Local Minimal Curves in Homogeneous Reductive Spaces of the Unitary Group of a Finite von Neumann Algebra

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

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Coordinates for analytic operator algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500007527 ◽

1989 ◽

Vol 31 (1) ◽

pp. 31-47

Author(s):

Baruch Solel

Keyword(s):

Abelian Group ◽

Von Neumann Algebra ◽

Operator Algebras ◽

Dual Group ◽

Positive Semigroup ◽

Analytic Operator ◽

Von Neumann ◽

Finite Von Neumann Algebra ◽

Analytic Algebra ◽

Image Position

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].

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Entropic upper bound for Bayes risk in the quantum case

Probability and Mathematical Statistics ◽

10.19195/0208-4147.38.2.9 ◽

2018 ◽

Vol 38 (2) ◽

pp. 429-440

Author(s):

Rafał Wieczorek ◽

Hanna Podsędkowska

Keyword(s):

Lower Bound ◽

Loss Function ◽

Upper Bound ◽

Von Neumann Algebra ◽

General Setting ◽

Probability Of Detection ◽

Bayes Risk ◽

Quantum Case ◽

Von Neumann ◽

Finite Von Neumann Algebra

The entropic upper bound for Bayes risk in a general quantum case is presented. We obtained generalization of the entropic lower bound for probability of detection. Our result indicates upper bound for Bayes risk in a particular case of loss function – for probability of detection in a pretty general setting of an arbitrary finite von Neumann algebra. It is also shown under which condition the indicated upper bound is achieved.

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Local and 2-local derivations on noncommutative Arens algebras

Mathematica Slovaca ◽

10.2478/s12175-014-0215-9 ◽

2014 ◽

Vol 64 (2) ◽

Cited By ~ 9

Author(s):

Shavkat Ayupov ◽

Karimbergen Kudaybergenov ◽

Berdakh Nurjanov ◽

Amir Alauadinov

Keyword(s):

Von Neumann Algebra ◽

Inner Derivation ◽

Von Neumann ◽

Local Derivation ◽

Finite Von Neumann Algebra ◽

Finite Trace

AbstractThe paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.

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Gradient forms and strong solidity of free quantum groups

Mathematische Annalen ◽

10.1007/s00208-020-02109-y ◽

2020 ◽

Author(s):

Martijn Caspers

Keyword(s):

Quantum Groups ◽

Von Neumann Algebra ◽

Von Neumann Algebras ◽

Dirichlet Forms ◽

Von Neumann ◽

Finite Von Neumann Algebra ◽

Crucial Part

AbstractConsider the free orthogonal quantum groups $$O_N^+(F)$$ O N + ( F ) and free unitary quantum groups $$U_N^+(F)$$ U N + ( F ) with $$N \ge 3$$ N ≥ 3 . In the case $$F = \text {id}_N$$ F = id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $$L_\infty (O_N^+)$$ L ∞ ( O N + ) is strongly solid. Moreover, Isono obtains strong solidity also for $$L_\infty (U_N^+)$$ L ∞ ( U N + ) . In this paper we prove for general $$F \in GL_N(\mathbb {C})$$ F ∈ G L N ( C ) that the von Neumann algebras $$L_\infty (O_N^+(F))$$ L ∞ ( O N + ( F ) ) and $$L_\infty (U_N^+(F))$$ L ∞ ( U N + ( F ) ) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.

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RELATIVE TORSION

Communications in Contemporary Mathematics ◽

10.1142/s0219199701000287 ◽

2001 ◽

Vol 03 (01) ◽

pp. 15-85 ◽

Cited By ~ 8

Author(s):

DAN BURGHELEA ◽

LEONID FRIEDLANDER ◽

THOMAS KAPPELER

Keyword(s):

Fundamental Group ◽

Finite Type ◽

Von Neumann Algebra ◽

Hilbert Module ◽

Von Neumann ◽

Geometric Invariants ◽

Finite Dimensional ◽

Finite Von Neumann Algebra ◽

Special Cases ◽

Relative Torsion

This paper achieves, among other things, the following: • It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. • It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an [Formula: see text]-Hilbert module of finite type ([Formula: see text] a finite von Neumann algebra). The result of [3] corresponds to [Formula: see text]. • It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).

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Embeddings of ℓp into Non-commutative Spaces

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700003359 ◽

2003 ◽

Vol 74 (3) ◽

pp. 331-350 ◽

Cited By ~ 2

Author(s):

Narcisse Randrianantoanina

Keyword(s):

Symmetric Space ◽

Von Neumann Algebra ◽

Lorentz Spaces ◽

Natural Conditions ◽

Von Neumann ◽

Finite Von Neumann Algebra ◽

Measurable Operators

AbstractLet ℳ be a semi-finite von Neumann algebra equipped with a faithful normal trace τ. We prove a Kadec-Pelczyński type dichotomy principle for subspaces of symmetric space of measurable operators of Rademacher type 2. We study subspace structures of non-commutative Lorentz spaces Lp, q, (ℳ, τ), extending some results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, ℓp cannot be embedded into Lp, q (ℳ, τ). As applications, we prove that for 0 < p < ∞ with p ≠ 2, ℓp cannot be strongly embedded into Lp(ℳ, τ). This provides a non-commutative extension of a result of Kalton for 0 < p < 1 and a result of Rosenthal for 1 ≦ p < 2 on Lp [0, 1].

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