The odd spectral localiser via asymptotic morphisms and quasi-projections

TL;DR

This paper introduces a novel index pairing method for odd K-theory classes using asymptotic morphisms and quasi-projections, connecting spectral localisers with Kasparov modules.

Contribution

It presents a new approach to index pairing involving asymptotic morphisms and quasi-projections, providing a fresh perspective on spectral localisers.

Findings

Established a pairing between K-theory classes and Kasparov modules via asymptotic morphisms.

Interpreted the spectral localiser as an instance of the index pairing.

Supported the approach with a finite spectral truncation technique.

Abstract

We describe the index pairing between an odd K-theory class and an odd unbounded Kasparov module by a pair of quasi-projections, supported on a submodule obtained from a finite spectral truncation. We achieve this by pairing the K-theory class with an asymptotic morphism determined by the unbounded Kasparov module. We interpret the spectral localiser of Loring and Schulz-Baldes as an instance of such an index pairing.