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

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 $η$-forms and to define and study new $ρ$-invariants generalizing Lott’s higher $ρ$-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.

Keywords

Atiyah-Patodi-Singer index theorem, higher index theory, Dirac operator, $C*$-vector bundle

2010 Mathematics Subject Classification

Primary 58J22. Secondary 58J28, 58J32.

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Published 5 July 2013