The Connes character formula for locally compact spectral triples/ Fedor Sukochev & Dmitriy Zanin

Includes bibliography

Introduction -- Preliminaries -- Special triples: Basic properties and examples -- Asymptotic of the heat trace -- Residue of the ζ-function and the Connes character formula -- Appendix

A fundamental tool in noncommutative geometry is Connes's character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterization of manifolds. A non-compact space is modeled in noncommutative geometry by a non-unital spectral triple. The authors' aim is to establish Connes's character formula for non-unital spectral triples. This is significantly more difficult than in the unital case, and they achieve it with the use of recently developed double operator integration techniques. Previously, only partial extensions of Connes's character formula to the non-unital case were known. In the course of the proof, the authors 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 [Riemann zeta] function. The authors require certain assumptions on the underlying spectral triple and verify these assumptions in the case of spectral triples associated to arbitrary complete Riemannian manifolds and also in the case of Moyal planes.