  We generalise the 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. The proof is a variant of that for the odd case which appears
in Part I. To allow for algebras with a non-trivial centre we have to establish
a theory of unbounded Fredholm operators in a general semifinite von Neumann
algebra and in particular prove a generalised McKean-Singer formula.
