Rosenthal's theorem for subspaces of noncommutative Lp

Junge, Marius; Parcet, Javier

doi:10.1215/s0012-7094-08-14112-2

Cited by 28 publications

(62 citation statements)

References 40 publications

(66 reference statements)

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“…Some of the most important results of classical Banach space theory, as well as probability theory and harmonic analysis have found analogous versions in the noncommutative case [29,26,27]. …”

Section: Mathematical Tools and Connectionsmentioning

confidence: 95%

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

Junge

Palazuelos

Pérez-Garcı́a

et al. 2010

Commun. Math. Phys.

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In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order Ω √ n log 2 n when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative L p embedding theory.As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise.

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Section: Mathematical Tools and Connectionsmentioning

confidence: 95%

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

Junge

Palazuelos

Pérez-Garcı́a

et al. 2010

Commun. Math. Phys.

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“…As application of Theorem 3.1, we obtained in connection with a recent result of Junge and Parcet [19] the following structural property of reflexive subspaces of preduals of general von Neumann algebras: Theorem 3.4: Let R be a von Neumann algebra. There exists a finite von Neumann algebra M so that every reflexive subspace of R * Banach embeds isomorphically into M * .…”

Section: Lemma 33mentioning

confidence: 95%

“…Let X be a reflexive subspace of R * . According to [19], there exists 1 < p < 2 so that X embeds isomorphically into L p (R ⊕ ∞ R). Apply Theorem 3.1 to the von Neumann algebra R ⊕ ∞ R in order to get a finite von Neumann algebra M so that L p (R ⊕ ∞ R) embeds isomorphically into M * .…”

Section: Lemma 33mentioning

confidence: 99%

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Embeddings of non-commutative L p -spaces into preduals of finite von Neumann algebras

Randrianantoanina

2008

Isr. J. Math.

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Let R be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra N so that for every 1 < p < 2, the Haagerup L p -space associated with R embeds isomorphically into N * . We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if M is a semi-finite von Neumann algebra then every reflexive subspace of M * embeds isomorphically into L r (M) for some r > 1.

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“…It has attracted great attention of the well known specialists in functional analysis and operator theory as J. Arazy, V. I. Chilin, P. G. Dodds, T. K. Dodds, U. Haagerup, M. Junge, N. Kalton, F. Lust-Piquard, B. De Pagter, G. Pisier, F. Sukochev, Q. Xu [5,35,53,58,60,76,110,104,30], and others. The noncommutative L p (M, τ ) spaces, and more general non-commutative spaces of measurable operators E(M, τ ), share many properties with the usual L p spaces, or symmetric spaces E, but on the other hand they are very different.…”

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confidence: 99%

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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Rosenthal's theorem for subspaces of noncommutative Lp

Cited by 28 publications

References 40 publications

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

Embeddings of non-commutative L p -spaces into preduals of finite von Neumann algebras

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

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