{"id":"71c7700c-680e-4061-a99a-3c3c253dd627","arxiv_id":"2505.02338","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Geometric Banach property (T) is defined via Banach representations of Roe algebras, shown to be coarsely invariant, equivalent to Kazhdan projections in Banach-Roe algebras, and linked to Banach property (T) of limit groups and box spaces.","lead":"This paper introduces a Banach-space version of geometric property (T), a rigidity notion for metric spaces, and shows it is preserved under coarse equivalence and tied to the existence of Kazhdan projections. It connects the new property to fixed-point properties and to Lafforgue's strong Banach property (T) for groups via box spaces.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Spectral-gap transfer in the coarse-invariance proof relies on a quotient-norm bound that fails for the natural restriction map; a fiber-averaging replacement is missing.","rationale":"The reader correctly identifies the quotient-space issue in Sections 5 and 6: Proposition 5.5 and Theorem 6.7 assert that the inclusion of B into L^p(µ_φ,B) induces an isometric embedding of quotient spaces. That particular claim is in fact true and can be repaired by a short averaging argument: for any invariant vector F in L^p(µ,B), the Bochner integral ∫F dµ is ρ-invariant, so dist(1⊗ξ, L^p_π) equals dist(ξ, B_ρ). Thus the reader's weakest assumption is a genuine but repairable gap. The more serious instance of the same failure mode is Section 7. There, the restriction map χ_X does not, in general, preserve quotient norms; the two-point model shows that it can annihilate the quotient entirely. The text's Lemmas 7.2-7.4 do not supply the needed bounded-below transfer, and no averaging argument over an invariant measure is available in that setting. Since Theorem 7.1 is one of the paper's central advertised claims, the proof as written is incomplete. The overall architecture is plausible and likely correct, but the coarse-invariance theorem requires a real argumentative fix, not merely added details. This supports the reader's CONDITIONAL verdict: the authors should supply a correct transfer proof, and the current manuscript should not be accepted as fully verified.","tokens_in":44560,"tokens_out":38621,"duration_ms":535023,"concrete_test":"Compute the finite model: X={0}, Y={0,1}, B=C^2, with π_Y the standard representation of Cu[Y]=M2(C). Verify that B^{π_Y}=span{(1,1)}, B^{π_X}=span{e_0}, and that χ_X descends to the zero map BY/B^{π_Y}->BX/B^{π_X}. Then replace χ_X by the fiber-averaging map Q(ξ_0,ξ_1)=V(0)ξ_0+V(1)ξ_1 and check whether Q is bounded below on the quotient. If no bounded-below map is obtained, Theorem 7.1 requires a substantially different transfer argument rather than a clarification of the present one.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 7.1, the advertised coarse invariance of geometric Banach property (T), is proved by transferring spectral gaps between a space X and a net X embedded in Y. The transfer is effected through Lemma 7.2 and Corollary 7.4, which assert that the range projection χ_X induces a map between quotient spaces by invariant subspaces that preserves the information needed for spectral gaps. No bounded-below estimate is supplied. The natural restriction map can collapse the quotient: let X={0}, Y={0,1}, B=C^2, and let π_Y be the standard representation of Cu[Y]=M2(C). Then B^{π_Y}=span{(1,1)} and B^{π_X}=span{e_0}, so χ_X maps B^{π_Y} onto B^{π_X}, but the induced map on quotients BY/B^{π_Y}->BX/B^{π_X} is zero. This model does not disprove the theorem, but it shows the proof mechanism as written cannot transfer an almost-invariant vector from Y to X; a fiber-averaging or isometric-block argument is required and is absent. The same quotient-norm issue appears in Proposition 5.5 and Theorem 6.7, where it is repairable by averaging against an invariant probability measure; the coarse-invariance context has no analogous repair supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces geometric Banach property (T_B) for a bounded-geometry monogenic metric space X as the existence of a uniform spectral gap for all contractive representations of the algebraic uniform Roe algebra Cu[X] on spaces in a uniformly convex family B. It establishes an equivalence with the existence of a Kazhdan projection in the maximal B-Roe algebra (Theorem 4.7), proves that the property is a coarse invariant (Theorem 7.1), and studies the relation with Banach property (T) of limit groups and box spaces of residually finite groups (Theorems 5.9 and 5.14). It also shows that the coarse fixed point property implies geometric Banach property (T) but not conversely (Theorem 6.4 and Example 6.6), and gives an L^p lifting result from Hilbert geometric property (T) (Theorem 6.8).","tokens_in":44740,"tokens_out":12068,"duration_ms":112020,"significance":"The framework is a natural and timely unification of Banach property (T) for groups and geometric property (T) for metric spaces, and the route through Banach representations of Roe algebras is well chosen. The paper is largely self-contained, supplies a functional-calculus construction of Kazhdan projections (Lemma 4.2), and states falsifiable equivalences for box spaces. If the missing quotient-norm estimates are supplied, the results would provide the first geometric characterization of strong Banach property (T) and would transfer L^p property (T) to box spaces. The main advertised theorems are not yet proven as written, but the gaps are localized and appear repairable within the paper's framework.","major_comments":[{"comment":"The proof asserts that the constant inclusion I:B→L^p(μ_φ,B) 'induces an isometrical embedding by definition' \\(\\hat I: B/B_\\rho \\to L^p(\\mu_\\varphi,B)/L^p(\\mu_\\varphi,B)_{\\pi\\circ\\iota}\\). This is not automatic and is not proved. One must show that \\((1\\otimes B)\\cap L^p(\\mu_\\varphi,B)_{\\pi\\circ\\iota} = 1\\otimes B_\\rho\\) and that for every \\([\\xi]\\in B/B_\\rho\\) the quotient norm of \\([1\\otimes \\xi]\\) is equal to (or at least bounded below by a constant independent of ρ and ξ) the quotient norm \\(\\|[\\xi]\\|\\). Without such an estimate, an almost-invariant vector in \\(B_\\rho\\) need not produce an almost-invariant vector in the L^p quotient, so the contradiction in Proposition 5.5 and the implication (3)⇒(1) in Theorem 5.9 do not follow. The same missing estimate is used in the claim that the inclusion \\(C\\Gamma \\to C_{\\delta_\\ell,s,c}(X)\\) is an isometry in the proof of Theorem 5.14.","section":"§5.2, Proposition 5.5"},{"comment":"The statement that χ_X 'descends to an operator from B_X/B^{π_X}_X onto B_Y/B^{π_Y}_Y' has the domain and codomain reversed, and, more importantly, no lower bound for the induced map \\(B_Y/B^{\\pi_Y}_Y \\to B_X/B^{\\pi_X}_X\\) is proved. The natural restriction map can collapse the quotient: for X={0}, Y={0,1}, B=C^2, and π_Y the standard representation of M_2(C), one has \\(B^{\\pi_Y}=\\mathrm{span}\\{(1,1)\\}\\) and \\(B^{\\pi_X}=\\mathrm{span}\\{e_0\\}\\), so the induced map on quotients is zero. Hence the proof of the (⇒) direction of Theorem 7.1 cannot transfer an almost-invariant vector from Y to X; a fiber-averaging or isometric-block argument is required and is absent.","section":"§7, Lemma 7.2 and Corollary 7.4"},{"comment":"The proof uses the same unproved quotient identification in the sentence 'H_s is isomorphic to the quotient space (1⊗H_s + L^2(μ_φ,H_s)_{π_s})/L^2(μ_φ,H_s)_{π_s}'. The π_s-invariant subspace need not intersect 1⊗H_s exactly in 1⊗Fix(ρ_s); even if it does, the quotient norm of a constant vector can be much smaller than its norm in H_s. Since the contradiction with geometric property (T) of X relies on [1⊗ι_s(ξ)] having norm uniformly bounded below, this step is load-bearing. In this Hilbertian setting an orthogonal projection argument should repair it, but it is not supplied.","section":"§6.2, Theorem 6.7"}],"minor_comments":[{"comment":"There are numerous typos (e.g., 'represtation', 'annhilator', 'serveral', 'relavant', 'coordinay', 'wiht') that should be corrected in a revised version.","section":"Throughout"},{"comment":"The notation Bπ is used both for the invariant subspace and for the annihilator complement, which makes the proof hard to follow; please introduce B_π and B^π and use them consistently.","section":"§3, Lemma 3.4"},{"comment":"The sentence introducing the action of Γ∞ω on L^p(μ_φ,B) should explicitly recall that μ_φ is Γ∞ω-invariant, since the equivariance of the constant inclusion depends on it.","section":"§5.2, Proposition 5.5"},{"comment":"The conclusion that the orthogonal representation of Z_{2n+1} on R is trivial should cite the fact that 2n+1 is odd, so there is no nontrivial homomorphism to Z_2.","section":"§6.1, Example 6.6"},{"comment":"The statement that the Mazur map conjugation 'can extend to an isometric linear map' should refer explicitly to the Banach-Lamperti theorem at the point where linearity is used, rather than only after the statement.","section":"§6.3, Theorem 6.8"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely problem and the overall strategy is credible. The single recurring gap—the isometric embedding of quotient spaces by invariant subspaces under the constant inclusion into L^p(μ,B)—appears in Proposition 5.5, Theorem 5.14, Theorem 6.7, and, in a different form, in Lemma 7.2. I believe it is repairable, but it is load-bearing for most of the advertised applications, so the revision should be substantive rather than cosmetic. I also suggest the authors add a small lemma that isolates this quotient-norm estimate once and for all."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this paper is the first to define geometric Banach property (T) for metric spaces via Banach representations of Roe algebras, and it proves the expected equivalences: Kazhdan projection existence, coarse invariance for uniformly convex families, and a box-space characterization of Banach property (T_{L^p}) and of Lafforgue's strong Banach property (T). That is a real and useful step, and the architecture is sensible. Second, the proofs are not all at the same standard. The most exposed spot is the claim in Proposition 5.5 and Theorem 7.1 that certain inclusion or range maps become isometric, or at least bounded below, on the relevant quotient spaces. No argument is supplied, and in the coarse-invariance setting the natural restriction map can have zero quotient norm. The stress-test example with X = {0}, Y = {0,1}, and B = C^2 shows the mechanism in Theorem 7.1 cannot transfer an almost-invariant vector from Y to X as written. A fiber-averaging or isometric-block argument is needed, and it is absent. The same type of gap appears in Proposition 5.5, where the quotient embedding B/B_rho into L^p(mu,B)/L^p(mu,B)_pi is asserted but not proven; this is load-bearing for the limit-group implication and for Theorem 6.7.\n\nWhat is genuinely new: the definition of geometric Banach property (T_B) and its uniform and strong variants, the coarse invariance theorem, the Kazhdan projection characterization in maximal Banach-Roe algebras, and the box-space results. The Laplacian functional calculus in Lemma 4.2 is a real construction, not a fitted parameter. The paper builds honestly on Willett-Yu, Vergara, and BFGM, and the self-citations to earlier work are appropriate. The abstract does elide the uniformity requirement: Theorem 4.5 concerns uniform geometric property (T_B), while the abstract says just geometric property (T_B). That is a minor sloppiness, not a fatal flaw.\n\nWho is this for? Researchers working on coarse geometry, Banach property (T), or the L^p-coarse Baum-Connes conjecture will want to engage with it. The big-picture claims are plausible and likely correct, but I cannot certify the proofs at the flagged points. The appropriate outcome is peer review with the expectation of revisions: the authors should supply the missing quotient-norm estimates and either fix or explicitly reformulate Theorem 7.1. I would send it out for serious refereeing.","headline":"A credible and substantial Banach-space extension of geometric property (T), but the coarse-invariance and limit-group transfer steps have proof gaps that need real work before it is fully verified.","tokens_in":45333,"tokens_out":2371,"would_cite":true,"duration_ms":29787,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46B20","46L05","20F65","22D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces a Banach-space version of geometric property (T) for metric spaces and proves it is a coarse invariant, equivalent to the existence of Kazhdan projections in maximal Banach-Roe algebras and to the $L^p$-rigidity of…","keywords":["geometric Banach property (T)","Roe algebra","Kazhdan projection","box spaces","residually finite groups","Lp property (T)","coarse fixed point property","uniformly convex Banach spaces"],"falsifier":"Choose the constant sequence of finite groups $Z/3$, so the limit group is $Z/3$, and let $\\rho$ be the trivial representation of $Z/3$ on $R$. Let $\\varphi$ be an invariant mean on the boundary of the box space and compute the quotient norm of the constant function $1$ in $L^p(\\mu_\\varphi,R)$ modulo its invariant subspace for $p=3$; if this norm is smaller than the quotient norm of $1$ in $R$ modulo the invariant vectors of $\\rho$, the natural inclusion does not preserve quotient distances, and the proof of Proposition 5.5 breaks at that step.","tokens_in":44214,"feed_emoji":"📐","tokens_out":13185,"duration_ms":126020,"temperature":0.7,"pith_summary":"This paper defines a coarse-geometric version of Banach property (T) for metric spaces, based on representations of the algebraic uniform Roe algebra on uniformly convex Banach spaces. When the space is a Hilbert space, the definition recovers the classical geometric property (T), and when the space is a box space of a residually finite group (a disjoint union of finite quotients), it recovers Banach property (T) of the group. The paper proves that the new property is a coarse invariant and that it is equivalent to the existence of a Kazhdan projection in the maximal Banach-Roe algebra $C_{B,\\max}(X)$. For $L^p$-spaces it shows that a residually finite group has property $(T_{L^p})$ exactly when its box spaces have geometric property $(T_{L^p})$, and likewise for strong Banach property (T). A reader should care because this makes the rigidity of group representations and the coarse rigidity of metric spaces the same phenomenon, with finite quotients of a group certifying its Banach rigidity.","feed_headline":"Banach property (T) becomes a coarse invariant","feed_subtitle":"Metric spaces inherit a Kazhdan-type spectral gap, and boxes of residually finite groups certify it.","key_machinery":"The load-bearing mechanism is the invariant-subspace decomposition for Banach representations of the Roe algebra. For a uniformly convex Banach space $B$, a representation $\\pi$ of $C_u[X]$ gives $B \\cong B_\\pi \\oplus B^\\pi$, where $B_\\pi$ is the invariant subspace and $B^\\pi$ is the annihilator of the invariant subspace of the dual representation; this replaces the orthogonal complement that exists in Hilbert spaces. The proof then builds a Laplacian $A = \\frac{1}{n}\\sum_{i=1}^{n}\\frac{1+A_i}{2}$ from full partial translations $A_i$ that generate the coarse structure, and uses uniform convexity to show that a spectral gap for $\\pi$ is equivalent to $\\|\\pi(A)|_{B^\\pi}\\|<1$, so that the powers $A^k$ converge at a summable rate to an idempotent $p$, the Kazhdan projection, which acts as the projection onto $B_\\pi$ in every representation.","core_discovery":"The central discovery is that the right notion of geometric Banach property (T) is a spectral gap for representations of the Roe algebra on uniformly convex Banach spaces: a metric space has the property when every such representation has no approximately invariant vectors after quotienting out the invariant subspace. In a uniformly convex setting the quotient argument works because every representation admits a complemented decomposition into invariant and annihilator parts, and the paper shows that this is enough to obtain a Kazhdan projection in the maximal Banach-Roe algebra $C_{B,\\max}(X)$. The coarse invariance theorem states that if $X$ and $Y$ are coarsely equivalent and the family $B$ is uniformly convex and closed under subspaces and finite direct sums, then $X$ has geometric property $(T_B)$ if and only if $Y$ does. Applied to box spaces of a residually finite group, the paper derives equivalences between Banach property $(T_{L^p})$ of the group and geometric property $(T_{L^p})$ of its box spaces, and a similar characterization of strong Banach property (T).","pith_inferences":["Beyond the paper, the box-space equivalence suggests that Banach property (T_{L^p}) is a finite-quotient phenomenon: verifying a spectral gap on one filtration could certify the property for a group.","Beyond the paper, the Example 6.6 separation of geometric property (T_B) from the coarse fixed point property indicates that the geometric notion records representation rigidity while the coarse fixed point property records affine isometric rigidity; testing this separation in super-reflexive spaces would clarify the boundary between the two notions.","Beyond the paper, the norm-preserving map between L^p and L^2 spheres used in the proof of Theorem 6.8 is a metric-space analogue of the group-level implication property (T) implies property (T_{L^p}); a natural transfer would extend this implication to noncommutative L^p spaces."],"forward_implications":["Geometric Banach property (T_B) is a coarse invariant: coarsely equivalent metric spaces have it or fail it together, for any uniformly convex family B closed under subspaces and finite direct sums.","For every p in (1,∞), a residually finite group has property (T_{L^p}) if and only if every box space, equivalently some box space, has geometric property (T_{L^p}).","Strong Banach property (T) of a residually finite group is characterized by geometric strong Banach property (T) of its box spaces, so finite quotients can certify this stronger rigidity.","A metric space with geometric Hilbert property (T) automatically has uniform geometric property (T_{L^p}) for every p in (1,∞) different from 2, via the coarse fixed point result and duality.","Sequences of finite group extensions with an FCE-by-FCE structure cannot have geometric property (T) unless the sequence is uniformly bounded, so fibred coarse embeddability is incompatible with this rigidity."],"supporting_citations":[{"why":"Introduced geometric property (T) for metric spaces and its Roe algebra representation framework, which this paper generalizes to Banach spaces.","marker":"[WY14]"},{"why":"Introduced Banach property (T) for groups and the complemented-representation structure for uniformly convex spaces used in Lemma 3.4.","marker":"[BFGM07]"},{"why":"Gave the Kazhdan-projection characterization of uniform Banach property (T) for groups, the model for Theorem 4.7.","marker":"[DN19]"},{"why":"Established the Kazhdan-projection characterization of geometric property (T) and supplied the decomposition of partial translations into full ones used throughout.","marker":"[Ver24]"},{"why":"Developed the ultraproduct and limit-group machinery that turns a box space into its limit group in Section 5.","marker":"[GQW24]"},{"why":"Introduced the coarse fixed point property used in Section 6 to relate geometric property (T) to Banach coarse fixed point properties.","marker":"[TW22]"},{"why":"Introduced strong Banach property (T) and its Kazhdan projection formulation, which the paper carries over to box spaces.","marker":"[Laf08]"}],"fun_headline_variants":["Geometric Banach (T) is a coarse invariant","Roe algebras yield spectral gap for metric spaces","Kazhdan projections characterize geometric Banach (T)","Metric spaces get property (T) from Roe algebras","Box spaces bridge group and metric Banach (T)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For every representation of a group on a Banach space and its induced representation on the $L^p$-space of boundary functions, the box-space transfer assumes that factoring out invariant vectors in the larger space does not shrink distances compared with factoring out invariant vectors in the original space.","fun_headline_variants_meta":{"raw":{"variants":["Geometric Banach (T) is a coarse invariant","Roe algebras yield spectral gap for metric spaces","Kazhdan projections characterize geometric Banach (T)","Metric spaces get property (T) from Roe algebras","Box spaces bridge group and metric Banach (T)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000506,"raw_usage":{"total_tokens":2469,"prompt_tokens":946,"completion_tokens":1523,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":1446}},"tokens_in":562,"tokens_out":1523,"duration_ms":11471,"temperature":1.0,"reasoning_tokens":1446,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:58:16.277053+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose the constant sequence of finite groups $Z/3$, so the limit group is $Z/3$, and let $\\rho$ be the trivial representation of $Z/3$ on $R$. Let $\\varphi$ be an invariant mean on the boundary of the box space and compute the quotient norm of the constant function $1$ in $L^p(\\mu_\\varphi,R)$ modulo its invariant subspace for $p=3$; if this norm is smaller than the quotient norm of $1$ in $R$ modulo the invariant vectors of $\\rho$, the natural inclusion does not preserve quotient distances, and the proof of Proposition 5.5 breaks at that step.","supporting_citations":[],"review_version":1}