Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\boxplus$ be the free additive convolution, $\mathrm{sc}_{d,t}$ the semicircular law of variance $t$ and $\Phi^{*}$ the free Fisher information; the rays $(0,\infty)$, integrability on a Borel set and $\int_{B}f(t)\,dt$ are those of Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §rays and Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §integral, and $(0,1)$ is an open interval. Let $e=\exp(1)$ with $\exp$ the exponential function, $\pi$ the number $\pi$ and $\log$ the natural logarithm; $2\pi e>0$, since $\pi>0$ by The Number Pi §pi and The Least Positive Zero of the Cosine §least-zero, and $e>0$ by claim 5 of Basic Properties of the Exponential Function.

1. (Integrand)¶ For $\lambda\in\Sigma_{d}$ and real $t>0$, the law $\lambda\boxplus\mathrm{sc}_{d,t}$ has conjugate variables by Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information §conjugate, and

$$f_{\lambda}(t)=\Phi^{*}(\lambda\boxplus\mathrm{sc}_{d,t}),\qquad g_{\lambda}(t)=\frac{d}{1+t}-f_{\lambda}(t);$$

these are the functions of The Free Fisher Information along the Free Heat Flow: Bounds, Lower Semicontinuity in Time and Integrability, whose extensions by $0$ are measurable by The Free Fisher Information along the Free Heat Flow: Bounds, Lower Semicontinuity in Time and Integrability §semicontinuous.

2. (Domain)¶ $\mathcal{D}_{\chi}$ is the set of $\lambda\in\Sigma_{d}$ for which $f_{\lambda}$ is integrable on $(0,1)$.

3. (Free entropy)¶ For $\lambda\in\mathcal{D}_{\chi}$, the function $g_{\lambda}$ is integrable on $(0,\infty)$ by The Free Fisher Information along the Free Heat Flow: Bounds, Lower Semicontinuity in Time and Integrability §integrable, and the non-microstates free entropy of $\lambda$ is the real number

$$\chi^{*}(\lambda)=\frac{1}{2}\int_{(0,\infty)}g_{\lambda}(t)\,dt+\frac{d}{2}\log(2\pi e).$$