Construction of product $*$-probability spaces via free cumulants
Arup Bose, Soumendu Sundar Mukherjee
TL;DR
The paper addresses constructing the free product of a family of (A_i, varphi^(i)) by defining freeness via the vanishing of mixed free cumulants kappa_n. The authors first construct free cumulant functionals kappa_n on the free product algebra A that agree with the original cumulants on each A_i and ensure mixed cumulants vanish, then define the state varphi on A from these cumulants via phi_pi[a_1, ..., a_n] = sum_{sigma <= pi} kappa_sigma[a_1, ..., a_n]. They prove that the resulting (A, varphi) has freely independent subalgebras and satisfies the universal property of the free product; for *-algebras, they establish positivity of varphi by proving varphi(a^* a) >= 0 using kappa_2. This work clarifies and formalizes a cumulant-centered route to free products, connecting cumulant/moment duality with freeness and providing a constructive alternative to the traditional moment-based approach.
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.