pith. sign in

arxiv: 2411.02593 · v2 · submitted 2024-11-04 · 🧮 math.FA · math.NT· math.OA· math.QA

Noncommutative geometry on the Berkovich projective line

Masoud Khalkhali , Damien Tageddine This is my paper

Pith reviewed 2026-05-23 17:18 UTC · model grok-4.3

classification 🧮 math.FA math.NTmath.OAmath.QA
keywords Berkovich projective lineC*-algebrasspectral triplesKMS statesPatterson-Sullivan measureSchottky subgrouppartial isometriesnoncommutative geometry
0
0 comments X

The pith

C*-algebras and spectral triples on the Berkovich projective line produce KMS states that recover the Patterson-Sullivan measure.

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

The paper builds C*-algebras and spectral triples for the Berkovich projective line over the p-adic numbers. In the commutative setting it forms spectral triples by taking direct limits of finite real trees. It introduces more general algebras generated by partial isometries and shows that their representations yield a Perron-Frobenius operator together with projection-valued measures. The central result is that classical invariant measures appear as KMS states on the crossed product of the algebra with a Schottky subgroup of PGL_2 over C_p.

Core claim

We construct several C*-algebras and spectral triples associated to the Berkovich projective line P^1_Berk(C_p). In the commutative setting, we construct a spectral triple as a direct limit over finite R-trees. More general C*-algebras generated by partial isometries are also presented. We use their representations to associate a Perron-Frobenius operator and a family of projection valued measures. Finally, we show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of the crossed product algebra with a Schottky subgroup of PGL_2(C_p).

What carries the argument

The crossed product C*-algebra formed with a Schottky subgroup of PGL_2(C_p), whose KMS states correspond to invariant measures on the Berkovich projective line.

If this is right

  • C*-algebras can be constructed directly from the Berkovich projective line.
  • Spectral triples arise as direct limits over finite real trees in the commutative case.
  • Representations of algebras generated by partial isometries produce Perron-Frobenius operators and projection-valued measures.
  • Invariant measures such as the Patterson-Sullivan measure are realized as KMS states on the crossed product algebra.

Where Pith is reading between the lines

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

  • This construction suggests that thermodynamic formalism from dynamical systems can be applied in the noncommutative setting for p-adic curves.
  • Similar techniques might produce KMS states for other measures on Berkovich spaces beyond Schottky groups.
  • The direct limit approach to spectral triples could generalize to higher-dimensional Berkovich varieties.

Load-bearing premise

That representations of the C*-algebras generated by partial isometries can be used to associate a Perron-Frobenius operator and projection valued measures that match the classical invariants.

What would settle it

Computing the KMS states explicitly for a specific Schottky subgroup and checking whether they coincide with the Patterson-Sullivan measure on the Berkovich projective line.

Figures

Figures reproduced from arXiv: 2411.02593 by Damien Tageddine, Masoud Khalkhali.

Figure 1
Figure 1. Figure 1: The Berkovich affine line. Picture on the right is taken from [ [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

We construct several $C^*$-algebras and spectral triples associated to the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}({\mathbb{C}_p})$. In the commutative setting, we construct a spectral triple as a direct limit over finite $\mathbb{R}$-trees. More general $C^*$-algebras generated by partial isometries are also presented. We use their representations to associate a Perron-Frobenius operator and a family of projection valued measures. Finally, we show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of the crossed product algebra with a Schottky subgroup of $\mathrm{PGL}_2(\mathbb{C}_p)$.

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 / 1 minor

Summary. The paper constructs several C*-algebras and spectral triples associated to the Berkovich projective line P^1_Berk(C_p). In the commutative setting a spectral triple is obtained as a direct limit over finite R-trees; more general C*-algebras generated by partial isometries are introduced, their representations are used to define a Perron-Frobenius operator together with a family of projection-valued measures, and invariant measures (including the Patterson-Sullivan measure) are recovered as KMS states on the crossed-product algebra by a Schottky subgroup of PGL_2(C_p).

Significance. If the constructions are valid, the work supplies a noncommutative-geometric framework for the Berkovich projective line that recovers classical invariant measures via KMS states. The direct-limit spectral triple and the passage from partial-isometry representations to Perron-Frobenius operators and projection-valued measures constitute concrete technical contributions. The explicit link to the Patterson-Sullivan measure on a Schottky crossed product is a clear strength that could connect noncommutative geometry with p-adic dynamics.

minor comments (1)
  1. Notation for the direct-limit spectral triple (e.g., the precise indexing set of the finite R-trees and the compatibility maps) could be stated more explicitly in the opening paragraphs of the relevant section to aid readers unfamiliar with Berkovich geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. There are no major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a sequence of constructions: C*-algebras and spectral triples on the Berkovich projective line (via direct limits over R-trees in the commutative case), algebras generated by partial isometries, associated Perron-Frobenius operators and projection-valued measures from their representations, and finally KMS states on crossed products by Schottky subgroups yielding invariant measures such as Patterson-Sullivan. All steps are introduced by explicit definitions and verifications internal to the C*-algebra and spectral-triple framework; no fitted parameters are renamed as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and no equation reduces by construction to an input quantity. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms of C*-algebras, spectral triples, and Berkovich geometry; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of C*-algebras and unbounded spectral triples from noncommutative geometry
    The constructions of spectral triples and crossed-product KMS states presuppose Connes-style noncommutative geometry.
  • domain assumption Existence and basic properties of the Berkovich projective line over Cp and of Schottky subgroups of PGL2(Cp)
    The space and the group action are taken as given inputs to the constructions.

pith-pipeline@v0.9.0 · 5652 in / 1428 out tokens · 46238 ms · 2026-05-23T17:18:27.323448+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    45, American Mathematical Society, Providence, Rhode Island, August 2008 (en)

    Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran Kedlaya, and Jeremy Teitelbaum,p-adic Geometry, University Lecture Series, vol. 45, American Mathematical Society, Providence, Rhode Island, August 2008 (en)

    work page 2008

  2. [2]

    Rumely,Potential theory and dynamics on the Berkovich projective line, Mathematicalsurveysandmonographs, no.v.159, AmericanMathematicalSociety, Providence, R.I, 2010

    Matthew Baker and Robert S. Rumely,Potential theory and dynamics on the Berkovich projective line, Mathematicalsurveysandmonographs, no.v.159, AmericanMathematicalSociety, Providence, R.I, 2010

    work page 2010

  3. [3]

    1, 137–165 (en)

    Iztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, and Tina Sovič,Ważewski’s uni- versal dendrite as an inverse limit with one set-valued bonding function, Glasnik Matematicki48 (2013), no. 1, 137–165 (en)

    work page 2013

  4. [4]

    de Castro, Daniel Gonçalves, and Daniel W

    Giuliano Boava, Gilles G. de Castro, Daniel Gonçalves, and Daniel W. van Wyk,C*-Algebras of one-sided subshifts over arbitrary alphabets, December 2023, arXiv:2312.17644 [math]

    work page arXiv 2023

  5. [5]

    De Castro, Daniel Gonçalves, and Daniel W

    Giuliano Boava, Gilles G. De Castro, Daniel Gonçalves, and Daniel W. Van Wyk,Algebras of one- sided subshifts over arbitrary alphabets, Revista Matemática Iberoamericana40(2024), no. 3, 1045– 1088 (en)

    work page 2024

  6. [6]

    Ola Bratteli and Palle E. T. Jørgensen,Iterated function systems and permutation representations of the Cuntz algebra, American Mathematical Society, no. volume 139, number 663, American Mathe- matical Society, 1999 (eng)

    work page 1999

  7. [7]

    Charatonik and Anne Dilks,On self-homeomorphic spaces, Topology and its Appli- cations55(1994), no

    Wodzimierz J. Charatonik and Anne Dilks,On self-homeomorphic spaces, Topology and its Appli- cations55(1994), no. 3, 215–238 (en)

    work page 1994

  8. [8]

    Connes,Noncommutative differential geometry, Publications mathématiques de l’IHÉS62(1985), no

    A. Connes,Noncommutative differential geometry, Publications mathématiques de l’IHÉS62(1985), no. 1, 41–144 (en)

    work page 1985

  9. [9]

    ,Noncommutative geometry, Academic Press, San Diego, 1994 (eng)

    work page 1994

  10. [10]

    Connes and M

    A. Connes and M. Marcolli,Noncommutative Geometry, Quantum Fields and Motives, Colloquium Publications, vol. 55, American Mathematical Society, Providence, Rhode Island, December 2007 (en)

    work page 2007

  11. [11]

    Mathématique354(2015), no

    Alain Connes and Caterina Consani,The scaling site, Comptes Rendus. Mathématique354(2015), no. 1, 1–6 (en)

    work page 2015

  12. [12]

    ,Geometry of the arithmetic site, Advances in Mathematics291(2016), 274–329 (en)

    work page 2016

  13. [13]

    Alain Connes and Matilde Marcolli,From Physics to Number Theory via Noncommutative Geometry (en)

    work page

  14. [14]

    36, 1945 (en)

    Caterina Consani and Matilde Marcolli,Spectral triples from Mumford curves, International Math- ematics Research Notices2003(2003), no. 36, 1945 (en)

    work page 2003

  15. [15]

    2, 167–251 (en)

    ,Noncommutative geometry, dynamics, and∞-adic Arakelov geometry, Selecta Mathematica 10(2004), no. 2, 167–251 (en)

    work page 2004

  16. [16]

    2, 241–270 (fr)

    Michel Coornaert,Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific Journal of Mathematics159(1993), no. 2, 241–270 (fr). 25 REFERENCES REFERENCES

    work page 1993

  17. [17]

    Gunther Cornelissen and Matilde Marcolli,Graph reconstruction and quantum statistical mechanics, Journal of Geometry and Physics72(2013), 110–117, Noncommutative algebraic geometry and its applications to physics

    work page 2013

  18. [18]

    256, 1931–1970 (en)

    Dorin Dutkay and Palle Jorgensen,Iterated function systems, Ruelle operators, and invariant pro- jective measures, Mathematics of Computation75(2006), no. 256, 1931–1970 (en)

    work page 2006

  19. [19]

    Jorgensen,Atomic representations of Cuntz algebras, Journal of Mathematical Analysis and Applications421(2015), no

    Dorin Ervin Dutkay, John Haussermann, and Palle E.T. Jorgensen,Atomic representations of Cuntz algebras, Journal of Mathematical Analysis and Applications421(2015), no. 1, 215–243 (en)

    work page 2015

  20. [20]

    Exel and M

    R. Exel and M. Laca,Cuntz-Krieger algebras for infinite matrices, Journal für die reine und ange- wandte Mathematik (Crelles Journal)1999(1999), no. 512, 119–172

    work page 1999

  21. [21]

    Remus Floricel and Asghar Ghorbanpour,On inductive limit spectral triples, Proceedings of the American Mathematical Society (2017)

    work page 2017

  22. [22]

    Fremlin,Measure theory

    David H. Fremlin,Measure theory. 4,2: Topological measure spaces, 2. print ed., Torres Fremlin, Colchester, 2006 (en)

    work page 2006

  23. [23]

    Greenfield, M

    M. Greenfield, M. Marcolli, and K. Teh,Twisted spectral triples and quantum statistical mechanical systems, P-Adic Numbers, Ultrametric Analysis, and Applications6(2014), no. 2, 81–104 (en)

    work page 2014

  24. [24]

    Steven S. Gubser, Matthew Heydeman, Christian Jepsen, Matilde Marcolli, Sarthak Parikh, Ingmar Saberi, Bogdan Stoica, and Brian Trundy,Edge length dynamics on graphs with applications to p-adic AdS/CFT, Journal of High Energy Physics2017(2017), no. 6, 157 (en)

    work page 2017

  25. [25]

    4, 869–884 (en)

    Sa’Ar Hersonsky and John Hubbard,Groups of automorphisms of trees and their limit sets, Ergodic Theory and Dynamical Systems17(1997), no. 4, 869–884 (en)

    work page 1997

  26. [26]

    Matthew Heydeman, Matilde Marcolli, Ingmar Saberi, and Bogdan Stoica,Tensor networks, $p$- adic fields, and algebraic curves: arithmetic and the AdS$_3$/CFT$_2$ correspondence, Adv. Theor. Math. Phys.22(2018), no. 1 (en)

    work page 2018

  27. [27]

    P. E. T. Jorgensen and A. M. Paolucci,States on the Cuntz Algebras and p -adic random walks, Journal of the Australian Mathematical Society90(2011), no. 2, 197–211 (en)

    work page 2011

  28. [28]

    Palle E. T. Jorgensen,Use of operator algebras in the analysis of measures from wavelets and iterated function systems, arXiv: Operator Algebras (2005)

    work page 2005

  29. [29]

    Tsuyoshi Kajiwara and Yasuo Watatani,C∗-Algebras Associated with Complex Dynamical Systems and Backward Orbit Structure, January 2014, pp. 243–254

    work page 2014

  30. [30]

    4, 283– 326 (en)

    John Lott,Limit Sets as Examples in Noncommutative Geometry, K-Theory34(2005), no. 4, 283– 326 (en)

    work page 2005

  31. [31]

    1, 41–81 (en)

    Matilde Marcolli and Anna Maria Paolucci,Cuntz–Krieger Algebras and Wavelets on Fractals, Com- plex Analysis and Operator Theory5(2011), no. 1, 41–81 (en)

    work page 2011

  32. [32]

    1, 227–230 (en)

    Bernard Maskit,A characterization of Schottky groups, Journal d’Analyse Mathématique19(1967), no. 1, 227–230 (en)

    work page 1967

  33. [33]

    179–223, Springer International Publishing, Cham, 2021

    Jérôme Poineau and Daniele Turchetti,Berkovich curves and schottky uniformization i: The berkovich affine line, pp. 179–223, Springer International Publishing, Cham, 2021

    work page 2021

  34. [34]

    Qiuyu Ren,Linear Fractional Transformations on the Berkovich Projective Line(en)

    work page

  35. [35]

    Michael Stoll,p-adic Analysis in Arithmetic Geometry(en). 26

    work page