2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73063We 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.29 pages, to appear in Advances in MathematicsOperator AlgebrasK-Theory and Homologytext