Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Independences and partial $R$-transforms in bi-free probability

Paul Skoufranis

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In this paper, we examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove that bi-freeness is preserved under tensoring with matrices. Finally, via combinatorial arguments, we construct partial $R$-transforms in two settings relating the moments and cumulants of a left–right pair of operators.


Dans cet article, nous examinons comment diverses notions d’indépendance en théorie des probabilités non commutatives se traduisent en probabilités bi-libres. Nous montrons comment l’indépendance booléenne et monotone se produisent à partir de paires de faces bi-libres, et établissons un théorème de Kac/Loève pour l’indépendance bi-libre. En outre, nous prouvons que l’indépendance bi-libre est préservée par tensorisation avec des matrices. Enfin, par des arguments combinatoires, nous construisons deux types de $R$-transformations partielles, reliant les moments et les cumulants d’une paire gauche-droite des opérateurs.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1437-1473.

Received: 4 December 2014
Revised: 10 May 2015
Accepted: 31 May 2015
First available in Project Euclid: 28 July 2016

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Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras 46L53: Noncommutative probability and statistics

Bi-free probability Free independence Boolean independence Monotone independence Bi-free independence over matrices Partial $R$-transforms


Skoufranis, Paul. Independences and partial $R$-transforms in bi-free probability. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1437--1473. doi:10.1214/15-AIHP691.

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  • [1] S. T. Belinschi, T. Mai and R. Speicher. Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem. J. Reine Angew. Math. (2015). DOI: 10.1515/crelle-2014-0138. Available at arXiv:1303.3196.
  • [2] H. Bercovici. Multiplicative monotonic convolution. Illinois J. Math. 49 (3) (2005) 929–951.
  • [3] I. Charlesworth, B. Nelson and P. Skoufranis. On two-faced families of non-commutative random variables. Canad. J. Math. 67 (2015) 1290–1325.
  • [4] I. Charlesworth, B. Nelson and P. Skoufranis. Combinatorics of bi-freeness with amalgamation. Comm. Math. Phys. 338 (2015) 801–847.
  • [5] T. Hasebe and H. Saigo. The monotone cumulants. Ann. Inst. Henri Poincaré Probab. Stat. 47 (4) (2011) 1160–1170.
  • [6] F. Lehner. Cumulants in noncommutative probability theory. I. Noncommutative exchangeability systems. Math. Z. 248 (1) (2004) 67–100.
  • [7] M. Mastnak and A. Nica. Double-ended queues and joint moments of left–right canonical operators on full Fock space. Internat. J. Math. 26 (2015) 1550016.
  • [8] N. Muraki. Monotonic independence, monotonic central limit theorem and monotonic law of small numbers. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001) 39–58.
  • [9] N. Muraki. The five independences as quasi-universal products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (1) (2002) 113–134.
  • [10] N. Muraki. The five independences as natural products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (3) (2003) 337–371.
  • [11] A. Nica. $R$-Transforms of free joint distributions and non-crossing partitions. J. Funct. Anal. 135 (2) (1996) 271–296.
  • [12] A. Nica, D. Shylakhtenko and R. Speicher. Operator-valued distributions. I. Characterizations of freeness. Int. Math. Res. Not. IMRN 29 (2002) 1509–1538.
  • [13] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. London Mathematics Society Lecture Notes Series 335. Cambridge Univ. Press, Cambridge, 2006.
  • [14] M. Popa. A combinatorial approach to monotonic independence over a $C^{*}$-algebra. Pacific J. Math. 237 (2) (2008) 299–325.
  • [15] M. Popa. A new proof for the multiplicative property of the Boolean cumulants with applications to operator-valued case. Colloq. Math. 117 (1) (2009) 81–93.
  • [16] R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 (1) (1994) 611–628.
  • [17] R. Speicher. On universal products. In Free Probability Theory 257–266. Fields Inst. Commun. 12. Amer. Math. Soc., Providence, RI, 1997.
  • [18] R. Speicher and R. Woroudi. Boolean convolution. In Free Probability Theory 267–279. Fields Inst. Commun. 12. Amer. Math. Soc., Providence, RI, 1997.
  • [19] D. Voiculescu. Symmetries of some reduced free product $C^{*}$-algebras. In Operator Algebras and Their Connection with Topology and Ergodic Theory 556–588. Lecture Notes in Mathematics 1132. Springer, Berlin, 1985.
  • [20] D. Voiculescu. Addition of certain non-commuting random variables. J. Funct. Anal. 66 (3) (1986) 323–346.
  • [21] D. Voiculescu. Free probability for pairs of faces. I. Comm. Math. Phys. 332 (2014) 955–980.
  • [22] D. Voiculescu. Free probability for pairs of faces. II: 2-variable bi-free partial $R$-transform and systems with rank $\leq1$ commutation. Preprint, 2013. Available at arXiv:1308.2035.