Integration on locally compact noncommutative spaces

TL;DR

This paper develops a comprehensive integration theory for nonunital spectral triples in semifinite noncommutative geometry, establishing key equalities and equivalences among various integrability notions.

Contribution

It introduces an ab initio approach to integration without local units and proves the equivalence of zeta function, heat kernel, and Dixmier trace methods.

Findings

Dixmier trace equals the generalized residue of the zeta function and heat kernel.

Zeta functions and heat kernels provide equivalent integrability notions.

The approach applies broadly in semifinite noncommutative geometry.

Abstract

We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of semifinite noncommutative geometry. The main result is an equality between the Dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators. We also examine definitions for integrable bounded elements of a spectral triple based on zeta function, heat kernel and Dixmier trace techniques. We show that zeta functions and heat kernels yield equivalent notions of integrability, which imply Dixmier traceability.