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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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and

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It is readily verified using the fact that the operator-valued bi-free moment
function is bi-multiplicative that
$E_{\pi|_{V_{6}}}(Z_{8},Z_{9})=E(Z_{8}Z_{9})$ and $E_{\pi|_{V_{4}\cup
V_{6}}}(Z_{6},Z_{7},Z_{10})=E(Z_{6}R_{E(Z_{7})}Z_{10})$. Thus, using the fact
that $\pi|_{V_{3}}=1_{{\chi|}_{V_{3}}}$, the first step in Definition 4.6
yields

$X=\widetilde{E}\left(Z_{4}R_{E(Z_{6}R_{E(Z_{7})}Z_{10})}Z_{5}L_{E(Z_{8}Z_{9})}Z_{11}\xi\right).$

Hence

$\displaystyle\Psi_{\pi}(Z_{1},\ldots,Z_{11},\xi)$
$\displaystyle=\widetilde{E}\left(Z_{2}\widetilde{E}\left(Z_{1}Z_{3}\widetilde{E}\left(Z_{4}R_{E\left(Z_{6}R_{E(Z_{7})}Z_{10}\right)}Z_{5}L_{E(Z_{8}Z_{9})}Z_{11}\xi\right)\right)\right)$
$\displaystyle=\widetilde{E}\left(Z_{1}Z_{3}\widetilde{E}\left(Z_{4}R_{E\left(Z_{6}R_{E(Z_{7})}Z_{10}\right)}Z_{5}L_{E(Z_{8}Z_{9})}Z_{11}\xi\right)\right)E(Z_{2}),$

with the last equality following from Proposition 3.9.

Before investigating the bi-multiplicative properties inherited by the
analytical bi-moment function, we note it is truly an extension of the
operator-valued bi-moment function.

######  Theorem 4.8.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space. For any $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$,
$\pi\in\operatorname{BNC}(\chi)$, and $Z_{k}\in A_{\chi(k)}$,

$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})=E_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n})+N_{\tau_{B}}.$

###### Proof.

Note that each step in the recursive definition of Definition 4.6 is a step
that can be performed to the operator-valued bi-free moment function as the
operator-valued bi-free moment function is bi-multiplicative (see Definition
2.10). Therefore, as Proposition 3.7 implies that

$\widetilde{E}(a)=E(a)+N_{\tau_{B}}$

for all $a\in A$, by applying the same recusive properties to
$E_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n})$ as used to compute
$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})$, the result follows. ∎

Like with the construction of the operator-valued bi-free moment function in
[CNS2015-1], although the construction of the analytical bi-moment function is
done using specific rules from the operator-valued bi-free moment function in
a specific order, we desire more flexibility in the reductions that can be
done and the order they can be done in. In particular, we desire to show that
the analytical bi-moment function is an analytic extension of the operator-
valued bi-free moment function.

The main ideas used to prove this are similar to those utilized in the proof
of [CNS2015-1]*Theorem 5.1.4 and hence we shall be concerned with
demonstrating that the inclusion of the $L_{2}(A,\tau)$ term and the slightly
modified recursive definition are not issue and pose next to no changes. In
particular, it may appear that $\Psi$ behaves differently than the operator-
valued bi-free moment function as entries in $L_{2}(A,\tau)$ can also act as
mixtures of left and right operators, which was not dealt with in [CNS2015-1].
However, using the properties of $\widetilde{E}$ as developed in Proposition
3.9, one familiar with [CNS2015-1] can easily see that the desired results
will hold with simple adaptations. We note that similar adaptations were done
in [GS2018] without issue.

When examining the proof of the following, Example 4.7 serves as a good
example to keep in mind, just as Example 4.4 aided in comprehending why
analytic extensions of bi-multiplicative functions work.

######  Theorem 4.9.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and let

$\Psi:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in\\{\ell,r\\}^{n}}\operatorname{BNC}(\chi)\times
A_{\chi(1)}\times\ldots\times A_{\chi(n-1)}\times L_{2}(A,\tau)\to
L_{2}(B,\tau_{B})$

be the analytical bi-moment function. Then $\Psi$ is an analytically extension
of the operator-valued bi-free moment function.

Clearly the map $\Psi_{1_{\chi}}$ is $\mathbb{C}$-multilinear and it does not
matter whether $\chi(n)=\ell$ or $\chi(n)=r$. A straightforward induction
argument using the definition of $\Psi$ shows that the map $\Psi_{\pi}$ will
be $\mathbb{C}$-multilinear. Thus we focus on the remaining four properties.

###### Proof of Theorem 4.9 property (i).

This immediately follows from parts (ii) and (iii) of Proposition 3.9. ∎

###### Proof of Theorem 4.9 property (ii).

To see (ii), fix $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$,
$\pi\in\operatorname{BNC}(\chi)$, $\xi\in L_{2}(A,\tau)$, $b\in B$, and
$Z_{k}\in A_{\chi(k)}$, and let $p$ and $q$ be as in the statement of (ii). In
the case that $\chi(p)=\ell$, note that

$\Psi_{1_{\chi}}(Z_{1},\ldots,Z_{p-1},L_{b}Z_{p},Z_{p+1},\ldots,Z_{n-1},\xi)=\widetilde{E}(Z_{1}\cdots
Z_{p-1}L_{b}Z_{p}Z_{p+1}\cdots Z_{n-1}\xi).$

If $q\neq\infty$, then $Z_{q+1},\ldots,Z_{p-1}\in A_{r}$ and thus commute with
$L_{b}$. Hence

$\displaystyle\Psi_{1_{\chi}}(Z_{1},\ldots,Z_{p-1},L_{b}Z_{p},Z_{p+1},\ldots,Z_{n-1},\xi)$
$\displaystyle=\widetilde{E}(Z_{1}\cdots Z_{q-1}Z_{q}L_{b}Z_{q+1}\cdots
Z_{n-1}\xi)$
$\displaystyle=\Psi_{1_{\chi}}(Z_{1},\ldots,Z_{q-1},Z_{q}L_{b},Z_{q+1},\ldots,Z_{n-1},\xi)$

(and note $Z_{q}L_{b}\in A_{\ell}$).

If $q=\infty$, then $Z_{1},\ldots,Z_{p-1}\in A_{r}$ and thus commute with
$L_{b}$. Hence

$\displaystyle\Psi_{1_{\chi}}(Z_{1},\ldots,Z_{p-1},L_{b}Z_{p},Z_{p+1},\ldots,Z_{n-1},\xi)$
$\displaystyle=\widetilde{E}(L_{b}Z_{1}\cdots Z_{n-1}\xi)$
$\displaystyle=b\widetilde{E}(Z_{1}\cdots Z_{n-1}\xi)$
$\displaystyle=b\Psi_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},\xi)$

by Proposition 3.9. The case $\chi(p)=r$ is similar. ∎

###### Proof of Theorem 4.9 property (iii).

To highlight how the proof works, we begin with the case that $\pi$ consists
of exactly three blocks that are $\chi$-intervals and $n$ is contained in the
middle block under the $\chi$-ordering. Suppose that
$\pi=\\{V_{1},V_{2},V_{3}\\}$ and thus there exists $i\in\\{2,\ldots,n-1\\}$
and $j\in\\{0,\ldots,n-2\\}$ such that

$\displaystyle V_{1}$
$\displaystyle=\\{s_{\chi}(1),s_{\chi}(2),\ldots,s_{\chi}(i-1)\\},$
$\displaystyle V_{2}$
$\displaystyle=\\{s_{\chi}(i),s_{\chi}(i+1),\ldots,s_{\chi}(i+j)\\},\text{ and
}$ $\displaystyle V_{3}$
$\displaystyle=\\{s_{\chi}(i+j+1),s_{\chi}(i+j+2),\ldots,s_{\chi}(n)\\}.$

Thus $n=s_{\chi}(k)$ for some $i\leq k\leq i+j$. This implies that
$\chi(p)=\ell$ for all $p\in V_{1}$ and $\chi(p)=r$ for all $p\in V_{3}$. If
$W=V_{1}\cup V_{3}$, observe that the definition of the permutation $s_{\chi}$
yields that

$q=\max_{\leq}W=\max_{\leq}\\{s_{\chi}(i-1),s_{\chi}(i+j+1)\\}.$

Consider the case $q=s_{\chi}(i-1)$ so that $q\in V_{1}$, and let
$m=s_{\chi}(i+j+1)$. Notice that if

$X=\Psi_{\pi|_{V_{2}}}({(Z_{1},\ldots,Z_{n-1},\xi)|}_{V_{2}}),$

then by Definition 4.6 we obtain that

$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)=\Psi_{\pi|_{W}}({(Z_{1},\ldots,Z_{q-1},Z_{q}X,\ldots,Z_{n-1},\xi)|}_{W}).$

Thus, if

$Y=\Psi_{\pi|_{V_{1}}}({(Z_{1},\ldots,Z_{q-1},Z_{q}X,\ldots,Z_{n-1},\xi)|}_{V_{1}}),$

then again Definition 4.6 implies that

$\displaystyle\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)$
$\displaystyle=\Psi_{\pi|_{V_{3}}}({(Z_{1},\ldots,Z_{m-1},Z_{m}Y,\ldots,Z_{n-1},\xi)|}_{V_{3}})$
$\displaystyle=\widetilde{E}\left(Z_{s_{\chi}(n)}Z_{s_{\chi}(n-1)}\ldots
Z_{s_{\chi}(i+j+1)}Y\right).$

Therefore, since $Z_{s_{\chi}(n)}Z_{s_{\chi}(n-1)}\ldots
Z_{s_{\chi}(i+j+1)}\in A_{r}$, Proposition 3.9 implies that

$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)=YE\left(Z_{s_{\chi}(n)}Z_{s_{\chi}(n-1)}\ldots
Z_{s_{\chi}(i+j+1)}\right)=YE_{\pi|_{V_{3}}}((Z_{1},\ldots,Z_{n-1},\xi)|_{V_{3}}).$

Since $Z_{s_{\chi}(1)},Z_{s_{\chi}(2)},\ldots,Z_{s_{\chi}(i-1)}\in A_{r}$,
Proposition 3.9 implies that

$\displaystyle Y=E(Z_{s_{\chi}(1)}Z_{s_{\chi}(2)}\ldots
Z_{s_{\chi}(i-1)})X=E_{\pi|_{V_{1}}}((Z_{1},\ldots,Z_{n-1},\xi)|_{V_{1}})X,$

so the result follows. Note the case $q=s_{\chi}(i+j+1)$ is handled similarly
by interchanging the orders of the $\chi$\- intervals $V_{1}$ and $V_{3}$ .
This argument can be extended via induction to any bi-non-crossing partition
$\pi$ all of whose blocks are $\chi$-interval.

By the same argument as [CNS2015-1]*Lemma 5.2.1, one need only consider the
case in property (iii) that for each $\chi$-interval, the $\chi$-maximal and
$\chi$-minimal elements belong to the same block. When using the recursive
procedure in Definition 4.6 to reduce $\Psi_{\pi}$, one of the
$\chi$-intervals (which will either be entirely on the left or entirely on the
right) will have the $L_{2}(A,\tau)$ term added to the last entry as above.
This $L_{2}(A,\tau)$ entry can be pulled out on the appropriate side leaving
only the bi-moment function expression for the $\chi$-interval, which can be
undone as usual. By repetition, eventually all that remains is the expression
for the $\chi$-interval containing $n$ as desired. ∎

###### Proof of Theorem 4.9 property (iv).

The proof that property (iv) holds for the operator-valued bi-free moment
function is one of the longest of [CNS2015-1] consisting of [CNS2015-1]*Lemma
5.3.1, [CNS2015-1]*Lemma 5.3.2, [CNS2015-1]*Lemma 5.3.3, and [CNS2015-1]*Lemma
5.3.4. As such, we will only sketch the details here.

First one proceeds to show that properties (i) and (ii) of Definition 4.1 hold
for $\Psi$ when $1_{\chi}$ is replaced with an arbitrary bi-non-crossing
partition. This effectively makes use of the same arguments as in Lemma 4.5;
that is, one uses the recursive algorithm to reduce down and then note the
proofs of properties (i) and (ii) above still apply and lets one move elements
around as needed. In particular, the same arguments used in [CNS2015-1]*Lemma
5.3.2 and [CNS2015-1]*Lemma 5.3.3 transfer with the use of Proposition 3.9.

Next, using property (iii), we need only prove property (iv) under the
assumption that $s_{\chi}(1)$ and $s_{\chi}(n)$ are in the same block $W_{0}$
of $W$. One then follows many of the same ideas as [CNS2015-1]*Lemma 5.3.1 and
[CNS2015-1]*Lemma 5.3.4 by applying the recursive definition from Definition
4.6, moving around the appropriate $B$-elements using the more general (i) and
(ii), and combining the appropriate elements using (iii) as needed. ∎

### Analytical Operator-Valued Bi-Free Cumulant Function

By convolving the analytical bi-moment function with the bi-non-crossing
Möbius function, we obtain the following which is essential to our study of
conjugate variables in the subsequent section.

######  Definition 4.10.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and denote by $\Psi$ the analytical bi-moment function. The
analytical bi-cumulant function

$\widetilde{\kappa}:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in\\{\ell,r\\}^{n}}\operatorname{BNC}(\chi)\times
A_{\chi(1)}\times\ldots\times A_{\chi(n-1)}\times L_{2}(A,\tau)\to
L_{2}(B,\tau_{B})$

is defined by

$\widetilde{\kappa}_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\chi)\\\
\sigma\leq\pi\\\
\end{subarray}}\Psi_{\sigma}(Z_{1},\ldots,Z_{n-1},\xi)\mu_{\operatorname{BNC}}(\sigma,\pi),$

for all $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$,
$\pi\in\operatorname{BNC}(\chi)$, $\xi\in L_{2}(A,\tau)$, and $Z_{k}\in
A_{\chi(k)}$.

######  Remark 4.11.

  1. (i)

In the case when $\pi=1_{\chi}$, we will denote the map
$\widetilde{\kappa}_{1_{\chi}}$ simply by $\widetilde{\kappa}_{\chi}$. By
Möbius inversion, we obtain that

$\Psi_{\sigma}(Z_{1},\ldots,Z_{n})=\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\
\pi\leq\sigma\\\
\end{subarray}}\widetilde{\kappa}_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)$

for all $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$,
$\sigma\in\operatorname{BNC}(\chi)$, and $Z_{k}\in A_{\chi(k)}$.

  2. (ii)

In the case that $B=\mathbb{C}$, the analytical bi-cumulant function are
precisely the $L_{2}(A,\tau)$-valued bi-free cumulants that were used in
[CS2020].

Unsurprisingly, the analytic extension of the operator-valued bi-free cumulant
function lives upto its name.

######  Theorem 4.12.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space. Then the analytical bi-cumulant function is the analytic
extension of the operator-valued bi-free cumulant function.

###### Proof.

Recall by [CNS2015-1]*Theorem 6.2.1 that the convolution of a bi-
multiplicative function with a scalar-valued multiplicative function on the
lattice of non-crossing partitions (e.g. the bi-free Möbius function) produces
a bi-multiplicative function. As the properties of an analytic extension of a
bi-multiplicative function are analogous to those of a bi-multiplicative
function, we obtain that the convolution of an analytic extension of a bi-
multiplicative function with a scalar-valued multiplicative function on the
lattice of non-crossing partitions (e.g. the bi-free Möbius function) produces
the analytic extension of the corresponding bi-multiplicative function
obtained via the same convolution. Hence the result follows. ∎

######  Theorem 4.13.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space. For all $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$,
$\pi\in\operatorname{BNC}(\chi)$, and $Z_{k}\in A_{\chi(k)}$, we have that

$\widetilde{\kappa}_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})=\kappa^{B}_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n})+N_{\tau_{B}}.$

###### Proof.

By Theorem 4.8, we know that

$\widetilde{\Psi}_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})=E_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n})+N_{\tau_{B}}$

for all $\pi\in\operatorname{BNC}(\chi)$. Therefore as $\widetilde{\kappa}$
and $\kappa^{B}$ are the convolution of $\widetilde{\Psi}$ and $E$ against the
bi-free Möbius function respectively, the result follows. ∎

Of course, as [CNS2015-1]*Theorem 8.1.1 demonstrated that bi-freeness with
amalagmation over $B$ is equivalent to the mixed operator-valued bi-free
cumulants vanishing, Theorem 4.13 immediately implies the following.

######  Corollary 4.14.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space containing a family of pairs of $B$-algebras
$\\{(C_{k},D_{k})\\}_{k\in K}$. Consider the following two conditions:

  1. (1)

The family $\\{(C_{k},D_{k})\\}_{k\in K}$ is bi-free with amalgamation over
$B$ with respect to $E$.

  2. (2)

For all $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$, $Z_{1},\ldots,Z_{n}\in
A$, and non-constant maps $\gamma:\\{1,\ldots,n\\}\to K$ such that

$Z_{k}\in\begin{cases}C_{\gamma(k)}&\text{if }\chi(k)=\ell\\\
D_{\gamma(k)}&\text{if }\chi(k)=r\\\ \end{cases},$

it follows that

$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})=0.$

Then (1) implies (2). In the case that $\tau_{B}:B\to\mathbb{C}$ is faithful,
(2) implies (1).

###### Proof.

Note (1) implies (2) follows from [CNS2015-1]*Theorem 8.1.1 and Theorem 4.13.
In the case that $\tau_{B}$ is faithful, (2) immediately implies
$\kappa^{B}_{\pi}(Z_{1},\ldots,Z_{n-1},Z_{n})=0$ where $\\{Z_{k}\\}^{n}_{k=1}$
are as in (2) via Theorem 4.13. Hence [CNS2015-1]*Theorem 8.1.1 completes the
argument. ∎

### Vanishing Analytical Cumulants

However, something stronger than Corollary 4.14 holds. Indeed, note that the
analytic operator-valued bi-free cumulant function has the added benefit that
the last entry can be an element of $L_{2}(A,\tau)$ and thus the $L_{2}$-image
of a product of left and right operators. As such, it is possible to verify
that additional analytic bi-cumulants vanish.

The desired result is analogous to the scalar-valued result demonstrated in
[CS2020]*Proposition 2.30 and proved in a similar manner. Thus we begin with a
generalization of [CNS2015-1]*Theorem 9.1.5 where we can expand out a cumulant
involving products of operators. In [CNS2015-1]*Theorem 9.1.5 only products of
left and right operators were considered in the operator-valued setting
whereas [CS2020]*Lemma 2.29 expanded out scalar-valued cumulants involving a
product of left and right operator in the last entry.

To begin, fix $m<n\in\mathbb{N}$, $\chi\in{\\{\ell,r\\}}^{m}$, integers

$k(0)=0<k(1)<\ldots<k(m)=n,$

and any function $\widehat{\chi}\in{\\{\ell,r\\}}^{n}$ such that for all
$q\in\\{1,\ldots,n\\}$ for which there exists a (necessarily unique)
$p_{q}\in\\{1,\ldots,m-1\\}$ with $k(p_{q}-1)<q\leq k(p_{q})$, we have

$\widehat{\chi}(q)=\chi(p_{q}).$

Thus $\widehat{\chi}$ is constant from $k(p-1)+1$ to $k(p)$ whereas
$\widehat{\chi}$ does not need to be constant from $k(m-1)+1$ to $k(m)$.

We may embed $\operatorname{BNC}(\chi)$ into
$\operatorname{BNC}(\widehat{\chi})$ via $\pi\mapsto\widehat{\pi}$ where the
blocks of $\widehat{\pi}$ are formed by taking each block $V$ of $\pi$ and
forming a block

$\widehat{V}=\bigcup_{p\in V}\\{k(p-1)+1,\ldots,k(p)\\}$

of $\widehat{\pi}$. It is not difficult to see that
$\widehat{\pi}\in\operatorname{BNC}(\widehat{\chi})$ as $\widehat{\chi}$ is
constant on $\\{k(p-1)+1,\ldots,k(p)\\}$ for all $p\in V\setminus\\{m\\}$ and
although the block containing $\\{k(n-1)+1,\ldots,k(n)\\}$ has both left and
right entries, it occurs at the bottom of the bi-non-crossing diagram and thus
poses no problem. Alternatively, this map can be viewed as an analogue of the
map on non-crossing partitions from [NS2006]*Notation 11.9 after applying
$s^{-1}_{\chi}$.

It is easy to see that $\widehat{1_{\chi}}=1_{\widehat{\chi}}$,

$\widehat{0_{\chi}}=\bigcup_{p=1}^{m}\\{k(p-1)+1,\ldots,k(p)\\},$

and that the map $\pi\mapsto\widehat{\pi}$ is injective and order-preserving.
Furthermore, the image of $\operatorname{BNC}(\chi)$ under this map is

$\widehat{\operatorname{BNC}}(\chi)=\left[\widehat{0_{\chi}},\widehat{1_{\chi}}\right]=\left[\widehat{0_{\chi}},1_{\widehat{\chi}}\right]\subseteq\operatorname{BNC}(\widehat{\chi}).$

######  Remark 4.15.

Recall that since $\mu_{\operatorname{BNC}}$ is the bi-non-crossing Möbius
function, we have for each $\sigma,\pi\in\operatorname{BNC}(\chi)$ with
$\sigma\leq\pi$ that

$\sum_{\begin{subarray}{c}\upsilon\in\operatorname{BNC}(\chi)\\\
\sigma\leq\upsilon\leq\pi\end{subarray}}\mu_{\operatorname{BNC}}(\upsilon,\pi)=\left\\{\begin{array}[]{ll}1&\mbox{if
}\sigma=\pi\\\ 0&\mbox{otherwise }\end{array}\right..$

Since the lattice structure is preserved under the map defined above, we see
that
$\mu_{\operatorname{BNC}}(\sigma,\pi)=\mu_{\operatorname{BNC}}(\widehat{\sigma},\widehat{\pi})$.

It is also easy to see that the partial Möbius inversion from
[NS2006]*Proposition 10.11 holds in the bi-free setting; that is, if
$f,g:\operatorname{BNC}(\chi)\to\mathbb{C}$ are such that

$f(\pi)=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\chi)\\\
\sigma\leq\pi\end{subarray}}g(\sigma)$

for all $\pi\in\operatorname{BNC}(\chi)$, then for all
$\pi,\sigma\in\operatorname{BNC}(\chi)$ with $\sigma\leq\pi$, we have the
relation

$\sum_{\begin{subarray}{c}\upsilon\in\operatorname{BNC}(\chi)\\\
\sigma\leq\upsilon\leq\pi\end{subarray}}f(\upsilon)\mu_{\operatorname{BNC}}(\upsilon,\pi)=\sum_{\begin{subarray}{c}\omega\in\operatorname{BNC}(\chi)\\\
\omega\vee\sigma=\pi\end{subarray}}g(\omega)$

where $\pi\vee\sigma$ denotes the smallest element of
$\operatorname{BNC}(\chi)$ greater than $\pi$ and $\sigma$.

Thus, by following the proofs of either [NS2006]*Theorem 11.12,
[CNS2015-1]*Theorem 9.1.5, or [CS2020]*Lemma 2.29, we arrive at the following.

######  Proposition 4.16.

Let $(A,E,\varepsilon,\tau)$ be be an analytical $B$-$B$-non-commutative
probability space. Under the above notation, if
$\pi\in\operatorname{BNC}(\chi)$ and $Z_{k}\in A_{\widehat{\chi}(k)}$, then

$\widetilde{\kappa}_{\pi}(Z_{1}\cdots Z_{k(1)},\ldots,Z_{k(m-2)+1}\cdots
Z_{k(m-1)},Z_{k(m-1)+1}\cdots
Z_{k(m)}+N_{\tau})=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\widehat{\chi})\\\
\sigma\vee\widehat{0_{\chi}}=\widehat{\pi}\end{subarray}}\widetilde{\kappa}_{\sigma}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau}).$

In particular, when $\sigma=1_{\chi}$, we have

$\widetilde{\kappa}_{\chi}(Z_{1}\cdots Z_{k(1)},\ldots,Z_{k(m-2)+1}\cdots
Z_{k(m-1)},Z_{k(m-1)+1}\cdots
Z_{k(m)}+N_{\tau})=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\widehat{\chi})\\\
\sigma\vee\widehat{0_{\chi}}=1_{\widehat{\chi}}\\\
\end{subarray}}\widetilde{\kappa}_{\sigma}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau}),$

###### Proof.

First, it is not difficult to verify using the recursive definition of the
analytic operator-valued bi-moment function that

$\Psi_{\upsilon}(Z_{1}\cdots Z_{k(1)},\ldots,Z_{k(m-2)+1}\cdots
Z_{k(m-1)},Z_{k(m-1)+1}\cdots
Z_{k(m)}+N_{\tau})=\Psi_{\widehat{\upsilon}}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})$

for all $\upsilon\in\operatorname{BNC}(\chi)$. Therefore, we have

$\displaystyle\widetilde{\kappa}_{\pi}(Z_{1}\cdots
Z_{k(1)},\ldots,Z_{k(m-2)+1}\cdots Z_{k(m-1)},Z_{k(m-1)+1}\cdots
Z_{k(m)}+N_{\tau})$
$\displaystyle=\sum_{\begin{subarray}{c}\upsilon\in\operatorname{BNC}(\chi)\\\
\upsilon\leq\pi\end{subarray}}\Psi_{\upsilon}(Z_{1}\cdots
Z_{k(1)},\ldots,Z_{k(m-2)+1}\cdots Z_{k(m-1)},Z_{k(m-1)+1}\cdots
Z_{k(m)}+N_{\tau})\mu_{\operatorname{BNC}}(\upsilon,\pi)$
$\displaystyle=\sum_{\begin{subarray}{c}\upsilon\in\operatorname{BNC}(\chi)\\\
\upsilon\leq\pi\end{subarray}}\Psi_{\widehat{\upsilon}}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})\mu_{\operatorname{BNC}}(\widehat{\upsilon},\widehat{\pi})$
$\displaystyle=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\widehat{\chi})\\\
\widehat{0_{\chi}}\leq\sigma\leq\widehat{\pi}\end{subarray}}\Psi_{\sigma}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})\mu_{\operatorname{BNC}}(\sigma,\widehat{\pi})$
$\displaystyle=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\widehat{\chi})\\\
\sigma\vee\widehat{0_{\chi}}=\widehat{\pi}\end{subarray}}\widetilde{\kappa}_{\sigma}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})$

where the last line following from Remark 4.15. ∎

######  Theorem 4.17.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space containing a family of pairs of $B$-algebras
$\\{(C_{k},D_{k})\\}_{k\in K}$ that are bi-free with amalgamation over $B$
with respect to $E$. For each $k\in K$, let $L_{2}(A_{k},\tau)$ be the closed
subspace of $L_{2}(A,\tau)$ generated by

$\operatorname{alg}(C_{k},D_{k})+N_{\tau}.$

Then for all $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$, non-constant maps
$\gamma:\\{1,\ldots,n\\}\to K$, $\xi\in L_{2}(A_{\gamma(n)},\tau)$, and
$Z_{1},\ldots,Z_{n-1}\in A$ such that

$Z_{k}\in\begin{cases}C_{\gamma(k)}&\text{if }\chi(k)=\ell\\\
D_{\gamma(k)}&\text{if }\chi(k)=r\\\ \end{cases},$

it follows that

$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{n-1},\xi)=0.$

###### Proof.

Fix an $m\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{m}$, non-constant map
$\gamma:\\{1,\ldots,m\\}\to K$, and $Z_{1},\ldots,Z_{m-1}\in A$. For any
$n\geq m$, if $Z_{m},\ldots,Z_{n}\in C_{\gamma(m)}\cup D_{\gamma(m)}$ and
$\widehat{\chi}$ is defined by

$\widehat{\chi}(k)=\begin{cases}\chi(k)&\text{if }k\leq m\\\ \ell&\text{if
}k>m\text{ and }Z_{k}\in C_{\gamma(m)}\\\ r&\text{if }k>m\text{ and }Z_{k}\in
D_{\gamma(m)}\\\ \end{cases},$

then by Proposition 4.16 and Theorem 4.8 implies that

$\displaystyle\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{m-1},Z_{m}\cdots
Z_{n}+N_{\tau})$
$\displaystyle=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\widehat{\chi})\\\
\sigma\vee\widehat{0_{\chi}}=1_{\widehat{\chi}}\\\
\end{subarray}}\widetilde{\kappa}_{\sigma}(Z_{1},\ldots,Z_{n-1},Z_{n}+N_{\tau})$
$\displaystyle=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\widehat{\chi})\\\
\sigma\vee\widehat{0_{\chi}}=1_{\widehat{\chi}}\\\
\end{subarray}}\kappa^{B}_{\sigma}(Z_{1},\ldots,Z_{n-1},Z_{n})+N_{\tau_{B}}.$

As the conditions $\sigma\in\operatorname{BNC}(\widehat{\chi})$ and
$\sigma\vee\widehat{0_{\chi}}=1_{\widehat{\chi}}$ automatically imply that
each block of $\sigma$ containing one of $\\{1,2,\ldots,m-1\\}$ must also
contain an element of $\\{m,\ldots,n\\}$, the bi-multiplicative properties of
the operator-valued bi-free cumulant function imply that each cumulant
$\kappa^{B}_{\sigma}(Z_{1},\ldots,Z_{n-1},Z_{n})$ appearing in the sum above
can be reduced down to an expression involving a mixed $\kappa^{B}$ term which
must be 0 by [CNS2015-1]*Theorem 8.1.1. Hence

$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{m-1},Z_{m}\cdots
Z_{n}+N_{\tau})=0.$

Since $\widetilde{E}$ is a continuous function and left multiplication of $A$
on $L_{2}(A,\tau)$ yields bounded operators, due to the recursive nature of
$\Psi$ we see that $\Psi$ is continuous in the $L_{2}(A,\tau)$ entry.
Therefore, by Möbius inversion, $\widetilde{\kappa}$ is continuous in the
$L_{2}(A,\tau)$ entry. Hence the result follows. ∎

######  Corollary 4.18.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and let $n\geq 2$, $\chi\in\\{\ell,r\\}^{n}$, $\xi\in
L_{2}(A,\tau)$, $b\in B$, and $Z_{k}\in A_{\chi(k)}$. Suppose that either
there exists $p\in\\{1,\ldots,n-1\\}$ such that

$Z_{p}=\begin{cases}L_{b}&\text{if }\chi(p)=\ell\\\ R_{b}&\text{if
}\chi(p)=r\end{cases}$

or that $\xi\in L_{2}(B,\tau_{B})$. Then

$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{n-1},\xi)=0.$

###### Proof.

If $Z_{p}=L_{b}$ or $Z_{p}=R_{b}$ for some $p$, then we may proceed as in the
proof of Theorem 4.17 by assuming that $\xi$ is an element of $A$, expanding
out the analytic operator-valued bi-free cumulant function with the aid of
Proposition 4.16, and using the fact that non-singleton operator-valued bi-
free cumulants involving $L_{b}$ or $R_{b}$ terms are zero by
[CNS2015-1]*Proposition 6.4.1 and then taking a limit at the end.

In the case where $\xi\in L_{2}(B,\tau_{B})$, $\xi$ is a limit of terms of the
form $L_{b}+N_{\tau}$. As

$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{n-1},L_{b}+N_{\tau})=\kappa_{\chi}(Z_{1},\ldots,Z_{n-1},L_{b})+N_{\tau_{B}}=0+N_{\tau_{B}},$

by Theorem 4.13 and [CNS2015-1]*Proposition 6.4.1 the result follows by taking
a limit. ∎

##  5\. Bi-Free Conjugate Variables with respect to Completely Positive Maps

In this section we develop the appropriate notion of conjugate variables in
order to define bi-free Fisher information and entropy with respect to
completely positive maps. This can be viewed as both an extension of the bi-
free conjugate variables developed in [CS2020] and of the free conjugate
variables with respect to a completely positive map developed in [S2000]. We
will focus on both the moment and cumulant characterizations of these
conjugate variables, whereas [S2000] focused on the moment and derivation
characterizations of free conjugate variables. Although [CS2020] analyzed the
moment, cumulant, and bi-free difference quotient characterizations of bi-free
conjugate variables, we will forgo trying to generalize the bi-free difference
quotient characterization in this setting as it was the cumulant
characterization that was found most effective and as the bi-module structures
of [S2000] necessary for the derivation characterization using adjoints are
less clear in this setting.

We refer the reader to [NSS1999]*Definition 2.7 as an equivalent description
to [S2000] of the free conjugate variables with respect to a completely
positive map that we mimic below.

######  Definition 5.1.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, let $Y\in A_{r}$, and let $\eta:B\to B$ be a completely
positive map.

An element $\xi\in L_{2}(A,\tau)$ is said to satisfy the _left bi-free
conjugate variable relations for $X$ with respect to $\eta$ and $\tau$ in the
presence of $(C_{\ell},C_{r})$_ if for all $n\in\mathbb{N}\cup\\{0\\}$,
$Z_{1},\ldots,Z_{n}\in\\{X\\}\cup C_{\ell}\cup C_{r}$ we have

$\tau(Z_{1}\cdots Z_{n}\xi)=\sum_{\begin{subarray}{c}1\leq k\leq n\\\
Z_{k}=X\end{subarray}}\tau\left(\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n+1\\}}Z_{p}\right)L_{\eta\left(E\left(\prod_{p\in
V_{k}}Z_{p}\right)\right)}\right),$

where $V_{k}=\\{k<m<n+1\,\mid\,Z_{m}\in\\{X\\}\cup C_{\ell}\\}$ and where all
products are taken in numeric order (with the empty product being 1). If, in
addition,

$\xi\in\overline{\operatorname{alg}(X,C_{\ell},C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}},$

we call $\xi$ the _left bi-free conjugate variable for $X$ with respect to
$\eta$ and $\tau$ in the presence of $(C_{\ell},C_{r})$_ and denote $\xi$ by
$J_{\ell}(X:(C_{\ell},C_{r}),\eta)$.

Similarly, an element $\nu\in L_{2}(A,\tau)$ is said to satisfy the _right bi-
free conjugate variable relations for $Y$ with respect to $\eta$ and $\tau$ in
the presence of $(C_{\ell},C_{r})$_ if for all $n\in\mathbb{N}\cup\\{0\\}$,
$Z_{1},\ldots,Z_{n}\in\\{Y\\}\cup C_{\ell}\cup C_{r}$ we have

$\tau(Z_{1}\cdots Z_{n}\nu)=\sum_{\begin{subarray}{c}1\leq k\leq n\\\
Z_{k}=X\end{subarray}}\tau\left(\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n+1\\}}Z_{p}\right)R_{\eta\left(E\left(\prod_{p\in
V_{k}}Z_{p}\right)\right)}\right),$

where $V_{k}=\\{k<m<n+1\,\mid\,Z_{m}\in\\{Y\\}\cup C_{r}\\}$. If, in addition,

$\nu\in\overline{\operatorname{alg}(Y,C_{\ell},C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}},$

we call $\nu$ the _right bi-free conjugate variable for $X$ with respect to
$\eta$ and $\tau$ in the presence of $(C_{\ell},C_{r})$_ and denote $\nu$ by
$J_{r}(Y:(C_{\ell},C_{r}),\eta)$.

######  Example 5.2.

For an example of Definition 5.1, consider $X\in A_{\ell}$, $Y\in A_{r}$,
$Z_{2},Z_{3}\in C_{\ell}$, and $Z_{1},Z_{4}\in C_{r}$. If
$\xi=J_{\ell}(X:(C_{\ell},\operatorname{alg}(C_{r},Y)),\eta)$, then

$\displaystyle\tau(XZ_{1}Z_{2}YXYZ_{3}XZ_{4}\xi)$
$\displaystyle=\tau\left(Z_{1}YYZ_{4}L_{\eta(E(Z_{2}XZ_{3}X))}\right)$
$\displaystyle\qquad+\tau\left(XZ_{1}Z_{2}YYZ_{4}L_{\eta(E(Z_{3}X))}\right)$
$\displaystyle\qquad+\tau\left(XZ_{1}Z_{2}YXYZ_{3}Z_{4}L_{\eta(E(1))}\right)$

This can be observed diagrammatically by drawing
$X,Z_{1},Z_{2},Y,X,Y,Z_{3},X,Z_{4}$ as one would in a bi-non-crossing diagram
(i.e. drawing two vertical lines and placing the variables on these lines
starting at the top and going down with left variables on the left line and
right variables on the right line), drawing all pictures connecting the centre
of the bottom of the diagram to any $X$, taking the product of the elements
starting from the top and going down in each of the two isolated components of
the diagram, taking the expectation of the bounded region and applying $\eta$
to the result to obtain a $b\in B$, appending $L_{b}$ to the end of the
product of operators from the unbounded region, and applying $\tau$ to the
result.

This is analogous to applying the left bi-free difference quotient
$\partial_{\ell,X}$ defined in [CS2020] on a suitable algebraic free product
to $XZ_{1}Z_{2}YXYZ_{3}XZ_{4}$ to obtain

$Z_{1}YYZ_{4}\otimes Z_{2}XZ_{3}Z+XZ_{1}Z_{2}YYZ_{4}\otimes
Z_{3}X+XZ_{1}Z_{2}YXYZ_{3}Z_{4}\otimes 1,$

applying $\mathrm{Id}\otimes(\eta\circ E)$, collapsing the tensor, and
applying $\tau$ to the result.

Similarly, if $\nu=J_{r}(Y:(\operatorname{alg}(C_{\ell},X),C_{r}),\eta)$, then

$\displaystyle\tau(XZ_{1}Z_{2}YXYZ_{3}XZ_{4}\nu)$
$\displaystyle=\tau\left(XZ_{1}Z_{2}XZ_{3}XR_{\eta(E(YZ_{4}))}\right)+\tau\left(XZ_{1}Z_{2}YXZ_{3}XR_{\eta(E(Z_{4}))}\right).$

This can be observed diagrammatically in a similar fashion by drawing all
pictures connecting the centre of the bottom of the diagram to any $Y$ on the
right.

This is analogous to applying the right bi-free difference quotient
$\partial_{r,Y}$ defined in [CS2020] on a suitable algebraic free product to
$XZ_{1}Z_{2}YXYZ_{3}XZ_{4}$ to obtain

$XZ_{1}Z_{2}XZ_{3}X\otimes YZ_{4}+XZ_{1}Z_{2}YXZ_{3}X\otimes Z_{4},$

applying $\mathrm{Id}\otimes(\eta\circ E)$, collapsing the tensor, and
applying $\tau$ to the result.

######  Remark 5.3.

  1. (i)

As $\tau(aL_{b})=\tau(aR_{b})$ for all $a\in A$ and $b\in B$, one may use
either $L_{\eta\circ E}$ or $R_{\eta\circ E}$ in either part of Definition
5.1. In fact, one may simply use $\eta\circ E$ if one views the resulting
element of $B$ as an element of $L_{2}(B,\tau_{B})\subseteq L_{2}(A,\tau)$,
since $\tau(aL_{b})=\tau(a(b+N_{\tau}))$ for all $a\in A$ and $b\in B$ by
construction.

  2. (ii)

The element $J_{\ell}(X:(C_{\ell},C_{r}),\eta)$ is unique in the sense that if
$\xi_{0}\in\overline{\operatorname{alg}(X,C_{\ell},C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}}$
satisfies the left bi-free conjugate variable relations for $X$ with respect
to $(C_{\ell},C_{r})$, then $\xi_{0}=J_{\ell}(X:(C_{\ell},C_{r}),\eta)$ as the
left bi-free conjugate variable relations causes the inner products in
$L_{2}(A,\tau)$ of both $\xi_{0}$ and $J_{\ell}(X:(C_{\ell},C_{r}),\eta)$
against any element of $\operatorname{alg}(X,C_{\ell},C_{r})$ to be equal.

  3. (iii)

In the case where $B=\mathbb{C}$, $E$ reduces down to a unital, linear map
$\varphi:A\to\mathbb{C}$ and, as $\tau$ is compatible with $E$, one obtains
that $\tau=\varphi$. As $\varphi$ is linear, Definition 5.1 immediately
reduces down to the left and right conjugate variables with respect to
$\varphi$ in the presence of $(C_{\ell},C_{r})$ as in [CS2020], provided
$\eta$ is unital.

  4. (iv)

In the setting of Example 2.6, we note that
$J_{\ell}(X:(B_{\ell},B_{r}),\eta)$ exists if and only if the free conjugate
variable of $X$ with respect to $(B,\eta)$ from [S2000] exists. This
immediately follows as $B_{r}$ commutes with $X$ and $B_{\ell}$ and $\tau$ is
tracial, so the expressions for either conjugate variable can be modified into
the expressions of the other conjugate variable.

As with the bi-free conjugate variables in [CS2020], any moment expression
should be equivalent to certain cumulant expressions via Möbius inversion.
Thus we obtain the following equivalent characterization of conjugate
variables. Note in that which follows, it does not matter whether the last
entry in the analytical operator-valued bi-free cumulant function is treated
as a left or as a right operator by Definition 4.1.

######  Theorem 5.4.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, let $Y\in A_{r}$, and let $\eta:B\to B$ be a completely
positive map. For $\xi\in L_{2}(A,\tau)$, the following are equivalent:

  1. (i)

$\xi$ satisfies the left bi-free conjugate variables relations for $X$
(respectively $\xi$ satisfies the right bi-free conjugate variables relations
for $Y$) with respect to $\eta$ and $\tau$ in the presence of
$(C_{\ell},C_{r})$,

  2. (ii)

the following four cumulant conditions hold:

    1. (a)

$\widetilde{\kappa}_{1_{(\ell)}}(\xi)=0+N_{\tau_{B}}$,

    2. (b)

$\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{b},\xi)=\eta(b)+N_{\tau_{B}}$
(respectively
$\widetilde{\kappa}_{1_{(r,r)}}(YR_{b},\xi)=\eta(b)+N_{\tau_{B}}$) for all
$b\in B$,

    3. (c)

$\widetilde{\kappa}_{1_{(\ell,\ell)}}(c_{1},\xi)=\widetilde{\kappa}_{1_{(r,\ell)}}(c_{2},\xi)=0+N_{\tau_{B}}$
for all $c_{1}\in C_{\ell}$ and $c_{2}\in C_{r}$,

    4. (d)

for all $n\geq 3$, $\chi\in\\{\ell,r\\}^{n}$, and all
$Z_{1},Z_{2},\ldots,Z_{n-1}\in A$ such that

$Z_{k}\in\begin{cases}\\{X\\}\cup C_{\ell}&\text{if }\chi(k)=\ell\\\
C_{r}&\text{if }\chi(k)=r\\\ \end{cases}\qquad\left(\text{respectively}\quad
Z_{k}\in\begin{cases}C_{\ell}&\text{if }\chi(k)=\ell\\\ \\{Y\\}\cup
C_{r}&\text{if }\chi(k)=r\\\ \end{cases}\right)$

we have that

$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{n-1},\xi)=0+N_{\tau_{B}}.$

###### Proof.

We will prove the result for the left bi-free conjugate variable as the proof
for the right bi-free conjugate variable is analogous.

Suppose that $\xi$ satisfies (ii). To see that $\xi$ satisfies the left bi-
free conjugate variables relations, let $n\in\mathbb{N}\cup\\{0\\}$ and let
$Z_{1},\ldots,Z_{n}\in\\{X\\}\cup C_{\ell}\cup C_{r}$. Fix
$\chi\in\\{\ell,r\\}^{n+1}$ such that

$\displaystyle\chi(k)=\begin{cases}\ell&\text{if }Z_{k}\in\\{X\\}\cup
C_{\ell}\\\ r&\text{if }Z_{k}\in C_{r}\end{cases}$

(note the value of $\chi(n+1)$ does not matter in that which follows). By the
relation between the analytic extensions of the bi-moment and bi-cumulant
functions, we obtain that

$\displaystyle\widetilde{E}(Z_{1}\cdots
Z_{n}\xi)=\sum_{\pi\in\operatorname{BNC}(\chi)}\widetilde{\kappa}_{\pi}(Z_{1},\ldots,Z_{n},\xi).$

Due to the cumulant conditions in (ii), the only way
$\widetilde{\kappa}_{\pi}(Z_{1},\ldots,Z_{n},\xi)$ is non-zero is if the block
of $\pi$ containing $n+1$ contains a single other index $k$ with $Z_{k}=X$.
Moreover, there is a bijection between such partitions and partitions of the
form

$\pi=\\{k,n+1\\}\cup\pi_{1}\cup\pi_{2}$

where $\pi_{1}$ is a bi-non-crossing partition on
$V_{k}=\\{k<m<n+1\,\mid\,Z_{m}\in\\{X\\}\cup C_{\ell}\\}$ with respect to
$\chi|_{V_{k}}$ and where $\pi_{2}$ is a bi-non-crossing partition on
$W_{k}=V_{k}\setminus\\{k,n+1\\}$ with respect to $\chi|_{W_{k}}$. Using this
decomposition, the properties of bi-analytic extensions of bi-multiplicative
functions and the moment-cumulant formulas yield that

$\displaystyle\widetilde{E}(Z_{1}\cdots Z_{n}\xi)$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq k\leq n\\\
Z_{k}=X\end{subarray}}\sum_{\pi_{2}\in\operatorname{BNC}(\chi|_{W_{k}})}\sum_{\pi_{1}\in\operatorname{BNC}(\chi|_{V_{k}})}\widetilde{\kappa}_{\\{k,n+1\\}\cup\pi_{1}\cup\pi_{2}}(Z_{1},\ldots,Z_{n},\xi)$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq k\leq n\\\
Z_{k}=X\end{subarray}}\sum_{\pi_{2}\in\operatorname{BNC}(\chi|_{W_{k}})}\widetilde{\kappa}_{\\{k,n+1\\}\cup\pi_{1}}\left(\left.\left(Z_{1},\ldots,Z_{k-1},Z_{k}L_{E\left(\prod_{p\in
V_{k}}Z_{p}\right)}Z_{k+1},\ldots,Z_{n},\xi\right)\right|_{W_{k}\cup\\{k,n+1\\}}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq k\leq n\\\
Z_{k}=X\end{subarray}}\widetilde{\kappa}_{\pi_{1}}\left(\left.\left(Z_{1},\ldots,Z_{\max_{\leq}(W_{k})-1},Z_{\max_{\leq}(W_{k})}\eta\left(E\left(\prod_{p\in
V_{k}}Z_{p}\right)\right)\right)\right|_{W_{k}}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq k\leq n\\\
Z_{k}=X\end{subarray}}\tilde{E}\left(\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n+1\\}}Z_{p}\right)\eta\left(E\left(\prod_{p\in
V_{k}}Z_{p}\right)\right)\right).$

Hence, by applying $\tau_{B}$ to both sides of this equation, the left bi-free
conjugate variables relations from Definition 5.1 are obtained via part (i) of
Proposition 3.9.

For the converse direction, suppose $\xi$ satisfies the left bi-free conjugate
variables relations for $X$. Thus for all $b\in B$, $\tau(L_{b}\xi)=0$ by the
conjugate variable relations. Hence
$\widetilde{\kappa}_{1_{(\ell)}}(\xi)=\widetilde{E}(\xi)=0$ by part (vi) of
Proposition 3.9 and therefore (a) holds.

To see that (b) holds, note for all $b_{0},b\in B$ that

$\displaystyle
b_{0}\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{b},\xi)=\widetilde{\kappa}_{1_{(\ell,\ell)}}(L_{b_{0}}XL_{b},\xi)$
$\displaystyle=\Psi_{1_{(\ell,\ell)}}(L_{b_{0}}XL_{b},\xi)-\Psi_{0_{(\ell,\ell)}}(L_{b_{0}}XL_{b},\xi)$
$\displaystyle=\widetilde{E}(L_{b_{0}}XL_{b}\xi)-E(L_{b_{0}}XL_{b})\widetilde{E}(\xi)=\widetilde{E}(L_{b_{0}}XL_{b}\xi).$

Therefore, by applying $\tau_{B}$ to both sides, we obtain that

$\tau_{B}\left(b_{0}\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{b},\xi)\right)=\tau_{B}\left(\widetilde{E}(L_{b_{0}}XL_{b}\xi)\right)=\tau(L_{b_{0}}XL_{b}\xi).$

By the left bi-free conjugate variable relations we obtain that

$\displaystyle\tau_{B}\left(b_{0}\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{b},\xi)\right)=\tau\left(L_{b_{0}}L_{\eta(E(L_{b}))}\right)=\tau(L_{b_{0}}L_{\eta(b)})=\tau(L_{b_{0}\eta(b)})=\tau_{B}(b_{0}\eta(b)).$

As this holds for all $b_{0}\in B$, we obtain that
$\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{b},\xi)=\eta(b)+N_{\tau_{B}}$ as
desired.

To see that (c) holds, note for all $b\in B$ and $c_{1}\in C_{\ell}$ that

$\tau_{B}\left(b\widetilde{\kappa}_{1_{(\ell,\ell)}}(c_{1},\xi)\right)=\tau(L_{b}c_{1}\xi)=0$

by similar computations as above. Since this holds for all $b\in B$, we see
that $\widetilde{\kappa}_{1_{(\ell,\ell)}}(c_{1},\xi)=0+N_{\tau_{B}}$.
Similarly, for all $c_{2}\in C_{r}$ we see that

$\displaystyle\widetilde{\kappa}_{1_{(r,\ell)}}(c_{2},\xi)b=\widetilde{\kappa}_{1_{(r,\ell)}}(R_{b}c_{2},\xi)$
$\displaystyle=\Psi_{1_{(r,\ell)}}(R_{b}c_{2},\xi)-\Psi_{0_{(r,\ell)}}(R_{b}c_{2},\xi)$
$\displaystyle=\widetilde{E}(R_{b}c_{2}\xi)-\widetilde{E}(\xi)E(R_{b}c_{2})=\widetilde{E}(R_{b}c_{2}\xi).$

Therefore, by applying $\tau_{B}$ to both sides, we obtain that

$\tau_{B}\left(\widetilde{\kappa}_{1_{(r,\ell)}}(c_{2},\xi)b\right)=\tau_{B}\left(\widetilde{E}(R_{b}c_{2}\xi)\right)=\tau(R_{b}c_{2}\xi).$

By the left bi-free conjugate variable relations we obtain that

$\displaystyle\tau_{B}\left(\widetilde{\kappa}_{1_{(r,\ell)}}(c_{2},\xi)b\right)=0.$

Therefore, as $\tau_{B}$ is tracial and the above holds for all $b\in B$, we
obtain that $\widetilde{\kappa}_{1_{(r,\ell)}}(c_{2},\xi)=0+N_{\tau_{B}}$ as
desired.

For (d), we proceed by induction on $n$. To do so, we will prove the base case
$n=3$ and the inductive step simultaneously. Fix $n\geq 3$ and suppose when
$n>3$ that (d) holds for all $m<n$. Let $\chi\in\\{\ell,r\\}^{n}$ and let
$Z_{1},Z_{2},\ldots,Z_{n-1}\in A$ be as in the assumptions of (d). We will
assume that $\chi(1)=r$ as the case $\chi(1)=\ell$ will be handled similarly.
Thus for all $b\in B$ we know that

$\displaystyle\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{n-1},\xi)b$
$\displaystyle=\widetilde{\kappa}_{\chi}(R_{b}Z_{1},Z_{2},\ldots,Z_{n-1},\xi)$
$\displaystyle=\widetilde{E}(R_{b}Z_{1}Z_{2}\cdots
Z_{n-1}\xi)-\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\ \pi\neq
1_{\chi}\\\
\end{subarray}}\widetilde{\kappa}_{\pi}(R_{b}Z_{1},Z_{2},\ldots,Z_{n-1},\xi)$
$\displaystyle=\widetilde{E}(R_{b}Z_{1}Z_{2}\cdots
Z_{n-1}\xi)-\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\ \pi\neq
1_{\chi}\\\
\end{subarray}}\widetilde{\kappa}_{\pi}(Z_{1},Z_{2},\ldots,Z_{n-1},\xi)b.$

Using the fact that (a), (b), (c) hold and that (d) holds for all $m<n$, we
obtain using the same arguments used in the other direction of the proof that

$\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\ \pi\neq
1_{\chi}\\\
\end{subarray}}\widetilde{\kappa}_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)b=\sum_{\begin{subarray}{c}1\leq
k<n\\\ Z_{k}=X\end{subarray}}\tilde{E}\left(R_{b}\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}\right)\eta\left(E\left(\prod_{p\in
V_{k}}Z_{p}\right)\right)\right),$

where $V_{k}=\\{k<m<n\,\mid\,Z_{m}\in\\{X\\}\cup C_{\ell}\\}$. Therefore, by
applying $\tau_{B}$ to both sides of our initial equation, we obtain that

$\displaystyle\tau_{B}\left(\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{n-1},\xi)b\right)$
$\displaystyle=\tau_{B}\left(\widetilde{E}(R_{b}Z_{1}Z_{2}\cdots
Z_{n-1}\xi)-\sum_{\begin{subarray}{c}1\leq k<n\\\
Z_{k}=X\end{subarray}}\tilde{E}\left(R_{b}\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}\right)\eta\left(E\left(\prod_{p\in
V_{k}}Z_{p}\right)\right)\right)\right)$
$\displaystyle=\tau(R_{b}Z_{1}Z_{2}\cdots
Z_{n-1}\xi)-\sum_{\begin{subarray}{c}1\leq k<n\\\
Z_{k}=X\end{subarray}}\tau\left(R_{b}\left(\prod_{p\in
V_{k}^{c}\setminus\\{k,n\\}}Z_{p}\right)\eta\left(E\left(\prod_{p\in
V_{k}}Z_{p}\right)\right)\right)=0,$

by the left bi-free conjugate variable relations. Therefore, as the above
holds for all $b\in B$ and $\tau_{B}$ is tracial, the result follows. ∎

The cumulant approach to conjugate variables has merits as it is very simple
to check that most cumulants vanish and the values of others. For instance, an
observant reader might have noticed that the operators $X$ and $Y$ in
Definition 5.1 were not required to be self-adjoint. This is for later use in
the paper and can be converted to studying self-adjoint operators as follows.

######  Lemma 5.5.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, and let $\eta:B\to B$ be a completely positive map. The
left bi-free conjugate variables

$J_{\ell}(X:(\operatorname{alg}(C_{\ell},X^{*}),C_{r}),\eta)\qquad\text{and}\qquad
J_{\ell}(X^{*}:(\operatorname{alg}(C_{\ell},X),C_{r}),\eta)$

exist if and only if

$J_{\ell}(\Re(X):(\operatorname{alg}(C_{\ell},\Im(X)),C_{r}),\eta)\qquad\text{and}\qquad
J_{\ell}(\Im(X):(\operatorname{alg}(C_{\ell},\Re(X)),C_{r}),\eta)$

exist where $\Re(X)=\frac{1}{2}(X+X^{*})$ and $\Im(X)=\frac{1}{2i}(X-X^{*})$.
Furthermore,

$\displaystyle
J_{\ell}(\Re(X):(\operatorname{alg}(C_{\ell},\Im(X)),C_{r}),\eta)$
$\displaystyle=J_{\ell}(X:(\operatorname{alg}(C_{\ell},X^{*}),C_{r}),\eta)+J_{\ell}(X^{*}:(\operatorname{alg}(C_{\ell},X),C_{r}),\eta)$
$\displaystyle
J_{\ell}(\Im(X):(\operatorname{alg}(C_{\ell},\Re(X)),C_{r}),\eta)$
$\displaystyle=iJ_{\ell}(X:(\operatorname{alg}(C_{\ell},X^{*}),C_{r}),\eta)-iJ_{\ell}(X^{*}:(\operatorname{alg}(C_{\ell},X),C_{r}),\eta).$

A similar result holds for right bi-free conjugate variables.

###### Proof.

Suppose

$\xi_{1}=J_{\ell}(X:(\operatorname{alg}(C_{\ell},X^{*}),C_{r}),\eta)\qquad\text{and}\qquad\xi_{2}=J_{\ell}(X^{*}:(\operatorname{alg}(C_{\ell},X),C_{r}),\eta)$

exist. Hence $\xi_{1}$ and $\xi_{2}$ satisfy the appropriate analytic cumulant
equations from Theorem 5.4. Let

$h_{1}=\xi_{1}+\xi_{2}\qquad\text{and}\qquad h_{2}=i\xi_{1}-i\xi_{2}.$

As

$\xi_{1}\in\overline{\operatorname{alg}(X,\operatorname{alg}(C_{\ell},X^{*}),C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}}\qquad\text{and}\qquad\xi_{2}\in\overline{\operatorname{alg}(X^{*},\operatorname{alg}(C_{\ell},X),C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}},$

we easily see that

$h_{1}\in\overline{\operatorname{alg}(\Re(X),\operatorname{alg}(C_{\ell},\Im(X)),C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}}\qquad\text{and}\qquad\xi_{2}\in\overline{\operatorname{alg}(\Im(X),\operatorname{alg}(C_{\ell},\Re(X)),C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}}.$

Thus, by Theorem 5.4, it suffices to show that $h_{1}$ and $h_{2}$ satisfy the
appropriate conjugate variable formulae. Indeed, property (a) of Theorem 5.4
holds as

$\widetilde{\kappa}_{1_{(\ell)}}(h_{1})=\widetilde{\kappa}_{1_{(\ell)}}(h_{2})=0+N_{\tau_{B}}.$

Next, notice for all $b\in B$ that

$\displaystyle\widetilde{\kappa}_{1_{(\ell,\ell)}}(\Re(X)L_{b},h_{1})$
$\displaystyle=\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}XL_{b},\xi_{1}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}X^{*}L_{b},\xi_{1}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}XL_{b},\xi_{2}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}X^{*}L_{b},\xi_{2}\right)$
$\displaystyle=\frac{1}{2}\eta(b)+0+0+\frac{1}{2}\eta(b)=\eta(b),$

and

$\displaystyle\widetilde{\kappa}_{1_{(\ell,\ell)}}(\Im(X)L_{b},h_{2})$
$\displaystyle=\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2i}XL_{b},i\xi_{1}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(-\frac{1}{2i}X^{*}L_{b},i\xi_{1}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2i}XL_{b},-i\xi_{2}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(-\frac{1}{2i}X^{*}L_{b},-i\xi_{2}\right)$
$\displaystyle=\frac{1}{2i}i\eta(b)-0+0-\frac{1}{2i}(-i)\eta(b)=\eta(b).$

Hence property (b) of Theorem 5.4 holds.

To see properties (c) and (d) of Theorem 5.4 hold, note for all $b\in B$ that

$\displaystyle\widetilde{\kappa}_{1_{(\ell,\ell)}}(\Re(X)L_{b},h_{2})$
$\displaystyle=\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}XL_{b},i\xi_{1}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}X^{*}L_{b},i\xi_{1}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}XL_{b},-i\xi_{2}\right)+\widetilde{\kappa}_{1_{(\ell,\ell)}}\left(\frac{1}{2}X^{*}L_{b},-i\xi_{2}\right)$
$\displaystyle=\frac{1}{2}i\eta(b)+0+0+\frac{1}{2}(-i)\eta(b)=0,$

and similarly $\widetilde{\kappa}_{1_{(\ell,\ell)}}(\Im(X)L_{b},h_{1})=0$.
Therefore, Proposition 4.16 along with the linearity of the cumulants in each
entry yield properties (c) and (d).

The converse direction is proved analogously. ∎

Of course, many other results follow immediately from the cumulant definition
of the conjugate variables.

######  Lemma 5.6.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, and let $\eta:B\to B$ be a completely positive map. If

$\xi=J_{\ell}(X:(C_{\ell},C_{r}),\eta)$

exists, then for all $\lambda\in\mathbb{C}\setminus\\{0\\}$ the conjugate
variable $J_{\ell}(\lambda X:(C_{\ell},C_{r}),\eta)$ exists and is equal to
$\frac{1}{\lambda}\xi$.

A similar result holds for right bi-free conjugate variables.

######  Proposition 5.7.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, and let $\eta_{1},\eta_{2}:B\to B$ be completely positive
maps. If

$\xi_{1}=J_{\ell}(X:(C_{\ell},C_{r}),\eta_{1})\qquad\text{and}\qquad\xi_{2}=J_{\ell}(X:(C_{\ell},C_{r}),\eta_{2})$

exist, then $\xi=J_{\ell}(X:(C_{\ell},C_{r}),\eta_{1}+\eta_{2})$ exists and
$\xi=\xi_{1}+\xi_{2}$.

A similar result holds for right bi-free conjugate variables.

######  Proposition 5.8.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, and let $\eta:B\to B$ be a completely positive map. For
fixed $b_{1},b_{2}\in B$, define $\eta_{\ell,b_{1},b_{2}}:B\to B$ by

$\eta_{\ell,b_{1},b_{2}}(b)=\eta(bb_{2})b_{1}$

for all $b\in B$. If $\xi=J_{\ell}(X:(C_{\ell},C_{r}),\eta)$ exists and
$\eta_{\ell,b_{1},b_{2}}$ is completely positive, then
$J_{\ell}(X:(C_{\ell},C_{r}),\eta_{\ell,b_{1},b_{2}})$ exists and

$J_{\ell}(X:(C_{\ell},C_{r}),\eta_{\ell,b_{1},b_{2}})=R_{b_{1}}L_{b_{2}}\xi.$

Similarly, if $Y\in A_{r}$ and $\eta_{r,b_{1},b_{2}}:B\to B$ is defined by

$\eta_{r,b_{1},b_{2}}=b_{2}\eta(b_{1}b)$

for all $b\in B$ is completely positive, and $J_{r}(Y:(C_{\ell},C_{r}),\eta)$
exists, then $J_{r}(Y:(C_{\ell},C_{r}),\eta_{r,b_{1},b_{2}})$ exists and

$J_{r}(Y:(C_{\ell},C_{r}),\eta_{r,b_{1},b_{2}})=R_{b_{1}}L_{b_{2}}J_{r}(Y:(C_{\ell},C_{r}),\eta).$

###### Proof.

By Theorem 5.4, it suffices to show that $R_{b_{1}}L_{b_{2}}\xi$ satisfies the
appropriate analytical operator-valued bi-free cumulant formula. Indeed,
clearly

$\widetilde{\kappa}_{1_{(\ell)}}(R_{b_{1}}L_{b_{2}}\xi)=b_{2}\widetilde{\kappa}_{1_{(\ell)}}(\xi)b_{1}=0$

and for all $b\in B$

$\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{b},R_{b_{1}}L_{b_{2}}\xi)=\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{b}L_{b_{2}},\xi)b_{1}=\widetilde{\kappa}_{1_{(\ell,\ell)}}(XL_{bb_{2}},\xi)b_{1}=\eta(bb_{2})b_{1}=\eta_{b_{1},b_{2}}(b).$

To show that the other analytical operator-valued bi-free cumulants from
Theorem 5.4 vanish, one simply needs to use the analytical extension
properties of bi-multiplicative functions together with Proposition 4.16. The
result for right bi-free conjugate variables is analogous. ∎

Similarly, many results pertaining to conjugate variables from [V1998-2,
S2000, CS2020] immediately generalize to the conjugate variables in Definition
5.1. However, one result from [S2000] requires additional set-up. In the
context of Example 2.6, one can always consider a further von Neumann
subalgebra $D$ of $B$ and ask how the conjugate variables react. To analyze
the comparable situation in our setting, we need the following example.

######  Example 5.9.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and let $D$ be a unital $*$-subalgebra of $B$ (with
$1_{D}=1_{B}$). If $F:B\to D$ is a conditional expectation in the sense that
$F(d)=d$ for all $d\in D$ and $F(d_{1}bd_{2})=d_{1}F(b)d_{2}$ for all
$d_{1},d_{2}\in D$ and $b\in B$, then $(A,F\circ E,\varepsilon|_{D\otimes
D^{{\text{op}}}})$ is a $D$-$D$-non-commutative probability space by
[S2016-2]*Section 3.

Note $\tau_{D}=\tau_{B}|_{D}:D\to\mathbb{C}$ is a tracial state being the
restriction of a tracial state. Moreover, if $\tau_{B}$ is compatible with $F$
in the sense that $\tau_{B}(F(b))=\tau_{B}(b)$ for all $b\in B$, we easily see
that $\tau$ is compatible with $F\circ E$, as for all $a\in A$ we have that

$\tau(a)=\tau\left(L_{E(a)}\right)=\tau_{B}(E(a))=\tau_{B}(F(E(a)))=\tau\left(L_{F(E(a))}\right)$

and similarly $\tau(a)=\tau\left(R_{F(E(a))}\right)$. Hence $(A,F\circ
E,\varepsilon|_{D\otimes D^{{\text{op}}}},\tau)$ is an analytical $D$-$D$-non-
commutative probability space.

######  Proposition 5.10.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and $X\in A_{\ell}$. In addition, let $D$ and $F$ be as in
Example 5.9 and let $(C_{\ell},C_{r})$ be a pair of $B$-algebras (and thus
automatically a pair of $D$-algebras) and $\eta:D\to D$ be a completely
positive map. Moreover, suppose that $F$ is completely positive (and hence
$\eta\circ F:B\to D\subseteq B$ is also completely positive).

Then, the conjugate variable $J_{\ell}(X:(C_{\ell},C_{r}),\eta\circ F)$ exists
in the analytical $B$-$B$-non-commutative probability space
$(A,E,\varepsilon,\tau)$ if and only if the conjugate variable
$J_{\ell}(X:(C_{\ell},C_{r}),\eta)$ exists in the analytical $D$-$D$-non-
commutative probability space $(A,F\circ E,\varepsilon|_{D\otimes
D^{{\text{op}}}},\tau)$, in which case they are the same element of
$L_{2}(A,\tau)$.

A similar result holds for right bi-free conjugate variables.

###### Proof.

As $(\eta\circ F)\circ E=\eta\circ(F\circ E)$, the bi-free conjugate variable
relations from Definition 5.1 are precisely the same and thus there is nothing
to prove. ∎

With Proposition 5.10 out of the way, we turn our attention to proving that
the expected generalizations of conjugate variable properties from [V1998-2,
S2000, CS2020] hold.

######  Lemma 5.11.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, and let $\eta:B\to B$ be a completely positive map.
Suppose further that $D_{\ell}\subseteq C_{\ell}$ and $D_{r}\subseteq C_{r}$
are such that $(D_{\ell},D_{r})$ is a pair of $B$-algebras in $A$. If

$\xi=J_{\ell}(X:(C_{\ell},C_{r}),\eta)$

exists then

$\xi^{\prime}=J_{\ell}(X:(D_{\ell},D_{r}),\eta)$

exists. In particular, if $P$ is the orthogonal projection of $L_{2}(A,\tau)$
onto

$\overline{\operatorname{alg}(X,D_{\ell},D_{r})}^{\left\|\,\cdot\,\right\|_{\tau}},$

then $\xi^{\prime}=P(\xi)$.

A similar result holds for right bi-free conjugate variables.

###### Proof.

Notice if $\xi$ satisfies the left bi-free conjugate variable relations for
$X$ with respect to $E$ and $\eta$ in the presence of $(C_{\ell},C_{r})$,
$\xi$ satisfies the left bi-free conjugate variable relations for $X$ with
respect to $E$ and $\eta$ in the presence of $(D_{\ell},D_{r})$. Therefore,
since $\tau(ZP(\xi))=\tau(Z\xi)$ for all
$Z\in\operatorname{alg}(X,D_{\ell},D_{r})$, it follows that
$P(\xi)=J_{\ell}(X:(D_{\ell},D_{r}),\eta)$ as desired. ∎

The following generalizes [V1998-2]*Proposition 3.6, [S2000]*Proposition 3.8,
and [CS2020]*Proposition 4.3.

######  Proposition 5.12.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $(C_{\ell},C_{r})$ be a pair of $B$-algebras in $A$,
let $X\in A_{\ell}$, and let $\eta:B\to B$ be a completely positive map. If
$(D_{\ell},D_{r})$ is another pair of $B$-algebras such that

$(\operatorname{alg}(X,C_{\ell}),C_{r})\qquad\text{and}\qquad(D_{\ell},D_{r})$

are bi-free with amalgamation over $B$ with respect to $E$, then

$\xi=J_{\ell}(X:(C_{\ell},C_{r}),\eta)$

exists if and only if

$\xi^{\prime}=J_{\ell}(X:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),\eta)$

exists, in which case they are equal.

A similar result holds for right bi-free conjugate variables.

###### Proof.

Note by Lemma 5.11 that if $\xi^{\prime}$ exists then $\xi$ exists.

Conversely suppose that $\xi$ exists. Hence $\xi$ is an
$\left\|\,\cdot\,\right\|_{\tau}$-limit of elements from
$\operatorname{alg}(X,C_{\ell},C_{r})$. Since the analytical operator-valued
bi-free cumulants are $\left\|\,\cdot\,\right\|_{\tau}$-continuous in the last
entry, it follows that any analytical operator-valued bi-free cumulant
involving $\xi$ at the end and at least one element of $D_{\ell}$ or $D_{r}$
must be zero by Theorem 4.17 as

$(\operatorname{alg}(X,C_{\ell}),C_{r})\qquad\text{and}\qquad(D_{\ell},D_{r})$

are bi-free with amalgamation over $B$ with respect to $E$. Therefore, as

$\xi\in\overline{\operatorname{alg}(X,C_{\ell},C_{r})}^{\left\|\,\cdot\,\right\|_{\tau}}\subseteq\overline{\operatorname{alg}(X,\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r}))}^{\left\|\,\cdot\,\right\|_{\tau}},$

it follows that $\xi^{\prime}$ exists and $\xi=\xi^{\prime}$. ∎

The following generalizes [V1998-2]*Proposition 3.7, [S2000]*Proposition 3.11,
and [CS2020]*Proposition 4.4. In that which follows, we will use $\mathbf{Z}$
to denote a tuple of operators $(Z_{1},\ldots,Z_{k})$. Furthermore, given
another tuple $\mathbf{Z}^{\prime}=(Z^{\prime}_{1},\ldots,Z^{\prime}_{k})$, we
will use $\mathbf{Z}+\mathbf{Z}^{\prime}$ to denote the tuple
$(Z_{1}+Z^{\prime}_{1},\ldots,Z_{k}+Z^{\prime}_{k})$ and we will use
$\widehat{\mathbf{Z}_{p}}$ to denote the tuple
$(Z_{1},\ldots,Z_{p-1},Z_{p+1},\ldots,Z_{k})$ obtained by removing $Z_{p}$
from the list.

######  Proposition 5.13.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and let $\eta:B\to B$ be a completely positive map. Suppose
$\mathbf{X}$ and $\mathbf{X}^{\prime}$ are $n$-tuples of operators from
$A_{\ell}$, $\mathbf{Y}$ and $\mathbf{Y}^{\prime}$ are $m$-tuples of operators
from $A_{r}$, and $(C_{\ell},C_{r})$ and $(D_{\ell},D_{r})$ are pairs of
$B$-algebras such that

$(\operatorname{alg}(\mathbf{X},C_{\ell}),\operatorname{alg}(\mathbf{Y},C_{r}))\qquad\text{and}\qquad(\operatorname{alg}(\mathbf{X}^{\prime},D_{\ell}),\operatorname{alg}(\mathbf{Y}^{\prime},D_{r}))$

are bi-free with amalgamation over $B$ with respect to $E$. If

$\xi=J_{\ell}\left(X_{1}:\left(\operatorname{alg}\left(\widehat{\mathbf{X}_{1}},C_{\ell}\right),\operatorname{alg}(\mathbf{Y},C_{r})\right),\eta,\tau\right)$

exists then

$\xi^{\prime}=J_{\ell}\left(X_{1}+X^{\prime}_{1}:\left(\operatorname{alg}\left(\widehat{(\mathbf{X}+\mathbf{X}^{\prime})_{1}},C_{\ell},D_{\ell}\right),\operatorname{alg}(\mathbf{Y}+\mathbf{Y}^{\prime},C_{r},D_{r})\right)\right)$

exists. Moreover, if $P$ is the orthogonal projection of
$L_{2}(\mathcal{A},\varphi)$ onto

$\overline{\operatorname{alg}(\mathbf{X}+\mathbf{X}^{\prime},\mathbf{Y}+\mathbf{Y}^{\prime},C_{\ell},C_{r},D_{\ell},D_{r})}^{\left\|\,\cdot\,\right\|_{\tau}},$

then

$\xi^{\prime}=P(\xi).$

A similar result holds for right bi-free conjugate variables.

###### Proof.

Suppose $\xi$ exists. For notation purposes, let
$\mathcal{A}=\operatorname{alg}(\mathbf{X}+\mathbf{X}^{\prime},\mathbf{Y}+\mathbf{Y}^{\prime},C_{\ell},C_{r},D_{\ell},D_{r})$.

Since $\tau(L_{b}ZP(\xi))=\tau(L_{b}Z\xi)$ for all $Z\in\mathcal{A}$ and $b\in
B$ (as $B_{\ell}\subseteq C_{\ell}$), we obtain by Proposition 3.9 that
$\widetilde{E}(ZP(\xi))=\widetilde{E}(Z\xi)$ for all $Z\in\mathcal{A}$. Thus,
as $B_{\ell},B_{r}\subseteq\mathcal{A}$, we obtain for all
$\chi\in\\{\ell,r\\}^{p}$ with $\chi(p)=\ell$, for all
$\pi\in\operatorname{BNC}(\chi)$, and for all $Z_{k}\in\mathcal{A}$ with

$Z_{k}\in\begin{cases}\operatorname{alg}(\mathbf{X}+\mathbf{X}^{\prime},C_{\ell},D_{\ell})&\text{if
}\chi(k)=\ell\\\
\operatorname{alg}(\mathbf{Y}+\mathbf{Y}^{\prime},C_{r},D_{r})&\text{if
}\chi(k)=r\end{cases}$

that
$\Psi_{\pi}(Z_{1},\ldots,Z_{p-1}P(\xi))=\Psi_{\pi}(Z_{1},\ldots,Z_{p-1},\xi)$
and thus

$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{p-1},P(\xi))=\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{p-1},\xi).$

To show that $P(\xi)$ is the appropriate left bi-free conjugate variable, it
suffices to consider expressions of the form
$\widetilde{\kappa}_{\chi}(Z_{1},\ldots,Z_{p-1},P(\xi))$ and show they obtain
the correct values as dictated in Theorem 5.4. By the above, said cumulant is
equal to an analytic operator-valued bi-free cumulant involving elements from
$C_{\ell}$, $D_{\ell}$, $C_{r}$, $D_{r}$, $\mathbf{X}+\mathbf{X}^{\prime}$,
and $\mathbf{Y}+\mathbf{Y}^{\prime}$ (in the appropriate positions) and a
$\xi$ at the end. By expanding using linearity, said cumulant can be modified
to a sum of cumulants involving only elements from $C_{\ell}$, $D_{\ell}$,
$C_{r}$, $D_{r}$, $\mathbf{X}$, $\mathbf{X}^{\prime}$, $\mathbf{Y}$, and
$\mathbf{Y}^{\prime}$ with a $\xi$ at the end. By a similar argument to that
in Lemma 5.5, these cumulants then obtain the necessary values for $P(\xi)$ to
be the appropriate left bi-free conjugate variable due to Theorem 5.4 applied
to $\xi$ and the fact that

$(\operatorname{alg}(\mathbf{X},C_{\ell}),\operatorname{alg}(\mathbf{Y},C_{r}))\qquad\text{and}\qquad(\operatorname{alg}(\mathbf{X}^{\prime},D_{\ell}),\operatorname{alg}(\mathbf{Y}^{\prime},D_{r}))$

are bi-free with amalgamation over $B$ with respect to $E$, so mixed cumulants
vanish by Theorem 4.17. ∎

##  6\. Bi-Semicircular Operators with Completely Positive Covariance

One essential example of conjugate variables in [V1998-2, S1998, CS2020] comes
from central limit distributions. Thus this section is devoted to defining the
operator-valued bi-semicircular operators with covariance coming from a
completely positive map, showing that one may add in certain bi-semicircular
operators into analytical $B$-$B$-non-commutative probability spaces, and
showing the bi-free conjugate variables behave in the appropriate manner.

To begin, let $B$ be a unital ∗-algebra and let $K$ be a finite index set. For
each $k\in K$ let $Z_{k}$ be a symbol. Recall the _full Fock space_
$\mathcal{F}(B,K)$ is the algebraic free product of $B$ and $\\{Z_{k}\\}_{k\in
K}$; that is

$\mathcal{F}(B,K)=B\oplus H_{1}\oplus H_{2}\oplus\cdots$

where

$H_{m}=\\{b_{0}Z_{k_{1}}b_{1}\cdots
Z_{k_{m}}b_{m}\,\mid\,b_{0},b_{1},\ldots,b_{m}\in B,k_{1},\ldots,k_{m}\in
K\\}.$

Note $\mathcal{F}(B,K)$ is a $B$-$B$-bimodule with the obvious left and right
actions of $B$ on $B$ and $H_{m}$. Moreover, as $\mathcal{F}(B,K)$ is a direct
sum of $B$ and another $B$-$B$-bimodule, $\mathcal{F}(B,K)$ is a
$B$-$B$-bimodule with the specified vector state $p:\mathcal{F}(B,K)\to B$ (as
in the sense of [CNS2015-1]*Definition 3.1.1) defined by taking the $B$-term
in the above direct product. Therefore, the set $A$ of linear maps on
$\mathcal{F}(B,K)$ is a $B$-$B$-non-commutative probability space with respect
to the expectation $E:A\to B$ defined by $E(T)=p(T1_{B})$ (see
[CNS2015-1]*Remark 3.2.2).

Let $\\{\eta_{i,j}\\}_{i,j\in K}$ be linear maps on $B$. For each $k\in K$,
the _left creation and annihilation operators_ $l_{k}$ and $l^{*}_{k}$ are the
linear maps defined such that

$\displaystyle l_{k}b$ $\displaystyle=1_{B}Z_{k}b$ $\displaystyle
l_{k}(b_{0}Z_{k_{1}}b_{1}\cdots Z_{k_{m}}b_{m})$
$\displaystyle=1_{B}Z_{k}b_{0}Z_{k_{1}}b_{1}\cdots Z_{k_{m}}b_{m}$
$\displaystyle l^{*}_{k}b$ $\displaystyle=0$ $\displaystyle
l^{*}_{k}(b_{0}Z_{k_{1}}b_{1}\cdots Z_{k_{m}}b_{m})$
$\displaystyle=\eta_{k,k_{1}}(b_{0})b_{1}\cdots Z_{k_{m}}b_{m}$

and the _right creation and annihilation operators_ $r_{k}$ and $r_{k}^{*}$
are the linear maps defined such that

$\displaystyle r_{k}b$ $\displaystyle=bZ_{k}1_{B}$ $\displaystyle
r_{k}(b_{0}Z_{k_{1}}b_{1}\cdots Z_{k_{m}}b_{m})$
$\displaystyle=b_{0}Z_{k_{1}}b_{1}\cdots Z_{k_{m}}b_{m}Z_{k}1_{B}$
$\displaystyle r^{*}_{k}b$ $\displaystyle=0$ $\displaystyle
r^{*}_{k}(b_{0}Z_{k_{1}}b_{1}\cdots Z_{k_{m}}b_{m})$
$\displaystyle=b_{0}Z_{k_{1}}b_{1}\cdots b_{m-1}\eta_{k_{m},k}(b_{m})$

It is elementary to see that $l_{k},l_{k}^{*}\in A_{\ell}$ and
$r_{k},r_{k}^{*}\in A_{r}$. With these operators in hand, we make the
following definition.

######  Definition 6.1.

Using the above notation, write $K$ as the disjoint union of two sets $I$ and
$J$. For each $i\in I$ and $j\in J$, let

$S_{i}=l_{i}+l^{*}_{i}\qquad\text{and}\qquad D_{j}=r_{j}+r^{*}_{j}.$

The pair $(\\{S_{i}\\}_{i\in I},\\{D_{j}\\}_{j\in J})$ are called the
_operator-valued bi-semicircular operators with covariance
$\\{\eta_{i,j}\\}_{i,j\in K}$_.

In the case that $\eta_{k_{1},k_{2}}=0$ for all $k_{1},k_{2}\in K$ with
$k_{1}\neq k_{2}$, we say that $(\\{S_{i}\\}_{i\in I},\\{D_{j}\\}_{j\in J})$
is a collection of _$(\\{\eta_{i,i}\\}_{i\in I},\\{\eta_{j,j}\\}_{j\in J})$
bi-semicircular operators_.

######  Remark 6.2.

It is natural to ask what are the necessary conditions for operator-valued bi-
semicircular operators with covariance $\\{\eta_{i,j}\\}_{i,j\in K}$ to sit
inside an analytical $B$-$B$-non-commutative probability space. One may hope
that a condition similar to [S1998]*Theorem 4.3.1 would work; that is, the
answer is yes if $\tau_{B}$ is tracial and $\eta:M_{|K|}(B)\to M_{|K|}(B)$
defined by

$\eta([b_{i,j}]_{i,j\in K})=[\eta_{i,j}(b_{i,j})]_{i,j\in K}$

is completely positive. However, if $i\in I$, $j\in J$, and $b_{1},b_{2}\in
B$, it is not difficult to verify that

$\tau_{B}\left(E((S_{i}D_{j}L_{b_{1}}R_{b_{2}})^{*}(S_{i}D_{j}L_{b_{1}}R_{b_{2}})\right)=\tau_{B}\left(b_{2}^{*}\eta_{i,i}(1_{B})b_{1}b_{2}\eta_{j,j}(1_{B})b_{1}^{*}+b_{2}^{*}\eta_{i,j}(\eta_{i,j}(b_{1}b_{2}))b_{1}^{*}\right)$

and it is not clear if this is positive (even if the outer $\eta_{i,j}$ was a
$\eta_{j,i}$).

Only certain operator-valued bi-semicircular operators are required in this
paper. Indeed, we will need only the case where $(\\{S_{i}\\}_{i\in
I},\\{D_{j}\\}_{j\in J})$ are $(\\{\eta_{i,i}\\}_{i\in
I},\\{\eta_{j,j}\\}_{j\in J})$ bi-semicircular operators, as in this setting
the pairs of algebras

$\\{(\operatorname{alg}(B_{\ell},S_{i}),B_{r})\\}_{i\in
I}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},D_{j}))\\}_{j\in J}$

are bi-free with amalgamation over $B$ with respect to $E$, as the following
result shows.

######  Theorem 6.3.

Using the above notation, if $B$ is a $*$-algebra, $\tau_{B}$ is a tracial
state on $B$, $\\{\eta_{i}\\}_{i\in I}\cup\\{\eta_{j}\\}_{j\in J}$ are
completely positive maps from $B$ to $B$, $(\\{S_{i}\\}_{i\in
I},\\{D_{j}\\}_{j\in J})$ are $(\\{\eta_{i}\\}_{i\in I},\\{\eta_{j}\\}_{j\in
J})$ bi-semicircular operators,
$A_{\ell}=\operatorname{alg}(B_{\ell},\\{S_{i}\\}_{i\in I})$,
$A_{r}=\operatorname{alg}(B_{r},\\{D_{j}\\}_{j\in J})$, $A$ is generated as a
$*$-algebra by $A_{\ell}$ and $A_{r}$ and $\tau:A\to\mathbb{C}$ is defined by
$\tau=\tau_{B}\circ E$, then $(A,E,\varepsilon,\tau)$ is an analytical
$B$-$B$-non-commutative probability space. Moreover, all operator-valued bi-
free cumulants involving $\\{S_{i}\\}_{i\in I}$ and $\\{D_{j}\\}_{j\in J}$ of
order not two are zero and for all $i,i_{1},i_{2}\in I$ and $j,j_{1},j_{2}\in
J$,

$\displaystyle\kappa_{1_{(\ell,\ell)}}(S_{i_{1}}L_{b},S_{i_{2}})$
$\displaystyle=\delta_{i_{1},i_{2}}\eta_{i_{1}}(b)$
$\displaystyle\kappa_{1_{(r,r)}}(D_{j_{1}}R_{b},D_{j_{2}})$
$\displaystyle=\delta_{j_{1},j_{2}}\eta_{j_{1}}(b)$
$\displaystyle\kappa_{1_{(\ell,r)}}(S_{i}L_{b},D_{j})$ $\displaystyle=0$
$\displaystyle\kappa_{1_{(r,\ell)}}(D_{j}R_{b},S_{i})$ $\displaystyle=0.$

###### Proof.

As shown above, $(A,E,\varepsilon)$ is a $B$-$B$-non-commutative probability
space. Next, note that $E$ is completely positive when restricted $A_{\ell}$
and $A_{r}$ by [S1998]*Remark 4.3.2 as the expectations reduce to the free
case. In fact, this same idea can be used to show that $\tau$ is positive.
Indeed, first note that $S_{i}$ and $D_{j}$ commute. Thus every element of $A$
can be written as sum of elements of the form

$Z=L_{b_{0}}S_{i_{1}}L_{b_{1}}\cdots
S_{i_{n}}L_{b_{n}}R_{b_{n+1}}D_{i_{n+1}}R_{b_{n+2}}\cdots
D_{j_{n+m}}R_{b_{n+m}}$

where $b_{0},\ldots b_{n+m}\in B$. Moreover, we can write

$\mathcal{F}(B,K)\cong\mathcal{F}(B,I)\otimes_{B}\mathcal{F}(B,J)$

in such a way that $Z$ acts via

$L_{b_{0}}S_{i_{1}}L_{b_{1}}\cdots S_{i_{n}}L_{b_{n}}\otimes
L_{b_{n+m}}S_{j_{n+m}}L_{b_{n+m-1}}\cdots S_{j_{n+1}}L_{b_{n+1}},$

so that if $E_{I}:\mathcal{L}(\mathcal{F}(B,I))\to B$ and
$E_{J}:\mathcal{L}(\mathcal{F}(B,J))\to B$ are the corresponding expectations,
then

$E(Z)=E_{I}(L_{b_{0}}S_{i_{1}}L_{b_{1}}\cdots
S_{i_{n}}L_{b_{n}})E_{J}(L_{b_{n+m}}S_{j_{n+m}}L_{b_{n+m-1}}\cdots
S_{j_{n+1}}L_{b_{n+1}}).$

Therefore, if we have $Z=\sum^{d}_{k=1}X_{k}Y_{k}$ where $X_{k}$ is a product
of $L_{b}$’s and $S_{i}$’s and $Y_{k}$ is a product of $R_{b}$’s and
$D_{j}$’s, then

$\tau(Z^{*}Z)=\sum^{d}_{k_{1},k_{2}=1}\tau_{B}\left(E_{I}\left(X_{k_{1}}^{*}X_{k_{2}}\right)E_{J}\left(\widetilde{Y}_{k_{2}}\widetilde{Y}^{*}_{k_{1}}\right)\right),$

where $\widetilde{Y}$ represents the monomial obtained by reversing the order
and changing $R$’s to $L$’s and $D$’s to $S$’s. However, as $E_{I}$ and
$E_{J}$ are completely positive on the algebras generated by $L$’s and $S$’s,
we can find $b_{X,k_{1},k_{3}},b_{Y,k_{1},k_{4}}\in B$ such that

$E_{I}\left(X_{k_{1}}^{*}X_{k_{2}}\right)=\sum^{d_{2}}_{k_{3}=1}b^{*}_{X,k_{1},k_{3}}b_{X,k_{2},k_{3}}\quad\text{and}\quad
E_{J}\left(\widetilde{Y}_{k_{2}}\widetilde{Y}^{*}_{k_{1}}\right)=\sum^{d_{3}}_{k_{4}=1}b_{Y,k_{2},k_{4}}b^{*}_{Y,k_{1},k_{4}},$

thus

$\displaystyle\tau(Z^{*}Z)$
$\displaystyle=\sum^{d}_{k_{1},k_{2}=1}\sum^{d_{2}}_{k_{3}=1}\sum^{d_{3}}_{k_{4}=1}\tau_{B}(b^{*}_{X,k_{1},k_{3}}b_{X,k_{2},k_{3}}b_{Y,k_{2},k_{4}}b^{*}_{Y,k_{1},k_{4}})$
$\displaystyle=\sum^{d_{2}}_{k_{3}=1}\sum^{d_{3}}_{k_{4}=1}\tau_{B}\left(\left(\sum^{d}_{k_{1}=1}b_{X,k_{1},k_{3}}b_{Y,k_{1},k_{4}}\right)^{*}\left(\sum^{d}_{k_{2}=1}b_{X,k_{2},k_{3}}b_{Y,k_{2},k_{4}}\right)\right)\geq
0.$

Hence $\tau$ is positive.

Next, one can verify in $L_{2}(A,\tau)$ that $S_{i}$ and $D_{j}$ are the sum
of an isometry and its adjoint (see [S1998]*Proposition 4.6.9) and thus define
bounded linear operators. Hence $(A,E,\varepsilon,\tau)$ is an analytical
$B$-$B$-non-commutative probability space.

To see the cumulant condition, one can proceed in two ways. One can
immediately realize that

$(\operatorname{alg}(B_{\ell},\\{S_{i}\\}_{i\in
I}),B_{r})\qquad\text{and}\qquad(B_{\ell},\operatorname{alg}(B_{r},\\{D_{j}\\}_{j\in
J})$

are bi-free over $B$ due to the above tensor-product relation. This implies
mixed cumulants are zero. The other cumulants then follow from the free case
in [S1998]. Alternatively, one can analyze the actions of $L_{b}$, $R_{b}$,
$S_{i}$, and $D_{j}$ as one would on the operator-valued reduced free product
space in an identical way to the LR-diagrams of [CNS2015-2, CNS2015-1] to
obtain a diagrammatic description of the elements of $\mathcal{F}(B,K)$
produced, note that the ones that contribute to a $B$-element are exactly the
bi-non-crossing diagrams that correspond to pair bi-non-crossing partitions,
and use induction to deduce the values of the operator-valued cumulants. ∎

We immediately obtain the following using Theorem 6.3 and Theorem 5.4.

######  Lemma 6.4.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, and
let $(\\{S_{i}\\}^{n}_{i=1},\\{D_{j}\\}^{m}_{j=1})$ be
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ bi-semicircular
operators in $A$. Then

$J_{\ell}\left(S_{1}:\left(\operatorname{alg}\left(B_{\ell},\\{S_{i}\\}^{n}_{j=2}\right),\operatorname{alg}\left(B_{r},\\{D_{j}\\}^{m}_{j=1}\right)\right),\eta_{\ell,1}\right)=S_{1}.$

A similar result holds for the other left and the right conjugate variables.

In order to obtain more examples of bi-free conjugate variables, it would be
typical to perturb by bi-semicircular operators and use Proposition 5.13. To
do this we must have the collection of bi-semicircular operators in the same
analytical $B$-$B$-non-commutative probability space. Thus, it is natural to
ask whether given two analytical $B$-$B$-non-commutative probability spaces
there is a bi-free product which causes the pairs of left and right algebras
to be bi-freely independent over $B$ and preserve the analytical properties.

Unfortunately we do not have an answer to this question. The proof of
positivity in the operator-valued free case requires the characterization of
the vanishing of alternating centred moments in [S1998]*Proposition 3.3.3 to
ensure positivity in the end. One may attempt to use the bi-free analogue of
‘alternating centred moments vanish’ from [C2019], however the bi-free
formulae generalization of [S1998]*Proposition 3.3.3 is far more complicated.
In particular, the proof from [S1998] will not immediately generalize, as
Example 3.4 shows $E$ will not be positive and the traciality of $\tau_{B}$
will need to come into play.

Luckily, if we deal only with bi-semicircular operators, which is all that is
required for this paper, there is no issue. In fact, in the case one is
working with von Neumann factors as in Example 2.6, the following is trivial
as it we can add the corresponding collection of bi-semicircular operators
using factors by [S1998].

######  Theorem 6.5.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space with $A_{\ell}$ and $A_{r}$ generated by isometries, let
$\\{\eta_{\ell,i}\\}^{n}_{i=1}\cup\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely
positive maps from $B$ to $B$, and let
$(\\{S_{i}\\}^{n}_{i=1},\\{D_{j}\\}^{m}_{j=1})$ be
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ bi-semicircular
operators. Then, there exists an analytical $B$-$B$-non-commutative
probability space $(A^{\prime},E^{\prime},\varepsilon^{\prime},\tau^{\prime})$
with $A\subseteq A^{\prime}$, $E^{\prime}|_{A}=E$, $\tau^{\prime}|_{A}=\tau$,
$A_{\ell}\subseteq A^{\prime}_{\ell}$, $A_{r}\subseteq A^{\prime}_{r}$,
$\\{S_{i}\\}^{n}_{i=1}\subseteq A^{\prime}_{\ell}$,
$\\{D_{j}\\}^{m}_{j=1}\subseteq A^{\prime}_{r}$ and such that the pairs of
algebras

$(A_{\ell},A_{r})\qquad\text{and}\qquad\left(\operatorname{alg}(B_{\ell},\\{S_{i}\\}^{n}_{i=1}),\operatorname{alg}(B_{r},\\{D_{j}\\}^{m}_{j=1})\right)$

are bi-free with amalgamation over $B$ with respect to $E^{\prime}$.

As for any $(\\{\eta_{i,i}\\}_{i\in I},\\{\eta_{j,j}\\}_{j\in J})$ bi-
semicircular operators $(\\{S_{i}\\}_{i\in I},\\{D_{j}\\}_{j\in J})$ we know
that

$\\{(\operatorname{alg}(B_{\ell},S_{i}),B_{r})\\}_{i\in
I}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},D_{j}))\\}_{j\in J}$

are bi-free over $B$, to prove Theorem 6.5 it suffices to use the following
lemma and an analogous result on the right iteratively, or simply adapt the
proof to multiple operators simultaneously.

######  Lemma 6.6.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space with $A_{\ell}$ and $A_{r}$ generated by isometries, let
$\eta:B\to B$ be a completely positive map and let $S$ be an
$\eta$-semicircular operator. Then, there exists an analytical $B$-$B$-non-
commutative probability space
$(A^{\prime},E^{\prime},\varepsilon^{\prime},\tau^{\prime})$ such that
$A^{\prime}=\operatorname{alg}(A_{\ell},A_{r},S)$, $E^{\prime}|_{A}=E$,
$\tau^{\prime}|_{A}=\tau$, $A^{\prime}_{\ell}=\operatorname{alg}(A_{\ell},S)$,
$A^{\prime}_{r}=A_{r}$ and

$(A_{\ell},A_{r})\qquad\text{and}\qquad(\operatorname{alg}(B_{\ell},S),B_{r})$

are bi-free with amalgamation over $B$ with respect to $E^{\prime}$.

###### Proof.

By taking the operator-valued bi-free product of $B$-$B$-non-commutative
probability spaces, we obtain a $B$-$B$-non-commutative probability space
$(A^{\prime},E^{\prime},\varepsilon^{\prime})$ such that
$A^{\prime}=\operatorname{alg}(A_{\ell},A_{r},S)$,
$A^{\prime}_{\ell}=\operatorname{alg}(A_{\ell},S)$, $A^{\prime}_{r}=A_{r}$,
$E^{\prime}|_{A}=E$ and

$(A_{\ell},A_{r})\qquad\text{and}\qquad(\operatorname{alg}(B_{\ell},S),B_{r})$

are bi-free with amalgamation over $B$ with respect to $E^{\prime}$. Note
$E^{\prime}$ restricted to $A_{r}$ is trivially completely positive and
$E^{\prime}$ restricted to $A^{\prime}_{\ell}$ is completely positive by the
free result from [S1998]. Thus, to verify Definition 3.1 it suffices to verify
that if $\tau^{\prime}=\tau_{B}\circ E$, then $\tau^{\prime}$ is positive and
elements of $A^{\prime}_{\ell}$ and $A^{\prime}_{r}$ define bounded operators
on $L_{2}(A^{\prime},\tau^{\prime})$.

By analyzing the reduced free product construction, we can realize $A_{\ell}$,
$A_{r}$, and $S$ as operators acting on

$\mathcal{F}=B\oplus\mathcal{H}_{1}\oplus\mathcal{H}_{2}\oplus\cdots,$

where

$\mathcal{H}_{m}=\\{a_{0}Za_{1}\cdots
Za_{m}\,\mid\,a_{0},a_{1},\ldots,a_{m-1}\in A_{\ell},a_{m}\in A\\}$

and if $p:\mathcal{F}\to B$ is defined by taking the $B$-term in
$\mathcal{F}$, then

$E^{\prime}(T)=p(T1_{B}).$

Define a function
$\langle\,\cdot,\cdot\,\rangle:\mathcal{F}\times\mathcal{F}\to A$ by setting
$B$, $\mathcal{H}_{1}$, $\mathcal{H}_{2}$, $\ldots$. to be pairwise
orthogonal, $\langle b_{1},b_{2}\rangle=L_{b_{2}^{*}b_{1}}$, and

$\langle a^{\prime}_{0}Za^{\prime}_{1}\cdots Za^{\prime}_{m},a_{0}Za_{1}\cdots
Za_{m}\rangle=a_{m}^{*}L_{\eta}(a_{m-1}^{*}\cdots
L_{\eta}(a_{1}^{*}L_{\eta}(a_{0}^{*}a^{\prime}_{0})a^{\prime}_{1})\cdots
a^{\prime}_{m-1})a^{\prime}_{m},$

where $L_{\eta}(T)=L_{\eta(E(T))}$. As $\eta$ is completely positive and $E$
is completely positive when restricted to $A_{\ell}$, we obtain that
$\langle\,\cdot,\cdot\,\rangle$ is an $A$-valued inner product by the same
arguments as [S1998]*Proposition 4.6.6. To elaborate slightly, given
$\sum^{n}_{k=1}a_{k,0}Za_{k,1}\cdots Za_{k,m}$, the matrix
$[\eta(a^{*}_{i,0}a_{j,0})]$ is positive and thus can be written as
$[\sum^{n}_{k=1}b^{*}_{k,i}b_{k,j}]$ for some $b_{i,j}\in B$. One then
substitutes
$L_{\eta}(a^{*}_{i,0}a_{j,0})=\sum^{n}_{k=1}L_{b_{k,i}}^{*}L_{b_{k,j}}$ and
continues until one ends with a sum of products of elements of $A$ with their
adjoints.

As $\tau:A\to\mathbb{C}$ is positive and as for all $T\in A^{\prime}$,

$\tau^{\prime}(T^{*}T)=\tau(\langle T1_{B},T1_{B}\rangle),$

we obtain that $\tau^{\prime}$ is positive as desired. To see that elements of
$A^{\prime}_{\ell}$ and $A^{\prime}_{r}$ define bounded operators on
$L_{2}(A^{\prime},\tau^{\prime})$ note if $T\in A_{r}$, then, using the above
description,

$T(a_{0}Za_{1}\cdots Za_{m})=a_{0}Za_{1}\cdots ZTa_{m}.$

As any of the terms

$L_{\eta}(a_{m-1}^{*}\cdots
L_{\eta}(a_{1}^{*}L_{\eta}(a_{0}^{*}a^{\prime}_{0})a^{\prime}_{1})\cdots
a^{\prime}_{m-1})$

in the above $A$-valued inner product will be able to be written as sums
involving terms of the form $L_{b_{1}}^{*}L_{b_{2}}$ and will then produce
terms of the form

$a_{m}^{*}T^{*}L_{b_{1}}^{*}L_{b_{2}}Ta^{\prime}_{m}=a_{m}^{*}L_{b_{1}}^{*}T^{*}TL_{b_{2}}a^{\prime}_{m}$

in the $A$-valued inner product when $T$ acts on the left, the fact that
$A_{r}$ is generated by isometries yields that $A_{r}$ acts as bounded
operators on $L_{2}(A,\tau)$. The fact that $A_{\ell}$ is generated by
isometries immediately yields that $A_{\ell}$ acts as bounded operators on
$L_{2}(A,\tau)$, and it is not difficult to see that $S$ acts as the sum of an
isometry and its adjoint on $L_{2}(A,\tau)$ and thus is bounded. ∎

With Theorem 6.5 establishing we can always assume our $B$-$B$-non-commutative
probability spaces have
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ bi-semicircular
operators, we can proceed with the following.

######  Theorem 6.7.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, let
$\mathbf{X}\in A_{\ell}^{n}$ and $\mathbf{Y}\in A_{r}^{m}$ be tuples of self-
adjoint operators, let $(\\{S_{i}\\}^{n}_{i=1},\\{D_{j}\\}^{m}_{j=1})$ be
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ bi-semicircular
operators in $A$ and let $(C_{\ell},C_{r})$ be pairs of $B$-algebras of $A$
such that

$\\{(\operatorname{alg}(C_{\ell},\mathbf{X}),\operatorname{alg}(C_{r},\mathbf{Y})\rangle)\\}\cup\\{(\operatorname{alg}(B_{\ell},S_{i}),B_{r})\\}^{n}_{i=1}\cup\\{(B_{\ell},\operatorname{alg}(B_{r},D_{j}))\\}^{m}_{j=1}$

are bi-free. If $P$ is the orthogonal projection of $L_{2}(A,\varphi)$ onto

$\overline{\operatorname{alg}\left(C_{\ell},C_{r},\mathbf{X}+\sqrt{\epsilon}\mathbf{S},\mathbf{Y}+\sqrt{\epsilon}\mathbf{D}\right)}^{\left\|\,\cdot\,\right\|_{\tau}},$

then

$\xi=J_{\ell}\left(X_{1}+\sqrt{\epsilon}S_{1}:\left(\operatorname{alg}\left(C_{\ell},\widehat{(\mathbf{X}+\sqrt{\epsilon}\mathbf{S})}_{1}\right),\operatorname{alg}\left(C_{r},\mathbf{Y}+\sqrt{\epsilon}\mathbf{D}\right)\right),\eta_{\ell,1}\right)=\frac{1}{\sqrt{\epsilon}}P(S_{1}).$

Thus

$\left\|\xi\right\|_{\tau}\leq\frac{1}{\sqrt{\epsilon}}\sqrt{\tau_{B}(\eta(1_{B}))}.$

A similar computation holds for the other entries of the tuples and the right
conjugate variables.

###### Proof.

By Lemma 6.4 and Lemma 5.6, we have that

$\displaystyle
J_{\ell}\left(\sqrt{\epsilon}S_{1}:\left(\operatorname{alg}\left(B_{\ell},\sqrt{\epsilon}\hat{\mathbf{S}}_{1}\right),\operatorname{alg}\left(B_{r},\sqrt{\epsilon}\mathbf{D}\right)\right),\eta_{\ell,1}\right)=\frac{1}{\sqrt{\epsilon}}S_{1}.$

The conjugate variable result then follows from Propositions 5.12 and 5.13,
whereas the $\tau$-norm computation is trivial. ∎

##  7\. Bi-Free Fisher Information with Respect to a Completely Positive Map

With the above technology, the bi-free Fisher information with respect to
completely positive maps can be constructed and has similar properties to the
bi-free Fisher information from [CS2020] and the free Fisher information with
respect to a completely positive map from [S1998]. We highlight the main
results and properties in this section.

######  Definition 7.1.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, let
$\mathbf{X}\in A_{\ell}^{n}$ and $\mathbf{Y}\in A_{r}^{m}$ and let
$(C_{\ell},C_{r})$ be a pair of $B$-algebras of $A$. The _relative bi-free
Fisher information of $(\mathbf{X},\mathbf{Y})$ with respect to
$(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})$ in the presence
of $(C_{\ell},C_{r})$_ is

$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))=\sum^{n}_{i=1}\left\|\xi_{i}\right\|_{\tau}^{2}+\sum^{m}_{j=1}\left\|\nu_{j}\right\|_{\tau}^{2}$

where for $i\in\\{1,\ldots,n\\}$ and $j\in\\{1,\ldots,m\\}$

$\xi_{i}=J_{\ell}\left(X_{i}:\left(\operatorname{alg}\left(C_{\ell},\widehat{\mathbf{X}}_{i}\right),\operatorname{alg}\left(C_{r},\mathbf{Y}\right)\right),\eta_{\ell,i}\right)\quad\text{and}\quad\nu_{j}=J_{r}\left(Y_{j}:\left(\operatorname{alg}\left(C_{\ell},\mathbf{X}\right),\operatorname{alg}\left(C_{r},\widehat{\mathbf{Y}}_{j}\right)\right),\eta_{r,j}\right)$

provided these variables exist, and otherwise is defined as $\infty$.

In the case that $\eta_{\ell,i}=\eta_{r,j}=\eta$ for all $i$ and $j$, we use
$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),\eta)$ to denote the
above bi-free Fisher information. In the case that $C_{\ell}=B_{\ell}$ and
$C_{r}=B_{r}$, we use
$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))$.
In the case both occur, we use $\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:\eta)$.

Note the bi-free Fisher with respect to completely positive maps exists in
many settings due to Theorem 6.5 and Theorem 6.7. Furthermore, the properties
of the bi-free Fisher with respect to completely positive maps are in analogy
with those from [V1998-2, S1998, CS2020] as the following shows.

######  Remark 7.2.

  1. (i)

In the case that $B=\mathbb{C}$ and $\eta$ is unital, Definition 7.1
immediately reduces down to the bi-free Fisher information in
[CS2020]*Definition 5.1 by Remark 5.3.

  2. (ii)

In the case we are in the context of Example 2.6 with $m=0$,
$C_{\ell}=B_{\ell}$, and $C_{r}=B_{r}$, Definition 7.1 immediately reduces
down to the free Fisher information with respect to a complete positive map
from [S1998]*Definition 4.1.

  3. (iii)

Note

$\displaystyle\Phi^{*}($
$\displaystyle\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))$
$\displaystyle=\sum^{n}_{i=1}\Phi^{*}\left(X_{i}\sqcup\emptyset:\left(\operatorname{alg}\left(C_{\ell},\widehat{\mathbf{X}}_{i}\right),\operatorname{alg}\left(C_{r},\mathbf{Y}\right)\right),\eta_{\ell,i},\tau\right)$
$\displaystyle\quad+\sum^{m}_{j=1}\Phi^{*}\left(\emptyset\sqcup
Y_{j}:\left(\operatorname{alg}\left(C_{\ell},\mathbf{X}\right),\operatorname{alg}\left(C_{r},\widehat{\mathbf{Y}}_{j}\right)\right),\eta_{r,j},\tau\right).$

  4. (iv)

If $\mathbf{X}=(X_{1},X_{1}^{*},\ldots,X_{n},X_{n}^{*})$ and
$\mathbf{Y}=(Y_{1},Y_{1}^{*},\ldots,Y_{m},Y_{m}^{*})$, then Lemma 5.5 implies

$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))=\frac{1}{2}\Phi^{*}(\mathbf{X}^{\prime}\sqcup\mathbf{Y}^{\prime}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))$

where
$\mathbf{X}^{\prime}=(\Re(X_{1}),\Im(X_{1}),\ldots,\Re(X_{n}),\Im(X_{n}))$ and
$\mathbf{Y}^{\prime}=(\Re(Y_{1}),\Im(Y_{1}),\ldots,\Re(Y_{m}),\Im(Y_{m}))$.

  5. (v)

In the context of Proposition 5.10 (i.e. reducing the $B$-$B$-non-commutative
probability space to a $D$-$D$-non-commutative probability space),

$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))=\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\circ
F\\}^{n}_{i=1},\\{\eta_{r,j}\circ F\\}^{m}_{j=1})).$

  6. (vi)

By Lemma 5.6, for all $\lambda\in\mathbb{C}\setminus\\{0\\}$

$\Phi^{*}(\lambda\mathbf{X}\sqcup\lambda\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))=\frac{1}{|\lambda|^{2}}\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

  7. (vii)

In the context of Lemma 5.11 (i.e. $(D_{\ell},D_{r})$ a smaller pair of
$B$-algebras than $(C_{\ell},C_{r})$),

$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(D_{\ell},D_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))\leq\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

  8. (viii)

In the context of Proposition 5.12 (i.e. adding in a bi-free pair of
$B$-algebras),

$\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(\operatorname{alg}(D_{\ell},C_{\ell}),\operatorname{alg}(D_{r},C_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))=\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1})).$

  9. (ix)

If in addition to the assumptions of Definition 7.1 $\mathbf{X}^{\prime}\in
A_{\ell}^{n^{\prime}}$, $\mathbf{Y}^{\prime}\in A_{r}^{m^{\prime}}$,
$(\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1})$
is a collection of completely positive maps on $B$, and $(D_{\ell},D_{r})$ is
a pair of $B$-algebras then, by (iii) and (vii),

$\displaystyle\Phi^{*}($
$\displaystyle\mathbf{X},\mathbf{X}^{\prime}\sqcup\mathbf{Y},\mathbf{Y}^{\prime}:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1}\cup\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}\cup\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$
$\displaystyle\geq\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))+\Phi^{*}(\mathbf{X}^{\prime}\sqcup\mathbf{Y}^{\prime}:(D_{\ell},D_{r}),(\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$

  10. (x)

In the context of (ix) with the additional assumption that

$\left(\operatorname{alg}(C_{\ell},\mathbf{X}),\operatorname{alg}(C_{r},\mathbf{Y})\right)\qquad\text{and}\qquad\left(\operatorname{alg}(D_{\ell},\mathbf{X}^{\prime}),\operatorname{alg}(D_{r},\mathbf{Y}^{\prime})\right)$

are bi-free with amalgamation over $B$ with respect to $E$, Proposition 5.12
implies that

$\displaystyle\Phi^{*}($
$\displaystyle\mathbf{X},\mathbf{X}^{\prime}\sqcup\mathbf{Y},\mathbf{Y}^{\prime}:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1}\cup\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}\cup\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$
$\displaystyle=\Phi^{*}(\mathbf{X}\sqcup\mathbf{Y}:(C_{\ell},C_{r}),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))+\Phi^{*}(\mathbf{X}^{\prime}\sqcup\mathbf{Y}^{\prime}:(D_{\ell},D_{r}),(\\{\eta^{\prime}_{\ell,i}\\}^{n^{\prime}}_{i=1},\\{\eta^{\prime}_{r,j}\\}^{m^{\prime}}_{j=1}))$

Unsurprisingly, more complicated properties of free Fisher information extend.

######  Proposition 7.3 (Bi-Free Stam Inequality).

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\\{\eta_{\ell,i}\\}^{n}_{i=1}$ and
$\\{\eta_{r,j}\\}^{m}_{j=1}$ be completely positive maps from $B$ to $B$, let
$\mathbf{X},\mathbf{X}^{\prime}\in A_{\ell}^{n}$ and
$\mathbf{Y},\mathbf{Y}^{\prime}\in A_{r}^{m}$, and let $(C_{\ell},C_{r})$ and
$(D_{\ell},D_{r})$ be pairs of $B$-algebras of $A$ such that

$\left(\operatorname{alg}(C_{\ell},\mathbf{X}),\operatorname{alg}(C_{r},\mathbf{Y})\right)\qquad\text{and}\qquad\left(\operatorname{alg}(D_{\ell},\mathbf{X}^{\prime}),\operatorname{alg}(D_{r},\mathbf{Y}^{\prime})\right)$

are bi-free with amalgamation over $B$ with respect to $E$. Then

$\displaystyle\left(\Phi^{*}(\mathbf{X}+\mathbf{X}^{\prime}\sqcup\mathbf{Y}+\mathbf{Y}^{\prime}:(\operatorname{alg}(C_{\ell},D_{\ell}),\operatorname{alg}(C_{r},D_{r})),(\\{\eta_{\ell,i}\\}^{n}_{i=1},\\{\eta_{r,j}\\}^{m}_{j=1}))\right)^{-1}$