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SIGMA, 2010, Volume 6, 072, 36 pages (Mi sigma530)  

This article is cited in 8 scientific papers (total in 8 papers)

Measure Theory in Noncommutative Spaces

Steven Lorda, Fedor Sukochevb

a School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, Australia
b University of New South Wales, School of Mathematics and Statistics

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.

Keywords: Dixmier trace; singular trace; noncommutative integration; noncommutative geometry; Lebesgue integral; noncommutative residue

DOI: https://doi.org/10.3842/SIGMA.2010.072

Full text: PDF file (516 kB)
Full text: http://emis.mi.ras.ru/journals/SIGMA/2010/072/
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Bibliographic databases:

ArXiv: 1009.3095
MSC: 46L51; 47B10; 58B34
Received: March 25, 2010; in final form August 4, 2010; Published online September 16, 2010
Language:

Citation: Steven Lord, Fedor Sukochev, “Measure Theory in Noncommutative Spaces”, SIGMA, 6 (2010), 072, 36 pp.

Citation in format AMSBIB
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\by Steven Lord, Fedor Sukochev
\paper Measure Theory in Noncommutative Spaces
\jour SIGMA
\yr 2010
\vol 6
\papernumber 072
\totalpages 36
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Carey A., Sukochev F., “Measurable Operators and the Asymptotics of Heat Kernels and Zeta Functions”, J. Funct. Anal., 262:10 (2012), 4582–4599  crossref  mathscinet  zmath  isi  elib  scopus
    2. Sedaev, AA; Sukochev, FA, “Dixmier measurability in Marcinkiewicz spaces and applications”, Journal of Functional Analysis, 265:12 (2013), 3053–3066  crossref  mathscinet  zmath  isi  scopus
    3. Sukochev, F; Usachev, A; Zanin, D, “On the distinction between the classes of Dixmier and Connes–Dixmier traces”, Proceedings of the American Mathematical Society, 141:6 (2013), 2169–2179  crossref  mathscinet  zmath  isi  scopus
    4. Kanovei, V; Katz, MG; Mormann, T, “Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics”, Foundations of Science, 18:2 (2013), 259–296  crossref  mathscinet  isi  scopus
    5. Sukochev, F; Usachev, A; Zanin, D, “Generalized limits with additional invariance properties and their applications to noncommutative geometry”, Advances in Mathematics, 239 (2013), 164–189  crossref  mathscinet  zmath  isi  scopus
    6. Semenov, Evgeniy; Sukochev, Fedor; Usachev, Alexandr; Zanin, Dmitriy, “Banach limits and traces on L-1,L-infinity”, ADVANCES IN MATHEMATICS, 285 (2015), 568-628  crossref  mathscinet  zmath  scopus
    7. Sukochev F., “H?lder inequality for symmetric operator spaces and trace property of K-cycles”, Bull. London Math. Soc., 48:4 (2016), 637–647  crossref  mathscinet  zmath  isi  scopus
    8. Sukochev F., Usachev A., “Dixmier Traces and Non-Commutative Analysis”, J. Geom. Phys., 105 (2016), 102–122  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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