Voiculescu introduced two notions of free entropy for a given tuple X of self-adjoint operators: the microstates free entropy $\chi(X)$ and the non-microstates free entropy $\chi^*(X)$. In joint work with David Jekel, we give an elementary proof of the inequality $\chi(X) \leq \chi^*(X)$, originally proved by Biane, Capitaine, and Guionnet. We furthermore extend the inequality to conditional free entropy, conditioning upon any separable von Neumann subalgebra. The proof leverages relationships between the free entropy of a tuple X and the classical entropy of matrix approximations to X.
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