DIRICHLET FORMS AND MARKOVIAN SEMIGROUPS ON STANDARD FORMS OF VON-NEUMANN-ALGEBRAS

We characterize w-continuous, Markovian semigroups on a von Neumann a lgebra M, which are phi(o)-symmetric w.r.t. a faithful, normal state p hi(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 c ontinuous, contraction semigroups on a real Hilbert space H which leav e invariant a closed, convex set in H, in terms of a contraction prope rty of the associated quadratic forms. We apply the results to give cr iteria of essential selfadjointness for quadratic form sums and to giv e a characterization of w-continuous, Markovian semigroups on,tt, whi ch commute with the modular automorphism group sigma(t)(phi o). (C) 19 97 Academic Press.