Ultraproduct II_1 Factors — Problem 10

Definitions and mathematical statement

A type $\mathrm{II}_1$ factor is an infinite-dimensional finite von Neumann algebra with center ${\mathbb{C}} 1$ and normalized faithful normal trace. We interpret separability as separability of the predual. A type $\mathrm{II}_1$ factor $M$ is prime if there are no diffuse type $\mathrm{II}_1$ factors $P,Q$ such that \[ M\cong P\,\overline\otimes\,Q, \] where $\overline\otimes$ denotes the spatial von Neumann algebra tensor product.

We interpret an existential copy of $N$ in $M$ as a unital normal trace-preserving $*$-embedding $\iota:N\hookrightarrow M$ that is existential in the continuous first-order language of tracial von Neumann algebras. Explicitly, for every quantifier-free formula $\varphi(\bar x,\bar y)$ and every finite parameter tuple $\bar a$ from $N$, \[ \inf_{\bar b\in N}\varphi^N(\bar b,\bar a) = \inf_{\bar c\in M}\varphi^M(\bar c,\iota(\bar a)), \] with all variables ranging over the operator-norm ball sorts prescribed by the formula. The supplied record does not specify this language or its convention for existential embeddings; this is the standard model-theoretic interpretation used in this draft.

Question. For every separable type $\mathrm{II}_1$ factor $M$ and every nonamenable biexact type $\mathrm{II}_1$ factor $N$, does the existence of such an existential embedding $\iota:N\hookrightarrow M$ imply that $M$ is prime?

Here nonamenability is understood in the von Neumann algebra sense. The hypothesis that $N$ is biexact is retained exactly as supplied. The record gives no definition or convention for biexactness of a von Neumann factor, so this hypothesis requires clarification from the source before the formulation can be considered fully specified.

Short English statement

Must every separable type $\mathrm{II}_1$ factor containing an existential copy of a nonamenable biexact type $\mathrm{II}_1$ factor be prime? This draft uses the usual tracial model-theoretic interpretation of existential embedding; the intended definition of biexactness remains unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801109$, exact source problem code catalogue entry; title: Ultraproduct $\mathrm{II}_1$ Factors --- Problem 10. The registry attributes the problem to Elayavalli, with proposed year 2025, and labels it open. This label is supplied source metadata, not an independently verified mathematical status.
• Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 10, as identified in the supplied provenance: https://arxiv.org/abs/2511.20377. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of source consultation for this draft.
• Supplied registry revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on 2026-10-16.