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

Abstract

We prove a higher 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.