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On a Class of Type II$_1$ Factors with Betti Numbers Invariants | | Sorin Popa
; | | Date: |
11 Sep 2002 | | Subject: | Operator Algebras; Group Theory MSC-class: 46L10; 46L40; 22D40; 22D25 | math.OA math.GR | | 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 $M$, $eta^{^{HT}}_n(M), ngeq 0$. The class $Cal HCal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying $eta^{^{HT}}_n(M^t)= eta^{^{HT}}_n(M)/t, forall t>0$, and a K{ünneth type formula. An example of a factor in the class $Cal HCal T$ is given by the group von Neumann factor $M=L(Bbb Z^2
times SL(2, Bbb Z))$, for which $eta^{^{HT}}_1(M) = eta_1(SL(2, Bbb Z)) = 1/12$. Thus, $M^t
otsimeq M, forall t
eq 1$, 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. | | Source: | arXiv, math.OA/0209130 | | Services: | Forum | Review | PDF | Favorites | |
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