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 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].