Vol. 15, No. 4, 1965

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The Borel space of von Neumann algebras on a separable Hilbert space

Edward George Effros

Vol. 15 (1965), No. 4, 1153–1164

Let (S,𝒮) be a Borel space (see G. W. Mackey, Borel structures in groups and their duals, Trans. Amer. Math. Soc. 85, (1957) 134–165), a separable Hilbert space, L the bounded linear operators on with the Borel structure generated by the weak topology, and 𝒜 the collection of von Neumann algebras on . A field of von Neumann algebras on S is a map s A(s) of S into 𝒜. We prove that there is a unique standard Borel structures on 𝒜 with the property that s A(s) is Borel if and only if there exist countably many Borel functions s Ai(s) of S into L such that for each s, the operators Ai(s) generate A(s). This is a consequence of the more general result that when it is provided with a suitable Borel structure, the space of weakly closed subspaces of the dual of a separable Banach space has sufficiently many Borel choice functions.

We show that the commutant, join, and intersection operations on 𝒜 are Borel. It follows that the Borel space of factors is standard. The relevance of 𝒜 to the theory of group representations is also investigated.

Mathematical Subject Classification
Primary: 46.65
Received: 21 September 1964
Published: 1 December 1965
Edward George Effros