scholarly journals INTERSECTING JONES PROJECTIONS

2005 ◽  
Vol 16 (06) ◽  
pp. 687-691
Author(s):  
SEBASTIANO CARPI
Keyword(s):  
Hilbert Space ◽  
Normal State ◽  
Von Neumann Algebra ◽  
Unit Vector ◽  
Von Neumann ◽  
Conditional Expectations ◽  
Faithful Normal State

Let M be a von Neumann algebra on a Hilbert space ℋ with a cyclic and separating unit vector Ω and let ω be the faithful normal state on M given by ω(·) = (Ω,·Ω). Moreover, let {Ni : i ∈ I} be a family of von Neumann subalgebras of M with faithful normal conditional expectations Ei of M onto Ni satisfying ω = ω ◦ Ei for all i ∈ I and let N = ∩i∈I Ni. We show that the projections ei, e of ℋ onto the closed subspaces [Formula: see text] and [Formula: see text] respectively satisfy e = ∧i∈I ei. This proves a conjecture of Jones and Xu in [1].

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.

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-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.

Dense Subalgebras of Left Hilbert Algebras

1982 ◽  
Vol 34 (6) ◽  
pp. 1245-1250 ◽  
Author(s):  
A. van Daele
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Cyclic Vector ◽  
Quantum Field ◽  
Von Neumann ◽  
Hilbert Algebras

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

On Products of Symmetries

1966 ◽  
Vol 18 ◽  
pp. 897-900 ◽  
Author(s):  
Peter A. Fillmore
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Type Ii ◽  
Central Projections ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Principal Tool ◽  
The Subject

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

Correlations in a General Theory of Quantum Measurement

2007 ◽  
Vol 14 (04) ◽  
pp. 445-458 ◽  
Author(s):  
Hanna Podsędkowska
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
General Theory ◽  
Quantum Measurement ◽  
Von Neumann Algebra ◽  
Classical Case ◽  
Von Neumann ◽  
Full Algebra ◽  
Algebraic Formalism ◽  
Algebra Of Operators

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states — by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.

scholarly journals Generators of reflexive algebras

1975 ◽  
Vol 20 (2) ◽  
pp. 159-164
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Bounded Operators ◽  
Finitely Generated ◽  
Separable Space ◽  
Von Neumann ◽  
Underlying Space ◽  
Weakly Closed ◽  
Reflexive Algebras

For any collection of closed subspaces of a complex Hilbert space the set of bounded operators that leave invariant all the members of the collection is a weakly-closed algebra. The class of such algebras is precisely the class of reflexive algebras as defined for example in Radjavi and Rosenthal (1969) and contains the class of von Neumann algebras.In this paper we consider the problem of when such algebras are finitely generated as weakly-closed algebras. It is to be hoped that analysis of this problem may shed some light on the famous unsolved problem of whether every von Neumann algebra on a separable Hilbert space is finitely generated. The case where the underlying space is separable and the collection of subspaces is totally ordered is dealt with in Longstaff (1974). In the present paper the result of Longstaff (1974) is generalized to the case of a direct product of countably many totally ordered collections each on a separable space. Also a method of obtaining non-finitely generated reflexive algebras is given.

Short Geodesics of Unitaries in the L2 Metric

2005 ◽  
Vol 48 (3) ◽  
pp. 340-354 ◽  
Author(s):  
Esteban Andruchow
Keyword(s):  
Hilbert Space ◽  
Topological Group ◽  
Unitary Group ◽  
Von Neumann Algebra ◽  
Type Ii ◽  
Von Neumann ◽  
Smooth Curves

AbstractLet ℳ be a type II1 von Neumann algebra, τ a trace in ℳ, and L2 (ℳ, τ) the GNS Hilbert space of τ . We regard the unitary group Uℳ as a subset of L2(ℳ, τ) and characterize the shortest smooth curves joining two fixed unitaries in the L2 metric. As a consequence of this we obtain that Uℳ, though a complete (metric) topological group, is not an embedded riemannian submanifold of L2(ℳ, τ)

Commutators in Factors of Type III

1966 ◽  
Vol 18 ◽  
pp. 1152-1160 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Identity Operator ◽  
Von Neumann ◽  
Type Iii ◽  
Underlying Space ◽  
Special Cases ◽  
Weakly Closed ◽  
Algebra Of Operators

Let denote a separable, complex Hilbert space, and let R be a von Neumann algebra acting on . (A von Neumann algebra is a weakly closed, self-adjoint algebra of operators that contains the identity operator on its underlying space.) An element A of R is a commutator in R if there exist operators B and C in R such that A = BC — CB. The problem of specifying exactly which operators are commutators in R has been solved in certain special cases; e.g. if R is an algebra of type In (n < ∞) (2), and if R is a factor of type I∞ (1). It is the purpose of this note to treat the same problem in case R is a factor of type III. Our main result is the following theorem.

A UNIFIED APPROACH TO GENERALIZED CONDITIONAL EXPECTATIONS OPERATOR VALUED WEIGHTS AND RADON–NIKODYM DERIVATIVES ON VON NEUMANN ALGEBRAS

2002 ◽  
Vol 05 (03) ◽  
pp. 409-419
Author(s):  
STEFANO CAVALLARO ◽  
CARLO CECCHINI
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Unified Approach ◽  
Von Neumann ◽  
Canonical Map ◽  
Conditional Expectations

Given two von Neumann algebras, ℳ and [Formula: see text] with [Formula: see text], and two normal semifinite faithful weights, φ and ψ on ℳ and [Formula: see text] respectively, we define a canonical map from {b ∈ ℳ+ | φ(b)< ∞} to the set of positive forms on the Hilbert space of the GNS representation of [Formula: see text] associated to ψ. We show that generalized conditional expectations, operator valued weights and Radon–Nikodym derivatives on von Neumann algebras can be obtained from particular cases of this canonical map.

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