{"id":"1a61f3c5-e8f7-4c81-aeb9-b27ec11bcf2f","arxiv_id":"2411.15070","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For bounded-geometry spaces coarsely embeddable into ℓ^q, the ℓ^p-coarse Baum-Connes conjecture (for the paper's variant algebra) holds for every p ∈ [1,∞).","lead":"This paper proves the ℓ^p-coarse Baum-Connes conjecture for every bounded-geometry metric space that coarsely embeds into an ℓ^q-space, for all p and q in [1,∞). It extends Yu's Hilbert-space result to a strictly larger class of spaces, with consequences for the Novikov conjecture and positive scalar curvature obstructions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Mayer-Vietoris reduction in Section 2.4 relies on an unproved and generally false K-theoretic product decomposition for delta-disjoint covers; without it the twisted assembly isomorphism is not established.","rationale":"The Reader's weakest assumption is the dimension-independence of the Mazur map constant L. I checked Lemma 2.6 in detail: the homogeneous extension argument uses only the Lipschitz bound of the Mazur map on unit spheres, and the resulting constant L = 1 + q*2^q is independent of the dimension N. This is a standard result and is not where the proof is fragile. The genuinely load-bearing place is the K-theoretic glueing in Section 2.4. Proposition 2.21 is the bridge from the local isomorphism for bounded U to the full twisted assembly map, and its proof depends on Corollaries 2.26 and 2.27. The equality in Corollary 2.26 is asserted without proof; for a product-like controlled algebra it is not automatic and is generally false for infinite products of Banach algebras. Moreover, the paper then says the section-3 proof is completely similar, without supplying the p-analogue of these glueing arguments. This is not a mere exposition issue: if the decomposition fails, the Mayer-Vietoris step does not go through, and the theorem is unproved as written. The concern is concrete enough to test on a two-point example, and the test would settle exactly whether the asserted splitting holds. Since the reader already returned CONDITIONAL and this concern reinforces the need for clarification rather than demanding rejection, I keep the verdict UNCHANGED.","tokens_in":27728,"tokens_out":30903,"duration_ms":317773,"concrete_test":"Work out the two-point case X = {x,y}, U = U_1 disjoint union U_2 with dist(U_1,U_2) = delta and R > delta/2, with f(x) near U_1 and f(y) near U_2. Determine whether A(P_1(X),E)^R_U contains a nonzero element whose (x,y)-entry has E-support in the delta-corridor between U_1 and U_2, i.e., a block whose support is within R of both f(x) and f(y) and within epsilon of U for arbitrarily small epsilon. If such a block exists, the asserted equality K_*(A^R_U) = product_i K_*(A^R_{U_i}) fails and Corollary 2.26 needs a different proof; if no such block exists, identify the exact mechanism (e.g., an implicit bound on R relative to delta) that enforces the splitting. This single computation settles whether the Mayer-Vietoris reduction in Proposition 2.21 is valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Corollary 2.26, used in Proposition 2.21 and then inherited by the p >= 1 proof. For U = disjoint union of uniformly bounded, delta-disjoint sets U_i, it asserts K_*(A(P_1(X),E)^R_U) = product_i K_*(A(P_1(X),E)^R_{U_i}). No proof is given, and the assertion is not a formal consequence of delta-disjointness. In A^R_U, matrix entries T_s(x,y) are required only to have E-support eventually in an epsilon-neighborhood of U and within R of {f(x),f(y)}. If R > delta/2, an entry with f(x) near U_i and f(y) near U_j can have support in the corridor between the two components, so the algebra does not split as a direct sum or product over i. Even when the algebra does decompose as an infinite product, K-theory of Banach algebras does not commute with countable products in general; a Milnor lim^1 argument is required and is absent. The finite-multiplicity cover by balls B_q(f(x),R+1) is then handled by Corollary 2.27 via a standard Mayer-Vietoris argument that is also not written out. Since this is the only route from the local isomorphism for bounded U to the full twisted assembly map, the central theorem inherits this gap. The Mazur-map constant highlighted by the Reader is, by contrast, a standard dimension-free Lipschitz estimate; the computation in Lemma 2.6 does not appear to be the weak point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28083,"tokens_out":21245,"duration_ms":190944,"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":[{"comment":"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.","section":"§2.4, Corollary 2.26"},{"comment":"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.","section":"§2.4, Corollary 2.27"},{"comment":"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.","section":"§3.3, proof of Theorem 1.2"}],"minor_comments":[{"comment":"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.","section":"Lemma 2.7"},{"comment":"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.","section":"Lemma 2.6"},{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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→∞}'.","section":"Proposition 2.21 and proof of Theorem 2.2"},{"comment":"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.","section":"Definition 2.18(4)"},{"comment":"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.","section":"Question 3.6"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution whose main risk is not the strategy but the unproved product decomposition and Mayer–Vietoris steps in Section 2.4, which are inherited by Section 3.3. I judge these fixable within scope by standard techniques from [18, Section 12] with quantitative hypotheses on δ relative to R and explicit lim^1 arguments, hence major revision rather than rejection. Editors may also wish to flag the novelty wording around Question 3.6: the ℓ^p result concerns the authors' modified Roe algebras, and the relationship with the Zhang–Zhou algebras remains an open question. The paper's reliance on [18, 22] and on the authors' own [2] is appropriate and acknowledged."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The main theorem is genuinely new: full coarse Baum-Connes isomorphism (surjectivity, not just Novikov injectivity) for bounded geometry spaces coarsely embeddable into ℓ^q, and an ℓ^p analogue for all p. The p=2 argument follows Yu's Hilbert-space strategy, replacing the Hilbert norm with the ℓ^q norm via the homogeneous Mazur map. The dimension-free Lipschitz constant is explicit and the key estimates are written out. The authors are also honest about the relationship between their ℓ^p-Roe algebra and Zhang-Zhou's, flagged as Question 3.6. So credit where due.\n\nThe soft spot is exactly where the stress-test puts it: Corollary 2.26. It asserts K_*(A(P_1,E)^R_U) = ∏_i K_*(A(P_1,E)^R_{U_i}) for a δ-disjoint union U = ⊔ U_i. That is load-bearing: it is the only route from the local bounded-U lemma to the global twisted assembly map. As written, it is not justified. The algebra A^R_U does not obviously split over the U_i: even when the E-support is eventually in N_ε(U) with ε < δ/2, entries with f(x) near U_i and f(y) near U_j can survive as cross terms supported near one component only, so the algebra is not the product of the subalgebras. Even if the algebra did decompose as a countable product, K-theory of Banach algebras does not commute with countable products; you would need a Milnor lim^1 argument, and there is none. The subsequent Mayer-Vietoris step (Corollary 2.27) is also just asserted. This is not a cosmetic gap: the proof of Proposition 2.21 depends on it.\n\nThe Mazur-map constant the reader flagged is fine; Lemma 2.6 is a standard estimate and the dimension-free nature is correct. Lemma 2.7 has a typo, but harmless.\n\nThe ℓ^p section is more of a skeleton: the Eilenberg swindle and the Mayer-Vietoris are explicitly deferred as 'completely similar.' For a paper of this ambition, that is thin, especially since the ℓ^p-Roe algebra is a new variant and the vector-valued Marcinkiewicz-Zygmund theorem is doing real work.\n\nIf the gap in Corollary 2.26 is repairable, this is a solid advance. As it stands, the central theorem is not established. I would send it to a serious referee, but with a clear instruction to focus on the product decomposition and the Mayer-Vietoris argument. Conditional acceptance at best.","headline":"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.","tokens_in":28691,"tokens_out":10548,"would_cite":false,"duration_ms":99171,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L80","19K56"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["coarse Baum-Connes conjecture","ℓ^p-Roe algebra","ℓ^q coarse embeddability","Mazur map","Bott-Dirac operator","Novikov conjecture","K-theory of Banach algebras","Marcinkiewicz-Zygmund inequality"],"falsifier":"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.","tokens_in":27494,"feed_emoji":"📐","tokens_out":10478,"duration_ms":81358,"temperature":0.7,"pith_summary":"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.","feed_headline":"ℓ^p-coarse Baum–Connes conjecture proven for ℓ^q-embeddable spaces","feed_subtitle":"A Bott–Dirac twist proves it for every p,q≥1, making ℓ^p-Roe K-theory independent of p.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the Hilbert-space predecessor result and the coarse-disjoint-union reduction that this paper generalizes.","marker":"[22]"},{"why":"Supplies the standard definitions and framework for Roe algebras, localization algebras, and the coarse Baum–Connes conjecture.","marker":"[18]"},{"why":"Establishes the Novikov conjecture for groups coarsely embeddable into Banach spaces with property (H), which the descent principle recovers from the present theorem.","marker":"[9]"},{"why":"Proves the coarse Novikov conjecture for property (H) spaces, giving the injectivity result that this paper upgrades to a full isomorphism.","marker":"[2]"},{"why":"Contains the dimension-independent Lipschitz bound for the Mazur map used in Lemma 2.6.","marker":"[16]"},{"why":"Provides the factorization-theoretic result (Theorem 8.1) that inspires the proof of the vector-valued Marcinkiewicz–Zygmund inequality.","marker":"[12]"},{"why":"Introduces the ℓ^p-Roe algebras and the ℓ^p-coarse Baum–Connes conjecture; the paper modifies the definition and proves the conjecture for ℓ^q-embeddable spaces.","marker":"[24]"},{"why":"Shows coarse embeddability into ℓ^q for q≤2 is equivalent to Hilbert-space embeddability, justifying the reduction to q≥2.","marker":"[11]"}],"fun_headline_variants":["ℓ^p-coarse Baum-Connes proven for all ℓ^q-embeddable spaces","For all p,q≥1: ℓ^p-Baum-Connes for ℓ^q-embeddable","ℓ^p-coarse Baum-Connes conjecture proven for ℓ^q spaces","ℓ^p-Baum-Connes holds for all ℓ^q-embeddable spaces (p,q≥1)","ℓ^p-Roe K-theory independent of p for ℓ^q-embeddable spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["ℓ^p-coarse Baum-Connes proven for all ℓ^q-embeddable spaces","For all p,q≥1: ℓ^p-Baum-Connes for ℓ^q-embeddable","ℓ^p-coarse Baum-Connes conjecture proven for ℓ^q spaces","ℓ^p-Baum-Connes holds for all ℓ^q-embeddable spaces (p,q≥1)","ℓ^p-Roe K-theory independent of p for ℓ^q-embeddable spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000845,"raw_usage":{"total_tokens":3639,"prompt_tokens":869,"completion_tokens":2770,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":2637}},"tokens_in":485,"tokens_out":2770,"duration_ms":17173,"temperature":1.0,"reasoning_tokens":2637,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:33:03.920413+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"The coarse Baum-Connes conjecture for spaces which admit a uniform em- bedding into Hilbert space","cited_arxiv_id":null,"evidence_quote":"Provides the Hilbert-space predecessor result and the coarse-disjoint-union reduction that this paper generalizes."},{"cited_title":"The Novikov conjecture and geometry of Banach spaces","cited_arxiv_id":null,"evidence_quote":"Establishes the Novikov conjecture for groups coarsely embeddable into Banach spaces with property (H), which the descent principle recovers from the present theorem."},{"cited_title":"The coarse Novikov conj ecture and Banach spaces with Property (H)","cited_arxiv_id":null,"evidence_quote":"Proves the coarse Novikov conjecture for property (H) spaces, giving the injectivity result that this paper upgrades to a full isomorphism."},{"cited_title":"On the uniform classification of Lp(µ) spaces","cited_arxiv_id":null,"evidence_quote":"Contains the dimension-independent Lipschitz bound for the Mazur map used in Lemma 2.6."},{"cited_title":"Factorization of linear operators and geometry of Banach sp aces, volume 60 of CBMS Regional Conference Series in Mathematics","cited_arxiv_id":null,"evidence_quote":"Provides the factorization-theoretic result (Theorem 8.1) that inspires the proof of the vector-valued Marcinkiewicz–Zygmund inequality."},{"cited_title":"Lp coarse Baum-Connes conjecture and K-theory for Lp Roe algebras","cited_arxiv_id":null,"evidence_quote":"Introduces the ℓ^p-Roe algebras and the ℓ^p-coarse Baum–Connes conjecture; the paper modifies the definition and proves the conjecture for ℓ^q-embeddable spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows coarse embeddability into ℓ^q for q≤2 is equivalent to Hilbert-space embeddability, justifying the reduction to q≥2."}],"review_version":1}