DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON  ℤ<sub>2</sub>-GRADED VON NEUMANN ALGEBRAS

Bahn, Changsoo; Ko, Chul Ki; Park, Young Jun

doi:10.1142/s0129055x03001825

Cited by 8 publications

(11 citation statements)

References 13 publications

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“…Let M = π ω (A) ′′ and ξ 0 = Ω ω . Then for each n ∈ N, π ω (a(f n )) and π ω (a * (f n )) are σ t -entire analytic element [BKP2]. Thus one can apply Theorem 2.1 and Corollary 2.1 directly in this case.…”

Section: Notation Terminologies and Main Resultsmentioning

confidence: 93%

“…The next step in this research area would be the investigation of detailed properties of Markovian semigroups, such as ergodicity, mixing property and convergence to the equilibrium, etc. In the case of CCR and CAR algebras with respect to quasifree states, the spectrum of the generators of the Markovian semigroups constructed in [CFL,BKP1,BKP2,KP] has been analyzed. However, in general the detail properties of the Markovian semigroups associated to Dirichlet forms in [Par1,Par2] are hard to be established.…”

Section: Introductionmentioning

confidence: 99%

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Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

Park¹

2005

Journal of Mathematical Physics

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We give sufficient conditions for ergodicity of the Markovian semigroups associated to Dirichlet forms on standard forms of von Neumann algebras constructed by the method proposed in Refs. [Par1,Par2]. We apply our result to show that the diffusion type Markovian semigroups for quantum spin systems are ergodic in the region of high temperatures where the uniqueness of the KMS-state holds.Keywords : Standard forms of von Neumann Algebras; Dirichlet forms; Markovian semigroups; ergodicity; quantum spin systems; KMS-states.Markovian semigroup associated to (E, D(E)). Let N be the fixed point space of T t ;T t is ergodic if and only if M is a factor. We apply our result to the translation invariant Markovian semigroups for quantum spin systems [Par1], and show that the semigroups are ergodic in region of high temperatures where the uniqueness of the KMS-state holds. Let us describe the background of this study briefly. The need to construct Markovian semigroups on von Neumann algebras, which are (KMS) symmetric with respect to non-tracial states, is clear for various applications to open systems[Dav], quantum statistical mechanics[BR] and quantum probability[Acc, Part]. Although on the abstract level we have quite well-developed theory[Cip1, GL1, GL2], the progress in concrete applications is relatively slow. For construction of Dirichlet forms and associated Markovian semigroups, we refer to [BKP1, BKP2, MZ1, MZ2, Par1, Zeg] and the references there in. During the last ten years, systematic methods to construct Dirichlet forms and associated Markovian semigroups of jump and diffusion types have been developed. Nontrivial translation invariant symmetric semigroups of jump type for quantum spin systems have been constructed and the strong ergodicity of the semigroups has been established in Refs. [MZ1, MZ2]. See also [Zeg] and the references there in. In [Par1], we gave a general construction method of Dirichlet forms of diffusion type in the framework of the general theory of Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras developed by Cipriani[Cip1].The method has been used successfully to construct Dirichlet forms and associated Markovian semigroups for quantum spin systems [Par1], CCR and CAR algebras with respect to quasi-free states[BKP1, BKP2, KP], and quantum mechanical

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Section: Notation Terminologies and Main Resultsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

Park¹

2005

Journal of Mathematical Physics

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“…Turning to (infinite) quantum continuous systems, free Fermi or Bose gas, we note that Park and his school developed an approach based on quantum Dirichlet forms to study dynamics for such systems, see [7,8] and/or [14]. However, as was mentioned earlier in Introduction, these forms do not have simple connections to derivatives.…”

Section: Introductionmentioning

confidence: 99%

Quantum Fokker–Planck Dynamics

Labuschagne

Majewski

2021

Ann. Henri Poincaré

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The Fokker-Planck equation is a partial differential equation which is a key ingredient in many models in physics. This paper aims to obtain a quantum counterpart of Fokker-Planck dynamics, as a means to describing quantum Fokker-Planck dynamics. Given that relevant models relate to the description of large systems, the quantization of the Fokker-Planck equation should be done in a manner that respects this fact and is therefore carried out within the setting of non-commutative analysis based on general von Neumann algebras. Within this framework, we present a quantization of the generalized Laplace operator and then go on to incorporate a potential term conditioned to non-commutative analysis. In closing, we then construct and examine the asymptotic behaviour of the corresponding Markov semigroups. We also present a non-commutative Csiszar-Kullback inequality formulated in terms of a notion of relative entropy and show that for more general systems, good behaviour with respect to this notion of entropy ensures similar asymptotic behaviour of the relevant dynamics.

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“…One of the reasons is that the general structure of Dirichlet forms for non-tracial states is not well-understood compared to the tracial case [2,3,6,12]. For constructions of Dirichlet forms for non-tracial states, we refer to [8,9,11,18,20,21,25,23] and the references there in. In [23], we gave a general construction method of Dirichlet forms on standard forms of von Neumann algebras.…”

Section: Introductionmentioning

confidence: 99%

“…In [23], we gave a general construction method of Dirichlet forms on standard forms of von Neumann algebras. The method has been used to construct (translation invariant) symmetric Markovian semigroups for quantum spin systems [23], the CCR and CAR algebras with respect to quasi-free states [8,9] and quantum mechanical systems [7].…”

Section: Introductionmentioning

confidence: 99%

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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DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON ℤ₂-GRADED VON NEUMANN ALGEBRAS

Cited by 8 publications

References 13 publications

Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

Quantum Fokker–Planck Dynamics

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

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DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON ℤ2-GRADED VON NEUMANN ALGEBRAS

Cited by 8 publications

References 13 publications

Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

Quantum Fokker–Planck Dynamics

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

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Product

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DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON ℤ₂-GRADED VON NEUMANN ALGEBRAS