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.

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.

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.

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.

On the von Neumann algebras associated to Yang–Baxter operators

2020 ◽  
pp. 1-24
Author(s):  
Panchugopal Bikram ◽  
Rahul Kumar ◽  
Rajeeb Mohanta ◽  
Kunal Mukherjee ◽  
Diptesh Saha
Keyword(s):  
Hilbert Space ◽  
Special Form ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
Finite Von Neumann Algebra

Bożejko and Speicher associated a finite von Neumann algebra M T to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$ . We show that if dim $(\mathcal {H})$ ⩾ 2, then M T is a factor when T admits an eigenvector of some special form.

scholarly journals Boolean algebras of projections

1975 ◽  
Vol 19 (3) ◽  
pp. 287-289
Author(s):  
P. G. Spain
Keyword(s):  
Hilbert Space ◽  
Boolean Algebra ◽  
Von Neumann Algebra ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Von Neumann ◽  
Relative Compactness ◽  
Weakly Closed ◽  
Result Theorem ◽  
Bounded Sets

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).

Commutativity Preserving Maps of Factors

1988 ◽  
Vol 40 (1) ◽  
pp. 248-256 ◽  
Author(s):  
C. Robert Miers
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Von Neumann Algebra ◽  
Identity Operator ◽  
Closed Field ◽  
Von Neumann ◽  
Linear Map ◽  
One To One ◽  
Image Position ◽  
Commuting Pairs

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.

Operators of Rank One in Reflexive Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 19-23 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Closed Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Identity Operator ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Rank One ◽  
Weakly Closed ◽  
Reflexive Algebras

If H is a (complex) Hilbert space and is a collection of (closed linear) subspaces of H it is easily shown that the set of all (bounded linear) operators acting on H which leave every member of invariant is a weakly closed operator algebra containing the identity operator. This algebra is denoted by Alg . In the study of such algebras it may be supposed [4] that is a subspace lattice i.e. that is closed under the formation of arbitrary intersections and arbitrary (closed linear) spans and contains both the zero subspace (0) and H. The class of such algebras is precisely the class of reflexive algebras [3].

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.

Analytic Subalgebras of Von Neumann Algebras

1987 ◽  
Vol 39 (1) ◽  
pp. 74-99 ◽  
Author(s):  
Paul S. Muhly ◽  
Kichi-Suke Saito
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Identity Operator ◽  
Boundary Values ◽  
Von Neumann ◽  
Weakly Continuous ◽  
Closed Subalgebra ◽  
Weakly Closed ◽  
Upper Half Plane ◽  
Image Position

Let M be a von Neumann algebra and let {αt}t∊R be a σ-weakly continuous flow on M; i.e., suppose that {αt}t∊R is a one-parameter group of *-automorphisms of M such that for each ρ in the predual, M∗, of M and for each x ∊ M, the function of t, ρ(αt(x)), is continuous on R. In recent years, considerable attention has been focused on the subspace of M, H∞(α), which is defined to bewhere H∞(R) is the classical Hardy space consisting of the boundary values of functions bounded analytic in the upper half-plane. In Theorem 3.15 of [8] it is proved that in fact H∞(α) is a σ-weakly closed subalgebra of M containing the identity operator such thatis σ-weakly dense in M, and such that

scholarly journals ON PRODUCT SYSTEMS ARISING FROM SUM SYSTEMS

2005 ◽  
Vol 08 (01) ◽  
pp. 1-31 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
R. SRINIVASAN
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebra ◽  
Type I ◽  
Product System ◽  
Von Neumann ◽  
Type Iii ◽  
Product Systems ◽  
Weyl Representation ◽  
Uncountable Family ◽  
Type Spaces

B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy8). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson. By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.

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