Infinite divisibility and a non-commutative Boolean-to-free Bercovici–Pata bijection

We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue for the Bercovici–Pata bijection. An important tool is Voiculescuʼs subordination property for operator-valued free convolution.