Vol. 78, No. 2, 1978

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Normal expectations and integral decomposition of type III von Neumann algebras

Herbert Paul Halpern

Vol. 78 (1978), No. 2, 291–331

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 , 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)(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.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 1 October 1976
Revised: 10 March 1978
Published: 1 October 1978
Herbert Paul Halpern