On a Class of Type II$_1$ Factors with Betti Numbers Invariants
| dc.creator | Popa, Sorin | |
| dc.date | 2002-09-11 | |
| dc.date | 2003-12-19 | |
| dc.date.accessioned | 2026-07-07T04:50:47Z | |
| dc.date.available | 2026-07-07T04:50:47Z | |
| dc.description | 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 $M$, $β^{^{HT}}_n(M), n\geq 0$. The class $\Cal H\Cal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying $β^{^{HT}}_n(M^t)= β^{^{HT}}_n(M)/t, \forall t>0$, 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 $M=L(\Bbb Z^2 \rtimes SL(2, \Bbb Z))$, for which $β^{^{HT}}_1(M) = β_1(SL(2, \Bbb Z)) = 1/12$. Thus, $M^t \not\simeq M, \forall t \neq 1$, 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. | |
| dc.description | 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) | |
| dc.identifier | https://arxiv.org/abs/math/0209130 | |
| dc.identifier | http://arxiv.org/abs/math/0209130 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64915 | |
| dc.subject | Operator Algebras | |
| dc.subject | Group Theory | |
| dc.subject | 46L10; 46L40; 22D40; 22D25 | |
| dc.title | On a Class of Type II$_1$ Factors with Betti Numbers Invariants | |
| dc.type | text |