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An Analogue of Hinucin'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.