Infinitesimal Probability Theory with Amalgamation

Abstract

We introduce the notion of operator-valued infinitesimal independence for the free, Boolean, and monotone cases. We show that operator-valued infinitesimal free (respectively Boolean, monotone) independence is equivalent to the operator-valued free (respectively Boolean, monotone) independence over an algebra of $2\times 2$ upper triangular matrices. Then we construct the corresponding operator-valued infinitesimal cumulants for each notion of independence; moreover, we show that the infinitesimal free (respectively Boolean) independence in the operator-valued framework is equivalent to the vanishing of mixed free (respectively Boolean) cumulants and infinitesimal cumulants. In addition, for each notion of independence, we construct the corresponding operator-valued infinitesimal Central Limit Theorem. We establish formulas to obtain the operator-valued infinitesimal free (respectively Boolean, monotone) additive convolution. Furthermore, for the free case, we also provide the formula for the operator-valued infinitesimal multiplicative convolution.