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

Edward Effros
1965 Pacific Journal of Mathematics  
separable Hubert space, S the bounded linear operators on Sίf with the Borel structure generated by the weak topology, and S/ the collection of von Neumann algebras on §ίf. Afield of ^{f von Neumann algebras on S is a map s-»9ϊ(s) of S into j>/\ We prove that there is a unique standard Borel structures on S/ with the property that s -» %(s) is Borel if and only if there exist countably many Borel functions s-» Ai(s) of S into 8 such that for each s, the operators A*(s) generate 5ί(s). This is a
more » ... consequence of the more general result that when iί 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 jy are Borel. It follows that the Borel space of factors is standard. The relevance of S>/ to the theory of group representations is also investigated.
doi:10.2140/pjm.1965.15.1153 fatcat:xl3za6grdbgtpeze3gnmk4jzqq