Representations of von Neumann Algebras and Ultraproducts

We will introduce the concept of ergodicity of states with respect to some group of transformations on a von Neumann algebra and its properties are studied. A connection between the ergodic states on the von Neumann algebra and the representations of von Neumann algebras associated to them will be described. We also study the properties of ultraproducts of von Neumann algebras with ergodic states and corresponding representations. Here we use ultraproducts of von Neumann algebras by Groh (J. Operator Theory 11(2), 395–404 1984) and Raynaud (J. Operator Theory 48(1), 41–68 2002). In particular, we will show that the ultraproduct of irreducibles representations isn’t, generally speaking, irreducible.