Johann Langemets
We continue the investigation of the behaviour of diameter two properties in tensor products of Banach spaces. Our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products. We also prove a result for the symmetric strong diameter two property for the injective tensor product.
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The aim of this note is to provide several variants of the diameter two properties for Banach spaces. We study such properties looking for the abundance of diametral points, which holds in the setting of Banach spaces with the Daugavet property, for example, and we intro- duce the diametral diameter two properties in Banach spaces, showing for these new properties stability results, inheritance to subspaces and characterizations in terms of finite rank projections.
We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again this property.
We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c_0. We call X elastic if for some K < infinity for every Banach space Y which embeds into X, the space Y is K-isomorphic to a subspace of X. We also prove that if X is a separable Banach space such that for some K < infinity every isomorph of X is K-elastic then X is finite dimensional.
On the duality between Diameter Two Properties and Ideals in Banach Spaces
本文研究 Banach 空间中直径二性质与 M-理想的对偶关系，证明 M-理想的对偶继承各类 $w^*$ 直径二性质，并以反例说明逆命题不成立，进而考察 $C(K)$ 空间及其对偶。
We demonstrate the result stated in the title, thus answering an open question asked by Julio Becerra Guerrero, Ginés López-Pérez and Abraham Rueda Zoca in J. Conv. Anal. \textbf{25}, no. 3 (2018).
We prove that, if Banach spaces $X$ and $Y$ are $δ$-average rough, then their direct sum with respect to an absolute norm $N$ is $δ/N(1,1)$-average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$ the space $X\oplus_p Y$ is $2^{1-1/p}$-average rough, which is in general optimal. Another consequence is that for any $δ$ in $(1,2]$ there is a Banach space which is exactly $δ$-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diame…
Diameter, radius and Daugavet index of thickness of slices in Banach spaces
构造具有r-BSP性质的Banach空间，其单位球切片直径下确界为1，否定Ivakhno问题；并证明Daugavet性质空间可重赋范使厚度指数大于2-ε且存在小直径切片。
We study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in Banach spaces. We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that every slice of its unit ball has diameter 2 and satisfying that its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small, which answers an open problem and stresses the differences of diameter between slices and relatively weakly open subsets of the unit ball in Banach spaces.
The geometric notion of huskability initiated and developed in [B3], [BR] ,[EW], [GM] was subsequently extensively studied in the context of dentability and Radon Nikodym Property in [GGMS]. In this work, we introduce a new geometric property of Banach space, the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related geometric properties, $BSCSP$ namely, the unit ball has convex combination of slices of arbitrarily small diameter and $BDP$ namely, the closed unit ball has slices of arbitrar…
We study when diameter two properties pass down to subspaces. We obtain that the slice two property (respectively diameter two property, strong diameter two property) passes down from a Banach space $X$ to a subspace $Y$ whenever $Y$ is complemented by a norm one projection with finite-dimensional kernel (respectively the quotient $X/Y$ is finite dimensional, $X/Y$ is strongly regular). Also we study the same problem for dual properties of the above ones, as having octahedral, weakly octahedral or 2-rough norm.
In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in \cite{B3}, \cite{BR} ,\cite{EW}, \cite{GM} was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in \cite{GGMS}. We introduce the the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related properties, $…
Nonrough norms in spaces with small diameter
本文研究Banach空间中三类小直径性质与非粗糙范数的对偶关系，引入弱平均和平均非粗糙范数新概念，并证明其在特定条件下具有三空间性质。
A Banach space has the ball huskable property ($BHP$) if the closed unit ball has weakly open sets of arbitrarily small diameter. We can analogously define $w^*$-$BHP$ in the dual space. In this short note, we study these properties in the context of ideals in Banach spaces. The notion of an ideal, was introduced by Godefroy, Kalton and Saphar. We show that if a Banach space $X$ has $BHP,$ then any $M$-ideal of $X$ also has $BHP.$ We further show that if $Y$ is an $M$-ideal of $X,$ then $Y^*$ has $w^*$-$BHP$ implies $X^*$ has $w^{*}$-$BHP.$ We use this result to prove that for a compact Hausdo…
We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2 and, however, its unit ball still contains convex combinations of slices with diameter arbitrarily small, which improves in a optimal way the known results about the size of this kind of subsets in Banach spaces.
We prove that every separable Banach space containing $\ell_1$ can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its …
A Banach space is locally almost square if, for every $y$ in its unit sphere, there exists a sequence $(x_n)$ in its unit sphere such that $\lim\|y\pm x_n\|=1$. A Banach space is weakly almost square if, in addition, we require the sequence $(x_n)$ to be weakly null. It is known that these two properties are distinct, so we aim to investigate if local almost squareness implies a weaker version of the latter property by replacing the sequence with a net. In order to achieve this result, we restrict ourselves to Banach lattices and introduce a strengthening of local almost squareness by requirin…
Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions
该研究针对实值Lipschitz函数Banach空间的公开问题，证明直径二性质与弱星对应性质的差异，构造相关空间示例，引入新充分条件并明确已知示例状态，推进了相关空间几何性质的研究。
The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.
Banach spaces with the weak diametral diameter two property
引入并研究弱直径二性质（weak-DD2P），它介于直径二性质等之间。通过对偶单位球极点结构获其必要条件，建立相关稳定性结果，还给出特定向量值空间的完整描述，得到新例子并显示其与其他直径二性质不同。
Slice diameter two property in ultrapowers
本文研究巴拿赫空间超幂对切片直径二性质的继承性，给出该性质成立的刻画，同时构造反例证明该性质一般不被超幂继承。
We study transfinite analogues of the symmetric strong diameter two property. We investigate the stability of these properties under $c_0$, $\ell_\infty$ sums and under projective tensor products. Moreover, we characterize Banach spaces of the form $C_0(X)$, where $X$ is a T$_4$ locally compact space, which posses these transfinite properties via cardinal functions over $X$. As an application, we are able to produce a variety of examples of Banach spaces which enjoy or fail these properties.
We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of $w^*$-slices in the dual unit ball has diameter 2, which answer an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing $\ell_1$ and for every $\varepsilon >0$ there is an equivalent norm so that every convex combinat…
Big weak open radius versus big slice diameter
本文构造了一类非同构Banach空间，其中单位球非空弱开子集半径恒为最大值一，而切片直径下确界也为一，展现极端差异。
We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$ both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive $L$-embedded Banach space taking projective tensor products with an arbitrary Banach space.
We give two examples of polyhedral Banach spaces failing all the diameter two properties, showing that there is not any connection between polyhedrality and the diameter two properties.
A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P in short) if every nonempty relatively weakly open subset of its closed unit ball has diameter $2$. We construct an equivalent norm on $L_1[0,1]$ which is weakly midpoint locally uniformly rotund and has the D$2$P. We also prove that for Banach spaces admitting a norm-one finite-co-dimensional projection it is impossible to be uniformly rotund in every direction and at the same time have the D$2$P.
We prove a complex interpolation formula for the injective tensor product of vector-valued Banach function spaces satisfying certain geometric assumptions. This result unifies results of Kouba, and moreover, our approach offers an alternate proof of Kouba's interpolation formula for scalar-valued Banach function spaces.
We generalize results of Davie and Raeburn describing homotopy types of the group of invertible elements and of the set of idempotents of the projective tensor product of complex unital Banach algebras. We illustrate our results by specific examples.
In this article, the relationship between Birkhoff-James orthogonality of elementary tensors in certain tensor product spaces with the Birkhoff-James orthogonality of individual elements in their respective spaces is studied.
Let $A$ and $B$ be commutative semisimple Banach algebras. In this paper, the $BSE$ and $BED$- property of tensor Banach algebra $A{\otimes}_γ B$ with respect to the Banach algebras $A$ and $B$ are assesed. In particular, $BSE$ and $BED$- structure of vector-valued group algebra $L_{1}(G, A)$ and $C^{*}$- algebra $C_{0}(X, A)$ characterized.
We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice $E$ with itself and the 2-concavification of $E$.
In this paper, we show that four main diagonal spaces of injective tensor products are pairwise isometrically isomorphic. When E is a Banach lattice, we show that the tensor diagonal of E is a 1-unconditional basic sequence in both the n-fold injective tensor product of E and the n-fold symmetric injective tensor product of E.
We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(Ω,Σ)$ be a measurable space, let $E$ be a Banach lattice and let $ν:Σ\to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(ν)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue-Bochner space $L_1(μ,Y)$ is weakly almost square for any Banach space~$Y$ and for any non-atom…
We extend the (attaining of) strong diameter two property to infinite cardinals. In particular, a Banach space has the 1-norming attaining strong diameter two property with respect to $ω$ (1-ASD2P$_ω$ for short) if every convex series of slices of the unit ball intersects the unit sphere. We characterize $C(K)$ spaces and $L_1(μ)$ spaces having the 1-ASD2P$_ω$. We establish dual implications between the 1-ASD2P$_ω$, $ω$-octahedral norms and Banach spaces failing the $(-1)$-ball-covering property. The stability of these new properties under direct sums and tensor products is also investigated.
Series representation in tensor products of Banach spaces
本文通过极其简单的论证证明，在向量空间张量积上赋予任意范数时，其完备化空间中的每个元素都能表示为初等张量的收敛级数。
Remarks on the Daugavet Property for Complex Banach Spaces
本文研究复巴拿赫空间中Daugavet性质及其变体，通过刻画和关系证明sluacs空间无Δ点，并揭示向量值函数空间中多项式Daugavet性质的等价条件，最终统一无限维一致代数中的多种性质。
Projective and injective tensor products of Banach $L^0$-modules
本文研究$σ$-有限测度空间上Banach $L^0$-模的射影与内射张量积，并将商算子、可和族和Schauder基等技术工具推广到该框架。
We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $ω$. We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums.
In this paper, we investigate the relationship between the weak min-max property and the diameter uniformity of domains in Banach spaces with dimensions at least $2$. As an application, we show that diameter uniform domains are invariant under relatively quasimöbius mappings.
We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.
Suppose that $E$ is a uniformly complete vector lattice and $p_1,..., p_n$ are positive reals. We prove that the diagonal of the Fremlin projective tensor product of $E_{(p_1)},..., E_{(p_n)}$ can be identified with $E_{(p)}$ where $p = p_1+...+p_n$ and $E_{(p)}$ stands for the $p$-concavification of $E$. We also provide a variant of this result for Banach lattices. This extends the main result of [BBPTT].
We study almost square Banach spaces under a topological point of view. Indeed, we prove that the class of Banach spaces which admits an equivalent norm to be ASQ is that of those Banach spaces which contain an isomorphic copy of $c_0$. We also prove that the symmetric projective tensor products of an almost square Banach space have the strong diameter two property
The tensor product of the Fremlin tensor product of Banach lattice algebras
该论文研究巴拿赫格代数的Fremlin投射张量积的Arens正则性，引入L-弱几乎周期泛函并刻画其性质，证明其Arens正则性的蕴含关系、反例及充分条件，还得出巴拿赫f-代数的Fremlin投射张量积的Arens正则性结论。
We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a $1$-unconditional basis. A norm one element $x$ in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance $2$ from $x$. A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an uncon…
In this note, we study the Arens regularity of projective tensor product $A\hat{\otimes}B$ whenever $A$ and $B$ are Arens regular. We establish some new conditions for showing that the Banach algebras $A$ and $B$ are Arens regular if and only if $A\hat{\otimes}B$ is Arens regular. We also introduce some new concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property. Then if $A^{**}$ is weakly amenab…
We study almost squareness and the strong diameter two property in the setting of projective (symmetric) tensor product. We prove that almost squareness is preserved by taking projective tensor product, providing non-trivial examples of ASQ projective tensor product spaces. Furthermore, we give sufficient conditions for a projective symmetric tensor product to have the strong diameter two property which extend most of the known results and provide new examples of such spaces with the strong diameter two property.
Von Neumann-Jordan Constants in Banach Tensor Products
研究Banach张量积在注入和射影范数下的von Neumann-Jordan常数，建立下界、刻画极值情形并确定重赋范包络，进而阐明张量范数、维数和重赋范对该几何常数的影响。
This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural way to matricial Banach spaces, including products, direct sums, and completions. Also, while the minimal matrix-norm on a Banach space is well-known, this work characterizes the maximal matrix-norm on a Banach space from the work of Effros and Ruan as a dual operator space. Moreover, building from the work of Blecher, Ruan, and Sinclair, the Haagerup tensor product is merged with the direct sum to form a Haagerup tens…
Given unital Banach algebras $A$ and $B$ and elements $a\in A$ and $b\in B$, the Drazin spectrun of $a\otimes b\in A\overline{\otimes} B$ will be fully characterized, where $A\overline{\otimes} B$ is a Banach algebra that is the completion of $A\otimes B$ with respect to a uniform crossnorm. To this end, however, first the isolated points of the spectrum of $a\otimes b\in A\overline{\otimes} B$ need to be characterized. On the other hand, given Banach spaces $X$ and $Y$ and Banach space operators $S\in L(X)$ and $T\in L(Y)$, using similar arguments the Drazin spectrum of $τ_{ST}\in L(L(Y,X))$,…
Computing Borel complexity of some geometrical properties in Banach spaces
该研究计算了巴拿赫空间中直径二性质、道加瓦特定理空间等几何性质类别的博雷尔复杂度，多数情况达最优，完善了近年巴拿赫空间同构类相关研究。
The aim of this note is to study octahedrality in vector valued Lipschitz-free Banach spaces on a metric space under topological hypotheses on it. As a consequence, we get that the space of Lipschitz functions on a metric space valued in a dual Banach space satisfies the weak-star strong diameter two property, under natural topological hipothesess on the metric space. Also, we show an example proving that these hypotheses are optimal.
A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.
Fremlin tensor product of Banach lattices and unbounded convergences
本文研究了在局部固矢量格上的Fremlin张量积拓扑，探讨了其在Banach格中保持无界收敛性的条件。
Strictly convex norms and the local diameter two property
本文在实函数实心巴拿赫格中引入范数严格单调性以阻止局部直径二性质，证明$c_0(Γ)$上严格凸1-对称范数具此性质，并构造不可数$Γ$的$c_0(Γ)$上带1-无条件基且具局部直径二性质的严格凸范数。
For each positive integer $m$ and each real finite dimensional Banach space $X$, we set $β(X,m)$ to be the infimum of $δ\in (0,1]$ such that each set $A\subset X$ having diameter $1$ can be represented as the union of $m$ subsets of $A$ whose diameters are at most $δ$. Elementary properties of $β(X,m)$, including its stability with respect to $X$ in the sense of Banach-Mazur metric, are presented. Two methods for estimating $β(X,m)$ are introduced. The first one estimates $β(X,m)$ using the knowledge of $β(Y,m)$, where $Y$ is a Banach space sufficiently close to $X$. The second estimation uses…
We show that the tensor product of approximately amenable algebras need not be approximately amenable, and investigate conditions under which $A$ and $B$ being approximately amenable implies, or is implied by, $A\hat{\otimes}B$ or $A^{\#}\hat{\otimes} B^{\#}$ being approximately amenable. Our methods also enable us to prove that if $A\hat{\otimes}B$ is amenable, then so are $A$ and $B$, a result proved by Barry Johnson in 1996 under an additional assumption.
Galego and Samuel showed that if $K,L$ are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_πC(L)$ is $c_0$-saturated if and only if it is subprojective if and only if $K$ and $L$ are both scattered. We remove the hypothesis of metrizability from their result, and extend it from the case of the two-fold projective tensor product to the general $n$-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces $K_1, \ldots, K_n$, $\widehat{\otimes}_{π, i=1}^n C(K_i)$ is $c_0$-saturated if and only if it is subprojective if and only if each $…
We characterise the weak$^*$ symmetric strong diameter $2$ property in Lipschitz function spaces by a property of its predual, the Lipschitz-free space. We call this new property decomposable octahedrality and study its duality with the symmetric strong diameter $2$ property in general. For a Banach space to be decomposably octahedral it is sufficient that its dual space has the weak$^*$ symmetric strong diameter $2$ property. Whether it is also a necessary condition remains open.
A $Δ$-point $x$ of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance $2$ from $x$. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, $x$ is a Daugavet-point. A Banach space $X$ has the Daugavet property if and only if every norm one element is a Daugavet-point. We show that $Δ$- and Daugavet-points are the same in $L_1$-spaces, $L_1$-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the uni…
Birkhoff-James orthogonality in certain tensor products of Banach spaces II
本文研究Banach空间张量积中初等张量的Birkhoff-James正交性，通过刻画Lebesgue-Bochner空间中元素的正交性，建立了张量正交性与分量元素正交性之间的关系。
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