An elementary proof of the inequality χ≤χ* for conditional free entropy

Through the study of large deviations theory for matrix Brownian motion, Biane, Capitaine, and Guionnet proved the inequality χ(X)≤χ*(X) that relates two analogs of entropy in free probability defined by Voiculescu. We give a new proof of χ≤χ* that is elementary in the sense that it does not rely on stochastic differential equations and large deviations theory. Moreover, we generalize the result to conditional microstates and non-microstates free entropy.