Banach Space Theory Research Articles

Asymptotic Insights for Projection, Gordon–Lewis, and Sidon Constants in Boolean Cube Function Spaces

Abstract The main aim of this work is to study important local Banach space constants for Boolean cube function spaces. Specifically, we focus on $\mathcal{B}_{\mathcal{S}}^{N}$, the finite-dimensional Banach space of all real-valued functions defined on the $N$-dimensional Boolean cube $\{-1, +1\}^{N}$ that have Fourier–Walsh expansions supported on a fixed family $\mathcal{S}$ of subsets of $\{1, \ldots , N\}$. Our investigation centers on the projection, Sidon, and Gordon–Lewis constants of this function space. We combine tools from different areas to derive exact formulas and asymptotic estimates of these parameters for special types of families $\mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity characteristics of the support set $\mathcal{S}$. Using local Banach space theory, we establish the intimate relationship among these three important constants.

The Josefson–Nissenzweig theorem and filters on ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}

For a free filter F on ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}, endow the space NF=ω∪{pF}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F=\\omega \\cup \\{p_F\\}$$\\end{document}, where pF∉ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_F\ ot \\in \\omega $$\\end{document}, with the topology in which every element of ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is isolated whereas all open neighborhoods of pF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_F$$\\end{document} are of the form A∪{pF}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\cup \\{p_F\\}$$\\end{document} for A∈F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in F$$\\end{document}. Spaces of the form NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} carries a sequence ⟨μn:n∈ω⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\mu _n:n\\in \\omega \\rangle $$\\end{document} of normalized finitely supported signed measures such that μn(f)→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _n(f)\\rightarrow 0$$\\end{document} for every bounded continuous real-valued function f on NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} if and only if F∗≤KZ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^*\\le _K{\\mathcal {Z}}$$\\end{document}, that is, the dual ideal F∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^*$$\\end{document} is Katětov below the asymptotic density ideal Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {Z}}$$\\end{document}. Consequently, we get that if F∗≤KZ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^*\\le _K{\\mathcal {Z}}$$\\end{document}, then: (1) if X is a Tychonoff space and NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} is homeomorphic to a subspace of X, then the space Cp∗(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_p^*(X)$$\\end{document} of bounded continuous real-valued functions on X contains a complemented copy of the space c0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0$$\\end{document} endowed with the pointwise topology, (2) if K is a compact Hausdorff space and NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.

A novel analytical approach for solving partial differential equations via a tensor product theory of Banach spaces

In this paper, we aim to propose a novel analytical method for obtaining atomic solutions to partial differential equations. A theory of tensor products in Banach spaces is utilized and some properties of atom operators are studied to achieve our goals. The proposed scheme is demonstrated through illustrative examples of inhomogeneous partial differential equations. The approach is indeed quite direct and effective in exactly solving partial differential equations.

Wasserstein distance and metric trees

We study the Wasserstein (or earthmover) metric on the space P(X) of probability measures on a metric space X . We show that, if a finite metric space X embeds stochastically with distortion D in a family of finite metric trees, then P(X) embeds bi-Lipschitz into \ell^1 with distortion D . Next, we re-visit the closed formula for the Wasserstein metric on finite metric trees due to Evans–Matsen (2012).We advocate that the right framework for this formula is real trees, and we give two proofs of extensions of this formula: one making the link with Lipschitz-free spaces from Banach space theory, the other one algorithmic (after reduction to finite metric trees).

Atomic Solutions of Partial Differential Equations via Tensor Product Theory of Banach Spaces

This work purposes to establish a novel analytical method to obtain the atomic solutions of partial differential equations. This will be carried out by employing the tensor product theory in Banach spaces coupled with some properties of atoms operators. Some illustrative examples will be provided to validate our findings.

Approximation 性质 and operator dilation theory of Banach spaces and lattices

本文介绍近期发展的Banach空间扩张理论的主要结果, 在此基础上利用Banach空间有界逼近性质与Schauder框架的等价关系, 研究了有有界逼近性质的Banach空间上算子值测度的扩张和有限秩算子的扩张, 并得到算子空间上完全有界算子值测度的扩张结果. 最后, 我们还考虑了具有Schauder框架的Banach格上的算子值测度的扩张, 主要关注其基本扩张空间的偏序结构, 以及扩张中各映射对偏序结构的保持性质.

Transversality versus strong tangential transversality

The notions of transversality and strong tangential transversality are compared – it is proved that strong tangential transversality implies transversality and the reverse implication fails. Further, it is obtained that the two notions coincide for convex sets and some applications to Banach space theory are indicated. A sum rule for Clarke subdifferential for sum of functions “Lipschitz in one variable” is proved.

Stochastic integrodifferential models of fractional orders and Leffler nonsingular kernels: well-posedness theoretical results and Legendre Gauss spectral collocation approximations

Stochastic fractional integrodifferential models are widely employed to model several natural phenomena these days. This current work focuses on the well-posedness results and numerical solutions of a specific form of these models considering the Leffler nonsingular kernels operator wherein the stochastic term is driven by the standard Brownian motion. Accordingly, a combination of sufficient conditions, topological theorems, and Banach space theory are utilized to construct the well-posedness proof. For treating the numerical issue, a familiar spectral collocation technique relying upon shifted Legendre series expansion theory is proposed. The basic properties of Brownian motion and a linear spline interpolation method are used to simulate the standard Brownian motion at a fixed time value. In addition, the idea of the Gauss-Legendre numerical integration rule is implemented to approximate the finite integral. We also devote our attention to the concept of convergence of the proposed method and demonstrate its analysis. Ultimately, the obtained theoretical results and the presented method are examined with five numerous applications. The obtained results indicate the high accuracy and efficiency of applying this method in solving stochastic models of the above-mentioned form.

VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV

AbstractWe prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$ . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$ -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

Factorisation in stopping-time Banach spaces: Identifying unique maximal ideals

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L1 or C(Δ), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) Lp-spaces, BMO, SL∞, and others.

(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 the Hagler–Stegall characterisation of dual spaces containing complemented copies of L_{1}. As a corollary, we obtain the following quantitative version of the Hagler–Stegall theorem asserting that for a Banach space X, the following statements are equivalent:X contains almost isometric contains almost isometric copies of (bigoplus _{n=1}^{infty } ell _{infty }^{n})_{ell _1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of L_{1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of C[0,1]^{*}. Moreover, if X is separable, one may add the following assertion:for all varepsilon >0, there exists a (1+varepsilon )-quotient map T:Xrightarrow C(Delta ) so that T^{*}[C(Delta )^{*}] is (1+varepsilon )-complemented in X^{*}, where Delta is the Cantor set

A tensor norm approach to quantum compatibility

Measurement incompatibility is one of the most striking examples of how quantum physics is different from classical physics. Two measurements are incompatible if they cannot arise via classical post-processing from a third one. A natural way to quantify incompatibility is in terms of noise robustness. In the present article, we review recent results on the maximal noise robustness of incompatible measurements, which have been obtained by the present authors using free spectrahedra, and rederive them using tensor norms. In this way, we make them accessible to a broader audience from quantum information theory and mathematical physics and contribute to the fruitful interactions between Banach space theory and quantum information theory. We also describe incompatibility witnesses using the tensor norm and matrix convex set duality, emphasizing the relation between the different notions of witnesses.

The Grothendieck property from an ordered point of view

In this note, we consider several notions related to the Grothendieck property. Among them, we introduce the notion of the unbounded Grothendieck property for Banach lattices as an unbounded version of the known Grothedieck property in the Banach space theory. Beside other results, surprisingly, we show that spaces with the unbounded Grothendieck property are exactly the reflexive Banach lattices.

Point Degree Spectra of Represented Spaces

Abstract We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees and so on. The notion of point degree spectrum creates a connection among various areas of mathematics, including computability theory, descriptive set theory, infinite-dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite-dimensional Cantor manifolds with property C whose Borel structures at an arbitrary finite rank are mutually nonisomorphic. This resolves a long-standing question by Jayne and strengthens various theorems in infinite-dimensional topology such as Pol’s solution to Alexandrov’s old problem.

Macphail’s theorem revisited

In 1947, M.S. Macphail constructed a series in \(\ell _{1}\) that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach space theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky–Rogers theorem asserts that in every infinite-dimensional Banach space E, there exists an unconditionally convergent series \(\sum x^{\left( j\right) }\) such that \(\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty \) for all \(\varepsilon >0\). Their proof is non-constructive and Macphail’s result for \(E=\ell _{1}\) provides a constructive proof just for \(\varepsilon \ge 1\). In this note, we revisit Macphail’s paper and present two alternative constructions that work for all \(\varepsilon >0.\)

Grothendieck spaces: the landscape and perspectives

In 1973, Diestel published his seminal paper Grothendieck spaces and vector measures that drew a connection between Grothendieck spaces (Banach spaces for which weak- and weak*-sequential convergences in the dual space coincide) and vector measures. This connection was developed further in his book with J. Uhl Jr. Vector measures. Additionally, Diestel’s paper included a section with several open problems about the structural properties of Grothendieck spaces, and only half of them have been solved to this day. The present paper aims at synthetically presenting the state of the art at subjectively selected corners of the theory of Banach spaces with the Grothendieck property, describing the main examples of spaces with this property, recording the solutions to Diestel’s problems, providing generalisations/extensions or new proofs of various results concerning Grothendieck spaces, and adding to the list further problems that we believe are of relevance and may reinvigorate a better-structured development of the theory.

On Mazur rotations problem and its multidimensional versions

The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach’s famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraïssé theory and Ramsey theory, and led to development of concepts of interest independent of Mazur problem. This survey focuses on results that have been published after 2000, stressing two lines of research which were developed in the last 10 years. The first one is the study of approximate versions of Mazur rotations problem in its various aspects, most specifically in the case of the Lebesgue spaces \(L_p\). The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results. Some new results are also included.

Ramsey property and block oscillation stability on normalized sequences in Banach spaces

A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this application, as an intermediate step, we can talk about the notion of $(k,\varepsilon)-$oscillation stable sequence, which will be described and analyzed more generally in this article. Indeed, we introduce the notion $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequence where $(\mathcal{B}_i)_{i=1}^k$ is a finite sequence of barriers and using what we will call blocks of barriers. In particular, we prove that the Ramsey Theorem is equivalent to the statement ``for every finite sequence $(\mathcal{B}_i)_{i=1}^k$ of barriers, every $\varepsilon>0$ and every normalized sequence $(x_i)_{i\in\mathbb{N}}$ there is a subsequence $(x_i)_{i\in M}$ that is $((\mathcal{B}_i\cap\mathcal{P}(M))_{i=1}^k,\varepsilon)-$block oscillation stable'', where $\mathcal{P}(M)$ is the power set of the infinite set M. Besides, we introduce the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic model of a normalized basic sequence where $(\mathcal{B}_i)_{i\in\mathbb{N}}$ is a sequence of barriers. These models are a generalization of the spreading models and are related to the $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequences. We show that the Brunel-Sucheston is satisfied for the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models, and we also prove that this result is equivalent to the Ramsey Theorem. The difference between our theorem and the Brunel-Sucheston Theorem is based on the number of different models that are obtained from the same normalized basic sequence through them. This and other observations about $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models are noted in an example at the end of the article.

Unital Banach algebras not isomorphic to Calkin algebras of separable Banach spaces

Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may be found of the form $C(X)$ for a compact space $X$, $\ell_1(G)$ for some torsion-free Abelian group, and a~simple, unital AF $C^*$-algebra. Extensions to higher densities are also presented.

On Four Classical Measure Theorems

A subset B of an algebra A of subsets of a set Ω has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B-pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering {Bn:n∈N} of B contains a set Bp with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn1n2⋯nm:ni∈N,1≤i≤m,m∈N} of B contains a strand {Bp1p2⋯pm:m∈N} formed by elements Bp1p2⋯pm with property (N) [(G) or (VHS)] for every m∈N. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every σ-algebra has properties (N), (sN), (G) and (VHS). Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every σ-algebra has property (wN) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property (wN) of a σ-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (wWHS) if and only if B has property (wN) and A has property (G).