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Math-2719 · mathematics · Group theory

Strong Ulam stability characterizes amenability

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a countable discrete group G, does amenability follow from this property: for every $\varepsilon >0$ there exists $\delta >0$ such that, on every complex Hilbert space H, each normalized map $\mu :G\to U(H)$ with $sup_{g,h}\Vert \mu (gh)-\mu (g)\mu (h)\Vert \leq \delta $ is within $\varepsilon $ uniformly on G of a genuine unitary representation on that same H?

Short English statement

For a countable discrete group G, does amenability follow from this property: for every $\varepsilon >0$ there exists $\delta >0$ such that, on every complex Hilbert space H, each normalized map $\mu :G\to U(H)$ with $sup_{g,h}\Vert \mu (gh)-\mu (g)\mu (h)\Vert \leq \delta $ is within $\varepsilon $ uniformly on G of a genuine unitary representation on that same H?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2719-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2719

The paper defines strong Ulam stability with normalized unitary maps, the uniform operator-norm defect and all complex Hilbert spaces, and proves amenability iff this property for every countable group. lean/docs/251.md documents the different unitarizability companion and does not cover this new approximate-representation theorem. Within these assumptions, this is a claimed affirmative answer to the stated question.

Strong Ulam stability characterizes amenability

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For countable discrete groups, uniform stability of all unitary almost-representations is claimed equivalent to amenability, allowing infinite-dimensional spaces.

Mathematical statement of the claimed result

Result 1.
For countable discrete $G$, define the defect of $\mu:G\to\mathcal U(\mathcal H)$, $\mu(1)=I$, by \[ \operatorname{def}(\mu)=\sup_{g,h\in G}\|\mu(gh)-\mu(g)\mu(h)\|. \] Strong Ulam stability means: for every $\varepsilon>0$, some $\delta>0$ works for all complex Hilbert spaces and all $\mu$ of defect at most $\delta$, supplying a genuine unitary representation $\pi$ on the same space with $\sup_g\|\mu(g)-\pi(g)\|\le\varepsilon$. The claimed characterization is \[ G\text{ strongly Ulam stable}\quad\Longleftrightarrow\quad G\text{ amenable}. \] The quantifier includes infinite-dimensional Hilbert spaces.

Outline of the argument

Boundary actions supply finite flags and frame configurations. Metric-transfer and block-operator comparisons construct arbitrarily accurate unitary defects that remain a definite distance from exact representations when amenability fails; the amenable direction uses the established stability theorem.

Formal verification

For family 251, the documented theorem gives amenability--unitarizability equivalence for arbitrary discrete groups and almost-unitary nonunitarizable witnesses; countable groups have separable witnesses. The later strong-Ulam theorem is not selected. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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