Dynamical entropy for automorphisms of exact C∗-algebras

Some estimations of dynamical entropies are given by applying the entropy htφ(α) for an automorphism α of a C∗-algebra A with respect to α-invariant state φ. This htφ(α) coincides with the Connes–Narnhofer–Thirring (CNT) entropy hφ(α) when A is abelian. Let (A,φ) be the reduced free product of (Ai,φi)i∈I. If {Ai}i∈I are finite dimensional, then hφ(α)=htφ(α)=0 for all α∈Aut(A,φ). Concerning the free product α1∗α2 of αi∈Aut(Ai,φi), we have always that htφ1∗φ2(α1∗α2)=max{htφ1(α1),htφ2(α2)}, and the CNT entropy satisfies this formula for all automorphisms in the ergodic theory.