Ultraproduct II_1 Factors — Problem 31 (Solved by G. Patchell and A. Shiner, paper in preparation)

Definitions and mathematical statement

Context and definitions. The source concerns ultraproducts of type $\mathrm{II}_1$ factors, but does not explicitly specify $N$, its trace, or $\mathcal U$. Under the standard contextual interpretation, let $N$ be a type $\mathrm{II}_1$ factor with normalized faithful normal trace $\tau$, and let $\mathcal U$ be a nonprincipal ultrafilter on $\mathbb N$. Its tracial ultrapower is \[ N^{\mathcal U}=\ell^\infty(\mathbb N,N)\big/\mathcal I_{\mathcal U},\qquad \mathcal I_{\mathcal U}=\left\{(x_n):\lim_{n\to\mathcal U}\|x_n\|_{2,\tau}=0\right\}, \] where $\|x\|_{2,\tau}=\tau(x^*x)^{1/2}$ and $\tau_{\mathcal U}([(x_n)])=\lim_{n\to\mathcal U}\tau(x_n)$. These contextual hypotheses require confirmation from the source.

Unitaries $u,v\in N^{\mathcal U}$ are freely independent if their generated unital $*$-algebras $A_u$ and $A_v$ satisfy \[ \tau_{\mathcal U}(a_1\cdots a_m)=0 \] for every integer $m\geq 1$ and every choice of centered elements $a_j\in A_{\epsilon_j}$ with $\epsilon_j\in\{u,v\}$ and $\epsilon_j\ne\epsilon_{j+1}$. Here centered means $\tau_{\mathcal U}(a_j)=0$.

Source question. Suppose $u,v\in N^{\mathcal U}$ are freely independent unitaries. Can one find sequences of unitaries $(u_n)$ and $(v_n)$ in $N$ such that \[ u=[(u_n)]_{\mathcal U},\qquad v=[(v_n)]_{\mathcal U}, \] and the sequences have the source's stated property of being "pairwise $k$-independent unitaries in $N$ for all $k$"?

Unresolved formulation. The supplied record does not define $k$-independence or the meaning of "pairwise" for these sequences. It also does not specify the domain of $k$, whether one pair of representing sequences must work simultaneously for every $k$, or whether the required independence holds at every coordinate, eventually, or on a set belonging to $\mathcal U$. Consequently the quoted independence requirement cannot be replaced by a precise formula from the supplied evidence alone.

The source says that particular cases are established in the reference labeled [exoticCIKE]. Its contextual note attributes a solution to G. Patchell and A. Shiner, with a paper in preparation, whereas the registry status is open. These are supplied source qualifications, not independently verified mathematical conclusions.

Short English statement

The question asks whether freely independent unitaries in a tracial ultrapower can be represented by unitary sequences satisfying the source's pairwise $k$-independence requirement for all $k$. The meaning and quantifiers of that requirement are unspecified. The registry labels the problem open, while its contextual note reports a solution by G. Patchell and A. Shiner in a paper in preparation.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $801130$, exact source problem code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record attributes the problem to Elayavalli, with proposed year $2025$.
• Elayavalli, 50 Open Problems: Ultraproduct II_1 Factors ($2025$), Problem $31$: https://arxiv.org/abs/2511.20377. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these provenance claims have not been independently verified here.
• The supplied statement cites [exoticCIKE] for particular cases. No bibliographic identification or link for this citation is supplied.
• Supplied attribution/context: "Solved by G. Patchell and A. Shiner, paper in preparation." No title, manuscript link, or proof reference for that reported solution is supplied. The registry separately records status open.