Algebras of Analytic Operators associated with a Periodic Flow on a Von Neumann Algebra

1985 ◽  
Vol 37 (3) ◽  
pp. 405-429 ◽  
Author(s):  
Baruch Solel
Keyword(s):  
Von Neumann Algebra ◽  
Periodic Flow ◽  
Continuous Representation ◽  
Von Neumann ◽  
Weakly Continuous ◽  
Finite Von Neumann Algebra ◽  
Weakly Closed ◽  
Image Position ◽  
Special Case ◽  
Closed Subalgebras

Let M be a σ-finite von Neumann algebra and {σt}t∊T be a σ-weakly continuous representation of the unit circle, T, as *-automorphisms of M. Let H∞(σ) be the set of all x ∊ M such thatThe structure of H∞(σ) was studied by several authors (see [2-13]).The main object of this paper is to study the σ-weakly closed subalgebras of M that contain H∞(σ). In [12] this was done for the special case where H∞(σ) is a nonselfadjoint crossed product.Let Mn, for n ∊ Z, be the set of all x ∊ M such that

Analytic Subalgebras of Von Neumann Algebras

1987 ◽  
Vol 39 (1) ◽  
pp. 74-99 ◽  
Author(s):  
Paul S. Muhly ◽  
Kichi-Suke Saito
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Identity Operator ◽  
Boundary Values ◽  
Von Neumann ◽  
Weakly Continuous ◽  
Closed Subalgebra ◽  
Weakly Closed ◽  
Upper Half Plane ◽  
Image Position

Let M be a von Neumann algebra and let {αt}t∊R be a σ-weakly continuous flow on M; i.e., suppose that {αt}t∊R is a one-parameter group of *-automorphisms of M such that for each ρ in the predual, M∗, of M and for each x ∊ M, the function of t, ρ(αt(x)), is continuous on R. In recent years, considerable attention has been focused on the subspace of M, H∞(α), which is defined to bewhere H∞(R) is the classical Hardy space consisting of the boundary values of functions bounded analytic in the upper half-plane. In Theorem 3.15 of [8] it is proved that in fact H∞(α) is a σ-weakly closed subalgebra of M containing the identity operator such thatis σ-weakly dense in M, and such that

scholarly journals Coordinates for analytic operator algebras

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].

scholarly journals ON EQUALITY CONDITION FOR TRACE JENSEN INEQUALITY IN SEMI-FINITE VON NEUMANN ALGEBRAS

2008 ◽  
Vol 19 (04) ◽  
pp. 481-501 ◽  
Author(s):  
TETSUO HARADA ◽  
HIDEKI KOSAKI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Strict Convexity ◽  
Jensen Inequality ◽  
Von Neumann ◽  
Convexity Assumption ◽  
Finite Von Neumann Algebra ◽  
Equality Condition ◽  
Finite Von Neumann Algebras

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.

scholarly journals Entropic upper bound for Bayes risk in the quantum case

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.

scholarly journals Local and 2-local derivations on noncommutative Arens algebras

2014 ◽  
Vol 64 (2) ◽  
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.

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

scholarly journals Gradient forms and strong solidity of free quantum groups

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.

Ergodic theorems for semifinite von Neumann algebras: II

1980 ◽  
Vol 88 (1) ◽  
pp. 135-147 ◽  
Author(s):  
F. J. Yeadon
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Norm ◽  
Ergodic Theorems ◽  
Unbounded Operators ◽  
Von Neumann ◽  
The Mean ◽  
Image Position

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averagesconverge in norm.

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