Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras

We characterize w*-continuous, Markovian semigroups on a von Neumann algebra M, which are phi(o)-symmetric w.r.t. a faithful, normal state phi(o) in M*+, in terms of quadratic forms on the Hilbert space H of a standard form (M, H, P, J). We characterize also symmetric, strongly continuous, contraction semigroups on a real Hilbert space H which leave invariant a closed, convex set in H, in terms of a contraction property of the associated quadratic forms. We apply the results to give criteria of essential selfadjointness for quadratic form sums and to give a characterization of w*-continuous, Markovian semigroups on,tt, which commute with the modular automorphism group sigma(t)(phi o). (C) 1997 Academic Press.