The Free Entropy: the Semicircular Upper Bound, Semicircular Values, Smoothing, the de Bruijn Identity and Upper Semicontinuity

In the setting of Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation, let $\lambda\in\Sigma_{d}$ and let $s>0$ be real.

1. (Semicircular upper bound)¶ If $\lambda\in\mathcal{D}_{\chi}$, then $M(\lambda)>0$ and

$$\chi^{*}(\lambda)\le\frac{d}{2}\log\Bigl(\frac{2\pi e\,M(\lambda)}{d}\Bigr).$$

2. (Semicircular laws)¶ $\mathrm{sc}_{d,s}\in\mathcal{D}_{\chi}$ and $\chi^{*}(\mathrm{sc}_{d,s})=\frac{d}{2}\log(2\pi e\,s)$.

3. (Smoothing)¶ $\lambda\boxplus\mathrm{sc}_{d,s}\in\mathcal{D}_{\chi}$.

4. (de Bruijn identity)¶ If $\lambda\in\mathcal{D}_{\chi}$, then $f_{\lambda}$ is integrable on $(0,s)$ and

$$\chi^{*}(\lambda\boxplus\mathrm{sc}_{d,s})=\chi^{*}(\lambda)+\frac{1}{2}\int_{(0,s)}f_{\lambda}(t)\,dt.$$

5. (Upper semicontinuity)¶ Let $R>0$ and $c$ be real, and let $(\lambda_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{\chi}\cap\Sigma_{d,R}$ with $\chi^{*}(\lambda_{k})\ge c$ for every $k$ that converges weak-star to $\lambda$. Then $\lambda\in\mathcal{D}_{\chi}$ and $\chi^{*}(\lambda)\ge c$.