  In previous work we generalised both the odd and even local index formula of
Connes and Moscovici to the case of spectral triples for a *-subalgebra \A of a
general semifinite von Neumann algebra. Our proofs are novel even in the
setting of the original theorem and rely on the introduction of a function
valued cocycle (called the resolvent cocycle) which is `almost' a (b,B)-cocycle
in the cyclic cohomology of \A. In this paper we show that this resolvent
cocycle `almost' represents the Chern character, and assuming analytic
continuation properties for zeta functions, we show that the associated residue
cocycle, which appears in our statement of the local index theorem does
represent the Chern character.
