Remarks on the decomposition of Dirichlet forms on standard forms of von Neumann algebras

Abstract

For a bounded generator G of a weakly*-continuous, completely positive, KMS-symmetric Markovian semigroup on a von Neumann algebra M acting on a separable Hilbert space H, let H be the operator induced by G via the symmetric embedding of M into H. We decompose the Dirichlet form associated with H into a direct integral of forms whose associated generators are divergences of derivations. Moreover, if the derivations are inner, then the Dirichlet form can be written as the form given by Park [Infinite Dimen. Anal. Quantum Probab., Relat. Top. 3, 1 (2000); 8, 179 (2005)].

Decomposition of Dirichlet Forms Associated to Unbounded Dirichlet Operators

“…In case ω in (2.1) is a tracial state, the similar structure of bimodule in Proposition 2.1 and bimodule derivation in Theorem 2.2 have been introduced in [6], and the Dirichlet form (E, D(E)) has been decomposed. For a nontracial state and a bounded generator G, the decomposition of the Dirichlet form (E, D(E)) was established in [8].…”

“…In [10], for a general Lindblad type generator G of a KMS-symmetric quantum Markovian semigroup on a von Neumann algebra M acting on a Hilbert space H, the author gave the sufficient condition so that the operator H induced by G via the symmetric embedding of M into H is symmetric and the type of a Dirichlet operator given in [9]. Recently, in [8], the Dirichlet form associated to a bounded generator G of a weakly * -continuous, completely positive, KMS-symmetric Markovian semigroup on a von Neumann algebra M was decomposed.…”

“…We study a weakly * -continuous, completely positive, KMS-symmetric Markovian semigroup {S t } t≥0 , S t = e −tG with an unbounded generator G on a von Neumann algebra M acting on a separable Hilbert space H. Modifying the bimodule structure used in [6], we construct a vector valued derivation on the algebra M. Applying this derivation and the method used in [8], we decompose the Dirichlet form associated to the operator induced by G via the symmetric embedding into an integral of forms whose associated generators are the divergence of derivations.…”

“…Section 3 is devoted to the proofs of Proposition 2.1, Theorem 2.2 and Theorem 2.3. Using the similar method used in [6] and [8] (and also Section 1 of [11]), we construct a bimodule structure and a derivation, and decompose the associated Dirichlet form.…”

“…In [8], the author decomposed the Dirichlet form associated to a bounded generator G of a weakly * -continuous, completely positive, KMS-symmetric Markovian semigroup on a von Neumann algebra M. The aim of this paper is to extend G to the unbounded generator using the bimodule structure and derivations. …”