---
address: 'Department of Mathematics, UCLA, Los Angeles, CA 90095'
author:
- Dimitri Shlyakhtenko
title: A Microstates Approach to Relative Free Entropy
---

[^1]

Introduction.
=============

Let $ {X_{1},\dots ,X_{n}}\in (M,\tau ) $ be a family of non-commutative random variables in a tracial $ W^{*} $-probability space, and let $ B\subset M $ be a unital subalgebra. Voiculescu has introduced in [@dvv:entropy5] a free entropy quantity $$\chi ^{*}({X_{1},\dots ,X_{n}}:B).$$ His approach involved non-commutative Hilbert transform and is algebraic in nature. In the case that $ B=\mathbb C $, this quantity is denoted $ \chi ^{*}({X_{1},\dots ,X_{n}}) $, and its properties are very similar to those of the free entropy $ \chi ({X_{1},\dots ,X_{n}}) $ introduced by Voiculescu in [@dvv:entropy2] using microstates; in fact, it may very well be that the two quantities coinside.

Using the microstates approach to free entropy, we introduce in this paper a quantity $ \chi ({X_{1},\dots ,X_{n}}|B) $ which has several properties in common $ \chi ^{*}({X_{1},\dots ,X_{n}}:B) $.

The infinitesimal change of variables formula for $ \chi  $ involves conjugate variables, introduced by Voiculescu in order to define $ \chi ^{*} $. Both $ \chi  $ and $ \chi ^{*} $ have the same behavior 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 be embedded into the ultrapower of the hyperfinite II$ _{1} $ factor). Lastly, the infinitesimal change of variables formula for $ \chi (\cdots |B) $ involves conjugate variables used to define $ \chi ^{*}(\cdots :B) $ (see [@dvv:entropy5]).

It is interesting to note that $ \chi (\cdot |B) $ has an interpretation as a relative entropy, which suggests a similar interpretation for $ \chi ^{*}(\cdot :B) $. Indeed, we show that if $ {Y_{1},\dots ,Y_{m}}$ are generators of $ B $, then $ \chi ({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $, where the latter entropy has properties of a relative entropy of $ {X_{1},\dots ,X_{n}}$ and $ {Y_{1},\dots ,Y_{m}}$. We caution the reader that we use a definition of $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $ which may be different from the one used by Voiculescu in [@dvv:entropy2], although the two quantities are related.

Relative Free Entropy $ \chi (\cdot |B)\protect $.
==================================================

Let $ (M,\tau ) $ be a tracial non-commutative probability space, and consider self-adjoint non-commutative random variables $ X_{1},\dots ,X_{n} $, $ Y_{1},\dots ,Y_{m}\in M $. We denote by $ M_{k} $ the algebra of $ k\times k $ matrices, and by $ M_{k}^{{{\operatorname{sa}}}} $ the set of self-adjoint $ 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 *$$\begin{aligned}
\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}},l,\epsilon )= &  & \label{ean:defofentropy} \\
\lim _{k\to \infty }\frac{1}{k^{2}}\sup _{\begin{array}{c}
({y_{1},\dots ,y_{m}})\in \\
\Gamma _{R}({Y_{1},\dots ,Y_{m}};k,l,\epsilon )
\end{array}} &  & \\
\log \lambda {\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k ,l ,\epsilon  )}+\frac{n}{2}\log k & \end{aligned}$$*

$$\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=\inf _{l,\epsilon }\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,\epsilon )$$

$$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=\sup _{R}\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}).$$ The last quantity is called the relative free entropy of the $ n $-tuple $ ({X_{1},\dots ,X_{n}}) $ with respect to the $ m $-tuple $ ({Y_{1},\dots ,Y_{m}}) $.

If $ \omega  $ is a free ultrafilter on $ \mathbb N $, then one can also define $ \chi ^{\omega }({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $ exactly as in (\[ean:defofentropy\]), but replacing $ \limsup  $ by $ \lim _{k\to \omega } $.

\[rmq:relative\]It is not clear whether our definition of $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $ coninsides with that of Voiculescu (see [@dvv:entropy2]). His definition corresponds to defining $$\begin{aligned}
\chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,k,\epsilon )= &  & \\
\limsup _{k}\log \frac{\lambda \Gamma _{R}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};k,l,\epsilon )}{\lambda \Gamma _{R}({Y_{1},\dots ,Y_{m}};k,l,\epsilon )}+\frac{n}{2}\log k. &  & \end{aligned}$$ The connection to our definition can be made as follows: let $$f({y_{1},\dots ,y_{m}})=\lambda {\Gamma _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}[{y_{1},\dots ,y_{m}}];k ,l ,\epsilon  )}$$ Then Voiculescu’s definition corresponds to taking as $ \chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,k,\epsilon ) $ the average of $ f $ over $ \Gamma _{R}({Y_{1},\dots ,Y_{m}};l,k,\epsilon ) $. It follows that the quantity obtained in our definition is bigger than that of Voiculescu. We mention that it is possible to define, in the spirit of [@dvv:entropy2] the relative entropy as $$\chi '({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})-\chi ({Y_{1},\dots ,Y_{m}}:{X_{1},\dots ,X_{n}}).$$ Such a definition corresponds to defining $ \chi _{R}({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}};l,k,\epsilon ) $ as the average of $ f $ over $ \Gamma _{R}({Y_{1},\dots ,Y_{m}}:{X_{1},\dots ,X_{n}}) $. We don’t know whether $ \chi ' $ coincides with Voiculescu’s or our definition of $ \chi , $ and whether Voiculescu’s and our definitions are the same or different. Note, however, that we always have:

Let $ {X_{1},\dots ,X_{n}}$, $ {Y_{1},\dots ,Y_{m}}$ be non-commutative random variables. Then we have $$\begin{aligned}
\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})-\chi ({Y_{1},\dots ,Y_{m}})\leq  &  & \\
\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})-\chi ({Y_{1},\dots ,Y_{m}}:{X_{1},\dots ,X_{n}})\leq  &  & \\
\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}). &  & \end{aligned}$$ In particular, if $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=-\infty  $, then $ \chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})=-\infty  $.

In the case that $ \chi ({Y_{1},\dots ,Y_{m}}) $ is finite, the inequalities follow from the discussion in Remark \[rmq:relative\]. If $$\chi ({Y_{1},\dots ,Y_{m}})=-\infty ,$$ then $$\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})=-\infty .$$ If $ \chi ({Y_{1},\dots ,Y_{m}})\neq -\infty  $, then $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=-\infty$$ implies that $$\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})\leq -\infty +\chi ({Y_{1},\dots ,Y_{m}})=-\infty .$$

We shall write $$\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}};l,k,\epsilon )$$ for $${\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l ,k ,\epsilon  )}$$ when the $ {Y_{1},\dots ,Y_{m}}$ are understood.

If $ p<m $, then $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}|Y_{1},\dots ,Y_{p}) $, and a similar inequality holds for $ \chi _{R} $ and $ \chi ^{\omega } $.

We have the inclusion $$\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,k,\epsilon )\subset {\Gamma _{R}({X_{1},\dots ,X_{n}}|y_{1},\dots ,y_{p} [Y_{1},\dots ,Y_{p} ];l ,k ,\epsilon  )}$$ for all $ ({y_{1},\dots ,y_{m}})\in M_{k} $. It follows that $$\begin{aligned}
\sup _{({y_{1},\dots ,y_{m}})}\lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,k,\epsilon )\leq  &  & \\
\sup _{(y_{1},\dots ,y_{p})}\lambda {\Gamma _{R}({X_{1},\dots ,X_{n}}|y_{1},\dots ,y_{p} [Y_{1},\dots ,Y_{p} ];l ,k ,\epsilon  )} &  & \end{aligned}$$ which implies the desired inequality.

$ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}:{Y_{1},\dots ,Y_{m}}) $.

We clearly have $$\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,k,\epsilon )\subset \pi \Gamma _{r}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};l,k,\epsilon ),$$ where $ \pi  $ denotes the projection from $ (M_{k}^{{{\operatorname{sa}}}})^{n}\times (M^{{{\operatorname{sa}}}})^{m} $ onto $ (M_{k}^{{{\operatorname{sa}}}})^{n} $.

\[corr: max\]Let $ c^{2}=\frac{1}{n}\sum \tau (X_{i}^{2}) $. Then $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \frac{n}{2}\log 2\pi ec^{2}.$$ 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 $ \chi ({X_{1},\dots ,X_{n}}|D) $ we take the infimum over a smaller set. The equality between $ \chi ({X_{1},\dots ,X_{n}}|\mathbb {C}) $ and $ \chi ({X_{1},\dots ,X_{n}}) $ is left to the reader.

Let $ X_{i}^{j} $, $ i=1,\dots ,n $, $ j=1,2,\dots  $ be non-commutative random variables. Assume that $ {X_{1},\dots ,X_{n}}$ are such that as $ j\to \infty  $, the joint distribution of $ (X_{1}^{j},\dots ,X_{n}^{j}) $ and $ B $ converges to the joint distribution of $ B $ and $ {X_{1},\dots ,X_{n}}$. Let $ Y_{1}^{j},\dots ,Y_{m}^{j} $ be such that they converge in distribution to $ {Y_{1},\dots ,Y_{m}}$. Assume that there exists a finite constant $ R $, so that $ \sup _{i,j}\Vert X_{i}^{j}\Vert ,\Vert Y_{i}^{j}\Vert ,\Vert X_{i}\Vert ,\Vert Y_{i}\Vert <R $. Then $$\chi ({X_{1},\dots ,X_{n}}|B)\geq \limsup _{j}\chi (X_{1}^{j},\dots ,X_{n}^{j}|B)$$ and $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\geq \limsup _{j}\chi (X_{1}^{j},\dots ,X_{n}^{j}|Y_{1}^{j},\dots ,Y_{m}^{j}).$$ The same conclusion holds for $ \chi ^{\omega } $ instead of $ \chi  $.

Clearly, only the second inequality needs to be proved. It follows from the following inclusion, true for sufficiently large $ j $: $$\begin{aligned}
\Gamma _{R}(X_{1}^{j},\dots ,X_{n}^{j}|y_{1},\dots ,y_{m}[Y_{1}^{j},\dots ,Y_{n}^{j}];k,l,\epsilon /2)\subset  &  & \\
\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}};k,l,\epsilon ), &  & \end{aligned}$$ since it implies that $$\begin{aligned}
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R}(X_{1}^{j},\dots ,X_{n}^{j}|y_{1},\dots ,y_{m}[Y_{1}^{j},\dots ,Y_{n}^{j}];k,l,\epsilon /2)\leq  &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}};k,l,\epsilon ), &  & \end{aligned}$$ which in turn implies the desired inequality.

\[prop: generation\]Assume that $ {Z_{1},\dots ,Z_{r}}\in W^{*}({Y_{1},\dots ,Y_{m}}) $. Then $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}|{Z_{1},\dots ,Z_{r}}),$$ and similarly for $ \chi  $ replaced with $ \chi ^{\omega } $.

Fix $ \epsilon >0 $, $ l>0 $, $ R>0 $. Then there exists non-commutative polynomials $ p_{1},\dots ,p_{r} $ in $ m $ variables, such that $$\left( {X_{1},\dots ,X_{n}},p_{1}({Y_{1},\dots ,Y_{m}}),\dots ,p_{r}({Y_{1},\dots ,Y_{m}})\right)$$ approximate $$({X_{1},\dots ,X_{n}},{Z_{1},\dots ,Z_{r}})$$ strongly to any desired accuracy. It follows, that for a suitable choice of such polynomials, there exist $ \epsilon >\epsilon '>0 $, $ l'>l>0 $ and $ R'>R>0 $ such that whenever $$({y_{1},\dots ,y_{m}})\in \Gamma _{R}({Y_{1},\dots ,Y_{m}},{X_{1},\dots ,X_{n}};k,l',\epsilon ')$$ we have the inclusion $$\begin{aligned}
\Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')\subset  &  & \\
\Gamma _{R}({X_{1},\dots ,X_{n}}|p_{1}({y_{1},\dots ,y_{m}}),\dots ,p_{r}({y_{1},\dots ,y_{m}})[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ). &  & \end{aligned}$$ It follows that $$\begin{aligned}
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')\leq  &  & \\
\sup _{z_{1},\dots ,z_{r}}\log \lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|z_{1},\dots ,z_{r}[{Z_{1},\dots ,Z_{r}}];k,l,\epsilon ). &  & \end{aligned}$$ But this implies $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}|{Z_{1},\dots ,Z_{r}}) $.

$ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=\chi ({X_{1},\dots ,X_{n}}|W^{*}({Y_{1},\dots ,Y_{m}})) $.

By definition, $$\chi =\chi ({X_{1},\dots ,X_{n}}|W^{*}({Y_{1},\dots ,Y_{m}}))=\inf _{r}\inf _{{Z_{1},\dots ,Z_{r}}\in W^{*}({Y_{1},\dots ,Y_{m}})}\chi ({X_{1},\dots ,X_{n}}|{Z_{1},\dots ,Z_{r}}).$$ In particular, if $ r=m $ and $ ({Z_{1},\dots ,Z_{r}})=({Y_{1},\dots ,Y_{m}}) $, we have that $$\chi \leq \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}).$$ But by Proposition \[prop: generation\], we also have that, for $ Z_{1},\dots ,Z_{r}\in W^{*}({Y_{1},\dots ,Y_{m}}) $, $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}|{Z_{1},\dots ,Z_{r}})\leq \chi ,$$ so that $ \chi =\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $.

If $ {Y_{1},\dots ,Y_{m}}\in B $, then $ \chi ({X_{1},\dots ,X_{n}}|B)\leq \chi ({X_{1},\dots ,X_{n}}:{Y_{1},\dots ,Y_{m}}) $.

We have $ \chi ({X_{1},\dots ,X_{n}}|B)\leq \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}:{Y_{1},\dots ,Y_{m}}) $.

Let $ B=L^{\infty }[0,1]\subset M $ be a diffuse commutative von Neumann subalgebra. For each $ r=1,\dots ,n $ let $ \mu _{r} $ be a measure on $ [0,1]^{2} $ determined by $$\iint f(x)g(y)d\mu _{r}(x,y)=\tau (X_{r}fX_{r}g).$$ Assume that for at least one $ r\in \{1,\dots ,n\} $, Lebesgue measure on $ [0,1]^{2} $ is singular with respect to $ \mu _{r} $. Then $ \chi ({X_{1},\dots ,X_{n}}|B)=-\infty  $.

Let $ Y $ be a self-adjoint generator for $ B $. Then we have $$\chi ({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}|Y)\leq \chi ({X_{1},\dots ,X_{n}}:Y)=-\infty$$ because of [@dvv:entropy3 Corollary 7.7].

Note that the analogous theorem holds for $ \chi ^{*}(\cdots :B) $, see [@shlyakht:cpentropy].

Let $ p<n $. Then $$\chi ({X_{1},\dots ,X_{n}}|B)\leq \chi (X_{1},\dots ,X_{p}|B)+\chi (X_{m+1},\dots ,X_{p}|B)$$ and a similar inequality holds for $ \chi ^{\omega } $.

We clearly have $$\begin{aligned}
\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\subset \Gamma _{R}(X_{1},\dots ,X_{p}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\times  &  & \\
\Gamma _{R}(X_{p+1},\dots ,X_{n}|{y_{1},\dots ,y_{m}};k,l,\epsilon ) &  & \end{aligned}$$ so that $$\begin{aligned}
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\leq  &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \left( \Gamma _{R}(X_{1},\dots ,X_{p}|{y_{1},\dots ,y_{m}};k,l,\epsilon )+\right.  &  & \\
\left. \log \lambda \Gamma _{R}(X_{p+1},\dots ,X_{n}|{y_{1},\dots ,y_{m}};k,l,\epsilon \right) \leq  &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \left( \Gamma _{R}(X_{1},\dots ,X_{p}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\right) + &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \left( \Gamma _{R}(X_{p+1},\dots ,X_{nb}|{y_{1},\dots ,y_{m}};k,l,\epsilon \right) . &  & \end{aligned}$$ This implies the proposition.

\[thrm: whenfreefromB\]Let $ {X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}\in (M,\tau ) $ be self-adjoint non-commutative random variables. Assume that the family $ {X_{1},\dots ,X_{n}}$ is free from the von Neumann algebra generated by $ B $ and $ X_{n+1},\dots ,X_{p} $, and assume that $ B $ is embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor. Then $$\chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B)=\chi ({X_{1},\dots ,X_{n}})+\chi (X_{n+1},\dots ,X_{p}|B).$$ In particular, $$\chi (X_{1},\dots \, X_{n}|B)=\chi ({X_{1},\dots ,X_{n}}).$$ The same statements hold for $ \chi ^{\omega } $.

It is sufficient to show that $$\chi ({X_{1},\dots ,X_{n}})+\chi (X_{n+1},\dots ,X_{p}|B)\geq \chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B).$$ Fix $ m>0 $ and elements $ {Y_{1},\dots ,Y_{m}}\in B $. Let $ {y_{1},\dots ,y_{m}}\in \Gamma _{R}({Y_{1},\dots ,Y_{m}};k,l,\epsilon ) $; such an $ m $-tuple exists because $ B $ can be embedded into the ultrapower of the hyperfinite II$ _{1} $ factor. By Voiculescu’s result in [@dvv:improvedrandom], we have that the ratio $$\liminf _{k\to \infty }\frac{\lambda \Gamma _{R}({X_{1},\dots ,X_{n}};k,l,\epsilon )\times \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )}{\lambda \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )}\geq 1.$$ Indeed, given a $ \delta >0 $, there is a $ k_{0} $, for all [<span style="font-variant:small-caps;">$ k>k_{0} $</span>]{} and for each choice of an approximant $$(x_{n+1},\dots ,x_{p})\in \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ),$$ there exists an open subset $$\Gamma _{k,x_{n+1},\dots ,x_{p}}\subset \Gamma _{R}({X_{1},\dots ,X_{n}};k,l,\epsilon ),$$ so that $$\frac{\lambda \Gamma _{k,x_{n+1},\dots ,x_{p}}}{\Gamma _{R}(X_{1},\dots ,X_{n};k,l,\epsilon )}>1-\delta$$ satisfying $$\Gamma _{k,x_{n+1},\dots ,x_{p}}\times \{(x_{n+1},\dots ,x_{p})\}\subset \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\frac{\epsilon }{2}).$$ Let $ O=O(x_{n+1},\dots ,x_{p}) $ be an open ball of radius $ \epsilon ' $ for the operator norm on $ M_{k}^{n-p} $, centered at $ x_{n+1},\dots ,x_{p} $. Then for sufficiently small $ \epsilon ' $ (depending only on $ k $ and $ l $), we have $$\Gamma _{k,x_{n+1},\dots ,x_{p}}\times O\subset \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ).$$

Let $$\Gamma _{k}=\bigcup _{x_{n+1},\dots ,x_{n}}\Gamma _{k,x_{n+1},\dots \, x_{p}}\times O(x_{n+1},\dots ,x_{n}).$$ Then $ \Gamma _{k}\subset \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ), $ so that $$\begin{aligned}
\lambda \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )\geq \lambda \Gamma _{k} &  & \\
\geq \inf _{x_{n+1,\dots \, x_{p}}}\lambda \Gamma _{k,x_{n+1},\dots ,x_{p}}\times \lambda \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ) &  & \\
\geq (1-\delta )\cdot \lambda \Gamma _{R}({X_{1},\dots ,X_{n}};k,l,\epsilon )\times \lambda \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ). &  & \end{aligned}$$ The statement of the theorem follows. The proof for $ \chi ^{\omega } $ is identical.

We note that the preceding theorem implies that $ \chi ({X_{1},\dots ,X_{n}}|B) $ is not always $ -\infty  $. For example, if $ S_{1},\dots ,S_{n} $ is a free semicircular family free from a unital von Neumann algebra $ B $, which can be embedded into the ultrapower of the hyperfinite II$ _{1} $ factor, then $ \chi (S_{1},\dots ,S_{n}|B)=\frac{n}{2}\log 2\pi e>-\infty  $.

Let $ X_{1},\dots ,X_{p} $ be random variables. Assume that $ {X_{1},\dots ,X_{n}}$ are free from the algebra generated by $ X_{n+1},\dots ,X_{p} $ and $ B $. Then $$\label{eqn:additivity}
\chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B)=\chi ({X_{1},\dots ,X_{n}}|B)+\chi (X_{n+1},\dots ,X_{p}|B).$$

If $ B $ is embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor, we have by Theorem \[thrm: whenfreefromB\] that $$\chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B)=\chi ({X_{1},\dots ,X_{n}})+\chi (X_{n+1},\dots ,X_{p}|B),$$ and also that $$\chi ({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}).$$ If $ B $ is not embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor, then the quantities on both sides of (\[eqn:additivity\]) are equal to $ -\infty  $.

The analogy between $ \chi ({X_{1},\dots ,X_{n}}|B) $ and $ \chi ^{*}({X_{1},\dots ,X_{n}}:B) $ makes it tempting to conjecture that (\[eqn:additivity\]) holds under the weaker assumption that $ {X_{1},\dots ,X_{n}}$ and $ X_{n+1},\dots ,X_{p} $ are free with amalgamation over $ B $; however, we were unable to prove this.

Separate change of variables formulas.
======================================

\[thrm: onechangeofvar\]Let $ f_{i}:\mathbb {R}\to \mathbb {R} $ be diffeomorphisms, and let $ \mu _{i} $ be the distribution of $ X_{i} $. Then $$\chi (f_{1}(X_{1}),\dots ,f_{n}(X_{n})|B)=\chi ({X_{1},\dots ,X_{n}}|B)+\sum ^{n}_{i=1}\int \int \frac{\log |f_{i}(s)-f_{i}(t)|}{\log |s-t|}d\mu _{i}(s)d\mu _{i}(t),$$ and the same formula holds for $ \chi ^{\omega } $ in place of $ \chi  $.

It is sufficient to prove the statement assuming further that $ f_{i} $ are identity diffeomorphisms for $ i>1 $; we write $ f=f_{1} $. It is moreover sufficient to show that given $ {Y_{1},\dots ,Y_{m}}\in B $ $$\begin{aligned}
\chi (f_{1}(X_{1}),\dots ,f_{n}(X_{n})|{Y_{1},\dots ,Y_{m}})\geq \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})+ &  & \\
\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t), &  & \end{aligned}$$ since the reverse inequality follows by replacing $ f $ with its inverse. It is shown in [@dvv:entropy4 Proposition 3.1] that given $ \delta >0 $, $ \epsilon >0 $, $ l>0 $, $ R>0 $, there exist $ k_{0}>0 $, $ \epsilon >\epsilon _{0}>0 $, $ l_{0}>l>0 $, such that for all $ k>k_{0} $, $ 0<\epsilon '<\epsilon _{0} $ and $ l'>l_{0} $, the determinant of the map $$F:({x_{1},\dots ,x_{n}},{y_{1},\dots ,y_{m}})\mapsto (f(x_{1}),x_{2},\dots ,x_{n},{y_{1},\dots ,y_{m}})$$ is bounded below by $$\exp \left( k^{2}\left[ \int \int \log \frac{|f(s)-f(t)|}{|s-t|}d\mu _{1}(s)d\mu _{1}(t)-\delta \right] \right)$$ for $$({x_{1},\dots ,x_{n}},{y_{1},\dots ,y_{m}})\in \Gamma _{R'}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};k,l',\epsilon ').$$ Moreover, the image of $$\Gamma _{R'}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};k,l',\epsilon ')$$ under this map is contained in $$\Gamma _{R}(f(X_{1}),\dots ,X_{n},{Y_{1},\dots ,Y_{m}};k,l,\epsilon ).$$ Choose $ {y_{1},\dots ,y_{m}}$ such that $$\begin{aligned}
\log \sup _{z_{1},\dots ,z_{m}}\lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|z_{1},\dots ,z_{m}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')- &  & \\
\log \lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l;,\epsilon ')<\delta . &  & \end{aligned}$$ Then we have that $$\begin{aligned}
\sup _{z_{1},\dots ,z_{m}}\log \lambda \Gamma _{R'}(f(X_{1}),X_{2},\dots ,X_{n}|z_{1},\dots ,z_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )\geq  &  & \\
\log \lambda \Gamma _{R}(f(X_{1}),\dots X_{n}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )\geq  &  & \\
\log \lambda F\left( \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')\right) \geq  &  & \\
\log \lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ') &  & \\
+\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t)-\delta k^{2}\geq  &  & \\
\log \sup _{z_{1},\dots ,z_{m}}\lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|z_{1},\dots ,z_{m}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ') &  & \\
+\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t)-2\delta k^{2}. &  & \end{aligned}$$ Taking $ \limsup _{k\to \infty } $ gives us that $$\begin{aligned}
\chi _{R}(f(X_{1}),\dots ,X_{n}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,\epsilon )\geq  &  & \\
\chi _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l',\epsilon ') &  & \\
+\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t), &  & \end{aligned}$$ which implies the theorem. The proof for $ \chi ^{\omega } $ is exactly the same.

If $ \Vert X_{i}\Vert <R $ for all $ i $, then $$\chi _{R}({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}|B),$$ and similarly for $ \chi ^{\omega } $.

The proof is along the lines of that of Theorem \[thrm: onechangeofvar\], using the ideas of a similar Proposition in [@dvv:entropy2], and is therefore omitted.

General change of variables formula. 
=====================================

Let $ B\subset M $ be a unital subalgebra, and let $ {F_{1},\dots ,F_{n}}$ be non-commutative power series with coefficients from $ B $; i.e., $$F_{i}(t_{1},\dots ,t_{n})=\sum _{k}\sum _{i_{1},\dots ,i_{k}}b_{{i_{1},\dots ,i_{k}}}^{i,0}t_{i_{1}}b^{i,1}_{{i_{1},\dots ,i_{k}}}\dots t_{i_{k}}b_{{i_{1},\dots ,i_{k}}}^{i,k}.$$ Denote by $ B[t_{1},\dots ,t_{n}] $ the set of all such power series, which have the property that if $ F\in B[t_{1},\dots ,t_{n}] $ and $ {X_{1},\dots ,X_{n}}$ are self-adjoint, then $ F({X_{1},\dots ,X_{n}}) $ is also self-adjoint. Given $ F_{i}\in B[t_{1},\dots ,t_{n}] $ as above, denote by $ \hat{F}_{i} $ the power series $$\hat{F}_{i}(z_{1},\dots ,z_{n})=\sum _{k}\sum _{{i_{1},\dots ,i_{k}}}\prod ^{k}_{j=1}\Vert b_{{i_{1},\dots ,i_{k}}}^{i,j}\Vert y_{i_{1}}\dots y_{i_{k}}.$$ We say that $ (R_{1},\dots ,R_{n}) $ is a mutiradius of convergence of $ F_{i} $, if it is the multiradius of convergence of $ \hat{F}_{i} $ (as an ordinary commutative power series).

Let $ F\in B[t_{1},\dots ,t_{n}] $ be such a power series. Then by the derivative of $ F $ with respect to $ t_{i} $ we mean the formal power series $ D_{i}F\in B[t_{1},\dots ,t_{n})\otimes B[t_{1},\dots ,t_{n}] $. Here $ D_{i} $ is defined by the following properties; here we think of $ B[t_{1},\dots ,t_{n}] $ and $ B[t_{1},\dots ,t_{n})\otimes B[t_{1},\dots ,t_{n}] $ are viewed as bimodules over the algebra generated by $ B $ and $ t_{1},\dots ,t_{n} $ using its obvious left and right actions.

1.  $ D_{i} $ is bilinear over the algebra generated by $ B $ and $ t_{1},\dots ,t_{i-1},t_{i+1},\dots ,t_{n} $;

2.  $ D_{i}(t_{i})=1\otimes 1 $;

3.  $ D $ satisfies the Leibniz rule: $ D_{i}(FG)=(D_{i}F)G+F(D_{i}G) $.

As an example, $$D_{1}(b_{0}t_{1}b_{1}t_{2}b_{2}t_{1}b_{3}t_{4}b_{4})=b_{0}\otimes b_{1}t_{2}b_{2}t_{1}b_{3}t_{4}b_{4}+b_{0}t_{1}b_{1}t_{2}b_{2}\otimes b_{3}t_{4}b_{4}.$$

Given a family of non-commutative power series $ F_{1},\dots ,F_{n} $ with a common multiradius of convergence $ (R_{1},\dots ,R_{n}) $, we define for $ {X_{1},\dots ,X_{n}}\in M $, $ \Vert X_{i}\Vert <R_{i} $, its Jacobian at $ {X_{1},\dots ,X_{n}}$, to be the matrix $ D_{B}F(X_{1},\dots ,X_{n})\in M_{n}\otimes M\otimes M $ whose $ i,j $-th entry is equal to $ D_{i}(F_{j})({X_{1},\dots ,X_{n}}) $.

Note that if $ B\subset M_{k} $ and $ {x_{1},\dots ,x_{n}}\in M_{k} $, then $ D_{B}({x_{1},\dots ,x_{n}}) $ is precisely the Jacobian of the map $$(z_{1},\dots ,z_{n})\mapsto \left( F_{1}(z_{1},\dots ,z_{n}),\dots ,F_{n}(z_{1},\dots ,z_{n})\right) ,$$ evaluated at $ {x_{1},\dots ,x_{n}}$.

The proof of the following Theorem is almost identical to the proof of the change of variables formula given in [@dvv:entropy2], together with the line of the proof of Theorem \[thrm: onechangeofvar\], and is therefore omitted.

\[thrm: genchangeofvar\]Let $ F_{i}\in B[t_{1},\dots ,t_{n}] $, $ i=1,\dots ,n $ be non-commutative power series with common multiradius of convergence $ (R_{1},\dots ,R_{n}) $. Assume that there are non-commutative power series $ G_{i} $, $ i=1,\dots ,n $ with common multiradius of convergence $ (r_{1},\dots ,r_{n}) $, such that for all $ i=1,\dots ,n $, $$\begin{aligned}
F_{i}(G_{1}(t_{1},\dots ,t_{n}),\dots ,G_{n}(t_{1},\dots ,t_{n}))=t_{i}, &  & \\
G_{i}(F_{1}(t_{1},\dots ,t_{n}),\dots ,F_{n}(t_{1},\dots ,t_{n}))=t_{i},. &  & \end{aligned}$$ Assume that for each $ i $, $ \Vert X_{i}\Vert <\min (r_{i},R_{i}) $. Then $$\begin{aligned}
\chi (F_{1}(X_{1},\dots ,X_{n}),\dots ,F_{n}(X_{1},\dots ,X_{n})|B)= &  & \\
\chi (X_{1},\dots ,X_{n}|B)+{{\operatorname{Tr}}}\otimes \tau \otimes \tau \left( \log |D_{B}F({X_{1},\dots ,X_{n}})|\right) . &  & \end{aligned}$$ The same formulas hold for $ \chi ^{\omega } $ in place of $ \chi  $.

We deduce that “free Brownian motion” has a regularizing effect on free entropy (compare [@dvv:entropy5]). The following proposition follows also from the results of [@dvv:entropy3], but we could not find its exact statement there.

Let $ S_{1},\dots ,S_{n} $ be a free semicircular family, free from the algebra $ B=W^{*}({X_{1},\dots ,X_{n}}) $. Assume that $ B $ is embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor. Then for all $ t>0 $, we have $$\chi (X_{1}+\sqrt{t}S_{1},\dots ,X_{n}+\sqrt{t}S_{n})\geq n\log 2\pi et>-\infty .$$ The same estimate holds for $ \chi ^{\omega } $.

By the change of variables formula and Theorem \[thrm: whenfreefromB\], we have $$\begin{aligned}
\chi (X_{1}+\sqrt{t}S_{1},\dots ,X_{n}+\sqrt{t}S_{n})\geq \chi (X_{1}+\sqrt{t}S_{1},\dots ,X_{n}+\sqrt{t}S_{n}|B)= &  & \\
\chi (\sqrt{t}S_{1},\dots ,\sqrt{t}S_{n})=\frac{n}{2}\log 2\pi et. & \end{aligned}$$

Let $ P_{1},\dots ,P_{n} $ be non-commutative polynomials in $ n $ variables with coefficients from $ B $. Assume that $ \chi ({X_{1},\dots ,X_{n}}|B)>-\infty  $. Then $$\begin{aligned}
\frac{d}{d\epsilon }\chi (X_{1}+\epsilon P_{1}(X_{1},\dots ,X_{n}),\dots ,X_{n}+\epsilon P_{n}({X_{1},\dots ,X_{n}})|B)= &  & \\
\sum _{i}\langle J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n}),P_{i}\rangle , &  & \end{aligned}$$ where $ J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n})) $ is the first-order conjugate variable to $ X_{i} $ with respect to $ B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n}) $ (cf. [@dvv:entropy5]). The same equality holds for $ \chi ^{\omega } $.

For $ \epsilon >0 $ sufficiently small, the transformation $ F^{\epsilon } $ defined by $$F_{i}^{\epsilon }(X_{1},\dots ,X_{n})=X_{i}+\epsilon P_{i}(X_{1},\dots ,X_{n})$$ is a non-commutative power series in $ {X_{1},\dots ,X_{n}}$ with coefficients from $ B $ and satisfies the hypothesis of Theorem \[thrm: genchangeofvar\]. It follows that $$\begin{aligned}
\chi (X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}}),\dots ,X_{n}+P_{n}({X_{1},\dots ,X_{n}})|B)=\chi ({X_{1},\dots ,X_{n}}|B) &  & \\
+({{\operatorname{Tr}}}\otimes \tau \otimes \tau )\log |D_{B}F^{\epsilon }({X_{1},\dots ,X_{n}})|. &  & \end{aligned}$$ Hence the derivative in $ \epsilon  $ of $ \chi (X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}}),\dots ,X_{n}+P_{n}({X_{1},\dots ,X_{n}})|B) $ is equal to the derivative of $ (\tau \otimes \tau )\log |D_{B}F^{\epsilon }({X_{1},\dots ,X_{n}})| $. Notice that $$D_{i}F^{\epsilon }_{j}({X_{1},\dots ,X_{n}})=\delta _{ij}+\epsilon D_{j}(P_{i})({X_{1},\dots ,X_{n}}),$$ so that $$D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}})=I+\epsilon M,$$ where $ I $ is the identity matrix and $$M_{ij}=D_{i}P_{j}({X_{1},\dots ,X_{n}}).$$ Hence $$(D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}})^{*}D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}}))=I+\epsilon (M+M^{*}+\epsilon M^{*}M).$$ Since $ \log (1+t) $ has a power series expansion around zero, we have that $$\frac{1}{2}\log (D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}})^{*}D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}}))=I+\frac{\epsilon }{2}(M+M^{*})+O(\epsilon ^{2}).$$ It follows that the desired derivative is equal to $$\begin{aligned}
(\tau \otimes \tau )\left( \frac{1}{2}[M+M^{*}]\right) =\frac{1}{2}\sum _{i}\tau \otimes \tau ((D_{i}P_{i})({X_{1},\dots ,X_{n}})+(D_{i}P_{i})({X_{1},\dots ,X_{n}})^{*})= &  & \\
\sum _{i}\tau \otimes \tau (D_{i}P_{i})({X_{1},\dots ,X_{n}}), &  & \end{aligned}$$ since $ F_{i} $ maps self-adjoint variables to self-adjoint variables. Hence we have, by the definition of the conjugate variable, that $$\begin{aligned}
\frac{d}{d\epsilon }\chi (X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}}),\dots ,X_{n}+P_{n}({X_{1},\dots ,X_{n}})|B)= &  & \\
\sum _{i}\langle J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n}),P_{i}\rangle , & \end{aligned}$$ as 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 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 e $ if and only if $ {X_{1},\dots ,X_{n}}$ are a free semicircular family, which is free from $ B $. The same statement holds for $ \chi ^{\omega } $.

The condition that $ \chi ({X_{1},\dots ,X_{n}}|B) $ achieves a local algebraic maximum is weaker than the condition that it achieves its maximum, so if the maximum is achieved, the local algebraic maximum is achieved, and Proposition \[prop: maximality\] applies. Conversely, it was shown in Corollary \[corr: max\] that the given number is indeed a maximum.

We end with the following theorem, whose proof is identical to that of [@dvv:entropy4 Proposition 4.3].

If $ \chi (X|B)=\chi (X)\neq -\infty  $, then $ X $ is free from $ B $. The same statement holds for $ \chi ^{\omega } $.

$ \chi ({X_{1},\dots ,X_{n}}|B\otimes M_{N})\protect $. 
========================================================

Let $ X_{ij}^{k} $, $ i,j=1,\dots ,N $, $ k=1,\dots ,n $ be non-commutative random variables, such that $ X_{ij}=X_{ji}^{*} $, and let $ B $ be a unital subalgebra of $ (M,\tau ) $ . Then the joint $ * $-distribution of the family $ \{X_{ij}^{k}\}\cup B $ completely determines and is completely determined by, the joint distribution of the matrices $$Z_{k}=\left( \begin{array}{cccc}
X_{11}^{k} & X_{12}^{k} & \dots  & X_{1N}^{k}\\
X^{k}_{21} & X_{22}^{k} & \dots  & X_{2N}^{k}\\
\vdots  & \vdots  & \ddots  & \vdots \\
X_{N1}^{k} & X_{N2}^{k} & \dots  & X_{NN}^{k}
\end{array}\right) ,$$ the matrix units $ E_{ij}\in M_{N} $ (matrices whose only non-zero entry is in the position $ i,j $) and the algebra $ B\otimes M_{N} $, identified with those matrices that have entries from $ B $. Therefore, it is natural to expect a relationship between the free entropy of the entries of the matrix relative to $ B $ and the free entropy of the matrix relative to the algebra $ B\otimes M_{N} $ of $ B $-valued matrices. Such a property is enjoyed by $ \chi ^{*} $ (introduced by Voiculescu in [@dvv:entropy5]; this property for $ \chi ^{*} $ was proved in [@nss:entropy]).

Let us define $$Y_{ij}^{k}=\left\{ \begin{array}{cc}
\frac{1}{2}(X_{ij}^{k}+[X_{ij}^{k}]^{*}) & \textrm{if }i\geq j\\
\frac{1}{2\sqrt{-1}}(X_{ij}^{k}-[X_{ij}^{k}]^{*}) & \textrm{if }i<j.
\end{array}\right.$$ For $ \omega  $ a free ultrafilter (i.e., a homomorphism $ \omega :C(\mathbb {N})\to \mathbb {C}) $, from the algebra of continuous bounded functions on $ \mathbb N $), and $ n\in \mathbb N $, define $ n\omega  $ to be the free ultrafilter, which as a homomorphism from $ C(\mathbb {N}) $ is given by the composition of $ \omega  $ and the map $ n\cdot f $, given by $ (n\cdot f)(m)=f(nm) $.

Let $ Y_{\{ij\}} $Let $ \omega  $ be a free ultrafilter. Then $$\chi ^{N\omega }(\{Y_{ij}^{k}\}_{i,j,k}|B)=N^{2}\chi ^{\omega }(Z_{1},\dots ,Z_{n}|B\otimes M_{N})-N^{2}\frac{n}{2}\log N.$$ Moreover, $$\chi (\{Y_{ij}^{k}\}_{i,j,k})\leq N^{2}\chi (Z_{1},\dots ,Z_{n}|B\otimes M_{N})-N^{2}\frac{n}{2}\log N.$$

Let $ E_{ij} $ be as before. Then $$\chi ^{\omega }(Z_{1},\dots ,Z_{n}|B\otimes M_{N})=\inf _{P_{1},\dots ,P_{q}\in B}\chi ^{\omega }(Z_{1},\dots ,Z_{n}|P_{1},\dots ,P_{q},\{E_{ij}\}_{ij}).$$ Here $ E_{ij} $ are not self-adjoint; what we mean by the quantity on the right is the obvious extension of our quantity to such a non-selfadjoint case.

We first claim that $$\chi ^{N\omega }(\{Y_{ij}^{k}\}_{i,j,k}|B)\geq N^{2}\chi ^{\omega }(Z_{1},\dots ,Z_{n}|B\otimes M_{N})-N^{2}\frac{n}{2}\log N.$$

Assume that $ k=Nk' $. Fix $ \delta >0 $. Choose $ Q_{1},\dots ,Q_{s}\in B\otimes M_{N} $ and $ R>0 $ so that $$\chi _{R}^{\omega }(Z_{1},\dots ,Z_{n}|Q_{1},\dots ,Q_{q},\{E_{ij}\}_{ij})\geq \chi ^{\omega }(Z_{1},\dots \, Z_{n}|B\otimes M_{N})-\delta .$$ Let $ (e_{ij})\in \Gamma _{R}(\{E_{ij}\}:Z_{1},\dots \, Z_{n},Q_{1},\dots ,Q_{s};k,l,\epsilon ) $. Then by a suitable choice of $ l $, $ \epsilon  $, $ k $ and $ R $, we can guarantee that the exists a projection $ p\leq e_{11} $ of rank $ k' $, $ [e_{11},p]=0 $. Given $ \delta  $, choose $ q_{1},\dots ,q_{S} $ and $ e_{ij} $ so that $$\begin{aligned}
\log \lambda \Gamma _{R}(Z_{1},\dots ,Z_{n}|\{e_{ij}\},\{q_{i}\}[\{E_{ij}\},\{Q_{i}\}];k,l,\epsilon )\geq  &  & \\
\sup _{(f_{ij})\in \Gamma _{R}(\{E_{ij}\}:k,l,\epsilon )}\Gamma _{R}(Z_{1},\dots ,Z_{n}|\{f_{ij}\}[\{E_{ij}\}];k,l,\epsilon )-\delta k^{2} &  & \end{aligned}$$ Let $ p $ be as before, and identify $ pM_{k}p $ with $ M_{k'} $. For $ (z_{1},\dots ,z_{n})\in M^{n}_{k} $, let $ z_{ij}^{k}=pe_{1i}z_{k}e_{j1}p $ and $ q^{r}_{ij}=pe_{1i}q_{r}e_{j1}p $. Denote by $ T=T_{\{e_{ij}\}} $the map from $ M_{k}^{n} $ to $ M_{k'}^{nN^{2}} $ given by $ T(z_{1},\dots ,z_{n})=(y^{r}_{ij})_{ijr} $, where $$y^{r}_{ij}=\left\{ \begin{array}{cc}
\frac{1}{2}(z_{ij}^{r}+[z_{ij}^{r}]^{*}) & \textrm{if }i\geq j,\\
\frac{1}{2\sqrt{-1}}(z_{ij}^{r}-[z_{ij}^{r}]^{*}) & \textrm{if }i<j.
\end{array}\right.$$ It follows that for $ l $, $ k $, $ R $ sufficiently large and $ \epsilon  $ sufficiently small, we can assume that the logarithm of the Jacobian of $ T $ is at least $ -\delta k^{2} $. Moreover, given $ l', $ $ R' $ and $ \epsilon ' $ there exist $ l>l' $, $ R>R' $ and $ 0<\epsilon <\epsilon ' $, such that $$T(\Gamma _{R}(Z_{1},\dots ,Z_{n}|\{e_{ij}\},\{q_{i}\}[\{E_{ij}\},\{Q_{i}\}];k,l,\epsilon )\subset \Gamma _{R'}(\{Y_{is}^{r}\}|\{q_{ij}^{r}\}_{ijr}[\{Q_{ij}^{r}\}_{ijr}];k',l',\epsilon ').$$ It follows that $$\begin{aligned}
\log \lambda \Gamma _{R'}(\{Y_{is}^{r}\}|\{q_{ij}^{r}\}_{ijr}[\{Q_{ij}^{r}\}_{ijr}];k',l',\epsilon ')\geq  &  & \\
\log \lambda T(\Gamma _{R}(Z_{1},\dots ,Z_{n}|\{e_{ij}\},\{q_{i}\}[\{E_{ij}\},\{Q_{i}\}];k,l,\epsilon )\geq  &  & \\
\log \lambda \Gamma _{R}(Z_{1},\dots ,Z_{n}|\{e_{ij}\},\{q_{i}\}[\{E_{ij}\},\{Q_{i}\}];k,l,\epsilon )-\delta k^{2}\geq  &  & \\
\sup _{((f_{ij})_{ij},(p_{r})\in \Gamma _{R}(\{E_{ij}\},\{P_{i}\}:k,l,\epsilon )}\log \lambda \Gamma _{R}(Z_{1},\dots ,Z_{n}|\{f_{ij}\},\{p_{i}\}[\{E_{ij}\},\{Q_{i}\}];k,l,\epsilon )-2\delta k^{2}. & \end{aligned}$$ Therefore, remembering that $ k=Nk' $ and taking limits as $ k\to \omega  $, we get: $$\begin{aligned}
\chi ^{N\omega }_{R'}(\{Y_{is}^{r}\}|\{q_{ij}^{r}\}_{ijr}[\{Q_{ij}^{r}\}_{ijr}];l',\epsilon ')= &  & \\
\lim _{k\to \omega }\frac{1}{(k')^{2}}\log \lambda \Gamma _{R'}(\{Y_{is}^{r}\}\{q_{ij}^{r}\}_{ijr}[\{Q_{ij}^{r}\}_{ijr}];k',l',\epsilon ')+\frac{N^{2}n}{2}\log k'= &  & \\
\lim _{k\to \omega }N^{2}\left( \frac{1}{k^{2}}\log \lambda \Gamma _{R'}(\{Y_{is}^{r}\}\{q_{ij}^{r}\}_{ijr}[\{Q_{ij}^{r}\}_{ijr}];k',l',\epsilon ')+\frac{n}{2}\log k-\frac{n}{2}\log N\right) \geq  &  & \\
N^{2}\lim _{k\to \omega }\frac{1}{k^{2}}\sup _{(f_{ij})\in \Gamma _{R}(\{E_{ij}\}:k,l,\epsilon )}\log \lambda \Gamma _{R}(Z_{1},\dots ,Z_{n}|\{f_{ij}\},\{q_{i}\}[\{E_{ij}\},\{Q_{i}\}];k,l,\epsilon ) &  & \\
+N^{2}\frac{n}{2}\log k-N^{2}\frac{n}{2}\log N-2\delta = &  & \\
N^{2}\chi _{R}^{\omega }(Z_{1},\dots ,Z_{n}|\{E_{ij}\},\{Q_{i}\})-N^{2}\frac{n}{2}\log N-2\delta \geq  &  & \\
N^{2}\chi _{R}^{\omega }(Z_{1},\dots ,Z_{n}|B\otimes M_{N};l,\epsilon )-N^{2}\frac{n}{2}\log N-3\delta . &  & \end{aligned}$$ This implies the claimed inequality.

Next, we claim that $$\chi ^{N\omega }(\{Y_{ij}^{k}\}_{i,j,k}|B)\geq N^{2}\chi ^{\omega }(Z_{1},\dots ,Z_{n}|B\otimes M_{N})-N^{2}\frac{n}{2}\log N.$$

For this, choose $ Q_{1},\dots ,Q_{n}\in B\otimes M_{N} $ and $ R>0 $ in such a way that $$\chi _{R}^{\omega }(Z_{1},\dots ,Z_{n}|Q_{1},\dots \, Q_{n},\{E_{ij}\})\geq \chi ^{\omega }(Z_{1},\dots ,Z_{n}|B\otimes M_{N})-\delta .$$ Set $ Q_{ij}^{r}=E_{1i}Q_{r}E_{j1} $. Choose $ \{q_{ij}^{r}\}\in \Gamma _{R}(\{Q_{ij}^{r}\};m,k,\epsilon ) $. Assume that $$(y_{ij}^{r})\in \Gamma _{R}((Y_{ij}^{r})|\{q_{ij}^{r}\}[\{Q_{ij}^{r}\}];k,l,\epsilon ).$$ Then let $ x_{ij}^{r}=y_{ij}^{r}+\sqrt{-1}y_{ji}^{r} $. Set $ x_{r}\in M_{k}\otimes M_{N} $ to be $ x_{r}=\sum _{ij}x_{ij}^{r}\otimes E_{ij} $, and put $ q_{r}=\sum q_{ij}^{r}\otimes E_{ij}\in M_{k}\otimes M_{n} $. Then given $ R'>0 $, $ l'>0 $ and $ \epsilon '>0 $, there exist $ R>R'>0 $, $ l>l'>0 $ and $ 0<\epsilon <\epsilon ' $, such that $ (z_{1},\dots ,z_{r})\in \Gamma _{R'}(Z_{1},\dots ,Z_{n}|\{E_{ij}\},\{q_{i}\}[\{E_{ij}\},\{Q_{i}\}];nk,l',\epsilon ') $. Since the map assigning to $ (y_{ij}^{r}) $ the $ n $-tuple $ (z_{1},\dots ,z_{r}) $ is measure-preserving, we get that $$\begin{aligned}
\sup _{e_{ij},f_{i}}\log \lambda \Gamma _{R'}(Z_{1},\dots ,Z_{n}|\{e_{ij}\},\{f_{i}\}[\{E_{ij}\},\{Q_{i}\}];nk,l',\epsilon ')\geq  &  & \\
\log \lambda \Gamma _{R'}(Z_{1},\dots ,Z_{n}|\{E_{ij}\},\{q_{i}\}[\{E_{ij}\},\{Q_{i}\}];nk,l',\epsilon ')\geq  &  & \\
\log \lambda \Gamma _{R}((Y_{ij}^{r})|\{q_{ij}^{r}\}[Q_{ij}^{r}];k,l,\epsilon ). &  & \end{aligned}$$ This implies our claim.

The proof of the inequality for $ \chi  $ instead of $ \chi ^{\omega } $ is along the lines of the proof of the second inequality above, and is left to the reader.

[1]{}

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[^1]: Research supported in part by a National Science Foundation postdoctoral fellowship.
