Amenability decides when bounded actions can be made unitary and approximate actions repaired

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Family 251 · Group theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

A unitary operator preserves lengths and angles. A representation of a group by unitary operators therefore realizes every group element as an exact symmetry of a Hilbert space.

There are two natural ways to relax this ideal. One can keep the group law exact but allow operators that distort lengths by a uniformly bounded amount. Alternatively, one can keep every operator unitary but permit small errors in the multiplication law.

Family 251 asks when these defects can always be repaired. Its first manuscript claims that every uniformly bounded representation can be made unitary by a bounded change of coordinates exactly when the discrete group is amenable. Its second claims that, for countable discrete groups, sufficiently accurate unitary approximate representations can always be corrected exactly when the group is amenable.

The conclusions concern all Hilbert spaces, including infinite-dimensional ones. Restricting the second question to finite-dimensional matrices changes the answer and is not the theorem stated here.

2. Background and essential definitions

Bounded representations and changes of inner product

A complex Hilbert space $H$ is a complete inner-product space. It may be finite-dimensional, such as $\mathbb C^n$, or infinite-dimensional, such as the square-summable complex sequences.

A representation $\pi:G\to\mathrm{GL}(H)$ assigns a bounded invertible operator to each group element and satisfies

$$\pi(gh)=\pi(g)\pi(h),\qquad \pi(e)=I.$$

Here $e$ is the group identity and $I$ the identity operator. The operator norm measures the largest stretching of unit vectors.

The representation is uniformly bounded if

$$|\pi|=\sup_{g\in G}\|\pi(g)\|<\infty.$$

This bound is required over the whole group, not just over a generating set. Since inverses are also group elements, it controls the inverse operators as well.

The representation is unitarizable if there is one bounded invertible operator $S$ such that every $S\pi(g)S^{-1}$ is unitary. Equivalently, one can choose an equivalent Hilbert inner product preserved by the action. Equivalent means that the new and old norms bound each other by fixed positive constants.

A group is unitarizable when this is possible for every uniformly bounded representation. The change of coordinates may depend on the representation; the definition does not prescribe a common bound on its condition number $\|S\|\|S^{-1}\|$.

Amenability as an averaging property

A discrete group is amenable when bounded functions on it admit a positive normalized translation-invariant mean. This is a generalized averaging rule that assigns average one to the constant function one and respects translations.

Finite groups have an ordinary uniform average. The integers admit generalized averages obtained from long intervals. For amenable groups, such averaging can be used to turn a uniformly bounded family of distorted inner products into one invariant inner product.

The question is whether the ability to repair every representation already forces that averaging property to exist.

Approximate multiplication

Let $\mu:G\to\mathcal U(H)$ assign unitary operators, with $\mu(e)=I$, but without assuming exact multiplication. Its defect is

$$\operatorname{def}(\mu)= \sup_{g,h\in G}\|\mu(gh)-\mu(g)\mu(h)\|.$$

The group is strongly Ulam stable if, for every desired accuracy $\varepsilon>0$, some $\delta>0$ guarantees that any such map of defect at most $\delta$ is uniformly within $\varepsilon$ of a genuine unitary representation on the same $H$.

The tolerance must work for every Hilbert space. It can depend on the group and the desired accuracy, but not on the dimension of the chosen representation. Distance is measured uniformly over all group elements in operator norm.

3. Development history and previous partial results

Sz.-Nagy proved in 1947 that an invertible Hilbert-space operator with uniformly bounded positive and negative powers is similar to a unitary operator. This is the representation problem for the infinite cyclic group.

Day and Dixmier extended the averaging mechanism to amenable groups. Dixmier’s 1950 question asked for the converse: does unitarizability characterize amenability?

Early nonunitarizable representations showed that uniform boundedness alone was insufficient. Pytlik and Szwarc constructed families for free groups. Induction carries such an obstruction from a subgroup to its ambient group, covering groups with nonabelian free subgroups but leaving many other nonamenable groups untreated.

Pisier developed quantitative criteria. Amenability follows when every uniformly bounded representation admits a unitarizer with a suitable polynomial condition-number bound of exponent below three. Ordinary unitarizability yields a polynomial bound with some finite exponent, but not automatically below that threshold. The missing quantitative improvement was therefore substantive.

Epstein and Monod used random forests to obtain nonunitarizability from numerical invariants such as first $L^2$-Betti number and cost under additional hypotheses. Osin constructed nonunitarizable groups without nonabelian free subgroups. Monod and Ozawa, using measurable-action ideas of Gaboriau and Lyons, established results for wreath-product extensions. Those extensions are larger groups; an obstruction there does not by itself give one for the original group.

Vergara obtained another route under approximation-property assumptions, including classes related to actions on finite-dimensional cube complexes. M1 claims to remove the remaining restrictions for all discrete groups.

The approximate-representation problem has a separate history. Kazhdan’s 1982 theorem proves strong Ulam stability for amenable groups. Burger, Ozawa and Thom showed failure for groups containing nonabelian free subgroups and asked whether amenability is exactly the dividing line. Alpeev established instability for further wreath products using measured equivalence relations.

Meanwhile, Glebsky, Lubotzky, Monod and Rangarajan proved finite-dimensional stability for broad classes of nonamenable higher-rank lattices. Fournier-Facio and Rangarajan developed further finite-dimensional results, including examples related to Thompson groups. These positive theorems are compatible with M2 because strong stability also quantifies over infinite-dimensional Hilbert spaces.

4. What the new manuscripts claim

M1: the discrete Dixmier problem

M1, dated 23 September 2026, asserts that a discrete group is amenable if and only if it is unitarizable.

Its nonamenable direction is stronger than merely finding one bounded counterexample. For every nonamenable discrete $G$ and every $\varepsilon>0$, it supplies a representation $\pi$ with

$$|\pi|\leq1+\varepsilon$$

that admits no bounded similarity making it unitary. Thus the operators can have uniform norm bound arbitrarily close to one while still resisting every change of equivalent Hilbert norm.

If $G$ is countable, the witness Hilbert space can be separable, meaning it has a countable dense subset. For an uncountable group, the construction may use a nonseparable space. The definition in the theorem quantifies over all complex Hilbert spaces.

M2: strong Ulam stability

M2, dated 5 October 2026, asserts that a countable discrete group is strongly Ulam stable if and only if it is amenable.

For every nonamenable countable $G$ and every $\delta>0$, it constructs a separable Hilbert space and a normalized unitary-valued map $\mu$ with defect less than $\delta$, such that every genuine unitary representation $\rho$ on that same space satisfies

$$\sup_{g\in G}\|\mu(g)-\rho(g)\|>\frac1{10}.$$

The multiplication error can therefore tend to zero while the distance from every genuine representation stays bounded below by one fixed positive number.

The two papers use different constructions. M2 does not present its instability theorem as an immediate conversion of M1’s nonunitarizable representation. The underlying repair questions are related, but exact multiplication with nonunitary operators and approximate multiplication with unitary operators are distinct requirements.

5. Why the advance matters

Importance: 5/5, provisional. The family claims two general characterizations of amenability by representation-theoretic repair properties. M1 addresses Dixmier’s longstanding problem, and M2 supplies the converse to Kazhdan’s strong stability theorem.

The significance is the elimination of extra assumptions such as free subgroups, special approximation properties or passage to a larger wreath-product group. Each nonamenable group in the stated range would carry its own obstruction.

The quantitative forms sharpen the qualitative contrast. Arbitrarily small norm excess in M1 does not guarantee an invariant equivalent inner product. Arbitrarily small multiplication defect in M2 does not guarantee even moderately close correction on the same space.

These statements identify amenability as an exact structural dividing line for both problems, while preserving the essential difference between finite- and infinite-dimensional stability.

6. The main idea

M1: bounded changes that have no global implementer

Start with a unitary representation $U$. An operator cocycle is a family of bounded operators $D(g)$ satisfying

$$D(gh)=D(g)+U(g)D(h)U(g)^{-1}.$$

It is inner if a single bounded operator $B$ gives

$$D(g)=B-U(g)BU(g)^{-1}$$

for every $g$. In that case, the family of changes comes from one global choice of coordinates.

A bounded cocycle produces a triangular representation on $H\oplus H$:

$$\pi(g)=\begin{pmatrix} U(g)&D(g)U(g)\\ 0&U(g) \end{pmatrix}.$$

A similarity making this representation unitary would force the cocycle to be inner. The source therefore seeks a bounded cocycle with no bounded implementer.

It constructs operators whose changes under every group translation remain uniformly bounded, while their distance from all translation-commuting operators grows without bound. A direct sum packages these increasingly difficult finite stages into one obstruction. Scaling the cocycle gives the arbitrarily small norm excess in the theorem.

Nonamenability supplies a finite collection of group steps with a sparse assignment of their edges to endpoints. The paper weights those edges by positive rank-one matrix projections. Random-matrix estimates keep the relevant row and column norms under control even when many edges are present.

At the same time, the total squared matrix energy grows relative to the size of the matrix fibers. This forces the operator away from every translation-commuting operator. Repeated group products can coincide, so the construction retains edge labels as separate coordinates; positivity prevents those coincidences from canceling the energy needed for the obstruction.

For arbitrary discrete groups, a finitely generated nonamenable subgroup supplies the witness. Induction extends it to the full group without increasing the uniform bound. The established Day–Dixmier averaging argument supplies the amenable direction.

M2: almost compatible circuits that cannot come from an action

The second construction begins with a nontrivial boundary action of a nonamenable group on a compact space. Here the relevant boundary property means that group translates can concentrate probability measures toward point masses, and every orbit is dense.

From that action, the source constructs finite lists of group steps marked in two ways. Most steps leaving a fixed point have one mark, while most steps arriving at a fixed point have the other. A compatibility rule prevents a step of the first kind from being followed by one of the second kind.

These marked paths define unitary circuits on a Hilbert space with several levels. Comparisons made using one fixed collection of markings compose exactly. But the approximate representation uses comparisons from different pairs of markings, so the main estimate must control the change caused by adding a third marking.

Simply summing the errors over all levels would be too costly. The source chooses the isometries between slightly different coordinate metrics so that principal errors can be moved onto mutually orthogonal levels. Their norms then combine by a maximum rather than by a sum. This produces a multiplication defect as small as desired.

A separate quadratic-form test prevents correction to a true representation. Signed sums from two orthonormal bases make the test vanish on vectors coming from any genuine action. In the constructed approximate action, the initial inner product gives the same test the value $1/2$.

Most marked paths transport the relevant final vectors back toward these initial ones. If a genuine representation were uniformly within $1/10$, the two values of the quadratic form would have to be too close. The source chooses the parameters so that this contradicts the fixed discrepancy while the multiplication defect still tends to zero.

7. Scope, limitations, and open questions

M1 treats all discrete groups. M2’s stated characterization is for countable discrete groups. Neither statement here automatically extends to nondiscrete locally compact groups, where continuity requirements for representations enter.

Infinite-dimensional spaces are essential to the strong stability formulation. The theorem does not assert that finite-dimensional Ulam stability characterizes amenability; the historical positive results for some nonamenable groups show why that substitution would be wrong.

All approximation estimates use the operator norm and are uniform over the entire group, or over all pairs of group elements for the defect. Other norms, estimates only on generators, and correction after changing the Hilbert space define different stability questions.

M1 does not merely rule out well-conditioned similarities. It asserts that no bounded invertible similarity works at all for its witness. Conversely, its small bound $1+\varepsilon$ does not say that the representation is uniformly close to one fixed unitary representation.

M1 additionally discusses extensions to group operator algebras using a separate companion similarity theorem. That application has an extra input; it is not needed for the amenability characterization explained here.

The manuscripts leave room for further quantitative questions about optimal correction estimates in amenable cases and about restricted classes of representations. Their principal claims concern the existence or failure of uniform repair under the precise hypotheses above.

8. Sources and further reading

Day’s invariant-mean work and Dixmier’s Les moyennes invariantes dans les semi-groupes et leurs applications (1950) give the classical amenable direction and the original question. Pisier’s Are unitarizable groups amenable? explains the quantitative framework.

Epstein and Monod, Nonunitarizable representations and random forests, and Monod–Ozawa, The Dixmier problem, lamplighters and Burnside groups, give important preceding approaches.

Kazhdan, On ε-representations, proves amenable strong stability. Burger, Ozawa and Thom, On Ulam stability, formulates the converse question. Fournier-Facio and Rangarajan’s finite-dimensional results clarify why the dimensional convention matters.

Source manuscripts

• M1. OpenAI, Unitarizability implies amenability for discrete groups (September 23, 2026).
• M2. OpenAI, Strong Ulam Stability Characterizes Amenability (October 5, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.