The Chern-Connes character is the unique multiplicative transformation from Kasparov's bivariant K-theory to bivariant local cyclic cohomology. It provides a link between Kasparov theory, the basic cohomology theory for C*-algebras, and cyclic cohomology, a cohomology theory which is calculable by means of homological algebra. The Chern-Connes character is a far reaching generalization of the classical Chern character in topological K-theory. In the present talk we show that, contrary to the classical Chern character, the Chern-Connes character is not an isomorphism rationally (it is not even rationally injective). Counterexamples are provided by the reduced group C*-algebras of suitable discrete groups.