Von Neumann eta-invariants and C*-algebra K-theory

Let M be a smooth, compact, oriented, odd-dimensional Riemannian manifold and let Gamma --> M --> M be a normal covering of M. It is proved that the relative von Neumann eta-invariant rho ((2))(M) of Cheeger and Gromov is a homotopy invariant when Gamma is torsion-free, discrete and the Baum-Connes assembly map mu (max): K-0(B Gamma) --> K-0(C*Gamma) is an isomorphism.