Extreme points of convex fully symmetric sets of measurable operators

Chilin, Vladimir; Krygin, Andrei V.; Sukochev, Fedor

doi:10.1007/bf01204237

Cited by 43 publications

(40 citation statements)

References 12 publications

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“…The characterization of the complex extreme points is analogous to the results on extreme points in [16]. The relation between complex extreme and upper monotone points played an important role in proving that x inherits complex convexity from µ(x).…”

Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 90%

“…Chilin, A. Krygin and F. Sukochev in [16] extended J. Arazy's result to symmetric spaces of measurable operators E(M, τ ). Here the relations between extreme operators and their singular value functions become more complex.…”

Section: Extreme Points and Strict Convexitymentioning

confidence: 98%

“…The relation between complex extreme and upper monotone points played an important role in proving that x inherits complex convexity from µ(x). We observed in [16] that if µ(x) is a complex extreme point of B E then the functions from B E whose decreasing rearrangements majorize µ(x) must be equimeasurable with µ(x). Lemma 8.1.…”

Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 99%

“…He related the properties of the symmetric sequence space E to the corresponding properties of C E . His ideas influenced later V. Chilin, A. Krygin and F. Sukochev [16,17] and Q. Xu [112], who initiated investigation of the relation between the properties of the symmetric function space E and the properties of E (M, τ ).…”

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Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

Czerwińska

Kamińska

2017

commath

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Abstract. This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators E(M, τ ), where M is a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ , and E is a symmetric function space. If E ⊂ c0 is a symmetric sequence space then the analogous properties in the unitary matrix ideals CE are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Pólya, Köthe duality, the spaces Lp (M, τ ), 1 ≤ p < ∞, the identification between CE and G(B(H), tr) for some symmetric function space G, the commutative case when E is identified with E(N , τ ) for N isometric to L∞ with the standard integral trace, trace preserving * -isomorphisms between E and a * -subalgebra of E (M, τ ), and a general method of removing the assumption of non-atomicity of M. The main results on geometric properties are given in separate sections. We present the results on (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, k-extreme points and k-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec-Klee properties, Banach-Saks properties, Radon-Nikodým property and stability in the sense of Krivine-Maurey. We also state some open problems.In 1937, John von Neumann [83] pp. 205-218, observed that for a symmetric norm · in R n , it is possible to define a norm on the space of n × n matrices x by setting x = {s i (x)} n i=1 , where s i (x), i = 1, 2 . . . , n, are eigenvalues of the matrix |x| = (x * x) 1/2 ordered in a decreasing manner. Later on in the forties and fifties, J. von Neumann and R. Schatten developed analogous theory for infinite dimensional compact operators. They defined and studied unitary matrix ideals C E corresponding to a symmetric sequence Banach space (E, · E ). The space consists of all compact operators x on a Hilbert space such that {s n (x)} ⊂ E with the norm x = {s n (x)} E , where s n (x), n ∈ N, are singular numbers of x, that is eigenvalues of |x|. For E = ℓ 1 , the space C E is called the trace class of operators or the space of nuclear operators, while if E = ℓ 2 then it is called the class of Hilbert-Schmidt operators. The first monograph of these spaces was written by R. Schatten in 1960 [93], and later on in 1969 by I. C. Gohberg and M. G. Krein [49]. In 1967, C. McCarthy wrote an article on the now called the Schatten classes C p , 0 < p ≤ ∞, that is the spaces C E when E = ℓ p , and showed among others that this space is uniformly convex for 1 < p < ∞ [78]. The beginning of the theory of symmetric spaces of measurable operators can be traced back to the early fifties. It was then when I. Segal and J. Dixmier [94,29] laid out the foundation for noncommutative L p (M, τ ) spaces, 0 < p < ∞, by introducing t...

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Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 90%

Section: Extreme Points and Strict Convexitymentioning

confidence: 98%

Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 99%

mentioning

confidence: 99%

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Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

Czerwińska

Kamińska

2017

commath

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“…(i) следствия 3.2). Доказательство теоремы 2.2 опирается на один глубокий результат из [2]. Нами показано, что если оператор ∈̃︁ ℳ гипонормален и оператор 2 -компактен, то и оператор -компактен (п.…”

Section: Doi: 104213/mzm10311unclassified

О нормальных $\tau$-измеримых операторах, присоединенных к полуконечной алгебре фон Неймана

Бикчентаев¹

2014

Matematicheskie Zametki

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Lipschitz Continuity of the Absolute Value and Riesz Projections in Symmetric Operator Spaces

Dodds

Pagter

et al. 1997

Journal of Functional Analysis

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A principal result of the paper is that if E is a symmetric Banach function space on the positive half-line with the Fatou property then, for all semifinite von Neumann algebras (M, {), the absolute value mapping is Lipschitz continuous on the associated symmetric operator space E(M, {) with Lipschitz constant depending only on E if and only if E has non-trivial Boyd indices. It follows that if M is any von Neumann algebra, then the absolute value map is Lipschitz continuous on the corresponding Haagerup L p -space, provided 1

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Extreme points of convex fully symmetric sets of measurable operators

Cited by 43 publications

References 12 publications

Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

О нормальных $\tau$-измеримых операторах, присоединенных к полуконечной алгебре фон Неймана

Lipschitz Continuity of the Absolute Value and Riesz Projections in Symmetric Operator Spaces

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