Ultraproduct II_1 Factors — Problem 30

Definitions and mathematical statement

For a unital tracial von Neumann algebra $(M,\tau)$, an element $a\in M$ is centered if $\tau(a)=0$. Following the supplied record, unital von Neumann subalgebras $A,B\subseteq M$ are $k$-independent if \[ \tau(a_1\cdots a_k)=0 \] for every word whose factors are centered elements taken alternately from $A$ and $B$, with either algebra allowed to supply the first factor. Here the condition concerns words of length exactly $k$. The term independent means $2$-independent. Thus independence is equivalent to \[ \tau(ab)=\tau(a)\tau(b)\qquad(a\in A,\ b\in B). \] Unitaries $u,v\in\mathcal U(M)$ are independent when the unital von Neumann algebras $W^*(u)$ and $W^*(v)$ are independent.

The record does not explicitly specify $N$ or the ultrafilter $\mathcal U$. Under the conventional interpretation suggested by the source title, let $(N,\tau_N)$ be a $\mathrm{II}_1$ factor with its normalized trace, and let $\mathcal U$ be a nonprincipal ultrafilter on ${\mathbb{N}}$. Write \[ \|x\|_{2,\tau_N}=\tau_N(x^*x)^{1/2},\qquad N^{\mathcal U}=\ell^\infty({\mathbb{N}},N)\big/\mathcal I_{\mathcal U}, \] where \[ \mathcal I_{\mathcal U}=\{(x_n)\in\ell^\infty({\mathbb{N}},N):\lim_{n\to\mathcal U}\|x_n\|_{2,\tau_N}=0\}. \] Let $q$ be the quotient map and let $\tau_{\mathcal U}(q((x_n)))=\lim_{n\to\mathcal U}\tau_N(x_n)$.

Lifting question. If $u,v\in\mathcal U(N^{\mathcal U})$ satisfy \[ \tau_{\mathcal U}(ab)=\tau_{\mathcal U}(a)\tau_{\mathcal U}(b) \qquad(a\in W^*(u),\ b\in W^*(v)), \] must there exist sequences $(u_n)_{n\in{\mathbb{N}}}$ and $(v_n)_{n\in{\mathbb{N}}}$ in $\mathcal U(N)$ such that \[ q((u_n))=u,\qquad q((v_n))=v, \] and \[ \forall n\in{\mathbb{N}}\ \forall a\in W^*(u_n)\ \forall b\in W^*(v_n),\qquad \tau_N(ab)=\tau_N(a)\tau_N(b)? \] This formulation interprets the source phrase "sequences of pairwise independent unitaries" as requiring independence of $u_n$ and $v_n$ at each coordinate. Whether independence between different coordinates is intended is unresolved. The assumptions on $N$ and $\mathcal U$ above are explicit conventional interpretations, not hypotheses stated in the supplied excerpt; the intended scope of quantification over them requires source clarification.

Short English statement

Can two independent unitaries in a tracial ultrapower be represented by unitary sequences whose corresponding terms are exactly independent in the original algebra? Independence here means trace factorization between their generated von Neumann algebras. The assumptions on the original algebra and ultrafilter, and the meaning of pairwise independence across sequences, require clarification.

Sources

• Local UnsolvedMath registry, permanent ID 801129, source problem code catalogue entry, title: Ultraproduct $\mathrm{II}_1$ Factors --- Problem 30. The supplied record attributes the problem to Elayavalli, with proposed year 2025, and labels it open; that status is supplied registry metadata, not an independently verified mathematical assessment.
• Supplied source attribution: Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 30, https://arxiv.org/abs/2511.20377. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of independent source access or verification.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry record was created and updated on 2026-10-16T14:00:00Z.