On a Class of Type II${}_1$ Factors with Betti Numbers Invariants

Abstract

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 $\Cal H \Cal T$ of factors $M$ having such Cartan subalgebras $A\subset M$, the Betti numbers of the standard equivalence relation associated with $A\subset M$ ([G2]), are in fact isomorphism invariants for the factors $M$, ${\beta }_n^{{}^{HT}}(M),n\ge 0$. The class $\Cal H\Cal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying ${\beta }_n^{{}^{HT}}(M^t)={\beta }_n^{{}^{HT}}(M)/t,\forall t>0$, and a K{ü}nneth type formula. An example of a factor in the class $\Cal H\Cal T$ is given by the group von Neumann factor $M=L({\mathbb{Z}}^2⋊SL(2,\mathbb{Z}))$, for which ${\beta }_1^{{}^{HT}}(M)={\beta }_1(SL(2,\mathbb{Z}))=1/12$. Thus, $M^t\not\simeq M,\forall t\ne 1$, 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.

78 pages; minor revisions in text and ref. (Dec 1'st); more revisions in text and ref. (Jan 14 and 22, 2003); in Section 4 a notion of ``epsilon-rigidity'' for inclusions is introduced and is related with the initial notion of relative rigidity (Aug. 27, 2003), 82 pages. Minor changes (Dec. 2003)

Cited by in corpus (3)

• L^2-Homology for von Neumann Algebras
• L^2-Betti Numbers of Discrete Measured Groupoids
• Cohomology of property T groupoids and applications