The same estimate holds for $ \chi _{R} $ and $ \chi ^{\omega }. $

We have $ \chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}})\leq \frac{n}{2}\log 2\pi ec^{2} $, the last inequality by [@dvv:entropy2].

Let $ B\subset M $ be a unital subalgebra of $ M $. We define the free entropy of $ ({X_{1},\dots ,X_{n}}) $ relative to $ B $ to be $$\chi ({X_{1},\dots ,X_{n}}|B)=\inf _{m}\inf _{{Y_{1},\dots ,Y_{m}}\in B}\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}).$$

If $ \omega  $ is a free ultrafilter on the natural numbers, then we define $ \chi ^{\omega }({X_{1},\dots ,X_{n}}|B) $ in the obvious way.

For $ \chi ({X_{1},\dots ,X_{n}}|B) $ to be finite, we must have that for all $ {Y_{1},\dots ,Y_{m}}\in B $, $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\neq -\infty .$$ By Remark \[rmq: non-empty\], this subsumes that $ B $ is embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor.

If $ D\subset B $ is a unital subalgebra, then $$\chi ({X_{1},\dots ,X_{n}}|B)\leq \chi ({X_{1},\dots ,X_{n}}|D).$$ In particular, we have $ \chi ({X_{1},\dots ,X_{n}}|B)\leq \chi ({X_{1},\dots ,X_{n}}|\mathbb {C})=\chi ({X_{1},\dots ,X_{n}}) $. The same conclusion holds for $ \chi  $ replaced with $ \chi ^{\omega } $.

The first inequality is because in computing 