NONLINEAR MARKOV SEMIGROUPS ON C*-ALGEBRAS

2013 ◽  
Vol 16 (01) ◽  
pp. 1350004 ◽  
Author(s):  
P. ŁUGIEWICZ ◽  
R. OLKIEWICZ ◽  
B. ZEGARLINSKI
Keyword(s):  
Sufficient Conditions ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Quantum Dynamical Semigroup ◽  
C Algebra ◽  
C Algebras ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Nonlinear Quantum ◽  
Duality Map

A notion of a nonlinear quantum dynamical semigroup is introduced and discussed. Some sufficient conditions, expressed solely in terms of the duality map, in order that a multivalued mapping on a C*-algebra generates the nonlinear Markov semigroup are proposed.

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.

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.

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).

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.

The Approach to Equilibrium of a Class of Quantum Dynamical Semigroups

1998 ◽  
Vol 01 (04) ◽  
pp. 561-572 ◽  
Author(s):  
Franco Fagnola ◽  
Rolando Rebolledo
Keyword(s):  
Hilbert Space ◽  
Asymptotic Behavior ◽  
Conditional Expectation ◽  
Sufficient Conditions ◽  
Approach To Equilibrium ◽  
Bounded Operators ◽  
Quantum Dynamical Semigroup ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Necessary And Sufficient

This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .

scholarly journals Quantum Dynamical Semigroups and Decoherence

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Mario Hellmich
Keyword(s):  
Banach Spaces ◽  
Sufficient Conditions ◽  
Splitting Theorem ◽  
Quantum Dynamical Semigroup ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Quantum Dynamical Semigroups

We prove a version of the Jacobs-de Leeuw-Glicksberg splitting theorem for weak*continuous one-parameter semigroups on dual Banach spaces. This result is applied to give sufficient conditions for a quantum dynamical semigroup to display decoherence. The underlying notion of decoherence is that introduced by Blanchard and Olkiewicz (2003). We discuss this notion in some detail.

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

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.

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