G-Invariance and G-Traces

Abstract

We will assume in this paragraph that A admits an abelian cofinal sequence Ai. One may consult example 7°/ of the introduction for more explanations. Let A be ultraweakly closed, satisfying condition II, with a cofinal sequence Ai, and let M be the von Neumann algebra Aid. We denote by P the abelian von Neumann algebra generated by all A i -1 and by PU the group of unitary operators of P. Let G be the group of automorphisms of M of the form B ∈ M➙U B U-1 ∈ M, with U moving in PU. Let M G = P′ ∩ M be the von Neumann algebra of fixed points of G. Clearly, M G D ⊂ D and UD = D for U in PU. Replacing P by the von Neumann algebra (denoted always by P) generated by all A i -1 and the center Z of M, we get a slightly different definition of G-invariance (with coincidence of both notions for a normal f ≥ 0) and nothing is changed. Such a choice of P has to be taken in definition 9.2.