Introduction to Von Neumann Algebras and Continuous Geometry

1960 ◽  
Vol 3 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Israel Halperin
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Operators ◽  
Mathematical Work ◽  
Type I ◽  
Type Ii ◽  
Von Neumann ◽  
The Past ◽  
Projection Geometry ◽  
Continuous Geometry

What is a von Neumann algebra? What is a factor (i) of type I, (ii) of type II, (iii) of type III? What is a projection geometry? And finally, what is a continuous geometry?The questions recall some of the most brilliant mathematical work of the past 30 years, work which was done by John von Neumann, partly in collaboration with F. J. Murray, and which grew out of von Neumann1 s analysis of linear operators in Hilbert space.

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.

On the Unitary Equivalence of Certain Classes of Non-Normal Operators. I

1971 ◽  
Vol 23 (5) ◽  
pp. 849-856 ◽  
Author(s):  
P. K. Tam
Keyword(s):  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Equivalence Problem ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Type I ◽  
Unitary Equivalence ◽  
Bounded Linear ◽  
Von Neumann ◽  
Normal Operators

The following (so-called unitary equivalence) problem is of paramount importance in the theory of operators: given two (bounded linear) operators A1, A2 on a (complex) Hilbert space , determine whether or not they are unitarily equivalent, i.e., whether or not there is a unitary operator U on such that U*A1U = A2. For normal operators this question is completely answered by the classical multiplicity theory [7; 11]. Many authors, in particular, Brown [3], Pearcy [9], Deckard [5], Radjavi [10], and Arveson [1; 2], considered the problem for non-normal operators and have obtained various significant results. However, most of their results (cf. [13]) deal only with operators which are of type I in the following sense [12]: an operator, A, is of type I (respectively, II1, II∞, III) if the von Neumann algebra generated by A is of type I (respectively, II1, II∞, III).

scholarly journals LOCAL DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2011 ◽  
Vol 13 (04) ◽  
pp. 643-657 ◽  
Author(s):  
S. ALBEVERIO ◽  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV ◽  
B. O. NURJANOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Continuity Condition ◽  
Type I ◽  
Atomic Lattice ◽  
Von Neumann ◽  
Measurable Operators ◽  
Finite Trace ◽  
Necessary And Sufficient

The paper is devoted to local derivations on the algebra [Formula: see text] of τ-measurable operators affiliated with a von Neumann algebra [Formula: see text] and a faithful normal semi-finite trace τ. We prove that every local derivation on [Formula: see text] which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra [Formula: see text] for the algebra [Formula: see text] to admit local derivations which are not derivations.

scholarly journals A REMARK ON THE STRUCTURE OF SYMMETRIC QUANTUM DYNAMICAL SEMIGROUPS ON VON NEUMANN ALGEBRAS

2002 ◽  
Vol 05 (04) ◽  
pp. 571-579 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
DEBASHISH GOSWAMI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Type I ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Matrix Algebras ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Quantum Dynamical Semigroups ◽  
Symmetric Quantum

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with Abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, Høegh-Krohn and Olsen1 for the special case of the finite-dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.

PROBABILITY MEASURES ON PROJECTIONS IN VON NEUMANN ALGEBRAS

1989 ◽  
Vol 01 (02n03) ◽  
pp. 235-290 ◽  
Author(s):  
SHUICHIRO MAEDA
Keyword(s):  
Probability Measure ◽  
Linear Functional ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Probability Measures ◽  
Type I ◽  
Von Neumann ◽  
Positive Linear Functional ◽  
Finitely Additive Probability ◽  
Additive Probability

A state ϕ on a von Neumann algebra A is a positive linear functional on A with ϕ(1) = 1, and the restriction of ϕ to the set of projections in A is a finitely additive probability measure. Recently it was proved that if A has no type I 2 summand then every finitely additive probability measure on projections can be extended to a state on A. Here we give precise and complete arguments for proving this result.

Weakly Compact, Operator-Valued Derivations of Type I Von Neumann Algebras

1984 ◽  
Vol 36 (3) ◽  
pp. 436-457
Author(s):  
Steve Wright
Keyword(s):  
Von Neumann Algebras ◽  
Linear Operators ◽  
Type I ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Von Neumann ◽  
Weakly Compact Operator ◽  
Weak Operator ◽  
Image Position ◽  
Compact Extension

In [18], the author initiated an investigation of compact, Banach-module-valued derivations of C*-algebras. In collaboration with C. A. Akemann [3] and S.-K. Tsui [16], he determined the structure of all compact and weakly compact, A-valued derivations of a C*-algebra A, and of all compact, B(H)-valued derivations of a C*-subalgebra of B(H), the algebra of bounded linear operators on a Hilbert space H. In this paper, we begin the study of weakly compact, B(H)-valued derivations of C*-subalgebras of B(H).Let R be a C*-subalgebra of B(H), δ:R → B(H) a weakly compact derivation, i.e., a weakly compact linear map which hasSince δ has a unique weakly compact extension to a derivation of the closure of R in the weak operator topology (WOT) on B(H) (consult the proof of Theorem 3.1 of [16]), we may assume with no loss of generality that R is a von Neumann subalgebra of B(H). In this paper, we determine in Lemma 4.1 and Theorems 4.3 and 4.10 the structure of δ when R is type I, using I. E. Segal's multiplicity theory [14] for type I von Neumann algebras and results of E. Christensen [6], [7] on B(H)-valued derivations of von Neumann algebras.

scholarly journals Some aspects of quantum sufficiency for finite and full von Neumann algebras

2021 ◽  
Vol 111 (4) ◽  
Author(s):  
Andrzej Łuczak
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Pure States ◽  
Normal States ◽  
Full Algebra

AbstractSome features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.

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.

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.

scholarly journals ON EQUALITY CONDITION FOR TRACE JENSEN INEQUALITY IN SEMI-FINITE VON NEUMANN ALGEBRAS

2008 ◽  
Vol 19 (04) ◽  
pp. 481-501 ◽  
Author(s):  
TETSUO HARADA ◽  
HIDEKI KOSAKI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Strict Convexity ◽  
Jensen Inequality ◽  
Von Neumann ◽  
Convexity Assumption ◽  
Finite Von Neumann Algebra ◽  
Equality Condition ◽  
Finite Von Neumann Algebras

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

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