OP-00960 Open

Indices of irreducible subfactors of the hyperfinite II₁ factor

Let $R$ be the hyperfinite II$_1$ factor. Jones proved that the index of a subfactor of a II$_1$ factor lies in $\{4\cos^2(\pi/n) : n \ge 3\} \cup [4, \infty]$.

Problem. Does every number in $\{4\cos^2(\pi/n) : n \ge 3\} \cup [4, \infty]$ occur as the index $[R : N]$ of an irreducible subfactor $N \subseteq R$?

Source: V. F. R. Jones, Index for subfactors (1983); listed as O.7 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: OA — Operator Algebras
Keywords: hyperfinite II₁ factor, Jones index, Peterson's operator algebra problems, subfactors
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{jones1983,
  author  = {Jones, V. F. R.},
  title   = {Index for subfactors},
  journal = {Invent. Math.},
  volume  = {72},
  number  = {1},
  pages   = {1--25},
  year    = {1983},
  doi     = {10.1007/BF01389127},
}
@misc{peterson-o7,
  author       = {Peterson, Jesse},
  title        = {Open problems in operator algebras},
  howpublished = {Web page},
  year         = {2026},
  url          = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
  note         = {Problem O.7; version of 15 May 2026},
}
@misc{bisch2025,
  author        = {Bisch, Dietmar and C{\'a}ceres, Julio},
  title         = {New hyperfinite subfactors with infinite depth},
  year          = {2025},
  eprint        = {2509.00227},
  archivePrefix = {arXiv},
  note          = {v2, 21 Oct 2025},
}
@misc{bisch2024,
  author        = {Bisch, Dietmar and C{\'a}ceres, Julio},
  title         = {Graph planar algebra embeddings and infinite depth subfactors},
  year          = {2024},
  eprint        = {2410.14819},
  archivePrefix = {arXiv},
}
@misc{fang2026,
  author        = {Fang, Junsheng and Jiang, Chunlan and Ma, Minghui and Shi, Rui},
  title         = {Strongly irreducible operators, {$K_0$}-groups, and {J}ones indices in the hyperfinite type {II}$_1$ factor},
  year          = {2026},
  eprint        = {2610.22028},
  archivePrefix = {arXiv},
}
@article{shlyakhtenko2002,
  author  = {Shlyakhtenko, Dimitri and Ueda, Yoshimichi},
  title   = {Irreducible subfactors of {$L(\mathbb F_\infty)$} of index {$\lambda > 4$}},
  journal = {J. Reine Angew. Math.},
  volume  = {548},
  pages   = {149--166},
  year    = {2002},
  doi     = {10.1515/crll.2002.057},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00960, "Indices of irreducible subfactors of the hyperfinite II₁ factor", clopen.solutions.