The Classification of Factors is not Smooth

Abstract

There is a natural Borel structure on the set F of all factors on a separable Hilbert space [ 3 ]. Let denote the algebraic isomorphism classes in F together with the quotient Borel structure. Now that various non-denumerable families of mutually non-isomorphic factors are known to exist [ 1; 6; 8; 10; 11; 12; 13 ], the most obvious question to be resolved is whether or not is smooth (i.e. is there a countable family of Borel sets which separate points). We answer this question negatively by an explicit construction.