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

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.

💡 Research Summary

The paper establishes an Atiyah‑Patodi‑Singer (APS) index theorem for Dirac operators twisted by bundles of finitely generated projective modules over a C*‑algebra A. After recalling the necessary background on C*‑vector bundles, Hilbert‑A‑modules, and the Mishchenko‑Fomenko bundle associated with the reduced group C*‑algebra C*_r(π₁(M)), the author constructs a twisted Dirac operator D_E on an even‑dimensional compact manifold M with boundary ∂M. The APS boundary condition is formulated in the non‑commutative setting by using the spectral projection onto the non‑negative part of the boundary operator, now interpreted as an A‑module projection.

The main theorem states that the index of (D_E, APS) defines an element of K₀(A) given by
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