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 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.

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.

CONSTRUCTION OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS

2000 ◽  
Vol 03 (01) ◽  
pp. 1-14 ◽  
Author(s):  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Admissible Function ◽  
Standard Form ◽  
Dirichlet Forms ◽  
Quantum Spin ◽  
Quantum Spin Systems ◽  
Translation Invariant ◽  
Von Neumann ◽  
Markovian Semigroups

For a von Neumann algebra ℳ acting on a Hilbert space ℋ with a cyclic and separating vector ξ0, we give an explicit expression of Dirichlet forms on the natural standard form [Formula: see text] associated with the pair (ℳ, ξ0). For any self-adjoint analytic element x of ℳ and an admissible function f, we construct a (bounded) Dirichlet form which generates a symmetric Markovian semigroup on ℋ. We then apply our result to construct translation invariant, symmetric, Markovian semigroups for quantum spin systems with finite range interactions.

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.

scholarly journals Gradient forms and strong solidity of free quantum groups

2020 ◽  
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Crucial Part

AbstractConsider the free orthogonal quantum groups $$O_N^+(F)$$ O N + ( F ) and free unitary quantum groups $$U_N^+(F)$$ U N + ( F ) with $$N \ge 3$$ N ≥ 3 . In the case $$F = \text {id}_N$$ F = id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $$L_\infty (O_N^+)$$ L ∞ ( O N + ) is strongly solid. Moreover, Isono obtains strong solidity also for $$L_\infty (U_N^+)$$ L ∞ ( U N + ) . In this paper we prove for general $$F \in GL_N(\mathbb {C})$$ F ∈ G L N ( C ) that the von Neumann algebras $$L_\infty (O_N^+(F))$$ L ∞ ( O N + ( F ) ) and $$L_\infty (U_N^+(F))$$ L ∞ ( U N + ( F ) ) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.

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 The Spectral Scale of a Self-Adjoint Operator in a Semifinite von Neumann Algebra

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Christopher M. Pavone
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Von Neumann ◽  
Selfadjoint Operators ◽  
Normal Operators ◽  
Spectral Scale ◽  
Finite Setting

We extend Akemann, Anderson, and Weaver'sSpectral Scaledefinition to include selfadjoint operators fromsemifinitevon Neumann algebras. New illustrations of spectral scales in both the finite and semifinite von Neumann settings are presented. A counterexample to a conjecture made by Akemann concerning normal operators and the geometry of the their perspective spectral scales (in the finite setting) is offered.

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.

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.

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.

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