Entropy estimates for some C*-endomorphisms

In this paper we compute the non-commutative topological entropy in the sen se of Voiculescu for some endomorphisms of stationary inductive limits of c ircle algebras. These algebras are groupoid C*-algebras, and the endomorphi sms restricted to the canonical diagonal are induced by some expansive maps , whose entropies provide a lower bound. For the upper bound, we use a resu lt of Voiculescu, similar to the classical Kolmogorov-Sinai theorem. The sa me technique is used to compute the entropy of a non-commutative Markov shi ft.