An Analogue of Hinucin's Characterization of Infinite Divisibility for Operator-Valued Free Probability

Abstract

Let $B$ be a finite, separable von Neumann algebra. We prove that a $B$-valued distribution $\mu$ 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.

arXiv commentSeveral revisions, excised theorems incompatible with the final form of the paper