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A K-theoretic note on the spectral localiser

Koen van den Dungen

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.

A K-theoretic note on the spectral localiser

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.
Paper Structure (14 sections, 21 theorems, 54 equations)

This paper contains 14 sections, 21 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Spectral localisers and index pairings
  3. The even index pairing
  4. The even spectral localiser
  5. The spectral flow expression
  6. The odd index pairing
  7. The odd spectral localiser
  8. The spectral flow expression
  9. Spectral truncation of the index pairing
  10. The even index pairing
  11. The odd index pairing
  12. Fredholm operators and K-theory
  13. Relative index and spectral flow
  14. Computation of a Kasparov product

Key Result

Lemma 2.3

Assume $\kappa < g^2 \|[\mathcal{D},H]\|^{-1}$. Then the even spectral localiser $L^{\mathrm{ev}}_\kappa$ is invertible.

Theorems & Definitions (32)

  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Theorem 2.10
  • Remark 2.11
  • Definition 3.1
  • ...and 22 more