Automorphisms and equivalence in von neumann algebras

Abstract

Let R be a von Neumann algebra acting on a Hilbert space S. Let G be a group and let t->Ut be a unitary representation of G on S such that (formula presented) for all tEG. Two projections E and F in R are called G-equivalent, written E ˜G F, if there is for each teG an operator T* e R such that (formula presented) The main results in this paper state that this relation is indeed an equivalence relation (Thm. 1), that ‘semi-finiteness” is equivalent to the existence of a faithful normal semi-finite G-invariant trace on R+ (Thm. 2), and that “finiteness” together with countable decomposability of R is equivalent to the existence of a faithful normal finite G-invariant trace on R (Thm. 3). © 1973 Pacific Journal of Mathematics.