Nilpotency of Connes' periodicity operator and the idempotent conjectures

For certain classes of discrete groups we verify the idempotent conjectures for various group algebras by the method of cyclic cohomology. In particular, the Banach ¹ (G) of a torsion free word hyperbolic groupG of Gromov contains no nontrivial idempotents. Moreover, the range of any tracial state onK₀(¹(G)) is .Sponsored in part by a grant from the National Science Foundation.