$ k\times k $ matrices. Recall that the set $$\Gamma _{R}({X_{1},\dots ,X_{n}};k,l,\epsilon )\subset (M_{k}^{{{\operatorname{sa}}}})^{n}$$ was defined by Voiculescu in [@dvv:entropy2] as the set of those $ ({x_{1},\dots ,x_{n}})\in (M^{{{\operatorname{sa}}}}_{k})^{n} $, for which $ \Vert x_{i}\Vert \leq R $ and for any $ p\leq l $, and all $ i_{1},\dots ,i_{p} $ $$|\tau _{n}(x_{i_{1}},\dots ,x_{i_{p}})-\tau (X_{i_{1}}\dots X_{i_{p}})|<\epsilon .$$ Here $ \tau _{n} $ stands for the normalized trace on the matrices (so that $ \tau _{n}(1)=1 $).

For $ y_{1},\dots ,y_{n}\in M_{k} $, define $$\begin{aligned}
{\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k ,l ,\epsilon  )} &  & \\
=\{{x_{1},\dots ,x_{n}}\in (M_{k}^{{{\operatorname{sa}}}})^{n}:({x_{1},\dots ,x_{n}},{y_{1},\dots ,y_{m}}) &  & \\
\in {\Gamma _{R}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};k ,l ,\epsilon  )}\} &  & \end{aligned}$$

\[rmq: non-empty\]Note that for this set to be nonempty, we must have that $$({y_{1},\dots ,y_{m}})\in {\Gamma _{R}(({Y_{1},\dots ,Y_{m}}:{X_{1},\dots ,X_{n}};l ,k ,\epsilon  )}.$$ This set would be empty if $ W^{*}({Y_{1},\dots ,Y_{m}}) $ were not embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor.

Let $ \lambda  $ denote Lebesgue measure on $ (M_{k}^{{{\operatorname{sa}}}})^{n} $ corresponding to its Hilbert space structure coming from the non-normalized trace. Define **successively *$$\begin{aligned}
\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}},l,\epsilon )= &  & \label{ean:defofentropy} \\
\lim _{k\to \infty }\frac{1}{k^{2}}\sup _{\begin{array}{c}
({y_{1},\dots ,y_{m}})\in 