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Free Entropy χ: The Non-microstates Approach via Free Fisher Information

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Part of the Fields Institute Monographs book series (FIM, volume 35)

Abstract

In classical probability theory, there exist two important concepts which measure the amount of “information” of a given distribution. These are the Fisher information and the entropy. There exist various relations between these quantities, and they form a cornerstone of classical probability theory and statistics. Voiculescu introduced free probability analogues of these quantities, called free Fisher information and free entropy, denoted by Φ and χ, respectively. However, there remain some gaps in our present understanding of these quantities. In particular, there exist two different approaches, each of them yielding a notion of entropy and Fisher information. One hopes that finally one will be able to prove that both approaches give the same result, but at the moment this is not clear. Thus, for the time being, we have to distinguish the entropy χ and the free Fisher information Φ coming from the first approach (via microstates) and the free entropy χ and the free Fisher information Φ coming from the second non-microstates approach (via conjugate variables).

References

  1. 30.
    H. Bercovici, D. Voiculescu, Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 40.
    P. Biane, M. Capitaine, A. Guionnet, Large deviation bounds for matrix Brownian motion. Invent. Math. 152(2), 433–459 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 51.
    I. Charlesworth, D. Shlyakhtenko, Free entropy dimension and regularity of non-commutative polynomials. J. Funct. Anal. 271(8), 2274–2292 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 63.
    Y. Dabrowski, A note about proving non-Γ under a finite non-microstates free Fisher information assumption. J. Funct. Anal. 258(11), 3662–3674 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 109.
    P. Koosis, Introduction toH p Spaces. Cambridge Tracts in Mathematics, vol. 115, 2nd edn. (Cambridge University Press, Cambridge, 1998).Google Scholar
  6. 121.
    T. Mai, R. Speicher, M. Weber, Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension. Adv. Math. 304, 1080–1107 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 134.
    F.J. Murray, J. Von Neumann, On rings of operators. Ann. Math. (2) 37(1), 116–229 (1936)Google Scholar
  8. 139.
    A. Nica, D. Shlyakhtenko, R. Speicher, Operator-valued distributions. I. Characterizations of freeness. Int. Math. Res. Not. 2002(29), 1509–1538 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 150.
    M. Reed, B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis (Academic, New York/London, 1980)Google Scholar
  10. 168.
    E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1971)zbMATHGoogle Scholar
  11. 187.
    D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms. Invent. Math. 132(1), 189–227 (1998)MathSciNetzbMATHGoogle Scholar
  12. 189.
    D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information. Adv. Math. 146(2), 101–166 (1999)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada
  2. 2.FB MathematikUniversität des SaarlandesSaarbrückenGermany

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