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)

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.

An inner measure associated with a von Neumann algebra

1968 ◽  
Vol 64 (3) ◽  
pp. 645-650 ◽  
Author(s):  
G. de Barra
Keyword(s):  
Hilbert Space ◽  
Finite Type ◽  
Measure Space ◽  
Von Neumann Algebra ◽  
Dimension Function ◽  
Type Ii ◽  
Von Neumann ◽  
Linear Sets ◽  
Atomic Measure ◽  
Starting Point

An ‘inner measure’ analogous to Lebesgue inner measure and associated with a von Neumann algebra is constructed on the linear sets of a Hilbert space. We supposed to be of finite type and countably decomposable, as these restrictions will be necessary for some of the results obtained. As remarked by Dye in (2), Type II1 algebras have a structure analogous to that of a finite non-atomic measure space, the trace corresponding to the measure. We define a class of ‘measurable sets’ and obtain some of its properties. The development of the theory is indicated also from the starting point of a function-valued dimension function. The idea for the construction of such an inner measure comes from a remark in (3), paragraph 16·2, and the terminology of (3) and of (5) is used throughout.

The covariance algebra of an extended covariant system

1979 ◽  
Vol 85 (2) ◽  
pp. 271-280 ◽  
Author(s):  
Ronny Rousseau
Keyword(s):  
Hilbert Space ◽  
Compact Group ◽  
Unitary Group ◽  
Von Neumann Algebra ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Additional Axiom

Let M be a von Neumann algebra acting on a Hilbert space , and let G be a locally compact group. We consider an extension of G by , the unitary group of M. If the triple satisfies an additional axiom, we say that it is an extended covariant system. We define a Hilbert space and operators , acting on . The von Neumann algebra is then the covariance algebra of the extended covariant system , denoted by .

scholarly journals On completely singular von Neumann subalgebras

2009 ◽  
Vol 52 (3) ◽  
pp. 607-618 ◽  
Author(s):  
Junsheng Fang
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Type Ii ◽  
Von Neumann ◽  
Separable Predual

AbstractLet ℳ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $\mathcal{N}$ be a von Neumann subalgebra of ℳ. If $\mathcal{N}\operatorname{\bar{\otimes}}\mathcal{B}(\mathcal{K})$ is singular in $\mathcal{M}\operatorname{\bar{\otimes}}\mathcal{B}(\mathcal{K})$ for every Hilbert space $\mathcal{K}$, $\mathcal{N}$ is said to be completely singular in ℳ. We prove that if $\mathcal{N}$ is a singular abelian von Neumann subalgebra or if $\mathcal{N}$ is a singular subfactor of a type-II1 factor ℳ, then $\mathcal{N}$ is completely singular in ℳ. $\mathcal{H}$ is separable, we prove that $\mathcal{N}$ is completely singular in ℳ if and only if, for every θ∈Aut($\mathcal{N}$′) such that θ(X)=X for all X ∈ ℳ′, θ(Y)=Y for all Y∈$\mathcal{N}$′. As the first application, we prove that if ℳ is separable (with separable predual) and $\mathcal{N}$ is completely singular in ℳ, then $\mathcal{N}\operatorname{\bar{\otimes}}\mathcal{L}$ is completely singular in $\mathcal{M}\operatorname{\bar{\otimes}}\mathcal{L}$ for every separable von Neumann algebra $\mathcal{L}$. As the second application, we prove that if $\mathcal{N}$1 is a singular subfactor of a type-II1 factor ℳ1 and $\mathcal{N}$2 is a completely singular von Neumann subalgebra of ℳ2, then $\mathcal{N}_1\operatorname{\bar{\otimes}}\mathcal{N}_2$ is completely singular in $\mathcal{M}_1\operatorname{\bar{\otimes}}\mathcal{M}_2$.

GRASSMANNIANS OF A FINITE ALGEBRA IN THE STRONG OPERATOR TOPOLOGY

2006 ◽  
Vol 17 (04) ◽  
pp. 477-491 ◽  
Author(s):  
ESTEBAN ANDRUCHOW ◽  
LÁZARO RECHT
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Finite Algebra ◽  
Strong Operator Topology ◽  
Smooth Curve ◽  
Minimal Length ◽  
Type Ii ◽  
Von Neumann ◽  
Space Norm ◽  
Minimal Geodesic

If [Formula: see text] is a type II1 von Neumann algebra with a faithful trace τ, we consider the set [Formula: see text] of self-adjoint projections of [Formula: see text] as a subset of the Hilbert space [Formula: see text]. We prove that though it is not a differentiable submanifold, the geodesics of the natural Levi–Civita connection given by the trace have minimal length. More precisely: the curves of the form γ(t) = eitxpe-itx with x* = x, pxp = (1 - p)x(1 - p) = 0 have minimal length when measured in the Hilbert space norm of [Formula: see text], provided that the operator norm ‖x‖ is less or equal than π/2. Moreover, any two projections which are unitary equivalent are joined by at least one such minimal geodesic, and only unitary equivalent projections can be joined by a smooth curve. Finally, we prove that these geodesics have also minimal length if one measures them with the Schatten k-norms of τ, ‖x‖k = τ((x* x)k/2)1/k, for all k ∈ ℝ, k ≥ 0. We also characterize curves of unitaries which have minimal length with these k-norms.

Non-Isomorphic Non-Hyperfinite Factors

1969 ◽  
Vol 21 ◽  
pp. 1293-1308 ◽  
Author(s):  
Wai-Mee Ching
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Crossed Product ◽  
Type Ii ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Type Iii ◽  
Infinite Dimensional Hilbert Space ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space

A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II1 (4, Theorem 16’), and of type III (8, Theorem 1); all hyperfinite factors of type Hi are isomorphic (4, Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II1 (10), and of type III (11). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II1 (Theorem 2), and of type III (Theorem 3).Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.

Von Neumann algebras as complemented subspaces of $B({\mathscr{H}})$

2014 ◽  
Vol 25 (11) ◽  
pp. 1450107 ◽  
Author(s):  
Erik Christensen ◽  
Liguang Wang
Keyword(s):  
Hilbert Space ◽  
Fundamental Group ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Type Ii ◽  
Von Neumann ◽  
Algebraic Criterion ◽  
Complemented Subspaces ◽  
Complemented Subspace ◽  
Injective Von Neumann Algebra

Let [Formula: see text] be a von Neumann algebra of type II1 which is also a complemented subspace of [Formula: see text]. We establish an algebraic criterion, which ensures that [Formula: see text] is an injective von Neumann algebra. As a corollary we show that if [Formula: see text] is a complemented factor of type II1 on a Hilbert space [Formula: see text], then [Formula: see text] is injective if its fundamental group is nontrivial.

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]).

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