Measure Theory in Noncommutative Spaces
Steven Lord, Fedor Sukochev
TL;DR
This paper reviews measure-theoretic aspects of the noncommutative integral in noncommutative geometry, highlighting technical challenges and open problems for users of the field.
Contribution
It provides a comprehensive review of the measure-theoretic formulation of the noncommutative integral and discusses unresolved issues in the area.
Findings
Identifies key technical difficulties in defining the noncommutative integral
Summarizes known results on the measure-theoretic properties of the Dixmier trace
Outlines open problems for future research in noncommutative measure theory
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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