《Measure Theory in Noncommutative Spaces sigma10-072》.pdf

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 072, 36 pages Measure Theory in Noncommutative Spaces Steven LORD † and Fedor SUKOCHEV ‡ † School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, Australia E-mail: steven.lord@.au ‡ School of Mathematics and Statistics, University of New South Wales, Sydney, 2052, Australia E-mail: f.sukochev@.au Received March 25, 2010, in final form August 04, 2010; Published online September 16, 2010 doi:10.3842/SIGMA.2010.072 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 re- view 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 2010 Mathematics Subject Classification: 46L51; 47B10; 58B34 Contents 1 Introduction 2 2 Dixmier traces 5 2.1 Singular symmetric functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Measurability 11 3.1 Decomposition of measurable operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Closed symmetric s