Noncommutative geometry on the Berkovich projective line

In this paper, we construct several C^{*} -algebras 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}) .