Noncommutative geometry on the Berkovich projective line

Abstract

In this paper, we construct several $C^{\ast }$-algebras 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 ${\mathrm{PGL}}_2({\mathbb{C}}_p)$.