arXiv:2608.07421v1

The noncommutative topological factor theorem for rank-one product lattices

Cyril Houdayer, Corentin Le Bars

math.OAmath.DSmath.GR46L5522D2522E4037A55

Paper metadata

Original submission: August 7, 2026
Claimed MSC: 46L55, 22D25, 22E40, 37A55
arXiv categories: math.OA, math.DS, math.GR
Paper license: CC BY 4.0 ↗
Comments: 22 pages

Abstract

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that the corresponding classification for the full flag action of $\operatorname{SL}_3(\mathbb Z)$ would imply ordinary ITAP.

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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

01Statements4 reported findingsCorrect

The coordinate classification of intermediate reduced crossed products for abstract rank-one product boundaries, its real Lie, tree, and local-field specializations, and the equivalence between the scalar-expectation classification for the full flag action of $\operatorname{SL}_3(\mathbb Z)$ and ordinary ITAP are correct.

Theorem ACorrect

Intermediate crossed products are exactly the coordinate crossed products

Pages 2 and 11–12 · Theorem A and its proof · arXiv:2608.07421v1

The abstract factor theorem first forces $C^*(E(D))$ to equal one coordinate algebra $C(X_S)$ . Suzuki's Fourier reconstruction gives the upper bound $D\subset C(X_S)\rtimes_r\Gamma$ . Independently, $D\cap C(X_N)=C(X_T)$ for a coordinate set $T$ , and the upper bound gives $T\subset S$ . For every $j\in S$ , a coefficient depending on $j$ is sliced at a free point of a different coordinate; the Powers-averaging lemma places that slice in $D\cap C(X_N)$ while it still separates two points differing only at $j$ . Hence $j\in T$ , so $S=T$ . The diagonal $C(X_S)$ and the canonical group unitaries then generate the reverse inclusion, proving $D=C(X_S)\rtimes_r\Gamma,\qquad C^*(E(D))=D\cap C(X_N)=C(X_S).$

Full paper, version 1 ↗

Corollary BCorrect

Real rank-one product lattices satisfy the abstract hypotheses

Pages 3 and 12–13 · Corollary B and its proof · arXiv:2608.07421v1

On each real rank-one flag manifold, the Bruhat decomposition gives transitivity on ordered distinct pairs and a regular split-torus element has north–south dynamics. Irreducibility gives dense coordinate projections. Connectedness and center-freeness make every coordinate projection of the lattice injective; real analyticity and faithfulness then make the coordinate actions topologically free. Cowling–Haagerup weak amenability implies (AP), which is preserved by the finite product and inherited by its lattice. Theorem A therefore applies and indexes the whole interval by $2^N$ .

Cowling–Haagerup weak amenability theorem ↗

Theorems 5.1 and 5.4Correct

Tree and mixed local-field specializations

Pages 13–16 · Theorems 5.1 and 5.4 · arXiv:2608.07421v1

For tree boundaries, a hyperbolic automorphism supplies north–south dynamics and the assumed two-transitivity supplies the remaining rank-one boundary condition; compact vertex stabilizers give weak amenability and hence (AP). For adjoint rank-one algebraic groups over local fields, the rank-one Bruhat cell and a split-torus element give the same boundary dynamics. Irreducibility supplies density, Tits simplicity forces injective coordinate projections, the cited fixed-point theorem gives topological freeness, and the archimedean or tree arguments give (AP). Thus both families satisfy every hypothesis of Theorem A.

Theorem CCorrect

The scalar-expectation flag classification is equivalent to ordinary ITAP

Pages 4 and 17–21 · Theorem C and Propositions 6.2–6.3 · arXiv:2608.07421v1

For a topologically amenable compact action with a dense orbit, the orbit representation embeds $C(X)\rtimes_r\Lambda$ into the uniform Roe algebra and identifies elements with scalar Fourier coefficients with its intersection with $L(\Lambda)$ . The amenability kernels constructed in Proposition 6.2 show that this intersection is all of $C_u^*(\Lambda)\cap L(\Lambda)$ . Consequently, the assertion that every scalar-expectation intermediate algebra is $C_\lambda^*(\Lambda)$ is equivalent to $C_u^*(\Lambda)\cap L(\Lambda)=C_\lambda^*(\Lambda),$ namely ITAP. The full flag action of $\operatorname{SL}_3(\mathbb Z)$ is minimal and topologically amenable, so this applies. A full parabolic classification would force the scalar case to be the point quotient and hence would imply ITAP.

Full paper, version 1 ↗

02Proofs5 reported findingsCorrect

The central proofs are correct and complete. Recurrence supplies simultaneous contraction, the resulting closed-equivalence-relation theorem gives the coordinate commutative core, Fourier reconstruction gives the crossed-product upper bound, Powers averaging recovers coordinate slices, and the positive-definite Schur kernels establish the ITAP equivalence.

Proposition 2.1 and Theorems 2.3–2.5Correct and complete

Recurrence yields the coordinate topological factor theorem

Pages 5–8 · Proposition 2.1 and Theorems 2.3–2.5 · arXiv:2608.07421v1

Poincaré recurrence on $G/\Gamma$ supplies lattice elements arbitrarily close to the identity on a chosen subproduct and simultaneously contracting prescribed compact sets in all complementary boundaries. Applied twice, this lets any one coordinate of a related pair be varied arbitrarily while all other coordinates coalesce. Therefore every closed invariant equivalence relation forgets exactly a subset of coordinates. Passing to fibers of an equivariant quotient gives the coordinate factor theorem, and Gelfand duality gives the corresponding classification of unital invariant subalgebras of $C(X_N)$ .

Propositions 3.2–3.3Correct and complete

Fourier reconstruction supplies the upper bound

Pages 9–10 · Propositions 3.2–3.3 · arXiv:2608.07421v1

Suzuki's theorem states that, for a group with (AP), an element of a reduced crossed product whose Fourier coefficients all lie in a fixed closed subspace belongs to the closed span of that subspace's Fourier monomials. For an intermediate algebra $D$ , $C^*(E(D))$ is invariant because $D$ contains the implementing unitaries, so the commutative factor theorem makes it $C(X_S)$ . Every coefficient $E(du_\gamma^*)$ lies there, and Suzuki's result therefore gives $D\subset C(X_S)\rtimes_r\Gamma$ without a freeness assumption.

Suzuki, Proposition 3.4 ↗

Lemmas 4.1–4.2Correct and complete

Powers averaging recovers coordinate slices inside the intermediate algebra

Pages 10–11 · Lemmas 4.1–4.2 · arXiv:2608.07421v1

For each nontrivial Fourier mode $t$ , triviality of the stabilizer at the slicing point gives an open set $O$ with $tO\cap O=\varnothing$ . Recurrence produces conjugators $s_i$ that are close to the identity in the other coordinates and send $X_k\setminus O$ into pairwise disjoint open sets. The associated group partition satisfies the hypotheses of the support estimate, so the average of every nonzero Fourier term has norm at most $2\|f_t\|/\sqrt m$ . All but at most one diagonal translate are uniformly close to the desired slice. Letting the polynomial approximation error tend to zero places the exact slice in $D\cap C(X_{N\setminus\{k\}})$ .

Lemma 6.1 and Proposition 6.2Correct and complete

The dense-orbit representation saturates the invariant uniform Roe intersection

Pages 17–19 · Lemma 6.1 and Proposition 6.2 · arXiv:2608.07421v1

The orbit representation is faithful because its diagonal matrix coefficients evaluate the faithful conditional expectation on the dense orbit. Scalar Fourier coefficients are exactly the condition that its matrix entries are constant on right-invariant diagonals, hence that the image lies in $L(\Lambda)$ . Topological amenability provides continuous finitely supported probability fields. Their square roots define positive-definite kernels $k_i(r,t)$ of finite diagonal support; the associated unital completely positive Schur multipliers converge in norm on $C_u^*(\Lambda)$ . When $T\in L(\Lambda)$ , each Schur approximation has coefficients $c_s h_i(s,\cdot)$ and is the image of a finite Fourier polynomial in the crossed product. Closedness then proves $\pi_{x_0}(C(X)\rtimes_r\Lambda)\cap L(\Lambda)=C_u^*(\Lambda)\cap L(\Lambda).$

Proposition 6.3 and proof of Theorem CCorrect and complete

Scalar expectation reduces exactly to ITAP

Pages 19–20 · Proposition 6.3 and proof of Theorem C · arXiv:2608.07421v1

An intermediate algebra has $C^*(E(D))=\mathbb C1$ exactly when all its Fourier coefficients are scalar, equivalently when it is contained in $D_0=A\cap L(\Lambda)$ . Since $D_0$ itself is intermediate and has scalar expectation, uniqueness of the scalar intermediate algebra is equivalent to $D_0=C_\lambda^*(\Lambda)$ . Proposition 6.2 identifies $D_0$ with $C_u^*(\Lambda)\cap L(\Lambda)$ , so the resulting equality is precisely ordinary ITAP. The flag-action hypotheses cited immediately before Theorem C verify topological amenability and a dense orbit.

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Paper: arXiv:2608.07421v1
Authors listed: Cyril Houdayer, Corentin Le Bars
Audit date: August 18, 2026

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The noncommutative topological factor theorem for rank-one product lattices

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Intermediate crossed products are exactly the coordinate crossed products

Real rank-one product lattices satisfy the abstract hypotheses

Tree and mixed local-field specializations

The scalar-expectation flag classification is equivalent to ordinary ITAP

Recurrence yields the coordinate topological factor theorem

Fourier reconstruction supplies the upper bound

Powers averaging recovers coordinate slices inside the intermediate algebra

The dense-orbit representation saturates the invariant uniform Roe intersection

Scalar expectation reduces exactly to ITAP

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