On operator-valued monotone independence

TL;DR

This paper develops the theory of operator-valued monotone independence, introducing cumulants, proving a moment-cumulant formula, and applying these to establish a CLT and generalize Muraki's formula in noncommutative probability.

Contribution

It introduces operator-valued monotone cumulants, provides a moment-cumulant formula, and extends classical results like the CLT and Muraki's formula to the operator-valued setting.

Findings

Established a moment-cumulant formula for operator-valued monotone independence.

Proved a CLT for the operator-valued case.

Generalized Muraki's formula for sums of independent variables.

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 Central Limit Theorem for 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.