  We show that the Chern-Connes character from Kasparov's bivariant K-theory to
bivariant local cyclic cohomology is not always rationally injective.
Counterexamples are provided by the reduced group $C^*$-algebras of
word-hyperbolic groups with Kazhdan's property (T). The proof makes essential
use of Skandalis' work on K-nuclearity and of Lafforgue's recent demonstration
of the Baum-Connes conjecture with coefficients for word-hyperbolic groups.
