OP-00966 Open

Uniqueness of McDuff decompositions up to amplification

Here $R$ is the hyperfinite II$_1$ factor and $N^t$ the amplification of $N$ by $t > 0$.

Problem. Let $M$ and $N$ be II$_1$ factors, neither of them McDuff, with $M \mathbin{\overline\otimes} R \cong N \mathbin{\overline\otimes} R$. Must $M \cong N^t$ for some $t > 0$?

Source: Listed without attribution as S.8 in J. Peterson, Open problems in operator algebras (version of 15 May 2026); the uniqueness question for McDuff decompositions studied by Popa (2007) and by Ioana and Spaas (2019).

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

Classification

Primary: OA — Operator Algebras
Keywords: amplifications, McDuff decomposition, McDuff factors, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{peterson-s8,
  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 S.8; version of 15 May 2026},
}
@article{popa2007,
  author  = {Popa, Sorin},
  title   = {On {O}zawa's property for free group factors},
  journal = {Int. Math. Res. Not.},
  year    = {2007},
  number  = {11},
  pages   = {Art. ID rnm036},
  doi     = {10.1093/imrn/rnm036},
}
@article{ioana2019,
  author  = {Ioana, Adrian and Spaas, Pieter},
  title   = {A class of {II}$_1$ factors with a unique {M}c{D}uff decomposition},
  journal = {Math. Ann.},
  volume  = {375},
  number  = {1--2},
  pages   = {177--212},
  year    = {2019},
  doi     = {10.1007/s00208-019-01862-z},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00966, "Uniqueness of McDuff decompositions up to amplification", clopen.solutions.