An Elementary Proof of the Inequality $\chi \le {\chi }^{\ast }$ for Conditional Free Entropy

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