Dirichlet Forms on Noncommutative Spaces

  • Fabio Cipriani
Part of the Lecture Notes in Mathematics book series (LNM, volume 1954)


We show how Dirichlet forms provide an approach to potential theory of noncommutative spaces based on the notion of energy. The correspondence with KMS-symmetric Markovian semigroups is explained in details and applied to the dynamical approach to equilibria of quantum spin systems. Second part focuses on the differential calculus underlying a Dirichlet form. Applications are given in Riemannian Geometry to a potential theoretic characterization of spaces with positive curvature and to the construction of Fredholm modules in Noncommutative Geometry.


Dirichlet Form Continuous Semigroup Dirichlet Space Markovian Semigroup Cyclic Vector 
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  1. [Abk] W. Abikoff, The uniformization theorem, Am. Math. Montly, 88, n. 8, (1981), 574–592.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [AC] L. Accardi, C. Cecchini, Conditional expectations in von Neuman algebras and a theorem of Takesaki. J. Funct. Anal. 45 (1982), 245–273.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [AFLe] L. Accardi, A. Frigerio, J.T. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18, n. 1, (1982), 97–133.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [AFLu] L. Accardi, A. Frigerio, Y. G. Lu, The weak coupling limit as a quantum functional central limit, Comm. Math. Phys., 131, n. 3, (1990), 537570.MathSciNetCrossRefGoogle Scholar
  5. [AHK1] S. Albeverio, R. Hoegh-Krohn, Dirichlet forms and Markovian semigroups on C*-algebras, Comm. Math. Phys. 56 (1977), 173–187.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [AHK2] S. Albeverio, R. Hoegh-Krohn, Frobenius theory for positive maps on von Neumann algebras, Comm. Math. Phys. 64 (1978), 83–94.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [A1] R. Alicki, On the detailed balance condition for non-Hamiltonian systems, Rep. Math. Phys. 10 (1976), 249–258.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [Ara] H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math., 50 (1974), 309–354zbMATHMathSciNetGoogle Scholar
  9. [Arv] W. Arveson, “An invitation to C*-algebra”, Graduate Text in Mathematics 39, x+106 pages, Springer-Verlag, Berlin, Heidelber, New York, 1976.Google Scholar
  10. [At] M.F. Atiyah, Global theory of elliptic operators, Proc. Internat. Conf. on Proc. Internat. Conf. on Functional Analysis and Related Topics, (Tokyo, 1969) (1970), 21–30, Univ. of Tokyo Press, Tokyo.Google Scholar
  11. [AtS] M.F. Atiyah, G. B. Singer, The Index of elliptic operators on compact manifolds, Bull. A.M.S. 69 (1963), 422–433.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [BKP1] C. Bahn, C.K. Ko, Y.M. Park, Dirichlet forms and symmetric Markovian semigroups on ℤ2 von Neumann algebras. Rev. Math. Phys. 15 (2003), no. 8, 823–845.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [BKP2] C. Bahn, C.K. Ko, Y.M. Park, Dirichlet forms and symmetric Markovian semigroups on CCR Algebras with quasi-free states. Rev. Math. Phys. 44, (2003), 723–753.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [B] A. Barchielli, Continual measurements in quantum mechanics and quantum stochastic calculus, Lectures Notes Math. 1882 (2006), 207–292.MathSciNetCrossRefGoogle Scholar
  15. [BD1] A. Beurling and J. Deny, Espaces de Dirichlet I: le cas élémentaire, Acta Math. 99 (1958), 203–224.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [BD2] A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. 45 (1959), 208–215.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [B1] P. Biane, Logarithmic Sobolev Inequalities, Matrix Models and Free Entropy, Acta Math. Sinica, English Series. 19 (2003), 497–506.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [Boz] M. Bozejko, Positive definite functions on the free group and the noncommutative Riesz product, Bollettino U.M.I. 5-A (1986), 13–21.MathSciNetGoogle Scholar
  19. [BR1] Bratteli O., Robinson D.W., “Operator algebras and Quantum Statistical Mechanics 1”, Second edition, 505 pages, Springer-Verlag, Berlin, Heidelberg, New York, 1887.Google Scholar
  20. [BR2] Bratteli O., Robinson D.W., “Operator algebras and Quantum Statistical Mechanics 2”, Second edition, 518 pages, Springer-Verlag, Berlin, Heidelberg, New York, 1997.zbMATHGoogle Scholar
  21. [Cip1] F. Cipriani, Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal. 147 (1997), 259–300.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [Cip2] F. Cipriani, The variational approach to the Dirichlet problem in C*-algebras, Banach Center Publications 43 (1998), 259–300.MathSciNetGoogle Scholar
  23. [Cip3] F. Cipriani, Perron theory for positive maps and semigroups on von Neumann algebras, CMS Conf. Proc., Amer. Math. Soc., Providence, RI 29 (2000), 115–123.MathSciNetGoogle Scholar
  24. [Cip4] F. Cipriani, Dirichlet forms as Banach algebras and applications, Pacific J. Math. 223 (2006), no. 2, 229–249.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [CFL] F. Cipriani, F. Fagnola, J.M. Lindsay, Spectral analysis and Feller property for quantum Ornstein-Uhlenbeck semigroups, Comm. Math. Phys. 210 (2000), 85–105.zbMATHMathSciNetCrossRefGoogle Scholar
  26. [CS1] F. Cipriani, J.-L. Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. Anal. 201 (2003), no. 1, 78–120.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [CS2] F. Cipriani, J.-L. Sauvageot, Noncommutative potential theory and the sign of the curvature operator in Riemannian geometry, Geom. Funct. Anal. 13 (2003), no. 3, 521–545.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [CS3] F. Cipriani, J.-L. Sauvageot, Strong solutions to the Dirichlet problem for differential forms: a quantum dynamical semigroup approach, Contemp. Math, Amer. Math. Soc., Providence, RI 335 (2003), 109–117.MathSciNetGoogle Scholar
  29. [CS4] F. Cipriani, J.-L. Sauvageot, Fredholm modules on p.c.f. self-similar fractals and their conformal geometry, arXiv:0707.0840v 1 [math.FA] (5 Jul 2007), 16 pages.Google Scholar
  30. [Co1] A. Connes, Caracterisation des espaces vectoriels ordonnés sous-jacents aux algbres de von Neumann, Ann. Inst. Fourier (Grenoble) 24 (1974), 121–155.zbMATHMathSciNetGoogle Scholar
  31. [Co2] A. Connes, “Noncommutative Geometry”, Academic Press, 1994.Google Scholar
  32. [Co3] A. Connes, On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), no. 2, 248–253.zbMATHMathSciNetCrossRefGoogle Scholar
  33. [CST] A. Connes, D. Sullivan, N. Teleman, Quasconformal mappings, operators on Hilbert space, and local formulae for characteristic classes, Preprint I.H.E.S. M/38 (1993).Google Scholar
  34. [CE] E. Christensen, D.E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J. London. Math. Soc. 20 (1979), 358–368.zbMATHMathSciNetCrossRefGoogle Scholar
  35. [D1] E.B. Davies, (1976), “Quantum Theory of Open Systems”, 171 pages, Academic Press, London U.K.zbMATHGoogle Scholar
  36. [D2] E.B. Davies, Analysis on graphs and noncommutative geometry, J. Funct. Anal. 111 (1993), 398–430.zbMATHMathSciNetCrossRefGoogle Scholar
  37. [DL1] E.B. Davies, J.M. Lindsay, Non-commutative symmetric Markov semigroups, Math. Z. 210 (1992), 379–411.zbMATHMathSciNetCrossRefGoogle Scholar
  38. [DL2] E.B. Davies, J.M. Lindsay, Superderivations and symmetric Markov semigroups, Comm. Math. Phys. 157 (1993), 359–370.zbMATHMathSciNetCrossRefGoogle Scholar
  39. [DR1] E.B. Davies, O.S. Rothaus, Markov semigroups on C*-bundles, J. Funct. Anal. 85 (1989), 264–286.zbMATHMathSciNetCrossRefGoogle Scholar
  40. [DR2] E.B. Davies, O.S. Rothaus, A BLW inequality for vector bundles and applications to spectral bounds, J. Funct. Anal. 86 (1989), 390–410.zbMATHMathSciNetCrossRefGoogle Scholar
  41. [Del] G.F. DelľAntonio, Structure of the algebras of some free systems, Comm. Math. Phys. 9 (1968), 81–117.MathSciNetCrossRefGoogle Scholar
  42. [Den] J. Deny, Méthodes hilbertien en thorie du potentiel, Potential Theory (C.I.M.E., I Ciclo, Stresa), Ed. Cremonese Roma, 1970, 85, 121–201.Google Scholar
  43. [Dir] P.A.M. Dirac, The quantum theory of the electron, Pro. Roy. Soc. of London A 117 (1928), 610–624.CrossRefGoogle Scholar
  44. [Do] J.L. Doob, “Classical potential theory and its probabilistic counterpart”, Springer-Verlag, New York, 1984.zbMATHGoogle Scholar
  45. [Dix1] J. Dixmier, “Les C*-algèbres et leurs représentations”, Gauthier-Villars, Paris, 1969.Google Scholar
  46. [Dix2] J. Dixmier, “Les algébres ďoperateurs dans les espaces hilbertienne (algébres de von Neumann)”, Gauthier-Villars, Paris, 1969.Google Scholar
  47. [F] W. Faris, Invariant cones and uniqueness of the ground state for Fermion systems, J. Math. Phys. 13 (1972), 1285–1290.zbMATHMathSciNetCrossRefGoogle Scholar
  48. [FU] F. Fagnola, V. Umanita', Generators of detailed balance quantum Markov semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), 335–363.zbMATHMathSciNetCrossRefGoogle Scholar
  49. [FOT] M. Fukushima, Y. Oshima, M. Takeda, “Dirichlet Forms and Symmetric Markov Processes”, de Gruyter Studies in Mathematics, 1994.Google Scholar
  50. [GL1] S. Goldstein, J.M. Lindsay, Beurling-Deny conditions for KMS-symmetric dynamical semigroups, C. R. Acad. Sci. Paris, Ser. I 317 (1993), 1053–1057.zbMATHMathSciNetGoogle Scholar
  51. [GL2] S. Goldstein, J.M. Lindsay, KMS-symmetric Markov semigroups, Math. Z. 219 (1995), 591–608.zbMATHMathSciNetCrossRefGoogle Scholar
  52. [GL3] S. Goldstein, J.M. Lindsay, Markov semigroup KMS-symmetric for a weight, Math. Ann. 313 (1999), 39–67.zbMATHMathSciNetCrossRefGoogle Scholar
  53. [GKS] V. Gorini, A. Kossakowski, E.C.G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17 (1976), 821–825.MathSciNetCrossRefGoogle Scholar
  54. [GS] D. Goswami, K.B. Sinha, “Quantum stochastic processes and noncommutative geometry”, Cambridge Tracts in Mathematics 169, 290 pages, Cambridge University press, 2007.Google Scholar
  55. [G1] L. Gross, Existence and uniqueness of physical ground states, J. Funct. Anal. 10 (1972), 59–109.CrossRefGoogle Scholar
  56. [G2] L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form, Duke Math. J. 42 (1975), 383–396.zbMATHMathSciNetCrossRefGoogle Scholar
  57. [GIS] D. Guido, T. Isola, S. Scarlatti, Non-symmetric Dirichlet forms on semifinite von Neumann algebras, J. Funct. Anal. 135 (1996), 50–75.zbMATHMathSciNetCrossRefGoogle Scholar
  58. [H1] U. Haagerup, Standard forms of von Neumann algebras, Math. Scand. 37 (1975), 271–283.MathSciNetGoogle Scholar
  59. [H2] U. Haagerup, Lp-spaces associated with an arbitrary von Neumann algebra, Algebre ďopérateurs et leur application en Physique Mathematique. Colloques Internationux du CNRS 274, Ed. du CNRS, Paris 1979, 175–184.Google Scholar
  60. [H3] U. Haagerup, All nuclear C *-algebras are amenable, Invent. Math. 74 (1983), no. 2, 305–319.zbMATHMathSciNetCrossRefGoogle Scholar
  61. [HHW] R. Haag, N.M. Hugenoltz, M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215–236.zbMATHMathSciNetCrossRefGoogle Scholar
  62. [Io] B. Iochum, “Cones autopolaires et algebres de Jordan”, Lecture Notes in Mathematics, vol. 1049, Springer-Verlag, Berlin, 1984.Google Scholar
  63. [Ki] J. Kigami, “Analysis on Fractals,”, Cambridge Tracts in Mathematics vol. 143, Cabridge: Cambridge University Press, 2001.Google Scholar
  64. [KP] C.K. Ko, Y.M. Park, Construction of a family of quantum Ornstein-Uhlenbeck semigroups, J. Math. Phys. 45, (2004), 609–627.zbMATHMathSciNetCrossRefGoogle Scholar
  65. [Ko] H. Kosaki, Application of the complex interpolation method to a von Neumann algebra: non-commutative Lp-spaces, J. Funct. Anal. 56 (1984), 29–78.zbMATHMathSciNetCrossRefGoogle Scholar
  66. [KFGV] A. Kossakowski, A. Frigerio, V. Gorini, M. Verri, Quantum detailed balance and the KMS condition, Comm. Math. Phys. 57 (1977), 97–110.zbMATHMathSciNetCrossRefGoogle Scholar
  67. [Kub] R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12 (1957), 570–586.MathSciNetCrossRefGoogle Scholar
  68. [LM] H.B. Lawson JR., M.-L. Michelson, “Spin Geometry”, Princeton University Press, Princeton New Jersey, 1989.zbMATHGoogle Scholar
  69. [LJ] Y, Le Jan, Mesures associés a une forme de Dirichlet. Applications., Bull. Soc. Math. France 106 (1978), 61–112.Google Scholar
  70. [Lig] T.M. Liggett, “Interacting Particle Systems”, Grund. math. Wissen., 276. Springer-Verlag, Berlin, 1985.Google Scholar
  71. [Lin] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976), 119–130.zbMATHMathSciNetCrossRefGoogle Scholar
  72. [LR] E.H. Lieb, D.W. Robinson, The finite group velocity of quantum spin systems, Comm. Math. Phys. 28 (1972), 251–257.MathSciNetCrossRefGoogle Scholar
  73. [MZ1] A. Majewski, B. Zegarlinski, Quantum stochastic dynamics. I. Spin systems on a lattice, Math. Phys. Electron. J. 1 (1995), Paper 2, 1–37.MathSciNetGoogle Scholar
  74. [MZ2] A. Majewski, B. Zegarlinski, On Quantum stochastic dynamics on noncommutative L p-spaces, Lett. Math. Phys. 36 (1996), 337–349.zbMATHMathSciNetCrossRefGoogle Scholar
  75. [MOZ] A. Majewski, R. Olkiewicz, B. Zegarlinski, Dissipative dynamics for quantum spin systems on a lattice, J. Phys. A 31 (1998), no. 8, 2045–2056.zbMATHMathSciNetCrossRefGoogle Scholar
  76. [Mat1] T. Matsui, Markov semigroups on UHF algebras, Rev. Math. Phys. 5 (1993), no. 3, 587–600.zbMATHMathSciNetCrossRefGoogle Scholar
  77. [Mat2] T. Matsui, Markov semigroups which describe the time evolution of some higher spin quantum models, J. Funct. Anal. 116 (1993), no. 1, 179–198.zbMATHMathSciNetCrossRefGoogle Scholar
  78. [Mat3] T. Matsui, Quantum statistical mechanics and Feller semigroups, Quantum probability communications QP-PQ X 31 (1998), 101–123.MathSciNetGoogle Scholar
  79. [Mok] G. Mokobodzki, Fermabilité des formes de Dirichlet et inégalité de type Poincaré, Pot. Anal. 4 (1995), 409–413.zbMATHMathSciNetCrossRefGoogle Scholar
  80. [MS] D.C. Martin, J. Schwinger, Theory of many-particle systems. I., J. Phys. Rev. 115 (1959), 1342–1373.zbMATHMathSciNetCrossRefGoogle Scholar
  81. [MvN] F.J. Murray, J. von Neumann, On rings of operators, Ann. Math. 37 (1936), 116–229. On rings of operators II, Trans. Amer. Math. Soc. 41 (1937), 208–248. On rings of operators IV, Ann. Math. 44 (1943), 716–808.CrossRefGoogle Scholar
  82. [Ne] E. Nelson, Notes on non-commutative integration, Ann. Math. 15 (1974), 103–116.zbMATHGoogle Scholar
  83. [OP] M. Ohya, D. Petz, “Quantum Entropy and its use”, viii+335 pages, Springer-Verlag, Berlin, Heidelberg, New York, 1993.zbMATHGoogle Scholar
  84. [P1] Y.M. Park, Construction of Dirichlet forms on standard forms of von Neumann algebras. Infinite Dim. Anal., Quantum. Prob. and Related Topics 3 (2000), 1–14.zbMATHCrossRefGoogle Scholar
  85. [P2] Y.M. Park, Ergodic property of Markovian semigroups on standard forms of von Neumann algebras J. Math. Phys. 46 (2005), 113507.MathSciNetCrossRefGoogle Scholar
  86. [P3] Y.M. Park, Remarks on the structure of Dirichlet forms on standard forms of von Neumann algebras Infin. Dimens. Anal. Quantum Probab. Relat. 8 no. 2 (2005), 179–197.zbMATHMathSciNetCrossRefGoogle Scholar
  87. [Pa4r] K.R. Parthasarathy, “An introduction to quantum stochastic calculus”, Monographs in Mathematics, 85. Birkhuser Verlag, Basel, 1992.Google Scholar
  88. [Ped] G. Pedersen, “C*-algebras and their authomorphisms groups”, London Mathematical Society Monographs, 14. Academic Press, Inc., London-New York, 1979.Google Scholar
  89. [Pen] R.C. Penney, Self-dual cones in Hilbert space., J. Funct. Anal. 21 (1976), 305–315.zbMATHMathSciNetCrossRefGoogle Scholar
  90. [Pet] P. Petersen, “Riemannian Geometry”, Graduate Text in Mathematics 171, xii +432 pages, Springer-Verlag, Berlin, Heidelberg, New York, 1998.Google Scholar
  91. [Petz] D. Petz, A dual in von Neumann algebra Quart. J. Math. Oxford 35 (1984), 475–483.zbMATHMathSciNetCrossRefGoogle Scholar
  92. [RS] M. Reed, B. Simon, “Methods of modern mathematical physics. II. Fourier Analysis, Self-adjointness”, Academic Press, xi+361 pages, New York-London, 1975.Google Scholar
  93. [Ric] C.E. Rickart, “General Theory of Banach Algebras”, D. van Nostrand Company Inc. Princeton New Jersey, 1960.zbMATHGoogle Scholar
  94. [S1] J.-L. Sauvageot, Sur le produit tensoriel relatif ďespaces de Hilbert, J. Operator Theory. 9 (1983), 237–252.zbMATHMathSciNetGoogle Scholar
  95. [S2] J.-L. Sauvageot, Tangent bimodule and locality for dissipative operators on C*-algebras, Quantum Probability and Applications IV, Lecture Notes in Math. 1396 (1989), 322–338.MathSciNetCrossRefGoogle Scholar
  96. [S3] J.-L. Sauvageot, Quantum differential forms, differential calculus and semigroups, Quantum Probability and Applications V, Lecture Notes in Math. 1442 (1990), 334–346.MathSciNetCrossRefGoogle Scholar
  97. [S4] J.-L. Sauvageot, Semi-groupe de la chaleur transverse sur la C*-algèbre ďun feulleitage riemannien, C.R. Acad. Sci. Paris Sér. I Math. 310 (1990), 531–536.zbMATHMathSciNetGoogle Scholar
  98. [S5] J.-L. Sauvageot, Le probleme de Dirichlet dans les C*-algèbres, J. Funct. Anal. 101 (1991), 50–73.zbMATHMathSciNetCrossRefGoogle Scholar
  99. [S6] J.-L. Sauvageot, From classical geometry to quantum stochastic flows: an example, Quantum probability and related topics, QP-PQ, VII, 299–315, World Sci. Publ., River Edge, NJ, 1992.Google Scholar
  100. [S7] J.-L. Sauvageot, Semi-groupe de la chaleur transverse sur la C*-algèbre ďun feulleitage riemannien, J. Funct. Anal. 142 (1996), 511–538.zbMATHMathSciNetCrossRefGoogle Scholar
  101. [S8] J.-L. Sauvageot, Strong Feller semigroups on C*-algebras, J. Op. Th. 42 (1999), 83–102.zbMATHMathSciNetGoogle Scholar
  102. [Se] E.A. Segal, A non-commutative extension of abstract integration, Ann. of Math. 57 (1953), 401–457.MathSciNetCrossRefGoogle Scholar
  103. [SU] R. Schrader, D. A. Uhlenbrock, Markov structures on Clifford algebras, Jour. Funct. Anal. 18 (1975), 369–413.zbMATHMathSciNetCrossRefGoogle Scholar
  104. [Sew] G. Sewell, “Quantum Mechanics and its Emergent Macrophysics”, Princeton University Press, 292 pages, Princeton and Oxford, 2002.Google Scholar
  105. [Sti] W.F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1975), 211–216.MathSciNetCrossRefGoogle Scholar
  106. [T1] M. Takesaki, “Structure of factors and automorphism groups”, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, R.I. 51 (1983).Google Scholar
  107. [T2] M. Takesaki, “Theory of Operator Algebras I”, Encyclopedia of Mathematical Physics, 415 pages, Springer-Verlag, Berlin, Heidelberg, New York, 2000.Google Scholar
  108. [Te] M. Terp, Interpolation spaces between a von Neumann algebra and its predual, J. Operator Theory 8 (1982), 327–360.zbMATHMathSciNetGoogle Scholar
  109. [V] A. Van Daele, A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Funct. Anal. 15 (1974), 379–393.Google Scholar
  110. [Voi1] D.V. Voiculescu, Lectures on Free Probability theory., Lecture Notes in Math. 1738 (2000), 279–349.MathSciNetGoogle Scholar
  111. [Voi2] D.V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, Invent. Math. 132 (1998), 189–227.zbMATHMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Fabio Cipriani
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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