Homological index formulas for elliptic operators over C*-algebras

TL;DR

This paper develops index formulas for elliptic operators over C*-algebras, connecting K-theory, de Rham homology, and noncommutative geometry, with applications to coverings, foliations, and group actions.

Contribution

It introduces new index formulas involving Karoubi's Chern character for elliptic operators on C*-vector bundles, extending classical index theory to noncommutative settings.

Findings

Derived index formulas for elliptic operators over C*-algebras.

Applied formulas to higher index theorems for coverings and foliated bundles.

Established relations between K-theory pairings, spectral flow, and pseudodifferential operators.

Abstract

We prove index formulas for elliptic operators acting between sections of C*-vector bundles on a closed manifold. The formulas involve Karoubi's Chern character from K-theory of a C*-algebra to de Rham homology of smooth subalgebras. We show how they apply to the higher index theorem for coverings and to flat foliated bundles, and prove an index theorem for C*-dynamical systems associated to actions of compact Lie groups. In an Appendix we relate the pairing of odd K-theory and KK-theory to the noncommutative spectral flow and prove the regularity of elliptic pseudodifferential operators over C*-algebras.