Concavification of free entropy

We introduce a modification of Voiculescu’s free entropy which coincides with the lim inflim inf 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 et al. (2009) [8] to non-hyperfinite multivariables and prove freeness in the case of additivity of Voiculescu’s entropy (or vanishing of our extended orbital entropy).