Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 072, 36 pages      arXiv:1009.3095
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Measure Theory in Noncommutative Spaces

Steven Lord a and Fedor Sukochev b
a) School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, Australia
b) School of Mathematics and Statistics, University of New South Wales, Sydney, 2052, Australia

Received March 25, 2010, in final form August 04, 2010; Published online September 16, 2010

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.

Key words: Dixmier trace; singular trace; noncommutative integration; noncommutative geometry; Lebesgue integral; noncommutative residue.

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  1. Chamseddine A.H., Connes A., The spectral action principle, Comm. Math. Phys. 186 (1997), 731-750, hep-th/9606001.
  2. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  3. Connes A., Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), 155-176, hep-th/9603053.
  4. Segal I.E., A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401-457.
  5. Kunze R.A., Lp Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958), 519-540.
  6. Stinespring W.F., Integration theorems for gages and duality for unimodular groups, Trans. Amer. Math. Soc. 90 (1959), 15-56.
  7. Nelson E., Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103-116.
  8. Fack T., Kosaki H., Generalised s-numbers of τ-measurable operators, Pacific J. Math. 123 (1986), 269-300.
  9. Pisier G., Xu Q., Non-commutative Lp-spaces, in Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, 1459-1517.
  10. Pederson G.K., C*-algebras and their automorphism groups, London Mathematical Society Monographs, Vol. 14, Academic Press, London - New York, 1979.
  11. Simon B., Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, Vol. 120, American Mathematical Society, Providence, RI, 2005.
  12. von Neumann J., Some matrix inequalities and metrization of metric-space, Rev. Tomsk. Univ. 1 (1937), 286-300.
  13. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. 1. C*- and W*-algebras, symmetry groups, decomposition of states, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987.
  14. Cipriani F., Guido D., Scarlatti S., A remark on trace properties of K-cycles, J. Operator. Theory 35 (1996), 179-189, funct-an/9506003.
  15. Carey A.L., Rennie A., Sedaev A., Sukochev F., The Dixmier trace and asymptotics of zeta functions, J. Funct. Anal. 249 (2007), 253-283, math.OA/0611629.
  16. Benameuar M.-T., Fack T., Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math. 199 (2006), 29-87.
  17. Connes A., The action functional in noncommutative geometry, Comm. Math. Phys. 117 (1988), 673-683.
  18. Dunford N., Schwartz J.T., Linear operators. Part I. General theory, John Wiley & Sons, Inc., New York, 1988.
  19. Carey A.L., Sukochev F.A., Dixmier traces and some applications in noncommutative geometry, Russian Math. Surveys 61 (2006), 1039-1099, math.OA/0608375.
  20. Guido D., Isola T., Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal. 203 (2003), 362-400, math.OA/0202108.
  21. Pietsch A., About the Banach envelope of l1,∞, Rev. Mat. Complut. 22 (2009), 209-226.
  22. Dixmier J., Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1107-A1108.
  23. Lorentz G.G., A contribution to the theory of divergent sequences, Acta. Math. 80 (1948), 167-190.
  24. Sucheston L., Banach limits, Amer. Math. Monthly 74 (1967), 308-311.
  25. Carey A., Phillips J., Sukochev F., Spectral flow and Dixmier traces, Adv. Math. 173 (2003), 68-113. math.OA/0205076.
  26. Krein S.G., Petunin Yu.I., Semenov E.M., Interpolation of linear operators, Translations of Mathematical Monographs, Vol. 54, American Mathematical Society, Providence, R.I., 1982.
  27. Lord S., Sedaev A., Sukochev F., Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal. 244 (2005), 72-106, math.FA/0501131.
  28. Dodds P.G., de Pagter B., Semenov E.M., Sukochev F.A., Symmetric functional and singular traces, Positivity 2 (1998), 47-75.
  29. Dodds P.G., de Pagter B., Sedaev A.A., Semenov E.M., Sukochev F.A., Singular symmetric functionals, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 290 (2002), 42-71 (English transl.: J. Math. Sci. 2 (2004), 4867-4885).
  30. Dodds P.G., de Pagter B., Sedaev A.A., Semenov E.M., Sukochev F.A., Singular symmetric functionals and Banach limits with additional invariance properties, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 6, 111-136 (English transl.: Izv. Math. 67 (2003), 1187-1212).
  31. Guido D., Isola T., Singular traces on semifinite von Neumann algebras, J. Funct. Anal. 134 (1995), 451-485.
  32. Sedaev A.A., Generalized limits and related asymptotic formulas, Math. Notes 86 (2009), 577-590.
  33. Sedaev A.A., Sukochev F.A., Zanin D.V., Lidskii-type formulae for Dixmier traces, Integral Equations Operator Theory, to appear, arXiv:1003.1813.
  34. Semenov E.M., Sukochev F.A., Invariant Banach limits and applications, J. Funct. Anal. 259 (2010), 1517-1541.
  35. Azamov N.A., Sukochev F.A., A Lidskii type formula for Dximier traces, C. R. Math. Acad. Sci. Paris 340 (2005), 107-112.
  36. Prinzis R., Traces résiduelles et asymptotique du spectre d'opérateurs pseduo-différentiels, Ph.D. Thesis, Lyon, 1995.
  37. Calkin J.W., Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839-873.
  38. Kalton N.J., Sukochev F.A., Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81-121.
  39. Hardy G.H., Littlewood J.E., Pólya G., Inequalities, Cambridge University Press, Cambridge, 1934.
  40. Kalton N., Sukochev F., Rearrangement-invariant functionals with applications to traces on symmetrically normed ideals, Canad. Math. Bull. 51 (2008), 67-80.
  41. Albeverio S., Guido D., Ponosov A., Scarlatti S., Singular traces and compact operators, J. Funct. Anal. 137 (1996), 281-302, funct-an/9308001.
  42. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001.
  43. Wodzicki M., Local invariants of spectral asymmetry, Invent. Math. 75 (1984), 143-178.
  44. Shubin M.A., Pseudodifferential operators and spectral theory, 2nd ed., Springer-Verlag, Berlin, 2001.
  45. Landi G., An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics, New Series m: Monographs, Vol. 51, Springer-Verlag, Berlin, 1997.
  46. Hörmander L., The analysis of linear partial differential operators. III. Pseudo-differential operators, Grundlehren der Mathematischen Wissenschaften, Vol. 274, Springer-Verlag, Berlin, 1994.
  47. Connes A., Moscovici H., The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), 174-243.
  48. Carey A.L., Philips J., Rennie A., Sukochev F.A., The Hochschild class of the Chern character for semifinite spectral triples, J. Funct. Anal. 213 (2004), 111-153, math.OA/0312073.
  49. Carey A.L., Philips J., Rennie A., Sukochev F.A., The local index formula in semifinite von Neumann algebras. I. Spectral flow, Adv. Math. 202 (2006), 451-516, math.OA/0411019.
  50. Carey A.L., Philips J., Rennie A., Sukochev F.A., The local index formula in semifinite von Neumann algebras. II. The even case, Adv. Math. 202 (2006), 517-554, math.OA/0411021.
  51. Carey A., Potapov D., Sukochev F., Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators, Adv. Math. 222 (2009), 1809-1849, arXiv:0807.2129.
  52. Higson N., The residue index theorem of Connes and Moscovici, in Surveys in Noncommutative Geometry, Clay Math. Proc., Vol. 6, Amer. Math. Soc., Providence, RI, 2006, 71-126.
  53. Ponge R., Noncommutative geometry and lower dimensional volumes in Riemannian geometry, Lett. Math. Phys. 83 (2008), 19-32, arXiv:0707.4201.
  54. Guido D., Isola T., Noncommutative Riemann integration and Novikov-Shubin invariants for open manifolds, J. Funct. Anal. 176 (2000), 115-152, math.OA/9802015.
  55. Englis M., Guo K., Zhang G., Toeplitz and Hankel operators and Dixmier traces on the unit ball of Cn, Proc. Amer. Math. Soc. 137 (2009), 3669-3678, arXiv:0707.2025.
  56. Nicola F., Rodino L., Dixmier traceability for general pseudo-differential operators, in C*-Algebras and Elliptic Theory II, Trends Math., Birkhäuser, Basel, 2008, 227-237.
  57. Nest R., Schrohe E., Dixmier's trace for boundary value problems, Manuscripta Math 96 (1998), 203-218.
  58. Pearson J., Bellissard J., Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets, J. Noncommut. Geom. 3 (2009), 447-480, arXiv:0802.1336.
  59. Lapidus M.L., Toward a noncommutative fractal geometry, Laplacians and volume measures on fractals, in Harmonic Analysis and Nonlinear Differential Equations (Riverside, CA, 1995), Contemp. Math., Vol. 208, Amer. Math. Soc., Providence, RI, 1997, 211-252.
  60. Rudin W., Invariant means on L, in Collection of Articles Honoring the Completion by Antoni Zygmund of 50 Years of ScientifIc Activity. III, Studia Math. 44 (1972), 219-227.
  61. Lord S., Potapov D., Sukochev F., Measures from Dixmier traces and zeta functions, J. Funct. Anal. 259 (2010), 1915-1949, arXiv:0905.1172.
  62. Lindenstrauss J., Tzafriri L., Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 97, Springer-Verlag, Berlin - New York, 1979.
  63. Wodzicki M., Noncommutative residue. I. Fundamentals, in K-Theory, Arithmetic and Geometry (Moscow, 1984-1986), Lecture Notes in Math., Vol. 1289, Springer, Berlin, 1987, 320-399.
  64. Seeley R.T., Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, 288-307.
  65. Minakshisundaram S., Pleijel A., Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242-256.
  66. Higson N., Meromorphic continuation of zeta functions associated to elliptic operators, Operator Algebras, Quantization, and Noncommutative Geometry, Contemp. Math., Vol. 365, Amer. Math. Soc., Providence, RI, 2004, 129-142.
  67. Gilkey P.B., Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
  68. Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften, Vol. 298, Springer-Verlag, Berlin, 1992.
  69. Connes A., Geometry from the spectral point of view, Lett. Math. Phys. 34 (1995), 203-238.
  70. Lord S., Sukochev F.A., Noncommutative residues and a characterisation of the noncommutative integral, Proc. Amer. Math. Soc., to appear, arXiv:0905.0187.
  71. Kalton N.J., Spectral characterization of sums of commutators. I, J. Reine Angew. Math. 504 (1998), 115-125, math.FA/9709209.
  72. Dykema K.J., Figiel T., Weiss G., Wodzicki M., Commutator structure of operator ideals, Adv. Math. 185 (2004), 1-79.
  73. Donnelly H., Quantum unique ergodicity, Proc. Amer. Math. Soc. 131 (2003), 2945-2951.
  74. Sarnak P., Arithmetic quantum chaos, in The Schur Lectures (Tel Aviv, 1992), Israel Math. Conf. Proc., Vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, 183-236.
  75. Rieffel M.A., C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429.

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