The Connes character formula for locally compact spectral triples

TL;DR

This paper extends Connes' character formula to non-unital spectral triples in noncommutative geometry, using advanced operator techniques, and establishes related heat kernel and zeta function properties for these triples.

Contribution

It provides the first full proof of Connes' character formula for non-unital spectral triples, expanding its applicability in noncommutative geometry.

Findings

Extended Connes' character formula to non-unital spectral triples

Proved asymptotic behavior of heat semigroup in this setting

Demonstrated analyticity of the associated zeta function

Abstract

A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds. A non-compact space is modelled in noncommutative geometry by a non-unital spectral triple. Our aim is to establish the Connes character formula for non-unital spectral triples. This is significantly more difficult than in the unital case and we achieve it with the use of recently developed double operator integration techniques. Previously, only partial extensions of Connes' character formula to the non-unital case were known. In the course of the proof, we establish two more results of importance in noncommutative geometry: an asymptotic for the heat semigroup of a non-unital spectral triple, and the analyticity of the associated $\zeta$-function. We require certain assumptions on the underlying spectral triple, and we verify these assumptions in the case of spectral triples associated to arbitrary complete Riemannian manifolds and also in the case of Moyal planes.