Spectral localizers in $KK$-theory

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^{*} -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^{*} -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.