An elementary proof of the inequality $\chi\leq\chi^{*}$ for conditional free entropy
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Abstract
Through the study of large deviations theory for matrix Brownian motion, Biane, Capitaine, and 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 deviations theory. Moreover, we generalize the result to conditional microstates and non-microstates free entropy.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
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