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.