Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

Czerwińska, Małgorzata; Kamińska, Anna

doi:10.4064/sm201-3-3

Cited by 18 publications

(47 citation statements)

References 32 publications

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“…By E × = ℓ ∞ , the space C E × is well defined, and applying the analogous argument as in the proof of Proposition 2.3 in [21], we can show that y is an order continuous element of C E × . Finally, [21, Proposition 1.5] implies that y − y n C E × → 0.…”

Section: Strong Smoothnessmentioning

confidence: 93%

“…Complex extreme points of noncommutative symmetric spaces were only studied in [21]. The characterization of the complex extreme points is analogous to the results on extreme points in [16].…”

Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 95%

“…Suppose that x ± y, x ± iy belong to B E(M,τ ) , for some y ∈ B E(M,τ ) . Without loss of generality it can be assumed that y is a self-adjoint operator [21,Lemma 3.2]. Now by [102,Proposition 3], for all t > 0, µ(t, x) ≤ µ(t, x + iy).…”

Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 99%

“…One can also show that S(y)−S ((y + y n )/2) E × → 0. By the assumption E = ℓ 1 we have E × = ℓ ∞ , and by [21,Lemma 1.3], y n tr → y, that is y n converges to y in measure.…”

Section: Strong Smoothnessmentioning

confidence: 99%

“…It was demonstrated in [21] that the notions of C − LU R and C − M LU R points, and hence the notions of C − LU R and C − M LU R spaces, are equivalent in any complex normed space. Consequently, in complex normed spaces these complex properties are not distinguishable contrary to their corresponding "real" properties LU R and M LU R [72].…”

Section: Complex Local Uniform Convexitymentioning

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: 93%

Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 95%

Section: Complex Extreme Points and Convex Convexitymentioning

confidence: 99%

“…One can also show that S(y)−S ((y + y n )/2) E × → 0. By the assumption E = ℓ 1 we have E × = ℓ ∞ , and by [21,Lemma 1.3], y n tr → y, that is y n converges to y in measure.…”

Section: Strong Smoothnessmentioning

confidence: 99%

Section: Complex Local Uniform Convexitymentioning

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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k-Extreme Points in Symmetric Spaces of Measurable Operators

Czerwińska¹,

Kamińska²

2014

Integr. Equ. Oper. Theory

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Abstract. Let M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ and E be a strongly symmetric Banach function space on [0, τ (1)). We show that an operator x in the unit sphere of E(M, τ ) is k-extreme, k ∈ N, whenever its singular value function µ(x) is k-extreme and one of the following conditions hold (i) µ(∞, x) = limt→∞ µ(t, x) = 0 or (ii) n(x)Mn(x * ) = 0 and |x| ≥ µ(∞, x)s(x), where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever M is nonatomic. The global k-rotundity property follows, that is if M is non-atomic then E is k-rotund if and only if E(M, τ ) is k-rotund. As a consequence of the noncommutive results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement µ(f ) is k-extreme and |f | ≥ µ(∞, f ). We conclude with the corollary on orbits Ω(g) and Ω ′ (g). We get that f is a k-extreme point of the orbit Ω(g), g ∈ L 1 + L∞, or Ω ′ (g), g ∈ L 1 [0, α), α < ∞, if and only if µ(f ) = µ(g) and |f | ≥ µ(∞, f ). From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.Acknowledgement. The final publication is available at Springer via http://dx

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Smooth and strongly smooth points in symmetric spaces of measurable operators

2011

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We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ -measurable operators E(M, τ ) associated to a semifinite von Neumann algebra (M, τ ). We prove that x is a smooth point of the unit ball in E(M, τ ) if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f ) = 0, for the function f ∈ S E × supporting μ(x), or s(x * ) = 1. Under the assumption that the trace τ on M is σ -finite, we show that x is strongly smooth point of the unit ball in E(M, τ ) if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ( f ). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M, τ ).

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Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

Cited by 18 publications

References 32 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

k-Extreme Points in Symmetric Spaces of Measurable Operators

Smooth and strongly smooth points in symmetric spaces of measurable operators

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