On the singular abelian rank of ultraproduct II${}_1$ factors

Abstract

We prove that, under the continuum hypothesis $\mathfrak{c}={\aleph }_1$, any ultraproduct II${}_1$ factor $M={\prod }_{\omega }{M}_n$ of separable finite factors $M_n$ contains more than $\mathfrak{c}$ many mutually disjoint singular MASAs, in other words the {\it singular abelian rank of} $M$, $\text{r}(M)$, is larger than $\mathfrak{c}$. Moreover, if the strong continuum hypothesis $2^{\mathfrak{c}}={\aleph }_2$ is assumed, then $\text{r}(M)=2^{\mathfrak{c}}$. More generally, these results hold true for any II${}_1$ factor $M$ with unitary group of cardinality $\mathfrak{c}$ that satisfies the bicommutant condition $(A_0^{\prime }\cap M{)}^{\prime }\cap M=M$, for all $A_0\subset M$ separable abelian.

12 pages. The paper is dedicated to Jacques Dixmier on the occasion of his 100th anniversary