A K-theoretic note on the spectral localiser

TL;DR

This paper explores the spectral localiser using K-theory, providing a direct approach to spectral flow and index pairing, applicable without invertibility assumptions, and connecting infinite and finite volume perspectives.

Contribution

It offers a new K-theoretic framework for understanding the spectral localiser, unifying even and odd cases, and simplifying the computation of index pairings.

Findings

Spectral flow can be expressed via the spectral localiser in K-theoretic terms.

The approach treats even and odd cases uniformly, simplifying analysis.

Spectral truncation relates infinite volume spectral flow to finite volume signature.

Abstract

We review the construction of the spectral localiser (due to Loring and Schulz-Baldes) from a K-theoretic perspective. We first give a K-theoretic argument providing a spectral flow expression for the even or odd index pairing in terms of the "infinite volume" spectral localiser. Our approach towards this first step is more direct, treats the even and odd cases on an equal footing, and has the advantage that the construction of the spectral localiser becomes immediately apparent from the computation of the index pairing via a Kasparov product. In a second step of "spectral truncation", we then describe how this spectral flow expression can be computed in terms of the signature of the "finite volume" spectral localiser. Throughout, we do not require invertibility of the operator representing the K-homology class, and the even index pairing then obtains an additional contribution coming from the Fredholm index.