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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

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