The Annals of Probability

A continuous semigroup of notions of independence between the classical and the free one

Florent Benaych-Georges and Thierry Lévy

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Abstract

In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for noncommutative random variables. These notions are related to the liberation process introduced by Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.

Article information

Source
Ann. Probab., Volume 39, Number 3 (2011), 904-938.

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281728

Digital Object Identifier
doi:10.1214/10-AOP573

Mathematical Reviews number (MathSciNet)
MR2789579

Zentralblatt MATH identifier
1222.46049

Subjects
Primary: 46L54: Free probability and free operator algebras 15A52

Keywords
Free probability independence random matrices unitary Brownian motion convolution cumulants

Citation

Benaych-Georges, Florent; Lévy, Thierry. A continuous semigroup of notions of independence between the classical and the free one. Ann. Probab. 39 (2011), no. 3, 904--938. doi:10.1214/10-AOP573. https://projecteuclid.org/euclid.aop/1300281728


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References

  • [1] Benaych-Georges, F. (2009). Rectangular random matrices, related convolution. Probab. Theory Related Fields 144 471–515.
    Mathematical Reviews (MathSciNet): MR2496440
    Zentralblatt MATH: 1171.15022
    Digital Object Identifier: doi:10.1007/s00440-008-0152-z
  • [2] Benaych-Georges, F. and Cabanal-Duvillard, T. From classical to free infinite divisibility. Unpublished manuscript, UPMC Univ. Paris 6 and Univ. Paris 5.
  • [3] Benaych-Georges, F. and Chapon, F. Random right eigenvalues of random matrices with quaternionic entries. Unpublished manuscript, UPMC Univ. Paris 6.
  • [4] Biane, P. (1997). Free Brownian motion, free stochastic calculus and random matrices. In Free Probability Theory (Waterloo, ON, 1995). Fields Inst. Commun. 12 1–19. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1426833
    Zentralblatt MATH: 0873.60056
  • [5] Biane, P. (1997). Segal–Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems. J. Funct. Anal. 144 232–286.
    Mathematical Reviews (MathSciNet): MR1430721
    Zentralblatt MATH: 0889.47013
    Digital Object Identifier: doi:10.1006/jfan.1996.2990
  • [6] Biane, P. (2003). Free probability for probabilists. In Quantum Probability Communications, Vol. XI (Grenoble, 1998) 55–71. World Sci. Publ., River Edge, NJ.
    Mathematical Reviews (MathSciNet): MR2032363
    Zentralblatt MATH: 1105.46045
  • [7] Biane, P. and Speicher, R. (1998). Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Related Fields 112 373–409.
    Mathematical Reviews (MathSciNet): MR1660906
    Zentralblatt MATH: 0919.60056
    Digital Object Identifier: doi:10.1007/s004400050194
  • [8] Haagerup, U. and Larsen, F. (2000). Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176 331–367.
    Mathematical Reviews (MathSciNet): MR1784419
    Zentralblatt MATH: 0984.46042
    Digital Object Identifier: doi:10.1006/jfan.2000.3610
  • [9] Nelson, E. (1974). Notes on non-commutative integration. J. Funct. Anal. 15 103–116.
    Mathematical Reviews (MathSciNet): MR355628
    Digital Object Identifier: doi:10.1016/0022-1236(74)90014-7
  • [10] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR2266879
    Zentralblatt MATH: 1133.60003
  • [11] Speicher, R. (1997). On universal products. In Free Probability Theory (Waterloo, ON, 1995). Fields Inst. Commun. 12 257–266. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1426844
    Zentralblatt MATH: 0877.46044
  • [12] Voiculescu, D. (1999). The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information. Adv. Math. 146 101–166.
    Mathematical Reviews (MathSciNet): MR1711843
    Zentralblatt MATH: 0956.46045
    Digital Object Identifier: doi:10.1006/aima.1998.1819
  • [13] Voiculescu, D. V., Dykema, K. J. and Nica, A. (1992). Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1217253

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