Borel cocycles, approximation properties and relative property T

2000 ◽  
Vol 20 (2) ◽  
pp. 483-499 ◽  
Author(s):  
PAUL JOLISSAINT
Keyword(s):  
Von Neumann Algebra ◽  
Approximation Property ◽  
Compact Groups ◽  
Approximation Properties ◽  
Von Neumann ◽  
Rigidity Results ◽  
Relative Property ◽  
Finite Von Neumann Algebra ◽  
Property T ◽  
Standard Probability

Let $G$ and $H$ be locally compact groups. Assume that $G$ acts on a standard probability space $(S,\mu)$, $\mu$ being $G$-invariant. We prove that if there exists a Borel cocycle $\alpha:S\times G\longrightarrow H$ which is proper in an appropriate sense, then $G$ inherits some approximation properties of $H$, for instance amenability or the so-called Haagerup Approximation Property. On the other hand, if $G_{0}$ is a closed subgroup of $G$, if the pair $(G,G_{0})$ has the relative property (T) of Margulis [19] and if either $H$ has Haagerup Approximation Property, or if it is the unitary group of a finite von Neumann algebra with a similar property, then we give rigidity results analogous to that in [23] and [1].

scholarly journals A VON NEUMANN ALGEBRA CHARACTERIZATION OF PROPERTY (T) FOR GROUPOIDS

2018 ◽  
Vol 108 (3) ◽  
pp. 363-386
Author(s):  
MARTINO LUPINI
Keyword(s):  
Probability Measure ◽  
Von Neumann Algebra ◽  
Discrete Probability ◽  
Von Neumann ◽  
Relative Property ◽  
Property T ◽  
Algebra Characterization

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusions $L(H)\subset L(G)$.

Correspondences and Approximation Properties for von Neumann Algebras

2003 ◽  
Vol 14 (06) ◽  
pp. 619-665 ◽  
Author(s):  
Jon Kraus
Keyword(s):  
Approximation Property ◽  
Von Neumann Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Approximation Properties ◽  
Locally Compact ◽  
Von Neumann ◽  
Weak Amenability ◽  
Operator Approximation ◽  
Weak Operator

The notion of the amenability of a locally compact group has been extended in various ways. Two weaker versions of amenability, weak amenability and the approximation property, have been defined for locally compact groups (by Haagerup and Haagerup and Kraus, respectively) and Bekka has defined a notion of amenability for representations of locally compact groups. Correspondences can be viewed as a generalization of representations of such groups. Using this viewpoint, Ananthraman–Delaroche has defined a notion of (left) amenability for correspondences. In this paper, we define notions of weak amenability and the approximation property for correspondences (and representations of locally compact groups), and obtain various results concerning these notions. Ananthraman–Delaroche showed that if N ⊂ M is an inclusion of von Neumann algebras, and if the associated inclusion correspondence is left amenable, then various approximation properties of N (semidiscreteness, the weak* completely bounded approximation property, and the weak* operator approximation property) are shared by M. We show that if this correspondence has the (weaker) approximation property, then if N has the weak* operator approximation property, so does M. An application of this result to crossed products is also given.

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

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 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 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.

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.

scholarly journals RELATIVE TORSION

2001 ◽  
Vol 03 (01) ◽  
pp. 15-85 ◽  
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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