The Local Index Formula in Noncommutative Geometry

Abstract

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.KeywordsHilbert SpacePseudo Differential OperatorNoncommutative GeometryPrincipal SymbolChern CharacterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.