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

TL;DR

This paper explores the structure of Dirichlet forms on standard forms of von Neumann algebras, providing conditions for self-adjointness of operators induced by quantum dynamical semigroups and extending related formulas.

Contribution

It introduces new sufficient conditions for self-adjointness of operators associated with Lindblad generators on von Neumann algebras and extends the applicability of existing Dirichlet form formulas.

Findings

Operators induced by Lindblad generators can be self-adjoint under certain conditions.

Self-adjoint operators can be expressed as Dirichlet operators linked to specific Dirichlet forms.

The paper extends the application range of a known Dirichlet form formula.

Abstract

For a von Neumann algebra M acting on a Hilbert space H with a cyclic and separating vector v, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (M,v). For a general Lindblad type generator L of a conservative quantum dynamical semigroup on M, we give sufficient conditions so that the operator S induced by L via the symmetric embedding of M into H to be self-adjoint. It turns out that the self-adjoint operator S 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].