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

• Autorzy: Czerwińska M. , Kamińska A.

Identyfikatory

DOI

10.14708/cm.v57i1.3291

• Języki publikacji: EN

Abstrakty

EN

This is a review article of geometric properties of noncommutative symmetric spaces of measurable operators E(M., t), where M is a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ, and E is a symmetric function space. If E c_o 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 Polya, Kothe duality, the spaces L_p (M, τ), 1 ≤p < ∞, the identification of C_E and G(B(H),tr) for some symmetric function space G, the commutative case when E is identified with E(N, t) 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 for removing the assumption of non-atomicity of . 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-Nikodym property and stability in the sense of Krivine-Maurey. We also state some open problems.

Słowa kluczowe

EN

symmetric spaces of measurable operators; unitary matrix spaces; rearrangement invariant spaces;; k-extreme point; k-convexity; complex extreme points; complex convexity; strict and uniform monotonicity; uniform convexity; p-convexity; p-concavity; strong smoothness; strongly exposed points; uniform Kadec-Klee properties; Banach-Saks properties; Radon-Nikodym property; Krivine-Maurey stability

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2017
• Tom: Vol 57, No. 1
• Strony: 45--122
• Opis fizyczny: Bibliogr. 116 poz.,

Twórcy

autor

Czerwińska M.

• Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224

autor

Kamińska A.

• Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152

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Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).