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

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 by Park. 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. (c) 2005 American Institute of Physics.