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Dixmier's unitarizability problem for discrete groups

Theorem 1.1: a discrete group is amenable if and only if every uniformly bounded representation on every complex Hilbert space is similar to a unitary one. For every nonamenable $G$ and $\varepsilon>0$ there is a representation with $\sup_g\|\pi(g)\|\le1+\varepsilon$ not similar to a unitary one, on a separable space if $G$ is countable. This settles Dixmier's question for all discrete groups. Not shown: anything for nondiscrete locally compact groups, or a bound on similarity degree beyond what the theorem gives.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Dixmier's unitarizability problem for discrete groups · Dixmier's unitarizability problem

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Named evidence

VibeMathed record: Dixmier's unitarizability problem for discrete groups

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Unitarizability implies amenability for discrete groups' was read against Dixmier's question as cited (1950, section 5). The proof (a bounded non-inner operator cocycle built from sparse assignments and random-sign matrix weights) was not refereed. Lean: lean/formalization.yaml lists comparator config ComparatorChallenges/DixmierAllDiscrete.json, declaration OAI.Dixmier.current_main_theorem in OAI/Analysis/Unitarizability/AllDiscrete.lean, permitted axioms propext, Classical.choice and Quot.sound. The statement file was read: for every discrete group, amenability (a positive normalized left-invariant mean on bounded functions) is equivalent to every uniformly bounded representation on every complete complex inner product space being similar to an isometric one, and every nonamenable group has a nonunitarizable representation with bound at most 1 + epsilon, on a separable space when the group is countable. That is the headline claim. A second challenge, ComparatorChallenges/Dixmier.json (countable groups, separable witness with bound 101, solution module OAI/Analysis/Unitarizability/UniverseTransport.lean, present at the commit), is not in the formalization catalogue; its statement was read here. Neither was rebuilt here.

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Same family: Unitarizability implies amenability for discrete groups

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean proof: OAI.Dixmier.current_main_theorem

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean comparator statement: DixmierAllDiscrete

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.