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Noncommutative geometry on the Berkovich projective line

Masoud Khalkhali, Damien Tageddine

TL;DR

This work develops a noncommutative geometric framework for the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}(\mathbb{C}_p)$ by constructing several $C^*$-algebras and spectral triples, including a commutative triple as an inverse limit over finite $\mathbb{R}$-trees and a noncommutative model via the Wa\'zewski universal dendrite. It combines inverse-limit constructions with semibranching systems and graph-algebra techniques to build projections, partial isometries, and Perron–Frobenius operators, leading to representations on $L^2$-spaces equipped with a $\mathrm{PGL}_2(\mathbb{C}_p)$-invariant measure $\mu_G$, and to projection-valued measures and spectral integrals. A key result is the realization of invariant measures as KMS states for crossed-product algebras associated to Schottky subgroups, tying boundary dynamics to thermodynamic formalisms in noncommutative geometry. The framework provides a coherent, $p$-adic analogue of archimedean noncommutative geometry, enriching both operator-algebraic dynamics and number-theoretic perspectives with explicit constructions on the Berkovich line and its boundary.

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 $\mathrm{PGL}_2(\mathbb{C}_p)$.

Noncommutative geometry on the Berkovich projective line

TL;DR

This work develops a noncommutative geometric framework for the Berkovich projective line by constructing several -algebras and spectral triples, including a commutative triple as an inverse limit over finite -trees and a noncommutative model via the Wa\'zewski universal dendrite. It combines inverse-limit constructions with semibranching systems and graph-algebra techniques to build projections, partial isometries, and Perron–Frobenius operators, leading to representations on -spaces equipped with a -invariant measure , and to projection-valued measures and spectral integrals. A key result is the realization of invariant measures as KMS states for crossed-product algebras associated to Schottky subgroups, tying boundary dynamics to thermodynamic formalisms in noncommutative geometry. The framework provides a coherent, -adic analogue of archimedean noncommutative geometry, enriching both operator-algebraic dynamics and number-theoretic perspectives with explicit constructions on the Berkovich line and its boundary.

Abstract

We construct several -algebras and spectral triples associated to the Berkovich projective line . In the commutative setting, we construct a spectral triple as a direct limit over finite -trees. More general -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 .

Paper Structure

This paper contains 22 sections, 43 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Berkovich projective line
  3. The multiplicative seminorms and Berkovich spaces
  4. Berkovich's classification theorem
  5. The Berkovich affine line $\mathbb{A}^1_{\mathrm{Berk}}(\mathbb{C}_p)$
  6. The Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}(\mathbb{C}_p)$ and the $\mathbb{R}$-tree structure
  7. A commutative spectral triple on $\mathbb{P}^1_{\mathrm{Berk}}(\mathbb{C}_p)$
  8. Spectral triples on finite $\mathbb{R}$-trees
  9. Inverse limit of spectral triples
  10. $\mathbb{P}^1_{\mathrm{Berk}}(\mathbb{C}_p)$ as the universal Ważewski dendrite
  11. Two equivalent topologies on $\mathbb{P}^1_{\mathrm{Berk}}(\mathbb{C}_p)$
  12. Noncommutative Geometry on $\mathbb{P}^1_{\mathrm{Berk}}(\mathbb{C}_p)$
  13. $C^*$-algebras of countable subshifts
  14. Semibranching systems on $\mathbb{P}^1_{\mathrm{Berk}}(\mathbb{C}_p)$
  15. Invariant measure
  16. ...and 7 more sections

Key Result

Theorem 2.1

Every point $x\in \mathbb{A}^1_{\mathrm{Berk}}(K)$ corresponds to a nested sequence $D(a_1,r_1)\supseteq D(a_2,r_2)\supseteq D(a_3,r_3)\supseteq \cdots$ of closed disks, in the sense Two such nested sequences define the same point of $\mathbb{A}^1_{\mathrm{Berk}}(K)$ if and only if

Figures (1)

  • Figure 1: The Berkovich affine line. Picture on the right is taken from Poineau2021

Theorems & Definitions (91)

  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Definition 2.3
  • Theorem 2.1: Berkovich's Classification Theorem
  • Remark 2.1
  • Proposition 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: $\mathbb{R}$-tree
  • ...and 81 more