The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebras

TL;DR

This paper extends the Atiyah-Patodi-Singer index theorem to Dirac operators twisted by C*-vector bundles, enabling new insights into eta-forms, rho-invariants, and their applications in noncommutative geometry.

Contribution

It proves a generalized index theorem for Dirac operators over C*-algebras and introduces new rho-invariants, unifying and extending previous results in the field.

Findings

Derived a product formula for eta-forms

Defined new rho-invariants generalizing Lott's higher rho-form

Reproduced the higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza

Abstract

We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C*-vector bundles. We use it to derive a general product formula for eta-forms and to define and study new rho-invariants generalizing Lott's higher rho-form. The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be recovered by applying the theorem to Dirac operators twisted by the Mishenko-Fomenko bundle associated to the reduced C*-algebra of the fundamental group.