Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 335-347.

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

Authors: Patrick Hiatt (1), Sorin Popa (2)
Author institution: (1) Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, U.S.A.
(2) Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, U.S.A.

Summary: We prove that, under the continuum hypothesis $\mathfrak{c}=\aleph_1$, any ultraproduct II$_1$ factor $M= \prod\limits_{\omega} M_n$ of separable finite factors $M_n$ contains more than $\mathfrak{c}$ many mutually disjoint singular MASAs, in other words the singular abelian rank of $M$, ${\mathrm r}(M)$, is larger than $ \mathfrak{c}.$ Moreover, if the strong continuum hypothesis $2^{\mathfrak{c}}=\aleph_2$ is assumed, then ${\text{\rm r}}(M) = 2^{\mathfrak{c}}$. More generally, these results hold true for any II$_1$ factor $M$ with unitary group of cardinality $\mathfrak{c}$ that satisfies the bicommutant condition $(A_0'\cap M)'\cap M=M$, for all $A_0\subset M$ separable abelian.

DOI: http://dx.doi.org/10.7900/jot.2024mar11.2449
Keywords: II$_1$ factor, ultraproduct factors, singular MASA, singular abelian rank

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