Dirichlet Forms and Markovian Semigroups on Standard Forms of von Neumann Algebras

Abstract

We characterize w*-continuous, Markovian semigroups on a von Neumann algebra M, which areφ0-symmetric w.r.t. a faithful, normal stateφ0in 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 M, which commute with the modular automorphism groupσφ0t.