On operator-valued infinitesimal Boolean and monotone independence

TL;DR

This paper introduces operator-valued infinitesimal independence for Boolean and monotone cases, establishing their equivalence to existing notions over matrix algebras, and develops related convolutions, cumulants, and limit theorems.

Contribution

It defines OVI independence for Boolean and monotone cases, connects it to matrix algebra representations, and extends cumulant and limit theorem frameworks.

Findings

OVI Boolean and monotone independence are equivalent to operator-valued independence over 2x2 upper triangular matrices.

Formulas for OVI additive convolutions are derived via reduction to operator-valued cases.

Constructed OVI Central Limit Theorems and analyzed relations among cumulants.

Abstract

We introduce the notion of operator-valued infinitesimal (OVI) independence for the Boolean and monotone cases. Then show that OVI Boolean (resp. monotone) independence is equivalent to the operator-valued Boolean (resp. monotone) independence over an algebra of $2\times 2$ upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (resp. monotone) additive convolution by reducing it to the operator-valued case. We also define OVI Boolean and monotone cumulants and study its basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence. Finally we study two random matrix models that are asymptotically Boolean independent but turn out to not be infinitesimally Boolean independent.