Ultraproduct II_1 Factors — Problem 26

Definitions and mathematical statement

A type $\mathrm{II}_1$ factor is a von Neumann algebra $N$ with center ${\mathbb{C}} 1$, no nonzero minimal projections, and a faithful normal tracial state $\tau$ satisfying $\tau(1)=1$. No separability assumption is stated in the supplied record. Write $\mathfrak c=2^{\aleph_0}$ for the cardinality of the continuum.

The supplied record does not define the invariant $h(N)$ or the term "sequential commutation orbit." Consequently, the following is a conditional formulation requiring those definitions from the source. Let $\mathscr O_{\mathrm{sc}}(N)$ denote the family of sequential commutation orbits intended by the source, assuming these are subsets of $N$. For a subset $S\subseteq N$, let $W_N^*(S)$ denote the smallest unital von Neumann subalgebra of $N$ containing $S$.

Interpreting "continuum many" as exactly $\mathfrak c$, the two questions are:

1. Does there exist a type $\mathrm{II}_1$ factor $N$ such that \[ h(N)=0 \quad\text{and}\quad \lvert\mathscr O_{\mathrm{sc}}(N)\rvert=\mathfrak c? \]
2. Does there exist a type $\mathrm{II}_1$ factor $N$ such that \[ h(N)=0 \quad\text{and}\quad \forall O\in\mathscr O_{\mathrm{sc}}(N),\qquad W_N^*(O)\ne N? \]

The existential witnesses in the two questions need not coincide. The meaning of $h$, the underlying elements and ambient algebra used to define the orbits, and any source-specific hypotheses remain unresolved; this formulation does not supply those missing definitions.

Short English statement

The source asks two separate existence questions: can a type $\mathrm{II}_1$ factor with $h(N)=0$ have continuum many sequential commutation orbits, and can such a factor fail to be generated by any single such orbit? The invariant and orbit terminology require source verification before these questions become fully specified mathematical statements.

Sources

• Local UnsolvedMath registry: permanent ID $801125$, exact source problem code catalogue entry, title "Ultraproduct $\mathrm{II}_1$ Factors --- Problem 26." The supplied record attributes the problem to Elayavalli, with proposed year 2025, and labels it open; this status is supplied metadata, not an independently verified mathematical assessment.
• Supplied source list: Elayavalli, 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025), Problem 26. Supplied URL: https://arxiv.org/abs/2511.20377. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.