Smooth and strongly smooth points in symmetric spaces of measurable operators

Czerwińska, Małgorzata; Kamińska, Anna; Kubiak, Damian

doi:10.1007/s11117-010-0108-2

Cited by 5 publications

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“…The analogous result on strongly smooth points in C E can be proved using similar techniques as in the case of E(M, τ ). The proof of the next theorem is a good overview of various strategies employed in [23]. By Lemma 13.4 applied for M = B(H) with the canonical trace tr we have that |tr(x)| ≤ tr(|x|) for x ∈ C 1 and tr(|xy|) ≤ ∞ n=1 s n (x)s n (y) for x, y ∈ B(H).…”

Section: Strong Smoothnessmentioning

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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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: Strong Smoothnessmentioning

confidence: 99%

“…The characterization of smooth points of B E(M,τ ) was done in [23], for order continuous symmetric function spaces E.…”

Section: Smoothnessmentioning

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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“…It is known that there is a unital * -isomorphism from S (M p , τ p ) onto pS (M, τ ) p. Moreover, the decreasing rearrangement µ τp computed with respect to the von Neumann algebra (M p , τ p ) is given by µ τp (y) = µ(pyp), y ∈ S (M p , τ p ). See [23,35] for details.…”

Section: Trace Preserving Isomorphismsmentioning

confidence: 99%

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

Czerwińska¹,

Kamińska²

2017

Preprint

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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 the concep...

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“…Let M be a non-atomic von Neumann algebra with a faithful, normal, semifinite trace τ . Given a positive operator x ∈ S 0 (M, τ ), such that τ (s(x)) = τ (1), the von Neumann algebra M s(x) = {s(x)y| s(x)(H) : y ∈ M} has σ -finite trace τ s(x) (for details see [6,7]). The singular value function µ τ s(x) computed with respect to the von Neumann…”

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Complex uniform rotundity in symmetric spaces of measurable operators

Czerwińska¹

2012

Journal of Mathematical Analysis and Applications

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Keywords:Symmetric spaces of measurable operators Unitary matrix spaces Complex uniform rotundity Uniform monotonicity of a norm Uniform Kadec-Klee property with respect to a local convergence in measure a b s t r a c t Let M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ and E be a symmetric Banach function space on [0, τ (1)). We show that E is complex uniformly rotund if and only if E(M, τ ) + is complex uniformly rotund. Moreover, under the assumption that E is p-convex for some p > 1, complex uniform rotundity of E implies complex uniform rotundity of E(M, τ ). Therefore if E has non-trivial convexity, complex uniform convexity of E is equivalent with complex uniform convexity of E(M, τ ).We obtain an analogous result for the unitary matrix space C E and a symmetric Banach sequence space E. From the above we conclude that E(M, τ ) + is complex uniformly rotund if and only if its norm ∥ · ∥ E(M,τ ) is uniformly monotone.Complex uniform rotundity of complex normed spaces was introduced by Globevnik in [1] as a natural generalization of uniform convexity. In addition to showing that the complex space L 1 is complex uniformly convex, Globevnik gave an application of complex uniform convexity to analytic functions.Complex uniform convexity of the Schatten class C 1 = C ℓ 1 was first discovered by Haagerup, as a consequence of the more general result included in [2], stating that the dual of C * -algebra is uniformly complex convex. Later, Mattila in [3], gave an alternative proof of the complex uniform convexity of C 1 .In 1990, Xu observed that if concavity and convexity constants of a symmetric Banach space E are both equal to 1, then E(M, τ ) is a complex uniformly rotund space, for any semifinite von Neumann algebra M. Moreover, he showed that if E is q-concave for some q < ∞ then the space E(M, τ ) is complex uniformly convexifiable, [4].The goal of this paper is to show that the symmetric Banach function space E is complex uniformly rotund if and only if E(M, τ ) + is complex uniformly rotund, where M is a semifinite von Neumann algebra, with a faithful, normal, semifinite trace τ . We obtain also that complex uniform convexity of E is inherited by E(M, τ ), if E has non-trivial p-convexity. The analogous results follow for the unitary matrix space C E . The paper is organized as follows. In the preliminary section we explain definitions and notations used throughout the paper. In the second section we investigate the relation between complex uniform rotundity of E and E(M, τ ). In the last section we present the direct proof of complex uniform rotundity of L 1 (M, τ ). PreliminariesWe begin by recalling some necessary terminology. We refer the reader to [5][6][7], for example, for further details.Let M be a semifinite von Neumann algebra on the Hilbert space H with the unit element 1, equipped with a faithful normal semifinite trace τ . Given a self-adjoint (possibly unbounded) linear operator, we denote by e x (·) the spectral measure of x.

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Smooth and strongly smooth points in symmetric spaces of measurable operators

Cited by 5 publications

References 21 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

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

Complex uniform rotundity in symmetric spaces of measurable operators

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