A K-theoretic note on the spectral localiser

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.