An analogue of Hinčin's characterization of infinite divisibility for operator-valued free probability

Let B be a finite, separable von Neumann algebra. We prove that a B-valued distribution μ that is the weak limit of an infinitesimal array is infinitely divisible. The proof of this theorem utilizes the Steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories. We also develop weak topologies for this theory and prove the corresponding compactness and convergence results.