Ultraproduct II_1 Factors — Problem 15

Definitions and mathematical statement

The title places the question in the setting of type $\mathrm{II}_1$ factors. A type $\mathrm{II}_1$ factor is a von Neumann algebra with center ${\mathbb{C}} 1$, no nonzero minimal projections, and a faithful normal tracial state $\tau$ normalized by $\tau(1)=1$. Here separability is understood as separability of the predual. Write $\|x\|_{2,\tau}=\tau(x^*x)^{1/2}$.

Under the finite-set convention for Property Gamma, a tracial factor $(M,\tau)$ has Property Gamma if, for every finite subset $F\subseteq M$ and every $\varepsilon>0$, there is a unitary $u\in M$ such that \[ \tau(u)=0,\qquad \|ux-xu\|_{2,\tau}<\varepsilon\quad\text{for every }x\in F. \] A non-Gamma factor is one that does not have this property. The supplied record does not define Property Gamma for the potentially nonseparable ultraproduct; the finite-set convention used here requires confirmation against the source.

Let $\mathcal U$ be a nonprincipal ultrafilter on ${\mathbb{N}}=\{1,2,\ldots\}$. For a sequence of type $\mathrm{II}_1$ factors $(N_i,\tau_i)$, define their tracial ultraproduct by \[ \prod_{i\to\mathcal U}N_i = \frac{\{(x_i):x_i\in N_i,\ \sup_i\|x_i\|<\infty\}} {\{(x_i):x_i\in N_i,\ \sup_i\|x_i\|<\infty,\ \lim_{i\to\mathcal U}\|x_i\|_{2,\tau_i}=0\}}, \] with trace $\tau_{\mathcal U}([(x_i)])=\lim_{i\to\mathcal U}\tau_i(x_i)$.

For this ultrafilter parameter, the source question is whether there exists a sequence $(N_i)_{i\in{\mathbb{N}}}$ of separable type $\mathrm{II}_1$ factors such that every $N_i$ fails Property Gamma, while $\prod_{i\to\mathcal U}N_i$ has Property Gamma. The source record does not specify whether $\mathcal U$ is fixed in advance or is to be chosen existentially; this quantifier remains unresolved. Nonprincipality and the tracial interpretation of the ultraproduct are explicit conventions of this draft, consistent with the stated setting but not explicitly given in the extracted question.

Short English statement

Can a sequence of separable type $\mathrm{II}_1$ factors, each without Property Gamma, have a tracial ultraproduct with Property Gamma? The ultrafilter quantifier and the convention for Property Gamma on the potentially nonseparable ultraproduct require confirmation. The registry labels this question open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801114$, exact source problem code catalogue entry, titled Ultraproduct II$ {}_1 $ Factors --- Problem 15. The record attributes the proposal to Elayavalli, gives the year 2025, and labels its status as open.
• Supplied source attribution: Elayavalli, 50 Open Problems: Ultraproduct II$ {}_1 $ Factors (2025), Problem 15, https://arxiv.org/abs/2511.20377. The supplied background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source access or verification.