Ultraproduct II_1 Factors — Problem 3

Definitions and mathematical statement

For an integer $n\geq 2$, let $\mathbb F_n$ be the free group on $n$ generators, and let $L(\mathbb F_n)$ be its group von Neumann algebra, generated by the left regular representation on $\ell^2(\mathbb F_n)$, equipped with its canonical trace $\tau(x)=\langle x\delta_e,\delta_e\rangle$. A $\mathrm{II}_1$ factor is an infinite-dimensional finite von Neumann algebra with trivial center and a faithful normal normalized trace. Such a factor has property $\Gamma$ if, for every finite subset $F\subseteq N$ and every $\varepsilon>0$, there is a unitary $u\in N$ with $\tau(u)=0$ and $\|ux-xu\|_2<\varepsilon$ for every $x\in F$, where $\|y\|_2=\tau(y^*y)^{1/2}$. A non-$\Gamma$ factor fails this property.

The supplied source calls a non-$\Gamma$ factor $N$ pseudocompact if there exists an ultrafilter $\mathcal U$ such that \[ N\equiv\mathbb M^{\mathcal U}. \] Here elementary equivalence conventionally means equality of the values of all sentences in the continuous first-order language of tracial von Neumann algebras. However, the supplied record does not define $\mathbb M$, the construction $\mathbb M^{\mathcal U}$, or the allowed ultrafilters. This definition is therefore conditional on recovering those conventions from the source; no particular matrix family or ultraproduct construction is asserted here.

The source asks whether $L(\mathbb F_n)$ is pseudocompact for $n\geq 2$. For the finite integer parameters, its target can be written, subject to the unresolved conventions above, as \[ \forall n\in\mathbb Z,\ n\geq 2,\qquad L(\mathbb F_n)\text{ is pseudocompact}. \] The witnessing ultrafilter may depend on $n$. The record does not state whether the countably infinite rank case $n=\infty$ is also intended.

The registry labels the record open, while the supplied statement begins with a solved annotation attributing a solution to J. Peterson and citing the key peterson2026ultraproductapproximationspropertyt. These are conflicting supplied status indications. Neither a resolution nor its direction is independently established by this draft.

Short English statement

The question asks whether every finite-rank free group factor $L(\mathbb F_n)$, $n\geq 2$, is pseudocompact in the source's sense. The comparison object $\mathbb M^{\mathcal U}$ and the permitted ultrafilters require clarification. The source also contains a Peterson solution annotation despite the registry's open label.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $801102$, exact source code catalogue entry, title "Ultraproduct II$_1$ Factors --- Problem 3." The registry attributes the problem to Elayavalli, with proposed year 2025.
• Supplied source list: Elayavalli, 50 Open Problems: Ultraproduct II$_1$ Factors (2025), Problem 3, https://arxiv.org/abs/2511.20377. The provenance records access on 2026-10-16 and extraction by arxiv_latex. This access information is supplied provenance, not an independently verified visit.
• The supplied statement includes "Solved by J. Peterson" with reference key peterson2026ultraproductapproximationspropertyt. No corresponding bibliographic entry, theorem statement, or link is supplied. The registry separately records status open; neither status indication has been independently verified.