Commutation properties of automorphism groups and mappings of von Neumann algebras

AbstractWe prove a commutation theorem for point ultraweakly continuous oneparameter groups of automorphisms of von Neumann algebras. If αt, is such a group in Aut(R) for a von Neumann algebra R, we show the equivalence of the following three conditions on an ultraweakly continuous linear transformation μ: R → R: (a) μ commutes weakly with the infinitesimal generator for αt; (b) μ ° αt = αt ° μ, t ∈ R; and (c) μ leaves invariant each of the spectral subspaces associated with αt. A simple condition which is applicable when μ is an automorphism is pointed out