On a class of type II1factors with Betti numbers invariants

We prove that a type II 1 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 ⊂ M , the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for the factors M , β HT n (M ), n ≥ 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying β HT n (M t ) = β HT n (M )/t, ∀t > 0, and a Künneth type formula.An example of a factor in the class HT is given by the group von Neumann factorM, ∀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.