Ultraproduct II_1 Factors — Problem 44

Definitions and mathematical statement

A type $\mathrm{II}_1$ factor is a von Neumann algebra $M$ with center $Z(M)={\mathbb{C}} 1$, no nonzero minimal projections, and a faithful normal tracial state $\tau_M$ satisfying $\tau_M(1)=1$. In a common continuous first-order language $L$ for tracial von Neumann algebras, write $\sigma^M$ for the value of an $L$-sentence $\sigma$ in $M$. Elementary equivalence means \[ M\equiv_L N \quad\Longleftrightarrow\quad \sigma^M=\sigma^N\text{ for every }L\text{-sentence }\sigma. \] The source asks whether two type $\mathrm{II}_1$ factors can have the same "2-quantifier theory" without being elementarily equivalent. The supplied record does not define the language or the intended quantifier-complexity convention. Accordingly, let $\mathcal S_2$ denote the class of sentences intended by that phrase, pending clarification. Relative to a specified $L$ and $\mathcal S_2$, the requested existence statement is \[ \begin{gathered} \exists\, M,N\text{ of type }\mathrm{II}_1:\\ \left(\forall\sigma\in\mathcal S_2,\ \sigma^M=\sigma^N\right) \quad\text{and}\quad \left(\exists\text{ an }L\text{-sentence }\varphi,\ \varphi^M\ne\varphi^N\right). \end{gathered} \] This is a formulation schema until the meaning of "2-quantifier theory" is fixed: the record does not say whether it counts individual quantifiers, quantifier blocks, quantifier depth, or alternations, nor whether it includes both quantifier orders. No separability assumption or requirement that $M$ and $N$ themselves be ultraproducts is stated. The supplied source points to Proposition 7 in [OzawaSolidActa] and also to [HURigid], and describes the problem as still open; these are supplied source qualifications, not independently verified conclusions.

Short English statement

Do there exist two type $\mathrm{II}_1$ factors that agree on every sentence in the source's intended 2-quantifier fragment but disagree on some first-order sentence? The fragment and language require clarification before this becomes a uniquely specified mathematical question. The supplied record labels the problem open.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801142$, exact source problem code catalogue entry, titled "Ultraproduct $\mathrm{II}_1$ Factors --- Problem 44." Attribution: Elayavalli, proposed year 2025; registry status: open.
• Supplied source-list attribution: Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 44. Supplied URL: https://arxiv.org/abs/2511.20377. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not an independent source visit or verification.
• The supplied statement directs the reader to Proposition 7 in [OzawaSolidActa], with an additional reference to [HURigid]. No bibliographic entries or links resolving these two reference labels are supplied, so their identities, hypotheses, and relevance remain unverified.