Types of von Neumann algebras associated with extremal invariant states

Abstract

A globalized version of the following is proved. Let ℛ be a factor acting on a Hilbert space ℋ,G a group of unitary operators on ℋ inducing automorphisms of ℛ,x a vector separating and cyclic for ℛ which is up to a scalar multiple the unique vector invariant under the unitaries inG. Then either ℛ is of type III or ω x is a trace of ℛ. The theorem is then applied to study the representations due to invariant factors state of asymptotically abelianC*-algebras, and to show that in quantum field theory certain regions in the Minkowski space give type III factors.