Dirichlet Forms and Markovian Semigroups on Standard Forms of von Neumann Algebras

For a quantum Markov semigroup T on the algebra B(h) with a faithful invariant state ρ, we can define an adjoint T with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterising adjoints T that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H,We study the adjoint semigroup with respect to both scalar products a, b = tr(ρa * b) and a, b = tr(ρ 1/2 a * ρ 1/2 b).

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“…The 0-detailed balance is stronger than the symmetric detailed balance (see also [5] Th. 6.6 p.296).…”

Section: Thent Is a Qms If And Only If The Special Lindblad Representmentioning

confidence: 99%

Generators of Detailed Balance Quantum Markov Semigroups

Fagnola

Umanità

2007

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…Then E is a Dirichlet form if and only if E(ξ, ξ 0 ) ≥ 0 for all ξ ∈ D(E) ∩ P. The above result follows from Proposition 4.5 (b) and Proposition 4.10 (ii) of [13]. The following is Theorem 4.11 of [13] : Let {T t } t≥0 be a J-real, strongly continuous, symmetric semigroup on H and let (E, D(E)) be the associated densely defined J-real, real valued quadratic form.…”

Section: Notation Definitions and Main Resultsmentioning

confidence: 70%

“…Next, we collect main results of [13]. Let (E, D(E)) be a J-real, real valued, densely defined closed form.…”

Section: Notation Definitions and Main Resultsmentioning

confidence: 99%

“…Thus we believe that the conditions (1.5) and (1.6) are very close to necessary conditions for H = H * for any non-tracial ξ 0 . We organize the paper as follows: In Section 2, we introduce notation, definitions and necessary terminologies in the theory of noncommutative Dirichlet forms in the sense of Cipriani [13]. We then list our main results(Theorem 2.1, Proposition 2.1 and Theorem 2.2).…”

Section: Introductionmentioning

confidence: 99%

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Remarks on the Structure of Dirichlet Forms on Standard Forms of Von Neumann Algebras

Park

2005

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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For a von Neumann algebra M acting on a Hilbert space H with a cyclic and separating vector ξ 0 , we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (M, ξ 0 ). For a general Lindblad type generator L of a conservative quantum dynamical semigroup on M, we give sufficient conditions so that the operator H induced by L via the symmetric embedding of M 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 [23]. In order to make the connection possible, we also extend the range of applications of the formula in [23].

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“…However, such a general construction of non-commutative L p -spaces is rather involved (see [22,23,27,29,30,31]). Nevertheless, such a general scheme has proved to be very useful for the description of concrete models with quantum Markov dynamics (see [22,30,31,32,33,34]). …”

Section: A Well Defined Norm On B(h) the Banach Space Defined As Thementioning

confidence: 99%

Implementability of Liouville Evolution, Koopman and Banach-Lamperti Theorems in Classical and Quantum Dynamics

Antoniou

Majewski

Suchanecki

2002

Open Syst. Inf. Dyn.

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Dirichlet Forms and Markovian Semigroups on Standard Forms of von Neumann Algebras

Cited by 90 publications

References 42 publications

Generators of Detailed Balance Quantum Markov Semigroups

Generators of Detailed Balance Quantum Markov Semigroups

Remarks on the Structure of Dirichlet Forms on Standard Forms of Von Neumann Algebras

Implementability of Liouville Evolution, Koopman and Banach-Lamperti Theorems in Classical and Quantum Dynamics

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