$K$-theory of ghostly ideals for $\ell^p$-coarsely embeddable spaces

Kang Li; Liang Guo; Qin Wang

REVIEW 4 major objections 6 minor 5 references

$K$-theory of ghostly ideals for $\ell^p$-coarsely embeddable spaces

T0 review · 4 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read In ℓ^p-embeddable spaces, geometric and ghostly ideals always have the same K-theory.

desk verdict A genuinely new K-theory isomorphism for ℓ^p-embeddable spaces, built on the ℓ^p Bott–Dirac machine; the main idea is right, but the pivotal Corollary 3.15 is asserted rather than proved and should be fixed before this is accepted as written. read the letter →

arxiv 2511.22438 v3 pith:6XGLPSSR submitted 2025-11-27 math.KT math.FAmath.OA

Liang Guo , Kang Li , Qin Wang This is my paper · ORCID

classification math.KTmath.FAmath.OA MSC 19K5647L20

keywords coarsegeometryRoealgebraghostidealK-theoryℓ^p-spacesBaum-ConnesconjectureoperatornormlocalizationMazurmap

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that for any metric space with bounded geometry that admits a coarse embedding into some ℓ^p-space (1 ≤ p < ∞), the ghostly ideal in the Roe algebra carries no extra K-theory beyond its geometric sub-ideal: the inclusion induces an isomorphism in K-theory, equivalently the quotient has zero K-theory. Previously this was known only for spaces coarsely embeddable into Hilbert space, via K-amenability of the coarse groupoid; here that tool is replaced by an ℓ^p Bott-Dirac construction, which is restricted to the geometric and ghostly ideals inside a twisted Roe algebra. A sympathetic reader should care because the theorem, if correct, settles a natural family of higher-index conjectures for all ℓ^p-embeddable bounded-geometry spaces: the relative coarse Baum-Connes conjecture, the maximal coarse Baum-Connes conjecture, and the operator norm localization property for finite-rank projections. The argument reduces the problem to sparse subspaces and then shows that in the twisted setting the ghostly and geometric ideals have the same asymptotic behavior, forcing the K-theory isomorphisms.

What carries the argument

The machinery is the ℓ^p Dirac-dual-Dirac construction, transplanted from Hilbert-space coarse index theory to a twisted Roe algebra. On each finite-dimensional ℓ^p approximation E_n, one uses the extended Mazur map Ψ between the ℓ^p and Euclidean unit spheres to define a Bott generator C_v and then a Bott-Dirac operator B_{s,v} = s^{-1}D + C_v on L^2(E_n) with Clifford coefficients. Its bounded transform Φ(B_{s,v}) is packaged into a twisted Roe algebra A(X,E) of functions of the zoom parameter s; the crucial input is a set of dimension-uniform estimates on the propagation and Lipschitz continuity of Φ(B_{s,v}) in the ℓ^p direction, which let the index map descend to ideals and eventually e

What would settle it

Compute the Lipschitz constant of the extended Mazur map on finite-dimensional ℓ^p_n for p>2 and large n: if the ratio ||Ψ(v)-Ψ(u)||_2 / ||v-u||_p ever exceeds the bound asserted in Lemma 3.2 (or grows with n), the dimension-uniform estimates in Proposition 3.4 break and the index map into the twisted Roe algebra is not defined. Alternatively, find a bounded-geometry ℓ^p-embeddable space (p>2) not embeddable into Hilbert space and check whether K_*(G(X,U)/I(X,U)) vanishes for some invariant open U; any nonzero group would contradict the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 2.6: if a bounded-geometry metric space X coarsely embeds into an ℓ^p-space (p ∈ [1,∞)), then for every invariant open U ⊆ βX the inclusion I(X,U) → G(X,U) induces a K-theory isomorphism; equivalently, G(X,U)/I(X,U) has trivial K-theory. The proof generalizes the ℓ^p Dirac-dual-Dirac construction: after reducing to sparse subspaces, it builds a twisted Roe algebra and shows the geometric and ghostly ideals coincide there asymptotically. Consequences drawn in the paper: the relative coarse Baum-Connes conjecture for all subspaces, the finite-rank operator norm localization property, and maximal-versus-reduced Roe algebra K-theory isomorphisms.

Load-bearing premise

The proof depends on a dimension-uniform bound quantifying how much the ℓ^p-to-Euclidean sphere map stretches distances; the paper asserts this bound but the key formula is not proved here, and without it the main construction does not go through.

Editorial extensions

If this is right

For any ℓ^p-embeddable bounded-geometry space, the relative coarse Baum-Connes conjecture holds for every subspace Y ⊆ X, including the boundary coarse Baum-Connes conjecture.
Such spaces have the operator norm localization property for equi-approximable finite-rank projections (ONL_PFin), meaning ghost projections on sparse subspaces are forced to be compact.
The maximal coarse Baum-Connes conjecture holds for these spaces, and the canonical quotient maps from maximal to reduced Roe algebras (and their relative versions) induce K-theory isomorphisms.
For every invariant open U, the K-theory of the ghostly quotient G(X,U)/I(X,U) vanishes, so ghosts are invisible to K-theory in all directions.
The main result also extends from ℓ^p to general L^p-spaces, including non-separable ones (Theorem 3.17).

Reading between the lines

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

We infer that the twisted Bott-Dirac framework may give a direct route to K-amenability of the coarse groupoid for ℓ^p-embeddable spaces; the paper leaves this as an open question, but its maximal-versus-reduced K-theory isomorphisms are the usual signature of K-amenability.
Going beyond the paper, the equivalence it proves for sparse subspaces suggests ONL_PFin might itself be characterized by vanishing of K_*(G(X)/I(X)) for the full ghostly ideal, giving a testable higher-index criterion for spaces of unknown embeddability.
Because ℓ^p- and Hilbert-space embeddability are equivalent for p ≤ 2, the theorem is only genuinely new for p > 2; it therefore predicts the desired K-theory vanishing in the one regime where no separating examples are yet known.

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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 result is real and worth taking seriously. For metric spaces of bounded geometry that coarsely embed into some ℓ^p (or L^p) space, the paper proves that every geometric ideal I(X,U) and its ghostly counterpart G(X,U) have isomorphic K-theory. This genuinely extends the Hilbert-space case of Finn-Sell and of Wang–Fu–Zhang, and the full-Roe-algebra case treated by WXYZ24. It also brings the relative and maximal coarse Baum–Connes statements and ONL_{P_Fin} as corollaries. If the proof is solid, this is a substantial step, especially because K-amenability of the coarse groupoid is still open for ℓ^p-embeddable spaces. The architecture is sensible: reduce to sparse spaces, restrict the ℓ^p Dirac-dual-Dirac construction to the ideals, then compare the twisted geometric and ghostly ideals. The authors give real credit where the machinery comes from and do not overclaim novelty.

The soft spots are real but mostly local. The biggest is Corollary 3.15. Lemma 3.14 shows that every element of A_G(U) can be approximated, asymptotically in s, by an element of A_I(U). But the jump from that approximation to a K-theory isomorphism is made in a single sentence, citing WY20 Lemma 6.4.11, with details left to the reader. That is the pivotal new step in Theorem 2.6, and it is not obvious: one needs a homotopy or an evaluation-at-infinity argument that actually identifies the K-theory of the two twisted ideals, and the E-propagation conditions in Definition 3.5(2) have to be checked. This should be written out.

The other issues are minor but should be cleaned up. Lemma 3.2 has a visible typo: the extended Mazur map Ψ is evaluated at v/‖v‖_2 even though the domain is the ℓ^p sphere, and the global Lipschitz bound L = 1 + p·2^p is stated without proof. That is likely repairable by citing the corresponding estimates from WXYZ24, but as printed it is a hole in a lemma the whole Bott-Dirac estimate depends on. Proposition 2.12 also relies on an external sparse-cover lemma without a precise citation, and Theorem 4.7 on the maximal conjecture is more sketch than proof. None of these look fatal; they are gaps in presentation and verification, not obvious contradictions.

Who is this for? Anyone working in coarse index theory, ghost ideals, or the coarse Baum–Connes family of conjectures. The paper deserves a serious referee, not a desk rejection, provided the referee is willing to chase down the WXYZ24 machinery. I would want the Corollary 3.15 argument supplied, Lemma 3.2 repaired, and the maximal-version proof expanded before I would call it settled.

Referee Report

4 major / 6 minor

Summary. The paper claims that for a bounded-geometry metric space X admitting a coarse embedding into an ℓ^p-space (1≤p<∞), the inclusion i:I(X,U)→G(X,U) from any geometric ideal to the associated ghostly ideal induces an isomorphism on K-theory (Theorem 2.6). The proof follows the ℓ^p Bott–Dirac machinery of [WXYZ24] and [WY20]: it reduces to sparse spaces, constructs a twisted Roe algebra A(X,E), defines index maps with a dimension-dependent scale shift s_n=L(dim E_n)^2, restricts the construction to ideals, and attempts to show that the twisted geometric and ghostly ideals have the same asymptotic behaviour. From this, the paper derives the relative coarse Baum–Connes conjecture, the operator norm localization property ONL_PFin, and a maximal coarse Baum–Connes statement.

Significance. If the main theorem is correct, it is a substantial extension of the Hilbert-space result of [WFZ25] and of earlier ghost-ideal K-theory results, and it provides a uniform route to several corollaries that are currently open in general. The paper is honest about its reliance on prior ℓ^p machinery, and the high-level strategy is coherent. The main weakness is that the pivotal new step (Corollary 3.15) is only sketched, and some supporting technical claims (Lemma 3.2, Proposition 3.13) are either misprinted or insufficiently proved. The result is plausible and likely repairable, but the manuscript as it stands leaves load-bearing details to the reader.

major comments (4)

[§3.4, Corollary 3.15] This is the pivotal new step of the paper, but its proof is one sentence: the K-theory of A_I(U)(X,E) and A_G(U)(X,E) 'only depends on their asymptotic behavior as s→∞', citing [WY20, Lemma 6.4.11], and Lemma 3.14 is said to show they have the same asymptotic behaviour. Lemma 3.14 only produces, for each element T and ε>0, a tail approximation by an element T' of A_I(U)[X,E] with sup norm ≤ε. It does not supply a homotopy, a mapping cone argument, or a verification that the hypotheses of the cited lemma are satisfied for these ideals, which are simultaneously constrained by the E-propagation condition in Definition 3.5(2). The diagram chase after (3) depends entirely on this implication. This needs to be proved explicitly.
[§3.1, Lemma 3.2] As printed, the formula for the extended Mazur map is not well-defined: ψ is defined on the ℓ^p unit sphere S_p(E), but the displayed formula evaluates ψ at v/||v||_2, which lies in S_2(E). Also, the asserted Lipschitz bound L=1+p·2^p for p≥2 is not proved in this paper and is only referred to [WXYZ24, Lemma 2.6]. This lemma is load-bearing: it enters Proposition 3.4, Definition 3.6 (via s_n=L(dim E_n)^2), and the spectral estimates. The authors should either give the correct formula and proof or quote the precise statement from the literature.
[§3.3, Proposition 3.13] The proof of Proposition 3.13, which asserts that the evaluation map after the index map is an isomorphism on ideals, is only sketched and contains a serious technical issue in the 'stacking argument'. The path P_N(t) defined near the end of the proof is written as a difference of two diagonal block projection matrices; the individual diagonal entries of this difference are differences of projections, and as written P_N(t) is not a projection-valued path. Consequently [P_N(t)] is not defined in K-theory. Since this proposition is used to identify the vertical maps in diagram (3), it must be repaired or replaced by a complete argument.
[§4.2, Theorem 4.7 and Corollary 1.4] The maximal coarse Baum–Connes conjecture is asserted after a paragraph of informal discussion, with statements such as 'one can verify that the proofs in [WY20, Section 12.3 & 12.4] also hold' and 'combining these observations, we obtain'. No detailed proof is supplied. Corollary 1.4 depends on Theorem 4.7. If the maximal result is to be claimed, it needs a real proof; otherwise it should be explicitly marked as conditional or deferred.

minor comments (6)

[Throughout] The internal cross-references are inconsistent: Theorem 2.6 is called 'Definition 2.6', Proposition 3.13 is called 'Definition 3.13', and Lemma 3.14 is called 'Definition 3.14'. This makes the paper difficult to read.
[§3.4, Lemma 3.14] The equality A[X,E]∩A_G(U)(X,E)=A_I(U)[X,E] is asserted without proof in the last lines of the lemma. It is plausible but requires justification, since A_I(U)[X,E] was defined by generators with supports in U and the intersection statement is not immediate.
[§3.1, display before Proposition 3.4] The homogeneous extension of the Mazur map is written with Ψ:(E,||·||_p)→(E,||·||_2), but the display evaluates ψ at v/||v||_2. In addition, the normalization factor ||v||_p appears to be inconsistent with the claimed Lipschitz estimate for ||Ψ(v)-Ψ(u)||_2 with respect to ||v-u||_p. This should be cleaned up.
[§3.3, proof of Proposition 3.13] The sentence 'Note that f_n(X_n)⊂E_n is uniformly bounded by M∈N' is unclear: M is used both as a natural number and as a bound, and the conclusion χ_K p_n^(k)=χ_K p_n^(∞) for k≥M depends on this bound. The notation should be clarified.
[§3, Remark 3.18] The argument for non-separable L^p-spaces is informal. The claim that 'the family of all finite-dimensional subspaces of any L^p-space admits a uniformly coarse embedding into ℓ^p' is not proved and is not an immediate consequence of finite representability as stated. This remark is not used in the main proof, but the wording should be softened or the argument expanded.
[References] There are typographical issues in the references and text, e.g. 'P . Nowak' and the use of 'Definition 2.6' for the main theorem. The references [WZ25], [WFZ25], [GWZ25], and [WXYZ24] should be cited with precise theorem/lemma numbers where they are used.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the central K-theory isomorphism is built from external ℓ^p Bott-Dirac results and standard asymptotic K-theory lemmas, not from its own conclusion.

full rationale

The derivation chain for Theorem 2.6 is not circular. The target statement — that i_*: K_*(I(X,U)) → K_*(G(X,U)) is an isomorphism for ℓ^p-coarsely embeddable X — is first reduced to sparse subspaces by general Mayer-Vietoris/pushout facts (Proposition 2.12, citing [HRY93]), then to the twisted ideals A_I(U)(X,E) and A_G(U)(X,E) via the commuting diagram (3). The maps Ind and ι_{s*} are shown to be isomorphisms in Proposition 3.13 using the Bott-Dirac machinery imported from [WXYZ24] and [WY20]; neither source assumes the ideal-isomorphism conclusion of this paper. The decisive comparison is Lemma 3.14: for T ∈ A_G(U)(X,E) and ε > 0 it produces T′ ∈ A_I(U)[X,E] and S > 0 with sup_{s∈[S,∞),n∈N} ‖T_{s,n} − T′_{s,n}‖ ≤ ε. Corollary 3.15 converts this into a K-theory isomorphism by citing [WY20, Lemma 6.4.11]. That step is terse and is the main default risk: asymptotic closeness in sup norm is not by itself the K-theory statement, and the hypotheses of the cited lemma are not verified in the text. But this is a proof-gap/correctness issue, not circularity: the cited lemma is external and is not equivalent to the paper's input or conclusion. The same can be said of the evident misprint in Lemma 3.2 (ψ is defined on S_p(E) but the formula writes v/‖v‖_2, and the global Lipschitz bound is asserted without proof); these affect the imported estimates, but they do not make the argument self-referential. The self-citations ([WZ25], [GWZ25], [GLWZ24]) provide background ideal theory and the relative Baum-Connes framework, while the load-bearing ℓ^p coarse Baum-Connes results come from [WXYZ24], whose authors do not overlap with the present paper. No fitted parameters, no uniqueness theorem imported from the authors, and no self-justifying ansatz force the result. Consequently, no specific circular step can be exhibited; the score reflects a few benign self-citations and one externally-cited but under-verified bridging lemma, not structural circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or dimensions are postulated. The twisted Roe algebra A(X,E) and its ideals A_I(U), A_G(U) are new mathematical constructions rather than invented empirical entities; they are defined explicitly and do not require independent evidence. The main external inputs are prior theorems and lemmas, listed as axioms.

free parameters (1)

s_n (Bott-Dirac scale shift) = L·(dim E_n)^2
Introduced in Definition 3.6 to make the support estimates of Prop 3.4(4) hold uniformly in n. It is chosen by hand, not fitted to data; its exact value should not affect the final K-theory statement, but the proof depends on the scaling regime s+s_n ≥ L(dim E_n)^2.

assumptions (6)

domain assumption X has bounded geometry (uniformly bounded ball sizes)
Hypothesis of Theorem 2.6; used to identify C*(X) with the reduced crossed product ℓ∞(X,K)⋊_r G(X) (Lemma 2.3) and to control cardinalities in Prop 2.8.
domain assumption X admits a coarse embedding into ℓ^p (or L^p), 1≤p<∞
Central hypothesis; used to construct the ℓ^p Bott-Dirac operators E_n and the twisted Roe algebra A(X,E) in Section 3.2.
domain assumption Every bounded geometry X has an ω-excisive cover {Y,Z} with Y,Z,Y∩Z sparse
Invoked at the start of Section 2.4 with citation [CWY13, Section 4]; it is the basis of Prop 2.12's reduction to sparse subspaces. No proof or precise statement is given in this paper.
domain assumption The extended Mazur map Ψ is Lipschitz with constant L=1+p·2^p (Lemma 3.2)
Imported from [WXYZ24, Lemma 2.6]; powers the support estimates in Prop 3.4. The displayed formula in the present paper appears to contain a typo (ψ(v/∥v∥_2) while ψ is defined on S_p), so the exact statement needs verification.
domain assumption Uniform-in-dimension support estimates for Φ(B_{s,v}) (Prop 3.4)
Imported from [WY20, Prop 12.1.10] and [WXYZ24, Prop 2.17]; these estimates are needed in Definition 3.6/3.11 and Lemma 3.14 to control E-propagation and to localize entries.
standard math K-theory of the twisted algebras is determined by their asymptotic behaviour as s→∞ (WY20 Lemma 6.4.11)
Used in Corollary 3.15 to infer K_*(A_I) ≅ K_*(A_G) from Lemma 3.14. The applicability to the specific ideals A_I(U), A_G(U) is not demonstrated in the text.

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Cite this review

Pith. "Pith review of $K$-theory of ghostly ideals for $\ell^p$-coarsely embeddable spaces." pith.science (2026). https://pith.science/paper/6XGLPSSR

@misc{pith2026251122438,
  author       = {Pith},
  title        = {Pith review of: $K$-theory of ghostly ideals for $\ell^p$-coarsely embeddable spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6XGLPSSR}},
  note         = {Machine review of arXiv:2511.22438}
}

abstract

Ghostly ideals are among the most mysterious objects in coarse index theory. In this paper, we show that if a metric space $X$ with bounded geometry admits a coarse embedding into an $\ell^p$-space ($1 \le p < \infty$), then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces an isomorphism in $K$-theory. As consequences, we deduce that such spaces satisfy the relative coarse Baum-Connes conjectures, as well as the operator norm localization property for finite rank projections ($ONL_{\mathcal P_{Fin}}$).

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Reference graph

Works this paper leans on

5 extracted references · 2 linked inside Pith

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28 L. GUO, K. LI, AND Q. WANG [WXYZ24] J. Wang, Z. Xie, G. Yu, and B. Zhu.ℓ p-coarse Baum-Connes conjecture forℓ q-coarse embeddable spaces.arXiv e-prints, page arXiv:2411.15070, November

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[Yu00] G. Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embed- ding into Hilbert space.Invent. Math., 139(1):201–240, 2000

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[GWZ25] L. Guo, Q. Wang, and C. Zhang. Relative higher index theory on quotients of Roe al- gebras and positive scalar curvature at infinity.arXiv e-prints, page arXiv:2509.23380, September

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[DG24] J. Deng and L. Guo. Twisted Roe algebras and theirK-theory.arXiv e-prints, page arXiv:2409.16556, September

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[GWY08] G

©2022. [GWY08] G. Gong, Q. Wang, and G. Yu. Geometrization of the strong Novikov conjecture for residually finite groups.J. Reine Angew. Math., 621:159–189,

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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28 L. GUO, K. LI, AND Q. WANG [WXYZ24] J. Wang, Z. Xie, G. Yu, and B. Zhu.ℓ p-coarse Baum-Connes conjecture forℓ q-coarse embeddable spaces.arXiv e-prints, page arXiv:2411.15070, November

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[Yu00] G. Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embed- ding into Hilbert space.Invent. Math., 139(1):201–240, 2000

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[GWZ25] L. Guo, Q. Wang, and C. Zhang. Relative higher index theory on quotients of Roe al- gebras and positive scalar curvature at infinity.arXiv e-prints, page arXiv:2509.23380, September

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Deng and L

[DG24] J. Deng and L. Guo. Twisted Roe algebras and theirK-theory.arXiv e-prints, page arXiv:2409.16556, September

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