Dixmier’s unitarizability problem for countable groups
Let $G$ be a countable discrete group. A representation $\pi : G \to GL(H)$ on a complex Hilbert space $H$, by bounded invertible operators, is uniformly bounded if $\sup_{g \in G} \|\pi(g)\| < \infty$. The group $G$ is amenable if $\ell^\infty(G)$ has a positive linear functional $m$ with $m(1) = 1$ and $m(f(g^{-1}\,\cdot)) = m(f)$ for all $g \in G$ and $f \in \ell^\infty(G)$. A group is unitarizable if every uniformly bounded representation of it on a complex Hilbert space is similar to a unitary one. Day and Dixmier proved that every amenable group is unitarizable. Nonabelian free groups are not, and unitarizability passes to subgroups, so no group containing a nonabelian free subgroup is unitarizable. Monod and Ozawa proved that a group $G$ is amenable if and only if the wreath product $A \wr G$ is unitarizable for some infinite abelian group $A$.
Problem. Suppose that for every complex Hilbert space $H$ and every representation $\pi : G \to GL(H)$ with $\sup_{g \in G} \|\pi(g)\| < \infty$, there exists a bounded invertible $S$ such that $S^{-1}\pi(g)S$ is unitary for every $g \in G$. Must $G$ be amenable?
Source: Dixmier (1950), who asked whether unitarizability characterizes amenability; for countable groups as stated by Castillo and Ferenczi (2023). Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- GR, OA
- Keywords
- AIM Open Applied Problems, amenability, uniformly bounded representations, unitarizable groups
References
@article{dixmier1950,
author = {Dixmier, Jacques},
title = {Les moyennes invariantes dans les semi-groupes et leurs applications},
journal = {Acta Sci. Math. (Szeged)},
volume = {12},
year = {1950},
pages = {213--227},
}
@article{day1950,
author = {Day, Mahlon M.},
title = {Means for the bounded functions and ergodicity of the bounded representations of semi-groups},
journal = {Trans. Amer. Math. Soc.},
volume = {69},
number = {2},
year = {1950},
pages = {276--291},
doi = {10.2307/1990358},
}
@article{monod2010,
author = {Monod, Nicolas and Ozawa, Narutaka},
title = {The {D}ixmier problem, lamplighters and {B}urnside groups},
journal = {J. Funct. Anal.},
volume = {258},
number = {1},
year = {2010},
pages = {255--259},
doi = {10.1016/j.jfa.2009.06.029},
eprint = {0902.4585},
archivePrefix = {arXiv},
}
@article{castillo2023,
author = {Castillo, Jes{\'u}s M. F. and Ferenczi, Valentin},
title = {Group actions on twisted sums of {B}anach spaces},
journal = {Bull. Malays. Math. Sci. Soc.},
volume = {46},
number = {4},
year = {2023},
pages = {135},
doi = {10.1007/s40840-023-01531-0},
eprint = {2003.09767},
archivePrefix = {arXiv},
}
@misc{aim-dixmier-unitarizability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Dixmier’s unitarizability problem for countable groups}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/038-dixmier-unitarizability.md},
note = {Entry 038 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
Yes: a discrete group (countable or not) is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary one; every nonamenable group has a nonunitarizable representation of bound at most 1 + eps, on a separable space when the group is countable..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00676, "Dixmier’s unitarizability problem for countable groups", clopen.solutions.