Minus the Free Entropy Is Bounded Below on Bounded Laws and Has Weak-Star Closed Sublevel Sets

In the setting of Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation, let $-\chi^{*}:\mathcal{D}_{\chi}\to\mathbb{R}$ be the map $\lambda\mapsto-\chi^{*}(\lambda)$.

1. (Lower bound)¶ For every real $R>0$ and every $\lambda\in\mathcal{D}_{\chi}\cap\Sigma_{d,R}$,

$$-\chi^{*}(\lambda)\ge-\frac{d}{2}\log(2\pi e\,R^{2}).$$

Hence the pair $(\mathcal{D}_{\chi},-\chi^{*})$ satisfies the lower bound clause of a free entropy penalty.

2. (Closed sublevel sets)¶ The pair $(\mathcal{D}_{\chi},-\chi^{*})$ satisfies the closed sublevel set clause of a free entropy penalty: for all reals $R>0$ and $c$, and every sequence $(\lambda_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}_{\chi}\cap\Sigma_{d,R}$ with $-\chi^{*}(\lambda_{k})\le c$ for every $k$ that converges weak-star to some $\lambda\in\Sigma_{d}$, one has $\lambda\in\mathcal{D}_{\chi}$ and $-\chi^{*}(\lambda)\le c$.