Vol. 44, No. 1, 1973

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Automorphisms and equivalence in von Neumann algebras

Erling Stormer

Vol. 44 (1973), No. 1, 371–383

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 UtRUt = 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 = t𝜃TtTt,F = tGUtTtTtUt. 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).

Mathematical Subject Classification 2000
Primary: 46L10
Received: 27 September 1971
Published: 1 January 1973
Erling Stormer