# Bi-Free Entropy with Respect to Completely Positive Maps

Georgios Katsimpas  Department of Mathematics and Statistics, York University,
4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada<EMAIL_ADDRESS>and  Paul Skoufranis  Department of Mathematics and Statistics, York
University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada
<EMAIL_ADDRESS>
###### Abstract.

In this paper, a notion of non-microstate bi-free entropy with respect to
completely positive maps is constructed thereby extending the notions of non-
microstate bi-free entropy and free entropy with respect to a completely
positive map. By extending the operator-valued bi-free structures to allow for
more analytical arguments, a notion of conjugate variables is constructed
using both moment and cumulant expressions. The notions of free Fisher
information and entropy are then extended to this setting and used to show
minima of the Fisher information and maxima of the non-microstate bi-free
entropy at bi-R-diagonal elements.

###### Key words and phrases:

bi-free probability, entropy, completely positive maps, bi-free cumulants, bi-
R-diagonals.

###### 2010 Mathematics Subject Classification:

46L54, 46L53

The funding for the first author was partially supported and the research of
the second author was supported in part by NSERC (Canada) grant
RGPIN-2017-05711.

##  1\. Introduction

Free entropy was introduced in a series of papers by Voiculescu including
[V1998-2, V1999] that cemented the foundations of free probability and its
applications to operator algebras. Of note is the non-microstate approach of
[V1998-2] that generalized the notions of Fisher information and entropy to
the non-commutative random variables studied in free probability by using a
conjugate variable system and free Brownian motions. These ideas were further
extended to the operator-valued setting by Shlyakhtenko in [S1998] by
modifying the conjugate variable formulae to involve a completely positive map
on the algebra of amalgamation. One immediate application was
[S1998]*Proposition 7.14 that obtained a formula for the Jones index of a
subfactor. Furthermore, free entropy with respect to a completely positive map
was essential to the work in [NSS1999] which demonstrated that minimal values
for the free Fisher information and maximal values for the non-microstate free
entropy existed and were obtained at R-diagonal elements.

More recently in [V2014], Voiculescu extended the notion of free independence
to simultaneously study the left and right actions of algebras on reduced free
product spaces. In particular, this permits a notion of independence, called
bi-free independence, that contains both free and classical independence (see
[S2016-1]) and a free probability construction to simultaneously study both a
von Neumann algebra and its commutant (see Example 2.6). Significant effort
has gone into enhancing results from free probability to the bi-free setting
and examining potential applications. In terms of entropy, Charlesworth and
the second author recently developed [CS2020, CS2019], thereby extending both
the microstate and non-microstate free entropies to the bi-free setting.

The purpose of this paper is to extend the notion of non-microstate bi-free
entropy to incorporate the existence of a completely positive map and examine
applications of said theory. In particular, our main applications are Theorem
9.6 which examines the minimal value of the bi-free Fisher information for
collections of pairs of operators with similarities in their distributions,
and the following theorem which cumulates the first author’s work on bi-R-
diagonal elements from [K2019].

######  Theorem 1.

Let $(\mathcal{A},\varphi)$ be a C∗-non-commutative probability space and let
$x,y\in\mathcal{A}$ be such that $x^{*}x$ and $xx^{*}$ have the same
distribution with respect to $\varphi$ and $y^{*}y$ and $yy^{*}$ have the same
distribution with respect to $\varphi$. With $\tau_{2}:\mathcal{A}\otimes
M_{2}(\mathbb{C})\otimes M_{2}(\mathbb{C})^{{\text{op}}}\to\mathbb{C}$ defined
by

$\tau_{2}(T\otimes b_{1}\otimes
b_{2})=\varphi(T)\operatorname{tr}_{2}(b_{1}b_{2})$

and

$X=x\otimes E_{1,2}\otimes I_{2}+x^{*}\otimes E_{2,1}\otimes
I_{2}\quad\text{and}\quad Y=y\otimes I_{2}\otimes E_{1,2}+y^{*}\otimes
I_{2}\otimes E_{2,1},$

we have that

$\chi^{*}(\\{x,x^{*}\\}\sqcup\\{y,y^{*}\\})\leq 2\chi^{*}(X\sqcup Y)$

and equality holds whenever the pair $(x,y)$ is bi-R-diagonal and alternating
adjoint flipping.

This paper is structured as follows. After reviewing some preliminaries and
notation pertaining to bi-free probability in Section 2, Section 3 will extend
the structures used in operator-valued bi-free probability. This is necessary
as expectations in operator-valued bi-free probability need not be positive
and thus to perform analytical computations additional structures are
required. These structures occur and are modelled based on the left and right
actions of a II1 factor on its $L_{2}$-space (see Example 3.6). By adding a
tracial state on the algebra of amalgamation that satisfies certain
compatibility conditions, the appropriate $L_{2}$-spaces can be constructed
and used to study operator-valued bi-free probability. It is these structures
that the authors believe will be vital to future applications of operator-
valued bi-free probability.

In Section 4 the operator-valued bi-free cumulant functions are extended to
allow for the last entry to be an element of the corresponding $L_{2}$-space.
This is thematic in bi-free probability where the last entry corresponds to
the location where left and right operators mix and therefore it is natural to
extend the bi-free moment and cumulant functions to have a mixture of left and
right operators in the last entry. These analytical extensions of the
operator-valued bi-free cumulants are shown to have the appropriate bi-
multiplicative properties by methods similar to [CNS2015-1].

In Section 5 the bi-free conjugate variables with respect to completely
positive maps are defined via moment relations. It is then shown that these
moment relations transfer to cumulant relations using the results of Section 4
and thus the natural properties of free conjugate variables extend to this
setting. One technical detail in developing the bi-free entropy with respect
to a completely positive map is to show that if one perturbs operators by
operator-valued bi-free central limit distributions, then the resulting
operators have bi-free conjugate variables. Thus, it is necessary to show that
one can always add in the appropriate operator-valued bi-free central limit
distributions into the structures from Section 3 and remain within that
context. Results along these line sufficient for this paper are developed in
Section 6.

In Sections 7 and 8 the bi-free Fisher information and entropy with respect to
a completely positive map are defined and shown to have the desired properties
from [V1998-2, S1998, CS2020] with very similar proofs. Sections 9 and 10
extend the techniques from [NSS1999] taking into account the differences in
bi-free probability to obtain the minimal value of the bi-free Fisher
information of a collection of pairs of operators with similarities in their
distributions and the above theorem. Finally, Section 11 outlines some other
results that immediately extend from [NSS1999] to the bi-free setting using
the results and techniques from this paper.

##  2\. Preliminaries

In this section we will remind the reader on the basic combinatorial and
operator-valued structures that have been used in previous papers on bi-free
independence. For a more in-depth reminder of these concepts, we refer the
readers to the original papers [CNS2015-1, CNS2015-2].

### Combinatorics on the Lattice of Bi-Non-Crossing Partitions

For $n\in\mathbb{N}$, the collection of all partitions on $\\{1,\ldots,n\\}$
is denoted by $\mathcal{P}(n)$. The elements of any $\pi\in\mathcal{P}(n)$ are
called the _blocks_ of $\pi$. A partial ordered on $\mathcal{P}(n)$ is defined
via refinement where for $\pi,\sigma\in\mathcal{P}(n)$ we write
$\pi\leq\sigma$ if every block of $\pi$ is contained in a block of $\sigma$.
The maximal element of $\mathcal{P}(n)$ with respect to this partial order is
the partition consisting of one block and is denoted by $1_{n}$, while the
minimal element is the partition consisting of $n$-blocks and denoted by
$0_{n}$. Note that $\mathcal{P}(n)$ becomes a lattice under this partial
ordering. For $\pi,\sigma\in\mathcal{P}(n)$ the _join_ of $\pi$ and $\sigma$,
denoted $\pi\vee\sigma$, is the minimum element of the non-empty set
$\\{\upsilon\in\mathcal{P}(n)\,\mid\,\upsilon\geq\pi,\sigma\\}$. A partition
$\pi$ on $\\{1,\ldots,n\\}$ is said to be _non-crossing_ if whenever $V,W$ are
blocks of $\pi$ and $v_{1},v_{2}\in V,w_{1},w_{2}\in W$ are such that

$v_{1}<w_{1}<v_{2}<w_{2},$

then $V=W$. The lattice of non-crossing partitions on $\\{1,\ldots,n\\}$ is
denoted $\operatorname{NC}(n)$.

In the bi-free setting, all operators are implicitly imbued with a direction;
either left or right. Given a sequence of $n$ operators, we will use a map
$\chi\in{\\{\ell,r\\}}^{n}$ to distinguish whether the $k^{\text{th}}$
operator is a left or a right operator. Such a map automatically gives rise to
a permutation $s_{\chi}$ on $\\{1,\ldots,n\\}$ defined as follows: if
${\chi}^{-1}(\\{\ell\\})=\\{i_{1}<\ldots<i_{p}\\}$ and
${\chi}^{-1}(\\{r\\})=\\{j_{1}<\ldots<j_{n-p}\\}$, then

$s_{\chi}(k)=\begin{cases}i_{k}&\text{if }k\leq p\\\ j_{n+1-k}&\text{if
}k>p\\\ \end{cases}.$

From a combinatorial point of view, the main difference between free and bi-
free probability arise from dealing with $s_{\chi}$. The permutation
$s_{\chi}$ naturally induces a total order $\prec_{\chi}$ on
$\\{1,\ldots,n\\}$ henceforth referred to as the $\chi$-order as follows:

$i\prec_{\chi}j\iff s_{\chi}^{-1}(i)<s_{\chi}^{-1}(j).$

Instead of reading $\\{1,\ldots,n\\}$ in the traditional order, this
corresponds to first reading the elements of $\\{1,\ldots,n\\}$ labelled
“$\ell$” in increasing order followed by reading the elements labelled “$r$”
in decreasing order. Note that if $V$ is any non-empty subset of
$\\{1,\ldots,n\\}$, the map $\chi|_{V}$ naturally gives rise to a map
$s_{{\chi}|_{V}}$, which should be thought of as a permutation on
$\\{1,\ldots,|V|\\}$.

######  Definition 2.1.

Let $n\in\mathbb{N}$ and $\chi\in{\\{\ell,r\\}}^{n}$. A partition
$\pi\in\mathcal{P}(n)$ is called bi-non-crossing with respect to $\chi$ if the
partition $s_{\chi}^{-1}\circ\pi$ (i.e. the partition obtained by applying the
permutation $s_{\chi}^{-1}$ to each entry of every block of $\pi$) is non-
crossing. Equivalently, $\pi$ is bi-non-crossing with respect to $\chi$ if
whenever $V,W$ are blocks of $\pi$ and $v_{1},v_{2}\in V,w_{1},w_{2}\in W$ are
such that

$v_{1}\prec_{\chi}w_{1}\prec_{\chi}v_{2}\prec_{\chi}w_{2},$

then $V=W$. The collection of bi-non-crossing partitions with respect to
$\chi$ is denoted by $\operatorname{BNC}(\chi)$. It is clear that

$\operatorname{BNC}(\chi)=\left\\{\pi\in\mathcal{P}(n)\,\mid\,s_{\chi}^{-1}\circ\pi\in\operatorname{NC}(n)\right\\}=\left\\{s_{\chi}\circ\sigma\,\mid\,\sigma\in\operatorname{NC}(n)\right\\}.$

In the context when the map $\chi$ is clear, we will refer to an element of
$\operatorname{BNC}(\chi)$ simply as bi-non-crossing. To each partition
$\pi\in BNC(\chi)$ we can associate a “bi-non-crossing diagram” (with respect
to $\chi$) by placing nodes along two vertical lines, labelled $1$ to $n$ from
top to bottom, such that the nodes on the left line correspond to those values
for which $\chi(k)=\ell$ (similarly for the right), and connecting those nodes
which are in the same block of $\pi$ in a non-crossing manner. In particular,
a partition $\pi\in\mathcal{P}(n)$ is in $\operatorname{BNC}(\chi)$ if and
only if it can be drawn in this non-crossing way.

######  Example 2.2.

If $\chi\in{\\{\ell,r\\}}^{6}$ is such that
$\chi^{-1}(\\{\ell\\})=\\{1,2,3,6\\}$ and $\chi^{-1}(\\{r\\})=\\{4,5\\}$, then

$(s_{\chi}(1),\ldots,s_{\chi}(6))=(1,2,3,6,5,4)$

and the partition given by

$\pi=\\{\\{1,4\\},\\{2,5\\},\\{3,6\\}\\}$

is bi-non-crossing with respect to $\chi$ even though
$\pi\notin\operatorname{NC}(6)$. This may also be seen via the following
diagrams:

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The set of bi-non-crossing partitions with respect to a map
$\chi\in{\\{\ell,r\\}}^{n}$ automatically inherits the lattice structure from
$\mathcal{P}(n)$ via the partial order of refinement. The minimal and maximal
elements of $\operatorname{BNC}(\chi)$ will be denoted by $0_{\chi}$ and
$1_{\chi}$ respectively, and note that $0_{\chi}=s_{\chi}(0_{n})=0_{n}$ and
$1_{\chi}=s_{\chi}(1_{n})=1_{n}$. For $\emptyset\neq
V\subseteq\\{1,\ldots,n\\}$, we denote by $\min_{\leq}V$ and
$\min_{\preceq_{\chi}}V$ the minimum element of $V$ with respect to the
natural order and the ${\chi}$-order of $\\{1,\ldots,n\\}$ respectively, with
similar notation used for the maximum elements. For
$\pi,\sigma\in\operatorname{BNC}(\chi)$ with $\sigma\leq\pi$ we will denote by
$[\sigma,\pi]$ the interval with respect to the partial order of refinement.

######  Definition 2.3.

The _bi-non-crossing Möbius function_ is the map

$\mu_{\operatorname{BNC}}:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in{\\{\ell,r\\}}^{n}}\operatorname{BNC}(\chi)\times\operatorname{BNC}(\chi)\to\mathbb{Z}$

defined recursively by

$\sum_{\begin{subarray}{c}\nu\in\operatorname{BNC}(n)\\\
\pi\leq\tau\leq\sigma\\\
\end{subarray}}\mu_{\operatorname{BNC}}(\pi,\nu)=\sum_{\begin{subarray}{c}\tau\in\operatorname{BNC}(n)\\\
\pi\leq\tau\leq\sigma\\\
\end{subarray}}\mu_{\operatorname{BNC}}(\nu,\sigma)=\begin{cases}1&\text{if
}\pi=\sigma\\\ 0&\text{if }\pi<\sigma\\\ \end{cases}$

whenever $\pi\leq\sigma$, while taking the zero value otherwise.

The connection between the bi-non-crossing Möbius function and the Möbius
function on the lattice of non-crossing partitions $\mu_{\operatorname{NC}}$
is given by the formula

$\mu_{\operatorname{BNC}}(\pi,\sigma)=\mu_{\operatorname{NC}}(s_{\chi}^{-1}\circ\pi,s_{\chi}^{-1}\circ\sigma)$

for all $\pi\leq\sigma\in\operatorname{BNC}(\chi)$. Hence
$\mu_{\operatorname{BNC}}$ inherits permuted analogues of the multiplicative
properties of $\mu_{\operatorname{NC}}$ (see [CNS2015-2]*Section 3). In
particular, if $n\in\mathbb{N}$, if $\chi\in\\{\ell,r\\}^{n}$, if
$\pi,\sigma\in\operatorname{BNC}(\chi)$ such that $\pi\leq\sigma$, and if
$V_{1},\ldots,V_{m}$ are unions of blocks of $\pi$ that partition
$\\{1,\ldots,n\\}$, then the natural map

$\displaystyle[\sigma,\pi]$
$\displaystyle\longrightarrow\prod_{k=1}^{m}[\sigma|_{V_{k}},\pi|_{V_{k}}]$
$\displaystyle\tau$ $\displaystyle\mapsto\qquad{({\tau|}_{V_{k}})}_{k=1}^{m}$

is a bijection and

$\mu_{\operatorname{BNC}}(\sigma,\pi)=\mu_{\operatorname{BNC}}({\sigma|}_{V_{1}},\pi|_{V_{1}})\cdots\mu_{\operatorname{BNC}}({\sigma|}_{V_{m}},\pi|_{V_{m}}).$

### B-B-Non-Commutative Probability Spaces and Bi-Freeness

To study operator-valued bi-free independence, certain structures are
required. Thus we shall remind the reader of the general structures as
developed in [CNS2015-1] and refer the reader there for more details.

######  Definition 2.4.

Let $B$ be a unital $*$-algebra and let $B^{{\text{op}}}$ denote the unital
$*$-algebra with the same elements as $B$ with the opposite multiplication. A
B-B-non-commutative probability space consists of a triple $(A,E,\varepsilon)$
where $A$ is a unital $*$-algebra, $\varepsilon:B\otimes B^{{\text{op}}}\to A$
is a unital $*$-homomorphism such that the restrictions
${\varepsilon|}_{B\otimes 1_{B}}$ and ${\varepsilon|}_{1_{B}\otimes
B^{{\text{op}}}}$ are both injective, and $E:A\to B$ is a unital linear map
that such that

$E(\varepsilon(b_{1}\otimes b_{2})a)=b_{1}E(a)b_{2}\qquad\text{and}\qquad
E(a\varepsilon(b\otimes 1_{B}))=E(a\varepsilon(1_{B}\otimes b)),$

for all $b,b_{1},b_{2}\in B$ and $a\in A$. In addition, consider the unital
$*$-subalgebras $A_{\ell}$ and $A_{r}$ of $A$ given by

$A_{\ell}=\\{a\in A\,\mid\,a\varepsilon(1_{B}\otimes
b)=\varepsilon(1_{B}\otimes b)a\text{ for all }b\in B\\}$

and

$A_{r}=\\{a\in A\,\mid\,a\varepsilon(b\otimes 1_{B})=\varepsilon(b\otimes
1_{B})a\text{ for all }b\in B\\}.$

We call $A_{\ell}$ and $A_{r}$ the left and right algebras of $A$
respectively.

Note one can always assume that a $B$-$B$-non-commutative probability space is
generated as a $*$-algebra by $A_{\ell}$ and $A_{r}$.

######  Example 2.5.

Let $\mathcal{A}$ and $B$ be unital $*$-algebras and let
$\varphi:\mathcal{A}\to\mathbb{C}$ be a unital, linear map. If
$A=\mathcal{A}\otimes B\otimes B^{{\text{op}}}$, if $\varepsilon:B\otimes
B^{{\text{op}}}\to A$ is defined by $\varepsilon(b_{1}\otimes
b_{2})=1_{\mathcal{A}}\otimes b_{1}\otimes b_{2}$ for all $b_{1},b_{2}\in B$,
and $E:A\to B$ is defined by

$E(a\otimes b_{1}\otimes b_{2})=\varphi(a)b_{1}b_{2}$

for all $a\in A$ and $b_{1},b_{2}\in B$, then $(A,E,\varepsilon)$ is a
$B$-$B$-non-commutative probability space. Indeed, clearly $\varepsilon$ is a
unital injective $*$-homomorphism. Furthermore, note for all $Z\in\mathcal{A}$
and $b,b_{1},b_{2},b_{3},b_{4}\in B$ that

$E((1_{\mathcal{A}}\otimes b_{1}\otimes b_{2})(Z\otimes b_{3}\otimes
b_{4}))=\varphi(Z)b_{1}b_{3}b_{4}b_{2}=b_{1}E(Z\otimes b_{3}\otimes
b_{4})b_{2}$

and

$E((Z\otimes b_{1}\otimes b_{2})(1_{\mathcal{A}}\otimes b\otimes
1_{B}))=\varphi(Z)b_{1}bb_{2}=E((Z\otimes b_{1}\otimes
b_{2})(1_{\mathcal{A}}\otimes 1_{B}\otimes b)).$

Hence $E$ satisfies the required properties.

For future use, notice that

$\mathcal{A}\otimes B\otimes
1_{B}\subseteq\mathcal{A}_{\ell}\qquad\text{and}\qquad\mathcal{A}\otimes
1_{B}\otimes B^{{\text{op}}}\subseteq\mathcal{A}_{r}.$

Moreover, in the case $B=\mathbb{C}$, $(A,E,\varepsilon)$ efficiently reduces
down to $(\mathcal{A},\varphi)$; the usual notion of a non-commutative
probability space.

######  Example 2.6.

Let $\mathfrak{M}$ be a finite von Neumann algebra with a tracial state
$\tau:\mathfrak{M}\to\mathbb{C}$ and let $L_{2}(\mathfrak{M},\tau)$ be the GNS
Hilbert space generated by $(\mathfrak{M},\tau)$. For $T\in\mathfrak{M}$, let
$L_{T}$ denote the left action of $T$ on $L_{2}(\mathfrak{M},\tau)$, and let
$R_{T}$ denote the right action of $T$ on $L_{2}(\mathfrak{M},\tau)$.
Furthermore, let $A$ be the algebra generated by
$\\{L_{T},R_{T}\,\mid\,T\in\mathfrak{M}\\}$.

Let $B$ be a unital von Neumann subalgebra of $\mathfrak{M}$ and let
$E_{B}:\mathfrak{M}\to B$ be the conditional expectation of $\mathfrak{M}$
onto $B$. Recall that if $P:L_{2}(\mathfrak{M},\tau)\to L_{2}(B,\tau)$ is the
orthogonal projection of $L_{2}(\mathfrak{M},\tau)$ onto $L_{2}(B,\tau)$, then
$E_{B}(Z)=P(Z1_{\mathfrak{M}})$ for all $Z\in\mathfrak{M}$.

Define $\varepsilon:B\otimes B^{{\text{op}}}\to A$ by
$\varepsilon(b_{1}\otimes b_{2})=L_{b_{1}}R_{b_{2}}$ and define $E:A\to B$ by

$E(Z)=P(Z1_{\mathfrak{M}})$

for all $Z\in A$. Elementary von Neumann algebra theory implies that the range
of $E$ is indeed contained in $B$. To see that $(A,E,\varepsilon)$ is a
$B$-$B$-non-commutative probability space, first note that $\varepsilon$ is
clearly a unital $*$-homomorphism that is injective when restricted to
$B\otimes 1_{B}$ and when restricted to $1_{B}\otimes B^{\text{op}}$.
Moreover, note for all $Z\in A$ and $b,b_{1},b_{2}\in B$ that

$E(L_{b_{1}}R_{b_{2}}Z)=P(b_{1}(Z1_{\mathfrak{M}})b_{2})=b_{1}P(Z1_{\mathfrak{M}})b_{2}=b_{1}E(Z)b_{2}$

and

$E(TL_{b})=P(TL_{b}1_{\mathfrak{M}})=P(Tb)=P(TR_{b}1_{\mathfrak{M}})=E(TR_{b}).$

Hence $E$ satisfies the required properties.

The map $\varepsilon:B\otimes B^{{\text{op}}}\to A$ encodes the left and right
elements of $B$ in $A$. For notational purposes, for each $b\in B$ we will
denote $\varepsilon(b\otimes 1_{B})$ and $\varepsilon(1_{B}\otimes b)$ by
$L_{b}$ and $R_{b}$ respectively and we denote

$B_{\ell}=\varepsilon(B\otimes 1_{B})=\\{L_{b}\,\mid\,b\in
B\\}\quad\text{and}\quad B_{r}=\varepsilon(1_{B}\otimes
B^{{\text{op}}})=\\{R_{b}\,\mid\,b\in B\\}.$

To examine bi-free independence with amalgamation over $B$, it is necessary
that left operators are contained in $A_{\ell}$ (i.e. commute with the right
copy of $B$) and right operators are contained in $A_{r}$ (i.e. commute with
the left copy of $B$).

######  Definition 2.7 ([CNS2015-1]).

Let $(A,E,\varepsilon)$ be a $B$-$B$-non-commutative probability space.

  1. (i)

A pair of $B$-algebras is a pair $(C,D)$ consisting of unital subalgebras of
$A$ such that

$B_{\ell}\subseteq C\subseteq A_{\ell}\quad\text{and}\quad B_{r}\subseteq
D\subseteq A_{r}.$

  2. (ii)

A family ${\\{(C_{k},D_{k})\\}}_{k\in K}$ of pairs of $B$-algebras in $A$ is
called bi-free with amalgamation over $B$ if there exist $B$-$B$-bimodules
with specified $B$-vector states
${\\{(\mathcal{X}_{k},\overset{\circ}{\mathcal{X}}_{k},p_{k})\\}}_{k\in K}$
and unital homomorphisms

$l_{k}:C_{k}\to\mathcal{L}_{\ell}(\mathcal{X}_{k})\quad\text{and}\quad
r_{k}:D_{k}\to\mathcal{L}_{r}(\mathcal{X}_{k}),$

such that the joint distribution of the family ${\\{(C_{k},D_{k})\\}}_{k\in
K}$ with respect to $E$ coincides with the joint distribution of the images

${\\{((\lambda_{k}\circ l_{k})(C_{k}),(\rho_{k}\circ r_{k})(D_{k}))\\}}_{k\in
K}$

in the space $\mathcal{L}(*_{k\in K}\mathcal{X}_{k})$, with respect to
$E_{\mathcal{L}(*_{k\in K}\mathcal{X}_{k})}$, where $*_{k\in
K}\mathcal{X}_{k}$ is the reduced free product of
${\\{(\mathcal{X}_{k},\overset{\circ}{\mathcal{X}}_{k},p_{k})\\}}_{k\in K}$
with amalgamation over $B$.

######  Remark 2.8.

Let $\mathcal{A}$ and $B$ be unital $*$-algebras and let
$\varphi:\mathcal{A}\to\mathbb{C}$ be a unital linear map. Let
$(A,E,\varepsilon)$ be as in Example 2.5. By [S2016-1, CS2019], if
$\\{(C_{k},D_{k})\\}$ are $*$-subalgebras of $\mathcal{A}$ that are bi-free
with respect to $\varphi$, then $\\{(C_{k}\otimes B\otimes 1_{B},D_{k}\otimes
1_{B}\otimes B^{\text{op}})\\}_{k\in K}$ are bi-free with amalgamation over
$B$ with respect to $E$. Thus Example 2.5 is the correct notion of “inflating
$(\mathcal{A},\varphi)$ by $B$” in the bi-free setting.

######  Example 2.9.

Let $\mathfrak{M}_{1}$ and $\mathfrak{M}_{2}$ be finite von Neumann algebras
with a common von Neumann subalgebra $B$ and tracial states $\tau_{1}$ and
$\tau_{2}$ respectively such that $\tau_{1}|_{B}=\tau_{2}|_{B}$. Let
$\mathfrak{M}=\mathfrak{M}_{1}\ast_{B}\mathfrak{M}_{2}$ be the reduced free
product von Neumann algebra with amalgamation over $B$, let
$E_{B}:\mathfrak{M}\to B$ be the conditional expectation of $\mathfrak{M}$
onto $B$, and let $\tau=\tau_{1}\ast\tau_{2}=\tau_{1}|_{B}\circ E_{B}$ be the
tracial state on $\mathfrak{M}$. If $E$ and $\varepsilon$ are as in Example
2.6 for $(\mathfrak{M},\tau)$, then

$\left\\{\left(\\{L_{X}\,\mid\,X\in\mathfrak{M}_{1}\\},\\{R_{Y}\,\mid\,Y\in\mathfrak{M}_{1}\\}\right)\right\\}\qquad\text{and}\qquad\left\\{\left(\\{L_{X}\,\mid\,X\in\mathfrak{M}_{2}\\},\\{R_{Y}\,\mid\,Y\in\mathfrak{M}_{2}\\}\right)\right\\}$

are bi-free with amalgamation over $B$.

In order to study bi-free independence with amalgamation, the operator-valued
bi-free moment and cumulant functions are key. These functions have specific
properties that are described via the following concept. In what follows and
for the remainder of the paper, given an $n$-tuple $(Z_{1},\ldots,Z_{n})$ and
$V\subseteq\\{1,\ldots,n\\}$, we will use $(Z_{1},\ldots,Z_{n})|_{V}$ to
denote the $|V|$-tuple where only the entries $Z_{k}$ where $k\in V$ remain.

######  Definition 2.10 ([CNS2015-1]*Definition 4.2.1).

Let $(A,E,\varepsilon)$ be a $B$-$B$-non-commutative probability space. A map

$\Phi:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in\\{\ell,r\\}^{n}}\operatorname{BNC}(\chi)\times
A_{\chi(1)}\times A_{\chi(2)}\times\ldots\times A_{\chi(n)}\to B$

is called bi-multiplicative if it is $\mathbb{C}$-linear in each of the
$A_{\chi(k)}$ entries and for all $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$,
$\pi\in\operatorname{BNC}(\chi)$, $b\in B$, and $Z_{k}\in A_{\chi(k)}$ the
following four conditions hold:

  1. (i)

Let

$q=\max_{\leq}\\{k\in\\{1,\ldots,n\\}\,\mid\,\chi(k)\neq\chi(n)\\}.$

If $\chi(n)=\ell,$ then

$\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n}L_{b})=\begin{cases}\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{q-1},Z_{q}R_{b},Z_{q+1},\ldots,Z_{n})&\text{if
}q\neq-\infty,\\\ \Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n})b&\text{if
}q=-\infty.\\\ \end{cases}$

If $\chi(n)=r,$ then

$\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n}R_{b})=\begin{cases}\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{q-1},Z_{q}L_{b},Z_{q+1},\ldots,Z_{n})&\text{if
}q\neq-\infty,\\\ b\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n})&\text{if
}q=-\infty.\\\ \end{cases}$

  2. (ii)

Let $p\in\\{1,\ldots,n\\}$ and let

$q=\max_{\leq}\left\\{k\in\\{1,\ldots,n\\}\,\mid\,\chi(k)=\chi(p),k<p\right\\}.$

If $\chi(p)=\ell,$ then

$\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{p-1},L_{b}Z_{p},Z_{p+1}\ldots,Z_{n})=\begin{cases}\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{q-1},Z_{q}L_{b},Z_{q+1},\ldots,Z_{n})&\text{if
}q\neq-\infty,\\\ b\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n})&\text{if
}q=-\infty.\\\ \end{cases}$

If $\chi(p)=r,$ then

$\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{p-1},R_{b}Z_{p},Z_{p+1}\ldots,Z_{n})=\begin{cases}\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{q-1},Z_{q}R_{b},Z_{q+1},\ldots,Z_{n})&\text{if
}q\neq-\infty,\\\ \Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n})b&\text{if
}q=-\infty.\\\ \end{cases}$

  3. (iii)

Suppose $V_{1},\ldots,V_{m}$ are unions of blocks of $\pi$ that partition
$\\{1,\ldots,n\\}$ with each being a $\chi$-interval (i.e. an interval in the
$\chi$-ordering) and the sets $V_{1},\ldots,V_{m}$ are ordered by
$\preceq_{\chi}$ (i.e.
$(\min_{\preceq_{\chi}}V_{k})\prec_{\chi}(\min_{\preceq_{\chi}}V_{k+1})$ for
all $k$). Then

$\Phi_{\pi}(Z_{1},\ldots,Z_{n})=\Phi_{\pi|_{V_{1}}}({(Z_{1},\ldots,Z_{n})|}_{V_{1}})\cdots\Phi_{\pi|_{V_{m}}}({(Z_{1},\ldots,Z_{n})|}_{V_{m}}).$

  4. (iv)

Suppose that $V$ and $W$ are unions of blocks of $\pi$ that partition
$\\{1,\ldots,n\\}$, $V$ is a $\chi$-interval, and $s_{\chi}(1),s_{\chi}(n)\in
W$. Let

$p=\max_{\preceq_{\chi}}\left\\{k\in
W\,\left|\,k\prec_{\chi}\min_{\preceq_{\chi}}V\right.\right\\}\qquad\text{and}\qquad
q=\min_{\preceq_{\chi}}\left\\{k\in
W\,\left|\,\max_{\preceq_{\chi}}V\prec_{\chi}k\right.\right\\}.$

Then, we have that

$\displaystyle\Phi_{\pi}(Z_{1},\ldots,Z_{n})$
$\displaystyle=\begin{cases}\Phi_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{p-1},Z_{p}L_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})},Z_{p+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(p)=\ell,\\\
\Phi_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{p-1},R_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})}Z_{p},Z_{p+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(p)=r,\\\ \end{cases}$
$\displaystyle=\begin{cases}\Phi_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{q-1},L_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})}Z_{q},Z_{q+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(q)=\ell,\\\
\Phi_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{q-1},Z_{q}R_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})},Z_{q+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(q)=r.\\\ \end{cases}$

Given a $B$-$B$-non-commutative probability space $(A,E,\varepsilon)$, the
moment and cumulant functions are well-defined bi-multiplicative functions.

######  Definition 2.11.

Let $(A,E,\varepsilon)$ be a $B$-$B$-non-commutative probability space.

  1. (i)

The operator-valued bi-free moment function

$E:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in\\{\ell,r\\}^{n}}\operatorname{BNC}(\chi)\times
A_{\chi(1)}\times\ldots\times A_{\chi(n)}\to B$

is the bi-multiplicative function (see [CNS2015-1]*Theorem 5.1.4) that
satisfies

$E_{1_{\chi}}(Z_{1},Z_{2},\ldots,Z_{n})=E(Z_{1}Z_{2}\cdots Z_{n}),$

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

  2. (ii)

The operator-valued bi-free cumulant function

$\kappa^{B}:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in\\{\ell,r\\}^{n}}\operatorname{BNC}(\chi)\times
A_{\chi(1)}\times\ldots\times A_{\chi(n)}\to B$

is the bi-multiplicative function (see [CNS2015-1]*Corollary 6.2.2) defined by

$\kappa^{B}_{\pi}(Z_{1},\ldots,Z_{n})=\sum_{\begin{subarray}{c}\sigma\in\operatorname{BNC}(\chi)\\\
\sigma\leq\pi\\\
\end{subarray}}E_{\sigma}(Z_{1},\ldots,Z_{n})\mu_{\operatorname{BNC}}(\sigma,\pi),$

for each $n\in\mathbb{N}$, $\chi\in\\{\ell,r\\}^{n}$,
$\pi\in\operatorname{BNC}(\chi)$, and $Z_{k}\in A_{\chi(k)}$. In the special
case when $\pi=1_{\chi},$ the map $\kappa^{B}_{1_{\chi}}$ is simply denoted by
$\kappa^{B}_{\chi}$. An instance of Möbius inversion yields that the equality

$E_{\sigma}(Z_{1},\ldots,Z_{n})=\sum_{\begin{subarray}{c}\pi\in\operatorname{BNC}(\chi)\\\
\pi\leq\sigma\\\ \end{subarray}}\kappa^{B}_{\pi}(Z_{1},\ldots,Z_{n})$

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

The condition of bi-freeness with amalgamation over $B$ for a family of pairs
of $B$-faces is equivalent to the vanishing of their mixed operator-valued bi-
free cumulants, as the following result indicates.

######  Theorem 2.12 ([CNS2015-1]*Theorem 8.1.1).

Let $(A,E,\varepsilon)$ be a $B$-$B$-non-commutative probability space and let
${\\{(C_{k},D_{k})\\}}_{k\in K}$ be a family of pairs of $B$-algebras in $A$.
The following are equivalent:

  1. (i)

the family ${\\{(C_{k},D_{k})\\}}_{k\in K}$ is bi-free with amalgamation over
$B$,

  2. (ii)

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

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

we have that

$\kappa^{B}_{\chi}(Z_{1},\ldots,Z_{n})=0.$

##  3\. Analytical B-B-Non-Commutative Probability Spaces

The notion of a $B$-$B$-non-commutative probability space is purely an
algebraic construction. In order perform the more analytical computations
necessary in this paper, additional structure is needed. These structures are
analogous to those observed in Example 2.6 and will be seen to be the correct
enhancement of a $B$-$B$-non-commutative probability space to perform
functional analysis.

Given a unital $*$-algebra $A$, by a state $\tau:A\to\mathbb{C}$ we will
always mean a unital, linear functional with the property that
$\tau(a^{*}a)\geq 0$ for all $a\in A$. If $N_{\tau}=\\{a\in
A:\tau(a^{*}a)=0\\}$, then $L_{2}(A,\tau)$ will denote the Hilbert space
completion of the quotient space $A/N_{\tau}$ with respect to the inner
product induced by $\tau$ given by

$\left\langle
a_{1}+N_{\tau},a_{2}+N_{\tau}\right\rangle=\tau(a_{2}^{*}a_{1}),$

for all $a_{1},a_{2}\in A$ and $\left\|\,\,\cdot\,\,\right\|_{\tau}$ will
denote the Hilbert space norm on $L_{2}(A,\tau)$.

######  Definition 3.1.

Given a unital $*$-algebra $B$, an analytical $B$-$B$-non-commutative
probability space consists of a tuple $(A,E,\varepsilon,\tau)$ such that

  1. (i)

$(A,E,\varepsilon)$ is a $B$-$B$-non-commutative probability space,

  2. (ii)

$\tau:A\to\mathbb{C}$ is a state that is compatible with $E$; that is,

$\tau(a)=\tau\left(L_{E(a)}\right)=\tau\left(R_{E(a)}\right)$

for all $a\in A$,

  3. (iii)

the canonical state $\tau_{B}:B\to\mathbb{C}$ defined by
$\tau_{B}(b)=\tau(L_{b})$ for all $b\in B$ is tracial,

  4. (iv)

left multiplication of $A$ on $A/N_{\tau}$ are bounded linear operators and
thus extend to bounded linear operators on $L_{2}(A,\tau)$, and

  5. (v)

$E$ is completely positive when restricted to $A_{\ell}$ and when restricted
to $A_{r}$.

######  Remark 3.2.

Given an analytical $B$-$B$-non-commutative probability space
$(A,E,\varepsilon,\tau)$, note the following.

  1. (i)

The that fact that $\tau_{B}$ is a state immediately follows from the fact
that $\tau$ is a state and $\varepsilon$ is a $*$-homomorphism. Specifically,
for positivity, notice for all $b\in B$ that

$\tau_{B}(b^{*}b)=\tau(L_{b^{*}b})=\tau((L_{b})^{*}L_{b})\geq 0.$

  2. (ii)

Note for all $b\in B$ that

${\left\|b+N_{\tau_{B}}\right\|}_{\tau_{B}}^{2}=\tau_{B}(b^{*}b)=\tau(L_{b^{*}b})={\left\|L_{b}+N_{\tau}\right\|}_{\tau}^{2}.$

Hence the map from $B/N_{\tau_{B}}$ to $L_{2}(A,\tau)$ defined by

$b+N_{\tau_{B}}\mapsto L_{b}+N_{\tau}$

for all $b\in B$ is a well-defined, linear isometry. Therefore, a standard
density argument yields that

$L_{2}(B,\tau_{B})\cong{\overline{\\{L_{b}+N_{\tau}\,\mid\,b\in
B\\}}}^{{\left\|\cdot\right\|}_{\tau}}\subseteq L_{2}(A,\tau).$

Henceforth, we shall only be making reference to the space $L_{2}(B,\tau_{B})$
via this identification.

  3. (iii)

The state $\tau$ naturally extends to a linear functional on $L_{2}(A,\tau)$
by defining

$\tau(\xi)={\left\langle\xi,1_{A}+N_{\tau}\right\rangle}_{L_{2}(A,\tau)}$

for all $\xi\in L_{2}(A,\tau)$. Similarly, the scalar
$\tau_{B}(\zeta)=\tau(\zeta)$ is well-defined for any $\zeta\in
L_{2}(B,\tau_{B})$.

  4. (iv)

As left multiplication by $A$ on $A/N_{\tau}$ is bounded, we immediately
extend the left multiplication map to obtain a unital $*$-homomorphism from
$A$ into $\mathcal{B}(L_{2}(A))$. Thus $a\xi$ is a well-defined element of
$L_{2}(A,\tau)$ for all $a\in A$ and $\xi\in L_{2}(A,\tau)$.

  5. (v)

The requirement of the left multiplication inducing bounded operators is
immediate in the case when $A$ is a $\mathrm{C}^{*}$-algebra, however it also
holds in more general situations. For instance, when $A$ is a unital $*$\-
generated its partial isometries, the left multiplication map is automatically
bounded (see [NS2006]*Exercise 7.22).

  6. (vi)

Since the state $\tau_{B}$ is assumed to be tracial, right multiplication of
$B$ on $B/N_{\tau_{B}}$ is also bounded. Thus, for any $b_{1},b_{2}\in B$ and
$\zeta\in L_{2}(B,\tau_{B})$, we have that $b_{1}\zeta b_{2}$ is a well-
defined element of $L_{2}(B,\tau_{B})$ and, in $L_{2}(A,\tau)$,
$L_{b_{1}}R_{b_{2}}\zeta=b_{1}\zeta b_{2}$. Furthermore, note that left and
right multiplication of $B$ on $L_{2}(B,\tau_{B})$ are commuting
$*$-homomorphisms.

  7. (vii)

For all $a\in A$ and $b\in B$, we automatically have
$\tau(aL_{b})=\tau(aR_{b})$, as $\tau$ is compatible with $E$. Indeed

$\tau(aL_{b})=\tau(L_{E(aL_{b})})=\tau(L_{E(aR_{b})})=\tau(aR_{b}),$

as desired. Hence $L_{b}+N_{\tau}=R_{b}+N_{\tau}$ for all $b\in B$.

In some cases, property (v) of Definition 3.1 is redundant.

######  Lemma 3.3.

Let $(A,E,\varepsilon,\tau)$ satisfy assumptions (i), (ii), (iii), and (iv) of
Definition 3.1. If $B$ is a C∗-algebra and $\tau_{B}$ is faithful, then
property (v) of Definition 3.1 holds.

###### Proof.

To see that $E$ is completely positive on $A_{\ell}$, let $d\in\mathbb{N}$ and
$A=[a_{i,j}]\in M_{d}(A_{\ell})$. To verify that $E_{d}(A^{*}A)\geq 0$ in $B$,
as $B$ is a C∗-algebra and $\tau_{B}$ is faithful, it suffices to show for all
$h=(b_{1},\ldots,b_{d})\in B^{d}$ that

$\langle E_{d}(A^{*}A)h,h\rangle_{L_{2}(B,\tau_{B})^{\oplus d}}\geq 0.$

Note that

$\displaystyle\left\langle
E_{d}(A^{*}A)h,h\right\rangle_{L_{2}(B,\tau_{B})^{\oplus d}}$
$\displaystyle=\sum^{d}_{i,j,k=1}\tau_{B}\left(b_{i}^{*}E(a^{*}_{k,i}a_{k,j})b_{j}\right)$
$\displaystyle=\sum^{d}_{i,j,k=1}\tau_{B}\left(E(R_{b_{j}}L_{b_{i}^{*}}a^{*}_{k,i}a_{k,j})\right)$
$\displaystyle=\sum^{d}_{i,j,k=1}\tau_{B}\left(E(L_{b_{i}^{*}}a^{*}_{k,i}a_{k,j}R_{b_{j}})\right)$
$\displaystyle=\sum^{d}_{i,j,k=1}\tau_{B}\left(E(L_{b_{i}}^{*}a^{*}_{k,i}a_{k,j}L_{b_{j}})\right)$
$\displaystyle=\sum^{d}_{i,j,k=1}\tau\left(L_{b_{i}}^{*}a^{*}_{k,i}a_{k,j}L_{b_{j}}\right)$
$\displaystyle=\sum^{d}_{k=1}\tau\left(c_{k}^{*}c_{k}\right)$

where $c_{k}=\sum^{d}_{j=1}a_{k,j}L_{b_{j}}$. Hence, as $\tau$ is positive and
the computation for $A_{r}$ is similar, the result follows. ∎

At this point, let us revisit Examples 2.5 and 2.6 to provide the canonical
examples of analytical $B$-$B$-non-commutative probability spaces.

######  Example 3.4.

Let $\mathcal{A}$ and $B$ be unital $\mathrm{C}^{*}$-algebras and let
$\varphi:\mathcal{A}\to\mathbb{C}$ be a state. Recall from Example 2.5 that
$(A,E,\varepsilon)$ is a $B$-$B$-non-commutative probability space where
$A=\mathcal{A}\otimes B\otimes B^{{\text{op}}}$, $\varepsilon:B\otimes
B^{{\text{op}}}\to A$ is the natural embedding, and $E:A\to B$ is defined by

$E(Z\otimes b_{1}\otimes b_{2})=\varphi(Z)b_{1}b_{2},$

for all $Z\in\mathcal{A}$ and $b_{1},b_{2}\in B$.

Let $\tau_{B}:B\to\mathbb{C}$ be any tracial state. Extend $\tau_{B}$ to a
linear map $\tau:A\to\mathbb{C}$ by defining

$\tau(Z\otimes b_{1}\otimes b_{2})=\tau_{B}(E(Z\otimes b_{1}\otimes
b_{2}))=\varphi(Z)\tau_{B}(b_{1}b_{2}),$

for all $Z\otimes b_{1}\otimes b_{2}\in A$. We claim that
$(A,E,\varepsilon,\tau)$ is an analytical $B$-$B$-non-commutative probability
space. To see this, it suffices to prove that $\tau$ is a state that is
compatible with $E$, since $\mathcal{A}$ and $B$ being unital C∗-algebras
automatically implies that left multiplication will be bounded on
$L_{2}(A,\tau)$, and Lemma 3.3 implies that $E$ is completely positive when
restricted to $A_{\ell}$ or $A_{r}$ (or one may simply use the fact that
states are completely positive).

Clearly $\tau$ is a unital, linear map that is compatible with $E$. To see
that $\tau$ is positive, let ${(Z_{i})}_{i=1}^{n}\subseteq\mathcal{A}$,
${(b_{k})}_{k=1}^{n},{(c_{k})}_{k=1}^{n}\subseteq B$, and

$a=\sum_{k=1}^{n}Z_{k}\otimes b_{k}\otimes c_{k}\in A.$

To see that $\tau(a^{*}a)\geq 0$, note that

$\displaystyle\tau(a^{*}a)$
$\displaystyle=\sum_{i,j=1}^{n}\tau(Z_{i}^{*}Z_{j}\otimes
b_{i}^{*}b_{j}\otimes c_{j}c_{i}^{*})$
$\displaystyle=\sum_{i,j=1}^{n}\varphi(Z_{i}^{*}Z_{j})\tau_{B}(b_{i}^{*}b_{j}c_{j}c_{i}^{*})$
$\displaystyle=\sum_{i,j=1}^{n}\varphi(Z_{i}^{*}Z_{j})\tau_{B}(c_{i}^{*}b_{i}^{*}b_{j}c_{j})$
$\displaystyle=\sum_{i,j=1}^{n}\varphi(Z_{i}^{*}Z_{j})\tau_{B}({(b_{i}c_{i})}^{*}(b_{j}c_{j})),$

with the third equality being due to the fact that $\tau_{B}$ is tracial.
Observe that the matrices

${[Z_{i}^{*}Z_{j}]}\qquad\text{and}\qquad{[{(b_{i}c_{i})}^{*}(b_{j}c_{j})]}$

are positive in $M_{n}(\mathcal{A})$ and $M_{n}(B)$ respectively. Therefore,
as states on C∗-algebras are completely positive, this implies that the
matrices

${[\varphi(Z_{i}^{*}Z_{j})]}\qquad\text{and}\qquad{[{\tau_{B}((b_{i}c_{i})}^{*}(b_{j}c_{j}))]}$

are positive in $M_{n}(\mathbb{C})$. Consequently

${[\varphi(Z_{i}^{*}Z_{j}){\tau_{B}((b_{i}c_{i})}^{*}(b_{j}c_{j}))]}$

is also positive being the Schur product of positive matrices (see, for
instance, [NS2006]*Lemma 6.11). Therefore, as the sum of all entries of a
positive matrix equals a positive scalar, we obtain that $\tau(a^{*}a)\geq 0$.
Hence $(A,E,\varepsilon,\tau)$ is an analytical $B$-$B$-non-commutative
probability space.

######  Remark 3.5.

Note Example 3.4 demonstrates $E$ need not be a positive map on $A$ since the
product of two positive matrices need not be positive. Thus, even if
$\tau_{B}:B\to\mathbb{C}$ is defined to be a state, $\tau_{B}\circ E$ may not
be for an arbitrary $A$.

######  Example 3.6.

For a finite von Neumann algebra $\mathfrak{M}$ with a unital von Neumann
subalgebra $B$ and tracial state $\tau$, let $(A,E,\varepsilon)$ be the
$B$-$B$-non-commutative probability space as in Example 2.6. Note that $\tau$
extends to a unital linear map $\tau_{A}:A\to\mathbb{C}$ defined by

$\tau_{A}(T)=\langle
T1_{\mathfrak{M}},1_{\mathfrak{M}}\rangle_{L_{2}(\mathfrak{M},\tau)}$

for all $T\in A$. Clearly $\tau_{A}$ is a state as
$A\subseteq\mathcal{B}(L_{2}(\mathfrak{M},\tau))$ and $\tau_{A}$ is a vector
state. Furthermore, notice that

$\tau_{A}(T)=\langle
P(T1_{\mathfrak{M}}),1_{\mathfrak{M}}\rangle_{L_{2}(\mathfrak{M},\tau)}=\langle
L_{E(T)}1_{\mathfrak{M}},1_{\mathfrak{M}}\rangle_{L_{2}(\mathfrak{M},\tau)}=\tau_{A}(L_{E(T)})$

for all $T\in A$ and $\tau_{A}(T)=\tau_{A}(R_{E(T)})$ by a similar
computation. Finally as

$\tau_{A}(L_{b})=\langle
b,1_{\mathfrak{M}}\rangle_{L_{2}(\mathfrak{M},\tau)}=\tau(b)$

for all $b\in B$, we see that $\tau_{B}=\tau_{A}\circ E$ is tracial on $B$ as
$\tau$ is tracial. Again, we automatically have that left multiplication will
be bounded on $L_{2}(A,\tau)$ and that $E$ is completely positive when
restricted (as they are the conditional expectation of a copy of
$\mathfrak{M}$ onto $B$). Hence $(A,E,\varepsilon,\tau_{A})$ is an analytical
$B$-$B$-non-commutative probability space.

As motivated by Example 3.6, it is natural in an analytical $B$-$B$-non-
commutative probability space to extend the expectation $E:A\to B$ to a map
from $L_{2}(A,\tau)$ to $L_{2}(B,\tau_{B})$ via orthogonal projection. From
this point onwards, for $a\in A$ we will often denote the coset $a+N_{\tau}$
simply by $a$ and, for $b\in B$ we will often denote the coset
$b+N_{\tau_{b}}$ by $\widehat{b}$. Note that if $\tau_{B}$ is faithful, then
the map $b\mapsto\widehat{b}$ is a bijection.

######  Proposition 3.7.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space. If $\widetilde{E}:L_{2}(A,\tau)\to L_{2}(B,\tau_{B})$
denotes the orthogonal projection, then

$\widetilde{E}(a)=\widehat{E(a)}$

for all $a\in A$. In particular, when $\tau_{B}$ is faithful, $\widetilde{E}$
extends $E$.

###### Proof.

Notice for all $a\in A$ and $b\in B$ that

$\displaystyle{\left\langle
a-\widehat{E(a)},L_{b}\right\rangle}_{L_{2}(B,\tau_{B})}$
$\displaystyle=\langle
L_{b^{*}}(a-L_{E(a)})1_{A},1_{A}\rangle_{L_{2}(A,\tau)}$
$\displaystyle=\tau\left(L_{b^{*}}(a-L_{E(a)})\right)$
$\displaystyle=\tau(L_{b^{*}}a)-\tau\left(L_{b*}L_{E(a)}\right)$
$\displaystyle=\tau\left(L_{E(L_{b^{*}}a)}\right)-\tau\left(L_{b^{*}E(a)}\right)=0.$

Since $b$ was arbitrary, the element $a-\widehat{E(a)}$ is orthogonal to
$L_{2}(B,\tau_{B})$ and hence $\widetilde{E}(a)=\widehat{E(a)}$. ∎

######  Remark 3.8.

Notice in Proposition 3.7 that if $B$ is finite-dimensional and the trace
$\tau_{B}:B\to\mathbb{C}$ is faithful, then $L_{2}(B,\tau_{B})\cong B$, so
$E:A\to B$ extends to a map from $L_{2}(A,\tau)$ into $B$.

Of course $\widetilde{E}$ inherits many properties that $E$ is required to
have.

######  Proposition 3.9.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and let $\widetilde{E}:L_{2}(A,\tau)\to L_{2}(B,\tau_{B})$
denote the orthogonal projection. For $a\in A$, $b,b_{1},b_{2}\in B$,
$\xi,\xi_{1},\xi_{2}\in L_{2}(A,\tau)$, and $\zeta\in L_{2}(B,\tau_{B})$, the
following hold:

  1. (i)

$\tau(\xi)=\tau_{B}\left(\widetilde{E}(\xi)\right)$,

  2. (ii)

$\widetilde{E}(aL_{b})=\widetilde{E}(aR_{b})$,

  3. (iii)

$\widetilde{E}(L_{b_{1}}R_{b_{2}}\xi)=b_{1}\widetilde{E}(\xi)b_{2}$,

  4. (iv)

if $a\in A_{\ell}$, then $\widetilde{E}(a\zeta)=E(a)\zeta$,

  5. (v)

if $a\in A_{r}$, then $\widetilde{E}(a\zeta)=\zeta E(a)$, and

  6. (vi)

if $\tau(L_{b}\xi_{1})=\tau(L_{b}\xi_{2})$ for all $b\in B$, then
$\widetilde{E}(\xi_{1})=\widetilde{E}(\xi_{2})$.

  7. (vii)

if $\tau(R_{b}\xi_{1})=\tau(R_{b}\xi_{2})$ for all $b\in B$, then
$\widetilde{E}(\xi_{1})=\widetilde{E}(\xi_{2})$.

###### Proof.

For (i), since $L_{1_{B}}=1_{A}$ as $\varepsilon$ is unital, note that

$\tau_{B}\left(\widetilde{E}(\xi)\right)=\left\langle\widetilde{E}(\xi),1_{B}\right\rangle_{L_{2}(B,\tau_{B})}=\left\langle\xi,\widetilde{E}(1_{B})\right\rangle_{L_{2}(A,\tau)}=\left\langle\xi,1_{A}\right\rangle_{L_{2}(A,\tau)}=\tau(\xi)$

as desired.

For (ii), note for all $b^{\prime}\in B$ that

$\displaystyle\left\langle\widetilde{E}(aL_{b}),\widehat{b^{\prime}}\right\rangle_{L_{2}(B,\tau_{B})}$
$\displaystyle=\left\langle
aL_{b},L_{b^{\prime}}\right\rangle_{L_{2}(A,\tau)}=\tau\left(L_{(b^{\prime})^{*}}aL_{b}\right)=\tau\left(L_{(b^{\prime})^{*}}aR_{b}\right)=\left\langle\widetilde{E}(aR_{b}),\widehat{b^{\prime}}\right\rangle_{L_{2}(B,\tau_{B})}.$

Hence $\widetilde{E}(aL_{b})=\widetilde{E}(aR_{b})$.

For (iii), let $(a_{n})_{n\geq 1}$ be a sequence of elements of $A$ that
converge to $\xi$ in $L_{2}(A,\tau)$. Since left multiplication in
$L_{2}(A,\tau)$ by elements of $A$ are bounded and thus continuous, and since
left and right multiplication in $L_{2}(B,\tau_{B})$ by elements of $B$ are
bounded and thus continuous, we obtain that

$\displaystyle\widetilde{E}(L_{b_{1}}R_{b_{2}}\xi)=\lim_{n\to\infty}E(L_{b_{1}}R_{b_{2}}a_{n})+N_{\tau_{B}}=\lim_{n\to\infty}b_{1}E(a_{n})b_{2}+N_{\tau_{B}}=b_{1}\widetilde{E}(\xi)b_{2}$

as desired.

For (iv) and (v), let $(c_{n})_{n\geq 1}$ be a sequence of elements of $B$
that converge to $\zeta$ in $L_{2}(B,\tau_{B})$. Thus, by the inclusion of
$L_{2}(B,\tau_{B})$ into $L_{2}(A,\tau)$, we have that $(L_{c_{n}})_{n\geq 1}$
is a sequence of elements of $L_{2}(A,\tau)$ that converge to $\zeta$ in
$L_{2}(A,\tau)$. Thus, if $a\in A_{\ell}$, then

$\displaystyle\widetilde{E}(a\zeta)$
$\displaystyle=\lim_{n\to\infty}E(aL_{c_{n}})+N_{\tau_{B}}$
$\displaystyle=\lim_{n\to\infty}E(aR_{c_{n}})+N_{\tau_{B}}$
$\displaystyle=\lim_{n\to\infty}E(R_{c_{n}}a)+N_{\tau_{B}}$
$\displaystyle=\lim_{n\to\infty}E(a)c_{n}+N_{\tau_{B}}=E(a)\zeta$

thereby proving (iv). Note (v) is similar using $(R_{c_{n}})_{n\geq 1}$ in
place of $(L_{c_{n}})_{n\geq 1}$.

As (vi) and (vii) are similar, we prove (vii). Note by (iii) and the fact that
$\tau_{B}$ is tracial that

$\displaystyle\left\langle\widetilde{E}(\xi_{1})-\widetilde{E}(\xi_{2}),\widehat{b}\right\rangle_{L_{2}(B,\tau_{B})}$
$\displaystyle=\left\langle\widetilde{E}(\xi_{1})b^{*},\widetilde{E}(1_{A})\right\rangle_{L_{2}(B,\tau_{B})}-\left\langle\widetilde{E}(\xi_{2})b^{*},\widetilde{E}(1_{A})\right\rangle_{L_{2}(B,\tau_{B})}$
$\displaystyle=\left\langle\widetilde{E}(R_{b^{*}}\xi_{1}),\widetilde{E}(1_{A})\right\rangle_{L_{2}(B,\tau_{B})}-\left\langle\widetilde{E}(R_{b^{*}}\xi_{2}),\widetilde{E}(1_{A})\right\rangle_{L_{2}(B,\tau_{B})}$
$\displaystyle=\left\langle
R_{b^{*}}\xi_{1},1_{A}\right\rangle_{L_{2}(A,\tau)}-\left\langle
R_{b^{*}}\xi_{2},1_{A}\right\rangle_{L_{2}(A,\tau)}$
$\displaystyle=\tau(R_{b^{*}}\xi_{1})-\tau(R_{b^{*}}\xi_{2})=0$

As the above holds for all $b\in B$, (vii) follows. ∎

##  4\. Analytical Bi-Multiplicative Functions

In this section, we extend the notion of bi-multiplicative functions on
analytical $B$-$B$-non-commutative probability spaces in order to permit the
last entry to be an element of $L_{2}(A,\tau)$. This is possible as the last
entry can be treated as a left or right operator as [S2016-2] shows, or can be
treated as a mixture of left and right operators as [CS2020] shows. Extending
the operator-valued bi-free cumulant function to permit the last entry to be
an element of $L_{2}(A,\tau)$ is necessary in order to permit the simple
development of conjugate variable systems in the next section.

We advise the reader that familiarity with specifics of bi-multiplicative
functions, the construction of the operator-valued bi-free moment function,
and the construction of the operator-valued bi-free cumulant function from
[CNS2015-1] would be of great aid in comprehension of this section. As the
proofs are nearly identical, to avoid clutter we will focus on that which is
different and why the results of [CNS2015-1] extend.

######  Definition 4.1.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space and let $\Phi$ be a bi-multiplicative function on
$(A,E,\varepsilon)$. A function

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

is said to be _analytical extension of $\Phi$_ if $\widetilde{\Phi}_{\pi}$ is
$\mathbb{C}$-multilinear function that does not change values if the last
entry of $\chi$ is changed from an $\ell$ to an $r$ and satisfies the
following three properties: For all $n\in\mathbb{N}$,
$\chi\in\\{\ell,r\\}^{n}$, $\pi\in\operatorname{BNC}(\chi)$, $\xi\in
L_{2}(A,\tau)$, $\zeta\in L_{2}(B,\tau_{B})$, $b\in B$, and $Z_{k}\in
A_{\chi(k)}$:

  1. (i)

If $\chi(k)=\ell$ for all $k\in\\{1,2,\ldots,n\\}$ then

$\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n}\zeta)=\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n})\zeta,$

and if $\chi(k)=r$ for all $k\in\\{1,2,\ldots,n\\}$, then

$\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},Z_{n}\zeta)=\zeta\Phi_{1_{\chi}}(Z_{1},\ldots,Z_{n}).$

In particular, by setting $\zeta=1_{B}=1_{A}$, we see $\widetilde{\Phi}$ does
extend $\Phi$.

  2. (ii)

Let $p\in\\{1,\ldots,n\\}$ and let

$q=\max_{\leq}\left\\{k\in\\{1,\ldots,n\\}\,\mid\,\chi(k)=\chi(p),k<p\right\\}.$

If $\chi(p)=\ell$, then

$\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{p-1},L_{b}Z_{p},Z_{p+1},\ldots,Z_{n-1},\xi)=\begin{cases}\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{q-1},Z_{q}L_{b},Z_{q+1},\ldots,Z_{n-1},\xi)&\text{if
}q\neq-\infty,\\\
b\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},\xi)&\text{if }q=-\infty\\\
\end{cases},$

and if $\chi(p)=r,$ then

$\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{p-1},R_{b}Z_{p},Z_{p+1},\ldots,Z_{n-1},\xi)=\begin{cases}\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{q-1},Z_{q}R_{b},Z_{q+1},\ldots,Z_{n-1},\xi)&\text{if
}q\neq-\infty,\\\
\widetilde{\Phi}_{1_{\chi}}(Z_{1},\ldots,Z_{n-1},\xi)b&\text{if }q=-\infty.\\\
\end{cases}$

  3. (iii)

Suppose $V_{1},\ldots,V_{m}$ are unions of blocks of $\pi$ that partition
$\\{1,\ldots,n\\}$, with each being a $\chi$-interval. Moreover, assume that
the sets $V_{1},\ldots,V_{m}$ are ordered by $\preceq_{\chi}$ (i.e.
$(\min_{\preceq_{\chi}}V_{k})\prec_{\chi}(\min_{\preceq_{\chi}}V_{k+1})$). Let
$q\in\\{1,\ldots,m\\}$ be such that $n\in V_{q}$ and for each $k\neq q$ let

$b_{k}=\Phi_{\pi|_{V_{k}}}\left({(Z_{1},\ldots,Z_{n-1},\xi)|}_{V_{k}}\right).$

Then $b_{k}\in B$ for $k\neq q$ and

$\widetilde{\Phi}_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)=b_{1}b_{2}\cdots
b_{q-1}\widetilde{\Phi}_{\pi|_{V_{i}}}({(Z_{1},\ldots,Z_{n-1},\xi)|}_{V_{q}})b_{q+1}\cdots
b_{m}.$

  4. (iv)

Suppose that $V$ and $W$ are unions of blocks of $\pi$ that partition
$\\{1,\ldots,n\\}$ such that $V$ is a $\chi$-interval and
$s_{\chi}(1),s_{\chi}(n)\in W$. Let

$p=\max_{\preceq_{\chi}}\left\\{k\in
W\,\left|\,k\preceq_{\chi}\min_{\preceq_{\chi}}V\right.\right\\}\qquad\text{and}\qquad
q=\min_{\preceq_{\chi}}\left\\{k\in
W\,\left|\,\max_{\preceq_{\chi}}V\preceq_{\chi}k\right.\right\\}.$

Then one of the following four cases hold:

    1. a)

If $n\in V$ and $k=\max_{\leq}W$, then

$\widetilde{\Phi}_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)=\widetilde{\Phi}_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{k-1},Z_{k}\widetilde{\Phi}_{\pi|_{V}}{({(Z_{1},\ldots,Z_{n-1},\xi)|}_{V})},\ldots,Z_{n-1},\xi)|}_{W}\right).$

    2. b)

If $n\in W$ then

$\displaystyle\widetilde{\Phi}_{\pi}(Z_{1},\ldots,Z_{n})$
$\displaystyle=\begin{cases}\widetilde{\Phi}_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{p-1},Z_{p}L_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})},Z_{p+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(p)=\ell,\\\
\widetilde{\Phi}_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{p-1},R_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})}Z_{p},Z_{p+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(p)=r,\\\ \end{cases}$
$\displaystyle=\begin{cases}\widetilde{\Phi}_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{q-1},L_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})}Z_{q},Z_{q+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(q)=\ell,\\\
\widetilde{\Phi}_{\pi|_{W}}\left({(Z_{1},\ldots,Z_{q-1},Z_{q}R_{\Phi_{\pi|_{V}}({(Z_{1},\ldots,Z_{n})|}_{V})},Z_{q+1},\ldots,Z_{n})|}_{W}\right)&\text{if
}\chi(q)=r.\\\ \end{cases}$

(Recall we can set $\chi(n)=\ell$ or $\chi(n)=r$.)

######  Remark 4.2.

Note that the pair of a bi-multiplicative function and its extension are very
reminiscent of the two expectation extensions of bi-multiplicative functions
used for operator-valued conditional bi-free independence from [GS2018]. The
main difference is that the notion in [GS2018] looks at interior versus
exterior blocks of the partition whereas Definition 4.1 looks at the blocks
containing the last entry. This is due to the fact that the $L_{2}(A,\tau)$
element is always the last entry and must be treated differently being a
generalization of a mixture of left and right operators.

It is worth pointing out that treating the last entry as an element of
$L_{2}(A,\tau)$ is no issue. In particular, the properties in Definition 4.1
are well-defined. Indeed, properties (i) and (ii) of Definition 4.1 are
clearly well-defined and properties (iii) and (iv) in Definition 4.1 are well-
defined as all terms where $\Phi$ is used over $\widetilde{\Phi}$ never
involve an element of $L_{2}(A,\tau)$ and as elements from $B$ have left and
right actions on $L_{2}(B,\tau_{B})$.

######  Remark 4.3.

Note that property (i) of Definition 4.1 is clearly the correct generalization
of property (i) from Definition 2.10, as an element of $L_{2}(B,\tau_{B})$ is
playing the role of $L_{b}$ and $R_{b}$ in this generalization and thus should
be able to escape these expressions if only left operators or right operators
are present. The absence of the full property (i) from Definition 2.10 causes
no issues when attempting to reduce or rearrange the value of
$\widetilde{\Phi}_{\pi}$ to an expression involving only
$\widetilde{\Phi}_{1_{\chi}}$’s, as the last entry of any sequence input into
$\widetilde{\Phi}$ is always in $L_{2}(A,\tau)$, is reduced to an element of
$L_{2}(B,\tau)$, and an element of $A_{\ell}$ or $A_{r}$ then acts on it via
the left action of $A$ on $L_{2}(A,\tau)$. Thus there is never any need to
move the $L_{2}(A,\tau)$ entry to another position.

If property (i) is ever used, we note that if $\zeta\in L_{2}(B,\tau_{B})$ is
viewed as an element of $L_{2}(A,\tau)$, then $L_{b}\zeta$ is simply the
element $b\zeta\in L_{2}(B,\tau_{B})$ and $R_{b}\zeta$ is simply the element
$\zeta b\in L_{2}(B,\tau_{B})$. Thus, using (i) does not pose problems when
trying to “move around $L_{b}$ and $R_{b}$ elements” in proofs when trying to
show the equivalence of any reductions as the following example demonstrates.

######  Example 4.4.

Let $\chi\in\\{\ell,r\\}^{8}$ be such that $\chi^{-1}(\\{\ell\\})=\\{5,6\\}$,
let $\xi\in L_{2}(A,\tau)$, let $Z_{k}\in A_{\chi(k)}$, and let
$\pi\in\operatorname{BNC}(\chi)$ be the partition

$\pi=\\{\\{1,2\\},\\{3,5\\},\\{4,7\\},\\{6,8\\}\\}.$

Note the bi-non-crossing diagram of $\pi$ can be represented as the following
(with the convention now that the last entry is at the bottom instead of on
its respective side):

When reducing $\widetilde{\Phi}_{\pi}(Z_{1},Z_{2},\ldots,Z_{7},\xi)$, we can
clearly use property (iii) of Definition 4.1 first to obtain with
$U=\\{3,4,\ldots,8\\}$ that

$\widetilde{\Phi}_{\pi}(Z_{1},Z_{2},\ldots,Z_{7},\xi)=\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)\Phi_{1_{(r,r)}}(Z_{1},Z_{2}).$

To reduce the expression fully, we have to simply reduce
$\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$ using property
(iv) of Definition 4.1 of which there are three ways to do so.

The first way to reduce is to use $V=\\{4,6,7,8\\}$ and $W=\\{3,5\\}$. By
applying property (iv) of Definition 4.1 we obtain that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{\pi|_{W}}\left(Z_{3},Z_{5}\widetilde{\Phi}_{\pi|_{V}}(Z_{4},Z_{6},Z_{7},\xi)\right).$

Finally, by applying property (iii) of Definition 4.1 to
$\widetilde{\Phi}_{\pi|_{V}}(Z_{4},Z_{6},Z_{7},\xi)$ we obtain that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}\left(\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\Phi_{1_{(r,r)}}(Z_{4},Z_{7})\right)\right)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right).$

The second way to reduce is to use $V=\\{6,8\\}$ and $W=\\{3,4,5,7\\}$. By
applying property (iv) of Definition 4.1 we obtain that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{\pi|_{W}}\left(Z_{3},Z_{4},Z_{5},Z_{7}\widetilde{\Phi}_{\pi|_{V}}(Z_{6},\xi)\right).$

By applying property (iv) of Definition 4.1 again as $\\{4,7\\}$ is now a
$\chi|_{W}$-interval), we obtain that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{1_{r,\ell}}\left(Z_{3},Z_{5}\left(\widetilde{\Phi}_{1_{(r,r)}}(Z_{4},Z_{7}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)\right).$

However, as $Z_{4},Z_{7}\in A_{r}$, we obtain by property (i) that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}\left(\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\Phi_{1_{(r,r)}}(Z_{4},Z_{7})\right)\right)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right),$

thereby agreeing with the above expression.

The third way to reduce is to use $V=\\{4,7\\}$ and $W=\\{3,5,6,8\\}$. By
applying property (iv) of Definition 4.1 we obtain that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{\pi|_{W}}\left(Z_{3},Z_{5},Z_{6},R_{\Phi_{\pi|_{V}}(Z_{4},Z_{7})}\xi\right)$
$\displaystyle=\widetilde{\Phi}_{\pi|_{W}}\left(Z_{3}R_{\Phi_{\pi|_{V}}(Z_{4},Z_{7})},Z_{5},Z_{6},\xi\right),$

by using the two expressions in property (iv). Using either expression, we
will now again property (iv) of Definition 4.1 as $\\{6,8\\}$ is now
$\chi|_{W}$-interval. For the first, we obtain that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}\widetilde{\Phi}_{1_{(\ell,\ell)}}\left(Z_{6},R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\xi\right)\right)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}\left(\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\Phi_{1_{(r,r)}}(Z_{4},Z_{7})\right)\right)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)$

where the second equality follows from applying property (ii) of Definition
4.1, as $Z_{6}\in A_{\ell}$. For the second expression, we obtain that

$\displaystyle\widetilde{\Phi}_{\pi|_{U}}(Z_{3},Z_{4},\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})},Z_{5}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}Z_{5}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right),$

where the second equality follows from applying property (ii) of Definition
4.1 as the last entry is now the $L_{2}(A,\tau)$ entry, and the third equality
holds as $Z_{5}\in A_{\ell}$ and thus commutes with $R_{b}$.

Hence Definition 4.1 is consistent in this example (and will be in all
examples due to similar computations).

Using similar reductions for arbitrary expressions, one can prove the
following.

######  Lemma 4.5.

Let $(A,E,\varepsilon,\tau)$ be an analytical $B$-$B$-non-commutative
probability space, let $\Phi$ be a bi-multiplicative function on
$(A,E,\varepsilon)$, and let $\widetilde{\Phi}$ be an analytic extension of
$\Phi$. Then, properties (i) and (ii) of Definition 4.1 hold when $1_{\chi}$
is replaced with any $\pi\in\operatorname{BNC}(\chi)$.

###### Proof.

The proof is essentially the same as the proof that properties (i) and (ii) of
Definition 2.10 hold for $\Phi$ when $1_{\chi}$ is replaced with any
$\pi\in\operatorname{BNC}(\chi)$ as in [CNS2015-1]*Proposition 4.2.5. To see
that property (i) of Definition 4.1 extends, note when using (iii) and (iv) to
reduce the expression for
$\widetilde{\Phi}_{\pi}(Z_{1},Z_{2},\ldots,Z_{n-1},Z_{n}\zeta)$ that one is
effectively using the bi-multiplicative properties of $\Phi$ and including
$\zeta$ in the appropriate spot. To see that property (ii) of Definition 4.1
extends, indices that are always adjacent in the $\chi$-ordering will remain
in the correct ordering so that when $L_{b}$ or $R_{b}$ operators are
considered, we can always move them outside the $\Phi$\- and
$\widetilde{\Phi}$-expressions on the correct side to move them to the next
operator (that is, things will always move around as they do in the free
multiplicative functions from [S1998] after reordering by the $\chi$-order).
For example, in Example 4.4, we showed that

$\widetilde{\Phi}_{\pi}(Z_{1},Z_{2},\ldots,Z_{7},\xi)=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2}).$

If $Z_{3}$ were replaced with $R_{b}Z_{3}$, we would have

$\displaystyle\widetilde{\Phi}_{\pi}(Z_{1},Z_{2},R_{b}Z_{3},Z_{4}\ldots,Z_{7},\xi)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(R_{b}Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2})$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)b\Phi_{1_{(r,r)}}(Z_{1},Z_{2})$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2}R_{b})$
$\displaystyle=\widetilde{\Phi}_{\pi}(Z_{1},Z_{2}R_{b},Z_{3},Z_{4}\ldots,Z_{7},\xi).$

If $\xi$ were replaced with $L_{b}\xi$, then clearly the $L_{b}$ can be moved
to give $Z_{6}L_{b}$ via (ii) with a $1_{\chi}$ as the expression
$\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)$ is present. If $\xi$ were
replaced with $R_{b}\xi$, then

$\displaystyle\widetilde{\Phi}_{\pi}(Z_{1},Z_{2},\ldots,Z_{7},R_{b}\xi)$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},R_{b}\xi)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2})$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\left(\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)b\right)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2})$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}R_{b}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2})$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{b\Phi_{1_{(r,r)}}(Z_{4},Z_{7})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2})$
$\displaystyle=\widetilde{\Phi}_{1_{(r,\ell)}}\left(Z_{3},Z_{5}R_{\Phi_{1_{(r,r)}}(Z_{4},Z_{7}R_{b})}\widetilde{\Phi}_{1_{(\ell,\ell)}}(Z_{6},\xi)\right)\Phi_{1_{(r,r)}}(Z_{1},Z_{2})$
$\displaystyle=\widetilde{\Phi}_{\pi}(Z_{1},\ldots,Z_{6},Z_{7}R_{b},\xi),$

as desired. Thus the result follows. ∎

### Analytical Operator-Valued Bi-Moment Function

We will now construct the analytical extension of the operator-valued bi-
moment function via recursion and the map $\widetilde{E}:L_{2}(A,\tau)\to
L_{2}(B,\tau_{B})$ from Section 3. Note that the recursive process in the
following definition is different than that from [CNS2015-1]*Definition 5.1
and [GS2018]*Definition 4.4, in order to facilitate the introduction of the
$L_{2}(A,\tau)$ element. The same recursive process could have been used in
[CNS2015-1]*Definition 5.1 and [GS2018]*Definition 4.4, as these processes are
equivalent in those settings. Note we use $\Psi$ in the following to avoid
confusion with $\widetilde{E}$ in Section 3, although $\Psi$ is a multi-entry
extension of $\widetilde{E}$.

######  Definition 4.6.

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

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

is defined recursively as follows: Let $n\in\mathbb{N}$,
$\chi\in\\{\ell,r\\}^{n}$, $\pi\in\operatorname{BNC}(\chi)$, $\xi\in
L_{2}(A,\tau)$, and $Z_{k}\in A_{\chi(k)}$.

  * •

If $\pi=1_{\chi}$, then

$\Psi_{1_{\chi}}(Z_{1},Z_{2},\ldots,Z_{n-1},\xi)=\widetilde{E}(Z_{1}Z_{2}\cdots
Z_{n-1}\xi).$

  * •

If $\pi\neq 1_{\chi}$, let $V$ be the block in $\pi$ such that $n\in V$. We
divide discussion into two cases:

    * –

Suppose that $\min_{\preceq_{\chi}}V=s_{\chi}(1)$ and
$\max_{\preceq_{\chi}}V=s_{\chi}(n)$ and let

$p=\min_{\preceq_{\chi}}\\{i\in\\{1,\ldots,n\\}\,\mid\,i\notin V\\},\quad
q=\min_{\preceq_{\chi}}\\{j\in V\,\mid\,p\prec_{\chi}j\\}\quad\text{and}\quad
m=\max_{\preceq_{\chi}}\\{i\in V\,\mid\,i\prec_{\chi}p\\}.$

Set

$W=\\{i\in\\{1,\ldots,n\\}\,\mid\,p\preceq_{\chi}i\prec_{\chi}q\\}.$

Note by construction and the fact that $\pi\in\operatorname{BNC}(\chi)$ that
$W$ is equal to a union of blocks of $\pi$ and $\chi(p)=\chi(j)$ for all $j\in
W$. Thus we define

$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)=\begin{cases}\Psi_{\pi|_{W^{\mathsf{c}}}}\left({(Z_{1},\ldots,Z_{p-1},Z_{m}L_{E_{\pi|_{W}}({(Z_{1},\ldots,Z_{n-1},\xi)|}_{W})},\ldots,Z_{n-1},\xi)|}_{W^{\mathsf{c}}}\right)&\text{if
}\chi(p)=\ell,\\\
\Psi_{\pi|_{W^{\mathsf{c}}}}\left({(Z_{1},\ldots,Z_{q-1},Z_{q}R_{E_{\pi|_{W}}({(Z_{1},\ldots,Z_{n-1},\xi)|}_{W})},\ldots,Z_{n-1},\xi)|}_{W^{\mathsf{c}}}\right)&\text{if
}\chi(p)=r.\\\ \end{cases}$

[Note as $n\notin W$, the quantity
$E_{\pi|_{W}}({(Z_{1},\ldots,Z_{n-1},\xi)|}_{W})$ is always a well-defined
element of $B$ in this case. Note in the case that $\chi(p)=\ell$ that
$m\prec_{\chi}p\prec_{\chi}n$ and thus $Z_{m}\neq\xi$, so
$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)$ is well-defined. Also, in the case when
$\chi(p)=r$ note that $n\prec_{\chi}q$ and thus $Z_{q}\neq\xi$, so
$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)$ is well-defined.]

    * –

Otherwise, set

$\widetilde{V}=\left\\{i\in\\{1,\ldots,n\\}\,\left|\,\min_{\preceq_{\chi}}V\preceq_{\chi}i\preceq_{\chi}\max_{\preceq_{\chi}}V\right.\right\\}.$

Note $\widetilde{V}$ is a proper subset of $\\{1,\ldots,n\\}$ that is a union
of blocks of $\pi$ and is such that $n\in V\subseteq\widetilde{V}$. For
$q=\max_{\leq}{\widetilde{V}^{\mathsf{c}}}$ and define

$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)=\Psi_{\pi|_{\widetilde{V}^{\mathsf{c}}}}\left({(Z_{1},\ldots,Z_{q-1},Z_{q}\Psi_{\pi|_{\widetilde{V}}}{({(Z_{1},\ldots,Z_{n-1},\xi)|}_{\widetilde{V}})},\ldots,Z_{n-1},\xi)|}_{\widetilde{V}^{\mathsf{c}}}\right).$

[Note that the quantity
$\Psi_{\pi|_{\widetilde{V}}}{({(Z_{1},\ldots,Z_{n-1},\xi)|}_{\widetilde{V}})}$
is a well-defined element of $L_{2}(B,\tau_{B})$ due to the recursive nature
of our definition. Moreover,the last element of the sequence

${(Z_{1},\ldots,Z_{q-1},Z_{q}\Psi_{\pi|_{\widetilde{V}}}{({(Z_{1},\ldots,Z_{n-1},\xi)|}_{\widetilde{V}})},\ldots,Z_{n-1},\xi)|}_{\widetilde{V}^{\mathsf{c}}}$

is equal to
$Z_{q}\Psi_{\pi|_{\widetilde{V}}}{({(Z_{1},\ldots,Z_{n-1},\xi)|}_{\widetilde{V}})}$,
which is an element of $L_{2}(A,\tau)$ so
$\Psi_{\pi}(Z_{1},\ldots,Z_{n-1},\xi)$ is well-defined.]

To aid in the comprehension of Definition 4.6, we provide an example using of
bi-non-crossing diagrams to show the recursive construction. We note that
$\xi$ will always appear last in a sequence of operators and is an element of
$L_{2}(A,\tau)$ and thus neither a left nor right operator. As such, we treat
it as neither. This is reminiscent of [S2016-2]*Lemma 2.17 where it does not
matter whether we treat the last operator in a list as a left or as a right
operator, and of [CS2020]*Lemma 2.29 and Proposition 2.30 where the last entry
can be a mixture of left and right operators.

######  Example 4.7.

Let $\chi\in\\{\ell,r\\}^{12}$ be such that
$\chi^{-1}(\\{\ell\\})=\\{1,5,8,9,11,12\\}$, let $\xi\in L_{2}(A,\tau)$, let
$Z_{k}\in A_{\chi(k)}$, and let $\pi\in\operatorname{BNC}(\chi)$ be the
partition with blocks

$V_{1}=\\{1,3\\},\quad V_{2}=\\{2\\},\quad V_{3}=\\{4,5,11,12\\},\quad
V_{4}=\\{6,10\\},\quad V_{5}=\\{7\\},\quad\text{and}\quad V_{6}=\\{8,9\\}.$

To compute $\Psi_{\pi}(Z_{1},\ldots,Z_{11},\xi)$, we note the second part of
the second step of the recursive definition from Definition 4.6 applies first.
In particular

$\widetilde{V}=\bigcup_{k=3}^{6}V_{k}.$

Thus, if

$X=\Psi_{\pi|_{\widetilde{V}}}(Z_{4},Z_{5},Z_{6},Z_{7},Z_{8},Z_{9},Z_{10},Z_{11},\xi),$

then

$\Psi_{\pi}(Z_{1},\ldots,Z_{11},\xi)=\Psi_{\pi|_{{\widetilde{V}^{\mathsf{c}}}}}(Z_{1},Z_{2},Z_{3}X).$

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} \pgfsys@invoke{\lxSVG@closescope
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{-105.27512pt}\pgfsys@moveto{-12.80367pt}{-105.27512pt}\pgfsys@curveto{-12.80367pt}{-104.48936pt}{-13.44063pt}{-103.8524pt}{-14.22638pt}{-103.8524pt}\pgfsys@curveto{-15.01213pt}{-103.8524pt}{-15.6491pt}{-104.48936pt}{-15.6491pt}{-105.27512pt}\pgfsys@curveto{-15.6491pt}{-106.06087pt}{-15.01213pt}{-106.69783pt}{-14.22638pt}{-106.69783pt}\pgfsys@curveto{-13.44063pt}{-106.69783pt}{-12.80367pt}{-106.06087pt}{-12.80367pt}{-105.27512pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{-105.27512pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-14.22638pt}{65.44142pt}\pgfsys@lineto{25.6073pt}{65.44142pt}\pgfsys@lineto{25.6073pt}{31.2982pt}\pgfsys@lineto{65.44142pt}{31.2982pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-14.22638pt}{-105.27512pt}\pgfsys@lineto{25.6073pt}{-105.27512pt}\pgfsys@lineto{25.6073pt}{-122.34694pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\longrightarrow\qquad\leavevmode\hbox
to123.15pt{\vbox to107.64pt{\pgfpicture\makeatletter\hbox{\hskip
31.43407pt\lower-34.54321pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{68.28644pt}\pgfsys@lineto{-14.22638pt}{-34.14322pt}\pgfsys@lineto{65.44142pt}{-34.14322pt}\pgfsys@lineto{65.44142pt}{68.28644pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$Z_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{65.44142pt}\pgfsys@moveto{-12.80367pt}{65.44142pt}\pgfsys@curveto{-12.80367pt}{66.22717pt}{-13.44063pt}{66.86414pt}{-14.22638pt}{66.86414pt}\pgfsys@curveto{-15.01213pt}{66.86414pt}{-15.6491pt}{66.22717pt}{-15.6491pt}{65.44142pt}\pgfsys@curveto{-15.6491pt}{64.65567pt}{-15.01213pt}{64.0187pt}{-14.22638pt}{64.0187pt}\pgfsys@curveto{-13.44063pt}{64.0187pt}{-12.80367pt}{64.65567pt}{-12.80367pt}{65.44142pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{65.44142pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{68.97443pt}{28.78377pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$Z_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{65.44142pt}{31.2982pt}\pgfsys@moveto{66.86414pt}{31.2982pt}\pgfsys@curveto{66.86414pt}{32.08395pt}{66.22717pt}{32.72092pt}{65.44142pt}{32.72092pt}\pgfsys@curveto{64.65567pt}{32.72092pt}{64.0187pt}{32.08395pt}{64.0187pt}{31.2982pt}\pgfsys@curveto{64.0187pt}{30.51245pt}{64.65567pt}{29.87549pt}{65.44142pt}{29.87549pt}\pgfsys@curveto{66.22717pt}{29.87549pt}{66.86414pt}{30.51245pt}{66.86414pt}{31.2982pt}\pgfsys@closepath\pgfsys@moveto{65.44142pt}{31.2982pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{68.97443pt}{-16.74081pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$Z_{3}X$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{65.44142pt}{-14.22638pt}\pgfsys@moveto{66.86414pt}{-14.22638pt}\pgfsys@curveto{66.86414pt}{-13.44063pt}{66.22717pt}{-12.80367pt}{65.44142pt}{-12.80367pt}\pgfsys@curveto{64.65567pt}{-12.80367pt}{64.0187pt}{-13.44063pt}{64.0187pt}{-14.22638pt}\pgfsys@curveto{64.0187pt}{-15.01213pt}{64.65567pt}{-15.6491pt}{65.44142pt}{-15.6491pt}\pgfsys@curveto{66.22717pt}{-15.6491pt}{66.86414pt}{-15.01213pt}{66.86414pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{65.44142pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{-14.22638pt}{65.44142pt}\pgfsys@lineto{25.6073pt}{65.44142pt}\pgfsys@lineto{25.6073pt}{-14.22638pt}\pgfsys@lineto{65.44142pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad$

Note

$\Psi_{\pi|_{{\widetilde{V}^{\mathsf{c}}}}}(Z_{1},Z_{2},Z_{3}X)=\Psi_{\pi|_{V_{2}}}(Z_{2}\Psi_{\pi|_{V_{1}}}(Z_{1},Z_{3}X))=\widetilde{E}\left(Z_{2}\widetilde{E}(Z_{1}Z_{3}X)\right),$

where the first equality holds by the same recursive idea, whereas the second
equality holds by the first step of Definition 4.6.

When computing the value of $X$, the minimal and maximal elements of
$\\{4,5,\ldots,11,12\\}$ in the ${\chi|}_{\widetilde{V}}$-order are $5$ and
$4$ respectively and the block that contains the index corresponding to $\xi$
contains both 5 and 4. Thus the first part of the second step of Definition
4.6 should be used. The algorithm in Definition 4.6 then calculates the value
of $X$ by “stripping out” the $\chi$-intervals $V_{6}$ and $V_{4}\cup V_{5}$
successively and this is seen via the following two diagrammatic reductions:

$\displaystyle\leavevmode\hbox to96.92pt{\vbox
to137.22pt{\pgfpicture\makeatletter\hbox{\hskip
34.23405pt\lower-64.1245pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{68.28644pt}\pgfsys@lineto{-14.22638pt}{-48.3696pt}\pgfsys@lineto{42.67914pt}{-48.3696pt}\pgfsys@lineto{42.67914pt}{68.28644pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$Z_{4}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{42.67914pt}{65.44142pt}\pgfsys@moveto{44.10185pt}{65.44142pt}\pgfsys@curveto{44.10185pt}{66.22717pt}{43.46489pt}{66.86414pt}{42.67914pt}{66.86414pt}\pgfsys@curveto{41.89339pt}{66.86414pt}{41.25642pt}{66.22717pt}{41.25642pt}{65.44142pt}\pgfsys@curveto{41.25642pt}{64.65567pt}{41.89339pt}{64.0187pt}{42.67914pt}{64.0187pt}\pgfsys@curveto{43.46489pt}{64.0187pt}{44.10185pt}{64.65567pt}{44.10185pt}{65.44142pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{65.44142pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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