claimed.

Recall (see [@dvv:entropy4]) that a function $ \phi ({X_{1},\dots ,X_{n}}) $ is said to attain a local algebraic maximum at $ {X_{1},\dots ,X_{n}}$ on the set $ S=\{{X_{1},\dots ,X_{n}}:\sum _{i}\tau (X_{i}^{2})=n\} $, if for all non-commutative polynomials $ P_{i}, $ with coefficients from $ B $ there exist $ \epsilon _{0}>0 $, such that for all $ 0<\epsilon <\epsilon _{0} $, $$\phi \left( \frac{X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}})}{\Vert X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}})\Vert _{2}},\dots ,\frac{X_{n}+\epsilon P_{n}({X_{1},\dots ,X_{n}})}{\Vert X_{n}+\epsilon P_{n}({X_{1},\dots ,X_{n}})\Vert _{2}}\right) \leq \phi ({X_{1},\dots ,X_{n}}).$$ Clearly this is a much weaker requirement than saying that $ \phi  $ attains a maximum on $ S $ at $ {X_{1},\dots ,X_{n}}$.

\[prop: maximality\]Let **$ B $** be a von Neumann algebra, embeddable into an ultrapower of the hyperfinite II$ _{1} $ factor. Then the function $ ({X_{1},\dots ,X_{n}})\mapsto \chi ({X_{1},\dots ,X_{n}}|B) $ attains a local algebraic maximum on the set $ \{{X_{1},\dots ,X_{n}}:\sum _{i}\tau (X_{i}^{2})=n\} $ exactly when $ {X_{1},\dots ,X_{n}}$ are $ n $ free $ (0,1) $ semicircular variables, free from $ B. $ The same statement holds for $ \chi ^{\omega } $.

Note that by Corollary \[corr: max\], we have that a global maximum (and hence a local algebraic maximum) is 