  We use the symbol calculus for foliations developed in our previous paper to
derive a cohomological formula for the Connes-Chern character of the
semi-finite spectral triple. The same proof works for the Type I spectral
triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern
characters induce the same map on the image of the maximal Baum-Connes map in
K-theory, thereby proving an Atiyah $L^2$ covering index theorem.

Geometric Topology;Differential Geometry