under compression by matrix units in the case that $ B=M_{n}\otimes D $, where $ M_{n} $ is the algebra of $ n\times n $ matrices. This behavior is a useful technical tool; for example, it was used to prove certain maximization results for matrices of non-commutative random variables in [@nss:entropy-micro].

If the $ {X_{1},\dots ,X_{n}}$ are free from $ B $, we have $$\chi ({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}),$$ provided that $ B $ can be embedded into the ultrapower of the hyperfinite II$ _{1} $ factor; a similar fact holds for $ \chi ^{*} $.

If $ {X_{1},\dots ,X_{n}}$ are free from the algebra generated by $ {Y_{1},\dots ,Y_{m}}$ and $ B $, then $$\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}}|B)=\chi ({X_{1},\dots ,X_{n}}|B)+\chi ({Y_{1},\dots ,Y_{m}}|B);$$ a similar fact holds for $ \chi ^{*} $.

We prove a maximization result for $ \chi  $ (which is essentially identical to the one for $ \chi ^{*} $), namely $ \chi ({X_{1},\dots ,X_{n}}|B) $ attains its maximum among all $ {X_{1},\dots ,X_{n}}$ with $ \sum _{i}\tau (X_{i}^{2})=n $ if and only if $ {X_{1},\dots ,X_{n}}$ is a free semicircular family, free from $ B $ (we need as an assumption that $ B $ can 