Relative Commutants and Möbius Nets for Finite-Group Inclusions

Definitions and mathematical statement

Let $N_0\subset M_0\subset B(\mathcal H_0)$ be von Neumann algebras and let $\Omega_0$ be a unit vector. Interpret a standard half-sided modular inclusion as follows: $\Omega_0$ is cyclic and separating for both $N_0$ and $M_0$, is cyclic for $N_0'\cap M_0$, and the modular operator $\Delta_{M_0,\Omega_0}$ satisfies \[ \Delta_{M_0,\Omega_0}^{it}N_0\Delta_{M_0,\Omega_0}^{-it}\subseteq N_0 \qquad\text{whenever }\varepsilon t\geq 0, \] for a fixed sign $\varepsilon\in\{+1,-1\}$. Here cyclicity means that the orbit of the vector under the algebra has dense linear span, and separating means that $x\Omega_0=0$ implies $x=0$ for elements of the algebra. The supplied record does not specify the half-sided sign convention; either convention must be used consistently throughout the construction.

Let $K$ be a finite nonempty indexing set, and for each $k\in K$ take a copy $(N_k\subset M_k,\Omega_k)$ of $(N_0\subset M_0,\Omega_0)$, with vector state $\varphi_k(x)=\langle\Omega_k,x\Omega_k\rangle$. Form the reduced von Neumann algebra free product with respect to these states, \[ (M,\varphi)=\mathop{*}_{k\in K}(M_k,\varphi_k), \] and let $\Omega$ be its cyclic vector in the free-product representation. Write $\lambda_k:M_k\to M$ for the canonical embeddings and set \[ N=\left(\bigcup_{k\in K}\lambda_k(N_k)\right)''. \] The free-product state restricts to $\varphi_k$ on each copy, and freeness means that \[ \varphi\left(\lambda_{k_1}(x_1)\cdots\lambda_{k_r}(x_r)\right)=0 \] whenever $r\geq1$, $x_j\in M_{k_j}$, $\varphi_{k_j}(x_j)=0$, and $k_j\ne k_{j+1}$ for adjacent indices.

Problem. For the supplied initial inclusion and finite family of copies, determine the relative commutant \[ R=N'\cap M=\{x\in M: xn=nx\text{ for every }n\in N\}. \] In particular, determine whether $R\ne{\mathbb{C}}1$. If it is nontrivial, identify the Möbius-covariant net associated with the inclusion, insofar as the necessary reconstruction hypotheses hold. Such a net assigns von Neumann algebras to proper nonempty open intervals of the circle and has isotony, locality, and positive-energy Möbius covariance with a vacuum vector.

Unresolved qualifications. The original wording calls $K$ a finite group, whereas the reconstructed statement uses a finite set. The construction above uses only its underlying indexing set; no group action or use of multiplication in $K$ is specified. Nonemptiness is imposed to describe a family of copies, since the source does not discuss an empty free product or give a lower bound on $|K|$. Moreover, nontriviality of $R$ alone does not state that $\Omega$ is cyclic for $R$, as required for standardness in the usual reconstruction from a standard half-sided modular inclusion. The record does not specify whether standardness of the resulting inclusion is part of the intended question, or whether reconstruction should instead occur on a suitable subspace. Thus the net-identification clause retains this unresolved qualification.

Short English statement

Take finitely many state-preserving copies of a standard half-sided modular inclusion and form their free-product inclusion. Determine which elements of the larger algebra commute with the smaller one. If this relative commutant is nontrivial, identify the associated Möbius-covariant net, subject to clarifying the reconstruction hypotheses. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $30003672$, source problem code catalogue entry, in the supplied complete source record associated with data/unsolvedmath/problems.json; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2017.
• Supplied report citation: Reflection Positivity (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/55, pp. 3263--3343. https://doi.org/10.4171/owr/2017/55. The supplied title year is 2018, while the report identifier and recorded proposed year refer to 2017; these details are retained as supplied.
• Supplied literature source: Free products in AQFT. https://aif.centre-mersenne.org/item/10.5802/aif.3269.pdf.
• Supplied literature source: OWR Report 55/2017. https://ems.press/content/serial-article-files/46717.
• Supplied literature assessment, dated exactly 2026-08-21: the relative commutant problem for finite free-product half-sided modular inclusions is described as still unknown, while nontrivial relative commutants are reported in related finite Möbius-net constructions. This is supplied dated curation, not an independently verified literature assessment; related constructions are not asserted here to settle the stated problem.
• The registry labels the statement corrected_verified and reports that the official report was checked and that the reconstruction restored the free-product inclusion and finite indexing set. These are supplied provenance claims, not source verification performed for this draft.