Crossed products and entropy of automorphisms

AbstractLet A be an exact C∗-algebra, let G be a locally compact group, and let (A,G,α) be a C∗-dynamical system. Each automorphism αg induces a spatial automorphism Adλg on the reduced crossed product A×αG. In this paper we examine the question, first raised by E. Størmer, of when the topological entropies of αg and Adλg coincide. This had been answered by N. Brown for the particular case of discrete abelian groups. Using different methods, we extend his result to preservation of entropy for αg when the subgroup of Aut(G) generated by the corresponding inner automorphism Adg has compact closure. This property is satisfied by all elements of a wide class of groups called locally [FIA]−. This class includes all abelian groups, both discrete and continuous, as well as all compact groups