1 · The essentials
These manuscripts connect three ways of understanding how a group can act on a space. The first is a property called amenability. The main result says that, for a discrete group, this property holds exactly when every action by uniformly bounded linear transformations can be changed into one that preserves lengths and angles. This result has a Lean formalization and resolves the discrete-group version of Dixmier’s unitarizability problem. The companion manuscript shows another characterization for countable discrete groups: sufficiently accurate actions that almost obey the group’s rules can always be uniformly corrected to exact actions. The correction must stay on the same space, and allowing spaces with infinitely many dimensions is essential.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a set of moves whose combinations are supposed to follow a fixed rulebook. The stability result resembles saying that, if every combination follows the rules closely enough, the moves can all be adjusted slightly so that they follow the rules exactly, without changing the setting in which they operate. This image does not capture the mathematical requirements on the space or on how errors are measured.
Why it matters
The main result identifies amenability with the ability to turn every uniformly bounded Hilbert-space representation into a unitary one by a similarity. The companion manuscript connects amenability to a different question: whether approximate representations can be corrected uniformly. Together, they give two characterizations of amenability, with the stability characterization restricted to countable discrete groups.
2 · Going further
The main problem concerns representations of a discrete group on a complex Hilbert space. Uniform boundedness means . A representation is similar to a unitary representation if an invertible bounded change of coordinates turns all its operators into unitary operators. The Lean-formalized main result states that is amenable if and only if every uniformly bounded representation of has this property. The manuscript describes this as an affirmative resolution of Dixmier’s unitarizability problem for discrete groups.
The companion manuscript concerns unitary almost representations: maps into unitary operators that satisfy the group multiplication rule only approximately. Strong Ulam stability requires every sufficiently accurate such map to be uniformly close, in operator norm, to a genuine representation on the same complex Hilbert space. The manuscript shows that, for countable discrete groups, this condition is equivalent to amenability. Its abstract identifies the new direction as the converse to Kazhdan’s amenable stability theorem and says that it answers the question of Burger, Ozawa, and Thom.
The two manuscripts thus relate amenability to two different requirements: changing coordinates in an exact, uniformly bounded representation, and correcting an approximate unitary representation. The supplied abstracts state the results but do not describe the arguments, so no proof mechanism can be extracted from them. Only the main manuscript is identified here as Lean-formalized; the companion explicitly emphasizes that infinite-dimensional Hilbert spaces are essential to its characterization.
Useful notions
- Discrete groups and their multiplication
- Complex Hilbert spaces
- Linear operators and operator norm
- Group representations
- Unitary operators and similarity
What was known before / what is new
Before
The source identifies Dixmier’s unitarizability problem as the problem resolved by the main manuscript for discrete groups. It also names Kazhdan’s amenable stability theorem as the existing direction to which the companion supplies a converse. The companion describes its result as answering a question of Burger, Ozawa, and Thom.
New
The Lean-formalized main result establishes that unitarizability implies amenability for discrete groups, completing the stated equivalence. The companion manuscript shows that strong Ulam stability characterizes amenability for countable discrete groups. Its formulation allows any complex Hilbert space, including infinite-dimensional ones, and requires correction on that same space.
Limits and assumptions
The main equivalence concerns discrete groups and uniformly bounded representations on complex Hilbert spaces. The companion equivalence is stated only for countable discrete groups, uses uniform operator-norm closeness, and requires the exact representation to act on the same space as the approximate one. Infinite-dimensional spaces are explicitly essential; the source does not give a finite-dimensional substitute or a stability characterization for uncountable groups. The main result has a Lean formalization, but no formalization is specified for the companion. The supplied abstracts give neither quantitative error bounds nor details of the arguments, and they identify no further open problems.
Mini glossary
- Amenability —
- The group property characterized here by unitarizability for discrete groups and by strong Ulam stability for countable discrete groups.
- Uniformly bounded representation —
- An action of a group by linear operators whose operator norms share a finite bound over all group elements.
- Unitary representation —
- A representation whose operators preserve the Hilbert space’s inner product, and hence lengths and angles.
- Unitarizability —
- The property that every uniformly bounded Hilbert-space representation can be made unitary by an invertible bounded change of coordinates.
- Strong Ulam stability —
- The requirement that every sufficiently accurate unitary almost representation be uniformly close in operator norm to a genuine representation on the same Hilbert space.
Keywords : Amenability · Unitarizability · Strong Ulam stability
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Unitarizability implies amenability for discrete groups
23/09/2026We 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
05/10/2026A 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.