Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

2007 ◽  
Vol 1 (3) ◽  
pp. 367-398 ◽  
Author(s):  
Ilwoo Cho
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann

Further results on the relative entropy

1987 ◽  
Vol 101 (2) ◽  
pp. 363-373 ◽  
Author(s):  
Matthew J. Donald
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebra ◽  
Measurement Problem ◽  
Bounded Operators ◽  
Many Worlds ◽  
Von Neumann ◽  
Normal States ◽  
Injective Von Neumann Algebra ◽  
Made In

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.

scholarly journals On characterization of integrable sesquilinear forms

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
A. Sherstnev ◽  
O. Tikhonov
Keyword(s):  
Normal State ◽  
Von Neumann Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Sesquilinear Forms ◽  
Von Neumann ◽  
Sesquilinear Form ◽  
Faithful Normal State ◽  
Necessary And Sufficient

AbstractWe give a necessary and sufficient condition for a sesquilinear form to be integrable with respect to a faithful normal state on a von Neumann algebra.

Isometries of non-commutative Lp-spaces

1981 ◽  
Vol 90 (1) ◽  
pp. 41-50 ◽  
Author(s):  
F. J. Yeadon
Keyword(s):  
Von Neumann Algebra ◽  
Function Spaces ◽  
Main Step ◽  
Main Result Theorem ◽  
Von Neumann ◽  
Lp Spaces ◽  
Result Theorem

The spaces Lp(, φ) for 1 ≤ p ≤ ∞, where φ is a faithful semifinite normal trace on a von Neumann algebra , are defined in (10),(2),(14). The problem of determining the general form of an isometry of one such space into another has been studied in (i), (6), (9), (12), (5). Our main result, Theorem 2, is a characterization of such isometries for 1 ≤ p ≤ ∞, ≠ 2. The method of proof is based on that of (7), where isometries between Lp function spaces are characterized. The main step in the proof is Theorem 1, which gives the conditions under which equality holds in Clarkson's inequality.

Voiculescu amenable Hopf von Neumann algebras with applications

2016 ◽  
Vol 15 (06) ◽  
pp. 1650079 ◽  
Author(s):  
Fatemeh Akhtari ◽  
Rasoul Nasr-Isfahani
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Continuous Functions ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann

For a Hopf von Neumann algebra [Formula: see text], we give a fixed point characterization of Voiculescu amenability of [Formula: see text] in terms of modules over [Formula: see text]. As a consequence, we present some descriptions for amenability of locally compact groups in terms of certain associated Hopf von Neumann algebras. We finally apply this result to some modules of continuous functions on a multiplicative subsemigroup of [Formula: see text].

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 $L(H)\subset L(G)$.

scholarly journals Spectral gap characterization of full type III factors

2019 ◽  
Vol 2019 (753) ◽  
pp. 193-210 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Discrete Group ◽  
Von Neumann Algebra ◽  
Spectral Gap ◽  
Crossed Product ◽  
Von Neumann ◽  
Type Iii ◽  
Outer Action ◽  
Usual Topology ◽  
Sigma G

AbstractWe give a spectral gap characterization of fullness for type {\mathrm{III}} factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if M is a full factor and {\sigma:G\rightarrow\mathrm{Aut}(M)} is an outer action of a discrete group G whose image in {\mathrm{Out}(M)} is discrete, then the crossed product von Neumann algebra {M\rtimes_{\sigma}G} is also a full factor. We apply this result to prove the following conjecture of Tomatsu–Ueda: the continuous core of a type {\mathrm{III}_{1}} factor M is full if and only if M is full and its τ invariant is the usual topology on {\mathbb{R}}.

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.

Sign in / Sign up

Export Citation Format

Share Document