Ultraproduct II_1 Factors — Problem 2

Definitions and mathematical statement

A type $\mathrm{II}_1$ factor is an infinite-dimensional finite von Neumann algebra $N$ with center ${\mathbb{C}} 1_N$, equipped with its normalized faithful normal trace $\tau_N$. Write $\|x\|_{2,\tau_N}=\tau_N(x^*x)^{1/2}$. Using the finite-set formulation, $N$ has property $\Gamma$ if, for every finite subset $F\subseteq N$ and every $\varepsilon>0$, there exists a unitary $u\in N$ such that \[ \tau_N(u)=0,\qquad \|ux-xu\|_{2,\tau_N}<\varepsilon\quad(x\in F). \] The term non-$\Gamma$ means failure of this property. The supplied record does not state a separability hypothesis or specify its convention for property $\Gamma$ outside the separable setting.

Under the usual tracial-ultrapower convention, Connes embeddability means that $N$ admits a unital trace-preserving injective $*$-homomorphism into $R^\omega$, where $R$ is the hyperfinite type $\mathrm{II}_1$ factor and $\omega$ is a free ultrafilter on ${\mathbb{N}}$. Here \[ R^\omega=\ell^\infty({\mathbb{N}},R)\big/\{(x_n):\lim_{n\to\omega}\|x_n\|_{2,\tau_R}=0\}. \] This explains the standard terminology; the supplied record does not specify its precise embeddability convention or the intended separability scope.

Let $A$ temporarily denote the ambient algebra written $\mathbb{M}^{\mathcal U}$ in the source. With embeddings understood as unital normal trace-preserving injective $*$-homomorphisms, the requested assertion is the following schema: \[ \begin{gathered} \text{For every Connes embeddable non-$\Gamma$ type $\mathrm{II}_1$ factor $N$,}\\ \text{there exists an embedding }\pi:N\hookrightarrow A\text{ such that}\\ \pi(N)'\cap A={\mathbb{C}} 1_A. \end{gathered} \] The relative commutant is \[ \pi(N)'\cap A=\{a\in A:a\pi(x)=\pi(x)a\text{ for every }x\in N\}. \] Thus the source's notation $N'\cap\mathbb{M}^{\mathcal U}$ is interpreted using the image of the requested embedding. The record does not define $\mathbb M$, the ultrafilter $\mathcal U$, or their quantifiers. Consequently, it does not determine a complete standalone assertion about a specified target: whether the target is fixed, universally quantified, or allowed to depend on $N$ remains unresolved. No particular matrix ultraproduct or other ambient factor is selected here.

Short English statement

The supplied question asks whether every Connes embeddable type $\mathrm{II}_1$ factor without property $\Gamma$ admits an embedding into the source's $\mathbb M^{\mathcal U}$ whose relative commutant consists only of scalars. The target and its quantifiers are unspecified, so this remains a conditional formulation requiring source clarification. The registry labels the problem open; that label is supplied curation, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $801101$, source problem code catalogue entry, title "Ultraproduct $\mathrm{II}_1$ Factors --- Problem 2." The record attributes the problem to Elayavalli, with proposed year $2025$, and supplies status open.
• Supplied source attribution: Elayavalli, "50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors" (2025), Problem 2, https://arxiv.org/abs/2511.20377. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not claims of independent access or verification.
• Registry provenance: upstream ID $801101$, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82, updated 2026-10-16T14:00:00Z. No separate dated literature assessment is supplied.