## Nagoya Mathematical Journal

### On operator-valued monotone independence

#### Abstract

We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of the central limit theorem for the operator-valued case. Moreover, we prove a generalization of Muraki’s formula for the sum of independent random variables and a relation between generating functions of moments and cumulants.

#### Article information

Source
Nagoya Math. J., Volume 215 (2014), 151-167.

Dates
First available in Project Euclid: 22 July 2014

https://projecteuclid.org/euclid.nmj/1406039894

Digital Object Identifier
doi:10.1215/00277630-2741151

Mathematical Reviews number (MathSciNet)
MR3263527

Zentralblatt MATH identifier
1291.81350

#### Citation

Hasebe, Takahiro; Saigo, Hayato. On operator-valued monotone independence. Nagoya Math. J. 215 (2014), 151--167. doi:10.1215/00277630-2741151. https://projecteuclid.org/euclid.nmj/1406039894

#### References

• [1] S. T. Belinschi, M. Popa, and V. Vinnikov, On the operator-valued analogues of the semicircle, arcsine and Bernoulli laws, J. Operator Theory 70 (2013), 239–258.
• [2] P. Biane, Processes with free increments, Math. Z. 227 (1998), 143–174.
• [3] K. J. Dykema, Multilinear function series and transforms in free probability theory, Adv. Math. 258 (2007), 351–407.
• [4] U. Franz, Monotone and Boolean convolutions for non-compactly supported probability measures, Indiana Univ. Math. J. 58 (2009), 1151–1185.
• [5] T. Hasebe and H. Saigo, Joint cumulants for natural independence, Electron. Commun. Probab. 16 (2011), 491–506.
• [6] T. Hasebe and H. Saigo, The monotone cumulants, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), 1160–1170.
• [7] F. Lehner, Cumulants in noncommutative probability theory, I: Noncommutative exchangeability systems, Math. Z. 248 (2004), 67–100.
• [8] N. Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 337–371.
• [9] N. Muraki, Monotonic convolution and monotonic Lévy-Hinčin formula, preprint, 2000.
• [10] M. Popa, A combinatorial approach to monotonic independence over a $C^{*}$-algebra, Pacific J. Math. 237 (2008), 299–325.
• [11] M. Popa, A new proof for the multiplicative property of the Boolean cumulants with applications to the operator-valued case, Colloq. Math. 117 (2009), 81–93.
• [12] M. Skeide, “Independence and product systems” in Recent Developments in Stochastic Analysis and Related Topics, World Scientific, Hackensack, N.J., 2004, 420–438.
• [13] R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627.
• [14] R. Speicher and R. Woroudi, “Boolean convolution” in Free Probability Theory (Waterloo, Ontario, 1995), Fields Inst. Commun. 12, Amer. Math. Soc., 1997, 267–280.
• [15] D. Voiculescu, “Symmetries of some reduced free product $C^{*}$-algebras” in Operator Algebras and Their Connections with Topology and Ergodic Theory (Busteni, 1983), Lecture Notes in Math. 1132, Springer, Berlin, 1985, 556–588.
• [16] D. Voiculescu, “Operations on certain non-commutative operator-valued random variables” in Recent Advances in Operator Algebras (Orléans, 1992), Astérisque 232, Soc. Math. France, Paris, 1995, 243–275.