Let N be a tracial von Neumann algebra, let U be a free ultrafilter, and let u,v∈ N^ U be freely independent unitaries. Is it true that, for every k∈ N, there exist sequences of unitaries (u_n^(k))_n∈ N and (v_n^(k))_n∈ N in N representing u and v in N^ U such that (u_n^(k),v_n^(k)) are k-independent in N; that is, every alternating product of at most k trace-zero words in the u_n^(k)-variables and v_n^(k)-variables has trace zero?

Problem 31 of Kunnawalkam Elayavalli on lifting N-independence

Let $N$ be a tracial von Neumann algebra, let $\mathcal U$ be a free ultrafilter, and let $u,v\in N^{\mathcal U}$ be freely independent unitaries. Is it true that, for every $k\in\mathbb N$ , there exist sequences of unitaries $(u_n^{(k)})_{n\in\mathbb N}$ and $(v_n^{(k)})_{n\in\mathbb N}$ in $N$ representing $u$ and $v$ in $N^{\mathcal U}$ such that $(u_n^{(k)},v_n^{(k)})$ are $k$ -independent in $N$ ; that is, every alternating product of at most $k$ trace-zero words in the $u_n^{(k)}$ -variables and $v_n^{(k)}$ -variables has trace zero?

References

Primary source

arXiv

Additional references

Lifting for N-independent sets in II1 factors — arXiv

operator-algebras von-neumann-algebras

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A new unrefereed preprint claims to settle the lifting question, but the claimed solution has not been independently verified.

Problem 31 of Kunnawalkam Elayavalli asks whether freely independent unitaries $u,v\in N^{\mathcal U}$ can be lifted to sequences of pairwise $k$ -independent unitaries in $N$ for every $k$ .

Known results

The case $k=2$ is known: $2$ -independent unitaries can be lifted to independent unitaries. The all- $k$ statement was recorded as open before the new claim.

September 9, 2026 claimed solution

The preprint Lifting for N-independent sets in II1 factors claims a perturbative lifting theorem that upgrades approximate independence to exact independence in $\mathrm{II}_1$ factors, thereby applying to Problem 31. This is a claimed resolution, not a verified one.

Current status (as of September 2026): the $k=2$ case is known, while the general all- $k$ problem has a claimed solution whose correctness remains unverified.

Sources

No solutions have been posted yet.