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An Elementary Proof of the Inequality $χ\leq χ^*$ for Conditional Free Entropy (2305.02574v4)
Published 4 May 2023 in math.OA and math.PR
Abstract: Through the study of large deviations theory for matrix Brownian motion, Biane-Capitaine-Guionnet proved the inequality $\chi(X) \leq \chi*(X)$ that relates two analogs of entropy in free probability defined by Voiculescu. We give a new proof of $\chi \leq \chi*$ 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.