Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 072, 36 pages      arXiv:1009.3095      https://doi.org/10.3842/SIGMA.2010.072
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Measure Theory in Noncommutative Spaces

Steven Lord a and Fedor Sukochev b
a) School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, Australia
b) School of Mathematics and Statistics, University of New South Wales, Sydney, 2052, Australia

Received March 25, 2010, in final form August 04, 2010; Published online September 16, 2010

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.

Key words: Dixmier trace; singular trace; noncommutative integration; noncommutative geometry; Lebesgue integral; noncommutative residue.

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