On the singular abelian rank of ultraproduct II₁ factors

2024/02/28 by Patrick Hiatt, Hiatt, Patrick, Sorin Popa +1 · 1 citation
Mathematics · #46L10 #46L36 #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA) #Rings, Modules, and Algebras #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.18559

openalex publication_date 2024/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, under the continuum hypothesis \frak c=ℵ₁, any ultraproduct II₁ factor M= ∏_ω M_n of separable finite factors M_n contains more than \frak c many mutually disjoint singular MASAs, in other words the \it singular abelian rank of M, \rm r(M), is larger than \frak c. Moreover, if the strong continuum hypothesis 2^{\frak c}=ℵ₂ is assumed, then \rm r(M) = 2^{\frak c}. More generally, these results hold true for any II₁ factor M with unitary group of cardinality \frak c that satisfies the bicommutant condition (A₀'∩ M)'∩ M=M, for all A₀⊂ M separable abelian.

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