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), ℋ a separable Hilbert space, L the
bounded linear operators on ℋ with the Borel structure generated by the weak
topology, and 𝒜 the collection of von Neumann algebras on ℋ. A field of ℋ vonNeumann 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.
