Free entropy

  • Dan Voiculescu
Lectures On Free Probability Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1738)


Variable Formula Free Convolution Free Probability Theory Free Entropy Markov Quantum Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anshelevich, M., The linearization of the central limit operator in free probability theory, preprint.Google Scholar
  2. 2.
    Avitzour, D., Free products of C*-algebras, Trans. Amer. Math. Soc. 271 (1982), 423–465.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bercovici, H. and Pata, V. (with an appendix by P. Biane), Stable laws and domains of attraction in free probability theory, Annals of Math., to appear.Google Scholar
  4. 4.
    Bercovici, H. and Voiculescu, D., Levy-Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math. 153 (1992), no. 2, 217–248.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bercovici, H. and Voiculescu, D., Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), no. 3, 733–773.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bercovici, H. and Voiculescu, D., Superconvergence to the central limit and failure of the Cramer Theorem for free random variables, Probab. Th. and Rel. Fields 102 (1995), 215–222.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bercovici, H. and Voiculescu, D., Regularity questions for free convolution, Preprint, Berkeley (1996).Google Scholar
  8. 8.
    Bhat, B.V.R. and Parthasarathy, K.R., Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory, Ann. Inst. H. Poincaré 31 (1995), no. 4, 601–651.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Biane, P., Representations of unitary groups and free convolution, Publ. RIMS Kyoto Univ. 31 (1995), 63–79.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Biane, P., Processes with free increments, Math. Z. (1998), no. 1, 143–174.Google Scholar
  11. 11.
    Biane, P., Free Brownian motion, free stochastic calculus and random matrices, in [66], 1–19.Google Scholar
  12. 12.
    Biane, P., Free hypercontractivity, Comm. Math. Phys. 184 (1997), 457–474.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Biane, P., On the free convolution with a semicircular distribution, Indiana Univ. Math. J. 46 (1997), no. 3, 705–718.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Biane, P., Representations of symmetric groups and free probability, Preprint (1998).Google Scholar
  15. 15.
    Biane, P. and Speicher, R., Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Preprint ENS (1997).Google Scholar
  16. 16.
    Bozejko, M. and Speicher, R., An example of generalized Brownian motion, Commun. Math. Phys. 137 (1991), 519–531.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bozejko, M. and Speicher, R., An example of generalized Brownian motion II, Quantum Probability and Related Topics (L. Accardi, ed.), vol. VI, World Scientific, Singapore, 1991, pp. 219–236.CrossRefGoogle Scholar
  18. 18.
    Dixmier, J., Les C*-Algebres et leur Représentations, Gauthier-Villars, Paris, 1964.zbMATHGoogle Scholar
  19. 19.
    Dixmier, J., Les Algebres d'Opérateurs dans l'Espace Hilbertien, Gauthier-Villars, Paris, 1969.zbMATHGoogle Scholar
  20. 20.
    Douglas, M.R., Stochastic master fields, Phys. Lett. B344, 117–126.Google Scholar
  21. 21.
    Dykema, K.J., On certain free product factors via an extended matrix model, J. Funct. Anal. 112, 31–60.Google Scholar
  22. 22.
    Fagnola, F., On quantum stochastic integration with respect to “free” noises, Quantum Probability and Related Topics (L. Accardi, ed.), vol. VI, World Scientific, Singapore, 1991, pp. 285–304.CrossRefGoogle Scholar
  23. 23.
    Gopakumar, R. and Gross, D.J., Mastering the master field, Nucl. Phys. B451, 379–415.Google Scholar
  24. 24.
    Gromov, M. and Milman, V.D., A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), 843–854.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Haagerup, U., On Voiculescu's R-and S-transforms for free noncommuting random variables, in [66], 127–148.Google Scholar
  26. 26.
    Kadison, R. and Ringrose, J., Fundamentals of the Theory of Operator Algebras, (3 Volumes) Birkhäuser, Boston.Google Scholar
  27. 27.
    Kummerer, B., Markov dilations on W*-algebras, J. Funct. Anal. 63 (1985), 139–177.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kummerer, B. and Speicher, R., Stochastic integration on the Cuntz algebra, J. Funct. Anal. 103 (1992), 372–408.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Maassen, H., Addition of freely independent random variables, J. Funct. Anal. 106 (1992), 409–438.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nica, A., Asymptotically free families of random unitaries in symmetric groups, Pacific J. Math. 157 (1993), no. 2, 295–310.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Nica, A., R-transforms of free joint distributions and non-crossing partitions, J. Funct. Anal. 135 (1996), 271–296.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nica, A. and Speicher, R. (with an appendix by D. Voiculescu), On the multiplication of free N-tuples of noncommutative random variables, Amer. J. Math. 118 (1996), 799–837.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nica, A. and Speicher, R., A “Fourier transform” for multiplicative functions on noncrossing partitions, J. of Algebraic Combinatorics 6 (1997), 141–160.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Nica, A. and Speicher, R., Commutators of free random variables, Duke Math. J., to appear.Google Scholar
  35. 35.
    Nica, A.; Shlyakhtenko, D. and Speicher, R., Some minimization problems for the free analogue of the Fisher information, Preprint (1998).Google Scholar
  36. 36.
    Sauvageot, J.L., Markov quantum semigroups admit covariant Markov C*-dilations, Commun. Math. Phys. 106 (1986), 91–103.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Shannon, C.E. and Weaver, W., The Mathematical Theory of Communications, University of Illinois Press, 1963.Google Scholar
  38. 38.
    Shlyakhtenko, D., Limit distributions of matrices with bosonic and fermionic entries, in [66], 241–252.Google Scholar
  39. 39.
    Shlyakhtenko, D., Free entropy with respect to a completely positive map, Preprint (1998).Google Scholar
  40. 40.
    Shlyakhtenko, D., Random Gaussian band matrices and freeness with amalgamation, International Math. Res. Notices (1996), no. 20, 1013–1025.Google Scholar
  41. 41.
    Shiryayev, A.N., Probability, Springer, 1984.Google Scholar
  42. 42.
    Singer, I.M., On the master field in two dimensions, Functional Analysis on the Eve of the 21st Century in Honor of the 80th Birthday of I.M. Gelfand, Progress in Mathematics, vol. 131, pp. 263–283.Google Scholar
  43. 43.
    Speicher, R., A new example of “independence” and “white noise”, Prob. Th. Rel. Fields 84 (1990), 141–159.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Speicher, R., Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Memoirs of the AMS 627 (1998).Google Scholar
  45. 45.
    Speicher, R., Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), 611–628.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Speicher, R., Free probability theory and non-crossing partitions, Seminaire Lotharingien de Combinatoire B39c (1997).Google Scholar
  47. 47.
    Stratila, S. and Zsido, L., Lectures on von Neumann Algebras, Editura Academia and Abacus Press, 1979.Google Scholar
  48. 48.
    Szarek, S.V. and Voiculescu, D., Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality, Comun. Math. Phys. 178 (1996), 563–570.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Voiculescu, D., Symmetries of some reduced free product C*-algebras, Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math. 1132 (1985), Springer Verlag, 556–588.Google Scholar
  50. 50.
    Voiculescu, D., Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), 323–346.MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Voiculescu, D., Multiplication of certain non-commuting random variables, J. Operator Theory 18 (1987), 223–235.MathSciNetzbMATHGoogle Scholar
  52. 52.
    Voiculescu, D., Limit laws for random matrices and free products, Invent. Math. 104 (1991), 201–220.MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Voiculescu, D., Circular and semicircular systems and free product factors, Progr. Math. 92 (1990), Birkhäuser, Boston, 45–60.MathSciNetzbMATHGoogle Scholar
  54. 54.
    Voiculescu, D., Free non-commutative random variables, random matrices and the II1-factors of free groups, Quantum Probability and Related Topics (L. Accardi, ed.), vol. VI, World Scientific, Boston, 1991, pp. 473–487.CrossRefGoogle Scholar
  55. 55.
    Voiculescu, D., Free probability theory: random matrices and von Neumann algebras, Proceedings of the International Congress of Mathematicians, Zürich 1994, Birkhäuser, Boston (1995), 227–241.Google Scholar
  56. 56.
    Voiculescu, D., Operations on certain non-commuting operator-valued random variables, Astérisque (1995), no. 232, 243–275.Google Scholar
  57. 57.
    Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory I, Commun. Math. Phys. 155 (1993), 71–92.MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory II, Invent. Math. 118 (1994), 411–440.MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory III: the absence of Cartan subalgebras, Geometric and Funct. Anal. 6 (1996), no. 1, 172–199.MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory IV: Maximum entropy and freeness, in [66], 293–302.Google Scholar
  61. 61.
    Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory V: Noncommutative Hilbert transforms, Invent. Math. 132 (1998), 182–227.MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory VI: liberation and mutual free information,, preprint 1998.Google Scholar
  63. 63.
    Voiculescu, D., The derivative of order 1/2 of a free convolution by a semicircle distribution, Indiana Univ. Math. J. 46 (1997), no. 3, 697–703.MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Voiculescu, D., A strengthened asymptotic freeness result for random matrices with applications to free entropy, International Math. Res. Notices (1998), no. 1, 41–63.Google Scholar
  65. 65.
    Voiculescu, D., A note on free Markovianity, (in preparation).Google Scholar
  66. 66.
    Voiculescu, D. (editor), Free Probability Theory, Fields Institute Communications, Vol. 12, American Math. Soc., 1997.Google Scholar
  67. 67.
    Voiculescu, D.; Dykema, K.J. and Nica, A., Free Random Variables, CRM Monograph Series, Vol. 1, American Math. Soc., 1992.Google Scholar
  68. 68.
    Wigner, E., Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62 (1955), 548–564.MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Wigner, E., On the distribution of the roots of certain symmetric matrices, Ann. Math. 67 (1958), 325–327.MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Xu, F., A random matrix model from two-dimensional Yang-Mills theory, Commun. Math. Phys. 190 (1997), 287–307.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2000

Authors and Affiliations

  • Dan Voiculescu

There are no affiliations available

Personalised recommendations