An Elementary Proof of the Free-additivity of Voiculescu's Free Entropy

TL;DR

This paper provides a simple, elementary proof of Voiculescu's free-additivity of free entropy, a fundamental concept in free probability theory, demonstrating the asymptotic freeness of certain random matrices.

Contribution

It introduces a more accessible proof of the free-additivity of free entropy, simplifying the understanding of Voiculescu's original complex argument.

Findings

Elementary proof of asymptotic freeness

Confirmation of free entropy additivity

Simplification of Voiculescu's original proof

Abstract

D. Voiculescu [2] proved that a standard family of independent random unitary k by k matrices and a constant k by k unitary matrix is asymtotically free as k goes to infinity. This result was a key ingredient in Voiculescu's proof [3] that his free entropy is additive when the variables are free. In this paper, we give a very elementary proof of a more detailed version of this result [2].