arXiv:2608.11662v1

Whirliness beyond amenability

Yannik Höll, Friedrich Martin Schneider

math.FAmath.DSmath.GR37A1522A0528D1543A07

Paper metadata

Original submission: August 12, 2026
Claimed MSC: 37A15, 22A05, 28D15, 43A07
arXiv categories: math.FA, math.DS, math.GR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 25 pages, no figures

Abstract

Given a non-atomic standard probability space $(Ω,μ)$ , we exhibit examples of non-amenable closed topological subgroups of $\mathrm{Aut}(Ω,μ)$ whose natural near-action on $(Ω,μ)$ is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The faithful ergodic Gaussian realization theorem, the whirliness and inertness results for the three stated families, and the resulting negative answer to Pestov's question are correct. The external structural inputs are used within their hypotheses, and the few notation slips in auxiliary proofs have unique harmless corrections.

Theorem 1.3Correct

Every second-countable unitarily representable group has a faithful ergodic near-action

Pages 2 and 5–7 · Theorem 1.3 and its proof · arXiv:2608.11662v1

A countable family of strong-operator seminorms reduces the faithful representation to a separable invariant Hilbert space. Its finite-dimensional part has compact closure and therefore an ergodic translation action, while the orthogonal complement contains no nonzero finite-dimensional invariant subspace and hence gives a weakly mixing Gaussian near-action. The product representation is still a topological embedding, and the product of an ergodic action with a weakly mixing action is ergodic. This proves all parts of the stated conclusion, including topological faithfulness.

Theorem 1.4Correct

The three families are whirly and inert as claimed

Pages 3 and 20–22 · Theorem 1.4 and its proof · arXiv:2608.11662v1

For $L^0(\mu,G)$ , each simple function lies in a continuous image of $L^0(\mu,\mathbb Z)$ , so the cited whirliness result for non-elliptic diffuse submeasures and extreme amenability for diffuse submeasures propagate by density and quotients. Every unitary in a $\mathrm{II}_1$ factor lies in a diffuse maximal abelian von Neumann subalgebra, whose unitary group is an $L^0$ -group. For a full group $[E]$ , every torsion element lies in a continuous image of $L^0([0,1],\lambda;\mathbb Z)$ and torsion is dense; the alternative hyperfinite-subrelation argument independently reaches the same conclusion. These deductions match the persistence lemmas proved in Sections 4–5.

Schneider–Solecki, groups without unitary representations ↗

Corollary 1.5Correct

The non-amenable closed whirly subgroup exists

Pages 3 and 22 · Corollary 1.5 and its proof · arXiv:2608.11662v1

Each displayed choice of $G$ is Polish, unitarily representable, whirly, and non-amenable by the cited $L^0$ , von Neumann algebra, or full-group characterization. Theorem 1.3 embeds $G$ into $\operatorname{Aut}(\Omega',\mu')$ with ergodic image. Ergodicity and infinitude rule out a finite atomic probability space, a standard-space isomorphism transports the action to the prescribed non-atomic space, and Raikov completeness makes the embedded image closed. Proposition 4.3 then converts group whirliness into whirliness of the induced ergodic near-action.

02Proofs5 reported findingsCorrect

The central proofs are correct and complete. The Gaussian decomposition, permanence properties, $L^0$ factorization, maximal-abelian-subalgebra reduction, and full-group density argument close all substantive obligations. Three local notation slips have unique corrections and do not change any argument or conclusion.

Proof of Theorem 1.3Correct and complete

The compact and weakly mixing components are handled completely

Pages 5–7 · proof of Theorem 1.3 · arXiv:2608.11662v1

The closure of the finite-dimensional component is compact because its restriction map embeds it into a product of compact orthogonal groups and the closure is Raikov complete. Haar translation supplies the compact ergodic factor. Any finite-dimensional invariant subspace of the complementary representation would belong simultaneously to the finite-dimensional span and its orthogonal complement, so that representation is weakly mixing. The Gaussian Koopman construction is faithful and continuous, and the two factors jointly retain the original representation's topology.

Sections 4–7Correct and complete

The persistence and example arguments are complete

Pages 8–22 · Lemmas 4.2–6.12 and proof of Theorem 1.4 · arXiv:2608.11662v1

Whirliness and inertness are correctly shown to pass through dense subgroups, continuous quotients, and the generated dense union of suitable subgroups. Lemma 6.12 constructs a $1$ -Lipschitz homomorphism from step functions, then extends it to the Raikov completion; this places every torsion element of $[E]$ in a whirly inert subgroup. The cited equivalences for amenability of $L^0$ -groups and hyperfinite full groups, together with the standard amenability characterization of unitary groups of von Neumann algebras, are invoked with the required diffuseness, non-atomicity, separability, or factor hypotheses.

Le Maître, Theorem 8.4 ↗

Lemma 4.4(b)Typo

The invariant measure is assigned to the wrong group before extension

Page 9 · proof of Lemma 4.4(b), reverse implication · arXiv:2608.11662v1

The proof begins with an action of the dense subgroup $H$ and prints that $\mu$ is ' $G$ -invariant' before the action has been extended to $G$ . It must say ' $H$ -invariant.' The following paragraph then proves exactly that this $H$ -invariant measure is $G$ -invariant after extension, so the intended correction is unique and the argument is unchanged.

Lemma 6.7Typo

Dominated convergence uses the original measure

Page 16 · proof of Lemma 6.7 · arXiv:2608.11662v1

In the displayed pairing $\langle\chi_{B_n},f\rangle$ , both integrals are printed with $d\mu$ . They must use $d\nu$ , because the pairing is between $L^\infty(\Omega,\mathcal B,\nu)$ and its predual $L^1(\Omega,\mathcal B,\nu)$ , while countable additivity of the newly defined $\mu(B)=\tau_M(\iota(\chi_B))$ is precisely what this convergence proves. Replacing $d\mu$ by $d\nu$ twice makes the dominated-convergence step valid and leaves the remainder unchanged.

Remark 6.8Typo

The domain symbol changes from $X$ to $\Omega$

Page 16 · Remark 6.8 · arXiv:2608.11662v1

The remark starts with a finite measure space $(X,\mathcal B,\mu)$ but writes $L^0(\Omega,\mathcal B,\mu;\mathbb T)$ in the displayed equality. The unique consistent correction is $L^0(X,\mathcal B,\mu;\mathbb T)$ . This notation mismatch has no downstream effect.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.11662v1
Authors listed: Yannik Höll, Friedrich Martin Schneider
Audit date: August 18, 2026

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Whirliness beyond amenability

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Every second-countable unitarily representable group has a faithful ergodic near-action

The three families are whirly and inert as claimed

The non-amenable closed whirly subgroup exists

The compact and weakly mixing components are handled completely

The persistence and example arguments are complete

The invariant measure is assigned to the wrong group before extension

Dominated convergence uses the original measure

The domain symbol changes from XXX to Ω\OmegaΩ

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The domain symbol changes from $X$ to $\Omega$