Spectral localizers in $KK$-theory

Abstract

We study the index homomorphism of even $K$-groups arising from a class in even $KK$-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the $C^{\ast }$-algebras in question, such a class in $KK$-theory can always be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even $K$-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator $D$, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the $K$-theoretic input and the abstract Dirac operator. Since the spectral projections for $D$ are not available in the general context of Hilbert $C^{\ast }$-modules, we instead rely on certain continuous compactly supported functions applied to $D$ to construct the spectral localizer. In the special case where even $KK$-theory coincides with even $K$-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.