Invariant normal positive functionals

Abstract

Let (02, S, a) be a system consisting of a von Neumann algebra a, a semigroup S and an action 01: S x a+ @. We continue in this paper the study of the existence of invariant (invariant with respect to a) normal positive functionals on 6Y begun in [3]. We shall freely make use of the notations of [3]. In [3], we considered the case when 5’ was a left amenable semigroup and (Y. an antirepresentation of S as normal positive contractions on G?. In this paper we consider S to be amenable and 01 an antirepresentation of S as normal *homomorphisms on 0Z into GZ (i.e., for each s E S, 01~ is a normal *homomorphism and ast = ata,). For a given normal positive linear functional 9s on G?, we investigate conditions for the existence of an S-invariant normal positive linear functional y on a such that p,, < v (or q0 N v). As a consequence, we get conditions for G! to be S-finite.