Noncommutative geometry on the Berkovich projective line

TL;DR

This paper develops noncommutative geometric structures, including $C^*$-algebras and spectral triples, on the Berkovich projective line, linking them to measures and dynamical systems in $p$-adic geometry.

Contribution

It constructs new $C^*$-algebras and spectral triples on the Berkovich projective line, extending noncommutative geometry to non-Archimedean settings.

Findings

Constructed spectral triples as limits over finite $ ext{R}$-trees.

Presented $C^*$-algebras generated by partial isometries.

Connected invariant measures to KMS-states of crossed product algebras.

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