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$\ell^p$-coarse Baum-Connes conjecture for $\ell^{q}$-coarse embeddable spaces

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Pith's one-line read The ℓ^p-coarse Baum–Connes conjecture holds for every bounded-geometry space that coarsely embeds into an ℓ^q-space, for all p,q≥1.

desk verdict A real advance is claimed, but the proof's load-bearing product decomposition in Corollary 2.26 is unjustified; the paper deserves a careful referee, not a desk reject. read the letter →

arxiv 2411.15070 v2 pith:JIY4ZH4H submitted 2024-11-22 math.KT math.OA

Jinmin Wang , Zhizhang Xie , Guoliang Yu , Bo Zhu This is my paper · ORCID
classification math.KTmath.OA MSC 46L8019K56
keywords coarseBaum-Connesconjectureℓ^p-Roealgebraℓ^qembeddabilityMazurmapBott-DiracoperatorNovikovK-theoryofBanachalgebrasMarcinkiewicz-Zygmundinequality
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 the ℓ^p-coarse Baum–Connes conjecture is true for any discrete metric space of bounded geometry that coarsely embeds into ℓ^q, for every p and q in [1,∞). The conjecture says that a canonical evaluation map from the K-theory of an ℓ^p-localization algebra to the K-theory of an ℓ^p-Roe algebra is an isomorphism, so it gives a way to compute the higher index classes that obstruct positive scalar curvature. Previous results covered spaces coarsely embeddable into Hilbert space or Banach spaces with property (H); this paper removes that restriction using a Mazur-map construction of a Bott–Dirac operator and a vector-valued Marcinkiewicz–Zygmund inequality to control the mixed ℓ^p-$ℓ^{2}$ norms. The result implies the Novikov conjecture for discrete groups coarsely embeddable into any ℓ^q, and it shows the K-theory of the ℓ^p-Roe algebra is independent of p for such spaces.

What carries the argument

The load-bearing mechanism is the homogeneous Mazur map Ψ:(E,‖·‖_q)→(E,‖·‖_2), extended from the Mazur map on spheres, whose Lipschitz constant L=1+q·2^q is independent of the dimension of E (Lemma 2.6). This dimension independence is what makes the scale s_n=2L(dim E_n)^2 in equation (2.4) produce uniform norm bounds on the rescaled Bott–Dirac operators B_{s,v}=$s^{{-1}}$D+C_v and their functional-calculus images Φ_{s,v}; the commutator and propagation estimates of Proposition 2.17 all depend on it. The other essential ingredient is Theorem 3.7, a vector-valued Marcinkiewicz–Zygmund inequality proved via the Grothendieck inequality, which controls operators on the mixed ℓ^p-$ℓ^{2}$ module ℓ^p(Z,ℓ^p(N,H)) so that the $ℓ^{2}$-theoretic estimates survive in the ℓ^p setting.

What would settle it

A concrete check would be to compute the Lipschitz constant of the homogeneous Mazur map on a sequence of finite-dimensional subspaces E_n of ℓ^q with dim E_n→∞. If these constants grow without bound, the uniform estimates in Corollary 2.14 and Proposition 2.17(5)–(6) fail, and the Mayer–Vietoris proof of the twisted assembly isomorphism cannot proceed; alternatively, finding a bounded-geometry space that coarsely embeds into some ℓ^q but whose ℓ^p-coarse Baum–Connes evaluation map is not an isomorphism would directly refute the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: if (X,d) is a discrete metric space with bounded geometry and X coarsely embeds into ℓ^q for some q∈[1,∞), then for every p∈[1,∞) the evaluation map ev : C^p_L(P_r(X)) → C^p(X) induces an isomorphism ev_* : lim_{r→∞} K_*(C^p_L(P_r(X))) → K_*(C^p(X)), where C^p(X) is the paper's modified ℓ^p-Roe algebra (Definition 3.2). The proof constructs a twisted ℓ^p-Roe algebra A^p(P,E) by tensorially attaching finite-dimensional Euclidean spaces through the coarse embedding, and uses a Bott–Dirac operator built from the homogeneous Mazur map Ψ:(E,‖·‖_q)→(E,‖·‖_2) to twist K-theory classes. It then shows that the twisted assembly map is an isomorphism by a Mayer–Vietoris argument over ℓ^q-balls, using the vector-valued Marcinkiewicz–Zygmund inequality to pass from $ℓ^{2}$ estimates. For p=2 the argument recovers the classical coarse Baum–Connes conjecture for ℓ^q-embeddable spaces, which is Theorem 1.1.

Load-bearing premise

The proof rests on the assumption that the homogeneous Mazur map from ℓ^q to $ℓ^{2}$ has a Lipschitz constant that does not grow with the dimension of the finite-dimensional subspaces; if that constant grew with dimension, the uniform norm and propagation estimates that make the twisted assembly map an isomorphism would collapse.

Editorial extensions

If this is right

  • For every bounded-geometry space coarsely embeddable into ℓ^q, the ℓ^p-coarse Baum–Connes conjecture holds for all p≥1, so in particular the coarse assembly map is an isomorphism on K-theory.
  • The K-theory K_*(C^p(X)) of the modified ℓ^p-Roe algebra is independent of p for ℓ^q-embeddable spaces, since it is isomorphic to the K-homology of X through the assembly map.
  • By the descent principle, the Novikov conjecture holds for any discrete group whose Cayley graph coarsely embeds into some ℓ^q, recovering and extending the known Banach-space property (H) result.
  • The theorem covers spaces that do not coarsely embed into Hilbert space, such as those coarsely embeddable into ℓ^q with q>2, which was the obstruction in earlier Hilbert-space approaches.
  • The modified definition of ℓ^p-Roe algebra (with coefficients in ℓ^p(N,H)) is the right one for this argument; the question of whether its K-theory agrees with the earlier Zhang–Zhou ℓ^p-Roe algebra is left open.

Reading between the lines

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

  • The same Bott–Dirac twist ought to work for any Banach space whose unit sphere admits a dimension-independent Lipschitz (or Hölder) map to a Hilbert space; the Mazur map is one instance, so the proof may generalize to other uniformly convex targets.
  • The p-independence of K_*(C^p(X)) might hold for a wider class of spaces: if one ℓ^p-coarse Baum–Connes isomorphism holds for a space, the others likely follow by the same twisted-algebra comparison.
  • The vector-valued Marcinkiewicz–Zygmund inequality of Theorem 3.7 could be useful for other ℓ^p-index problems where mixed norms appear, beyond the specific coarse-geometric setting.
  • A natural test case is a coarsely ℓ^q-embeddable box space of a group that does not have finite asymptotic dimension; verifying the conjecture there would give a concrete non-Hilbert example.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proves (Theorem 1.2) that if a bounded-geometry discrete metric space X coarsely embeds into ℓ^q for some q ∈ [1,∞), then for every p ∈ [1,∞) the ℓ^p-coarse Baum–Connes conjecture holds for X, where the ℓ^p-Roe algebra is the modified one of Definition 3.2 acting on ℓ^p(Z, ℓ^p(N, H)). Section 2 treats the C*-case (p = 2): a Bott–Dirac operator is built from the homogeneous Mazur map Ψ, twisted Roe algebras A(P,E) and their localization algebras are defined, an index map Ind : K_*(C^*(P)) → K_*(A(P,E)) is constructed (Propositions 2.19–2.20), and the twisted assembly map is claimed to be an isomorphism via a localization/Mayer–Vietoris argument (Proposition 2.21, Corollaries 2.26–2.27). Theorem 2.2 (the C*-case) and Theorem 1.2 follow by diagram chasing. Section 3 introduces ℓ^p-Roe algebras with an extra Hilbert-space factor H, proves a vector-valued Marcinkiewicz–Zygmund inequality (Theorem 3.7) using the Grothendieck inequality, and asserts that the same twisted-algebra argument works verbatim for all p.

Significance. If the missing technical steps are supplied, this is a significant result: it extends Yu's coarse Baum–Connes theorem from Hilbert-space-embeddable spaces to the strictly larger class of ℓ^q-embeddable spaces (for q > 2, by Johnson–Randrianarivony), gives a uniform family of ℓ^p-versions, and yields the coarse Novikov conjecture for ℓ^q-embeddable groups. The paper is a fresh derivation with no circularity: the constants L = 1 + q·2^q (Lemma 2.6) and s_n = 2L(dim E_n)^2 in (2.4) are derived explicitly rather than fitted, and the Grothendieck constant K_G in Theorem 3.7 is explicit. The authors honestly flag Question 3.6, which limits comparison with the Zhang–Zhou ℓ^p-Roe algebras. The overall architecture follows [18, 22] faithfully, which makes the unproved product/Mayer–Vietoris steps the main risk rather than the overall strategy.

major comments (3)
  1. [§2.4, Corollary 2.26] The proof consists of the single assertion K_*(A(P_1(X),E)^R_U) = ∏_{i≥1} K_*(A(P_1(X),E)^R_{U_i}), with no argument. Definition 2.24 requires the E-support of each matrix entry T_s(x,y) to lie eventually in the ε-neighborhood of U and, by Definition 2.23, within the (R+ε)-neighborhood of {f(x), f(y)}; it does not require the support to lie in the neighborhood of the component U_i associated with x or y. When the separation δ satisfies δ < 2(R+ε), a nonzero entry with f(x) near U_i and f(y) near U_j may have E-support in the corridor between the two components, so the Banach algebra A(P_1(X),E)^R_U need not be the product algebra ∏_i A(P_1(X),E)^R_{U_i}. Even when the algebra does decompose, K-theory of Banach algebras does not commute with countable products in general, and a Milnor lim^1 argument would be required; none is given. Since Corollary 2.27, Proposition 2.21, and hence Theorem 2.2 rest on this step, this is load-bearing and must be repaired, for instance by proving the decomposition under a hypothesis relating δ to R (e.g., δ > 2(R+1)) and controlling the leftover corridor terms, together with a lim^1 argument.
  2. [§2.4, Corollary 2.27] The 'standard Mayer–Vietoris argument' is not written out. The finite-multiplicity cover U_R = ⋃_{x∈X} B_q(f(x), R+1) appearing in the proof of Proposition 2.21 is the only route from the bounded-set isomorphism of Lemma 2.25 to the full twisted assembly map, and it must pass through the unproved product decomposition of Corollary 2.26 and through Mayer–Vietoris sequences for the twisted Banach algebras A(P,E)^R_U and their localization algebras. The intersection terms and the interchange of the limits (r → ∞, R → ∞, and the tail behavior in s) require explicit verification in the ℓ^q-metric setting; as written, Corollary 2.27 is a restatement of the required result rather than a proof.
  3. [§3.3, proof of Theorem 1.2] The general-p case is dispatched with 'the proof is completely similar' and 'we shall be brief', and Proposition 3.11 is said to follow from the 'same computation' as Proposition 2.19. This omits: the ℓ^p analogue of the Eilenberg swindle (Proposition 2.20), which requires the homotopy and almost-idempotent estimates to hold uniformly with the Marcinkiewicz–Zygmund constant K_G from Theorem 3.7 in the mixed-norm module M_{P,E}; the ℓ^p analogues of Lemma 2.25 and Corollaries 2.26–2.27, which inherit the product-decomposition gap of Corollary 2.26; and the ℓ^p versions of the uniform-in-s estimates of Proposition 2.17(5)–(7) for the Banach-module action. These steps are load-bearing for Theorem 1.2 and should be stated and proved rather than referenced by analogy.
minor comments (6)
  1. [Lemma 2.7] The displayed estimate ‖B_s(ρ_R φ)‖² ≥ (1/4)‖R²ρ_R φ‖² has the wrong dimension in R (it scales as R⁴), and it is inconsistent with the subsequent bound ‖ρ_R(1+B_s²)^{-1/2}σ‖ ≤ (2/R)(1+s^{-1})‖σ‖, which requires ‖B_s ψ‖ ≳ R‖ψ‖ outside B_q(R). The proof should be corrected to match the cleaner estimate derived in Lemma 2.11.
  2. [Lemma 2.6] The dimension-free constant is correct: the computation uses the unit-sphere Lipschitz bound ‖ψ(u)−ψ(v)‖₂ ≤ q2^{q−1}‖u−v‖_q together with ‖v_s−u‖_q ≤ 2δ, yielding L = 1 + q·2^q, and I found no hidden dependence on dim E.
  3. [Abstract and Introduction] The phrase 'coarsely embedds' should read 'coarsely embeds' (it also appears in the statements of Theorem 1.1 and Definition 2.1), and 'Atiya h–Singer' contains a stray space.
  4. [Proposition 2.21 and proof of Theorem 2.2] The codomain of ev_* in Proposition 2.21 should be K_*(A(P_1(X),E)) rather than K_*(A(P_r(X),E)) to match its use in the diagram, and the limit 'lim_{d→∞}' in the diagram should be 'lim_{r→∞}'.
  5. [Definition 2.18(4)] The E-support condition on T_s(x,y) is stated only with respect to the ball centered at f(y); since Definition 2.23 later uses the symmetric condition with {f(x), f(y)}, the intended convention in Definition 2.18 should be clarified or symmetrized.
  6. [Question 3.6] Question 3.6 is honestly posed but leaves open whether K_*(C^p(X)) agrees with the K-theory of the Zhang–Zhou ℓ^p-Roe algebra; the introduction should state explicitly that Theorem 1.2 concerns the modified algebra, so that the comparison with [24] is not overstated.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the ℓq-embeddable ℓp-coarse Baum–Connes isomorphism is derived from explicit non-fitted estimates (Mazur map, Grothendieck inequality) plus an Eilenberg–swindle index map; the flagged Corollary 2.26 product-decomposition step is an omitted proof (correctness risk), not a self-referential reduction.

full rationale

The derivation chain is not circular. Theorem 1.2 (the ℓp-coarse Baum–Connes conjecture for bounded-geometry spaces coarsely embedding into ℓq) is never assumed as an input. All constants are explicit and either derived in the paper or taken from external theorems: the Mazur-map Lipschitz constant L = 1 + q·2^q is proved in Lemma 2.6 from the external Lipschitz bound for the Mazur map ψ ([16], Weston); the scale s_n = 2L(dim E_n)^2 in line (2.4) is built directly from that L; and the Grothendieck constant K_G enters through the externally stated Grothendieck inequality (Theorem 3.8) and bounds T⊗1 in Theorem 3.7. No parameter is fitted to any subset of the target data; the claimed ev_*-isomorphism is attacked head-on by constructing the twisted index map Ind (Propositions 2.19 and 3.11), the Eilenberg–swindle trivialization of ı_*∘Ind (Proposition 2.20), and the localization/Mayer–Vietoris decomposition of Section 2.4 (Lemmas 2.25, Corollaries 2.26–2.27, Proposition 2.21). Self-citations ([2], [9], [18], [20], [22]) support only strictly weaker or standard statements — coarse Novikov for spaces with property (H), the Hilbert-space-embeddable case, localization-algebra K-theory, and the coarse-disjoint-union reduction; none of them asserts the target ℓq-embeddable ℓp-Baum–Connes isomorphism, so no self-citation chain forces the conclusion. Per the reviewing rule, I flag the following missing support as correctness risks, not circularity: Corollary 2.26 asserts 'K_*(A(P_1(X),E)^R_U) = ∏_{i≥1} K_*(A(P_1(X),E)^R_{U_i})' with no proof that the twisted algebra splits over δ-disjoint components and no Milnor lim^1 check that K-theory commutes with the countable product; Corollary 2.27 invokes a 'standard Mayer–Vietoris argument' without writing it out; and the proof of Theorem 1.2 says the ℓp twisted ev_*-isomorphism 'follows from the same construction and the Mayer–Vietoris argument as the ℓ2-case,' inheriting those gaps. None of these equates the theorem's conclusion to its own hypotheses by construction, so they do not constitute circularity under the stated standards. Score 1: no load-bearing circularity; minor, non-load-bearing self-citations are present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central claim rests on standard tools of coarse geometry and operator algebras (Roe algebras, localization algebras, Rips complexes), plus two external theorems: Nowak's equivalence of ℓ^q and Hilbert embeddability for q ≤ 2, and Grothendieck's inequality. The paper introduces no fitted numerical parameters; the constants (L, s_n, KG) are explicit and derived. The main mathematical object introduced is the modified ℓ^p-Roe algebra, whose K-theory relation to the prior Zhang-Zhou algebra is left open in Question 3.6.

assumptions (6)
  • domain assumption Nowak's theorem: for 1 ≤ q ≤ 2, coarse embeddability into ℓ^q is equivalent to coarse embeddability into Hilbert space.
    Used to reduce the proof to the case q ≥ 2, stated immediately after Theorem 2.2.
  • standard math Yu's localization algebra theorem: K_*(C^*_L(P)) is naturally isomorphic to the K-homology of P.
    Used in Section 2.1 to define the coarse assembly map and in the reduction to coarse disjoint unions; cited from [20].
  • standard math Grothendieck inequality (Theorem 3.8).
    Key input for the vector-valued Marcinkiewicz-Zygmund inequality (Theorem 3.7), which underlies the ℓ^p index map construction.
  • standard math Mazur map Lipschitz and Hölder estimates from Weston [16].
    Used in Lemma 2.6 to obtain the dimension-independent Lipschitz constant L = 1 + q·2^q for q ≥ 2.
  • domain assumption The reduction to coarse disjoint unions of finite metric spaces from [18, Section 12.5].
    Used at the start of Section 2.3 to restrict the proof to X = ⊔ X_n with finite X_n and finite-dimensional E_n.
  • standard math Roe algebra K-theory is invariant under coarse equivalence, and Rips complexes are coarsely equivalent to X.
    Used to pass between X and its Rips complexes and to identify K-groups; cited from [18, Section 5.1].
invented entities (2)
  • Modified ℓ^p-Roe algebra C^p(X) (Definition 3.2), acting on ℓ^p(Z, ℓ^p(N,H)) instead of ℓ^p(Z, ℓ^p(N)).
    purpose: Makes the tensor product map id_* : K_*(C^p(P)) → K_*(C^p(P,E)) an isomorphism, which is essential for the index-theoretic proof.
    The paper itself raises Question 3.6 asking whether K_*(C^p(X)) is isomorphic to K_*(C_p(X)) of Zhang-Zhou. Without this isomorphism, the theorem does not directly prove the original ℓ^p-coarse Baum-Connes conjecture.
  • Twisted Roe algebras A(P,E) and A^p(P,E), and the Bott-Dirac twisted index map.
    purpose: Auxiliary algebras used to factor the K-theory isomorphism through a twisted algebra where the coarse Baum-Connes map can be shown to be an isomorphism by a Mayer-Vietoris argument.
    These are mathematical constructions introduced for the proof; they have no independent falsifiable content outside the paper.

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Pith. "Pith review of $\ell^p$-coarse Baum-Connes conjecture for $\ell^{q}$-coarse embeddable spaces." pith.science (2026). https://pith.science/paper/JIY4ZH4H

@misc{pith2026241115070,
  author       = {Pith},
  title        = {Pith review of: $\ell^p$-coarse Baum-Connes conjecture for $\ell^q$-coarse embeddable spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JIY4ZH4H}},
  note         = {Machine review of arXiv:2411.15070}
}
abstract

We prove an $\ell^p$-version of the coarse Baum-Connes conjecture for spaces that coarsely embedds into $\ell^q$-spaces for any $p$ and $q$ in $[1,\infty)$.

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Forward citations

Cited by 2 Pith papers

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  2. Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture

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    If a proper Γ-space with equivariant bounded geometry equivariantly and coarsely embeds into an admissible Hilbert-Hadamard space, then the rational analytic equivariant coarse Novikov conjecture holds.

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