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.

scholarly journals Structure of block quantum dynamical semigroups and their product systems

2020 ◽  
Vol 23 (01) ◽  
pp. 2050001
Author(s):  
B. V. Rajarama Bhat ◽  
U. Vijaya Kumar
Keyword(s):  
Von Neumann Algebra ◽  
Markov Semigroup ◽  
Map Building ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Quantum Dynamical ◽  
Product Systems ◽  
Dynamical Semigroup ◽  
Quantum Markov Semigroup ◽  
Unit Formula

Paschke’s version of Stinespring’s theorem associates a Hilbert [Formula: see text]-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a [Formula: see text]-algebra [Formula: see text] one may associate an inclusion system [Formula: see text] of Hilbert [Formula: see text]-[Formula: see text]-modules with a generating unit [Formula: see text]. Suppose [Formula: see text] is a von Neumann algebra, consider [Formula: see text], the von Neumann algebra of [Formula: see text] matrices with entries from [Formula: see text]. Suppose [Formula: see text] with [Formula: see text] is a QDS on [Formula: see text] which acts block-wise and let [Formula: see text] be the inclusion system associated to the diagonal QDS [Formula: see text] with the generating unit [Formula: see text] It is shown that there is a contractive (bilinear) morphism [Formula: see text] from [Formula: see text] to [Formula: see text] such that [Formula: see text] for all [Formula: see text] We also prove that any contractive morphism between inclusion systems of von Neumann [Formula: see text]-[Formula: see text]-modules can be lifted as a morphism between the product systems generated by them. We observe that the [Formula: see text]-dilation of a block quantum Markov semigroup (QMS) on a unital [Formula: see text]-algebra is again a semigroup of block maps.

MINIMAL QUANTUM DYNAMICAL SEMIGROUP ON A VON NEUMANN ALGEBRA

1999 ◽  
Vol 02 (02) ◽  
pp. 221-239 ◽  
Author(s):  
DEBASHISH GOSWAMI ◽  
KALYAN B. SINHA
Keyword(s):  
State Space ◽  
Von Neumann Algebra ◽  
Sufficient Conditions ◽  
Markov Semigroup ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Arbitrary State ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Necessary And Sufficient

Given a formal unbounded generator, the minimal quantum dynamical semigroup on a von Neumann algebra is constructed. A set of equivalent necessary and sufficient conditions for the conservativity of the minimal semigroup is given and in the case when it is not conservative, a distinguished family of conservative perturbations of the semigroup is studied. Finally, some of these results are applied to the classical Markov semigroup with arbitrary state space.

STOCHASTIC DILATION OF A QUANTUM DYNAMICAL SEMIGROUP ON A SEPARABLE UNITAL C* ALGEBRA

2000 ◽  
Vol 03 (01) ◽  
pp. 177-184 ◽  
Author(s):  
DEBASHISH GOSWAMI ◽  
ARUP KUMAR PAL ◽  
KALYAN B. SINHA
Keyword(s):  
Von Neumann Algebra ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
C Algebra ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Algebraic Framework ◽  
Uniformly Continuous

Given a uniformly continuous quantum dynamical semigroup on a separable unital C* algebra, we construct a canonical Evans–Hudson (E-H) dilation. Such a result was already proved by Goswami and Sinha6 in the von Neumann algebra setup, which has been extended to the C* algebraic framework in this paper. The authors make use of the coordinate-free calculus and results of Ref. 6, but the proof of the existence of structure maps differs from that of Ref. 6.

DILATION OF ARBITRARY SYMMETRIC QUANTUM DYNAMICAL SEMIGROUPS ON $B({\mathcal H})$

2012 ◽  
Vol 15 (03) ◽  
pp. 1250017 ◽  
Author(s):  
BISWARUP DAS
Keyword(s):  
Quantum Dynamical Semigroup ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Quantum Dynamical Semigroups ◽  
Symmetric Quantum

We prove the existence of Hudson–Parthasarathy dilation of a quantum dynamical semigroup on [Formula: see text], which is symmetric with respect to the canonical normal trace on it.

scholarly journals REMARKS ON THE STRUCTURE OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS

2005 ◽  
Vol 08 (02) ◽  
pp. 179-197 ◽  
Author(s):  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Bounded Operator ◽  
Standard Form ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Dirichlet Forms ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Dynamical Semigroup

For a von Neumann algebra ࡕ acting on a Hilbert space ℋ with a cyclic and separating vector ξ0, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (ࡕ, ξ0). For a general bounded Lindblad type generator L of a conservative quantum dynamical semigroup on ࡕ, we give sufficient conditions so that the bounded operator H induced by L via the symmetric embedding of ࡕ into ℋ to be self-adjoint. It turns out that the self-adjoint operator H can be written in the form of a Dirichlet operator associated to a Dirichlet form given in Ref. 23. In order to make the connection possible, we also extend the range of applications of the formula in Ref. 23.

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 Subalgebras of C*-algebras and von Neumann algebras

1984 ◽  
Vol 25 (1) ◽  
pp. 19-25 ◽  
Author(s):  
Charles A. Akemann
Keyword(s):  
Recent Work ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Weierstrass Theorem ◽  
Von Neumann ◽  
Matrix Algebras ◽  
C Algebras ◽  
Weakly Closed

Recent work [2, 6] on subalgebras of matrix algebras leads naturally to the following situation. Let A be a C*-subalgebra of the C*-algebra B andM be a weakly closed *-subalgebra of the von Neumann algebra N. Consider the following Conditions.Condition 1. For every b≠ 0 in B there exists a ∈ A such that O≠ab ∈ A.Condition 2. For every b∈B there exists a ≠ 0 in A such that ab ∈ A.If we replace A by M and B by N in Conditions 1 and 2 we get von Neumann algebra versions which we shall call Condition 1'and Condition 2'. Clearly Condition 1 implies Condition 2, and both conditions suggest that A is some kind of weak ideal of B. This paper explores the extent to which this is true. The paper grew out of the author's attempts [1, 3] to generalize the Stone-Weierstrass theorem to C*-algebras.

scholarly journals REMARKS ON SUFFICIENT CONDITIONS FOR CONSERVATIVITY OF MINIMAL QUANTUM DYNAMICAL SEMIGROUPS

2005 ◽  
Vol 17 (07) ◽  
pp. 745-768 ◽  
Author(s):  
CHANGSOO BAHN ◽  
CHUL KI KO ◽  
YONG MOON PARK
Keyword(s):  
Sufficient Conditions ◽  
Heavy Ion ◽  
Elliptic Operators ◽  
Heavy Ion Collision ◽  
Quantum Dynamical Semigroup ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Quantum Dynamical Semigroups ◽  
Ion Collision

We have obtained sufficient conditions for conservativity of minimal quantum dynamical semigroup by modifying and extending the method used in [1]. Our criterion for conservativity can be considered as a complement to Chebotarev and Fagnola's conditions [1]. In order to show that our conditions are useful, we apply our results to concrete examples (models of heavy ion collision and noncommutative elliptic operators).

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.

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