Ultraproduct II_1 Factors — Problem 40

Definitions and mathematical statement

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

For an index set $I$ and an ultrafilter $\mathcal U$ on $I$, its tracial ultrapower is \[ N^{\mathcal U}=\ell^\infty(I,N)\big/\mathcal I_{\mathcal U},\qquad \mathcal I_{\mathcal U}=\left\{(x_i)_{i\in I}:\lim_{i\to\mathcal U}\|x_i\|_{2,\tau}=0\right\}, \] with trace $\tau_{\mathcal U}([(x_i)])=\lim_{i\to\mathcal U}\tau(x_i)$. Let $h$ denote the invariant called 1-bounded entropy in the supplied source.

With the intended ultrafilter $\mathcal U$ and entropy convention fixed, the question is whether there exists a type $\mathrm{II}_1$ factor $N$ such that \[ h(N)=0\qquad\text{and}\qquad h(N^{\mathcal U})=+\infty. \] The source does not specify the index set, the properties or quantification of $\mathcal U$, a separability hypothesis on $N$, or the convention defining $h$ on the potentially nonseparable ultrapower. These qualifications remain unresolved; no separability restriction or ultrafilter quantifier is added here.

As background, the supplied source attributes to arguments of S. Popa, under the reference label [PopaOrthoPairs], the assertion that ultrapowers can never be nontrivial tensor products, crossed products, or amalgamated free products, and cannot have diffuse regular abelian subalgebras. Here an abelian von Neumann subalgebra $A\subseteq M$ is diffuse if it has no nonzero minimal projections, and is regular if its unitary normalizer generates $M$. The excerpt supplies no precise scope or hypotheses for these structural assertions, including what makes the listed decompositions nontrivial. They are retained as attributed background, not asserted here as independently verified theorems.

Short English statement

The question asks whether a type $\mathrm{II}_1$ factor can have 1-bounded entropy zero while its tracial ultrapower has 1-bounded entropy $+\infty$. The supplied record labels the problem open. The intended ultrafilter, separability assumptions, and entropy convention require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $801139$, exact source problem code catalogue entry, title Ultraproduct $\mathrm{II}_1$ Factors --- Problem 40. The record attributes the problem to Elayavalli, with proposed year 2025, and supplies the status open; that status is not an independently verified literature assessment.
• Supplied source list: Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 40. Supplied URL: https://arxiv.org/abs/2511.20377. The record reports access on 2026-10-16 and extraction by arxiv_latex. This is supplied dated provenance, not an independently verified access or literature assessment.
• The background attributes structural claims about ultrapowers to S. Popa under the exact reference label [PopaOrthoPairs]. No bibliographic details or link resolving that reference are supplied.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82; registry creation and update timestamps: 2026-10-16T14:00:00Z.