Omid Zabeti
In this note, we consider several notions related to the Grothendieck property. Among them, we introduce the notion "unbounded Grothendieck property" in a Banach lattice 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.
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A Banach space $X$ is Grothendieck if the weak and the weak$^*$ convergence of sequences in the dual space $X^*$ coincide. The space $\ell^\infty$ is a classical example of a Grothendieck space due to Grothendieck. We introduce a quantitative version of the Grothendieck property, we prove a quantitative version of the above-mentioned Grothendieck's result and we construct a Grothendieck space which is not quantitatively Grothendieck. We also establish the quantitative Grothendieck property of $L^\infty(μ)$ for a $σ$-finite measure $μ$.
The main pupose of this paper is to fully characterize continuous concave functions $ψ$ such that the corresponding Marcinkiewicz Banach function space $M_{ψ}$ is a Grothendieck space.
Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the continuum. It follows that the classical result stating that ``no nontrivial complemented subspace of a Grothendieck $C(K)$ space is separable'' cannot be strengthened by replacing ``is separable'' by ``has density less than that of $l_\infty$'', without using an additional set-theoretic assumption. Such a strengthening was proved by Haydon, Levy and Odell, assuming Martin's axiom and the negation of th…
For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. If $X$ is compact, the classic result of A.~Grothendieck states that $C_k(X)$ has the Dunford-Pettis property and the sequential Dunford--Pettis property. We extend Grothendieck's result by showing that $C_k(X)$ has both the Dunford-Pettis property and the sequential Dunford-Pettis property if $X$ satisfies one of the following conditions: (i) $X$ is a hemicompact space, (ii) $X$ is a cosmic space (…
Suppose $E$ is a Banach lattice. Recently, there have been some motivating contexts regarding the known Banach-Saks property and the Grothendieck property from an order point of view. In this paper, we establish these results for operators that enjoy different types considered for the Banach-Saks property as well as for different notions related to the Grothendieck property. In particular, beside other results, we characterize order continuity and reflexivity of Banach lattices in terms of the corresponding bounded operators defined on them.
A Banach space $E$ has the Grothendieck property if every (linear bounded) operator from $E$ into $c_0$ is weakly compact. It is proved that, for an integer $k>1$, every $k$-homogeneous polynomial from $E$ into $c_0$ is weakly compact if and only if the space ${\cal P}(^kE)$ of scalar valued polynomials on $E$ is reflexive. This is equivalent to the symmetric $k$-fold projective tensor product of $E$ (i.e., the predual of ${\cal P}(^kE)$) having the Grothendieck property. The Grothendieck property of the projective tensor product $E\widehat{\bigotimes}F$ is also characterized. Moreover, the Gr…
A Banach space is said to be Grothendieck if weak and weak$^*$ convergent sequences in the dual space coincide. This notion has been quantificated by H. Bendová. She has proved that $\ell_\infty$ has the quantitative Grothendieck property, namely, it is 1-Grothendieck. Our aim is to show that Banach spaces from a certain wider class are 1-Grothendieck, precisely, $C(K)$ is 1-Grothendieck provided $K$ is a totally disconnected compact space such that its algebra of clopen subsets has the so called Subsequential completeness property.
Grothendieck ideals of $\ell_\infty$
研究包含$c_0$的$\ell_\infty$理想的格罗滕迪克性质相关问题，通过有限可加测度等给出刻画，证明存在解析理想使$c_{0,\mathcal I}$是格罗滕迪克空间，同时表明多数博雷尔理想不具有该性质，且格罗滕迪克性质在相关序中不向下封闭。
Construction under Martin's axiom of a Boolean algebra with the Grothendieck property but without the Nikodym property
该研究在弱Martin公理假设下构造特殊完全不连通紧Hausdorff空间，改进了Talagrand的结果，证明Martin公理蕴含存在具Grothendieck性质但无Nikodym性质的布尔代数。
This article deals with properties of spectra of operators on C(K)-spaces with the Grothendieck property (e.g. l^{\infty}) and application to so called J-class operators introduced by A. Manoussos and G. Costakis. We will show that C(K) has the Grothendieck property if and only if the boundary of the spectrum of every operator on C(K) consists entirely of eigenvalues of its adjoint. As a consequence we will see that there does not exist invertible J-class operators on C(K) with the Grothendieck property. In the third section we will give a quantitative and qualitative characterization of all J…
Grothendieck $C(K)$-spaces and the Josefson--Nissenzweig theorem
本文研究紧空间上连续函数空间的 $\ell_1$-Grothendieck 性质，证明其等价于不存在满足 Josefson--Nissenzweig 定理的序列，构造了可分紧空间使该性质成立但标准 Grothendieck 性质不成立，并指出经典 Efimov 空间缺乏该性质。
For $0\leqslant ξ\leqslant ω_1$, we define the notion of $ξ$-weakly precompact and $ξ$-weakly compact sets in Banach spaces and prove that a set is $ξ$-weakly precompact if and only if its weak closure is $ξ$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\leqslant ξ\leqslant ω_1$, we prove that a Banach space $X$ has the $ξ$-Schur property if and only if every $ξ$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.
Several recent papers were devoted to various modifications of limited, Grothendieck, and Dunford--Pettis operators, etc., through involving the Banach lattice structure. In the present paper, it is shown that many of these operators appear as operators affiliated to well known properties of Banach lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck property, the property (d), and the sequential w*-continuity of the lattice operations. It is proved that the spaces consisting of regularly versions of the above operators are all Banach spaces. The domination problem for …
Grothendieck's compactness principle for the absolute weak topology
本文在Banach格中建立绝对弱拓扑的格罗滕迪克紧性原理，证明绝对弱紧集与绝对弱序列紧集的等价条件，并刻画绝对弱紧子集可由绝对弱零序列的闭凸包包含当且仅当格具有正Schur性质。
The Nikodym and Grothendieck properties of Boolean algebras and rings related to ideals
研究布尔代数和理想相关环的Nikodym和Grothendieck性质，证明若布尔代数不具有Nikodym性质则不具有Grothendieck性质，同时构造了具有Nikodym性质但不具有Grothendieck性质的布尔子代数和非分析布尔代数。
In this paper, we provide a sublinear function $p$ on ordered Banach spaces, which depends on the order structure of the space. With respect to this $p$, we study the relation between $p$-contractivity of positive semigroups and the $p$-dissipativity of its generators. The positive off-diagonal property of generators is also studied in ordered vector spaces.
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 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…
Epic math battle of history: Grothendieck vs Nikodym
本文通过定义σ-中心力迫概念，构造出具有格罗滕迪克性质却无尼科迪姆性质的布尔代数，证明其存在与连续统假设否定一致，且可简化塔拉格兰德在连续统假设下的构造。
In this note we present the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ and functions of the harmonic oscillator in terms of the global symbol.
In this paper, we introduce a generalized notion of monotone property and prove some results regarding existence and uniqueness of multi-tupled fixed points for nonlinear contraction mappings satisfying monotone property in ordered complete metric spaces. Our results unify several classical and well known n-tupled (including coupled, tripled and quadruple ones) fixed point results existing in literature.
In this paper, we describe the Grothendieck groups $K_1(V)$ and $K(V)$ of an absolute matrix order unit space $V$ for unitary and partial unitary elements respectively. For this purpose, we study some basic properties of unitary and partial unitary elements, and define their path homotopy equivalence. The construction of $K(V)$ follows in a almost similar manner as that of $K_1(V).$ We prove that $K_1(V)$ and $K(V)$ are ordered abelian groups. We also prove that $K_1(V)$ and $K(V)$ are functors from the category of absolute matrix order unit spaces with morphisms as unital completely $\vert \c…
We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Crit{è}res de compacit{é}" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.
It is shown that an ordered vector space $X$ is Archimedean if and only if $\inf\limits_{τ\in\{τ\}, y\in L}(x_τ-y) \ = 0$ for any bounded decreasing net $x_τ\downarrow$ in $X$, where $L$ is the collection of all lower bounds of $\{x_τ\}_τ$. We give also a characterization of the almost Archimedean property of $X$ in terms of existence of a linear extension of an additive mapping $T:Y_+\to X_+$ of the positive cone $Y_+$ of an ordered vector space $Y$ into $X_+$.
In the present article, we introduce a unified notion of multi-tupled fixed points and utilize the same to prove some existence and uniqueness unified multi-tupled fixed point theorems for Boyd-Wong type nonlinear contractions satisfying generalized mixed monotone property in ordered metric spaces. Our results unify several classical and well-known n-tupled (including coupled, tripled and quadrupled ones) fixed point results existing in the literature.
Dunford--Pettis type properties of locally convex spaces
本文在局部凸空间中引入严格DP_p性质和序列DP_{(p,q)}性质，借助Grothendieck拓扑τ_{Σ'}将其与弱Glicksberg性质和p-Schur性质建立等价刻画，并给出首个严格DP但非DP的局部凸空间反例。
In this paper, we define and study absolute matrix order ideals in absolute matrix order unit spaces. As an application of absolute matrix order unit property, we construct some kinds of absolute matrix order ideals in absolute matrix order unit spaces. Later, we show that the Grothendieck group of a such kind of absolute matrix order ideal for order projections is a subgroup of Grothendieck group of corresponding absolute matrix order unit space for order projections.
The celebrated Josefson-Nissenzweig theorem implies that for a Banach space $C(K)$ of continuous real-valued functions on an infinite compact space $K$ there exists a sequence of Radon measures $\langleμ_n\colon\ n\inω\rangle$ on $K$ which is weakly* convergent to the zero measure on $K$ and such that $\big\|μ_n\big\|=1$ for every $n\inω$. We call such a sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we study the situation when the space $K$ admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others…
Using free random varaibles we find an embedding of the operator space $OH$ in the predual of a von Neumann algebra. The properties of this embedding allow us to determined the projection constant of $OH_n$, i.e. there exists a projection $P:B(\ell_2)\to OH_n$ whose completely bounded norm behaves as n^{1/2}/(1+ln n)^{1/2}. According to recent results of Pisier/Shlyahtenko, the lower bound holds for every projection. Improving a previous estimate of order $(1+ ln n)$ of the author, Pisier/Shlyahtenko obtained a `logarithmic little Grothendieck inequality'. We find a second proof of this inequa…
Relative uniform convergence and Archimedean property in pre-ordered vector spaces and Abelian groups
本文研究了预序向量空间中 quotient 空间作为 Archimedean 化的性质，通过正楔的闭包和 quotient 集合的构造，探讨了均匀收敛与阿基米德性质的关系。
Approximation property in terms of Lipschitz maps via tensor product approach
本文通过张量积方法研究了Banach空间的Lipschitz近似性质及其与p-近似性质的关系，引入了Lipschitz p-近似性质并建立了相关分解定理。
Increasing sequences in ordered Banach spaces -- new theorems and open problems
本文围绕有序Banach空间的Levi性质证明四个新定理，包括从更弱条件推出Levi性质、可分正规锥情形的推广、一般化Dini定理及其对紧算子空间的应用，并给出反例与未决问题。
A Banach space $X$ has \textit{property $(K)$}, whenever every weak* null sequence in the dual space admits a convex block subsequence $(f_{n})_{n=1}^\infty$ so that $\langle f_{n},x_{n}\rangle\to 0$ as $n\to \infty$ for every weakly null sequence $(x_{n})_{n=1}^\infty$ in $X$; $X$ has \textit{property $(μ^{s})$} if every weak$^{*}$ null sequence in $X^{*}$ admits a subsequence so that all of its subsequences are Cesàro convergent to $0$ with respect to the Mackey topology. Both property $(μ^{s})$ and reflexivity (or even the Grothendieck property) imply property $(K)$. In the present paper we…
Extreme points, positive Grothendieck constants and tensor product norms
研究极值点理论、格罗滕迪克型不等式和张量积范数相互作用产生的问题，建立刻画有限维巴拿赫空间间正收缩集极值点的通用框架，明确相关结果，用于精确评估常数，如证明正格罗滕迪克常数等特定常数的值。
Let $(E,F)$ be a pair of Fréchet spaces. In this paper, we discuss whether a certain property $P$ enjoyed by both $E$ and $F$ is also satisfied by the complete tensor product $E \widehat{\otimes}_π F$. Specifically we focus on the two properties generalizing Köthe spaces: the normability condition called the property $(y)$, and the property of smallness of bounded subsets of Fréchet spaces up to a complemented Banach subspace in connection with the problem of topologies of A. Grothendieck. We also consider the stability of some well known equivalences of operator classes with an application in…
We show that Sobczyk's Theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta.
We study the $H^1$-projective Banach spaces. We prove that they have the Analytic Radon-Nikodym Property, and that they are cotype 2 spaces which satisfy Grothendieck's Theorem. We show also that the ultraproduct of $H^1$-projective spaces is $H^1$-projective. Other results are also discussed.
The main purpose of this paper is to study some geometric and topological properties on $ c_0$ sum of the finite dimensional Banach lattice $\ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattices $c_0(\ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{\infty}(\ell_2^n)$ is an almost limited set.
The purpose of this seminar, which was presented at the Universitat Politècnica de València in late 2012, is to explain several results concerning the bounded approximation property for Fréchet spaces. We give a full detailed proof of an important result due to Pełczyński that asserts that every separable Fréchet space with the bounded approximation property is isomorphic to a complemented subspace of a Fréchet space with a Schauder basis. We also explain Vogt's example of a nuclear Fréchet space without the bounded approximation property. This example is simpler than the original counterexamp…
In the present article, it is proved that every nuclear operator in a locally convex space E has a well-defined trace if E possesses the approximation property. However, even if a space possesses the approximation property this still does not guarantee a positive solution of A. Grothendieck's uniqueness problem for this space. Below, we present an example of a quasi-complete space with the approximation property in which it is not possible to define the trace for all Fredholm operators (in the sense of A. Grothendieck). We prove that the uniqueness problem has a positive solution if E possesse…
In this paper, we study the connections between the normality, regularity, full regularity, and chain-complete property in partially ordered Banach spaces. Then, by applying these properties, we prove some fixed point theorems on partially ordered Banach spaces. As applications of these fixed point theorems, we prove the existence of solutions of some integral equations, such as Hammerstein integral equations, in Banach spaces.
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。
In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the Grothendieck-Lidskii formula. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ in terms of global symbols.
On a variant of the Grothendieck inequality and estimates on tensor product norms
本文研究了有限维 Banach 空间对的 Grothendieck 型不等式及常数，证明了余型相关性质，并给出了收缩 little Parrott 同态的精确范数估计，为已有定理提供了新证明。
In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.
A Banach space X has Pelczynski's property (V) if for every Banach space Y every unconditionally converging operator T: X -> Y is weakly compact. H. Pfitzner proved that C*-algebras have Pelczynski's property (V). In the preprint "H. Krulisova: Quantification of Pelczynski's property (V)" the author explores possible quantifications of the property (V) and shows that C(K) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner's theorem. Moreover, we prove that in dual Banach spaces a quantitative …
Order-to-ring ideals via submultiplicative lattice seminorms in lattice-ordered algebras
本文刻画了具有序到环理想性质的格序代数，证明阿基米德格序代数具有该性质当且仅当它是f-代数且具有零半范数元素为幂零元的子乘法格半范数，并推广到半素f-代数的情形。
This is a companion to recent papers of the authors; here we construct the `noncommutative Shilov boundary' of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider `maximal' and `minimal' unitizations of such ordered operator spaces.
In this paper, we prove several fixed point theorems on both of normal partially ordered Banach spaces and regular partially ordered Banach spaces by using the normality, regularity, full regularity, and chain -complete property. Then, by applying these theorems, we provide some existence and uniqueness of solutions to some integral equations. We also prove the solvability of some equilibrium problems in Banach spaces.
We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Ban…
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