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Unitarizability implies amenability for discrete groups

OpenAI · September 23, 2026

Lean

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Abstract

We prove that a discrete group is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary representation. This resolves Dixmier's unitarizability problem affirmatively for discrete groups.

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Notes

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Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope note 251: Amenability, unitarizability, and strong Ulam stability (lean/docs/251.md, October 2026). The release's Lean scope note states that the formalization proves amenability ⇔ unitarizability for every discrete group and gives nonunitarizable representations with bound at most 1+ε, with comparator statements DixmierAllDiscrete.lean and Dixmier.lean (the latter a separable witness with bound 101 for countable groups). [AI-generated per source]

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

  • Vadim Alekseev, A geometric account of the solution to the Dixmier unitarisability problem (GitHub note, 8 October 2026). A re-exposition of this paper's proof via a positive divergence flow on a non-amenable group, replacing the randomized Markov-operator construction by one coming from free groups and recording a remark, credited to Alon Dogon, that non-amenable groups are not deformation rigid; the note states it was developed through interaction with an AI model ("GPT-6 Astra") and edited by the author (TeX source).
  • Claude Opus 5.5 / Open Mathematics Courses, Amenability and unitarizability (online course, undated). Four lessons with exercises present the manuscript's route through contracting averages, sparse edge assignments, random sign frames and the operator-cocycle obstruction; the source describes the course as self-checked with independent review incomplete. [AI-generated per source]

News and blogs

  • Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (cert-machine audit of release commit fd4aeeb2, 10 October 2026). The audit lists family 251's strong Ulam stability assertion among claims from manuscripts not linked by their family's Lean scope note, distinguishing that assertion from the unitarizability formalization. [AI-generated per source] [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026,
  author = {{OpenAI}},
  title = {{Unitarizability implies amenability for discrete groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf}{OAI:Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/251.md. The Lean statements have not been checked by people against the paper's claims.

Family 251: Amenability, unitarizability, and strong Ulam stability

Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.

Other papers in this family: