Vania Mascioni, Lajos Molnar
The aim of this paper is to characterize those linear maps from a von Neumann factor $\A$ into itself which preserve the extreme points of the unit ball of $\A$. For example, we show that if $\A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital *-homomorphism or a unital *-antihomomorphism.
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Let $E$ be a $(\mathrm{IV})$-polyhedral Banach space. We show that, for each $ε>0$, $E$ admits an $ε$-equivalent $\mathrm{(V)}$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points. In particular, we obtain that every separable isomorphically polyhedral Banach space, for each $ε>0$, admits an $ε$-equivalent $(\mathrm{V})$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points.
A Study of the Extreme Points in the Unit Ball of $JT$
本文研究了$JT$空间单位球的极值点，通过分离向量概念分析其几何结构，并为正向量建立等和性质。
We give a structural characterisation of linear operators from one $C^\ast$% -algebra into another whose adjoints map extreme points of the dual ball onto extreme points. We show that up to a $\ast$-isomorphism, such a map admits of a decomposition into a degenerate and a non-degenerate part, the non-degenerate part of which appears as a Jordan $\ast$-morphism followed by a ``rotation'' and then a reduction. In the case of maps whose adjoints preserve pure states, the degenerate part does not appear, and the ``rotation'' is but the identity. In this context the results concerning such pure sta…
On extreme points of the unit ball of a Hardy-Lorentz space
本文针对Semenov于1978年提出的Hardy-Lorentz空间单位球端点刻画问题，给出了规范化函数属于端点集的新的充要条件，并在外解析函数与Blaschke因子乘积情形下获得了最完整的结果。
The Extreme Points of the Unit Ball of the James space $J$ and its dual spaces
该研究给出James拟自反空间$J$单位球端点刻画的新证明，得到$J^{**}$单位球端点的类似刻画，还描述了$J^*$单位球的端点集及其范数闭包，揭示了两类端点描述的紧密关联。
On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces
本文刻画了有限度量空间到严格凸Banach空间的Lipschitz函数单位球的极值点，并提出表示定理，其所需极值点数量不依赖维度，优于Carathéodory定理。
We study the geometry of $\mathcal{L}(X)_w,$ the space of all bounded linear operators on a Banach space $X,$ endowed with the numerical radius norm, whenever the numerical radius defines a norm. We obtain the form of the extreme points of the unit ball of the dual space of $\mathcal{L}(X)_w.$ Using this structure, we explore Birkhoff-James orthogonality, best approximation and deduce distance formula in $\mathcal{L}(X)_w.$ A special attention is given to the case of operators satisfying a notion of smoothness. Finally, we obtain an equivalence between Birkhoff-James orthogonality in $\mathcal…
On the set of extreme points of the unit ball of a Hardy-Lorentz space
本文证明在满足特定条件的Lorentz空间中，模为1的函数是单位球极值点，并补充了de Leeuw-Rudin定理，指出H^1是唯一具有此性质的Hardy-Lorentz空间，同时给出非切向极限为常数的函数为极值点的充分条件。
We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of $H^\infty(\mathbb{B}_d)$ has no extreme points in its unit ball.
Plasticity of the unit ball of the real Banach space $\ell_\infty$
研究实巴拿赫空间$\ell_\infty$单位球的可塑性问题，通过证明单位球上非扩张双射将极点映射到极点，利用半径为一的球的基本覆盖及Fakhoury定理得出单位球是可塑的。
We prove that all extreme points of the unit ball of a Lipschitz-free space over a compact metric space have finite support. Combined with previous results, this completely characterizes extreme points and implies that all of them are also extreme points in the bidual ball. For the proof, we develop some properties of an integral representation of functionals on Lipschitz spaces originally due to K. de Leeuw.
This paper is a revision and an enlargement of the previous version titled "Extreme points of the unit ball of a quasi-multiplier space" which had been circulated since 2004. We study extreme points of the unit ball of an operator space by introducing the new notion (approximate) "quasi-identities". Then we characterize an operator algebra with a contractive approximate quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points. Furthermore, we give a very neat necessary and sufficient condition for a given operator space to become a $C^*$-algebra o…
Extreme points of the unit balls and isometries of noncommutative quasi-Banach Marcinkiewicz spaces
本文刻画了Marcinkiewicz空间单位球的极值点，证明非交换情形下正等距为初等形式，并研究了弱$L_p$空间等上的等距及其单参数群。
We define an $AW^*$-TRO as an off-diagonal corner of an $AW^*$-algebra, and show that the unit ball of an $AW^*$-TRO has an extreme point. In particular, the unit ball of an injective operator space has an extreme point, which answers a question raised in the author's previous work [Journal of Operator Theory, 76(2) (2016), 219-248] affirmatively. We also show that an $AW^*$-TRO (respectively, an injective operator space) has an ideal decomposition, that is, it can be decomposed into the direct sum of a left ideal, a right ideal, and a two-sided ideal in an $AW^*$-algebra (respectively, an inj…
Almost preserved extreme points
引入几乎保留端点概念并系统研究，证明Banach空间有Radon-Nikodým性质当且仅当其闭凸有界子集有该点，还研究了经典Banach空间等的单位球的情况，部分解决了开放问题。
Norming and dense sets of extreme points of the unit ball in spaces of bounded Lipschitz functions
本文在有界Lipschitz函数空间中，通过McShane扩展算子构造规范极端函数子集，证明其在单位球极端点中弱星稠密，为Dudley范数的计算开辟了新途径。
Geometry of the unit ball of ${\mathcal L}(X,Y^*)$
本文以射影张量积为前对偶研究算子空间单位球几何，刻画了秩一算子及对偶空间中弱*-强端点与Namioka点的结构关系，并部分解决了Namioka点张量积的公开问题。
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 obtain all extreme and exposed points of the closed unit ball of the space of bilinear forms $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow \mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty }^{2}\times\ell_{\infty}^{2}\rightarrow\mathbb{R}$ for which the optimal constant of the Littlewood's $4/3$ inequality is achieved is necessarily an extreme point. In the case of complex scalars we combine analytical and numerical evidences supporting that, at least for complex bilinear forms with real coefficients, the optimal constant of Littlewood's $4/3$ inequality s…
Inner functions as strongly extreme points: stability properties
本文研究了H∞空间的有限约束子空间中强极点的稳定性性质，证明了内函数仍为强极点，并探讨了非内函数与强极点的差异。
We consider the "order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn-Banach type separation result is obtained, which allows us to establish an "order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining functionals. Finally,we introduce the "solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikodym Property.
We study the relationship between the point-wise symmetry of Birkhoff-James orthogonality and the geometry of the space of operators $\mathbb{B}(\ell_\infty^n,\ell_1^m)$. We show that any non-zero left-symmetric point in this space is a smooth point. We also show that for $n\geq4$, any unit norm right-symmetric point of this space is an extreme point of the closed unit ball. This marks the first step towards characterizing the extreme points of these unit balls and finding the Grothendieck constants $G(m,n)$ using Birkhoff-James orthogonality techniques.
We provide an elementary proof of a result by V.P.~Fonf and C.~Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J.~Lindenstrauss and R.R.~Phelps \cite{LP} to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result: Let $X$ be an infinite-dimensional Hilbert space satisfying $\mathrm{dens}(X)<2^{\aleph_0}$, then $X$ does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper,…
Weakly Compact Operators Whose Adjoints Preserve $C^*$-Structure
该文研究二次对偶与$C^*$-代数等距的复Banach空间相关的弱紧算子，证明伴随算子保持特定端点结构的算子集为一致强端点集，并拓展到全体算子空间的情形。
We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say $x$ and $y$, which define it satisfy that the metric segment $[x, y]$ only contains $x$ and $y$. The most notable among them is the case when the free space admits an isometric predual with some additional p…
The Hardy space $H^1$ consists of the integrable functions $f$ on the unit circle whose Fourier coefficients $\widehat f(k)$ vanish for $k<0$. We are concerned with $H^1$ functions that have some additional (finitely many) holes in the spectrum, so we fix a finite set $\mathcal K$ of positive integers and consider the "punctured" Hardy space $$H^1_{\mathcal K}:=\{f\in H^1:\,\widehat f(k)=0\,\,\,\text{for all }\, k\in\mathcal K\}.$$ We then investigate the geometry of the unit ball in $H^1_{\mathcal K}$. In particular, the extreme points of the ball are identified as those unit-norm functions i…
In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates $f\_r(z):=f(rz)~(r \textless{} 1)$. We show that this is \emph{not} the case for the de Branges--Rovnyak spaces $\cH(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function $f\in\cH(b)$ such that $\lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty$. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in $\cH(b)$. We give the first constructive proof of this…
We extend the result of B. Cascales at al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.
Extreme points, strongly extreme points and exposed points in Orlicz--Lorentz spaces
本文研究最一般类Orlicz--Lorentz空间单位球的极值结构，在不假设φ为N函数且ω严格递减的情况下，刻画了端点、强端点和暴露点，还给出了对偶空间泛函达到Luxemburg范数的充要条件及支撑泛函的刻画。
We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also constru…
Isometries of James-Schreier and Lorentz spaces
本文通过极值点保持方法研究实 Banach 序列空间上的满射线性等距群，刻画了 James-Schreier 空间 $V_1$ 对偶单位球的极值点，证明其等距映射仅为 $\pm Id$，并给出 Lorentz 序列空间的 Banach-Stone 型结果。
We discuss the geometry of the unit ball -- specifically, the structure of its extreme points (if any) -- in subspaces of $L^1$ and $L^\infty$ on the circle that are formed by functions with prescribed spectral gaps. A similar issue is considered for kernels of Toeplitz operators in $H^\infty$.
Transfinite contractive propagation and the ball fixed point property in \texorpdfstring{$C(K)$}{C(K)}
本文证明紧豪斯多夫空间$K$的$C(K)$具有球不动点性质当且仅当$K$极不连通，通过构造超限压缩传播系统解决该问题。
A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces
论文证明了Lipschitz-free空间中每个元素均可表示为具有紧支撑的元素的凸级数，解决了极点问题，并推导出具有Radon-Nikodým性质的Lipschitz-free空间元素可表示为分子的凸积分。
Joint numerical radius of Tuples: Extreme points, subdifferential set and Gateaux derivative
本文研究有限维 Banach 空间上算子元组在联合数值半径范数下的几何性质，通过确定对偶单位球的端点结构，给出联合数值半径的次微分集表达式，并据此刻画光滑元组、Birkhoff-James 正交性及 Gateaux 导数。
Sub-Hardy Hilbert spaces in the non-commutative unit row-ball
本文将经典 Hardy 空间理论中的 Szegö 定理与 Toeplitz 算子分解等思想推广至全 Fock 空间的多变量非交换设定，并证明了非交换有理函数的 Fejér-Riesz 型定理。
For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}(M)$ are precisely the elementary molecules $(δ(p)-δ(q))/d(p,q)$ defined by pairs of points $p,q$ in $M$ such that the triangle inequality $d(p,q)<d(p,r)+d(q,r)$ is strict for any $r\in M$ different from $p$ and $q$. To this end, we show that the class of Lipschitz-free spaces over closed subsets of $M$ is closed under arbitrary intersections when $M$ has finite diameter, and that this allows a natural definition of the support of elements of $\mathcal{F…
Extreme and exposed points of shift-invariant spaces generated by Gaussian kernel and hyperbolic secant
本文研究了由高斯核和双曲正割函数生成的移不变空间中单位球的极值和暴露点特性。
We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, each preserved extreme point corresponds to a pair of points $p,q$ in $X$ such that the triangle inequality $d(p,q)\leq d(p,r)+d(q,r)$ is uniformly strict for $r$ away from $p,q$. For compact $X$, this condition reduces to the triangle inequality being strict. This result gives an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points.
A note on extreme points of the unit of Hardy-Lorentz spaces
本文研究 Hardy-Lorentz 空间单位球的极点问题，证明在 $φ$ 严格递增且严格凹时，内函数是 $H(Λ(φ))$ 单位球的极点。
We introduce a family of natural normalized Loewner chains in the unit ball, which we call "geräumig"---spacious---which allow to construct, by means of suitable variations, other normalized Loewner chains which coincide with the given ones from a certain time on. We apply our construction to the study of support points, extreme points and time-$\log M$-reachable functions in the class $S^0$ of mappings admitting parametric representation.
Given a subset $Λ$ of $\mathbb Z_+:=\{0,1,2,\dots\}$, let $H^\infty(Λ)$ denote the space of bounded analytic functions $f$ on the unit disk whose coefficients $\widehat f(k)$ vanish for $k\notinΛ$. Assuming that either $Λ$ or $\mathbb Z_+\setminusΛ$ is finite, we determine the extreme points of the unit ball in $H^\infty(Λ)$.
In this paper we characterize surjective isometries on certain classes of non-commutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$, as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$. The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.
Extreme points in quotients of Hardy spaces
研究Hardy空间商H^1/E和H^∞/E中的极值点特征，结合Toeplitz算子核的底层空间，推导经典定理的扩展形式。
On geometry of the unit ball of Paley-Wiener space over two symmetric intervals
本文研究两个对称区间上 Paley-Wiener 空间 $PW_S^1$ 单位球的几何，并在 $ρ>σ/2$ 时完整刻画其极点与暴露点。
Nonlocal perimeters and variations: Extremality and decomposability for finite and infinite horizons
本文研究了非局部周长的极值性和可分解性，通过非局部变换和视界参数，建立了非局部分解定理，并展示了非局部到局部的过渡过程。
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