On the entropy and index of the winding endomorphisms of p-adic ring C∗-algebras

For p ≥ 2, the p-adic ring C∗-algebra Qp is the universal C∗-algebra generated by a unitary U and an isometry Sp such that (Equation presented) and (Equation presented). For any k coprime to p we define an endomorphism χk ∈ End(Qp) by setting χk(U):= Uk and χk(Sp):= Sp. We then compute the entropy of χk, which turns out to be log |k|. Finally, for selected values of k we also compute the Watatani index of χk showing that the entropy is the natural logarithm of the index.