Result 251, Group theory

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.

Lean formalization Proof

The bigger picture

Why it matters

Amenability means that a group admits a consistent averaging rule unchanged by its symmetries. Two unreviewed manuscripts claim this property determines when certain linear actions can be made length-preserving and, for countable discrete groups, when approximate actions can be corrected.

What changes?

For all discrete groups, the first claim equates amenability with unitarizability: every representation, a multiplication-respecting action by linear operators on a complex Hilbert space, whose operator norms have a common bound becomes length-preserving after a bounded invertible change of coordinates. For countable discrete groups, the second equates amenability with strong Ulam stability: every sufficiently accurate unitary approximate action is uniformly close in operator norm to an exact action on the same complex Hilbert space. Infinite-dimensional spaces must be included.

What does that help mathematicians do?

In particular, the claims would force every nonamenable discrete group to have a uniformly bounded action that no such coordinate change makes unitary. For a countable nonamenable group, they also imply approximate unitary actions with arbitrarily small multiplication errors that remain a fixed positive distance from every exact action on the same space. This identifies amenability as the precise boundary for these two correction principles.

Are there practical applications?

The immediate value is foundational: these characterizations link an averaging property of groups to the geometry and robustness of their Hilbert-space actions. Researchers could use a failure of either correction property to detect nonamenability, respecting the countability restriction for stability. The abstracts do not establish a practical correction algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Unitarizability implies amenability for discrete groups

September 23, 2026 19 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Strong Ulam Stability Characterizes Amenability

October 5, 2026 19 pages

A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026,
  author = {{OpenAI}},
  title = {{Strong Ulam Stability Characterizes Amenability}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf}{OAI:Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/251.md.

Amenability, unitarizability, and strong Ulam stability

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Dixmier's unitarizability problem asks whether a discrete group is amenable exactly when every uniformly bounded Hilbert-space representation is similar to a unitary one. The formalization establishes this equivalence for every discrete group.

For every nonamenable group and every ε>0\varepsilon>0, it also gives a nonunitarizable representation with uniform operator bound at most 1+ε1+\varepsilon. The Hilbert space can be chosen separable for countable groups; a separate linked statement records a separable witness with uniform bound 101101 in that case.

Comparator links

Result Comparator statement
Unitarizability for all discrete groups DixmierAllDiscrete.lean
Separable nonunitarizable witnesses for countable nonamenable groups Dixmier.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.