Ultraproduct II_1 Factors — Problem 38

Definitions and mathematical statement

In the type $\mathrm{II}_1$ setting indicated by the source title, let $N$ be a factor with normalized faithful normal trace $\tau$, and let $\mathcal U$ be an ultrafilter on an index set $I$. Define $\|x\|_{2,\tau}=\tau(x^*x)^{1/2}$ and the tracial ultrapower \[ N^{\mathcal U}=\ell^\infty(I,N)\big/\{(x_i)_{i\in I}:\lim_{i\to\mathcal U}\|x_i\|_{2,\tau}=0\}. \] Here $\ell^\infty(I,N)$ consists of the operator-norm-bounded families. A subfactor is a unital von Neumann subalgebra that is itself a factor, and $[M:P]$ denotes the Jones index of a subfactor inclusion $P\subset M$.

With the ambient parameters fixed, the source asks whether every subfactor $N_0\subset N^{\mathcal U}$ satisfying \[ [N^{\mathcal U}:N_0]=2 \] admits a family of subfactors $(N_i\subset N)_{i\in I}$ such that \[ [N:N_i]=2\quad\text{for every }i\in I, \qquad N_0\cong\prod_{i\to\mathcal U}(N_i,\tau|_{N_i}). \] The product on the right is the tracial ultraproduct, defined by the analogous quotient of bounded families using the restricted traces. The symbol $\cong$ denotes an isomorphism of von Neumann algebras; the supplied statement does not require this isomorphism to respect the inclusions into $N^{\mathcal U}$.

The supplied statement does not explicitly specify the hypotheses on $N$, the index set $I$, or whether $\mathcal U$ must be nonprincipal. The type $\mathrm{II}_1$ interpretation comes from the source title; any further intended restrictions, including separability or countability, remain unresolved.

Short English statement

Must every Jones-index-$2$ subfactor of a tracial ultrapower of a type $\mathrm{II}_1$ factor be isomorphic to a tracial ultraproduct of index-$2$ subfactors of the original factor? The supplied record leaves the precise ambient and ultrafilter hypotheses unspecified.

Sources

• UnsolvedMath local registry, permanent ID 801137, source problem code catalogue entry, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry attributes the problem to Elayavalli (2025) and labels it open; these are supplied metadata, not an independently verified status assessment.
• Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 38. Supplied source URL: https://arxiv.org/abs/2511.20377. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source consultation.