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

Abstract

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].

On two quantum versions of the detailed balance condition

“…We stress that for s = 1/2 this is exactly the definition of KMS-symmetry with respect to the faithful state ω ρ = tr(ρ ·) (see for example [8], [18] and [23]). We refer to the lecture notes [9] for a discussion on this point.…”

“…satisfying L = L , investigated by Accardi and Mohari [2], Goldstein and Lindsay [18], Cipriani [8], Park [23] and the references therein.…”

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

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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)).

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“…Let {x k : k ∈ I} be a (finite or countable) family of elements in M 1/2 which generates M. Let (E, D(E)) be the Dirichlet form constructed with {x k : k ∈ I} and an admissible function by means of Refs. [Par1,Par2]. For the details, see Section 2.…”

Generators of KMS Symmetric Markov Semigroups on $${\mathcal{B}({\rm h})}$$ Symmetry and Quantum Detailed Balance

“…Their non-commutative counterpart has also been deeply investigated (Albeverio and Goswami [1], Cipriani [6], Davies and Lindsay [8], Goldstein and Lindsay [15], Guido, Isola and Scarlatti [17], Park [23], Sauvageot [26] and the references therein).…”