A noncommutative Atiyah-Patodi-Singer index theorem in KK-theory

We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Kreiger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras.

💡 Research Summary

The paper extends the classical Atiyah‑Patodi‑Singer (APS) index theorem to the realm of non‑commutative geometry by working entirely within Kasparov’s KK‑theory. The authors begin by recalling the APS index theorem for Dirac‑type operators on manifolds with boundary and then introduce the mapping‑cone construction for a *‑homomorphism φ : A → B. The mapping cone C_φ, defined as the algebra of pairs (a, f) with f∈C₀((0,1], B) and f(1)=φ(a), plays the role of a non‑commutative “boundary” algebra.

The central technical achievement is a precise “odd‑to‑even” index correspondence. Given an even Kasparov A–B module (E, F), the authors construct a canonical even Kasparov module (Ē, Ĥ) over the cone C_F. They prove that for any class