Invariance of free entropy dimension under change of generators
Here $\delta$ is Voiculescu's free entropy dimension of a finite tuple of self-adjoint elements of a tracial von Neumann algebra.
Problem. Let $M$ be a II$_1$ factor, and let $x_1, \dots, x_n \in M$ and $y_1, \dots, y_m \in M$ be self-adjoint elements such that each of the two tuples generates $M$ as a von Neumann algebra. Is $\delta(x_1, \dots, x_n) = \delta(y_1, \dots, y_m)$?
Source: D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory II (1994); listed as O.10 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
- Secondary
- PR
- Keywords
- free entropy dimension, free probability, II₁ factors, Peterson's operator algebra problems
References
@article{voiculescu1994,
author = {Voiculescu, Dan},
title = {The analogues of entropy and of {F}isher's information measure in free probability theory, {II}},
journal = {Invent. Math.},
volume = {118},
number = {1},
pages = {411--440},
year = {1994},
doi = {10.1007/BF01231539},
}
@misc{peterson-o10,
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.10; version of 15 May 2026},
}
@article{jung2007,
author = {Jung, Kenley},
title = {Strongly 1-bounded von {N}eumann algebras},
journal = {Geom. Funct. Anal.},
volume = {17},
number = {4},
pages = {1180--1200},
year = {2007},
doi = {10.1007/s00039-007-0624-9},
}
@article{jungshlyakhtenko2007,
author = {Jung, Kenley and Shlyakhtenko, Dimitri},
title = {Any generating set of an arbitrary property {T} von {N}eumann algebra has free entropy dimension {$\le 1$}},
journal = {J. Noncommut. Geom.},
volume = {1},
number = {2},
pages = {271--279},
year = {2007},
doi = {10.4171/JNCG/7},
}
Resolution claims
We solve the free group factor isomorphism problem affirmatively by proving that $L(\F_2)$ and $L(\F_3)$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $L(\F_\infty)$, are isomorphic and have fundamental group $\R_{>0}$.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: math @ 7ef5c0c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01052, "Invariance of free entropy dimension under change of generators", clopen.solutions.