Spectral localizers in KK-theory

TL;DR

This paper introduces a new explicit formula for the index homomorphism in KK-theory using spectral localizers, which depend only on the spectrum's intersection with a compact interval, simplifying calculations.

Contribution

It provides an explicit spectral localizer-based formula for the index homomorphism in KK-theory, extending previous work to a broader context involving Hilbert C*-modules.

Findings

The index homomorphism can be expressed via spectral localizers.

The formula depends on the spectrum's intersection with a compact interval.

The approach generalizes previous results in K-homology and index pairing.

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^*-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.