Quadrilaterals of Subfactors with Two Intermediate Factors

Definitions and mathematical statement

A factor is a von Neumann algebra whose center consists of the scalar multiples of its identity. The supplied configuration is a quadruple of factors $(N,P,Q,M)$ with inclusions \[ N\subseteq P\subseteq M,\qquad N\subseteq Q\subseteq M. \] Thus $P$ and $Q$ are intermediate subfactors of $N\subseteq M$. The cleaned source formulation imposes finite index, written symbolically as \[ [M:N]<\infty. \] The factor type and the index convention are not specified in the supplied record; this notation records the stated requirement without resolving those choices. The record also does not explicitly specify properness of the inclusions, distinctness of $P$ and $Q$, irreducibility, or the conditions $P\cap Q=N$ and $P\vee Q=M$, where $P\vee Q$ denotes the von Neumann algebra generated by $P$ and $Q$. These additional conditions are therefore not imposed here.

Problem, as a research programme. Develop a planar-algebra theory for finite-index configurations $N\subseteq P,Q\subseteq M$ that accounts for the relative position of the two intermediate subfactors, extending the treatment beyond the commuting-square and restricted cocommuting cases mentioned in the cleaned source formulation.

Unresolved scope. The supplied verbatim excerpt says only that the situation with two intermediate subfactors remains an open problem. It does not specify a theorem, classification criterion, invariant, or other concrete output that would constitute a solution. The precise planar-algebra framework, its required hypotheses, and the restricted cases being referenced are also unspecified. Consequently, the supplied evidence supports this research programme but does not determine a unique fully quantified mathematical conjecture.

Short English statement

Study how two intermediate subfactors $P$ and $Q$ sit inside a finite-index inclusion $N\subseteq M$, using planar-algebra methods beyond the cases identified by the supplied curation. The exact hypotheses and the intended mathematical output remain unspecified, so this is a statement-only research-programme draft requiring review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006356$, source problem code catalogue entry, title Quadrilaterals of Subfactors with Two Intermediate Factors. Project registry references: data/unsolvedmath/problems.json and the supplementary catalog/problems.json. No individual proposer is identified; the supplied proposed year is 2025.
• Supplied report citation: Subfactors and Applications (2026), Oberwolfach Reports, pp. 1823--1886, DOI 10.4171/owr/2025/34. https://doi.org/10.4171/owr/2025/34. The supplied verbatim excerpt states that the situation for a subfactor with two intermediate subfactors remains an open problem.
• Supplied literature source: Oberwolfach Report 34/2025. https://ems.press/content/serial-article-files/52239.
• Supplied literature source: Classification of noncommuting quadrilaterals of factors. https://arxiv.org/abs/0704.1121.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general two-intermediate-subfactor configuration open, while reporting substantial results for restricted irreducible and structured cases. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from its surrounding exposition, and its notation repaired. These are supplied provenance claims; no source visit or independent verification is asserted here.