OP-01001 Open

Popa's free flip problem

Problem. Let $M$ be a II$_1$ factor such that the free flip $x * y \mapsto y * x$ is connected to the identity by a path in $\mathrm{Aut}(M * M)$. Must $M$ be isomorphic to an interpolated free group factor?

Source: S. Popa, Some open problems in W*-rigidity (2013), for nonamenable M; listed as F.3 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: automorphism groups, free products, interpolated free group factors, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{popa2013problems-f3,
  author       = {Popa, Sorin},
  title        = {Some open problems in {W}$^*$-rigidity},
  howpublished = {Slides, Paris, June 2013},
  year         = {2013},
  url          = {https://www.math.ucla.edu/~popa/ProblemsJune2013.pdf},
}
@misc{peterson-f3,
  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 F.3; version of 15 May 2026},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-01001, "Popa's free flip problem", clopen.solutions.