Amenability, unitarizability, and strong Ulam stability

A discrete group is amenable exactly when every uniformly bounded representation can be made unitary.

2 papers September 23, 2026 Explained by AI

Some groups have a fair way to average over all their elements; these are called amenable. In 1950 Dixmier asked if this matches a property of their actions by matrices. Every bounded action becomes length-keeping after one change of ruler. The first paper proves yes for every discrete group, with a Lean formalization listed. A second paper claims a similar answer for approximate actions of countable groups.

First step

1950

Jacques Dixmier, who asked whether unitarizability characterizes amenable groups

Significance

Major

Dixmier's unitarizability problem has been a central open question in representation theory since 1950.

Trust

Formal proof listed

Field

Group theory

2 papers

The explanation

Start with one 2×2 matrix, T = [[1, 1], [0, −1]]. Multiply it by itself and you get the identity matrix. So its powers just alternate between T and the identity. Its largest stretch factor, the operator norm, is (1 + √5)/2 ≈ 1.618. A unitary matrix keeps every length, so its norm is 1. T is not unitary, but it is close to one. Take S = [[1, −0.5], [0, 1]]. Then S⁻¹TS is the diagonal matrix with entries 1 and −1, which is unitary. Changing the ruler, by the matrix S, turned T into a length-keeping matrix.

That small case is a group at work. A representation sends each group element to an invertible operator, respecting multiplication. It is uniformly bounded when all those operators have norms below one fixed number. It is unitarizable when one change of ruler S makes every operator unitary at onceSource 1. Sz.-Nagy showed in 1947 that an invertible operator with uniformly bounded powers is always similar to a unitary oneSource 2. That covers the integers.

A group is amenable when it has an invariant mean: a fair average over all bounded functions on the group, unchanged by shifting. The integers are amenable. The free group on two letters is not. Day and Dixmier showed amenability implies unitarizability, and Dixmier asked whether the converse holdsSource 3. Counterexamples for free groups were known. Nonunitarizability passes from a subgroup to a larger groupSource 4. So groups with free subgroups were covered. The obstacle was nonamenable groups without free subgroups.

The first paper proves the converse: a discrete group is amenable if and only if it is unitarizableSource 5. Take any nonamenable group and any ε > 0. The paper builds a representation with norm bound at most 1 + ε that no ruler change makes unitarySource 6. The construction uses scalar kernels whose rows and columns have bounded sums, weighted by positive rank-one piecesSource 7.

A second paper treats almost representations. These are maps to unitary operators that respect multiplication only up to a small error, called the defect. A group is strongly Ulam stable when every small-defect map is close to a genuine representation on the same spaceSource 8. Kazhdan proved that amenable groups have this propertySource 9. The paper claims the converse for countable groups: strong Ulam stability holds exactly for amenable groupsSource 10. Infinite-dimensional spaces are essential; with finite dimensions only, some nonamenable groups are stableSource 11.

How sure is this? A computer proof checker called Lean has a formal proof of Dixmier's equivalence for every discrete group. The Ulam stability paper has no Lean formalization. No specialist review of either paper has been published.

What it doesn't show. This does not settle Dixmier's question for non-discrete groups, and the strong Ulam stability result has no Lean formalization.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds, and the Dixmier equivalence has a Lean formalization. The strong Ulam stability claim has no Lean formalization. No specialist review is published.

Words used

Representation: A rule sending each group element to an invertible operator on a space, so that products match products.
Operator norm: The largest factor by which an operator can stretch the length of a vector.
Unitary: An operator that keeps every length and angle the same.
Uniformly bounded: All operators of the representation have norm below one fixed number.
Unitarizable: Some single change of coordinates makes every operator of the representation unitary.
Amenable group: A group with an invariant mean: a fair average of bounded functions on it that shifting does not change.
Hilbert space: A space of vectors, possibly infinite-dimensional, with lengths and angles like ordinary space.
Strongly Ulam stable: Every map that is nearly a unitary representation is uniformly close to a true one on the same space.

The original papers (2)

Unitarizability implies amenability for discrete groups (PDF, opens in a new tab)
September 23, 2026 Formal proof listed
Strong Ulam Stability Characterizes Amenability (PDF, opens in a new tab)
October 5, 2026

Lean notes

All quoted lines (11)

The representation is unitarizable if some bounded invertible $S$ makes every $S\pi(g)S^{-1}$ unitary.
Unitarizability implies amenability for discrete groups Back to text
The one-operator case goes back to Sz.-Nagy: an invertible operator whose integer powers are uniformly bounded is similar to a unitary operator.
Unitarizability implies amenability for discrete groups Back to text
Dixmier explicitly asked whether this property characterizes groups admitting an invariant mean.
Unitarizability implies amenability for discrete groups Back to text
Nonunitarizability passes from a subgroup to the ambient group by induction.
Unitarizability implies amenability for discrete groups Back to text
A discrete group is amenable if and only if it is unitarizable.
Unitarizability implies amenability for discrete groups Back to text
The proposition therefore gives a representation on $\mathcal K= \left(\bigoplus_{\ell\ge1}\mathcal H_\ell\right) \oplus\left(\bigoplus_{\ell\ge1}\mathcal H_\ell\right)$ with uniform bound at most $1+\varepsilon$ that is not similar to a unitary representation.
Unitarizability implies amenability for discrete groups Back to text
Positive rank-one weights preserve the row and column bounds while increasing the kernel's Hilbert–Schmidt energy relative to the fibre dimension.
Unitarizability implies amenability for discrete groups Back to text
The number $\delta$ must work simultaneously for all Hilbert spaces, including infinite-dimensional ones.
Strong Ulam Stability Characterizes Amenability Back to text
Kazhdan proved the implication from amenability to strong Ulam stability; see also.
Strong Ulam Stability Characterizes Amenability Back to text
A countable discrete group is strongly Ulam stable if and only if it is amenable.
Strong Ulam Stability Characterizes Amenability Back to text
Nonamenable groups can satisfy that property.
Strong Ulam Stability Characterizes Amenability Back to text

Other sources

openai/math, Lean scope note for family 251 The formalization establishes this equivalence for every discrete group.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

249A finitely generated Eilenberg–Ganea counterexampleGroup theory
250Boone–Higman embeddings with higher finitenessGroup theory
252A torsion-free hyperbolic group that is neither residually finite nor linear over any fieldGroup theory
253An infinite finitely presented simple amenable groupGroup theory
248Thompson's group F is nonamenableGroup theory

Browse Group theory Open since the 1950s Lean-checked results