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Communications in Mathematical Physics

, Volume 359, Issue 2, pp 449–466 | Cite as

Connes Integration Formula for the Noncommutative Plane

  • F. Sukochev
  • D. Zanin
Article

Abstract

Our aim is to prove the integration formula on the noncommutative (Moyal) plane in terms of singular traces a la Connes.

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References

  1. 1.
    Benameur M., Fack T.: Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras. Adv. Math. 199(1), 29–87 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carey A., Gayral V., Rennie A., Sukochev F.: Integration on locally compact noncommutative spaces. J. Funct. Anal. 263(2), 383–414 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carey, A., Gayral, V., Rennie, A., Sukochev, F.: Index theory for locally compact noncommutative geometries. Mem. Am. Math. Soc. 231(1085) (2014)Google Scholar
  4. 4.
    Connes A.: Noncommutative Geometry. Academic Press, San Diego (1994)zbMATHGoogle Scholar
  5. 5.
    Connes A.: The action functional in noncommutative geometry. Commun. Math. Phys. 117(4), 673–683 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dixmier J.: Existence de traces non normales. (French) . C. R. Acad. Sci. Paris Ser. A B 262, A1107–A1108 (1966)Google Scholar
  7. 7.
    Dykema K., Figiel T., Weiss G., Wodzicki M.: Commutator structure of operator ideals. Adv. Math. 185(1), 1–79 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Estrada R., Gracia-Bondia J., Varilly J.: On summability of distributions and spectral geometry. Commun. Math. Phys. 191(1), 219–248 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gayral V., Gracia-Bondia J., Iochum B., Schücker T., Varilly J.: Moyal planes are spectral triples. Commun. Math. Phys. 246(3), 569–623 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gracia-Bondia J., Varilly J., Figueroa H.: Elements of noncommutative geometry. Basel Textbooks, Birkhauser Boston Inc (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kadison, R., Ringrose, J.: Fundamentals of the theory of operator algebras. Vol. I. Elementary theory. Reprint of the 1983 original. Graduate Studies in Mathematics, vol. 15. American Mathematical Society, Providence, RI, (1997)Google Scholar
  12. 12.
    Kalton N., Lord S., Potapov D., Sukochev F.: Traces of compact operators and the noncommutative residue. Adv. Math. 235, 1–55 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Levitina, G., Sukochev, F., Zanin, D.: Cwikel estimates revisited. submitted manuscript. arXiv:1703.04254
  14. 14.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97. Springer, Berlin-New York, (1979)Google Scholar
  15. 15.
    Lord S., Potapov D., Sukochev F.: Measures from Dixmier traces and zeta functions. J. Funct. Anal. 259(8), 1915–1949 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lord, S., Sukochev, F., Zanin, D.: Singular traces. Theory and applications. De Gruyter Studies in Mathematics, vol. 46, De Gruyter, Berlin, (2013)Google Scholar
  17. 17.
    Pietsch A.: Traces and shift invariant functionals. Math. Nachr. 145, 7–43 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pietsch A.: About the Banach envelope of \({l_{1,\infty}}\). Rev. Mat. Comput. 22(1), 209–226 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Semenov E., Sukochev F., Usachev A., Zanin D.: Banach limits and traces on \({\mathcal{L}_{1,\infty}}\). Adv. Math. 285, 568–628 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sukochev F., Zanin D.: Fubini theorem in noncommutative geometry. J. Funct. Anal. 272(3), 1230–1264 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia

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