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

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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Algebraic structures and combinatorial models
