attained by such a semicircular family. Assume that the maximum is attained by some family $ {X_{1},\dots ,X_{n}}$. Then $ \chi ({X_{1},\dots ,X_{n}}|B) $ attains a local algebraic maximum at $ {X_{1},\dots ,X_{n}}$. Therefore, we have that for all non-commutative polynomials $ P_{i} $ with coefficients from $ B $ $$\frac{d}{d\epsilon }\chi \left( \left. \frac{X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}})}{\Vert X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}})\Vert _{2}},\dots ,\frac{X_{n}+\epsilon P_{n}({X_{1},\dots ,X_{n}})}{\Vert X_{n}+\epsilon P_{n}({X_{1},\dots ,X_{n}})\Vert _{2}}\right| B\right) =0.$$ But this is equal to $$\begin{aligned}
\frac{d}{d\epsilon }\chi ({X_{1},\dots ,X_{n}}|B)-\frac{d}{d\epsilon }\sum _{i}\log \Vert X_{i}+\epsilon P_{i}({X_{1},\dots ,X_{n}})\Vert _{2}= &  & \\
\sum _{i}\langle J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},X_{n})),P_{i}\rangle  &  & \\
-\sum _{i}\langle X_{i},P_{i}({X_{1},\dots ,X_{n}})\rangle . &  & \end{aligned}$$ It follows that for all non-commutative polynomials $ P_{i} $ with coefficients from $ B $, $$\langle J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},X_{n})),P_{i}\rangle =\langle X_{i},P_{i}({X_{1},\dots ,X_{n}})\rangle ,$$ which implies that $$X_{i}=J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},X_{n}))$$ for all $ i=1,\dots ,n $. But by [@dvv:entropy5] and [@shlyakht:cpentropy], this implies that $ {X_{1},\dots ,X_{n}}$ are a free semicircular family, free from $ B $.

Assume that $ B $ is embeddable into an ultrapower of the hyperfinite II$ _{1} $ factor, and $ \sum _{i=1}^{n}\tau (X_{i}^{2})=n $. Then $ \chi ({X_{1},\dots ,X_{n}}|B) $ attains its maximal value of $ n\log 2\pi 