scholarly journals Von Neumann Algebra Preduals Satisfy the Linear Biholomorphic Property

2016 ◽  
Vol 118 (2) ◽  
pp. 277
Author(s):  
Antonio M. Peralta ◽  
László L. Stachó
Keyword(s):  
Von Neumann Algebra ◽  
Symmetric Part ◽  
Von Neumann ◽  
Infinite Dimensional

We prove that for every $\mathrm{JBW}^*$-triple $E$ of rank $>1$, the symmetric part of its predual reduces to zero. Consequently, the predual of every infinite dimensional von Neumann algebra $A$ satisfies the linear biholomorphic property, that is, the symmetric part of $A_*$ is zero.

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.

ENERGY REPRESENTATIONS OF INFINITE DIMENSIONAL GAUGE GROUPS IN NONCOMMUTATIVE GEOMETRY

1994 ◽  
Vol 05 (03) ◽  
pp. 329-348
Author(s):  
JEAN MARION
Keyword(s):  
Gauge Group ◽  
Noncommutative Geometry ◽  
Von Neumann Algebra ◽  
Linear Operators ◽  
Consistent Theory ◽  
Bounded Linear ◽  
Von Neumann ◽  
Gauge Groups ◽  
Infinite Dimensional ◽  
Involutive Algebra

Let M be a compact smooth manifold, let [Formula: see text] be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let [Formula: see text] be the unital involutive algebra [Formula: see text], let [Formula: see text] be an hermitian projective right [Formula: see text]-module of finite type, and let [Formula: see text] be the gauge group of unitary elements of the unital involutive algebra [Formula: see text] of right [Formula: see text]-linear endomorphisms of [Formula: see text]. We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations [Formula: see text] can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group [Formula: see text], [Formula: see text] being the group of unitary elements of [Formula: see text], associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over [Formula: see text]. In the case where [Formula: see text] is a von Neumann algebra of type II 1 a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over [Formula: see text] always exist, and that each of them induces an energy representation.

Every Invertible Hilbert Space Operator is a Product of Seven Positive Operators

1995 ◽  
Vol 38 (2) ◽  
pp. 230-236 ◽  
Author(s):  
N. Christopher Phillips
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Invertible Operator ◽  
Positive Operators ◽  
Space Operator ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Hilbert Space Operator

AbstractWe prove that every invertible operator in a properly infinite von Neumann algebra, in particular in L(H) for infinite dimensional H, is a product of 7 positive invertible operators. This improves a result of Wu, who proved that every invertible operator in L(H) is a product of 17 positive invertible operators.

scholarly journals Oscillation Theory for the Density of States of High Dimensional Random Operators

2017 ◽  
Vol 2019 (15) ◽  
pp. 4579-4602
Author(s):  
Julian Groß mann ◽  
Hermann Schulz-Baldes ◽  
Carlos Villegas-Blas
Keyword(s):  
Density Of States ◽  
Von Neumann Algebra ◽  
Jacobi Operator ◽  
Oscillation Theory ◽  
Integrated Density Of States ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Sturm Liouville ◽  
Finite Trace

Abstract Sturm–Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

Best approximation in von Neumann algebras

1977 ◽  
Vol 81 (2) ◽  
pp. 233-236 ◽  
Author(s):  
A. Guyan Robertson
Keyword(s):  
Best Approximation ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Identity Operator ◽  
Chebyshev Subspace ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Continuous Complex ◽  
The One

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension > 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.

Denseness of norm-attaining operators into strictly convex spaces

1999 ◽  
Vol 129 (6) ◽  
pp. 1107-1114 ◽  
Author(s):  
M. D. Acosta
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Strictly Convex ◽  
Infinite Dimensional Banach Space ◽  
Strictly Convex Norm ◽  
Property B ◽  
Norm Attaining Operators ◽  
Norm Attaining ◽  
Convex Spaces

We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C0 and by Gowers for ℓp (1 < p < ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H1, any infinite-dimensional complex L1(μ), and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.

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.

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