View source for Dixmier's unitarizability problem

'''Dixmier's unitarizability problem''' asks whether a group is amenable exactly when every uniformly bounded representation on a Hilbert space is similar to a unitary representation. Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September and 5 October 2026, claim the answer for discrete groups: amenability is equivalent to unitarizability, and for countable groups also to strong Ulam stability. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the two manuscripts:
** Sz.-Nagy showed that an invertible operator with uniformly bounded integer powers is similar to a unitary operator. Day and Dixmier showed with invariant means that amenable groups are unitarizable, and Dixmier asked in 1950 whether the converse holds.
** Ehrenpreis and Mautner gave non-unitarizable uniformly bounded representations of SL<sub>2</sub>('''R'''), and Pytlik and Szwarc gave explicit families for free groups; by induction, every group containing a non-abelian free subgroup is non-unitarizable. Pisier characterized amenability by a quantitative similarity bound. Epstein and Monod related unitarizability to ℓ<sup>2</sup>-Betti numbers and cost through invariant random forests, and Osin used their criterion to obtain non-unitarizable groups without non-abelian free subgroups. Monod and Ozawa, using the Gaboriau–Lyons theorem, showed that the wreath product ''A'' ≀ ''G'' with an infinite abelian group ''A'' is unitarizable exactly when ''G'' is amenable; Alpeev extended this to finite lamp groups.
** Kazhdan proved that amenable groups are strongly Ulam stable. Burger, Ozawa and Thom showed that groups with a non-abelian free subgroup are not, and asked whether strong Ulam stability characterizes amenability.

== The original results ==
* '''Unitarizability.''' Theorem 1.1 of ''Unitarizability implies amenability for discrete groups'' states that a discrete group is amenable if and only if it is unitarizable. For every non-amenable discrete group and every ε &gt; 0 it gives a non-unitarizable representation with uniform bound at most 1 + ε, on a separable Hilbert space when the group is countable.
** ''Method, as described:'' a bounded operator cocycle that is not inner yields the non-unitarizable representation. The cocycle is built from operators on ℓ<sup>2</sup>(''G''; '''C'''<sup>''k''</sup>) whose rows and columns are controlled by a contracting finite average and by matrix weights from random unit vectors.
** ''Stated scope:'' the conclusion states that the argument concerns discrete groups and does not extend the theorem to non-discrete locally compact groups, where representations are normally required to be strongly continuous.
* '''Strong Ulam stability.''' Theorem 1.1 of ''Strong Ulam Stability Characterizes Amenability'' states that a countable discrete group is strongly Ulam stable if and only if it is amenable. Strong Ulam stability means that unitary almost representations with small defect, on every Hilbert space including infinite-dimensional ones, are uniformly close to unitary representations on the same space. The manuscript presents this as the converse to Kazhdan's theorem, answering the question of Burger, Ozawa and Thom.
* '''Sources.''' At commit <code>1581d4c</code>: [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026 Unitarizability implies amenability for discrete groups] (23 September 2026) and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026 Strong Ulam Stability Characterizes Amenability] (5 October 2026).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/251.md lean/docs/251.md] lists only the unitarizability manuscript. It states that the Lean formalization establishes the equivalence for every discrete group and gives non-unitarizable representations with uniform bound at most 1 + ε, with a separate statement of a separable witness with bound 101 for countable groups. The Comparator challenges are <code>DixmierAllDiscrete</code> and <code>Dixmier</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 23 October 2026)
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>bb5fdb6</code>, 18 October 2026) || Review (fidelity of the formal statements) || Rates this family "partial" and records sandboxed Comparator passes for <code>Dixmier</code> and <code>DixmierAllDiscrete</code>. Its notes state that the equivalence of amenability with strong Ulam stability, named in the family's summary, is not among the formal statements. || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|-
| 2026-10-08 (latest commit <code>501439c</code>, 15 October 2026) || V. Alekseev, [https://github.com/certusoid/unitarisability-and-amenability/tree/501439cd4de23e7df7da54abdbcca6e73b898388 ''A geometric account of the solution to the Dixmier unitarisability problem''] (GitHub, TeX and PDF) || Reading (new account of the proof) and extension (a corollary) || Gives a more geometric account of the unitarizability proof for countable discrete groups. It reinterprets parts of the argument through a flow of positive divergence on a non-amenable group, and replaces the randomized matrix coefficients by a construction from free products of finite groups. It also states, crediting A. Dogon, that the construction shows that every non-amenable group is not deformation rigid in the sense of Burger, Ozawa and Thom. The note states that the exposition was developed with GPT-6 Astra and edited by the author, who takes responsibility for the mathematical claims. || Note on GitHub.
|-
| 2026-10-10 || cert-machine, openai/math lane K: run 38019899278 for [https://github.com/carlostoledo1891/cert-machine/blob/61aae8eb42f1967e16d37d78cfe8a5c1be6d0d72/corpus/openai-math/kernel-runs/38019899278/Dixmier.json Dixmier] and [https://github.com/carlostoledo1891/cert-machine/blob/61aae8eb42f1967e16d37d78cfe8a5c1be6d0d72/corpus/openai-math/kernel-runs/38019899278/DixmierAllDiscrete.json DixmierAllDiscrete] || Verification (two kernels) || Records that Lean's default kernel and the nanoda kernel accept both solutions, in 103 s and 193 s. || Repository.
|-
| 2026-10-13 || M. Colbrook and contributors, [https://github.com/MColbrook/AIM AIM — Open Applied Problems] (GitHub), [https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/038-dixmier-unitarizability.md problem 038] || Catalogue || The collection records its problem 038, Dixmier's problem for countable discrete groups, as solved by the unitarizability manuscript. || The status review is described as "an AI statement-level review of a published manuscript, not a fresh independent audit of its complete proof or a Lean kernel replay". The collection is not affiliated with the American Institute of Mathematics.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]