A C*-Dynamical Entropy and Applications to Canonical Endomorphisms

For an automorphism α of a unital C*-algebra A, we give a definition of an entropy htφ(α) with respect to an α-invariant state φ of A. For Connes– Narnhofer–Thirring entropy hφ(α) and Voiculescuʹs topological entropy ht(α), in general hφ(α)⩽htφ(α)⩽ht(α), but the equalities do not always hold. We compute entropies of an endomorphism ρ with respect to the state ϕ defined from a left inverse of ρ. Cuntzʹs canonical inner endomorphism Φ of On satisfies hϕ(Φ)=htϕ(Φ), which is determined by the mean entropy of ϕ on the UHF (uniformly hyperfinite) algebra. If γ is Longoʹs canonical endomorphism for an irreducible graded standard AFD (approximately finite dimensional) inclusion N⊂M of infinite factors with finite index, then hϕ(γ)=(1/2) log(Ind Eγ), for the conditional expectation Eγ on γ(M).