The local index formula in noncommutative geometry revisited

In this review we discuss the local index formula in noncommutative geomety from the viewpoint of two new proofs are partly inspired by the approach of Higson especially that in but they differ in several fundamental aspedcts, in particular they apply to semifinite spectral triples for a *s-subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and reduce the hypotheses of the theorem to those necessary for its statement. These proofs rely on the introduction of a function valued cocycle which is 'almost' a (b, B)-cocycle in the cyclic cohomology of A. They do not need the 'discrete dimension spectrum' assumption of jthe original Connes-Moscovici proof only a much weaker condition on the analytic continuation of certain zeta functions, and this only for part of the statement. In this article we also explain the relationship of the pairing between k-theory amd semifinite spectral triples to KK-theory and the Kasparov product. This discussion shows that simifinite spectral triples are a specific kind of representative of a K K-class and the analytically defined index is compatible with the Kasparov product.