Integration on locally compact noncommutative spaces

Authors

A L. Carey, Australian National University
V Gayral, Universite Reims Champagne-Ardenne, France
Adam C. Rennie, University of WollongongFollow
F A. Sukochev, University of New South Wales

RIS ID

58100

Publication Details

Carey, A. L., Gayral, V., Rennie, A. C. & Sukochev, F. A. (2012). Integration on locally compact noncommutative spaces. Journal of Functional Analysis, 263 (2), 383-414.

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.