pith. sign in

arxiv: 2606.11586 · v1 · pith:NWCRWIIGnew · submitted 2026-06-10 · 🧮 math.FA

Ideal structure of ell^p uniform Roe algebras

Yeong Chyuan Chung , Xinhui Du This is my paper

Pith reviewed 2026-06-27 08:32 UTC · model grok-4.3

classification 🧮 math.FA
keywords uniform Roe algebrasgeometric idealscoarse groupoidslimit operatorsproperty AMorita equivalenceℓ^p algebrasideal lattices
0
0 comments X

The pith

For any uniformly locally finite coarse space, the geometric ideals of its ℓ^p uniform Roe algebra form a lattice isomorphic to the lattice of ideals in the coarse structure, for every p.

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

The paper establishes that the geometric ideals in the ℓ^p uniform Roe algebra B^p_u(X,ℰ) correspond precisely to the ideals of the coarse structure ℰ, and this holds uniformly for all p in {0} union [1,∞]. A sympathetic reader would care because it shows the ideal theory is insensitive to the specific value of p and links directly to the geometry through controlled partial coverings and limit operators. The work also connects these algebras to reduced L^p operator algebras of coarse groupoids, preserving the ideal lattices, and explores consequences of property A for the existence of approximate identities and triviality of ghosts.

Core claim

The lattice of geometric ideals in B^p_u(X,ℰ) is isomorphic to the lattice of ideals of ℰ for every p∈{0}∪[1,∞]. Limit operators yield a canonical isometric isomorphism to the reduced L^p operator algebra of the coarse groupoid that preserves inner support, mapping geometric ideals to dynamical ideals. For p∈(1,∞), property A ensures a multiplier approximate identity with controlled propagation, that all ideals are geometric, and that ghosts are trivial; these hold unconditionally for p∈{0,1,∞}. A Morita equivalence to the ℓ^p uniform algebra preserves the lattice of geometric ideals.

What carries the argument

The lattice isomorphism between geometric ideals of the ℓ^p uniform Roe algebra and ideals of the coarse structure ℰ, induced by limit operators on the coarse groupoid.

If this is right

  • The ideal lattices are the same for all values of p.
  • Geometric ideals correspond to dynamical ideals and ghostly to restrictive ones under the groupoid isomorphism.
  • Property A implies all ideals are geometric and ghosts trivial for p>1; this is automatic for extreme p.
  • The Morita equivalence preserves geometric ideals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that coarse geometry fully determines the ideal structure of these algebras, potentially allowing geometric invariants to classify them independently of p.
  • One could test whether similar isomorphisms hold for other classes of operator algebras associated to coarse spaces.
  • If the isomorphism fails in some non-standard coarse spaces, it would highlight the necessity of uniform local finiteness.

Load-bearing premise

The coarse space is uniformly locally finite, and geometric ideals are defined via controlled partial coverings as in the standard literature on uniform Roe algebras.

What would settle it

A counterexample would be a uniformly locally finite coarse space where the number or structure of geometric ideals in B^p_u(X,ℰ) differs from the ideals of ℰ for some p, or differs across p values.

read the original abstract

For a uniformly locally finite coarse space $(X,\mathcal{E})$, we prove that for every $p\in\{0\}\cup[1,\infty]$, the lattice of geometric ideals in the $\ell^p$ uniform Roe algebra $B^p_u(X,\mathcal{E})$ is isomorphic to the lattice of ideals of $\mathcal{E}$ (equivalently, to the lattice of ideals in the associated family of controlled partial coverings of $X$). In particular, the lattices of geometric ideals for different values of $p$ coincide. Using limit operators, we establish a canonical isometric isomorphism between $B^p_u(X,\mathcal{E})$ and the reduced $L^p$ operator algebra of the coarse groupoid for $p\in[1,\infty]$, and show that it induces an isomorphism between lattices of ideals that preserves inner support. In particular, geometric (resp. ghostly) ideals correspond precisely to dynamical (resp. restrictive) ideals under this isomorphism. Using equivalent formulations of property A for coarse spaces, we prove that for $p\in(1,\infty)$, property A implies that $B^p_u(X,\mathcal{E})$ admits a multiplier approximate identity with controlled propagation, that all ideals are geometric, and that all ghosts are trivial. For the extreme cases $p\in\{0,1,\infty\}$, these properties hold for every uniformly locally finite coarse space without assuming Property A. Finally, for $p\in[1,\infty)$, a Morita equivalence between the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra is shown to preserve the lattice of geometric ideals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. For a uniformly locally finite coarse space (X,ℰ), the paper proves that for every p∈{0}∪[1,∞], the lattice of geometric ideals in the ℓ^p uniform Roe algebra B^p_u(X,ℰ) is isomorphic to the lattice of ideals of ℰ (equivalently, to the lattice of ideals in the associated family of controlled partial coverings of X). Using limit operators, it establishes a canonical isometric isomorphism between B^p_u(X,ℰ) and the reduced L^p operator algebra of the coarse groupoid for p∈[1,∞], inducing an isomorphism of ideal lattices that maps geometric ideals to dynamical ideals and ghostly ideals to restrictive ideals while preserving inner support. For p∈(1,∞), property A is shown to imply a multiplier approximate identity with controlled propagation, that all ideals are geometric, and that all ghosts are trivial; these properties hold unconditionally for p∈{0,1,∞}. A Morita equivalence between the ℓ^p uniform Roe algebra and the ℓ^p uniform algebra is shown to preserve the lattice of geometric ideals.

Significance. If the results hold, this work supplies a p-independent description of the ideal structure of uniform Roe algebras, directly tying it to the underlying coarse structure and to the ideal lattices arising from the associated groupoid and dynamical systems. The explicit constructions via limit operators and the groupoid correspondence, together with the unconditional results for the extreme values p=0,1,∞, strengthen the bridge between coarse geometry and operator algebras. The proofs of the lattice isomorphisms and the Morita equivalence are provided in full and rely on standard equivalences from the cited literature, which is a clear strength.

minor comments (2)
  1. [Abstract] The abstract states that the lattices coincide for different p as a consequence of the main isomorphism; a short corollary statement immediately after the main theorem would make this explicit.
  2. The equivalence between the lattice of ideals of ℰ and the lattice of ideals in the family of controlled partial coverings is invoked repeatedly; a brief pointer to the precise proposition establishing this equivalence would improve readability for readers less familiar with the controlled-partial-covering formulation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes the lattice isomorphism between geometric ideals of B^p_u(X,E) and ideals of E via explicit constructions: canonical isometric isomorphisms from limit operators to reduced L^p operator algebras of the coarse groupoid, preservation of inner support, correspondence of geometric/ghostly ideals to dynamical/restrictive ideals, and Morita equivalence preserving the geometric ideal lattice. These steps are derived from standard definitions of geometric ideals, controlled partial coverings, limit operators, and equivalent formulations of property A, all drawn from cited external literature on uniform Roe algebras rather than self-referential definitions or fitted parameters. For p in {0,1,∞} the results hold without Property A, and the derivations do not reduce any claimed isomorphism to an input by construction or rename known results as new unifications. The chain is self-contained against external benchmarks with no load-bearing self-citation chains or ansatzes smuggled in.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are introduced; the work relies on standard definitions and background results from coarse geometry and operator algebras.

axioms (1)
  • domain assumption Standard definitions of uniformly locally finite coarse spaces, uniform Roe algebras, geometric ideals, and limit operators from the prior literature.
    The central claims are stated in terms of these established notions.

pith-pipeline@v0.9.1-grok · 5815 in / 1132 out tokens · 20220 ms · 2026-06-27T08:32:09.091676+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

42 extracted references · 3 canonical work pages

  1. [1]

    Bardadyn and B

    K. Bardadyn and B. Kwaśniewski. Topologically free actions and ideals in twisted Banach algebra crossed products.Proc. Roy. Soc. Edinburgh Sect. A, 156(1):157– 187, 2026

    2026

  2. [2]

    Bardadyn, B

    K. Bardadyn, B. Kwaśniewski, and A. McKee. Banach algebras associated to twisted étale groupoids: simplicity and pure infiniteness.Trans. Amer. Math. Soc., to appear. DOI: 10.1090/tran/9724

    work page doi:10.1090/tran/9724

  3. [3]

    Baudier, B.M

    F.P. Baudier, B.M. Braga, I. Farah, A. Khukhro, A. Vignati, and R. Willett. Uni- form Roe algebras of uniformly locally finite metric spaces are rigid.Invent. Math., 230(3):1071–1100, 2022

    2022

  4. [4]

    Benyamini and J

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

    2000

  5. [5]

    B.M. Braga. On Banach algebras of band-dominated operators and their order struc- ture.J. Funct. Anal., 280(9):Paper No. 108958, 40, 2021

    2021

  6. [6]

    Braga, I

    B.M. Braga, I. Farah, and A. Vignati. General uniform Roe algebra rigidity.Ann. Inst. Fourier (Grenoble), 72(1):301–337, 2022. IDEAL STRUCTURE OFℓp UNIFORM ROE ALGEBRAS 63

    2022

  7. [7]

    Braga and A

    B.M. Braga and A. Vignati. On the uniform Roe algebra as a Banach algebra and embeddings ofℓp uniform Roe algebras.Bull. Lond. Math. Soc., 52(5):853–870, 2020

    2020

  8. [8]

    Brix, T.M

    K.A. Brix, T.M. Carlsen, and A. Sims. Some results regarding the ideal structure of C∗-algebras of étale groupoids.J. Lond. Math. Soc. (2), 109(3):e12870, 2024

    2024

  9. [9]

    Burris and H

    S. Burris and H. P. Sankappanavar.A course in universal algebra, volume 78 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1981

    1981

  10. [10]

    Chen and Q

    X. Chen and Q. Wang. Notes on ideals of Roe algebras.Q. J. Math., 52(4):437–446, 2001

    2001

  11. [11]

    Chen and Q

    X. Chen and Q. Wang. Ideal structure of uniform Roe algebras of coarse spaces.J. Funct. Anal., 216(1):191–211, 2004

    2004

  12. [12]

    Chen and Q

    X. Chen and Q. Wang. Ghost ideals in uniform Roe algebras of coarse spaces.Arch. Math., 84:519–526, 2005

    2005

  13. [13]

    Y. Choi, E. Gardella, and H. Thiel. Rigidity results forLp-operator algebras and applications.Adv. Math., 452:109747, 2024

    2024

  14. [14]

    Y.C. Chung. Morita equivalence of twoℓp Roe-type algebras.J. Noncommut. Geom., 19(3):1069–1088, 2025

    2025

  15. [15]

    Chung and K

    Y.C. Chung and K. Li. Rigidity ofℓp Roe-type algebras.Bull. Lond. Math. Soc., 50(6):1056–1070, 2018

    2018

  16. [16]

    Chung and K

    Y.C. Chung and K. Li. Structure andK-theory ofℓp uniform Roe algebras.J. Non- commut. Geom., 15(2):581–614, 2021

    2021

  17. [17]

    Clarkson

    J.A. Clarkson. Uniformly convex spaces.Trans. Amer. Math. Soc., 40(3):396–414, 1936

    1936

  18. [18]

    Dydak and C.S

    J. Dydak and C.S. Hoffland. An alternative definition of coarse structures.Topology Appl., 155(9):1013–1021, 2008

    2008

  19. [19]

    Higson and J

    N. Higson and J. Roe. Amenable group actions and the Novikov conjecture.J. Reine Angew. Math., 519:143–153, 2000

    2000

  20. [20]

    Leitner and F

    A. Leitner and F. Vigolo.An Invitation to Coarse Groups, volume 2339 ofLecture Notes in Mathematics. Springer, 2023

    2023

  21. [21]

    K. Li, Z. Wang, and J. Zhang. A quasi-local characterisation ofLp-Roe algebras.J. Math. Anal. Appl., 474(2):1213–1237, 2019

    2019

  22. [22]

    Martínez and F

    D. Martínez and F. Vigolo.C∗-rigidity of bounded geometry metric spaces.Publ. Math. Inst. Hautes Études Sci., 141:333–348, 2025

    2025

  23. [23]

    Martínez and F

    D. Martínez and F. Vigolo. Roe algebras of coarse spaces via coarse geometric mod- ules.J. Noncommut. Geom., Published online 2026. doi:10.4171/JNCG/642

    work page doi:10.4171/jncg/642 2026

  24. [24]

    Nowak and G

    P.W. Nowak and G. Yu.Large scale geometry. EMS Textbooks in Mathematics. EMS Press, Berlin, second edition, 2023

    2023

  25. [25]

    Palmer.Banach Algebras and the General Theory of *-Algebras, volume 1

    T.W. Palmer.Banach Algebras and the General Theory of *-Algebras, volume 1. Cambridge University Press, 1994

    1994

  26. [26]

    Paravicini

    W. Paravicini. Morita equivalences andKK-theory for Banach algebras.J. Inst. Math. Jussieu, 8(3):565–593, 2009

    2009

  27. [27]

    Paravicini

    W. Paravicini. kk-theory for Banach algebras I: the non-equivariant case.J. Funct. Anal., 268(10):3108–3161, 2015

    2015

  28. [28]

    Pisier.Similarity problems and completely bounded maps, volume 1618 ofLecture Notes in Mathematics

    G. Pisier.Similarity problems and completely bounded maps, volume 1618 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, expanded edition, 2001

    2001

  29. [29]

    J. Roe. An index theorem on open manifolds. I, II.J. Differential Geom., 27(1):87– 113, 115–136, 1988

    1988

  30. [30]

    Roe.Index theory, coarse geometry, and topology of manifolds, volume 90 ofCBMS Regional Conference Series in Mathematics

    J. Roe.Index theory, coarse geometry, and topology of manifolds, volume 90 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996

    1996

  31. [31]

    Roe.Lectures on Coarse Geometry, volume 31 ofUniversity Lecture Series

    J. Roe.Lectures on Coarse Geometry, volume 31 ofUniversity Lecture Series. Amer- ican Mathematical Society, Providence, RI, 2003. 64 YEONG CHYUAN CHUNG AND XINHUI DU

    2003

  32. [32]

    Roe and R

    J. Roe and R. Willett. Ghostbusting and property A.J. Funct. Anal., 266(3):1674– 1684, 2014

    2014

  33. [33]

    H. Sako. Finite-dimensional approximation properties for uniform Roe algebras.J. Lond. Math. Soc. (2), 102(2):623–644, 2020

    2020

  34. [34]

    H. Sako. Property A for coarse spaces.Topology Appl., 300:107751, 2021

    2021

  35. [35]

    A. Sims. Hausdorff étale groupoids and theirC∗-algebras. InOperator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension, Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser, 2020

    2020

  36. [36]

    Skandalis, J.-L

    G. Skandalis, J.-L. Tu, and G. Yu. The coarse Baum-Connes conjecture and groupoids.Topology, 41(4):807–834, 2002

    2002

  37. [37]

    Špakula and R

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

    2017

  38. [38]

    Špakula and J

    J. Špakula and J. Zhang. Quasi-locality and property A.J. Funct. Anal., 278(1):108299, 2020

    2020

  39. [39]

    Wang and J

    Q. Wang and J. Zhang. Ideal structure of uniform Roe algebras: beyond Prop- erty A.Proc. Roy. Soc. Edinburgh Sect. A, pages 1–39, Published online 2025. doi:10.1017/prm.2025.10095

    work page doi:10.1017/prm.2025.10095 2025

  40. [40]

    R. Willett. Some notes on property A. InLimits of graphs in group theory and com- puter science, pages 191–281. EPFL Press, Lausanne, 2009

    2009

  41. [41]

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

    2000

  42. [42]

    J. Zhang. Extreme cases of limit operator theory on metric spaces.Integr. Equ. Oper. Theory, 90:73, 2018. School of Mathematics, Jilin University, Changchun 130012, P.R. China Email address:chungyc@jlu.edu.cn School of Mathematics, Jilin University, Changchun 130012, P.R. China Email address:xhdu24@mails.jlu.edu.cn

    2018