Let M be a σ-finite type III
von Neumann algebra with separating and cyclic vector ζ (on a not necessarily
separable Hilbert space), let C be the center of M, let e be the projection
corresponding to the subspace generated by Cζ, and let τ(x) be the unique
element in C with τ(x)e = exe for x in M. For χ in the spectrum X of C,
let ρχ be the canonical representation of the state τχ(x) = τ(x)(χ). The
integral ∫τχ(x)dν(χ) induces the central decomposition of M. A separable
C∗-algebra B of M is found so that ρχ(M)′′ has a σ-weakly continuous
projection of norm one on ρχ(B)′′, and ρχ(B)′′ is a type III factor on an
open dense set of X. It is shown that ρχ(M)′′ is type III and that τχ has a
decomposition (in the sense of Choquet-Bishop-de Leeuw) as an integral of type III
functionals quasi-supported by primary type III functionals for χ the open dense
set.
