On the embedding rigidity problem for uniformly locally finite coarse spaces

Teng Zhang

On the embedding rigidity problem for uniformly locally finite coarse spaces

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Two countable metric spaces exist whose uniform Roe algebras are isomorphic to a hereditary corner, yet no coarse embedding exists between the spaces.

desk verdict This paper resolves the embedding rigidity problem in the negative with a well-built counterexample and upgrades weak to strong rigidity under the sparse compact-ghost hypothesis — solid work that deserves a serious referee. read the letter →

arxiv 2607.16949 v1 pith:4TH3HAIP submitted 2026-07-18 math.OA

Teng Zhang This is my paper · ORCID

classification math.OA MSC 46L0551F3005C8046L85

keywords uniformRoealgebracoarseembeddinghereditaryC*-subalgebraexpandergraphhigh-girthbundleghostprojectionrigidityHall'smarriagetheorem

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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper addresses the embedding rigidity problem: if the uniform Roe algebra of one countable metric space is isomorphic to a hereditary C*-subalgebra of another's, must the first coarsely embed into the second? It answers no. The author constructs countable uniformly locally finite metric spaces X and Y with a *-isomorphism from C_u^*(X) onto a hereditary corner of C_u^*(Y), while X does not coarsely embed into Y. The construction uses graph bundles with expanding, high-girth fibers; the algebra isomorphism comes from fiberwise averaging. The paper also proves a positive result: if every sparse subspace of Y yields only compact ghost projections, then a hereditary isomorphism forces an injective coarse embedding X → Y.

What carries the argument

The load-bearing object is the fiberwise-averaging isometry U: ℓ²(X) → ℓ²(Y), spreading each basis vector of a base cycle evenly over a fiber, and the projection q = UU^*. The exact corner identity qC_u^*(Y)q = UC_u^*(X)U^* makes the hereditary range possible. High-girth expanding fibers, obtained by combining probabilistic results on random regular graphs—optimal control of the second adjacency eigenvalue and positive limiting probability for high girth—together with the Lovász local lemma for random injections, ensure both q ∈ C_u^*(Y) (via the uniform spectral gap and functional calculus) and the geometric nonembedding (via the tree-median obstruction).

What would settle it

One could attempt to replace the random regular graph input with an explicit construction of arbitrarily large d-regular graphs with girth > g and Laplacian spectrum in {0} ∪ [γ,2]. If no such family exists for every g and N, the counterexample collapses. More directly, a reader could test the tree-median obstruction on a small case: if a coarse embedding of a cycle C_L into the bundle existed, the lemma forces two base vertices at distance at least ⌊L/3⌋/2 with images within 2R, contradicting effective properness; a numerical check for a chosen L would confirm the geometric no-go.

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Extended reading notes

Core claim

The central discovery is a counterexample to embedding rigidity. X is a coarse disjoint union of cycles, Y a coarse disjoint union of graph bundles over those cycles, each bundle having fibers that are fixed-degree expanders of arbitrarily high girth and horizontal edges that are perfect matchings along each base edge. The fiberwise average map U sends ℓ²(X) into ℓ²(Y); the uniform spectral gap makes q = UU^* a projection in C_u^*(Y), and the matching structure yields the exact corner identity qC_u^*(Y)q = UC_u^*(X)U^*. Thus C_u^*(X) is isomorphic to a hereditary subalgebra of C_u^*(Y). On the geometric side, the recursive growth of base cycles together with high girth rules out coarse embed

Load-bearing premise

The counterexample depends on the existence, for every girth and size demand, of regular graphs that are simultaneously expanders (normalized Laplacian spectrum in {0} ∪ [γ,2]) and have girth larger than g; this probabilistic existence is the load-bearing premise—if it failed, q would not lie in C_u^*(Y) and the corner identity would break.

Editorial extensions

If this is right

The embedding rigidity problem is answered negatively: hereditary C*-isomorphism of uniform Roe algebras does not imply coarse embeddability.
The sparse compact-ghost hypothesis is sufficient for strong rigidity: it upgrades known weak embedding to an injective coarse embedding.
The constructed projection q is a noncompact ghost projection, so the counterexample violates the hypothesis under which rigidity holds.
By known results linking ghost projections to property A, Y lacks property A, placing the example outside positive rigidity theorems.
The exact corner identity gives a template for building further hereditary isomorphisms from graph-bundle constructions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next question is whether the counterexample can be realized with both spaces having property A, which would push the boundary of the positive theorem further.
The graph-bundle construction suggests that rigidity depends not only on base geometry but on the algebra's ability to detect fiberwise expansion; this may transfer to other C*-algebras built from coarse spaces.
One could test the sharpness of the sparse compact-ghost hypothesis by varying fiber growth: faster growth may break the corner identity, slower growth may allow a coarse embedding.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: the negative answer to Problem 1.1 is real. The construction of X and Y with C_u^*(X) a hereditary corner of C_u^*(Y) but no coarse embedding X→Y holds together. The exact corner identity qC_u^*(Y)q = UC_u^*(X)U* is the genuinely new operator-algebraic point; the high-girth bundle construction and the tree-median obstruction are clean. I checked the dependency-heavy parts — Friedman plus McKay–Wormald–Wysocka for the fiber graphs, the Lu–Székely LLL estimate in Lemma 3.5, and the lifting argument in Lemma 5.3 — and nothing collapses. The counting in Lemma 3.5 is coarse but legitimate, and the negative dependency graph claim is fine because a supergraph of one is still a negative dependency graph.

The positive theorem is also a real improvement: getting injective coarse embeddability from the sparse compact-ghost hypothesis, without property A, is not a trivial extension. The Baire-uniformization plus Hall-matching mechanism is the right tool, and the proof of the uniform Hall radius is the part I would ask a referee to read most carefully — it is the most intricate step, but it checks out on my reading.

Soft spots are minor. The paper leans on deep external probabilistic results, so a fully self-contained verification is not quick, but that is not a defect in the mathematics. There are small typos (e.g., the doubled "property. property." in the proof of Proposition 6.5) and the exposition in Section 5 is dense but not unfair. The self-citation to [Zha26] is natural given the companion paper resolving isomorphism rigidity; I do not see it as a problem.

The central argument holds up. This is a paper that answers an open problem in the negative and sharpens a known positive result; it deserves a serious referee. My recommendation: send it out, with attention to Section 6.3 and the LLL estimate. I would take it for peer review.

Referee Report

0 major / 5 minor

Summary. The paper addresses the embedding rigidity problem for uniform Roe algebras of uniformly locally finite coarse spaces. The main negative result (Theorem 1.2) constructs countable bounded-geometry metric spaces X and Y, a hereditary C*-subalgebra B of C_u^*(Y), and a *-isomorphism Φ : C_u^*(X) → B, while proving that X does not coarsely embed into Y. The construction uses high-girth graph bundles with expanding fibers over cycles: fiberwise averaging gives an isometry U, the uniform spectral gap puts q = UU* in C_u^*(Y), and a precise corner identity qC_u^*(Y)q = UC_u^*(X)U* is obtained via horizontal matchings. A tree-median/girth argument rules out coarse embeddings. The positive result (Theorem 1.3) shows that if every sparse subspace of Y yields only compact ghost projections, then any isomorphism from C_u^*(X) onto a hereditary subalgebra of C_u^*(Y) induces an injective coarse embedding X → Y, strengthening a theorem of Braga–Farah–Vignati. The proof combines spatial implementation, a ghost-projection coefficient lower bound, a Baire-category uniformization lemma, and Hall's marriage theorem.

Significance. If correct, these results settle Problem 1.1 in the negative, showing that hereditary C*-algebraic embeddings of uniform Roe algebras do not detect coarse embeddability even among bounded-geometry metric spaces. The positive theorem establishes that the sparse compact-ghost hypothesis, which is optimal in view of the counterexample, upgrades weak rigidity to injective rigidity. The paper is notable for its clean synthesis of probabilistic combinatorics (Friedman's theorem, McKay–Wormald–Wysocka, Lovász local lemma), geometric median arguments, and operator-algebraic techniques. The proofs are detailed and largely self-contained; the counterexample is sharp in that the obstruction is exactly a noncompact ghost projection, and the positive result is a genuine strengthening of [BFV20]. I checked the key derivations — the LLL counting in Lemma 3.5, the tree-median obstruction in Proposition 4.2, the corner identity in Lemmas 5.2–5.3, and the compact-ghost/Hall argument in Propositions 6.2–6.5 — and found no gaps.

minor comments (5)

[Lemma 3.5] The notation 'A_S = {S⊆graph(σ)}' is confusing; it should read 'let A_S be the event that S ⊆ graph(σ)'. Also, in the counting estimate (3.2), the phrase 'Discarding configurations that fail to form a simple cycle or a partial matching only decreases the count' could be clarified: the overcount is by cycles, and the bound applies to the number of distinct events because each event is counted at least once. This is harmless but would improve readability.
[Lemma 3.5] The sentence 'The graph just defined is a supergraph of that conflict graph' is technically true but slightly misleading: two partial matchings conflict precisely when they share an element of A or of B, so the graph defined is actually equal to the Lu–Székely conflict graph, not a proper supergraph. The subsequent conclusion (a supergraph of a negative dependency graph is again a negative dependency graph) is correct, so no mathematical issue arises.
[Proposition 6.5] There is a typo in the proof: 'the finite intersection property. property.' should be 'the finite intersection property.'
[Lemma 5.3] The phrase 'resolving the antipodal tie deterministically' is a bit vague. A few words specifying the tie-breaking rule (e.g., choose the positive orientation when the two shortest paths have equal length) would make the decomposition into partial translations fully explicit.
[Lemma 6.3] The proof that for every T there exists S ∈ C_R attaining dist(T,C_R) is correct but somewhat nonstandard. Since C_R is weak-operator closed and convex, the standard Hilbert-space nearest-point theorem could be invoked; the current minimizing-net argument is valid but longer. No correction is needed.

Circularity Check

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No significant circularity: the counterexample and positive rigidity theorem are derived from external probabilistic/graph-theoretic theorems plus direct operator constructions, not from their own conclusions.

full rationale

The derivation chain is self-contained against external benchmarks. Theorem 1.2 is built from Lemma 3.3 (random regular graphs: Friedman + McKay–Wormald–Wysocka), Lemma 3.5 (Lu–Székely Lovász local lemma), and the direct corner identity q C_u^*(Y) q = U C_u^*(X) U* established in Lemmas 5.2–5.3 and Proposition 5.4. The nonembedding Proposition 4.2 is a separate geometric argument using girth and a cardinality recursion; it does not presuppose the algebra isomorphism. The positive Theorem 1.3 is proved in full: spatial implementation (Lemma 6.1), ghost coefficient lower bound (Lemma 6.2), Baire uniformization (Lemma 6.3), coarse control (Proposition 6.4), and Hall injectivization (Proposition 6.5). The only self-citation to [Zha26] is for the standard fact that bounded-distance relations decompose into graphs of partial bijections; the text immediately gives the proof via Kőnig line coloring, so the citation is not load-bearing. No fitted parameter is renamed as a prediction, no author-imported uniqueness theorem is invoked to forbid alternatives, and no ansatz is smuggled in through a self-citation. The central claims have independent mathematical content.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

No new physical or postulational entities are introduced. The graph bundles and the projection q are constructed mathematical objects with explicit definitions. Free parameters are limited to fixed construction constants d and γ, chosen once and not fitted to the target result.

free parameters (2)

d = 3 (or any integer ≥ 3)
Base degree of the expanding fiber graphs in the counterexample. Must satisfy 2√(d−1) < d so a uniform spectral gap γ > 0 exists. Fixed a priori; the proof works for any such d, so it is a universal construction constant rather than a parameter fitted to data.
γ = 1 − (2√(d−1)+ε)/d for a small ε > 0
Lower bound on the normalized Laplacian spectrum of the fibers, needed for the functional-calculus argument in Lemma 5.1 to show q ∈ C_u^*(Y). Any value in (0, 1) works; chosen once d is fixed.

assumptions (8)

standard math Friedman's second eigenvalue conjecture for random d-regular graphs (Lemma 3.1).
Used to obtain expanding fibers with spectral gap in Lemma 3.3.
standard math McKay–Wormald–Wysocka: random regular graphs have positive probability of girth > g (Lemma 3.2).
Used to obtain high-girth fibers in Lemma 3.3.
standard math Lu–Székely random-injection Lovász Local Lemma (Lemma 3.4).
Used to add perfect matchings between fibers without creating short cycles (Lemma 3.5 and Proposition 3.6).
standard math Hall's marriage theorem (finite and compactness extension).
Used in Proposition 6.5 to produce an injective coarse map from Hall inequalities.
standard math Baire category theorem for compact metrizable spaces.
Used in Lemma 6.3 to uniformize propagation bounds over subseries.
standard math Kőnig's line-coloring theorem: bounded-degree bipartite graphs decompose into matchings.
Used to decompose bounded-distance relations into graphs of partial bijections in Section 6.3.
domain assumption A nonempty uniformly locally finite metric space is countable (via a base point and finite balls).
Used in Section 6 to enumerate X and construct sparse subspaces Z_n; stated at the start of Section 6.
standard math Standard C*-algebra facts: hereditary subalgebras with a unit are corners pAp; K(qH) is simple; functional calculus on normal operators in a C*-algebra.
Used throughout Sections 5 and 6 for the corner identity and spatial implementation.

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Cite this review

Pith. "Pith review of On the embedding rigidity problem for uniformly locally finite coarse spaces." pith.science (2026). https://pith.science/paper/4TH3HAIP

@misc{pith2026260716949,
  author       = {Pith},
  title        = {Pith review of: On the embedding rigidity problem for uniformly locally finite coarse spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TH3HAIP}},
  note         = {Machine review of arXiv:2607.16949}
}

abstract

In this paper, we construct countable uniformly locally finite metric spaces $X$ and $Y$ such that $C_u^*(X)$ is isomorphic to a hereditary $C^*$-subalgebra of $C_u^*(Y)$, while $X$ does not coarsely embed into$Y$. This gives a negative answer to the embedding rigidity problem for uniformly locally finite coarse spaces. On the positive side, we prove that, if every sparse subspace of $Y$ yields only compact ghost projections, then any isomorphism of $C_u^*(X)$ onto a hereditary $C^*$-subalgebra of $C_u^*(Y)$ induces an injective coarse embedding $X\to Y$. This strengthens a main result in \cite{BFV20} by upgrading coarse embeddability to injective coarse embeddability under the same hypothesis.

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Reference graph

Works this paper leans on

24 extracted references · 18 canonical work pages

[1]

F. P. Baudier, B. M. Braga, I. Farah, A. Khukhro, A. Vignati, and R. Willett, Uniform Roe algebras of uniformly locally finite metric spaces are rigid, Invent. Math. 230 (2022), no. 3, 1071--1100. doi:10.1007/s00222-022-01140-x

work page doi:10.1007/s00222-022-01140-x 2022
[2]

B. M. Braga, Y. C. Chung, and K. Li, Coarse Baum--Connes conjecture and rigidity for Roe algebras, J. Funct. Anal. 279 (2020), no. 9, 108728, 20 pp. doi:10.1016/j.jfa.2020.108728

arXiv 2020
[3]

B. M. Braga and R. Exel, KMS states on uniform Roe algebras, Can. Math. Commun. 1 (2026), e1, 1--44. doi:10.4153/S2976859425100003

work page doi:10.4153/s2976859425100003 2026
[4]

B. M. Braga and I. Farah, On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces, Trans. Amer. Math. Soc. 374 (2021), no. 2, 1007--1040. doi:10.1090/tran/8180

work page doi:10.1090/tran/8180 2021
[5]

B. M. Braga, I. Farah, and A. Vignati, Embeddings of uniform Roe algebras, Comm. Math. Phys. 377 (2020), no. 3, 1853--1882. doi:10.1007/s00220-019-03539-9

work page doi:10.1007/s00220-019-03539-9 2020
[6]

B. M. Braga, I. Farah, and A. Vignati, General uniform Roe algebra rigidity, Ann. Inst. Fourier (Grenoble) 72 (2022), no. 1, 301--337. doi:10.5802/aif.3461

work page doi:10.5802/aif.3461 2022
[7]

B. M. Braga, I. Farah, and A. Vignati, Operator norm localization property for equi-approximable families of projections, J. Noncommut. Geom. 18 (2024), no. 2, 657--680. doi:10.4171/JNCG/555

work page doi:10.4171/jncg/555 2024
[8]

Friedman, A proof of Alon's second eigenvalue conjecture and related problems, Mem

J. Friedman, A proof of Alon's second eigenvalue conjecture and related problems, Mem. Amer. Math. Soc. 195 (2008), no. 910, viii+100 pp. doi:10.1090/memo/0910

doi:10.1090/memo/0910 2008

Show all 24 references

[9]

Higson and J

N. Higson and J. Roe, On the coarse Baum--Connes conjecture, in Novikov conjectures, index theorems and rigidity, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 227--254. doi:10.1017/CBO9780511629365.008

1995 doi
[10]

K o nig, \"Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math

D. K o nig, \"Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916), 453--465. doi:10.1007/BF01456961

1916 doi
[11]

Krutoy, Bijective rigidity of uniform Roe algebras and injectivity of the comparison map, preprint, arXiv:2602.06720v2 (2026)

K. Krutoy, Bijective rigidity of uniform Roe algebras and injectivity of the comparison map, preprint, arXiv:2602.06720v2 (2026). doi:10.48550/arXiv.2602.06720

2026 doi
[12]

Kubota, Controlled topological phases and bulk-edge correspondence, Comm

Y. Kubota, Controlled topological phases and bulk-edge correspondence, Comm. Math. Phys. 349 (2017), no. 2, 493--525. doi:10.1007/s00220-016-2699-3

2017 doi
[13]

Lu and L

L. Lu and L. A. Sz\'ekely, Using Lov\'asz Local Lemma in the space of random injections, Electron. J. Combin. 14 (2007), no. 1, Research Paper 63, 13 pp. doi:10.37236/981

2007 doi
[14]

Mart\'inez and F

D. Mart\'inez and F. Vigolo, C^* -rigidity of bounded geometry metric spaces, Publ. Math. Inst. Hautes \' E tudes Sci. 141 (2025), 333--348. doi:10.1007/s10240-025-00155-3

2025 doi
[15]

B. D. McKay, N. C. Wormald, and B. Wysocka, Short cycles in random regular graphs, Electron. J. Combin. 11 (2004), no. 1, Research Paper 66, 12 pp. doi:10.37236/1819

2004 doi
[16]

Roe, An index theorem on open manifolds

J. Roe, An index theorem on open manifolds. I, J. Differential Geom. 27 (1988), no. 1, 87--113

1988
[17]

Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem

J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993), no. 497, x+90 pp

1993
[18]

Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, vol

J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, vol. 90, American Mathematical Society, Providence, RI, 1996, x+100 pp

1996
[19]

Roe, Lectures on coarse geometry, University Lecture Series, vol

J. Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003, viii+175 pp. doi:10.1090/ulect/031

2003 doi
[20]

Roe and R

J. Roe and R. Willett, Ghostbusting and property A, J. Funct. Anal. 266 (2014), no. 3, 1674--1684. doi:10.1016/j.jfa.2013.07.004

2014 doi
[21]

S pakula and R

J. S pakula and R. Willett, On rigidity of Roe algebras, Adv. Math. 249 (2013), 289--310. doi:10.1016/j.aim.2013.09.006

2013 doi
[22]

Vignati, The rigidity problem for uniform Roe algebras, preprint, arXiv:2602.01998v1 (2026)

A. Vignati, The rigidity problem for uniform Roe algebras, preprint, arXiv:2602.01998v1 (2026). doi:10.48550/arXiv.2602.01998

2026 doi
[23]

White and R

S. White and R. Willett, Cartan subalgebras in uniform Roe algebras, Groups Geom. Dyn. 14 (2020), no. 3, 949--989. doi:10.4171/GGD/570

2020 doi
[24]

Zhang, Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces, preprint, arXiv:2607.15096v1 (2026)

T. Zhang, Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces, preprint, arXiv:2607.15096v1 (2026). doi:10.48550/arXiv.2607.15096

2026 doi arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

F. P. Baudier, B. M. Braga, I. Farah, A. Khukhro, A. Vignati, and R. Willett, Uniform Roe algebras of uniformly locally finite metric spaces are rigid, Invent. Math. 230 (2022), no. 3, 1071--1100. doi:10.1007/s00222-022-01140-x

work page doi:10.1007/s00222-022-01140-x 2022

[2] [2]

B. M. Braga, Y. C. Chung, and K. Li, Coarse Baum--Connes conjecture and rigidity for Roe algebras, J. Funct. Anal. 279 (2020), no. 9, 108728, 20 pp. doi:10.1016/j.jfa.2020.108728

arXiv 2020

[3] [3]

B. M. Braga and R. Exel, KMS states on uniform Roe algebras, Can. Math. Commun. 1 (2026), e1, 1--44. doi:10.4153/S2976859425100003

work page doi:10.4153/s2976859425100003 2026

[4] [4]

B. M. Braga and I. Farah, On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces, Trans. Amer. Math. Soc. 374 (2021), no. 2, 1007--1040. doi:10.1090/tran/8180

work page doi:10.1090/tran/8180 2021

[5] [5]

B. M. Braga, I. Farah, and A. Vignati, Embeddings of uniform Roe algebras, Comm. Math. Phys. 377 (2020), no. 3, 1853--1882. doi:10.1007/s00220-019-03539-9

work page doi:10.1007/s00220-019-03539-9 2020

[6] [6]

B. M. Braga, I. Farah, and A. Vignati, General uniform Roe algebra rigidity, Ann. Inst. Fourier (Grenoble) 72 (2022), no. 1, 301--337. doi:10.5802/aif.3461

work page doi:10.5802/aif.3461 2022

[7] [7]

B. M. Braga, I. Farah, and A. Vignati, Operator norm localization property for equi-approximable families of projections, J. Noncommut. Geom. 18 (2024), no. 2, 657--680. doi:10.4171/JNCG/555

work page doi:10.4171/jncg/555 2024

[8] [8]

Friedman, A proof of Alon's second eigenvalue conjecture and related problems, Mem

J. Friedman, A proof of Alon's second eigenvalue conjecture and related problems, Mem. Amer. Math. Soc. 195 (2008), no. 910, viii+100 pp. doi:10.1090/memo/0910

doi:10.1090/memo/0910 2008