- Let A be an exact C*-algebra, let G be a locally compact group, and let (A, G, ) be a C*-dynamical system. Each automorphism g induces a spatial automorphism Adg on the reduced crossed product A G. In this paper we examine the question, first raised by E. Strmer, of when the topological entropies of g and Adg 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 g 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. 2003 Elsevier Science (USA). All rights reserved.