Concavification of free entropy

Philippe Biane (1) , Yoann Dabrowski (2)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 ICJ - Institut Camille Jordan

We introduce a modification of Voiculescu's free entropy which coincides with the lim 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) to non-hyperfinite multivariables and prove freeness in the case of additivity of Voiculescu's entropy (or vanishing of our extended orbital entropy).

Mots clés: Free entropy; Orbital entropy; Additivity of entropy; Poulsen simplex; Concentration of measure; Csiszar's projection

Domaines: Algèbres d'opérateurs [math.OA]; Probabilités [math.PR]

Philippe Biane, Yoann Dabrowski. Concavification of free entropy. Advances in Mathematics, 2013, 234, pp.667-696. ⟨10.1016/j.aim.2012.11.003⟩. ⟨hal-00867981⟩