Free Probability Theory: Random Matrices and von Neumann Algebras

  • Dan Voiculescu
Conference paper

Abstract

Independence in usual noncommutative probability theory (or in quantum physics) is based on tensor products. This lecture is about what happens if tensor products are replaced by free products. The theory one obtains is highly noncommutative: freely independent random variables do not commute in general. Also, at the level of groups, this means instead of ℤ n we will consider the noncommutative free group F(n) = ℤ* ⋯ *ℤ or, looking at the Cayler graphs, a lattice is replaced by a homogeneous tree.

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References

  1. [1]
    H. Bercovici and D. Voiculescu, Levy-Hincin type theorems for multiplicative and additive-free convolution, Pacific J. Math. 153 (1992), no. 2, 217–248.MathSciNetCrossRefGoogle Scholar
  2. [2]
    H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42, no. 3 (1993), 733–773.MathSciNetCrossRefGoogle Scholar
  3. [3]
    H. Bercovici and D. Voiculescu, Superconvergence to the central limit and failure of the Cramer theorem for free random variables, preprint 1994.Google Scholar
  4. [4]
    P. Biane, Permutation model for semicircular systems and quantum random walks, preprint 1993.Google Scholar
  5. [5]
    P. Biane, Representations of unitary groups and free convolution, Publ. RIMS Kyoto Univ. 31 (1995), 63–79.MathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Bozejko and R. Speicher, An example of generalized Brownian motion, Comm. Math. Phys. 137 (1991), 519–531; II in Quantum Probab. & Related Topics VII (1990), 67–77.MathSciNetCrossRefGoogle Scholar
  7. [7]
    A. Connes, Classification of injective factors, Ann. of Math. (2) 104 (1976), 73–115.MathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Connes, A factor of type IIi with countable fundamental group, J. Operator Theory 4 (1980), 151–153.MathSciNetMATHGoogle Scholar
  9. [9]
    A. Connes and V. F. R. Jones, Property T for von Neumann algebras, Bull. London Math. Soc. 17 (1985), 57–62.MathSciNetCrossRefGoogle Scholar
  10. [10]
    K. J. Dykema, On certain free product factors via an extended matrix model, J. Funct. Anal. 112 (1993), 31–60.MathSciNetCrossRefGoogle Scholar
  11. [11]
    K. J. Dykema, Free products of hyperfinite von Neumann algebras and free dimension, Duke Math. J. 69 (1993), 97–119.MathSciNetCrossRefGoogle Scholar
  12. [12]
    K. J. Dykema, Interpolated free group factors, Pacific J. Math. 163 (1994), 123–135.MathSciNetCrossRefGoogle Scholar
  13. [13]
    K. J. Dykema, Amalgamated free products of multi-matrix algebras and a construction of subfactors of a free group, preprint 1994.Google Scholar
  14. [14]
    U. Haagerup, Quasitraces on exact C*-algberas are traces, preprint 1991.Google Scholar
  15. [15]
    V. F. R. Jones, Index of subfactors, Invent. Math. 72 (1983), 1–25.MathSciNetCrossRefGoogle Scholar
  16. [16]
    H. Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (1992), 409–438.MathSciNetCrossRefGoogle Scholar
  17. [17]
    F. J. Murray and J. von Neumann, On rings of operators, IV, Ann. of Math. (2) 44 (1943), 716–808.MathSciNetCrossRefGoogle Scholar
  18. [18]
    A. Nica, Asymptotically free families of random unitaries in symmetric groups, Pacific J. Math. 157 no. 2, (1993), 295–310.MathSciNetCrossRefGoogle Scholar
  19. [19]
    A. Nica, Expected number of cycles of length c of a free word in k permutations, preprint.Google Scholar
  20. [20]
    A. Nica, R-transforms of free joint distributions, and non-crossing partitions, to appear in J. of Funct. Anal.Google Scholar
  21. [21]
    A. Nica, A one-parameter family of transforms linearizing convolution laws for probability distributions, Commun. Math. Phys. 168 (1995), 187–207.MathSciNetCrossRefGoogle Scholar
  22. [22]
    S. Popa, Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math. 111 (1993), 375–405.MathSciNetCrossRefGoogle Scholar
  23. [23]
    F. Radulescu, A one-parameter group of automorphism of L(F ∞ ) ⊗ B(H) scaling the trace, C. R. Acad. Sci. Paris, t.314, Serie I (1992), 1027–1032.MathSciNetMATHGoogle Scholar
  24. [24]
    F. Radulescu, Stable equivalence of the weak closures of free groups convolution algebras, Comm. Math. Phys. 156 (1993), 17–36.MathSciNetCrossRefGoogle Scholar
  25. [25]
    F. Radulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, Invent. Math. 115, no. 2 (1994), 347–389.MathSciNetCrossRefGoogle Scholar
  26. [26]
    F. Radulescu, A type IIIλfactor with core isomorphic to the free group von Neumann algebra of a free group tensor B(H), preprint.Google Scholar
  27. [27]
    F. Radulescu, On the von Neumann algebra of Toeplitz operators with automorphic symbol, preprint 1993.Google Scholar
  28. [28]
    R. Speicher, A new example of independence and ‘white noise’, Probab. Theory Related Fields 84 (1990), 141–159.MathSciNetCrossRefGoogle Scholar
  29. [29]
    R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), 611.MathSciNetCrossRefGoogle Scholar
  30. [30]
    R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Habilitationsschrift, Universität Heidelberg, 1994.MATHGoogle Scholar
  31. [31]
    E. Störmer, Entropy of some automorphisms of the II 1-factors of free groups in infinite number of generators, preprint.Google Scholar
  32. [32]
    D. Voiculescu, Symmetries of some reduced free product C*-algebras, in Lecture Notes in Math., vol. 1132 Springer-Verlag, Berlin and New York (1985), 556–588.Google Scholar
  33. [33]
    D. Voiculescu, Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), 323–346.MathSciNetCrossRefGoogle Scholar
  34. [34]
    D. Voiculescu, Multiplication of certain non-commuting random variables, J. Operator Theory 18 (1987), 223–235.MathSciNetMATHGoogle Scholar
  35. [35]
    D. Voiculescu, Dual algebraic structures on operator algebras related to free products, J. Operator Theory 17 (1987), 85–98.MathSciNetMATHGoogle Scholar
  36. [36]
    D. Voiculescu, Operations on certain noncommutative operator-valued random vari-ables, preprint, Berkeley, 1992.Google Scholar
  37. [37]
    D. Voiculescu, Circular and semicircular systems and free product, in Progr. Math. vol. 92 (1990), 45–60, Birkhäuser, Boston, MA.MathSciNetMATHGoogle Scholar
  38. [38]
    D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), 201–220.MathSciNetCrossRefGoogle Scholar
  39. [39]
    D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, I, Comm. Math. Phys. 155 (1993), 71–92; II, Invent. Math., 118 (1994), 411–440.MathSciNetCrossRefGoogle Scholar
  40. [40]
    D. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. I, American Mathematical Society, Providence, RI, 1992.CrossRefGoogle Scholar
  41. [41]
    E. Wigner, Characteristic vectors of bordered matrices with infinite dimension, Ann. of Math. (2) 62 (1955), 548–564.MathSciNetCrossRefGoogle Scholar
  42. [42]
    E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. 67 (1958), 325–327.MathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäser Verlag, Basel, Switzerland 1995

Authors and Affiliations

  • Dan Voiculescu
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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