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)
Manuscripts (2)
Unitarizability implies amenability for discrete groups
September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026}
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.
Strong Ulam Stability Characterizes Amenability
October 5, 2026 · source · cite as \cite{OAI:Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026}
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.
Result 251 · Group theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md