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On operator-valued monotone independence

Published online by Cambridge University Press:  11 January 2016

Takahiro Hasebe  and
Hayato Saigo
Takahiro Hasebe*
Affiliation:
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Hayato Saigo
Affiliation:
Nagahama Institute of Bio-Science and Technology, Nagahama 526-0829, Japan, h_saigoh@nagahama-i-bio.ac.jp
*
Department of Mathematics, Hokkaido University, Kita-ku, Sapporo 060-0810, Japan, thasebe@math.sci.hokudai
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Abstract

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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.

Keywords

46L5346L5413F2506A07
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 215 , September 2014 , pp. 151 - 167
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Belinschi, S. T., Popa, M., and Vinnikov, V., On the operator-valued analogues of the semicircle, arcsine and Bernoulli laws, J. Operator Theory 70 (2013), 239–258. MR 3085826. DOI 10.7900/jot.2011jun24.1963.Google Scholar
[2] Biane, P., Processes with free increments, Math. Z. 227 (1998), 143–174. MR 1605393. DOI 10.1007/PL00004363.Google Scholar
[3] Dykema, K. J., Multilinear function series and transforms in free probability theory, Adv. Math. 258 (2007), 351–407. MR 2304321. DOI 10.1016/j.aim.2006.02.011.CrossRefGoogle Scholar
[4] Franz, U., Monotone and Boolean convolutions for non-compactly supported probability measures, Indiana Univ. Math. J. 58 (2009), 1151–1185. MR 2541362. DOI 10.1512/iumj.2009.58.3578.Google Scholar
[5] Hasebe, T. and Saigo, H., Joint cumulants for natural independence, Electron. Commun. Probab. 16 (2011), 491–506. MR 2836756. DOI 10.1214/ECP.v16–1647.Google Scholar
[6] Hasebe, T. and Saigo, H., The monotone cumulants, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), 1160–1170. MR 2884229. DOI 10.1214/10-AIHP379.CrossRefGoogle Scholar
[7] Lehner, F., Cumulants in noncommutative probability theory, I: Noncommutative exchangeability systems, Math. Z. 248 (2004), 67–100. MR 2092722. DOI 10.1007/s00209–004-0653–0.CrossRefGoogle Scholar
[8] Muraki, N., The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 337–371. MR 2016316. DOI 10.1142/S0219025703001365.Google Scholar
[9] Muraki, N., Monotonic convolution and monotonic Lévy-Hincin formula, preprint, 2000.Google Scholar
[10] Popa, M., A combinatorial approach to monotonic independence over a C* -algebra, Pacific J. Math. 237 (2008), 299–325. MR 2421124. DOI 10.2140/pjm.2008.237.299.Google Scholar
[11] Popa, M., A new proof for the multiplicative property of the Boolean cumulants with applications to the operator-valued case, Colloq. Math. 117 (2009), 81–93. MR 2539549. DOI 10.4064/cm117–1-5.Google Scholar
[12] Skeide, M., “Independence and product systems” in Recent Developments in Stochastic Analysis and Related Topics, World Scientific, Hackensack, N.J., 2004, 420–438. MR 2200526.Google Scholar
[13] Speicher, R., Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627. MR 1407898. DOI 10.1090/memo/0627.Google Scholar
[14] Speicher, R. and Woroudi, R., “Boolean convolution” in Free Probability Theory (Waterloo, Ontario, 1995), Fields Inst. Commun. 12, Amer. Math. Soc., 1997, 267–280. MR 1426845.Google Scholar
[15] Voiculescu, D., “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. MR 0799593. DOI 10.1007/BFb0074909.Google Scholar
[16] Voiculescu, D., “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. MR 1372537.Google Scholar