Measure Theory in Noncommutative Spaces

Abstract

The integral in noncommutative geometry (NCG) involves a non-standard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measure-theoretic view, however, the formulation contains several difficulties. We review results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG.

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Cited by in corpus (6)

• Riemannian manifolds in noncommutative geometry
• Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics
• Banach limits and traces on ${\mathcal{L}}_{1,\infty }$
• A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity
• On the distinction between the classes of Dixmier and Connes-Dixmier traces
• Measurable operators and the asymptotics of heat kernels and zeta functions