Concavification of free entropy

TL;DR

This paper introduces a modified version of Voiculescu's free entropy that is concave and upper semi-continuous, extending its applicability and analyzing conditions for freeness in non-hyperfinite multivariables.

Contribution

It proposes a new concave, upper semi-continuous variant of free entropy and extends orbital free entropy to non-hyperfinite multivariables, providing new insights into freeness conditions.

Findings

Modified free entropy coincides with liminf variant on extremal states

Extended orbital free entropy to non-hyperfinite multivariables

Proved freeness under additivity or vanishing of extended orbital entropy

Abstract

We introduce a modification of Voiculescu's free entropy which coincides with the liminf variant of Voiculescu's free entropy on extremal states, but is a concave upper semi-continuous function on the trace state space. We also extend the orbital free entropy of Hiai, Miyamoto and Ueda to non-hyperfinite multivariables and prove freeness in case of additivity of Voiculescu's entropy (or vanishing of our extended orbital entropy).