An Analogue of Hin\u{c}in's Characterization of Infinite Divisibility for Operator-Valued Free Probability

TL;DR

This paper extends Hinin's characterization of infinite divisibility to operator-valued free probability, proving that certain distributions are infinitely divisible using advanced topological and combinatorial methods.

Contribution

It introduces a new proof of infinite divisibility for operator-valued distributions via the Steinitz lemma and develops related weak topologies.

Findings

Distribution  a weak limit of an infinitesimal array is infinitely divisible.

Established compactness and convergence results in the developed weak topologies.

Provided a nonstandard proof approach adaptable to other probabilistic categories.

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.