Let R be a von Neumann
algebra acting on a Hilbert space H. Let G be a group and let t → Ut be a unitary
representation of G on H such that Ut∗RUt= R for all t ∈ G. Two projections E
and F in R are called G-equivalent, written E ∼GF, if there is for each t ∈ G an
operator Tt∈ R such that E =∑t∈𝜃TtTt∗,F =∑t∈GUt∗Tt∗TtUt. 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).
