larger than 2.

Complemented subspaces of homogeneous polynomials

Let $\mathcal{P}_{K} (^{n}E; F)$ (resp. $\mathcal{P}_{w} (^{n}E; F)$) the subspace of all $P\in \mathcal{P}(^{n}E; F)$ which are compact (resp. weakly continuous on bounded sets). We show that if $\mathcal{P}_{K} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{K} (^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$. Likewise we show that if $\mathcal{P}_{w} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{w}(^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$.

Complemented Subspaces of L_p Determined by Partitions and Weights

Many of the known complemented subspaces of L_p have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well-known complemented subspaces of L_p. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of L_p. Using this we define a space Y_n with norm…

1-complemented subspaces of spaces with 1-unconditional bases

We prove that if $X$ is a complex strictly monotone sequence space with $1$-unconditional basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or …

Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support

Let $λ$ be an infinite cardinal number and let $\ell_\infty^c(λ)$ denote the subspace of $\ell_\infty(λ)$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite dimensional complemented subspaces of $\ell_\infty^c(λ)$, proving that they are isomorphic to $\ell_\infty^c(κ)$ for some cardinal number $κ$. Then we show that the Banach algebra of all bounded linear operators on $\ell_\infty^c(λ)$ or $\ell_\infty(λ)$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we …

关于 $L_1[0,1]$ 的补子空间

On complemented subspaces of $L_1[0,1]$

本文构造了L_1[0,1]的两个补子空间，分别具有Schur性质但不具Radon-Nikodým性质，以及包含ℓ_2副本但不含L_1副本，均不同构于Banach格，否定了Lindenstrauss和Rosenthal的补子空间问题，并附Lean 4形式化证明。

Projecting Lipschitz functions onto spaces of polynomials

The Banach space $\mathcal{P}({}^2X)$ of $2$-homogeneous polynomials on the Banach space $X$ can be naturally embedded in the Banach space ${{\rm Lip}_0}(B_X)$ of real-valued Lipschitz functions on $B_X$ that vanish at $0$. We investigate whether $\mathcal{P}({}^2X)$ is a complemented subspace of ${{\rm Lip}_0}(B_X)$. This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that $\mathcal{P}({}^1X)=X^*$ is complemented in ${{\rm Lip}_0}(B_X)$ for every Banach space $X$. Our main result asserts that $\mathcal{P}({}^2X)$ is not c…

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 …

Range-Kernel Complementation

If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a reflexive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint

On representing and absolutely representing systems of subspaces in Banach spaces

We study properties of representing and absolutely representing systems of subspaces in Banach spaces. We also present sufficient conditions for the system of subspaces to be a representing system of subspaces.

Relations and radicals in abstract lattices and in lattices of subspaces of Banach spaces and ideals of Banach algebras. Amitsur's theory revisited

We refine Amitsur's theory of radicals in complete lattices and apply the obtained results to the theory of radicals in the lattices of subspaces of Banach spaces and in the lattices of ideals of Banach and C*-algebras and of Banach Lie algebras.

不含Banach子空间的引理

Quojections without Banach subspaces

构造了一个不含无穷维Banach子空间的双射（含满射连通映射的巴拿赫空间的投射极限）.这一结果回答了G.Metafune和V.B.Moscatelli提出的一个问题。

巴拿赫空间全体子空间集上的拓扑与巴拿赫空间几何的有关问题

Topologies on the set of all subspaces of a banach space and related questions of banach space geometry

对于一个巴拿赫空间$X$，我们用$G(X)$表示$X$的所有闭子空间的集合。在某些类型的问题，它被证明是有用的赋予$G(X)$的拓扑结构。本论文的主要目的是调查结果上的两个最常见的拓扑$G(X)$。

$L_{1/2}$的一个不嵌入$L_1$（等距型）的Banach子空间

A Banach subspace of $L_{1/2}$ which does not embed in $L_1$ (isometric version)

对于每个$n\geq 3,$，我们构造了一个$n$维Banach空间，它与$L_{1/2}$的子空间等距，但与$L_1.$的子空间不等距。1969年（昭和10年），仍在营业。另一个例子给出了$L_{1/4}$的一个Banach子空间，它不等距地嵌入$L_{1/2}.$中。注意，从同构的角度来看，所有具有$q<1$的空间$L_q$都有相同的Banach子空间。

On absolutely representing families of subspaces in Banach spaces

An absolutely representing family of subspaces is a natural generalization of an absolutely representing system of subspaces and absolutely representing system (of elements). We obtain necessary an (or) sufficient conditions for a family of subspaces to be an absolutely representing family of subspaces and study properties of absolutely representing families of subspaces in Banach spaces. As an example, we study families of subspaces spanned by exponents.

Remarks on rich subspaces of Banach spaces

We investigate rich subspaces of $L_1$ and deduce an interpolation property of Sidon sets. We also present examples of rich separable subspaces of nonseparable Banach spaces and we study the Daugavet property of tensor products.

具有不可补等距子空间的自反 Banach 空间

An example of a reflexive Banach space with an uncomplemented isometric subspace

本文构造了一个自反 Banach 空间，其中存在与全空间等距同构却不可补的子空间。

Banach spaces without minimal subspaces - examples

We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in the previous paper Banach spaces without minimal subspaces, by classifying them according to which side of the dichotomies they fall. This paper may be seen as a more empirical continuation of Banach spaces without minimal subspaces, in which our stress is on the study of examples for the new classes of Banach spaces considered in that work.

从最佳共逼近视角看Banach空间的若干特殊子空间

On some special subspaces of a Banach space, from the perspective of best coapproximation

该研究运用Birkhoff-James正交性技术研究Banach空间最佳共逼近问题，引入两类特殊子空间，推导其在不同Banach空间中的判定条件并刻画相关几何结构。

对偶巴拿赫空间中不赋范的全子空间

Total subspaces in dual Banach spaces which are not norming

主要结果：可分巴拿赫空间$X$的对偶包含一个在$X$的任意无穷维子空间上不赋范的全子空间当且仅当$X$存在一个具有严格奇异商映射的非拟自反商空间.

Nice Banach Modules and Invariant Subspaces

Let $\A$ be a semisimple unital commutative Banach algebra. We say that a Banach $\A$-module $M$ is nice if every proper closed submodule of $M$ is contained in a closed submodule of $M$ of codimension 1. We provide examples of nice and non-nice modules.

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.

Left introverted subspaces of duals of Banach algebras and $WEAK^*-$continuous derivations on dual Banach algebras

Let $X$ be a left introverted subspace of dual of a Banach algebra. We study $Z_t(X^*),$ the topological center of Banach algebra $X^*$. We fined the topological center of $(X\cA)^*$, when $\cA$ has a bounded right approximate identity and $\cA\subseteq X^*.$ So we introduce a new notation of amenability for a dual Banach algebra $\cal A$. A dual Banach algebra $\cal A$ is weakly Connes-amenable if the first $weak^*-$continuous cohomology group of $\cal A$ with coefficients in $\cal A$ is zero; i.e., $H^1_{w^*}(\cal A, \cal A)=\{o\}$. We study the weak Connes-amenability of some dual Banach al…

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Given a Banach space $E$, we ask which closed subspaces may be realised as the kernel of a bounded operator $E \rightarrow E$. We prove some positive results which imply in particular that when $E$ is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space $E$ which contains a closed subspace that cannot be realized as the kernel of any bounded operator on $E$. This implies that the Banach algebra $\mathcal{B}(E)$ of bounded operators on $E$ fails to be weak*-topologically left Noetherian. The Banach space $E$ that we use is the dual of Wark's non-separa…

Banach spaces of polynomials as "large" subspaces of $\mathbf{\ell^\infty}$-spaces

In this note we study Banach spaces of traces of real polynomials on $\mathbb R^n$ to compact subsets equipped with supremum norms from the point of view of Geometric Functional Analysis.

巴拿赫格正张量积中$c_{0}$的补副本

Complemented Copies of $c_{0}$ in Positive Tensor Products of Banach Lattices

该研究针对巴拿赫格张量积，建立了Cembranos定理的正类似结论，给出Wittstock张量积含$c_0$补副本、不具备正Grothendieck性质的条件，推进了巴拿赫格正张量积结构的研究。

遗传不可分解巴拿赫空间子空间上的算子

Operators on subspaces of hereditarily indecomposable Banach spaces

一个空间$X$被称为遗传不可分解的，如果没有两个（无限维）$X$的子空间是直和的。本文证明了：如果$X$是复遗传不可分解的巴拿赫空间，则从$X$的子空间$Y$到$X$的每个算子都具有$λI + S$的形式，其中$I$是包含映射，$S$是严格奇异的.

On contractively complemented subspaces of separable L_1-preduals

Let X be an L_1-predual space and let K be a countable linearly independent subset of the extreme points of its closed dual ball. It is shown that if the norm-closed linear span Y of K is w^*-closed in X^*, then Y is the range of a w^*-continuous contractive projection in X^*. This result is applied in order to provide new and simpler proofs of the results of Lazar, Lindenstrauss and Zippin on the embedding of C(K) spaces into X.

A note on closedness of the sum of two closed subspaces in a Banach space

Let $X$ be a Banach space, and $M,N$ be two closed subspaces of $X$. We present several necessary and sufficient conditions for the closedness of $M+N$ ($M+N$ is not necessarily direct sum).

Subspaces with a common complement in a Banach space

We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution S on X exchanging the two subspaces, such that I+S is bounded from below on their union. Moreover we…

K(X,Y) as subspace complemented of L(X,Y)

Let X,Y be two Banach spaces ; in the first part of this work, we show that K(X,Y) contains a complemented copy of c0 if Y contains a copy of c0 and each bounded sequence in Y has a subsequece which is w* convergente. Afterward we obtain some results of M.Feder and G.Emmanuele: Finally in this part we study the relation between the existence of projection from L(X,Y) on K(X,Y) and the existence of pro- jection from K(X,Y ) on K(X,Y) if Y has the approximation property. In the second part we study the Radon-Nikodym property in L(X,Y):

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…

Spectrum of Weighted Composition Operators. Part II. Weighted composition operators on subspaces of Banach lattices

We describe the spectrum of weighted $d$-isomorphisms of Banach lattices restricted on closed subspaces that are "rich" enough to preserve some "memory" of the order structure of the original lattice. The examples include (but are not limited to) weighted isometries of Hardy spaces on the polydisk and unit ball in $\mathds{C}^n$.