Noncommutative geometry on the Berkovich projective line

Abstract

We construct several $C^{\ast }$-algebras and spectral triples associated to the Berkovich projective line ${\mathbb{P}}_{\mathrm{Berk}}^1({\mathbb{C}}_p)$. In the commutative setting, we construct a spectral triple as a direct limit over finite $\mathbb{R}$-trees. More general $C^{\ast }$-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 ${\mathrm{PGL}}_2({\mathbb{C}}_p)$.