Entropy of Cuntz’s canonical endomorphism

Let {Si}ni=1 be generators of the Cuntz algebraOn and let Φ be the *-endomorphism of On defined by Φ(x) = ∑n i=1 SixS i. Then both of Connes–Narnhofer–Thirring’s entropy hφ(Φ) and Voiculescu’s topological entropy ht(Φ) are logn, where φ is the unique logn-KMS state of On. Also Longo’s canon-ical endomorphism for N ⊂ M have the same entropy logn, where the inclusion N ⊂M comes from On. 1. Introduction. Connes-Stφrmer entropy H(·) extended the entropy invariant of Kolmogo-rov-Sinai to trace preserving automorphisms of finite von Neumann algebras ([CS]). Replacing a finite trace to an invariant state φ, Connes-Narnhofer-Thirring entropy hφ(·) is defined for automorphisms of C∗-algebras as