OpenAI · 251-01 · 2026-09-23 · Group theory
Unitarizability implies amenability for discrete groups
preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026
Family 251. Amenability, unitarizability, and Ulam stability.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 251: Amenability, unitarizability, and strong Ulam stability. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.