Extensions and renormalized traces

TL;DR

This paper develops explicit formulas for index theory in non-commutative geometry, connecting cyclic homology and K-theory without excision, and applies these to classical index theorems using renormalization techniques.

Contribution

It introduces a method to derive explicit local index formulas in non-commutative geometry avoiding excision, based on renormalization and the bivariant Chern character.

Findings

Explicit formulas for index maps in cyclic homology.

Application to classical family index theorem.

Characteristic numbers expressed via Wodzicki residue.

Abstract

It has been shown by Nistor that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern character for quasihomomorphisms, leading to "local" index formulas in the sense of non-commutative geometry. We illustrate these principles with the example of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue.