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Geometric Banach property (T) for metric spaces via Banach representations of Roe algebras

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2505.02338 v2 pith:VT2M3FVZ submitted 2025-05-05 math.FA math.MGmath.OA

Liang Guo , Qin Wang This is my paper · ORCID
classification math.FAmath.MGmath.OA MSC 46B2046L0520F6522D10
keywords geometricBanachproperty(T)RoealgebraKazhdanprojectionboxspacesresiduallyfinitegroupsLpcoarsefixedpointuniformlyconvex
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 5 minor

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).

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 (3)
  1. [§5.2, Proposition 5.5] 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.
  2. [§7, Lemma 7.2 and Corollary 7.4] 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.
  3. [§6.2, Theorem 6.7] 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.
minor comments (5)
  1. [Throughout] There are numerous typos (e.g., 'represtation', 'annhilator', 'serveral', 'relavant', 'coordinay', 'wiht') that should be corrected in a revised version.
  2. [§3, Lemma 3.4] 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.
  3. [§5.2, Proposition 5.5] 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.
  4. [§6.1, Example 6.6] 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.
  5. [§6.3, Theorem 6.8] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the paper's main equivalences are proved mutually from the definitions, and self-citations are background rather than load-bearing reductions.

full rationale

The central derivation is self-contained on its main axes. Geometric Banach property (T_B) is defined through a spectral-gap condition on quotients B/B_π (Definition 2.3), and Lemma 4.2 proves the equivalence between a spectral gap, the bound ||π(A)|_{B^π}||<1, and the summable convergence of π(A^k) to the projection p_π. These implications are argued from uniform convexity and Lemma 3.4, not asserted as definitions of each other. The Kazhdan projection in Theorem 4.5 is constructed as the limit of A^k in the maximal B-Roe algebra, and both directions of the equivalence with uniform geometric property (T_B) are proved using the convergence of A^k, so the 'prediction' is not fitted or renamed from the conclusion. The transfer in Proposition 5.5 from geometric property of a disjoint union to Banach property (T) of a limit group is obtained by building an induced representation on L^p(μ_φ,B) and embedding B as constant functions; the proof does contain a delicate quotient-norm identification that is not fully justified, but this is a mathematical gap rather than a circular reduction, since the spectral gap of the induced representation is not assumed to equal the spectral gap of ρ by construction. Similarly, the coarse-invariance proof in Theorem 7.1 transfers invariant subspaces via χ_X and the diagonal embedding; the missing lower bound on the induced quotient maps is a correctness concern, not an instance of the conclusion being used as an input. The paper also cites the authors' prior works [GQW24] and [DGWY25] for limit-space theory and FCE-by-FCE structure; these are genuine background results with stated assumptions that do not include the present theorem, and the argument does not reduce its central claims to a self-citation chain or to an imported uniqueness assertion. Accordingly, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted parameters: the constant N from bounded geometry and the decompositions {A_i} from Lemma 3.1 are structural choices, not values fitted to data. The axioms are domain assumptions about the metric space and the Banach family, plus standard functional analysis facts (open mapping theorem, Hahn-Banach, Banach-Lamperti, duality map continuity) and one ad hoc technical premise (the quotient embedding of Proposition 5.5). The invented-entity ledger is empty because the paper introduces no new particles, forces, or mediators; the objects it defines (geometric Banach property (T), Kazhdan projections in Banach-Roe algebras) are definitions and constructions, not postulated entities.

assumptions (7)
  • domain assumption X is a countable, monogenic, extended metric space with bounded geometry, with a fixed symmetric generating entourage E_0.
    Stated at the start of Section 2 and used throughout: the Roe algebra norms, the partial translation decomposition (Lemma 3.1), and the definitions of the relevant properties all depend on bounded geometry and monogenicity.
  • domain assumption Representations of Cu[X] are contractive and unital, so full partial translations act as surjective isometries.
    Section 2, after Lemma 2.2; the norm estimates in Lemmas 2.5 and 4.2 and the comparison of spectral gaps depend on this convention.
  • domain assumption B is a uniformly convex family of Banach spaces, closed under the operations each theorem requires: subspaces and finite direct sums (Theorem 7.1), ultraproducts (Theorem 4.7), duality, conjugation, ultraproducts and L2-Bochner tensor products (Theorem 5.14).
    Assumed in Sections 3, 4, 5.4 and 7; uniform spectral gap bounds and the idempotence of Kazhdan projections rely on uniform convexity and these closures.
  • standard math The duality map from the unit sphere of a uniformly convex space to the dual sphere is uniformly continuous with modulus depending only on the convexity modulus, and the same holds equi-uniformly over a uniformly convex family.
    Invoked in Lemma 3.4(3) and Theorem 3.5 via [BL00, Prop. A.4/A.5]; it supplies the uniform lower bounds that replace orthogonal complement constants.
  • standard math Every isometric representation of a group on a uniformly convex Banach space is complemented in the sense of [BFGM07, Prop. 2.10], and the analogous direct sum decomposition B ≅ B_π ⊕ B^π holds for Roe algebra representations (Lemma 3.4(1)).
    Section 3; this decomposition substitutes for the orthogonal decomposition that is unavailable in Banach spaces.
  • domain assumption Limit space facts from the authors' prior work [GQW24]: limit spaces of unit afar elements are finitely generated groups, and all limit groups of a box space of Γ are Γ.
    Section 5.1 through 5.3; this is a substantive dependency on an arXiv preprint by the same authors, though the dependency is not circular since the facts are external to the Banach property (T) framework.
  • ad hoc to paper For a representation ρ of the limit group on B, the induced Roe algebra representation on L^p(μ_φ, B) has invariant subspace meeting the constants 1⊗B exactly in 1⊗B_ρ, with comparable quotient norms, the 'isometrical embedding' asserted in Proposition 5.5.
    Asserted in the proof of Proposition 5.5 without proof and relied on in the transfer of spectral gaps in Theorem 6.7; if it fails, the limit-group and fixed-point consequences weaken.

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Pith. "Pith review of Geometric Banach property (T) for metric spaces via Banach representations of Roe algebras." pith.science (2026). https://pith.science/paper/VT2M3FVZ

@misc{pith2026250502338,
  author       = {Pith},
  title        = {Pith review of: Geometric Banach property (T) for metric spaces via Banach representations of Roe algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VT2M3FVZ}},
  note         = {Machine review of arXiv:2505.02338}
}
read the original abstract

In this paper, we introduce a notion of geometric Banach property (T) for metric spaces, which jointly generalizes Banach property (T) for groups and geometric property (T) for metric spaces. Our framework is achieved by Banach representations of Roe algebras of metric spaces. We show that geometric Banach property (T) is a coarse geometric invariant, and it is equivalent to the existence of the Kazhdan projections in the Banach-Roe algebras. Further, we study the implications of this property for sequences of finite Cayley graphs, establishing two key results: 1. geometric Banach property (T) of such sequences implies Banach property (T) for their limit groups; 2. while the Banach coarse fixed point property implies geometric Banach property (T), the converse fails. Additionally, we provide a geometric characterization of V. Lafforgue's strong Banach property (T) for a residually finite group in terms of geometric Banach property (T) of its box spaces.

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Works this paper leans on

56 extracted references · 51 canonical work pages

  1. [1]

    C. A. Akemann and M. E. Walter. Unbounded negative definite functions. Canadian J. Math. , 33(4):862--871, 1981

    work page 1981

  2. [2]

    Bretagnolle, D

    J. Bretagnolle, D. Dacunha-Castelle , and J.-L. Krivine. Fonctions de type positif sur les espaces L p . C. R. Acad. Sci. Paris , 261:2153--2156, 1965

    work page 1965

  3. [3]

    Bader, A

    U. Bader, A. Furman, T. Gelander, and N. Monod. Property ( T ) and rigidity for actions on B anach spaces. Acta Math. , 198(1):57--105, 2007

    work page 2007

  4. [4]

    Bekka, P

    B. Bekka, P. de la Harpe, and A. Valette. Kazhdan's property ( T ) , volume 11 of New Mathematical Monographs . Cambridge University Press, Cambridge, 2008

    work page 2008

  5. [5]

    Beurling and A

    A. Beurling and A. E. Livingston. A theorem on duality mappings in B anach spaces. Ark. Mat. , 4:405--411, 1962

    work page 1962

  6. [6]

    Benyamini and J

    Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. V ol. 1 , volume 48 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2000

    work page 2000

  7. [7]

    Y. C. Chung and P. W. Nowak. Expanders are counterexamples to the ^p coarse B aum- C onnes conjecture. J. Noncommut. Geom. , 17(1):305--331, 2023

    work page 2023

  8. [8]

    X. Chen, Q. Wang, and X. Wang. Characterization of the H aagerup property by fibred coarse embedding into H ilbert space. Bull. Lond. Math. Soc. , 45(5):1091--1099, 2013

    work page 2013

Show all 56 references
  1. [9]

    X. Chen, Q. Wang, and G. Yu. The maximal coarse B aum- C onnes conjecture for spaces which admit a fibred coarse embedding into H ilbert space. Adv. Math. , 249:88--130, 2013

    2013

  2. [10]

    M. M. Day. Some more uniformly convex spaces. Bull. Amer. Math. Soc. , 47:504--507, 1941

    1941

  3. [11]

    de Laat and M

    T. de Laat and M. de la Salle . Actions of higher rank groups on uniformly convex Banach spaces . arXiv e-prints , page arXiv:2303.01405, March 2023

    2023 arXiv

  4. [12]

    P. Delorme. 1 -cohomologie des repr\'esentations unitaires des groupes de L ie semi-simples et r\'esolubles. P roduits tensoriels continus de repr\'esentations. Bull. Soc. Math. France , 105(3):281--336, 1977

    1977

  5. [13]

    J. Deng, L. Guo, Q. Wang, and G. Yu. Higher index theory for spaces with an FCE-by-FCE structure. J. Funct. Anal. , 288(1):110679, 2025

    2025

  6. [14]

    Dacunha-Castelle and J.-L

    D. Dacunha-Castelle and J.-L. Krivine. Ultraproduits d'espaces d' O rlicz et applications g\'eom\'etriques. C. R. Acad. Sci. Paris S\'er. A-B , 271:A987--A989, 1970

    1970

  7. [15]

    de la Salle

    M. de la Salle. Towards strong B anach property ( T ) for SL (3, R ) . Israel J. Math. , 211(1):105--145, 2016

    2016

  8. [16]

    de la Salle

    M. de la Salle. Strong property (T) for higher-rank lattices. Acta Math. , 223(1):151--193, 2019

    2019

  9. [17]

    Dru t u and P

    C. Dru t u and P. W. Nowak. Kazhdan projections, random walks and ergodic theorems. J. Reine Angew. Math. , 754:49--86, 2019

    2019

  10. [18]

    Goldbring

    I. Goldbring. Ultrafilters throughout mathematics , volume 220 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, [2022] 2022

    2022

  11. [19]

    L. Guo , J. Qian , and Q. Wang . An ultraproduct approach to limit space theory . arXiv e-prints , page arXiv:2412.08130, December 2024

    2024

  12. [20]

    M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, V ol.\ 2 ( S ussex, 1991) , volume 182 of London Math. Soc. Lecture Note Ser. , pages 1--295. Cambridge Univ. Press, Cambridge, 1993

    1991

  13. [21]

    M. Gromov. Random walk in random groups. Geom. Funct. Anal. , 13(1):73--146, 2003

    2003

  14. [22]

    Guichardet

    A. Guichardet. Sur la cohomologie des groupes topologiques. II . Bull. Sci. Math. (2) , 96:305--332, 1972

    1972

  15. [23]

    L. Guo , Q. Wang , and C. Zhang . The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure . arXiv e-prints , page arXiv:2408.06660, August 2024

    2024 arXiv

  16. [24]

    Haagerup

    U. Haagerup. An example of a nonnuclear C -algebra, which has the metric approximation property. Invent. Math. , 50(3):279--293, 1978/79

    1978

  17. [25]

    Higson, V

    N. Higson, V. Lafforgue, and G. Skandalis. Counterexamples to the B aum- C onnes conjecture. Geom. Funct. Anal. , 12(2):330--354, 2002

    2002

  18. [26]

    D. A. Ka z dan. On the connection of the dual space of a group with the structure of its closed subgroups. Funkcional. Anal. i Prilo zen. , 1:71--74, 1967

    1967

  19. [27]

    Lafforgue

    V. Lafforgue. K -th\'eorie bivariante pour les alg\`ebres de B anach et conjecture de B aum- C onnes. Invent. Math. , 149(1):1--95, 2002

    2002

  20. [28]

    Lafforgue

    V. Lafforgue. Un renforcement de la propri\'et\'e ( T ). Duke Math. J. , 143(3):559--602, 2008

    2008

  21. [29]

    Lafforgue

    V. Lafforgue. Propri\'et\'e ( T ) renforc\'ee et conjecture de B aum- C onnes. In Quanta of maths , volume 11 of Clay Math. Proc. , pages 323--345. Amer. Math. Soc., Providence, RI, 2010

    2010

  22. [30]

    Lamperti

    J. Lamperti. On the isometries of certain function-spaces. Pacific J. Math. , 8:459--466, 1958

    1958

  23. [31]

    B. Liao. Strong B anach property ( T ) for simple algebraic groups of higher rank. J. Topol. Anal. , 6(1):75--105, 2014

    2014

  24. [32]

    G. A. Margulis. Explicit constructions of expanders. Problemy Pereda ci Informacii , 9(4):71--80, 1973

    1973

  25. [33]

    Marrakchi and M

    A. Marrakchi and M. de la Salle. Isometric actions on L_p -spaces: dependence on the value of p . Compos. Math. , 159(6):1300--1313, 2023

    2023

  26. [34]

    von Neumann

    J. von Neumann. Zur allgemeinen theorie des masses. Fundamenta Mathematicae , 13:73--116, 1929

    1929

  27. [35]

    B. Olivier. Kazhdan's property (T) with respect to non-commutative L_p -spaces. Proc. Amer. Math. Soc. , 140(12):4259--4269, 2012

    2012

  28. [36]

    Oppenheim

    I. Oppenheim. Banach property ( T ) for SL _n( Z) and its applications. Invent. Math. , 234(2):893--930, 2023

    2023

  29. [37]

    G. K. Pedersen. Analysis now , volume 118 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1989

    1989

  30. [38]

    N. C. Phillips and M. G. Viola. Classification of spatial L^p AF algebras. Internat. J. Math. , 31(13):2050088, 41, 2020

    2020

  31. [39]

    J. Roe. Lectures on coarse geometry , volume 31 of University Lecture Series . American Mathematical Society, Providence, RI, 2003

    2003

  32. [40]

    I. J. Schoenberg. Metric spaces and positive definite functions. Trans. Amer. Math. Soc. , 44(3):522--536, 1938

    1938

  33. [41]

    S pakula and R

    J. S pakula and R. Willett. Maximal and reduced R oe algebras of coarsely embeddable spaces. J. Reine Angew. Math. , 678:35--68, 2013

    2013

  34. [42]

    S pakula and R

    J. S pakula and R. Willett. A metric approach to limit operators. Trans. Amer. Math. Soc. , 369(1):263--308, 2017

    2017

  35. [43]

    Tessera and J

    R. Tessera and J. Winkel. Coarse fixed point properties . arXiv e-prints , page arXiv:2212.04900, December 2022

    2022 arXiv

  36. [44]

    A. Valette. Minimal projections, integrable representations and property ( T ) . Arch. Math. (Basel) , 43(5):397--406, 1984

    1984

  37. [45]

    A. Valette. Projections in full C^ -algebras of semisimple L ie groups. Math. Ann. , 294(2):277--287, 1992

    1992

  38. [46]

    I. Vergara. Geometric property ( T ) and K azhdan projections. Bull. Lond. Math. Soc. , 56(9):2935--2950, 2024

    2024

  39. [47]

    J. Winkel. Geometric property ( T ) for non-discrete spaces. J. Funct. Anal. , 281(8):Paper No. 109148, 36, 2021

    2021

  40. [48]

    J. H. Wells and L. R. Williams. Embeddings and extensions in analysis , volume Band 84 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] . Springer-Verlag, New York-Heidelberg, 1975

    1975

  41. [49]

    Wang , Z

    J. Wang , Z. Xie , G. Yu , and B. Zhu . L^p -coarse Baum--Connes conjecture for ^ q -coarse embeddable spaces . arXiv e-prints , page arXiv:2411.15070, November 2024

    2024 arXiv

  42. [50]

    Willett and G

    R. Willett and G. Yu. Higher index theory for certain expanders and G romov monster groups, I . Adv. Math. , 229(3):1380--1416, 2012

    2012

  43. [51]

    Willett and G

    R. Willett and G. Yu. Higher index theory for certain expanders and G romov monster groups, II . Adv. Math. , 229(3):1762--1803, 2012

    2012

  44. [52]

    Willett and G

    R. Willett and G. Yu. Geometric property ( T ). Chinese Ann. Math. Ser. B , 35(5):761--800, 2014

    2014

  45. [53]

    G. Yu. Localization algebras and the coarse B aum- C onnes conjecture. K -Theory , 11(4):307--318, 1997

    1997

  46. [54]

    G. Yu. The coarse B aum- C onnes conjecture for spaces which admit a uniform embedding into H ilbert space. Invent. Math. , 139(1):201--240, 2000

    2000

  47. [55]

    G. Yu. Hyperbolic groups admit proper affine isometric actions on l^p -spaces. Geom. Funct. Anal. , 15(5):1144--1151, 2005

    2005

  48. [56]

    Zhang and D

    J. Zhang and D. Zhou. L^p coarse B aum- C onnes conjecture and K -theory for L^p R oe algebras. J. Noncommut. Geom. , 15(4):1285--1322, 2021

    2021

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Reviewed August 16, 2026 · model on record in the stance chip above.