Abstract
We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A io M, the Betti numbers of the standard equivalence relation associated with A io M ([G2]), are in fact isomorphism invariants for the factors M, ¥â HT n (M), n iA 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying ¥â HT n (Mt) = ¥â HT n (M)/t, ¢£t > 0, and a Ki§unneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2 SL(2,Z)), for which ¥â HT 1 (M) = ¥â1(SL(2,Z)) = 1/12. Thus, Mt M,¢£t = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R. V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.
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