The spectral shift function and spectral flow

TL;DR

This paper generalizes the spectral shift function theory to semifinite spectral triples, defining it via spectral averaging and demonstrating its role in computing spectral flow, thus advancing noncommutative geometry methods.

Contribution

It introduces a new framework for spectral shift functions in semifinite spectral triples and links it to spectral flow calculations.

Findings

Spectral shift function defined via Birman-Solomyak formula for semifinite spectral triples.

Spectral shift function computes spectral flow in this generalized setting.

Extends classical theory to noncommutative geometry contexts.

Abstract

This paper extends Krein's spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow.