Geodesic
An elementary proof of the inequality $\chi\leq\chi^{*}$ for conditional free entropy
Documenta mathematica, Tome 29 (2024) no. 5, pp. 1085-1124

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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.
DOI : 10.4171/dm/969
Classification : 46L54, 46L53, 60B20, 94A17
Mots-clés : free entropy, conditional entropy, random matrix, free Fisher information, tracial von Neumann algebra 
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David Jekel; Jennifer Pi. An elementary proof of the inequality $\chi\leq\chi^{*}$ for conditional free entropy. Documenta mathematica, Tome 29 (2024) no. 5, pp. 1085-1124. doi: 10.4171/dm/969

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