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We use the theory of fully matricial, or non-commutative, functions toinvestigate infinite divisibility and limit theorems in operator-valuednon-commutative probability. Our main result is an operator-valued analogue ofthe Bercovici-Pata bijection. An important tool is Voiculescu's subordinationproperty for operator-valued free convolution.

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