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

摘要

For a von Neumann algebra M acting on a Hilbert space W with a cyclic and separating vector xi(0), we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (M, xi(0)). For a general bounded Lindblad type generator L of a conservative quantum dynamical semigroup on M, we give sufficient conditions so that the bounded operator H induced by L via the symmetric embedding of M into 71 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 Ref. 23. In order to make the connection possible, we also extend the range of applications of the formula in Ref. 23.