Remarks on positive semigroups

Let ℳ be a von Neumann algebra with a cyclic and separating vector Ω and let ω(·) denote the corresponding vector state, i.e., ω(A)=(Ω, AΩ) A ∈ ℳ. We have proved that a positive semigroup τ on ℳ can induce the dynamical semigroup in the GNS representation associated with ω if the state ω is a τ-invariant one. Some applications are given.