Asian Journal of Mathematics

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

Charlotte Wahl

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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.

Article information

Asian J. Math. Volume 17, Number 2 (2013), 265-320.

First available in Project Euclid: 8 November 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20]
Secondary: 58J28: Eta-invariants, Chern-Simons invariants 58J32: Boundary value problems on manifolds

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


Wahl, Charlotte. The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebras. Asian J. Math. 17 (2013), no. 2, 265--320.

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