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Uspekhi Mat. Nauk, 2006, Volume 61, Issue 6(372), Pages 45–110 (Mi umn4262)  

This article is cited in 40 scientific papers (total in 40 papers)

Dixmier traces and some applications in non-commutative geometry

A. L. Careya, F. A. Sukochevb

a Australian National University
b Flinders University

Abstract: This is a discussion of recent progress in the theory of singular traces on ideals of compact operators, with emphasis on Dixmier traces and their applications in non-commutative geometry. The starting point is the book Non-commutative geometry by Alain Connes, which contains several open problems and motivations for their solutions. A distinctive feature of the exposition is a treatment of operator ideals in general semifinite von Neumann algebras. Although many of the results presented here have already appeared in the literature, new and improved proofs are given in some cases. The reader is referred to the table of contents below for an overview of the topics considered.

DOI: https://doi.org/10.4213/rm4262

Full text: PDF file (1070 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2006, 61:6, 1039–1099

Bibliographic databases:

UDC: 517.98+514.7
MSC: 46L10, 58B34, 58J30, 58J42, 47L20
Received: 10.11.2005

Citation: A. L. Carey, F. A. Sukochev, “Dixmier traces and some applications in non-commutative geometry”, Uspekhi Mat. Nauk, 61:6(372) (2006), 45–110; Russian Math. Surveys, 61:6 (2006), 1039–1099

Citation in format AMSBIB
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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. Astashkin S.V., Sukochev F.A., “Banach-Saks property in Marcinkiewicz spaces”, J. Math. Anal. Appl., 336:2 (2007), 1231–1258  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Kalton N., Sukochev F., “Rearrangement-invariant functionals with applications to traces on symmetrically normed ideals”, Canad. Math. Bull., 51:1 (2008), 67–80  crossref  mathscinet  zmath  isi  elib  scopus
    3. Yu. A. Kordyukov, “Index theory and non-commutative geometry on foliated manifolds”, Russian Math. Surveys, 64:2 (2009), 273–391  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. A. Sedaev, “Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces”, Russian Math. (Iz. VUZ), 53:12 (2009), 77–80  mathnet  crossref  mathscinet  zmath
    5. A. A. Sedaev, “Generalized Limits and Related Asymptotic Formulas”, Math. Notes, 86:4 (2009), 577–590  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Engliš M., Rochberg R., “The Dixmier trace of Hankel operators on the Bergman space”, J. Funct. Anal., 257:5 (2009), 1445–1479  crossref  mathscinet  zmath  isi  scopus
    7. Sukochev F., Zanin D., “Orbits in symmetric spaces”, J. Funct. Anal., 257:1 (2009), 194–218  crossref  mathscinet  zmath  isi  elib  scopus
    8. Steven Lord, Fedor Sukochev, “Measure Theory in Noncommutative Spaces”, SIGMA, 6 (2010), 072, 36 pp.  mathnet  crossref  mathscinet
    9. Lord S., Potapov D., Sukochev F., “Measures from Dixmier traces and zeta functions”, J. Funct. Anal., 259:8 (2010), 1915–1949  crossref  mathscinet  zmath  isi  elib  scopus
    10. Semenov E.M., Sukochev F.A., “Invariant Banach limits and applications”, J. Funct. Anal., 259:6 (2010), 1517–1541  crossref  mathscinet  zmath  isi  elib  scopus
    11. V. Zh. Sakbaev, “O dinamike mnozhestva sostoyanii kvantovoi sistemy s vyrozhdennym gamiltonianom”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(23) (2011), 200–220  mathnet  crossref
    12. Sukochev F., Zanin D., “$\zeta$-function and heat kernel formulae”, J. Funct. Anal., 260:8 (2011), 2451–2482  crossref  zmath  isi  elib  scopus
    13. Lord S., Sukochev F.A., “Noncommutative residues and a characterisation of the noncommutative integral”, Proc. Amer. Math. Soc., 139:1 (2011), 243–257  crossref  mathscinet  zmath  isi  elib  scopus
    14. Kalton N.J., Sedaev A.A., Sukochev F.A., “Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces”, Adv. Math., 226:4 (2011), 3540–3549  crossref  mathscinet  zmath  isi  elib  scopus
    15. V. Z. Sakbaev, “The set of quantum states and its averaged dynamic transformations”, Russian Math. (Iz. VUZ), 55:10 (2011), 41–50  mathnet  crossref  mathscinet  elib
    16. Semenov E.M., Sukochev F.A., Usachev A.S., “Structure properties of the Banach limit set”, Dokl. Math., 84:3 (2012), 802–803  crossref  mathscinet  zmath  isi  elib  scopus
    17. A.L. Carey, V. Gayral, A. Rennie, F.A. Sukochev, “Integration on locally compact noncommutative spaces”, Journal of Functional Analysis, 2012  crossref  mathscinet  isi  scopus
    18. Albrecht Pietsch, “Dixmier traces of operators on Banach and Hilbert spaces”, Math. Nachr, 2012, n/a  crossref  mathscinet  isi  scopus
    19. V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Journal of Mathematical Sciences, 213:3 (2016), 287–459  mathnet  crossref  mathscinet
    20. 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
    21. G. G. Amosov, V. Zh. Sakbaev, “On Analogs of Spectral Decomposition of a Quantum State”, Math. Notes, 93:3 (2013), 351–359  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    22. E. Semenov, F. Sukochev, “Extreme points of the set of Banach limits”, Positivity, 17:1 (2013), 163  crossref  mathscinet  zmath  isi  scopus
    23. Albrecht Pietsch, “Connes–Dixmier Versus Dixmier Traces”, Integr. Equ. Oper. Theory, 2013  crossref  mathscinet  isi  scopus
    24. A.A. Sedaev, F.A. Sukochev, “Dixmier measurability in Marcinkiewicz spaces and applications”, Journal of Functional Analysis, 2013  crossref  mathscinet  isi  scopus
    25. Ken Dykema, Anna Skripka, “Perturbation Formulas for Traces on Normed Ideals”, Commun. Math. Phys, 2013  crossref  mathscinet  isi  scopus
    26. A. M. Bikchentaev, “The Haagerup problem on subadditive weights on $W^*$-algebras. II”, Russian Math. (Iz. VUZ), 57:12 (2013), 66–69  mathnet  crossref
    27. Sukochev F., Usachev A., Zanin D., “Generalized Limits with Additional Invariance Properties and their Applications to Noncommutative Geometry”, Adv. Math., 239 (2013), 164–189  crossref  mathscinet  zmath  isi  elib  scopus
    28. Sukochev F., Zanin D., “Traces on Symmetrically Normed Operator Ideals”, J. Reine Angew. Math., 678 (2013), 163–200  crossref  mathscinet  zmath  isi  elib  scopus
    29. Pietsch A., “Shift-Invariant Functionals on Banach Sequence Spaces”, Studia Math., 214:1 (2013), 37–66  crossref  mathscinet  zmath  isi  elib  scopus
    30. Sukochev F., Usachev A., Zanin D., “On the Distinction Between the Classes of Dixmier and Connes-Dixmier Traces”, Proc. Amer. Math. Soc., 141:6 (2013), 2169–2179  crossref  mathscinet  zmath  isi  elib  scopus
    31. E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Geometric properties of the set of Banach limits”, Izv. Math., 78:3 (2014), 596–620  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    32. Sukochev F., Zanin D., “Dixmier Traces Are Weak Dense in the Set of All Fully Symmetric Traces”, J. Funct. Anal., 266:10 (2014), 6158–6173  crossref  mathscinet  zmath  isi  scopus
    33. Sukochev F., Usachev A., Zanin D., “Dixmier Traces Generated By Exponentiation Invariant Generalised Limits”, J. Noncommutative Geom., 8:2 (2014), 321–336  crossref  mathscinet  zmath  isi  scopus
    34. Potapov D., Sukochev F., Usachev A., Zanin D., “Singular Traces and Perturbation Formulae of Higher Order”, 269, no. 5, 2015, 1441–1481  crossref  mathscinet  zmath  isi  scopus
    35. Semenov E., Sukochev F., Usachev A., Zanin D., Adv. Math., 285 (2015), 568–628  crossref  mathscinet  zmath  isi  elib  scopus
    36. Jefferies B., “Traceability of Positive Integral Operators”, 84, no. 3, 2016, 419–428  crossref  mathscinet  zmath  isi  scopus
    37. Sukochev F., Usachev A., “Dixmier traces and non-commutative analysis”, J. Geom. Phys., 105 (2016), 102–122  crossref  mathscinet  zmath  isi  scopus
    38. Dykema K., Sukochev F., Zanin D., “Determinants Associated to Traces on Operator Bimodules”, J. Operat. Theor., 78:1 (2017), 119–134  crossref  mathscinet  zmath  isi  scopus
    39. Sukochev F., Zanin D., “A C-Algebraic Approach to the Principal Symbol. i”, J. Operat. Theor., 80:2 (2018), 481–522  crossref  isi
    40. E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Geometriya banakhovykh predelov i ikh prilozheniya”, UMN, 75:4(454) (2020), 153–194  mathnet  crossref
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