On operator-valued infinitesimal Boolean and monotone independence

Abstract

We introduce the notion of operator-valued infinitesimal (OVI) Boolean independence and OVI monotone independence. Then we show that OVI Boolean (respectively, monotone) independence is equivalent to the operator-valued (OV) Boolean (respectively, monotone) independence over an algebra of [Formula: see text] upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (respectively, monotone) additive convolution by reducing it to the OV case. We also define OVI Boolean and monotone cumulants and study their 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.