THE LOCAL INDEX FORMULA IN NONCOMMUTATIVE GEOMETRY

In noncommutative geometry a geometric space is described from a spectral vantage point, as a triple (A, H, D) consisting of a *-algebra A represented in a Hilbert space H together with an unbounded selfadjoint operator D, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for arbitrary spectral triples of finite summability degree, in terms of the Dixmier trace and its residue-type extension.