Construction of product $*$-probability spaces via free cumulants
Arup Bose, Soumendu Sundar Mukherjee
TL;DR
This paper introduces a method to construct free products of *-probability spaces by defining freeness through the vanishing of mixed free cumulants, offering an alternative to traditional moment-based approaches.
Contribution
It develops a new construction of free products of *-probability spaces using free cumulants directly, bypassing moments.
Findings
Provides a new framework for constructing free products using free cumulants
Shows equivalence between freeness and vanishing mixed free cumulants in this context
Simplifies the understanding of free independence in *-probability spaces
Abstract
It is well known that free independence is equivalent to the vanishing of mixed free cumulants. The purpose of this short note is to build free products of -probability spaces using this as the definition of freeness and relying on free cumulants instead of moments.
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Taxonomy
TopicsRandom Matrices and Applications · Advanced Combinatorial Mathematics · Mathematical Inequalities and Applications