Ultraproduct II_1 Factors — Problem 25

Definitions and mathematical statement

A $\mathrm{II}_1$ factor is an infinite-dimensional von Neumann algebra $N$ with center ${\mathbb{C}} 1$ and a faithful normal tracial state $\tau$, normalized by $\tau(1)=1$. A von Neumann algebra is diffuse if it has no nonzero minimal projections. Write $\|x\|_2=\tau(x^*x)^{1/2}$.

The Haagerup property for $(N,\tau)$ means that there is a net of normal unital completely positive, trace-preserving maps $\Phi_i:N\to N$ such that each induced operator on $L^2(N,\tau)$ is compact and $\|\Phi_i(x)-x\|_2\to0$ for every $x\in N$. A net is used here without imposing a separability assumption, which the supplied record does not state.

For purposes of recording the question, let $\mathcal A(N)$ denote the source condition that $N$ has "no diffuse relative $(T)$ subalgebras," and let $\mathcal S(N)$ denote the source condition that $N$ has "unique sequential commutation orbit." These are placeholders for the source terminology. The customary inclusion-based reading of the first condition is that there is no diffuse von Neumann subalgebra $P\subseteq N$ for which the inclusion $P\subseteq N$ has relative property $(T)$. The record does not specify the inclusion conventions or define relative property $(T)$. In particular, this reading must not silently be replaced by the assertion that $P$ itself has property $(T)$.

Existence questions (Elayavalli, Problem 25). Are the following assertions true? \[ \exists\,N\text{ a }\mathrm{II}_1\text{ factor}\quad \mathcal A(N)\ \land\ \mathcal S(N), \] \[ \exists\,N\text{ a }\mathrm{II}_1\text{ factor}\quad \mathcal A(N)\ \land\ \mathcal S(N)\ \land\ N\text{ has the Haagerup property}. \] The supplied record gives no definition of sequential commutation orbit: it specifies neither the objects forming the orbits nor the relation defining them. Thus these displays preserve the two existence questions, but do not constitute a fully specified mathematical formulation until that terminology and the relevant ambient setting are supplied. No separability, ultraproduct representation, or choice of ultrafilter is imposed by this draft.

Short English statement

Elayavalli asks whether a $\mathrm{II}_1$ factor can have no diffuse relative $(T)$ subalgebras while having a unique sequential commutation orbit, and whether such a factor can also have the Haagerup property. The supplied record does not define the orbit condition or fully specify the relative $(T)$ convention, so the formulation remains conditional on clarification of those terms.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $801124$, source problem code catalogue entry, title "Ultraproduct $\mathrm{II}_1$ Factors --- Problem 25." The record attributes the problem to Elayavalli, with proposed year 2025, and labels it open; this is supplied registry status, not an independently verified literature assessment. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source attribution: Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 25, https://arxiv.org/abs/2511.20377. The record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source was not independently accessed or verified for this draft.