Ultraproduct II_1 Factors — Problem 19

Definitions and mathematical statement

For tracial von Neumann algebras $M$ and $N$, write $M\equiv N$ when they have the same values for all sentences in the continuous first-order language of tracial von Neumann algebras. The supplied record does not specify the language explicitly; this is the usual interpretation of elementary equivalence in this setting.

For clarity about the terminology, under the usual separable convention, let $P$ be a separable $\mathrm{II}_1$ factor with normalized trace $\tau$, and fix a free ultrafilter $\mathcal U$ on $\mathbb N$. Put $\|x\|_{2,\tau}=\tau(x^*x)^{1/2}$ and define its tracial ultrapower by \[ P^{\mathcal U}=\ell^\infty(\mathbb N,P)\big/\left\{(x_n):\lim_{n\to\mathcal U}\|x_n\|_{2,\tau}=0\right\}. \] Identify $P$ with its diagonal image in $P^{\mathcal U}$. Its central sequence algebra is \[ P'\cap P^{\mathcal U}=\{x\in P^{\mathcal U}:xa=ax\text{ for every }a\in P\}. \] The usual separable interpretation of super McDuff is that $P'\cap P^{\mathcal U}$ is a $\mathrm{II}_1$ factor. The supplied record does not itself give this definition or specify separability and ultrafilter conventions; this explanation therefore does not settle the intended scope of the source question.

Let $\mathcal K$ denote the intended class of $\mathrm{II}_1$ factors, with its separability convention unresolved. Write $\operatorname{SMcD}(P)$ and $\operatorname{USMcD}(P)$ for the source properties "super McDuff" and "uniformly super McDuff", respectively. The source asks whether \[ \forall N,M\in\mathcal K,\qquad \left(\operatorname{SMcD}(N)\ \land\ N\equiv M\right) \Longrightarrow \operatorname{SMcD}(M). \] It then asks, with the wording "In particular", whether \[ \exists P\in\mathcal K\qquad \operatorname{SMcD}(P)\ \land\ \neg\operatorname{USMcD}(P). \] No definition of uniformly super McDuff is included in the supplied evidence. Consequently, the second display preserves the exact property queried but remains a predicate-level formulation requiring that definition. No equivalence or implication between the two questions is asserted here beyond recording the source's wording.

Short English statement

Elayavalli's Problem 19 asks whether elementary equivalence preserves the super McDuff property of $\mathrm{II}_1$ factors. It also asks whether a super McDuff factor can fail to be uniformly super McDuff. The intended separability convention and the definition of the uniform property require source clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $801118$, source problem code catalogue entry, title "Ultraproduct $\mathrm{II}_1$ Factors --- Problem 19". The record attributes the problem to Elayavalli, with proposed year 2025.
• Supplied source attribution: Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 19, https://arxiv.org/abs/2511.20377. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not evidence of independent access or verification here.
• The registry labels the problem open; its creation and update timestamps are both 2026-10-16T14:00:00Z. This is a supplied status label, not an independently verified mathematical or literature assessment. The supplied upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.