Let ~//be a yon Neumann algebra with a cyclic and separating vector ~2 and let co(.) denote the corresponding vector state, i.e., co(A) = (~2,A~2)A E JP[. We have proved that a positive semigroup r on ~/ c a n induce the dynamical semigroup in the GNS representation associated with co if the state co is a r-invariant one. Some applications are given.
0. Introduction
Recently the theory of completely positive semigroups on yon Neumann algebras has made very interesting progress [ 1 -3 ] . An often useful way of examinating these semigroups is to study the induced semigroup on the representing Hilbert space. On the other hand, the positive semigroups have found many applications in mathematical physics [4][5][6]. For that reason it is interesting to ask whether one can induce the semigroups on the Hllbert space for this more general class of semigroups.
Our aim is to give the affirmative answer for the abo:ee-posed question. Moreover, as an application, the generalization of some ergodic properties of W*-dynamical system will be given (for definitions and terminology see [8] ).
1. NOTATION AND THE FORMULATION OF THE THEOREM
Let Jr/denote the von Neumann algebra on the Hilbert space Jf with a cyclic and separating vector ~2 and let J / ' denote its commutant. ~/1 (Jg'l) means the unit ball in Jg (.A/' respectively). Throughout this note mapping r of J / i n t o itself is always assumed to be linear, bounded, positive, i.e., r(A) >1 0 forA E zg andA ~> 0. Denote by co the following vector state on ~/, co(A) = (~2, AYZ) forA E Jh'. We will assume that the state co is r-invariant one, i.e., co o 7" = co.
Finally, for a subset S C JC, JC z~ means the polar set of dr'and B(J4) means the set of all linearbounded operators on Jr. We wish to prove: