Conditionally Bi-Free Independence with Amalgamation

Abstract

In this article, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operator-valued conditionally bi-free moment and cumulant functions. It is demonstrated that conditionally bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial |$\mathcal{R}$|-transform is constructed and various operator-valued conditionally bi-free limit theorems are studied.

1 Introduction

The notion of conditionally free (c-free for short) independence was introduced in [3] as a generalization of the notion of free independence to two-state systems. In our previous paper [6], we introduced the notion of conditionally bi-free (c-bi-free for short) independence in order to study the non-commutative left and right actions of algebras on a reduced c-free product simultaneously. Thus conditional bi-freeness is an extension of the notion of bi-free independence [14] to two-state systems. Moreover [6] introduced c-|$(\ell, r)$|-cumulants and demonstrated that a family of pairs of algebras in a two-state non-commutative probability space is c-bi-free if and only if mixed |$(\ell, r)$|- and c-|$(\ell, r)$|-cumulants vanish.

In [13], Voiculescu generalized his own notion of free independence by replacing the scalars with an arbitrary algebra thereby obtaining the notion of free independence with amalgamation (see also [12] for the combinatorial aspects). For c-free independence, the generalization to an amalgamated setting over a pair of algebras was done by Popa [9] (see also [8]). On the other hand, the framework for generalizing bi-free independence to an amalgamated setting was suggested by Voiculescu [14, Section 8] and the theory was fully developed in [4].

The main goal of this article is to extend the notion of c-bi-free independence to an amalgamated setting over a pair of algebras. Furthermore, we demonstrate that the combinatorics of c-bi-free probability and bi-free probability with amalgamation, which are governed by the lattice of bi-non-crossing partitions, are specific instances of more general combinatorial structures.

Including this introduction this article contains nine sections which are structured as follows. Section 2 briefly reviews some of the background material pertaining to c-bi-free probability and bi-free probability with amalgamation from [4–6]. In particular, the notions bi-non-crossing partitions and diagrams, their lateral refinements and cappings, interior and exterior blocks, |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability spaces, operator-valued bi-multiplicative functions, and the operator-valued bi-free moment and cumulant functions are recalled.

Section 3 introduces the structures studied within c-bi-free independence with amalgamation. We define the notion of a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| (see Definition 3.4), demonstrate a representation of |${\mathcal{A}}$| as linear operators on a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states (see Theorem 3.5), and define the notion of c-bi-free independence with amalgamation over |$({\mathcal{B}}, {\mathcal{D}})$| thereby generalizing c-bi-free independence to the operator-valued setting and bi-free independence with amalgamation to the two-state setting.

Section 4 introduces the notion of an operator-valued conditionally bi-multiplicative pair of functions (see Definition 4.2). Each such pair consists of two functions where the first function is operator-valued bi-multiplicative (see [4, Definition 4.2.1]) and the second function is defined via a certain rule using the first function. Furthermore, operator-valued c-bi-free moment and cumulant pairs (see Definitions 4.4 and 4.7) are introduced and shown to be operator-valued conditionally bi-multiplicative.

Sections 5 and 6 provide alternate characterizations of c-bi-free independence with amalgamation. More precisely, Section 5 demonstrates through Theorem 5.5 that a family of pairs of |${\mathcal{B}}$|-algebras in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| if and only if certain moment expressions with respect to |${\mathbb{E}}$| and |${\mathbb{F}}$| are satisfied. On the other hand, Section 6 demonstrates through Theorem 6.4 that a family of pairs of |${\mathcal{B}}$|-algebras is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| if and only if their mixed operator-valued bi-free and c-bi-free cumulants vanish.

Section 7 provides additional properties such as the vanishing of operator-valued c-bi-free cumulants when a left or right |${\mathcal{B}}$|-operator is input, how c-bi-free independence over |$({\mathcal{B}}, {\mathcal{D}})$| can be deduced from c-free independence over |$({\mathcal{B}}, {\mathcal{D}})$| under certain conditions, and how operator-valued c-bi-free cumulants involving products of operators may be computed.

In Section 8, an operator-valued c-bi-free partial |$\mathcal{R}$|-transform is constructed as the operator-valued analogue of the c-bi-free partial |$\mathcal{R}$|-transform (see [6, Definition 5.3]). As with the operator-valued bi-free partial |$\mathcal{R}$|-transform (see [11, Section 5]), the said transform is also a function of three |${\mathcal{B}}$|-variables, and a formula relating it to the moment series is proved using combinatorics. Finally, in Section 9, operator-valued c-bi-free distributions are discussed and various operator-valued c-bi-free limit theorems are studied.

2 Preliminaries

In this section, we review the necessary background on c-bi-free probability and operator-valued bi-free probability required for this paper.

2.1 C-bi-free probability

We recall several definitions and results relating to c-bi-free probability. For more precision, see [6].

 

Definition 2.1.

Let |$({\mathcal{A}}, \varphi, \psi)$| be a two-state non-commutative probability space; that is, |${\mathcal{A}}$| is a unital algebra and |$\varphi, \psi: {\mathcal{A}} \to {\mathbb{C}}$| are unital linear functionals. A pair of algebras in |${\mathcal{A}}$| is an ordered pair |$(A_\ell, A_r)$| of unital subalgebras of |${\mathcal{A}}$|⁠. □

 

Definition 2.2.

A family |$\{(A_{k, \ell}, A_{k, r})\}_{k \in K}$| of pairs of algebras in a two-state non-commutative probability space |$({\mathcal{A}}, \varphi, \psi)$| is said to be conditionally bi-freely independent (or c-bi-free for short) with respect to |$(\varphi, \psi)$| if there is a family of two-state vector spaces with specified state-vectors |$\{({\mathcal{X}}_k, {\mathcal{X}}_k^\circ, \xi_k, \varphi_k)\}_{k \in K}$| and unital homomorphisms

$${\ell }_k:A_{k,\ell }\to \mathcal{L}({\mathcal{X}}_k)\text{and}{r}_k:A_{k,r}\to \mathcal{L}({\mathcal{X}}_k)$$

such that the joint distribution of |$\{(A_{k, \ell}, A_{k, r})\}_{k \in K}$| with respect to |$(\varphi, \psi)$| is equal to the joint distribution of the family

$$\{({\lambda }_k\circ {\ell }_k(A_{k,\ell }),{\rho }_k\circ r_k(A_{k,r})){\}}_{k\in K}$$

in |${\mathcal{L}}({\mathcal{X}})$| with respect to |$(\varphi_\xi, \psi_\xi)$|⁠, where |$({\mathcal{X}}, {\mathcal{X}}^\circ, \xi, \varphi) = *_{k \in K}({\mathcal{X}}_k, {\mathcal{X}}_k^\circ, \xi_k, \varphi_k)$|⁠. □

In general, a map |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| is used to designate whether the |$k^{\mathrm{th}}$| operator in a sequence of |$n$| operators is a left operator (when |$\chi(k) = \ell$|⁠) or a right operator (when |$\chi(k) = r$|⁠), a map |$\omega: \{1, \dots, n\} \to I \sqcup J$| is used to designate the index of the |$k^\mathrm{th}$| operator, and a map |$\omega: \{1, \dots, n\} \to K$| is used to designate from which collection of operators the |$k^\mathrm{th}$| operator hails from.

Given |$\omega: \{1, \dots, n\} \to I \sqcup J$| for non-empty disjoint index sets |$I$| and |$J$|⁠, we define the corresponding map |$\chi_\omega: \{1, \dots, n\} \to \{\ell, r\}$| by

$${\chi }_{\omega }(k)=\{\begin{array}{ll}\hfill \ell & \text{if }\omega (k)\in I\hfill \\ \hfill r& \text{if }\omega (k)\in J\hfill \end{array}.$$

Given a map |$\omega: \{1, \dots, n\} \to K$|⁠, we may view |$\omega$| as a partition of |$\{1, \dots, n\}$| with blocks |$\{\omega^{-1}(\{k\})\}_{k \in K}$|⁠. Thus |$\pi \leq \omega$| denotes |$\pi$| is a refinement of the partition induced by |$\omega$|⁠.

For the basic definitions and combinatorics of bi-free probability that will be used in this article, we refer the reader to [4, 5, 7, 14] or the summary given in [6, Section 2]. Particular attention should be paid to:

• the set |${\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| of bi-non-crossing partitions with respect to |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, and the minimal and maximal elements |$0_\chi$| and |$1_\chi$| of |${\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| (see [4, Definition 2.1.1]);

• for |$m, n \geq 0$| with |$m + n \geq 1$|⁠, |$1_{m, n}$| denotes |$1_{\chi_{m, n}}$| where |$\chi_{m, n}: \{1, \dots, m + n\} \to \{\ell, r\}$| is such that |$\chi_{m, n}(k) = \ell$| if |$k \leq m$| and |$\chi_{m, n}(k) = r$| if |$k > m$|⁠;

• the Möbius function |$\mu_{{\mathcal{B}}{\mathcal{N}}{\mathcal{C}}}$| on the lattice of bi-non-crossing partitions (see [5, Remark 3.1.4]);

• the total ordering |$\prec_\chi$| on |$\{1, \dots, n\}$| and the notion of |$\chi$|-interval induced by |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| (see [4, Definition 4.1.1]);

• the set |${\mathcal{L}}{\mathcal{R}}(\chi, \omega)$| of shaded |${\mathcal{L}}{\mathcal{R}}$|-diagrams corresponding to |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| and |$\omega: \{1, \dots, n\} \to K$|⁠, and the subsets |${\mathcal{L}}{\mathcal{R}}_k(\chi, \omega)$| (⁠|$1 \leq k \leq n$|⁠) of |${\mathcal{L}}{\mathcal{R}}(\chi, \omega)$| with exactly |$k$| spines reaching the top (see [5, Section 2.5]);

• the notion |$\leq_\mathrm{lat}$| of lateral refinement (see [5, Definition 2.5.5]);

• the family |$\{\kappa_\chi: {\mathcal{A}}^n \to {\mathbb{C}}\}_{n \geq 1, \chi: \{1, \dots, n\} \to \{\ell, r\}}$| of |$(\ell, r)$|-cumulants (see [7, Definition 5.2]).

Inspired by the “vanishing of mixed |$(\ell, r)$|-cumulants” characterization of bi-free independence and the “vanishing of mixed free and c-free cumulants” characterization of c-free independence, we introduced in [6, Subsection 3.3] the family of c-|$(\ell, r)$|-cumulants using bi-non-crossing partitions that are divided into two types. More precisely, a block |$V$| of a bi-non-crossing partition |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| is said to be interior if there exists another block |$W$| of |$\pi$| such that |$\min_{\prec_\chi}(W) \prec_\chi \min_{\prec_\chi}(V)$| and |$\max_{\prec_\chi}(V) \prec_\chi \max_{\prec_\chi}(W)$|⁠, where |$\min_{\prec_\chi}$| and |$\max_{\prec_\chi}$| denote the minimum and maximum elements with respect to |$\prec_\chi$|⁠. A block of |$\pi$| is said to be exterior if it is not interior. The family

$$\{{\mathcal{K}}_{\chi }:{\mathcal{A}}^n\to \mathbb{C}{\}}_{n\ge 1,\chi :\{1,\dots ,n\}\to \{\ell ,r\}}$$

of c-|$(\ell, r)$|-cumulants of a two-state non-commutative probability space |$({\mathcal{A}}, \varphi, \psi)$| is recursively defined by

$$\phi (a_1\cdots a_n)=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )}{\mathcal{K}}_{\pi }(a_1,\dots ,a_n),$$

where

$${\mathcal{K}}_{\pi }(a_1,\dots ,a_n)=\left(\prod _{\begin{array}{c}V\in \pi \\ V\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}\end{array}}{\kappa }_{\chi {|}_V}((a_1,\dots ,a_n){|}_V)\right)\left(\prod _{\begin{array}{c}V\in \pi \\ V\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}\end{array}}{\mathcal{K}}_{\chi {|}_V}((a_1,\dots ,a_n){|}_V)\right)$$

for all |$n \geq 1$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, and |$a_1, \dots, a_n \in {\mathcal{A}}$|⁠.

Furthermore, as noticed in [6, Section 4], in order to obtain a moment formula for c-bi-free independence, additional sets of shaded diagrams and terminology are required.

 

Definition 2.3.

Let |$n \geq 1$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, and |$\omega: \{1, \dots, n\} \to K$| be given.

• (1) For |$0 \leq k \leq n$|⁠, let |${\mathcal{L}}{\mathcal{R}}_k^\mathrm{lat}(\chi, \omega)$| denote the set of all diagrams that can be obtained from |${\mathcal{L}}{\mathcal{R}}_k(\chi, \omega)$| under later refinement (i.e., cutting spines that do not reach the top). For |$D' \in {\mathcal{L}}{\mathcal{R}}_k^\mathrm{lat}(\chi, \omega)$| and |$D \in {\mathcal{L}}{\mathcal{R}}_k(\chi, \omega)$|⁠, write |$D \geq_\mathrm{lat} D'$| if |$D'$| can be obtained by laterally refining |$D$|⁠. Moreover, let

  $$\mathcal{L}{\mathcal{R}}^{\mathrm{l}\mathrm{a}\mathrm{t}}(\chi ,\omega )=\bigcup _{k=0}^n\mathcal{L}{\mathcal{R}}_k^{\mathrm{l}\mathrm{a}\mathrm{t}}(\chi ,\omega ).$$
• (2) Let |$0 \leq k \leq n$| and |$D \in {\mathcal{L}}{\mathcal{R}}_k^\mathrm{lat}(\chi, \omega)$|⁠. A diagram |$D'$| is said to be a capping of |$D$|⁠, denoted |$D \geq_\mathrm{cap} D'$|⁠, if |$D' = D$| or |$D'$| can be obtained by removing spines from |$D$| that reach the top. Let |${\mathcal{L}}{\mathcal{R}}_m^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$| denote the set of all diagrams with |$m$| spines reaching the top that can be obtained by capping some |$D \in {\mathcal{L}}{\mathcal{R}}_k^\mathrm{lat}(\chi, \omega)$| with |$k \geq m$|⁠. Moreover, let

  $$\mathcal{L}{\mathcal{R}}^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )=\bigcup _{m=0}^n\mathcal{L}{\mathcal{R}}_m^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega ).$$

• (3) For |$D \in {\mathcal{L}}{\mathcal{R}}_m^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$|⁠, let |$|D| = (\text{number of blocks of } D) + m$|⁠.

• (4) Let |$0 \leq m \leq n$|⁠, |$k \geq m$|⁠, |$D \in {\mathcal{L}}{\mathcal{R}}_k(\chi, \omega)$|⁠, and |$D' \in {\mathcal{L}}{\mathcal{R}}_m^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$|⁠. We say that |$D$|laterally caps to |$D'$|⁠, denoted |$D \geq_{\mathrm{lat}\mathrm{cap}} D'$|⁠, if there exists |$D'' \in {\mathcal{L}}{\mathcal{R}}_k^\mathrm{lat}(\chi, \omega)$| such that |$D \geq_\mathrm{lat} D''$| and |$D'' \geq_\mathrm{cap} D'$|⁠.

□

Suppose |$a_1, \dots, a_n$| are elements in a two-state non-commutative probability space |$({\mathcal{A}}, \varphi, \psi)$|⁠, and |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$| with blocks |$V_1, \dots, V_p$| whose spines do not reach the top and |$W_1, \dots, W_q$| whose spines reach the top. Writing |$V_i = \{r_{i, 1} < \cdots < r_{i, s_i}\}$| and |$W_j = \{r_{j, 1} < \cdots < r_{j, t_j}\}$|⁠, we define

$${\phi }_D(a_1,\dots ,a_n)=\prod _{i=1}^p\psi \left(a_{r_{i,1}}\cdots a_{r_{i,s_i}}\right)\prod _{j=1}^q\phi (a_{r_{j,1}}\cdots a_{r_{j,t_j}}).$$

Under the above notation, the following moment type characterization and vanishing of mixed cumulants characterization were established in [6, Theorems 4.1 and 4.8].

 

Theorem 2.4.

A family |$\{(A_{k, \ell}, A_{k, r})\}_{k \in K}$| of pairs of algebras in a two-state non-commutative probability space |$({\mathcal{A}}, \varphi, \psi)$| is c-bi-free with respect to |$(\varphi, \psi)$| if and only if

$$\psi (a_1\cdots a_n)=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )}\left[\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \le \sigma \le \omega \end{array}}{\mu }_{\mathcal{B}\mathcal{N}\mathcal{C}}(\pi ,\sigma )\right]{\psi }_{\pi }(a_1,\dots ,a_n)$$

(1)

and

$$\phi (a_1\cdots a_n)=\sum _{D\in \mathcal{L}{\mathcal{R}}^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )}\left[\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}\mathcal{R}(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}D\end{array}}(-1{)}^{|D|-|D^{\prime }|}\right]{\phi }_D(a_1,\dots ,a_n)$$

(2)

for all |$n \geq 1$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\omega: \{1, \dots, n\} \to K$|⁠, and |$a_1, \dots, a_n \in {\mathcal{A}}$| with |$a_k \in A_{\omega(k), \chi(k)}$|⁠.

Equivalently, for all |$n \geq 2$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\omega: \{1, \dots, n\} \to K$|⁠, and |$a_k$| as above, we have

$${\kappa }_{\chi }(a_1,\dots ,a_n)={\mathcal{K}}_{\chi }(a_1,\dots ,a_n)=0$$

whenever |$\omega$| is not constant. □

2.2 Bi-free probability with amalgamation

Now we recall bi-free probability in an amalgamated setting. Since our constructions for operator-valued c-bi-free independence in Section 3 are very similar, we shall only present the essential concepts. Please refer to [4, Section 3] or the summary given in [11, Section 2] for complete details. In particular, the following definitions and results will be generalized:

• a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a specified |${\mathcal{B}}$|-valued state |$({\mathcal{X}}, {\mathcal{X}}^\circ, \mathfrak{p})$| (see [4, Definition 3.1.1]);

• the free product with amalgamation over |${\mathcal{B}}$| of a family |$\{({\mathcal{X}}_k, {\mathcal{X}}_k^\circ, \mathfrak{p}_k)\}_{k \in K}$| of |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodules with specified |${\mathcal{B}}$|-valued states (see [4, Construction 3.1.7]);

• a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space |$({\mathcal{A}}, {\mathbb{E}}, \varepsilon)$| with left and right algebras |${\mathcal{A}}_\ell$| and |${\mathcal{A}}_r$| (see [4, Definition 3.2.1]) and left and right |$B$|-operators |$L_b$| and |$R_b$|⁠;

• any |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability can be represented on a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a specified |${\mathcal{B}}$|-valued state (see [4, Theorem 3.2.4]).

Furthermore, in order to discuss operator-valued bi-free probability, one needs the correct notions for moment and cumulant functions, which we now review in greater depth.

 

Definition 2.5.

Let |$({\mathcal{A}}, \mathbb{E}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space and let

$$\mathrm{\Psi }:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{B}$$

be a function that is linear in each |${\mathcal{A}}_{\chi(k)}$|⁠. We say that |$\Psi$| is operator-valued bi-multiplicative if for every |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠, |$b \in {\mathcal{B}}$|⁠, and |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, the following four conditions hold.

• (1) Let

  $$q=max\{k\in \{1,\dots ,n\}\mid \chi (k)\ne \chi (n)\}.$$
  If |$\chi(n) = \ell$|⁠, then

  $${\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},Z_n{L}_b)=\{\begin{array}{ll}{\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{q-1},Z_q{R}_b,Z_{q+1},\dots ,Z_n)\hfill & \text{if }q\ne -\mathrm{\infty }\hfill \\ {\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},Z_n)b\hfill & \text{if }q=-\mathrm{\infty }\hfill \end{array}.$$
  If |$\chi(n) = r$|⁠, then

  $${\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},Z_n{R}_b)=\{\begin{array}{ll}{\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{q-1},Z_q{L}_b,Z_{q+1},\dots ,Z_n)\hfill & \text{if }q\ne -\mathrm{\infty }\hfill \\ b{\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},Z_n)\hfill & \text{if }q=-\mathrm{\infty }\hfill \end{array}.$$
• (2) Let |$p \in \{1, \dots, n\}$|⁠, and let

  $$q=max\{k\in \{1,\dots ,n\}\mid \chi (k)=\chi (p),kz<p\}.$$
  If |$\chi(p) = \ell$|⁠, then

  $$\begin{array}{cc}& {\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{p-1},L_b{Z}_p,Z_{p+1},\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\{\begin{array}{ll}{\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{q-1},Z_q{L}_b,Z_{q+1},\dots ,Z_n)\hfill & \text{if }q\ne -\mathrm{\infty }\hfill \\ b{\mathrm{\Psi }}_{1_{\chi }}(Z_1,Z_2,\dots ,Z_n)\hfill & \text{if }q=-\mathrm{\infty }\hfill \end{array}.\hfill \end{array}$$
  If |$\chi(p) = r$|⁠, then

  $$\begin{array}{cc}& {\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{p-1},R_b{Z}_p,Z_{p+1},\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\{\begin{array}{ll}{\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_{q-1},Z_q{R}_b,Z_{q+1},\dots ,Z_n)\hfill & \text{if }q\ne -\mathrm{\infty }\hfill \\ {\mathrm{\Psi }}_{1_{\chi }}(Z_1,Z_2,\dots ,Z_n)b\hfill & \text{if }q=-\mathrm{\infty }\hfill \end{array}.\hfill \end{array}$$
• (3) Suppose that |$V_1, \dots, V_m$| are |$\chi$|-intervals ordered by |$\prec_\chi$| which partition |$\{1, \dots, n\}$|⁠, each a union of blocks of |$\pi$|⁠. Then

  $${\mathrm{\Psi }}_{\pi }(Z_1,\dots ,Z_n)={\mathrm{\Psi }}_{\pi {|}_{V_1}}((Z_1,\dots ,Z_n){|}_{V_1})\cdots {\mathrm{\Psi }}_{\pi {|}_{V_m}}((Z_1,\dots ,Z_n){|}_{V_m}).$$
• (4) Suppose that |$V$| and |$W$| partition |$\{1, \dots, n\}$|⁠, each a union of blocks of |$\pi$|⁠, |$V$| is a |$\chi$|-interval, and

  $$\underset{{\prec }_{\chi }}{min}(\{1,\dots ,n\}),\underset{{\prec }_{\chi }}{max}(\{1,\dots ,n\})\in W.$$
  Let

  $$\begin{array}{cc}& p=\underset{{\prec }_{\chi }}{max}\left(\left\{k\in W\mid k{\prec }_{\chi }\underset{{\prec }_{\chi }}{min}(V)\right\}\right)\text{and}\hfill \\ & q=\underset{{\prec }_{\chi }}{min}\left(\left\{k\in W\mid \underset{{\prec }_{\chi }}{max}(V){\prec }_{\chi }k\right\}\right).\hfill \end{array}$$
  Then

  $$\begin{array}{cc}& {\mathrm{\Psi }}_{\pi }(Z_1,\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\{\begin{array}{ll}\hfill {\mathrm{\Psi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{p-1},Z_p{L}_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)},Z_{p+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (p)=\ell \hfill \\ \hfill {\mathrm{\Psi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{p-1},R_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)}{Z}_p,Z_{p+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (p)=r\hfill \end{array}\hfill \\ & \hspace{0.17em}=\{\begin{array}{ll}\hfill {\mathrm{\Psi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{q-1},L_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)}{Z}_q,Z_{q+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (q)=\ell \hfill \\ \hfill {\mathrm{\Psi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{q-1},Z_q{R}_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)},Z_{q+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (q)=r\hfill \end{array}.\hfill \end{array}$$

□

Given an operator-valued bi-multiplicative function, conditions |$(1)$|–|$(4)$| above are reduction properties which allows one to move |${\mathcal{B}}$|-operators around and, more importantly, to compute the values on arbitrary bi-non-crossing partitions based on its values on full non-crossing partitions.

Finally, the two most important operator-valued bi-multiplicative functions in the theory, called operator-valued bi-free moment and cumulant functions, are defined as follows.

 

Definition 2.6.

Let |$({\mathcal{A}}, \mathbb{E}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space. For |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, and |$Z_1, \dots, Z_n \in {\mathcal{A}}$|⁠, define |${\mathbb{E}}_\pi(Z_1, \dots, Z_n) \in {\mathcal{B}}$| recursively as follows: Let |$V$| be the block of |$\pi$| that terminates closest to the bottom, so |$\min(V)$| is largest among all blocks of |$\pi$|⁠.

• (1) If |$\pi$| contains exactly one block (that is, |$\pi = 1_\chi$|⁠), define |${\mathbb{E}}_{1_\chi}(Z_1, \dots, Z_n) = {\mathbb{E}}(Z_1\cdots Z_n)$|⁠.

• (2) If |$V = \{k + 1, \dots, n\}$| for some |$k \in \{1, \dots, n - 1\}$| (so |$\min(V)$| is not adjacent to any spine of |$\pi$|⁠), define

  $${\mathbb{E}}_{\pi }(Z_1,\dots ,Z_n)=\{\begin{array}{ll}\hfill {\mathbb{E}}_{\pi {|}_{V^{\complement }}}(Z_1,\dots ,Z_k{L}_{{\mathbb{E}}_{\pi {|}_V}(Z_{k+1},\dots ,Z_n)})& \text{if }\chi (min(V))=\ell \hfill \\ \hfill {\mathbb{E}}_{\pi {|}_{V^{\complement }}}(Z_1,\dots ,Z_k{R}_{{\mathbb{E}}_{\pi {|}_V}(Z_{k+1},\dots ,Z_n)})& \text{if }\chi (min(V))=r\hfill \end{array}.$$
• (3) Otherwise, |$\min(V)$| is adjacent to a spine. Let |$W$| denote the block of |$\pi$| corresponding to the spine adjacent to |$\min(V)$| and let |$k$| be the smallest element of |$W$| that is larger than |$\min(V)$|⁠. Define

  $$\begin{array}{cc}& {\mathbb{E}}_{\pi }(Z_1,\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\{\begin{array}{ll}\hfill {\mathbb{E}}_{\pi {|}_{V^{\complement }}}((Z_1,\dots ,Z_{k-1},L_{{\mathbb{E}}_{\pi {|}_V}((Z_1,\dots ,Z_n){|}_V)}{Z}_k,Z_{k+1},\dots ,Z_n){|}_{V^{\complement }})& \text{if }\chi (min(V))=\ell \hfill \\ \hfill {\mathbb{E}}_{\pi {|}_{V^{\complement }}}((Z_1,\dots ,Z_{k-1},R_{{\mathbb{E}}_{\pi {|}_V}((Z_1,\dots ,Z_n){|}_V)}{Z}_k,Z_{k+1},\dots ,Z_n){|}_{V^{\complement }})& \text{if }\chi (min(V))=r\hfill \end{array}.\hfill \end{array}$$

□

 

Definition 2.7.

Let |$({\mathcal{A}}, \mathbb{E}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space. The operator-valued bi-free moment and cumulant functions on |${\mathcal{A}}$| are

$$\mathcal{E},\kappa :\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{B}$$

defined by

$${\mathcal{E}}_{\pi }(Z_1,\dots ,Z_n)={\mathbb{E}}_{\pi }(Z_1,\dots ,Z_n)\text{and}{\kappa }_{\pi }(Z_1,\dots ,Z_n)=\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma \le \pi \end{array}}{\mathcal{E}}_{\sigma }(Z_1,\dots ,Z_n){\mu }_{\mathcal{B}\mathcal{N}\mathcal{C}}(\sigma ,\pi )$$

for all |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, and |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. □

A substantial amount of effort was taken in [4, Sections 5 and 6] to show that both |${\mathcal{E}}$| and |$\kappa$| are operator-valued bi-multiplicative.

3 C-Bi-Free Families with Amalgamation

In this section, we develop the structures to discuss c-bi-free independence with amalgamation. To begin, we need an analogue of a two-state vector space with a specified state-vector.

 

Definition 3.1.

A |$\mathcal{B}$|-|$\mathcal{B}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states is a quadruple |$(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$|⁠, where |$\mathcal{B}$| and |$\mathcal{D}$| are unital algebras such that |$1 := 1_{\mathcal{D}} \in \mathcal{B} \subset \mathcal{D}$|⁠, |$\mathcal{X}$| is a direct sum of |$\mathcal{B}$|-|$\mathcal{B}$|-bimodules |$\mathcal{X} = \mathcal{B} \oplus \mathcal{X}^\circ$|⁠, |$\mathfrak{p}: \mathcal{X} \to \mathcal{B}$| is the linear map |$\mathfrak{p}(b \oplus \eta) = b$|⁠, and |$\mathfrak{q}: \mathcal{X} \to \mathcal{D}$| is a linear |$\mathcal{B}$|-|$\mathcal{B}$|-bimodule map such that |$\mathfrak{q}(1 \oplus 0) = 1$|⁠. □

Given a |$\mathcal{B}$|-|$\mathcal{B}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states |$(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$|⁠, we have

$$\mathfrak{p}(b_1\cdot x\cdot b_2)=b_1\mathfrak{p}(x)b_2\text{and}\mathfrak{q}(b_1\cdot x\cdot b_2)=b_1\mathfrak{q}(x)b_2$$

for all |$b_1, b_2 \in {\mathcal{B}}$| and |$x \in {\mathcal{X}}$|⁠. Moreover, let |${\mathcal{L}}({\mathcal{X}})$| denote the set of linear operators on |${\mathcal{X}}$|⁠, and recall from [4, Definition 3.1.3] that the operators |$L_b, R_b \in {\mathcal{L}}({\mathcal{X}})$| are defined by

$$L_b(x)=b\cdot x\text{and}{R}_b(x)=x\cdot b$$

for all |$b \in {\mathcal{B}}$| and |$x \in {\mathcal{X}}$|⁠. In addition, the left and right algebras of |${\mathcal{L}}({\mathcal{X}})$| are the unital subalgebras |${\mathcal{L}}_\ell({\mathcal{X}})$| and |${\mathcal{L}}_r(X)$| defined by

$${\mathcal{L}}_{\ell }(X)=\{Z\in \mathcal{L}(\mathcal{X})\mid Z{R}_b=R_{b}Z\text{ for all }b\in \mathcal{B}\}$$

and

$${\mathcal{L}}_r(\mathcal{X})=\{Z\in \mathcal{L}(\mathcal{X})\mid Z{L}_b=L_{b}Z\text{ for all }b\in \mathcal{B}\},$$

respectively.

As we are interested in c-bi-free independence with amalgamation, we need two expectations on |${\mathcal{L}}({\mathcal{X}})$|⁠, one onto |${\mathcal{B}}$| and one to |${\mathcal{D}}$|⁠.

 

Definition 3.2.

Given a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states |$(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$|⁠, define the unital linear maps |$\mathbb{E}_{{\mathcal{L}}({\mathcal{X}})}: {\mathcal{L}}({\mathcal{X}}) \to {\mathcal{B}}$| and |$\mathbb{F}_{{\mathcal{L}}({\mathcal{X}})}: {\mathcal{L}}({\mathcal{X}}) \to \mathcal{D}$| by

$${\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(Z)=\mathfrak{p}(Z(1\oplus 0))\text{and}{\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z)=\mathfrak{q}(Z(1\oplus 0))$$

for all |$Z \in {\mathcal{L}}({\mathcal{X}})$|⁠. We call |$\mathbb{E}_{{\mathcal{L}}({\mathcal{X}})}$| and |$\mathbb{F}_{{\mathcal{L}}({\mathcal{X}})}$| the expectations of |${\mathcal{L}}({\mathcal{X}})$| to |${\mathcal{B}}$| and |$\mathcal{D}$|⁠, respectively. □

There are specific properties of these expectations we wish to model.

 

Proposition 3.3.

Let |$(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$| be a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states. We have

$${\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(L_{b_1}{R}_{b_2}Z)=b_1{\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(Z)b_2,{\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(Z{L}_b)={\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(Z{R}_b)$$

and

$${\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(L_{b_1}{R}_{b_2}Z)=b_1{\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z)b_2,{\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z{L}_b)={\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z{R}_b)$$

for all |$b_1, b_2, b \in \mathcal{B}$| and |$Z \in {\mathcal{L}}({\mathcal{X}})$|⁠. □

 

Proof.

The results regarding |${\mathbb{E}}_{{\mathcal{L}}({\mathcal{X}})}$| were shown in [4, Proposition 3.1.6]. Moreover, it is immediate that |$\mathbb{F}_{{\mathcal{L}}({\mathcal{X}})}(ZL_b) = \mathbb{F}_{{\mathcal{L}}({\mathcal{X}})}(ZR_b)$| for all |$b \in {\mathcal{B}}$| and |$Z \in {\mathcal{L}}({\mathcal{X}})$| as |$L_b(1 \oplus 0) = R_b(1 \oplus 0)$|⁠. Finally, since |$\mathfrak{q}$| is a linear |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule map, we have

$${\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(L_{b_1}{R}_{b_2}Z)=\mathfrak{q}(L_{b_1}{R}_{b_2}Z(1\oplus 0))=b_1\mathfrak{q}(Z(1\oplus 0))b_2=b_1{\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z)b_2$$

for all |$b_1, b_2 \in {\mathcal{B}}$| and |$Z \in {\mathcal{L}}({\mathcal{X}})$|⁠. ■

Given the above definition and proposition, we extend the notion of a two-state non-commutative probability space |$({\mathcal{A}}, \varphi, \psi)$| to the operator-valued setting as follows. Note this is also a natural extension of the notion of a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space |$({\mathcal{A}}, {\mathbb{E}}, \varepsilon)$| from [4, Definition 3.2.1] to the two-state setting. For the remainder of the article, given a unital algebra |${\mathcal{B}}$| we denote the algebra with the same elements as |${\mathcal{B}}$| but with the opposite multiplication (i.e. |$b_1 \cdot b_2 = b_2b_1)$| by |${\mathcal{B}}^{\mathrm{op}}$|⁠.

 

Definition 3.4.

A |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations is a quadruple |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$|⁠, where |${\mathcal{A}}$|⁠, |$\mathcal{B}$|⁠, and |$\mathcal{D}$| are unital algebras such that |$1 := 1_{\mathcal{D}} \in \mathcal{B} \subset \mathcal{D}$|⁠, |$\varepsilon: \mathcal{B} \otimes \mathcal{B}^{\mathrm{op}} \to {\mathcal{A}}$| is a unital homomorphism such that |$\varepsilon|_{\mathcal{B} \otimes 1}$| and |$\varepsilon|_{1 \otimes \mathcal{B}^{\mathrm{op}}}$| are injective, and |$\mathbb{E}: {\mathcal{A}} \to \mathcal{B}$| and |$\mathbb{F}: {\mathcal{A}} \to \mathcal{D}$| are unital linear maps such that

$$\mathbb{E}(\epsilon (b_1\otimes b_2)Z)=b_1\mathbb{E}(Z)b_2,\mathbb{E}(Z\epsilon (b\otimes 1))=\mathbb{E}(Z\epsilon (1\otimes b))$$

and

$$\mathbb{F}(\epsilon (b_1\otimes b_2)Z)=b_1\mathbb{F}(Z)b_2,\mathbb{F}(Z\epsilon (b\otimes 1))=\mathbb{F}(Z\epsilon (1\otimes b))$$

for all |$b_1, b_2, b \in \mathcal{B}$| and |$Z \in {\mathcal{A}}$|⁠. Moreover, the unital subalgebras |${\mathcal{A}}_\ell$| and |${\mathcal{A}}_r$| of |${\mathcal{A}}$| defined by

$${\mathcal{A}}_{\ell }=\{Z\in \mathcal{A}\mid Z\epsilon (1\otimes b)=\epsilon (1\otimes b)Z\text{ for all }b\in \mathcal{B}\}$$

and

$${\mathcal{A}}_r=\{Z\in \mathcal{A}\mid Z\epsilon (b\otimes 1)=\epsilon (b\otimes 1)Z\text{ for all }b\in \mathcal{B}\}$$

will be called the left and right algebras of |${\mathcal{A}}$| respectively. □

As with the bi-free case (see [4, Remark 3.2.2]), if |$({\mathcal{X}}, {\mathcal{X}}^\circ, \mathfrak{p}, \mathfrak{q})$| is a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states, then we see via Proposition 3.3 that |$({\mathcal{L}}({\mathcal{X}}), {\mathbb{E}}_{{\mathcal{L}}({\mathcal{X}})}, {\mathbb{F}}_{{\mathcal{L}}({\mathcal{X}})}, \varepsilon)$| is a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations where |$\varepsilon: {\mathcal{B}} \otimes {\mathcal{B}}^{\mathrm{op}} \to {\mathcal{L}}({\mathcal{X}})$| is defined by |$\varepsilon(b_1 \otimes b_2) = L_{b_1}R_{b_2}$|⁠. Moreover, the following result demonstrates that any |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations can be represented as linear operators on some |$({\mathcal{X}}, {\mathcal{X}}^\circ, \mathfrak{p}, \mathfrak{q})$|⁠. Hence Definition 3.4 is the natural extension of [4, Definition 3.2.1]. As such, we will write |$L_b$| and |$R_b$| instead of |$\varepsilon(b \otimes 1)$| and |$\varepsilon(1 \otimes b)$| and refer to these as left and right |${\mathcal{B}}$|-operators, respectively.

 

Theorem 3.5.

If |$({\mathcal{A}}, \mathbb{E}_{\mathcal{A}}, \mathbb{F}_{\mathcal{A}}, \varepsilon)$| is a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, then there exist a |$\mathcal{B}$|-|$\mathcal{B}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states |$(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$| and a unital homomorphism |$\theta: {\mathcal{A}} \to {\mathcal{L}}({\mathcal{X}})$| such that

$$\begin{array}{c}\hfill \theta (L_{b_1}{R}_{b_2})=L_{b_1}{R}_{b_2},\theta ({\mathcal{A}}_{\ell })\subset {\mathcal{L}}_{\ell }(\mathcal{X}),\theta ({\mathcal{A}}_r)\subset {\mathcal{L}}_r(\mathcal{X}),\\ \hfill {\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(\theta (Z))={\mathbb{E}}_{\mathcal{A}}(Z),\text{and}{\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(\theta (Z))={\mathbb{F}}_{\mathcal{A}}(Z)\end{array}$$

for all |$b_1, b_2 \in {\mathcal{B}}$| and |$Z \in {\mathcal{A}}$|⁠. □

 

Proof.

As shown in the proof of [4, Theorem 3.2.4], consider |${\mathcal{X}} = {\mathcal{B}} \oplus {\mathcal{Y}}$| as a vector space over |$\mathbb{C}$| where

$$\mathcal{Y}=\ker ({\mathbb{E}}_{\mathcal{A}})/\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{Z{L}_b-Z{R}_b\mid Z\in \mathcal{A},b\in \mathcal{B}\}.$$

Define |$\theta: {\mathcal{A}} \to {\mathcal{L}}({\mathcal{X}})$| by

$$\theta (Z)(b)={\mathbb{E}}_{\mathcal{A}}(Z{L}_b)\oplus \pi (Z{L}_b-L_{{\mathbb{E}}_{\mathcal{A}}(Z{L}_b)}),b\in \mathcal{B}$$

and

$$\theta (Z)(\pi (Y))={\mathbb{E}}_{\mathcal{A}}(ZY)\oplus \pi (ZY-L_{{\mathbb{E}}_{\mathcal{A}}(ZY)}),Y\in \ker ({\mathbb{E}}_{\mathcal{A}}),$$

where |$\pi: \ker({\mathbb{E}}_{\mathcal{A}}) \to {\mathcal{Y}}$| denotes the canonical quotient map. It was shown in [4, Theorem 3.2.4] that |$\theta$| is a unital homomorphism and |${\mathcal{X}}$| is a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule via

$$b\cdot \xi =\theta (L_b)(\xi )\text{and}\xi \cdot b=\theta (R_b)(\xi )$$

for all |$b \in {\mathcal{B}}$| and |$\xi \in {\mathcal{X}}$|⁠. Thus we can define a specified |${\mathcal{B}}$|-valued state |$\mathfrak{p}$| on |${\mathcal{X}}$| by |$\mathfrak{p}(b \oplus \pi(Y)) = b$| for all |$b \in {\mathcal{B}}$| and |$\pi(Y) \in {\mathcal{Y}}$|⁠. Using this specified |${\mathcal{B}}$|-valued state, we obtain that |$\theta({\mathcal{A}}_\ell) \subset {\mathcal{L}}_\ell({\mathcal{X}})$|⁠, |$\theta({\mathcal{A}}_r) \subset {\mathcal{L}}_r({\mathcal{X}})$|⁠, and |$\mathbb{E}_{{\mathcal{L}}({\mathcal{X}})}(\theta(Z)) = \mathbb{E}_{\mathcal{A}}(Z)$|⁠.

On the other hand, since |${\mathbb{F}}_{\mathcal{A}}(ZL_b - ZR_b) = 0$| for all |$Z \in {\mathcal{A}}$| and |$b \in {\mathcal{B}}$|⁠, there exists a unique linear map |$\widetilde{\mathfrak{q}}: {\mathcal{Y}} \to {\mathcal{D}}$| such that |${\mathbb{F}}_{\mathcal{A}}|_{\ker({\mathbb{E}}_{\mathcal{A}})} = \widetilde{\mathfrak{q}} \circ \pi$|⁠. Let |$\mathfrak{q}: {\mathcal{X}} \to {\mathcal{D}}$| be the linear map defined by

$$\mathfrak{q}(b\oplus \pi (Y))=b+\tilde{\mathfrak{q}}\circ \pi (Y),b\in \mathcal{B},\pi (Y)\in \mathcal{Y}.$$

Then |$\mathfrak{q}(1 \oplus 0) = 1$| and

$$\begin{array}{cc}\hfill \mathfrak{q}(b_1\cdot (b\oplus \pi (Y))\cdot b_2)& \hspace{0.17em}=\mathfrak{q}(\theta (L_{b_1})\theta (R_{b_2})(b\oplus \pi (Y)))\hfill \\ & \hspace{0.17em}=\mathfrak{q}(\theta (L_{b_1})(b{b}_2\oplus \pi (R_{b_2}Y)))\hfill \\ & \hspace{0.17em}=\mathfrak{q}(b_{1}b{b}_2\oplus \pi (L_{b_2}{R}_{b_2}Y))\hfill \\ & \hspace{0.17em}=b_{1}b{b}_2+{\mathbb{F}}_{\mathcal{A}}(L_{b_1}{R}_{b_2}Y)\hfill \\ & \hspace{0.17em}=b_1(b+{\mathbb{F}}_{\mathcal{A}}(Y))b_2\hfill \\ & \hspace{0.17em}=b_1\mathfrak{q}(b\oplus \pi (Y))b_2\hfill \end{array}$$

for all |$b_1, b_2, b \in {\mathcal{B}}$| and |$\pi(Y) \in {\mathcal{Y}}$|⁠. Therefore, the quadruple |$({\mathcal{X}}, {\mathcal{Y}}, \mathfrak{p}, \mathfrak{q})$| is a |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states. Finally, we have

$${\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(\theta (Z))=\mathfrak{q}(\theta (Z)(1\oplus 0))=\mathfrak{q}({\mathbb{E}}_{\mathcal{A}}(Z)\oplus \pi (Z-L_{{\mathbb{E}}_{\mathcal{A}}(Z)}))={\mathbb{E}}_{\mathcal{A}}(Z)+{\mathbb{F}}_{\mathcal{A}}(Z-L_{{\mathbb{E}}_{\mathcal{A}}(Z)})={\mathbb{F}}_{\mathcal{A}}(Z)$$

for all |$Z \in {\mathcal{A}}$|⁠. ■

The next step is to extend the construction of the free product with amalgamation over |${\mathcal{B}}$| of a family |$\{({\mathcal{X}}_k, {\mathcal{X}}_k^\circ, \mathfrak{p}_k)\}_{k \in K}$| of |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodules with specified |${\mathcal{B}}$|-valued states (see [4, Construction 3.1.7]) to the current framework.

 

Construction 3.6.

Let |$\{({\mathcal{X}}_k, {\mathcal{X}}_k^\circ, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$| be a family of |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodules with pairs of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states. The c-free product of |$\{({\mathcal{X}}_k, {\mathcal{X}}_k^\circ, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$| with amalgamation over |$({\mathcal{B}}, {\mathcal{D}})$| is defined to be the |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule with a pair of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states |$({\mathcal{X}}, {\mathcal{X}}^\circ, \mathfrak{p}, \mathfrak{q})$|⁠, where |${\mathcal{X}} = {\mathcal{B}} \oplus {\mathcal{X}}^\circ$|⁠, |${\mathcal{X}}^\circ$| is the |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule

$${\mathcal{X}}^{\circ }=\underset{n\ge 1}{⨁}\left(\underset{k_1\ne \cdots \ne k_n}{⨁}{\mathcal{X}}_{k_1}^{\circ }{\otimes }_{\mathcal{B}}\cdots {\otimes }_{\mathcal{B}}{\mathcal{X}}_{k_n}^{\circ }\right)$$

with left and right actions of |${\mathcal{B}}$| on |${\mathcal{X}}^\circ$| defined by

$$b\cdot (x_1\otimes \cdots \otimes x_n)=(b\cdot x_1)\otimes x_2\otimes \cdots \otimes x_n\text{and}(x_1\otimes \cdots \otimes x_n)\cdot b=x_1\otimes \cdots \otimes x_{n-1}\otimes (x_n\cdot b),$$

respectively, |$\mathfrak{p}: {\mathcal{X}} \to {\mathcal{B}}$| is the linear map |$\mathfrak{p}(b \oplus \eta) = b$|⁠, and |$\mathfrak{q}: {\mathcal{X}} \to {\mathcal{D}}$| is the linear |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule map such that |$\mathfrak{q}(1 \oplus 0) = 1$| and

$$\mathfrak{q}(x_1\otimes \cdots \otimes x_n)={\mathfrak{q}}_{k_1}(x_1)\cdots {\mathfrak{q}}_{k_n}(x_n)$$

for |$x_1 \otimes \cdots \otimes x_n \in {\mathcal{X}}_{k_1}^\circ \otimes_{{\mathcal{B}}} \cdots \otimes_{{\mathcal{B}}} {\mathcal{X}}_{k_n}^\circ$| (note |$\mathfrak{q}$| is well defined as each |$\mathfrak{q}_k$| is a linear |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule map).

For every |$k \in K$|⁠, let

$$V_k:\mathcal{X}\to {\mathcal{X}}_k{\otimes }_{\mathcal{B}}\left(\mathcal{B}\oplus \underset{n\ge 1}{⨁}\left(\underset{\begin{array}{c}{k}_1\ne \cdots \ne k_n\\ k_1\ne k\end{array}}{⨁}{\mathcal{X}}_{k_1}^{\circ }{\otimes }_{\mathcal{B}}\cdots {\otimes }_{\mathcal{B}}{\mathcal{X}}_{k_n}^{\circ }\right)\right)$$

be the standard |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule isomorphism, and define the left representation |$\lambda_k: {\mathcal{L}}({\mathcal{X}}_k) \to {\mathcal{L}}({\mathcal{X}})$| by

$${\lambda }_k(Z)=V_k^{-1}(Z\otimes I)V_k$$

for |$Z \in {\mathcal{L}}({\mathcal{X}}_k)$|⁠. Similarly, let

$$W_k:\mathcal{X}\to \left(\mathcal{B}\oplus \underset{n\ge 1}{⨁}\left(\underset{\begin{array}{c}{k}_1\ne \cdots \ne k_n\\ k_n\ne k\end{array}}{⨁}{\mathcal{X}}_{k_1}^{\circ }{\otimes }_{\mathcal{B}}\cdots {\otimes }_{\mathcal{B}}{\mathcal{X}}_{k_n}^{\circ }\right)\right){\otimes }_{\mathcal{B}}{\mathcal{X}}_k$$

be the standard |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule isomorphism, and define the right representation |$\rho_k: {\mathcal{L}}({\mathcal{X}}_k) \to {\mathcal{L}}({\mathcal{X}})$| by

$${\rho }_k(Z)=W_k^{-1}(I\otimes Z)W_k$$

for |$Z \in {\mathcal{L}}({\mathcal{X}}_k)$|⁠. For the exact formulae of how |$\lambda_k(Z)$| and |$\rho_k(Z)$| act on |${\mathcal{X}}$|⁠, we refer to [4, Construction 3.1.7]. Note also that

$${\mathbb{E}}_{\mathcal{L}(\mathcal{X})}({\lambda }_k(Z))={\mathbb{E}}_{\mathcal{L}(\mathcal{X})}({\rho }_k(Z))={\mathbb{E}}_{\mathcal{L}({\mathcal{X}}_k)}(Z)\text{and}{\mathbb{F}}_{\mathcal{L}(\mathcal{X})}({\lambda }_k(Z))={\mathbb{F}}_{\mathcal{L}(\mathcal{X})}({\rho }_k(Z))={\mathbb{F}}_{\mathcal{L}({\mathcal{X}}_k)}(Z)$$

for all |$Z \in {\mathcal{L}}({\mathcal{X}}_k)$|⁠. □

 

Remark 3.7.

It is clear that that all of the above discussions hold if |${\mathcal{B}} = {\mathcal{D}}$|⁠. However, the more general setting that |${\mathcal{B}} \subset {\mathcal{D}}$| is desired due to a result of Boca [2]. Indeed, suppose |$\{{\mathcal{A}}_k\}_{k \in K}$| is a family of unital |$C^*$|-algebras containing |${\mathcal{B}}$| as a common |$C^*$|-subalgebra with |$1_{{\mathcal{A}}_k} \in {\mathcal{B}}$|⁠, |${\mathcal{D}}$| is a unital |$C^*$|-algebra with |$1_{\mathcal{D}} \in {\mathcal{B}} \subset {\mathcal{D}}$|⁠, and each |${\mathcal{A}}_k$| is endowed with two positive conditional expectations |$\Psi_k: {\mathcal{A}}_k \to {\mathcal{B}}$| and |$\Phi_k: {\mathcal{A}}_k \to {\mathcal{D}}$| such that |${\mathcal{A}}_k = {\mathcal{B}} \oplus {\mathcal{A}}_k^\circ$|⁠, where |${\mathcal{A}}_k^\circ = \ker(\Psi_k)$|⁠, as a direct sum of |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodules.

Let |${\mathcal{A}} = (*_{\mathcal{B}})_{k \in K}{\mathcal{A}}_k$| be the free product of |$\{{\mathcal{A}}_k\}_{k \in K}$| with amalgamation over |${\mathcal{B}}$| (which can be identified as

$$\mathcal{A}=\mathcal{B}\oplus \underset{n\ge 1}{⨁}\left(\underset{k_1\ne \cdots \ne k_n}{⨁}{\mathcal{A}}_{k_1}^{\circ }{\otimes }_{\mathcal{B}}\cdots {\otimes }_{\mathcal{B}}{\mathcal{A}}_{k_n}^{\circ }\right)$$

as |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodules), let |$\Psi = (*_{{\mathcal{B}}})_{k \in K}\Psi_k$| be the amalgamated free product of |$\{\Psi_k\}_{k \in K}$|⁠, and let |$\Phi: {\mathcal{A}} \to {\mathcal{D}}$| be the unital linear |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodule map defined by

$$\mathrm{\Phi }(a_1\otimes \cdots \otimes a_n)={\mathrm{\Phi }}_{k_1}(a_1)\cdots {\mathrm{\Phi }}_{k_n}(a_n)$$

for |$a_1 \otimes \cdots \otimes a_n \in {\mathcal{A}}_{k_1}^\circ \otimes_{{\mathcal{B}}} \cdots \otimes_{{\mathcal{B}}} {\mathcal{A}}_{k_n}^\circ$|⁠. It is well known that |$\Psi$| is positive (see, e.g., [12, Theorem 3.5.6]). On the other hand, it follows from [2, Theorem 3.1] that |$\Phi$| is also positive, which is the main motivation for our setting. □

With Definition 3.4 and Construction 3.6 complete, we can define the notion of c-bi-free independence with amalgamation as follows.

 

Definition 3.8.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. A pair of |${\mathcal{B}}$|-algebras in |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| is a pair |$({\mathcal{C}}_\ell, {\mathcal{C}}_r)$| of unital subalgebras of |${\mathcal{A}}$| such that

$$\epsilon (\mathcal{B}\otimes 1)\subset {\mathcal{C}}_{\ell }\subset {\mathcal{A}}_{\ell }\text{and}\epsilon (1\otimes {\mathcal{B}}^{\mathrm{o}\mathrm{p}})\subset {\mathcal{C}}_r\subset {\mathcal{A}}_r.$$

A family |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| of pairs of |${\mathcal{B}}$|-algebras in |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| is said to be c-bi-free with amalgamation over |$({\mathcal{B}}, {\mathcal{D}})$| (or c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| for short) if there is a family of |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodules with pairs of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states |$\{(\mathcal{X}_k, \mathcal{X}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$| and unital homomorphisms

$${\ell }_k:{\mathcal{A}}_{k,\ell }\to {\mathcal{L}}_{\ell }({\mathcal{X}}_k)\text{and}{r}_k:{\mathcal{A}}_{k,r}\to {\mathcal{L}}_r({\mathcal{X}}_k)$$

such that the joint distribution of |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| with respect to |$(\mathbb{E}, \mathbb{F})$| is equal to the joint distribution of the family

$$\{({\lambda }_k\circ {\ell }_k({\mathcal{A}}_{k,\ell }),{\rho }_k\circ r_k({\mathcal{A}}_{k,r})){\}}_{k\in K}$$

in |${\mathcal{L}}({\mathcal{X}})$| with respect to |$(\mathbb{E}_{{\mathcal{L}}({\mathcal{X}})}, \mathbb{F}_{{\mathcal{L}}({\mathcal{X}})})$|⁠, where |$({\mathcal{X}}, {\mathcal{X}}^\circ, \mathfrak{p}, \mathfrak{q}) = (*_{\mathcal{B}})_{k \in K}(\mathcal{X}_k, \mathcal{X}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)$|⁠. □

It will be an immediate consequence of Theorem 5.5 below that the above definition does not depend on a specific choice of representations. Moreover, it follows immediately from the definition that if |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠, then the family |$\{{\mathcal{A}}_{k, \ell}\}_{k \in K}$| (respectively, |$\{{\mathcal{A}}_{k, r}\}_{k \in K}$|⁠) of left |${\mathcal{B}}$|-algebras (respectively, right |${\mathcal{B}}$|-algebras) is c-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠.

4 Operator-Valued C-Bi-Free Pairs of Functions

In order to study operator-valued conditional bi-free independence we must extend the notion of operator-valued bi-multiplicative functions to pairs of functions.

4.1 Operator-valued conditionally bi-multiplicative pairs of functions

We begin with an observation, which will be useful later.

 

Remark 4.1.

If |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| is a bi-non-crossing partition for some |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, then there exists a unique partition |$V_1, \dots, V_m$| of |$\{1,\ldots, n\}$| into |$\chi$|-intervals such that each |$V_k$| a union of blocks of |$\pi$| and such that |$\min_{\prec_\chi}(V_k)$| and |$\max_{\prec_\chi}(V_k)$| are in the same block of |$\pi$| for each |$k \in \{1, \dots, m\}$|⁠. Furthermore, by reordering if necessary, we may assume |$\max_{\prec_\chi}(V_k) \prec_\chi \min_{\prec_\chi} (V_{k+1})$| for all |$k$|⁠. For example, if |$\pi$| has the following bi-non-crossing diagram

then |$V_1 = \{\{1, 6\}, \{2, 4\}\}$|⁠, |$V_2 = \{\{7, 11\}, \{9, 12\}\}$|⁠, and |$V_3 = \{\{3, 8, 10\}, \{5\}\}$| where |$\min_{\prec_\chi}(V_1)=1$|⁠, |$\max_{\prec_\chi}(V_1) = 6$|⁠, |$\min_{\prec_\chi}(V_2)= 7$|⁠, |$\max_{\prec_\chi}(V_2)=11$|⁠, |$\min_{\prec_\chi}(V_3) = 10$|⁠, and |$\max_{\prec_\chi}(V_3)=3$|⁠. Note that the blocks |$V_k' \subset V_k$| containing |$\min_{\prec_\chi}(V_k)$| and |$\max_{\prec_\chi}(V_k)$| are the exterior blocks of |$\pi$|⁠. □

 

Definition 4.2.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, and let

$$\mathrm{\Psi }:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{B}$$

and

$$\mathrm{\Phi }:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{D}$$

be a pair of functions that are linear in each |${\mathcal{A}}_{\chi(k)}$|⁠. It is said that |$(\Psi, \Phi)$| is an operator-valued conditionally bi-multiplicative pair if for every |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠, |$b \in {\mathcal{B}}$|⁠, and |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, |$\Psi$| satisfies conditions |$(1)$| to |$(4)$| of Definition 2.5 (i.e., |$\Psi$| is operator-valued bi-multiplicative), and |$\Phi$| satisfies conditions |$(1)$| to |$(3)$| of Definition 2.5 and the following modification of condition |$(4)$|⁠: Under the same notation with the additional assumption that |$\min_{\prec_\chi}(\{1, \dots, n\})$| and |$\max_{\prec_\chi}(\{1, \dots, n\})$| are in the same block of |$\pi$|⁠, we have

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{\pi }(Z_1,\dots ,Z_n)& \hspace{0.17em}=\{\begin{array}{ll}\hfill {\mathrm{\Phi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{p-1},Z_p{L}_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)},Z_{p+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (p)=\ell \hfill \\ \hfill {\mathrm{\Phi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{p-1},R_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)}{Z}_p,Z_{p+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (p)=r\hfill \end{array}\hfill \\ & \hspace{0.17em}=\{\begin{array}{ll}\hfill {\mathrm{\Phi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{q-1},L_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)}{Z}_q,Z_{q+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (q)=\ell \hfill \\ \hfill {\mathrm{\Phi }}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{q-1},Z_q{R}_{{\mathrm{\Psi }}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)},Z_{q+1},\dots ,Z_n\right){|}_W\right)& \text{if }\chi (q)=r\hfill \end{array}.\hfill \end{array}$$

□

Note the additional assumption that |$\min_{\prec_\chi}(\{1, \dots, n\})$| and |$\max_{\prec_\chi}(\{1, \dots, n\})$| are in the same block of |$\pi$| guarantees that |$W$| contains an exterior block of |$\pi$| and |$V$| is a union of interior blocks of |$\pi$|⁠.

 

Example 4.3.

As with operator-valued bi-multiplicative functions, one may reduce |$\Phi_\pi(Z_1, \dots, Z_n)$| to an expression involving |$\Psi_{1_\chi}$| and |$\Phi_{1_\chi}$| for various |$\chi: \{1, \dots, m\} \to \{\ell, r\}$|⁠. For example, if |$\pi$| is the bi-non-crossing partition from Remark 4.1 and |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠, then

$${\mathrm{\Phi }}_{\pi }(Z_1,\dots ,Z_{12})={\mathrm{\Phi }}_{\pi {|}_{V_1}}(Z_1,Z_2,Z_4,Z_6){\mathrm{\Phi }}_{\pi {|}_{V_2}}(Z_7,Z_9,Z_{11},Z_{12}){\mathrm{\Phi }}_{\pi {|}_{V_3}}(Z_3,Z_5,Z_8,Z_{10})$$

by condition |$(3)$| of Definition 2.5, which can be further reduced to

$${\mathrm{\Phi }}_{1_{2,0}}\left(Z_1{L}_{{\mathrm{\Psi }}_{1_{2,0}}(Z_2,Z_4)},Z_6\right){\mathrm{\Phi }}_{1_{1,1}}\left(Z_7{L}_{{\mathrm{\Psi }}_{1_{1,1}}(Z_9,Z_{12})},Z_{11}\right){\mathrm{\Phi }}_{1_{0,3}}\left(Z_3,R_{{\mathrm{\Psi }}_{1_{0,1}}(Z_5)}{Z}_8,Z_{10}\right)$$

by the modified condition |$(4)$| of Definition 4.2. □

4.2 Operator-valued c-bi-free moment pairs

In this subsection, we define the operator-valued c-bi-free moment pair |$({\mathcal{E}}, {\mathcal{F}})$| and show that it is operator-valued conditionally bi-multiplicative.

 

Definition 4.4.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. The operator-valued c-bi-free moment pair on |${\mathcal{A}}$| is the pair of functions

$$\mathcal{E}:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{B}$$

and

$$\mathcal{F}:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{D},$$

where |${\mathcal{E}}$| is the operator-valued bi-free moment function on |${\mathcal{A}}$| and |${\mathcal{F}}_\pi(Z_1, \dots, Z_n)$| for |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, and |$Z_k \in {\mathcal{A}}_{\chi(k)}$| is defined as follows.

• (1) If |$\pi$| contains exactly one block (that is, |$\pi = 1_\chi$|⁠), define |${\mathcal{F}}_{1_\chi}(Z_1, \dots, Z_n) = {\mathbb{F}}(Z_1\cdots Z_n)$|⁠.

• (2) If |$V_1, \ldots, V_n$| are the blocks of |$\pi$|⁠, each |$V_k$| is a |$\chi$|-intervals (thus all exterior), and |$\max_{\prec_\chi}(V_k) \prec_\chi \min_{\prec_\chi}(V_{k+1})$| for all |$k$|⁠, define

  $${\mathcal{F}}_{\pi }(Z_1,\dots ,Z_n)={\mathcal{F}}_{\pi {|}_{V_1}}((Z_1,\dots ,Z_n){|}_{V_1})\cdots {\mathcal{F}}_{\pi {|}_{V_m}}((Z_1,\dots ,Z_n){|}_{V_m})$$

  and apply step |$(3)$| to each piece.

• (3) Apply a similar recursive process as in Definition 2.6 to the interior blocks of |$\pi$| as follows: Let |$V$| be the interior block of |$\pi$| that terminates closest to the bottom. Then

  • If |$V = \{k + 1, \dots, n\}$| for some |$k \in \{1, \dots, n - 1\}$|⁠, then |$\min(V)$| is not adjacent to any spine of |$\pi$| and define

    $${\mathcal{F}}_{\pi }(Z_1,\dots ,Z_n)=\{\begin{array}{ll}\hfill {\mathcal{F}}_{\pi {|}_{V^{\complement }}}(Z_1,\dots ,Z_k{L}_{{\mathcal{E}}_{\pi {|}_V}(Z_{k+1},\dots ,Z_n)})& \text{if }\chi (min(V))=\ell \hfill \\ \hfill {\mathcal{F}}_{\pi {|}_{V^{\complement }}}(Z_1,\dots ,Z_k{R}_{{\mathcal{E}}_{\pi {|}_V}(Z_{k+1},\dots ,Z_n)})& \text{if }\chi (min(V))=r\hfill \end{array}.$$
  • Otherwise, |$\min(V)$| is adjacent to a spine. Let |$W$| denote the block of |$\pi$| corresponding to the spine adjacent to |$\min(V)$| and let |$k$| be the smallest element of |$W$| that is larger than |$\min(V)$|⁠. Define

    $$\begin{array}{cc}& {\mathcal{F}}_{\pi }(Z_1,\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\{\begin{array}{l}\hfill {\mathcal{F}}_{\pi {|}_{V^{\complement }}}((Z_1,\dots ,Z_{k-1},L_{{\mathcal{E}}_{\pi {|}_V}((Z_1,\dots ,Z_n){|}_V)}{Z}_k,Z_{k+1},\dots ,Z_n){|}_{V^{\complement }})\text{ if }\chi (min(V))=\ell \\ \hfill {\mathcal{F}}_{\pi {|}_{V^{\complement }}}((Z_1,\dots ,Z_{k-1},R_{{\mathcal{E}}_{\pi {|}_V}((Z_1,\dots ,Z_n){|}_V)}{Z}_k,Z_{k+1},\dots ,Z_n){|}_{V^{\complement }})\text{ if }\chi (min(V))=r\end{array}.\hfill \end{array}$$

□

 

Example 4.5.

Again, let |$\pi$| be the bi-non-crossing partition from Remark 4.1 and |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. Then

$${\mathcal{F}}_{\pi }(Z_1,\dots ,Z_{12})=\mathbb{F}\left(Z_1{L}_{\mathbb{E}(Z_2{Z}_4)}{Z}_6\right)\mathbb{F}\left(Z_7{L}_{\mathbb{E}(Z_9{Z}_{12})}{Z}_{11}\right)\mathbb{F}\left(Z_3{R}_{\mathbb{E}(Z_5)}{Z}_8{Z}_{10}\right).$$

In general, the rule is ‘one uses |${\mathcal{E}}$| to reduce the interior blocks and then factors |${\mathcal{F}}_\pi$| according to the remaining exterior blocks.’ □

 

Theorem 4.6.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. The operator-valued c-bi-free moment pair |$({\mathcal{E}}, {\mathcal{F}})$| on |${\mathcal{A}}$| is operator-valued conditionally bi-multiplicative. □

 

Proof.

The fact that the operator-valued bi-free moment function |${\mathcal{E}}$| on |${\mathcal{A}}$| is operator-valued bi-multiplicative is the main result of [4, Section 5]. It is also clear that the function |${\mathcal{F}}$| satisfies conditions |$(1)$|⁠, |$(2)$|⁠, and |$(3)$| of Definition 2.5.

To demonstrate the modified condition |$(4)$| in Definition 4.2, the proof relies on the techniques observed in [4, Subsection 5.3] which show that the function |${\mathcal{E}}$| satisfies condition |$(4)$| of Definition 2.5. In particular, we refer the reader to the proofs of [4, Lemmata 5.3.1–5.3.4] for additional details in that which follows. Under the same assumptions and notation, first note that the special case of the assertion holds under the additional assumption of [4, Lemma 5.3.1]; that is, there exists a block |$W_0 \subset W$| of |$\pi$| such that

$$p,q,\underset{{\prec }_{\chi }}{min}(\{1,\dots ,n\}),\underset{{\prec }_{\chi }}{max}(\{1,\dots ,n\})\in W_0.$$

Indeed, suppose |$\chi(p) = \ell$| (the other case is similar), and note that |$W_0$| is the only exterior block of |$\pi$|⁠. By the same arguments as in the proof of [4, Lemma 5.3.1], we have

$${\mathcal{F}}_{\pi }(Z_1,\dots ,Z_n)={\mathcal{F}}_{\pi {|}_W}\left(\left(Z_1,\dots ,Z_{p-1},Z_p{L}_{{\mathcal{E}}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)},Z_{p+1},\dots ,Z_n\right){|}_W\right)$$

for all three possible cases, that is, |$\chi(q) = \ell$|⁠; |$\chi(q) = r$| and |$p < q$|⁠; |$\chi(q) = r$| and |$p > q$|⁠.

To verify the modified condition |$(4)$| in full generality, we examine the proof of [4, Lemma 5.3.4]. Suppose |$\chi(p) = \ell$| (the other case is similar), and note that under the additional assumption of the modified condition |$(4)$| that there exists a block |$W_0 \subset W$| of |$\pi$| such that |$\min_{\prec_\chi}(\{1, \dots, n\}), \max_{\prec_\chi}(\{1, \dots, n\}) \in W_0$|⁠, the block |$W_0$| is always the only exterior block of |$\pi$|⁠. Let

$$\alpha =\underset{{\prec }_{\chi }}{max}\left(\left\{k\in W_0\mid k{⪯}_{\chi }p\right\}\right),\beta =\underset{{\prec }_{\chi }}{min}\left(\left\{k\in W_0\mid q{⪯}_{\chi }k\right\}\right)$$

and let |$U = \{k \, \mid \, \alpha \prec_\chi k \prec_\chi \beta\}$|⁠. Thus |$U$| is a union of blocks of |$\pi$|⁠. Let |$\overline{W_0} = U^{\complement}$|⁠. Then, by the special case above (with |$U$| being the |$\chi$|-interval), we have

$${\mathcal{F}}_{\pi }(Z_1,\dots ,Z_n)={\mathcal{F}}_{\pi {|}_{\overline{W_0}}}\left(\left(Z_1,\dots ,Z_{\alpha -1},Z_{\alpha }{L}_{{\mathcal{E}}_{\pi {|}_U}\left((Z_1,\dots ,Z_n){|}_U\right)},Z_{\alpha +1},\dots ,Z_n\right){|}_{\overline{W_0}}\right).$$

Since |${\mathcal{E}}$| is operator-valued bi-multiplicative, we have

$$Z_{\alpha }{L}_{{\mathcal{E}}_{\pi {|}_U}\left((Z_1,\dots ,Z_n){|}_U\right)}=Z_p{L}_{{\mathcal{E}}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)}{L}_{{\mathcal{E}}_{\pi {|}_{U\setminus V}}\left((Z_1,\dots ,Z_n){|}_{U\setminus V}\right)}$$

if |$\alpha = p$|⁠, and

$${\mathcal{E}}_{\pi {|}_U}\left((Z_1,\dots ,Z_n){|}_U\right)={\mathcal{E}}_{\pi {|}_{U\setminus V}}\left(\left(Z_1,\dots ,Z_{p-1},Z_p{L}_{{\mathcal{E}}_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)},Z_{p+1},\dots ,Z_n\right){|}_{U\setminus V}\right)$$

otherwise. Since |$W = \overline{W_0} \cup (U \setminus V)$|⁠, the assertion follows from applying the special case above in the opposite direction. ■

4.3 Operator-valued c-bi-free cumulant pairs

In this subsection, we recursively define the operator-valued c-bi-free cumulant pair |$(\kappa, \mathcal{K})$| using the pair |$({\mathcal{E}}, {\mathcal{F}})$| from the previous subsection and show that it is also operator-valued conditionally bi-multiplicative.

 

Definition 4.7.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations and let |$({\mathcal{E}}, {\mathcal{F}})$| be the operator-valued c-bi-free moment pair on |${\mathcal{A}}$|⁠. The operator-valued c-bi-free cumulant pair on |${\mathcal{A}}$| is the pair of functions

$$\kappa :\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{B}$$

and

$$\mathcal{K}:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{D},$$

where |$\kappa$| is the operator-valued bi-free cumulant function on |${\mathcal{A}}$| and |${\mathcal{K}}$| is recursively defined as follows.

• (1) If |$n = 1$|⁠, then |${\mathcal{K}}_{1_{1,0}}(Z_\ell) = {\mathcal{F}}_{1_{1,0}}(Z_\ell)$| for |$Z_\ell \in {\mathcal{A}}_\ell$| and |${\mathcal{K}}_{1_{0,1}}(Z_r) = {\mathcal{F}}_{1_{0,1}}(Z_r)$| for |$Z_r \in {\mathcal{A}}_r$|⁠.

• (2) Fix |$n \geq 2$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, and |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. If |$\pi \neq 1_\chi$|⁠, then let |$V_1, \dots, V_m$| be the partition of |$\pi$| as described in Remark 4.1. We define

  $${\mathcal{K}}_{\pi }(Z_1,\dots ,Z_n)={\mathcal{K}}_{\pi {|}_{V_1}}((Z_1,\dots ,Z_n){|}_{V_1})\cdots {\mathcal{K}}_{\pi {|}_{V_m}}((Z_1,\dots ,Z_n){|}_{V_m}),$$

  where each |${\mathcal{K}}_{\pi|_{V_k}}((Z_1, \dots, Z_n)|_{V_k})$| is defined as follows. Let |$V'_k \subset V_k$| be the block containing |$\min_{\prec_\chi}(V_k)$| and |$\max_{\prec_\chi}(V_k)$|⁠, let |$V \subset V_k \setminus V_k'$| be the block which terminates closest to the bottom (compared to other blocks of |$V_k$|⁠). If |$p = \max_{\prec_\chi}\left(\left\{j \in V_k \, \mid \, j \prec_\chi \min_{\prec_\chi}(V)\right\}\right)$| define

  $$\begin{array}{cc}& {\mathcal{K}}_{\pi {|}_{V_k}}((Z_1,\dots ,Z_n){|}_{V_k})\hfill \\ & \hspace{0.17em}=\{\begin{array}{ll}\hfill {\mathcal{K}}_{\pi {|}_{V_k\setminus V}}\left(\left(Z_1,\dots ,Z_{p-1},Z_p{L}_{{\kappa }_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)},Z_{p+1},\dots ,Z_n\right){|}_{V_k\setminus V}\right)& \text{if }\chi (p)=\ell \hfill \\ \hfill {\mathcal{K}}_{\pi {|}_{V_k\setminus V}}\left(\left(Z_1,\dots ,Z_{p-1},R_{{\kappa }_{\pi {|}_V}\left((Z_1,\dots ,Z_n){|}_V\right)}{Z}_p,Z_{p+1},\dots ,Z_n\right){|}_{V_k\setminus V}\right)& \text{if }\chi (p)=r\hfill \end{array}.\hfill \end{array}$$

  Repeat this process until the only remaining block of |$V_k$| is |$V'_k$|⁠.

• (3) Otherwise |$\pi = 1_\chi$| and define

  $${\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_n)={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \ne 1_{\chi }\end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_n).$$

□

 

Theorem 4.8.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. The operator-valued c-bi-free cumulant pair |$(\kappa, {\mathcal{K}})$| on |${\mathcal{A}}$| is operator-valued conditionally bi-multiplicative. □

 

Proof.

The fact that the operator-valued bi-free cumulant function |$\kappa$| on |${\mathcal{A}}$| is operator-valued bi-multiplicative was proved in [4, Section 6]. Moreover, it is easy to see that the function |${\mathcal{K}}$| satisfies condition |$(3)$| of Definition 2.5.

For condition |$(1)$| of Definition 2.5, we will proceed by induction on |$n$| to show that condition |$(1)$| holds in greater generality. To be specific, we will demonstrate that condition |$(1)$| holds whenever |$1_\chi$| is replaced with |$\pi$|⁠. To proceed, note the base case where |$n = 1$| is trivial. For the inductive step, suppose the assertion holds for all |$1 \leq n_0 \leq n - 1$|⁠, |$\chi_0: \{1, \dots, n_0\} \to \{\ell, r\}$|⁠, and |$\pi_0 \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_0)$|⁠. Suppose |$\chi : \{1,\ldots, n\} \to \{\ell, r\}$| and that |$\chi(n) = \ell$| (as the other case is similar). If |$q = -\infty$|⁠, then |$\chi: \{1, \dots, n\} \to \{\ell\}$| is the constant map, and thus for each |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, |$n$| necessarily belongs to an exterior block of |$\pi$|⁠. Since |${\mathcal{K}}_\pi(Z_1, \dots, Z_{n - 1}, Z_nL_b)$| factors according to the exterior blocks of |$\pi$|⁠, we have |${\mathcal{K}}_\pi(Z_1, \dots, Z_{n - 1}, Z_nL_b) = {\mathcal{K}}_\pi(Z_1, \dots, Z_n)b$| if |$\pi \neq 1_\chi$| by the induction hypothesis. Thus

$$\begin{array}{cc}{\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},Z_n{L}_b)& \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},Z_n{L}_b)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \ne 1_{\chi }\end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{n-1},Z_n{L}_b)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)b-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \ne 1_{\chi }\end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_n)b\hfill \\ & \hspace{0.17em}={\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_n)b.\hfill \end{array}$$

If |$q \neq -\infty$| and |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| such that |$\pi \neq 1_\chi$|⁠, then as |$n$| and |$q$| are adjacent with respect to |$\prec_\chi$|⁠, we have the following possible cases:

• (i) |$n, q \in V$| such that |$V$| is an interior block of |$\pi$|⁠;

• (ii) |$n, q \in V$| such that |$V$| is an exterior block of |$\pi$|⁠;

• (iii) |$n \in V_1$| and |$q \in V_2$| such that |$n= \max_{\prec_\chi}(V_1) \prec_\chi \min_{\prec_\chi}(V_2) = q$|⁠, and both of |$V_1$| and |$V_2$| are interior blocks of |$\pi$|⁠;

• (iv) |$n \in V_1$| and |$q \in V_2$| such that |$n= \max_{\prec_\chi}(V_1) \prec_\chi \min_{\prec_\chi}(V_2)= q$|⁠, and both of |$V_1$| and |$V_2$| are exterior blocks of |$\pi$|⁠;

• (v) |$n \in V_1$| and |$q \in V_2$| such that |$\min_{\prec_\chi}(V_2) \prec_\chi \min_{\prec_\chi}(V_1)$| (thus |$V_1$| is interior with respect to |$V_2$|⁠), and |$V_2$| is an interior block of |$\pi$|⁠;

• (vi) |$n \in V_1$| and |$q \in V_2$| such that |$\min_{\prec_\chi}(V_2) \prec_\chi \min_{\prec_\chi}(V_1)$| (thus |$V_1$| is interior with respect to |$V_2$|⁠), and |$V_2$| is an exterior block of |$\pi$|⁠;

• (vii) |$n \in V_1$| and |$q \in V_2$| such that |$\max_{\prec_\chi}(V_2) \prec_\chi \max_{\prec_\chi}(V_1)$| (thus |$V_2$| is interior with respect to |$V_1$|⁠), and |$V_1$| is an interior block of |$\pi$|⁠;

• (viii) |$n \in V_1$| and |$q \in V_2$| such that |$\max_{\prec_\chi}(V_2) \prec_\chi \max_{\prec_\chi}(V_1)$| (thus |$V_2$| is interior with respect to |$V_1$|⁠), and |$V_1$| is an exterior block of |$\pi$|⁠.

Since |$\pi \neq 1_\chi$|⁠, cases |$(i)$|⁠, |$(ii)$|⁠, |$(iii)$|⁠, |$(v)$|⁠, |$(vi)$|⁠, |$(vii)$|⁠, and |$(viii)$| follow from the induction hypothesis and from the fact that |$\kappa$| is operator-valued bi-multiplicative. For case |$(iv)$|⁠, |$V_1 \subset \chi^{-1}(\{\ell\})$| and |$V_2 \subset \chi^{-1}(\{r\})$|⁠, so the result follows from the |$q = -\infty$| situation (and the proof where |$\chi(n) = r$| which must be run simultaneously with induction). Therefore, we have

$${\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{n-1},Z_n{L}_b)={\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{q-1},Z_q{R}_b,Z_{q+1},\dots ,Z_n)$$

for all |$\pi \neq 1_\chi$|⁠, and hence

$${\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},Z_n{L}_b)={\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_{q-1},Z_q{R}_b,Z_{q+1},\dots ,Z_n)$$

by the same calculation as the |$q = -\infty$| situation.

The verification for condition |$(2)$| of Definition 2.5 follows from essentially the same induction arguments and casework as above with |$p$| replacing |$n$|⁠. The only difference is that if |$q = -\infty$|⁠, then |$p$| is the smallest element with respect to |$\prec_\chi$|⁠, and hence necessarily belongs to an exterior block of |$\pi$|⁠. Note this shows that the function |${\mathcal{K}}$| actually satisfies the additional properties that conditions |$(1)$| and |$(2)$| of Definition 2.5 hold for all |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠.

Finally, for the modified condition |$(4)$| as given in Definition 4.2, the result follows from the extended conditions |$(1)$| and |$(2)$| of Definition 2.5 as stated above along with the recursive definition in Definition 4.4 and the fact that |$\kappa$| is operator-valued bi-multiplicative. ■

5 Universal Moment Expressions for C-Bi-Free Independence with Amalgamation

In this section, we will demonstrate that a family of pairs of |${\mathcal{B}}$|-algebras is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| if and only if certain operator-valued moment expressions hold. To do so, we note that the shaded diagrams from Definition 2.3 and [4, Lemma 7.1.3] will be useful.

 

Definition 5.1.

Let |$\{({\mathcal{X}}_k, {\mathcal{X}}_k^\circ, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$| be a family of |${\mathcal{B}}$|-|${\mathcal{B}}$|-bimodules with pairs of specified |$({\mathcal{B}}, {\mathcal{D}})$|-valued states, let |$\lambda_k$| and |$\rho_k$| be the left and right representations of |${\mathcal{L}}({\mathcal{X}}_k)$| on |${\mathcal{L}}({\mathcal{X}})$|⁠, and let |${\mathcal{X}} = (*_{\mathcal{B}})_{k \in K}{\mathcal{X}}_k$|⁠. Fix |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\omega: \{1, \dots, n\} \to K$|⁠, |$Z_k \in {\mathcal{L}}_{\chi(k)}({\mathcal{X}}_{\omega(k)})$|⁠, and let |$\mu_k(Z_k) = \lambda_{\omega(k)}(Z_k)$| if |$\chi(k) = \ell$| and |$\mu_k(Z_k) = \rho_{\omega(k)}(Z_k)$| if |$\chi(k) = r$|⁠.

For |$D \in {\mathcal{L}}{\mathcal{R}}^\mathrm{lat}(\chi, \omega)$|⁠, recursively define |${\mathbb{E}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$| as follows: If |$D \in {\mathcal{L}}{\mathcal{R}}_0^\mathrm{lat}(\chi, \omega)$|⁠, then

$${\mathbb{E}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))=({\mathcal{E}}_{\mathcal{L}(\mathcal{X})}{)}_{\pi }({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\in \mathcal{B},$$

where |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| is the bi-non-crossing partition corresponding to |$D$|⁠. If every block of |$D$| has a spine reaching the top, then enumerate the blocks from left to right according to their spines as |$V_1, \dots, V_m$| with |$V_j = \{k_{j, 1} < \cdots < k_{j, q_j}\}$|⁠, and set

$${\mathbb{E}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))=[(1-{\mathfrak{p}}_{\omega (k_{1,1})})Z_{k_{1,1}}\cdots Z_{k_{1,q_1}}(1\oplus 0)]\otimes \cdots \otimes [(1-{\mathfrak{p}}_{\omega (k_{m,1})})Z_{k_{m,1}}\cdots Z_{k_{m,q_m}}(1\oplus 0)],$$

which is an element of |${\mathcal{X}}^\circ$|⁠. Otherwise, apply the recursive process using |${\mathbb{E}}_{{\mathcal{L}}({\mathcal{X}})}$| as in Definition 2.6 until every block of |$D$| has a spine reaching the top. □

Under the above assumptions and notation, it was demonstrated in [4, Lemma 7.1.3] that

$${\mu }_1(Z_1)\cdots {\mu }_n(Z_n)(1\oplus 0)=\sum _{k=0}^n\sum _{D\in \mathcal{L}{\mathcal{R}}_k^{\mathrm{l}\mathrm{a}\mathrm{t}}(\chi ,\omega )}\left[\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}{\mathcal{R}}_k(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{l}\mathrm{a}\mathrm{t}}D\end{array}}(-1{)}^{|D|-|D^{\prime }|}\right]{\mathbb{E}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))$$

(3)

and, consequently,

$${\mathbb{E}}_{\mathcal{L}(\mathcal{X})}({\mu }_1(Z_1)\cdots {\mu }_n(Z_n))=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )}\left[\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \le \sigma \le \omega \end{array}}{\mu }_{\mathcal{B}\mathcal{N}\mathcal{C}}(\pi ,\sigma )\right]({\mathcal{E}}_{\mathcal{L}(\mathcal{X})}{)}_{\pi }({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n)).$$

(4)

For |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$|⁠, we define |${\mathbb{E}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$| by exactly the same recursive process that used to define |${\mathbb{E}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))$| for |$D' \in {\mathcal{L}}{\mathcal{R}}^\mathrm{lat}(\chi, \omega)$|⁠. Note that, unlike |${\mathbb{E}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))$|⁠, it is not necessarily true that |${\mathbb{E}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) \in {\mathcal{X}}$| for all |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$| as such diagrams may have spines reaching the top which do not alternate in colour.

If |${\mathbb{E}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) = X_1 \otimes \cdots \otimes X_m$|⁠, let

$$\mathfrak{q}({\mathbb{E}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n)))=\mathfrak{q}(X_1)\cdots \mathfrak{q}(X_m)\in \mathcal{D}.$$

Observe that although it is possible |$X_1 \otimes \cdots \otimes X_m \notin {\mathcal{X}}^\circ$|⁠, it is still true that every |$X_j$| belongs to some |${\mathcal{X}}_{k_j}^\circ$|⁠, and thus the above expression makes sense.

Finally for |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$|⁠, recursively define |${\mathbb{F}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$| as follows: If |$D \in {\mathcal{L}}{\mathcal{R}}_0^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$|⁠, then

$${\mathbb{F}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))={\mathbb{E}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\in \mathcal{B}\subset \mathcal{D}.$$

If every block of |$D$| has a spine reaching the top, then enumerate the blocks from left to right according to their spines as |$V_1, \dots, V_m$| with |$V_j = \{k_{j, 1} < \cdots < k_{j, q_j}\}$|⁠, and set

$${\mathbb{F}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))={\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}\cdots Z_{k_{1,q_1}})\cdots {\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{m,1}}\cdots Z_{k_{m,q_m}})\in \mathcal{D}.$$

Otherwise, apply the recursive process using |${\mathbb{E}}_{{\mathcal{L}}({\mathcal{X}})}$| as in Definition 2.6 until every block of |$D$| has a spine reaching the top.

Note the values of |${\mathbb{F}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$| depend only on the values of |${\mathbb{F}}_{{\mathcal{L}}({\mathcal{X}}_{k_j})}(Z_{k_{j, 1}}\cdots Z_{k_{j, q_j}})$| and the values of |${\mathbb{E}}_{{\mathcal{L}}({\mathcal{X}}_{k_j})}(Z_{k_{j, 1}}\cdots Z_{k_{j, q_j}})$| for some |$k_j \in K$|⁠. Hence |${\mathbb{F}}_D$| makes sense in any |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-vector expectations and will not depend on the representation of the pairs of |${\mathcal{B}}$|-algebras.

 

Lemma 5.2.

Under the above assumptions and notation, for all |$D \in {\mathcal{L}}{\mathcal{R}}^\mathrm{lat}(\chi, \omega)$|

$$\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}{\mathcal{R}}^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )\\ D^{\prime }{\le }_{\mathrm{c}\mathrm{a}\mathrm{p}}D\end{array}}\mathfrak{q}\left({\mathbb{E}}_{D^{\prime }}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)={\mathbb{F}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n)).$$

□

 

Proof.

If |$D \in {\mathcal{L}}{\mathcal{R}}^\mathrm{lat}_0(\chi, \omega)$|⁠, then the only diagram |$D' \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$| such that |$D' \leq_\mathrm{cap} D$| is |$D$| itself. Thus the equation is trivially true by definition in this case.

For |$D \in {\mathcal{L}}{\mathcal{R}}_m^\mathrm{lat}(\chi, \omega)$| with |$0 < m \leq n$|⁠, it suffices to prove the following claim: Let |$V_1, \dots, V_m$| be the blocks of |$D$| with spines reaching the top, ordered from left to right according to their spines, let |$V_1 = \{k_{1, 1} < \cdots < k_{1, q_1}\}$|⁠, and let |$V_{1, 1}, \dots, V_{1, m_1}$| be the blocks of |$D$| which reduce to appropriate |$L_b$| or |$R_b$| multiplied on the left and/or right of some |$Z_{k_{1, j}}$| in the recursive process. Suppose |$D', D'' \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$| are such that |$D' \leq_{\mathrm{cap}} D$|⁠, |$D'' \leq_{\mathrm{cap}} D$|⁠, the spine of the block |$V_1$| reaches the top in |$D'$| but not in |$D''$|⁠, and the spines of all other blocks in |$D'$| and |$D''$| agree. We claim that

$$\begin{array}{cc}& \mathfrak{q}\left({\mathbb{E}}_{D^{\prime }}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)+\mathfrak{q}\left({\mathbb{E}}_{D^{″}}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)\hfill \\ & \hspace{0.17em}={\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}^{\prime }\cdots Z_{k_{1,q_1}}^{\prime })\mathfrak{q}\left({\mathbb{E}}_{D^{\prime }\setminus (V_1\cup V_{1,1}\cup \cdots \cup V_{1,m_1})}\left(({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n)){|}_{D^{\prime }\setminus (V_1\cup V_{1,1}\cup \cdots \cup V_{1,m_1})}\right)\right),\hfill \end{array}$$

where |$Z'_{k_{1, j}}$| is |$Z_{k_{1, j}}$|⁠, potentially multiplied on the left and/or right by appropriate |$L_b$| and |$R_b$| such that the multiplications correspond to the blocks |$V_{1, 1}, \dots, V_{1, m_1}$|⁠.

Indeed, if the claim is true, then for a given |$D$| as above, the spine of |$V_1$| reaches the top in exactly half of the cappings of |$D$| and each such capping |$D'$| can be paired with another capping |$D''$| such that the only difference between |$D'$| and |$D''$| is that the spine of |$V_1$| does not reach the top in |$D''$|⁠. Adding up |$\mathfrak{q}\left({\mathbb{E}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)$| and |$\mathfrak{q}\left({\mathbb{E}}_{D''}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)$| for all pairs yield the result by induction.

To prove the claim, note if |$m = 1$| (that is, the only spine that reaches the top is the spine of |$V_1$|⁠), then |$V_1 \cup V_{1, 1} \cup \cdots \cup V_{1, m_1} = D'$| and we have

$$\begin{array}{cc}\hfill \mathfrak{q}\left({\mathbb{E}}_{D^{\prime }}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)& \hspace{0.17em}={\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}^{\prime }\cdots Z_{k_{1,q_1}}^{\prime })-{\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}^{\prime }\cdots Z_{k_{1,q_1}}^{\prime })\hfill \\ & \hspace{0.17em}={\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}^{\prime }\cdots Z_{k_{1,q_1}}^{\prime })-\mathfrak{q}\left({\mathbb{E}}_{D^{″}}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)\hfill \end{array}$$

as |$D''$| has no spine reaching the top and |$\mathfrak{q}(b) = b$|⁠. Thus the result follows when |$m = 1$|⁠.

Otherwise, |$m > 1$|⁠. Let |$V = V_1 \cup V_{1, 1} \cup \cdots \cup V_{1, m_1}$|⁠. Since left |${\mathcal{B}}$|-operators commute with elements of |${\mathcal{L}}_r({\mathcal{X}})$|⁠, right |${\mathcal{B}}$|-operators commute with elements of |${\mathcal{L}}_\ell({\mathcal{X}})$|⁠, and by the properties of |${\mathbb{E}}_{{\mathcal{L}}({\mathcal{X}})}$| and |${\mathbb{F}}_{{\mathcal{L}}({\mathcal{X}})}$| (i.e., there are bi-multiplicative-like properties implied by the recursive definition), it can be checked via casework that

$$\begin{array}{cc}\hfill \mathfrak{q}& \left({\mathbb{E}}_{D^{\prime }}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)\hfill \\ & \hspace{0.17em}=\left({\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}^{\prime }\cdots Z_{k_{1,q_1}}^{\prime })-{\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}^{\prime }\cdots Z_{k_{1,q_1}}^{\prime })\right)\mathfrak{q}\left({\mathbb{E}}_{D^{\prime }\setminus V}\left(({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n)){|}_{D^{\prime }\setminus V}\right)\right)\hfill \end{array}$$

and

$${\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(Z_{k_{1,1}}^{\prime }\cdots Z_{k_{1,q_1}}^{\prime })\mathfrak{q}\left({\mathbb{E}}_{D^{\prime }\setminus V}\left(({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n)){|}_{D^{\prime }\setminus V}\right)\right)=\mathfrak{q}\left({\mathbb{E}}_{D^{″}}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)$$

for all |$D'$| and |$D''$|⁠. Thus the claim and proof follows. ■

To keep track of some coefficients that occur, we make the following definition.

 

Definition 5.3.

For |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}_k(\chi, \omega)$|⁠, define |$C'_D$| as follows: First define

$$C_D=\{\begin{array}{ll}{\displaystyle \sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}{\mathcal{R}}_k(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{l}\mathrm{a}\mathrm{t}}D\end{array}}(-1{)}^{|D|-|D^{\prime }|}}& \text{if }D\in \mathcal{L}{\mathcal{R}}_k^{\mathrm{l}\mathrm{a}\mathrm{t}}(\chi ,\omega )\hfill \\ 0& \text{otherwise}\hfill \end{array}.$$

Recursively, starting with |$k = n$|⁠, define

$$C_D^{\prime }=C_D-\sum _{m=k+1}^n\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}{\mathcal{R}}_m^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{c}\mathrm{a}\mathrm{p}}D\end{array}}{C}_{D^{\prime }}^{\prime }.$$

□

With Lemma 5.2 complete, we obtain the following operator-valued analogue of [6, Lemma 4.6].

 

Lemma 5.4.

Under the above assumptions and notation,

$${\mathbb{F}}_{\mathcal{L}(\mathcal{X})}({\mu }_1(Z_1)\cdots {\mu }_n(Z_n))=\sum _{k=0}^n\sum _{D\in \mathcal{L}{\mathcal{R}}_k^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )}{C}_D^{\prime }{\mathbb{F}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))$$

and

$$C_D^{\prime }=\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}\mathcal{R}(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}D\end{array}}(-1{)}^{|D|-|D^{\prime }|}=\sum _{m=k}^n\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}{\mathcal{R}}_m(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}D\end{array}}(-1{)}^{|D|-|D^{\prime }|}$$

for |$D \in \mathcal{L}\mathcal{R}^{\mathrm{lat}\mathrm{cap}}_k(\chi, \omega)$|⁠. □

 

Proof.

For |$Z_1, \dots, Z_n$| as above, the expression |${\mathbb{F}}_{{\mathcal{L}}({\mathcal{X}})}(\mu_1(Z_1)\cdots\mu_n(Z_n))$| is obtained by applying |$\mathfrak{q}$| to the left-hand side of equation (3). Using Definition 5.3, we have

$$\begin{array}{cc}& {\mathbb{F}}_{\mathcal{L}(\mathcal{X})}({\mu }_1(Z_1)\cdots {\mu }_n(Z_n))\hfill \\ & \hspace{0.17em}=\mathfrak{q}\left({\mu }_1(Z_1)\cdots {\mu }_n(Z_n)(1\oplus 0)\right)\hfill \\ & \hspace{0.17em}=\sum _{k=0}^n\sum _{D\in \mathcal{L}{\mathcal{R}}_k^{\mathrm{l}\mathrm{a}\mathrm{t}}(\chi ,\omega )}{C}_D\mathfrak{q}\left({\mathbb{E}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)\hfill \\ & \hspace{0.17em}=\sum _{k=0}^n\sum _{D\in \mathcal{L}{\mathcal{R}}_k^{\mathrm{l}\mathrm{a}\mathrm{t}}(\chi ,\omega )}{C}_D\left({\mathbb{F}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))-\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}{\mathcal{R}}^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )\\ D^{\prime }{\le }_{\mathrm{c}\mathrm{a}\mathrm{p}}D\\ D^{\prime }\ne D\end{array}}\mathfrak{q}\left({\mathbb{E}}_{D^{\prime }}({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n))\right)\right)\hfill \\ & \hspace{0.17em}=\sum _{k=0}^n\sum _{D\in \mathcal{L}{\mathcal{R}}_k^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )}{C}_D^{\prime }{\mathbb{F}}_D({\mu }_1(Z_1),\dots ,{\mu }_n(Z_n)),\hfill \end{array}$$

where the third equality follows from Lemma 5.2 and the fourth equality follows from Definition 5.3 as the coefficient |$C'_D$| for |$D \in \mathcal{L}\mathcal{R}^{\mathrm{lat}\mathrm{cap}}_k(\chi, \omega)$| was specifically defined this way. The second result regarding |$C'_D$| is exactly the content of [6, Lemma 4.7]. ■

Combining these results, we have the following moment type characterization of c-bi-free independence with amalgamation.

 

Theorem 5.5.

A family |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| of pairs of |${\mathcal{B}}$|-algebras in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| if and only if

$$\mathbb{E}(Z_1\cdots Z_n)=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )}\left[\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \le \sigma \le \omega \end{array}}{\mu }_{\mathcal{B}\mathcal{N}\mathcal{C}}(\pi ,\sigma )\right]{\mathcal{E}}_{\pi }(Z_1,\dots ,Z_n)$$

(5)

and

$$\mathbb{F}(Z_1\cdots Z_n)=\sum _{D\in \mathcal{L}{\mathcal{R}}^{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}(\chi ,\omega )}\left[\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}\mathcal{R}(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}D\end{array}}(-1{)}^{|D|-|D^{\prime }|}\right]{\mathbb{F}}_D(Z_1,\dots ,Z_n)$$

(6)

for all |$n \geq 1$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\omega: \{1, \dots, n\} \to K$|⁠, and |$Z_1, \dots, Z_n \in {\mathcal{A}}$| with |$Z_k \in {\mathcal{A}}_{\omega(k), \chi(k)}$|⁠. □

 

Proof.

Under the above notation, if the family |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠, then there exists a family |$\{({\mathcal{X}}_k, {\mathcal{X}}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$| such that

$$\mathbb{E}(Z_1\cdots Z_n)={\mathbb{E}}_{\mathcal{L}(\mathcal{X})}({\mu }_1(Z_1)\cdots {\mu }_n(Z_n))\text{and}\mathbb{F}(Z_1\cdots Z_n)={\mathbb{F}}_{\mathcal{L}(\mathcal{X})}({\mu }_1(Z_1)\cdots {\mu }_n(Z_n)),$$

where each |$Z_k$| on the right-hand side of the above equations is identified as |$\ell_k(Z_k)$| if |$\chi(k) = \ell$| and |$r_k(Z_k)$| if |$\chi(k) = r$| acting on |${\mathcal{X}}_{\omega(k)}$|⁠. The fact that equation (5) holds is part of [4, Theorem 7.1.4], and the fact that equation (6) holds follows from Lemma 5.4.

Conversely, suppose equations (5) and (6) hold. By Theorem 3.5, there exist |$(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$| and a unital homomorphism |$\theta: {\mathcal{A}} \to {\mathcal{L}}({\mathcal{X}})$| such that

$$\begin{array}{c}\theta (L_{b_1}{R}_{b_2})=L_{b_1}{R}_{b_2},\theta ({\mathcal{A}}_{\ell })\subset {\mathcal{L}}_{\ell }(\mathcal{X}),\theta ({\mathcal{A}}_r)\subset {\mathcal{L}}_r(\mathcal{X}),\\ {\mathbb{E}}_{\mathcal{L}(\mathcal{X})}(\theta (Z))=\mathbb{E}(Z),\text{and}{\mathbb{F}}_{\mathcal{L}(\mathcal{X})}(\theta (Z))=\mathbb{F}(Z)\end{array}$$

for all |$b_1, b_2 \in {\mathcal{B}}$| and |$Z \in {\mathcal{A}}$|⁠. For each |$k \in K$|⁠, let |$({\mathcal{X}}_k, {\mathcal{X}}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)$| be a copy of |$({\mathcal{X}}, {\mathcal{X}}^\circ, \mathfrak{p}, \mathfrak{q})$|⁠, and let |$\ell_k$| and |$r_k$| be copies of |$\theta: {\mathcal{A}} \to {\mathcal{L}}({\mathcal{X}}_k)$|⁠. By [4, Lemma 7.1.3] and Lemma 5.4, we have

$$\begin{array}{cc}& \mathbb{E}(Z_1\cdots Z_n)={\mathbb{E}}_{\mathcal{L}(({\ast }_{\mathcal{B}}{)}_{k\in K}{\mathcal{X}}_k)}({\mu }_1(Z_1)\cdots {\mu }_n(Z_n))\text{and}\hfill \\ & \mathbb{F}(Z_1\cdots Z_n)={\mathbb{F}}_{\mathcal{L}(({\ast }_{\mathcal{B}}{)}_{k\in K}{\mathcal{X}}_k)}({\mu }_1(Z_1)\cdots {\mu }_n(Z_n)),\hfill \end{array}$$

where each |$Z_k$| on the right-hand side of the above equations is identified as |$\theta(Z_k)$| acting on |${\mathcal{X}}_{\omega(k)}$|⁠. Hence, the family |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| by definition. ■

As |${\mathbb{F}}_D(Z_1, \ldots, Z_n)$| and |${\mathcal{E}}_\pi(Z_1, \ldots, Z_n)$| depend only on the distributions of individual pairs |$({\mathcal{A}}_{k,\ell}, {\mathcal{A}}_{k, r})$| inside our |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, we obtain that Definition 3.8 is well-defined in that the joint distributions do not depend on the representations.

6 Additivity of Operator-Valued C-Bi-Free Cumulant Pairs

The goal of this section is to prove the operator-valued analogue of [6, Theorem 4.1]; namely that c-bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and c-bi-free cumulants. To establish the result, we will need a method, analogous to [11, Lemma 3.8] for constructing a pair of |${\mathcal{B}}$|-algebras with any given operator-valued bi-free and c-bi-free cumulants. To this end, we discuss moment and cumulant series first.

Let |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| be a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, and let |$({\mathcal{C}}_\ell, {\mathcal{C}}_r)$| be a pair of |${\mathcal{B}}$|-algebras such that

$${\mathcal{C}}_{\ell }=\mathrm{a}\mathrm{l}\mathrm{g}(\{Z_i{\}}_{i\in I},\epsilon (\mathcal{B}\otimes 1))\text{and}{\mathcal{C}}_r=\mathrm{a}\mathrm{l}\mathrm{g}(\{Z_j{\}}_{j\in J},\epsilon (1\otimes {\mathcal{B}}^{\mathrm{o}\mathrm{p}}))$$

for some |$\{Z_i\}_{i \in I} \subset {\mathcal{A}}_\ell$| and |$\{Z_j\}_{j \in J} \subset {\mathcal{A}}_r$|⁠. By discussions in [11, Section 2] and by using the operator-valued conditionally bi-multiplicative properties, only certain operator-valued bi-free and c-bi-free moments/cumulants are required to study the joint distributions of elements in |$\mathrm{alg}({\mathcal{C}}_\ell, {\mathcal{C}}_r)$| with respect to |$({\mathbb{E}}, {\mathbb{F}})$|⁠. We make the following notation (in addition to [11, Notation 2.18] with some slight notational changes) and definition to describe the necessary moments and cumulants.

 

Notation 6.1.

Let |${\mathcal{Z}} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$| be as above, |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠.

• If |$\omega(k) \in I$| for all |$k$|⁠, define

  $$\begin{array}{cc}\hfill {\nu }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\mathbb{E}\left(Z_{\omega (1)}{L}_{b_1}{Z}_{\omega (2)}\cdots L_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{B},\hfill \\ \hfill {\mu }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\mathbb{F}\left(Z_{\omega (1)}{L}_{b_1}{Z}_{\omega (2)}\cdots L_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{D},\hfill \\ \hfill {\rho }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}={\kappa }_{1_{{\chi }_{\omega }}}\left(Z_{\omega (1)},L_{b_1}{Z}_{\omega (2)},\dots ,L_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{B},\text{ and}\hfill \\ \hfill {\eta }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}={\mathcal{K}}_{1_{{\chi }_{\omega }}}\left(Z_{\omega (1)},L_{b_1}{Z}_{\omega (2)},\dots ,L_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{D}.\hfill \end{array}$$
• If |$\omega(k) \in J$| for all |$k$|⁠, define

  $$\begin{array}{cc}\hfill {\nu }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\mathbb{E}\left(Z_{\omega (1)}{R}_{b_1}{Z}_{\omega (2)}\cdots R_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{B},\hfill \\ \hfill {\mu }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\mathbb{F}\left(Z_{\omega (1)}{R}_{b_1}{Z}_{\omega (2)}\cdots R_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{D},\hfill \\ \hfill {\rho }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}={\kappa }_{1_{{\chi }_{\omega }}}\left(Z_{\omega (1)},R_{b_1}{Z}_{\omega (2)},\dots ,R_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{B},\text{ and}\hfill \\ \hfill {\eta }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}={\mathcal{K}}_{1_{{\chi }_{\omega }}}\left(Z_{\omega (1)},R_{b_1}{Z}_{\omega (2)},\dots ,R_{b_{n-1}}{Z}_{\omega (n)}\right)\in \mathcal{D}.\hfill \end{array}$$
• Otherwise, let |$k_\ell = \min\{k \, \mid \, \omega(k) \in I\}$| and |$k_r = \min\{k \, \mid \, \omega(k) \in J\}$|⁠. Then |$\{k_\ell, k_r\} = \{1, k_0\}$| for some |$k_0$|⁠. Define |$\nu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$| and |$\mu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$| to be

  $$\begin{array}{cc}& \mathbb{E}(Z_{\omega (1)}{C}_{b_1}^{\omega (2)}{Z}_{\omega (2)}\cdots C_{b_{k_0-2}}^{\omega (k_0-1)}{Z}_{\omega (k_0-1)}{Z}_{\omega (k_0)}{C}_{b_{k_0-1}}^{\omega (k_0+1)}{Z}_{\omega (k_0+1)}\hfill \\ & \cdots C_{b_{n-3}}^{\omega (n-1)}{Z}_{\omega (n-1)}{C}_{b_{n-2}}^{\omega (n)}{Z}_{\omega (n)}{C}_{b_{n-1}}^{\omega (n)})\in \mathcal{B}\hfill \end{array}$$

  and

  $$\begin{array}{cc}& \mathbb{E}(Z_{\omega (1)}{C}_{b_1}^{\omega (2)}{Z}_{\omega (2)}\cdots C_{b_{k_0-2}}^{\omega (k_0-1)}{Z}_{\omega (k_0-1)}{Z}_{\omega (k_0)}{C}_{b_{k_0-1}}^{\omega (k_0+1)}{Z}_{\omega (k_0+1)}\hfill \\ & \cdots C_{b_{n-3}}^{\omega (n-1)}{Z}_{\omega (n-1)}{C}_{b_{n-2}}^{\omega (n)}{Z}_{\omega (n)}{C}_{b_{n-1}}^{\omega (n)})\in \mathcal{B}\hfill \end{array}$$

  respectively, and define |$\rho_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$| and |$\eta_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$| to be

  $$\begin{array}{cc}& {\kappa }_{1_{{\chi }_{\omega }}}(Z_{\omega (1)},C_{b_1}^{\omega (2)}{Z}_{\omega (2)},\dots ,C_{b_{k_0-2}}^{\omega (k_0-1)}{Z}_{\omega (k_0-1)},Z_{\omega (k_0)},C_{b_{k_0-1}}^{\omega (k_0+1)}{Z}_{\omega (k_0+1)},\hfill \\ & \dots ,C_{b_{n-3}}^{\omega (n-1)}{Z}_{\omega (n-1)},C_{b_{n-2}}^{\omega (n)}{Z}_{\omega (n)}{C}_{b_{n-1}}^{\omega (n)})\in \mathcal{B}\hfill \end{array}$$

  and

  $$\begin{array}{cc}& {\mathcal{K}}_{1_{{\chi }_{\omega }}}(Z_{\omega (1)},C_{b_1}^{\omega (2)}{Z}_{\omega (2)},\dots ,C_{b_{k_0-2}}^{\omega (k_0-1)}{Z}_{\omega (k_0-1)},Z_{\omega (k_0)},C_{b_{k_0-1}}^{\omega (k_0+1)}{Z}_{\omega (k_0+1)},\hfill \\ & \dots ,C_{b_{n-3}}^{\omega (n-1)}{Z}_{\omega (n-1)},C_{b_{n-2}}^{\omega (n)}{Z}_{\omega (n)}{C}_{b_{n-1}}^{\omega (n)})\in \mathcal{D}\hfill \end{array}$$

  respectively, where

  $$C_b^{\omega (k)}=\{\begin{array}{ll}\hfill L_b& \text{if }\omega (k)\in I\hfill \\ \hfill R_b& \text{if }\omega (k)\in J\hfill \end{array}.$$

□

 

Definition 6.2.

Let |${\mathcal{Z}} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$| be as above. The moment and cumulant series of |${\mathcal{Z}}$| with respect to |$({\mathbb{E}}, {\mathbb{F}})$| are the collections of maps

$$\begin{array}{cc}\hfill {\nu }^{\mathcal{Z}}& \hspace{0.17em}=\{{\nu }_{\omega }^{\mathcal{Z}}:{\mathcal{B}}^{n-1}\to \mathcal{B}\mid n\ge 1,\omega :\{1,\dots ,n\}\to I\bigsqcup J\},\hfill \\ \hfill {\mu }^{\mathcal{Z}}& \hspace{0.17em}=\{{\mu }_{\omega }^{\mathcal{Z}}:{\mathcal{B}}^{n-1}\to \mathcal{D}\mid n\ge 1,\omega :\{1,\dots ,n\}\to I\bigsqcup J\},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\rho }^{\mathcal{Z}}& \hspace{0.17em}=\{{\rho }_{\omega }^{\mathcal{Z}}:{\mathcal{B}}^{n-1}\to \mathcal{B}\mid n\ge 1,\omega :\{1,\dots ,n\}\to I\bigsqcup J\},\hfill \\ \hfill {\eta }^{\mathcal{Z}}& \hspace{0.17em}=\{{\eta }_{\omega }^{\mathcal{Z}}:{\mathcal{B}}^{n-1}\to \mathcal{D}\mid n\ge 1,\omega :\{1,\dots ,n\}\to I\bigsqcup J\},\hfill \end{array}$$

respectively. Note that if |$n = 1$|⁠, then |$\nu_\omega^{\mathcal{Z}} = \rho_\omega^{\mathcal{Z}} = {\mathbb{E}}(Z_{\omega(1)})$| and |$\mu_\omega^{\mathcal{Z}} = \eta_\omega^{\mathcal{Z}} = {\mathbb{F}}(Z_{\omega(1)})$|⁠. □

 

Lemma 6.3.

Let |$I$| and |$J$| be non-empty disjoint index sets, and let |${\mathcal{B}}$| and |${\mathcal{D}}$| be unital algebras such that |$1 := 1_{\mathcal{D}} \in {\mathcal{B}} \subset {\mathcal{D}}$|⁠. For every |$n \geq 1$| and |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, let |$\Theta_\omega: {\mathcal{B}}^{n - 1} \to {\mathcal{B}}$| and |$\Upsilon_\omega: {\mathcal{B}}^{n - 1} \to {\mathcal{D}}$| be linear in each coordinate. There exist a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| and elements |$\{Z_i\}_{i \in I} \subset {\mathcal{A}}_\ell$| and |$\{Z_j\}_{j \in J} \subset {\mathcal{A}}_r$| such that if |$\mathcal{Z} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$|⁠, then

$${\rho }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})={\mathrm{\Theta }}_{\omega }(b_1,\dots ,b_{n-1})\text{and}{\eta }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})={\mathrm{{\rm Y}}}_{\omega }(b_1,\dots ,b_{n-1})$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. □

 

Proof.

By the same construction presented in the proof of [11, Lemma 3.8], there exist a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space |$({\mathcal{A}}, {\mathbb{E}}, \varepsilon)$| and |$\mathcal{Z} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$| with |$\{Z_i\}_{i \in I} \subset {\mathcal{A}}_\ell$| and |$\{Z_j\}_{j \in J} \subset {\mathcal{A}}_r$| such that

$${\rho }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})={\mathrm{\Theta }}_{\omega }(b_1,\dots ,b_{n-1})$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. Thus we need only define an expectation |${\mathbb{F}}$| to produce the correct operator-valued c-bi-free cumulants.

For |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n + 1} \in {\mathcal{B}}$|⁠, let

$$C_b^{\omega (k)}=\{\begin{array}{ll}\hfill L_b& \text{if }\omega (k)\in I\hfill \\ \hfill R_b& \text{if }\omega (k)\in J\hfill \end{array}$$

and define

$${\widehat{\mathrm{{\rm Y}}}}_{1_{{\chi }_{\omega }}}\left(C_{b_1}^{\omega (1)}{Z}_{\omega (1)},\dots ,C_{b_{n-1}}^{\omega (n-1)}{Z}_{\omega (n-1)},C_{b_n}^{\omega (n)}{Z}_{\omega (n)}{C}_{b_{n+1}}^{\omega (n)}\right)\in \mathcal{D}$$

like how |$\widehat{\Theta}_{1_{\chi_\omega}}$| is defined in the proof of [11, Lemma 3.8] using |$\Upsilon_\omega$| instead of |$\Theta_\omega$|⁠. Subsequently, for |$\omega: \{1, \dots, n\} \to I \sqcup J$| and |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_\omega)$|⁠, define

$${\widehat{\mathrm{{\rm Y}}}}_{\pi }\left(C_{b_1}^{\omega (1)}{Z}_{\omega (1)},\dots ,C_{b_{n-1}}^{\omega (n-1)}{Z}_{\omega (n-1)},C_{b_n}^{\omega (n)}{Z}_{\omega (n)}{C}_{b_{n+1}}^{\omega (n)}\right)\in \mathcal{D}$$

by selecting one of the many possible ways to reduce an operator-valued conditionally bi-multiplicative function where |$\widehat{\Theta}_{1_\chi}$| is used for interior blocks and |$\widehat{\Upsilon}_{1_\chi}$| is used for exterior blocks.

As seen in the proof of [11, Lemma 3.8], every element in |${\mathcal{A}}$| is a linear combination of the form

$$C_{b_1}^{\omega (1)}{Z}_{\omega (1)}\cdots C_{b_n}^{\omega (n)}{Z}_{\omega (n)}{L}_b{R}_{b^{\prime }}+\mathcal{I},$$

where |$n \geq 0$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$| when |$n \geq 1$|⁠, |$b_1, \dots, b_n, b, b' \in {\mathcal{B}}$|⁠, and |$\mathcal{I}$| is some two-sided ideal. Define |${\mathbb{F}}: {\mathcal{A}} \to {\mathcal{D}}$| by

$$\mathbb{F}(L_b{R}_{b^{\prime }}+\mathcal{I})=b{b}^{\prime }$$

for all |$b, b' \in {\mathcal{B}}$|⁠, and

$$\mathbb{F}\left(C_{b_1}^{\omega (1)}{Z}_{\omega (1)}\cdots C_{b_n}^{\omega (n)}{Z}_{\omega (n)}{L}_b{R}_{b^{\prime }}+\mathcal{I}\right)=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}{\widehat{\mathrm{{\rm Y}}}}_{\pi }\left(C_{b_1}^{\omega (1)}{Z}_{\omega (1)},\dots ,C_{b_{n-1}}^{\omega (n-1)}{Z}_{\omega (n-1)},C_{b_n}^{\omega (n)}{Z}_{\omega (n)}{C}_{b{b}^{\prime }}\right)$$

for all |$n \geq 1$| and |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, where |$C_{bb'} = L_{bb'}$| if |$\omega(n) \in I$| and |$C_{bb'} = R_{bb'}$| if |$\omega(n) \in J$|⁠, and extend |${\mathbb{F}}$| by linearity. By construction and commutation in |${\mathcal{A}}$|⁠, one can verify that |${\mathbb{F}}$| is well-defined and

$$\mathbb{F}(L_b{R}_{b^{\prime }}Z+\mathcal{I})=b\mathbb{F}(Z+\mathcal{I})b^{\prime }\text{and}\mathbb{F}(Z{L}_b+\mathcal{I})=\mathbb{F}(Z{R}_b+\mathcal{I})$$

for all |$b, b' \in {\mathcal{B}}$| and |$Z + {\mathcal{I}} \in {\mathcal{A}}$|⁠. Finally, since Definition 4.7 completely determines the operator-valued c-bi-free cumulants and by our definition of |$\hat{\Upsilon}$| via a choice of operator-valued conditionally bi-multiplicative reduction, [11, Lemma 3.8] with an induction argument together imply that if |$\mathcal{Z} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$|⁠, then

$${\eta }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})={\mathrm{{\rm Y}}}_{\omega }(b_1,\dots ,b_{n-1})$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. ■

We are now ready to prove the main result of this section.

 

Theorem 6.4.

A family |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| of pairs of |${\mathcal{B}}$|-algebras in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| if and only if for all |$n \geq 2$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\omega: \{1, \dots, n\} \to K$|⁠, and |$Z_k \in {\mathcal{A}}_{\omega(k), \chi(k)}$|⁠, we have

$${\kappa }_{1_{\chi }}(Z_1,\dots ,Z_n)={\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_n)=0$$

whenever |$\omega$| is not constant. □

 

Proof.

If all mixed cumulants vanish, then |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is bi-free over |${\mathcal{B}}$| so equation (5) holds. To see that equation (6) also holds, recall from [6, Subsection 4.2] that |${\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi, ie)$| denotes the set of all pairs |$(\pi, \iota)$| where |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| is a bi-non-crossing partition and |$\iota: \pi \to \{i, e\}$| is a function on the blocks of |$\pi$|⁠. By Definitions 4.4 and 4.7, and the assumption that all mixed cumulants vanish, we have

$$\mathbb{F}(Z_1\cdots Z_n)={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)=\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \le \omega \end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_n).$$

By applying Definition 4.7 recursively, we obtain that

$$\mathbb{F}(Z_1\cdots Z_n)=\sum _{\begin{array}{c}(\pi ,\iota )\in \mathcal{B}\mathcal{N}\mathcal{C}(\chi ,ie)\\ \pi \le \omega \end{array}}c(\chi ,\omega ;\pi ,\iota ){\mathrm{\Theta }}_{(\pi ,\iota )}(Z_1,\dots ,Z_n),$$

(7)

where |$c(\chi, \omega; \pi, \iota)\Theta_{(\pi, \iota)}(Z_1, \dots, Z_n)$| for |$(\pi, \iota) \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi, ie)$| is defined as follows: If there is an interior block |$V$| of |$\pi$| such that |$\iota(V) = e$|⁠, then |$c(\chi, \omega; \pi, \iota) = 0$|⁠. Otherwise, apply the recursive process using |${\mathbb{E}}$| as in Definition 2.6 to the interior blocks of |$\pi$|⁠, order the remaining |$\chi$|-intervals by |$\prec_\chi$| as |$V_1, \dots, V_m$|⁠, and define

$${\mathrm{\Theta }}_{(\pi ,\iota )}(Z_1,\dots ,Z_n)={\mathrm{\Theta }}_{\pi {|}_{V_1}}((Z_1,\dots ,Z_n){|}_{V_1})\cdots {\mathrm{\Theta }}_{\pi {|}_{V_m}}((Z_1,\dots ,Z_n){|}_{V_m}),$$

where |$\Theta_{\pi|_{V_j}} = {\mathbb{E}}_{\pi|_{V_j}}$| if |$\iota(V_j) = i$| and |$\Theta_{\pi|_{V_j}} = {\mathbb{F}}_{\pi|_{V_j}}$| if |$\iota(V_j) = e$|⁠.

Notice that, as with the scalar-valued case (see [6, Remark 4.9]), |$\Theta_{(\pi, \iota)}(Z_1, \dots, Z_n)$| and |${\mathbb{F}}_D(Z_1, \dots, Z_n)$| agree for certain |$(\pi, \iota) \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi, ie)$| and |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$|⁠. Indeed, given |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$|⁠, defining |$\pi$| via the blocks of |$D$| and |$\iota$| via |$\iota(V) = e$| if the spine of |$V$| reaches the top and |$\iota(V) = i$| otherwise will produce such an equality.

If each |$(\pi, \iota) \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi, ie)$| with |$\pi \leq \omega$| and |$c(\chi, \omega; \pi, \iota) \neq 0$| corresponds to some |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$| in the sense as described above and

$$c(\chi ,\omega ;\pi ,\iota )=\sum _{\begin{array}{c}{D}^{\prime }\in \mathcal{L}\mathcal{R}(\chi ,\omega )\\ D^{\prime }{\ge }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}}D\end{array}}(-1{)}^{|D|-|D^{\prime }|}$$

for such |$(\pi, \iota)$|⁠, then equations (6) and (7) coincide implying that |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| by Theorem 5.5. Since the property that |$(\pi, \iota)$| corresponds to a |$D \in {\mathcal{L}}{\mathcal{R}}^{\mathrm{lat}\mathrm{cap}}(\chi, \omega)$| and the value of |$c(\chi, \omega; \pi, \iota)$| do not depend on the algebras |${\mathcal{B}}$| and |${\mathcal{D}}$|⁠, the result follows from the |${\mathcal{B}} = {\mathcal{D}} = {\mathbb{C}}$| case by [6, Lemma 4.13].

Conversely, if |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠, then equations (5) and (6) hold by Theorem 5.5. As shown in [4, Theorem 8.1.1]{CNS2015-2}, equation (5) is equivalent to the vanishing of mixed operator-valued bi-free cumulants. Thus we need only show that mixed operator-valued c-bi-free cumulants vanish. For fixed |$n \geq 2$|⁠, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\omega: \{1, \dots, n\} \to K$|⁠, and |$Z_k \in {\mathcal{A}}_{\omega(k), \chi(k)}$|⁠, construct a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}', {\mathbb{E}}_{{\mathcal{A}}'}, {\mathbb{F}}_{{\mathcal{A}}'}, \varepsilon')$|⁠, pairs of |${\mathcal{B}}$|-algebras |$\{({\mathcal{A}}'_{k, \ell}, {\mathcal{A}}'_{k, r})\}_{k \in K}$|⁠, and elements |$Z'_k \in {\mathcal{A}}'_{\omega(k), \chi(k)}$| such that

• for each |$k \in \{1, \dots, n\}$|⁠, |$\{Z'_j \, \mid \, \omega(j) = \omega(k), \chi(j) = \chi(k)\}$| generated |${\mathcal{A}}'_{\omega(k), \chi(k)}$|⁠,

• any joint operator-valued c-bi-free cumulant involving |$Z'_1, \dots, Z'_n$| containing a pair |$Z'_{k_1}, Z'_{k_2}$| with |$\omega(k_1) \neq \omega(k_2)$| is zero, and

• for each |$k \in \{1, \dots, n\}$|⁠, the joint distribution of |$\{Z'_j \, \mid \, \omega(j) = \omega(k)\}$| with respect to |$({\mathbb{E}}_{{\mathcal{A}}'}, {\mathbb{F}}_{{\mathcal{A}}'})$| equals the joint distribution of |$\{Z_j \, \mid \, \omega(j) = \omega(k)\}$| with respect to |$({\mathbb{E}}, {\mathbb{F}})$|⁠.

The above is possible via Lemma 6.3 by defining the operator-valued bi-free and c-bi-free cumulants appropriately.

By construction, |$Z'_1, \dots, Z'_n$| have vanishing mixed cumulants and hence satisfy equations (5) and (6) by the first part of the proof. However, since for each |$k \in \{1, \dots, n\}$|⁠, the joint distribution of |$\{Z'_j \, \mid \, \omega(j) = \omega(k)\}$| with respect to |$({\mathbb{E}}_{{\mathcal{A}}'}, {\mathbb{F}}_{{\mathcal{A}}'})$| equals the joint distribution of |$\{Z_j \, \mid \, \omega(j) = \omega(k)\}$| with respect to |$({\mathbb{E}}, {\mathbb{F}})$|⁠, equations (5) and (6) imply that the joint distribution of |$Z_1, \dots, Z_n$| with respect to |$({\mathbb{E}}, {\mathbb{F}})$| equals the joint distribution of |$Z'_1, \dots, Z'_n$| with respect to |$({\mathbb{E}}_{{\mathcal{A}}'}, {\mathbb{F}}_{{\mathcal{A}}'})$|⁠. Since the operator-valued bi-free and c-bi-free moments completely determine the operator-valued bi-free and c-bi-free cumulants, and since |$Z'_1, \dots, Z'_n$| have vanishing mixed cumulants, the result follows. ■

7 Additional Properties of C-Bi-Free Independence with Amalgamation

In this section, we collect a list of additional properties of c-bi-free independence with amalgamation and operator-valued c-bi-free cumulants. All of the results below are analogues of known results in the current framework with essentially the same proofs. We begin by recalling the following notation from [4, Notation 6.3.1].

 

Notation 7.1.

Let |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, and |$q \in \{1, \dots, n\}$|⁠. We denote by |$\chi|_{\setminus q}$| the restriction of |$\chi$| to the set |$\{1, \dots, n\} \setminus q$|⁠. If |$q \neq n$| and |$\chi(q) = \chi(q + 1)$|⁠, define |$\pi|_{q = q + 1} \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi|_{\setminus q})$| to be the bi-non-crossing partition which results from identifying |$q$| and |$q + 1$| in |$\pi$| (i.e., if |$q$| and |$q + 1$| are in the same block, then |$\pi|_{q = q + 1}$| is obtained from |$\pi$| by just removing |$q$| from the block in which |$q$| occurs, while if |$q$| and |$q + 1$| are in different blocks, then |$\pi|_{q = q + 1}$| is obtained from |$\pi$| by merging the two blocks and then removing |$q$|⁠). □

7.1 Vanishing of operator-valued cumulants

The following demonstrates that, like with many other kinds of cumulants, the operator-valued c-bi-free cumulants of order at least two vanish if at least one input is a |${\mathcal{B}}$|-operator.

 

Proposition 7.2.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| with |$n \geq 2$|⁠, and |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. If there exist |$q \in \{1, \dots, n\}$| and |$b \in {\mathcal{B}}$| such that |$Z_q = L_b$| if |$\chi(q) = \ell$| or |$Z_q = R_b$| if |$\chi(q) = r$|⁠, then

$${\kappa }_{1_{\chi }}(Z_1,\dots ,Z_n)={\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_n)=0.$$

□

 

Proof.

The assertion that |$\kappa_{1_\chi}(Z_1, \dots, Z_n) = 0$| was proved in [4, Proposition 6.4.1], and the other assertion will be proved by induction with the base case easily verified by direct computations.

For the inductive step, suppose the assertion is true for all |$\chi: \{1, \dots, m\} \to \{\ell, r\}$| with |$2 \leq m \leq n - 1$|⁠. Fix |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| and |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. Suppose there exist |$q \in \{1, \dots, n\}$| and |$b \in {\mathcal{B}}$| such that |$\chi(q) = \ell$| and |$Z_q = L_b$| (the case |$\chi(q) = r$| and |$Z_q = R_b$| is similar). Let

$$p=max\{k\in \{1,\dots ,n\}\mid \chi (k)=\ell ,k<q\}.$$

There are two cases. If |$p \neq -\infty$|⁠, then by the first assertion and the induction hypothesis,

$$\begin{array}{cc}{\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_n)& \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \ne 1_{\chi }\end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_n)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \{q\}\in \pi \end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{q-1},L_b,Z_{q+1},\dots ,Z_n)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi {|}_{\setminus q})}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_p{L}_b,Z_{p+1},\dots ,Z_{q-1},Z_{q+1},\dots ,Z_n)\hfill \end{array}$$

by properties of |$(\kappa, {\mathcal{K}})$|⁠. On the other hand, we have

$$\begin{array}{cc}\hfill {\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)& \hspace{0.17em}=\mathbb{F}(Z_1\cdots Z_{q-1}{L}_b{Z}_{q+1}\cdots Z_n)\hfill \\ & \hspace{0.17em}=\mathbb{F}(Z_1\cdots Z_p{L}_b{Z}_{p+1}\cdots Z_{q-1}{Z}_{q+1}\cdots Z_n)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_p{L}_b,Z_{p+1},\dots ,Z_{q-1},Z_{q+1},\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\sum _{\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi {|}_{\setminus q})}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_p{L}_b,Z_{p+1},\dots ,Z_{q-1},Z_{q+1},\dots ,Z_n),\hfill \end{array}$$

thus the assertion is true in this case. If |$p = -\infty$|⁠, then by the first assertion and the induction hypothesis,

$$\begin{array}{cc}{\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_n)& \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \pi \ne 1_{\chi }\end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_n)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \{q\}\in \pi \end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{q-1},L_b,Z_{q+1},\dots ,Z_n)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi {|}_{\setminus q})}b{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_{q-1},Z_{q+1},\dots ,Z_n)\hfill \end{array}$$

by properties of |$(\kappa, {\mathcal{K}})$| as |$q = \min_{\prec_\chi}(\{1, \dots, n\})$| in this case. On the other hand, we have

$$\begin{array}{cc}\hfill {\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)& \hspace{0.17em}=\mathbb{F}(Z_1\cdots Z_{q-1}{L}_b{Z}_{q+1}\cdots Z_n)\hfill \\ & \hspace{0.17em}=\mathbb{F}(L_b{Z}_1\cdots Z_{q-1}{Z}_{q+1}\cdots Z_n)\hfill \\ & \hspace{0.17em}=b{\mathcal{F}}_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_{q+1},\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\sum _{\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi {|}_{\setminus q})}b{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_{q-1},Z_{q+1},\dots ,Z_n),\hfill \end{array}$$

thus the assertion is true in this case as well. ■

7.2 Operator-valued cumulants of products

Next, we analyse operator-valued c-bi-free cumulants involving products of operators.

 

Lemma 7.3.

Let |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| be a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. If |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠, and |$q \in \{1, \dots, n - 1\}$| with |$\chi(q) = \chi(q + 1),$| then

$${\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)=\sum _{\begin{array}{c}\hfill \sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \hfill \sigma {|}_{q=q+1}=\pi \end{array}}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_n)$$

for all |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi|_{\setminus q})$|⁠. □

 

Proof.

We proceed by induction on |$n$|⁠. If |$n = 1$|⁠, there is nothing to check. If |$n = 2$|⁠, then

$${\mathcal{K}}_{0_{\chi }{|}_{1=2}}(Z_1{Z}_2)={\mathcal{K}}_{1_{\chi }{|}_{1=2}}(Z_1{Z}_2)={\mathcal{F}}_{1_{\chi }{|}_{1=2}}(Z_1{Z}_2)={\mathcal{F}}_{1_{\chi }}(Z_1,Z_2)={\mathcal{K}}_{0_{\chi }}(Z_1,Z_2)+{\mathcal{K}}_{1_{\chi }}(Z_1,Z_2)$$

as required. Suppose the assertion holds for |$n - 1$|⁠, and note from [4, Theorem 6.3.5] that the analogous result also holds for the operator-valued bi-free cumulant function |$\kappa$|⁠. Using the induction hypothesis and the operator-valued conditionally bi-multiplicativity of |$(\kappa, {\mathcal{K}})$|⁠, we see for all |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi|_{\setminus q}) \setminus \{1_{\chi|_{\setminus q}}\}$| that

$${\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)=\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma {|}_{q=q+1}=\pi \end{array}}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_n).$$

Hence,

$$\begin{array}{cc}& {\mathcal{K}}_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi {|}_{\setminus q})\\ \pi \ne 1_{\chi {|}_{\setminus q}}\end{array}}{\mathcal{K}}_{\pi }(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)\hfill \\ & \hspace{0.17em}={\mathcal{F}}_{1_{\chi }}(Z_1,\dots ,Z_n)-\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi {|}_{\setminus q})\\ \pi \ne 1_{\chi {|}_{\setminus q}}\end{array}}\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma {|}_{q=q+1}=\pi \end{array}}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\sum _{\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_n)-\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma {|}_{q=q+1}\ne 1_{\chi {|}_{\setminus q}}\end{array}}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma {|}_{q=q+1}=1_{\chi {|}_{\setminus q}}\end{array}}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_n)\hfill \end{array}$$

completing the inductive step. ■

Given two partitions |$\pi, \sigma \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, let |$\pi \vee \sigma$| denote the smallest partition in |${\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| greater than |$\pi$| and |$\sigma$|⁠. Furthermore, suppose |$m, n \geq 1$| with |$m < n$| are fixed, and consider a sequence of integers

$$0=k(0)<k(1)<\cdots <k(m)=n.$$

For |$\chi: \{1, \dots, m\} \to \{\ell, r\}$|⁠, define |$\widehat{\chi}: \{1, \dots, n\} \to \{\ell, r\}$| by

$$\hat{\chi }(q)=\chi (p_q),$$

where |$p_q$| is the unique number in |$\{1, \dots, m\}$| such that |$k(p_q - 1) < q \leq k(p_q)$|⁠. Let |$\widehat{0_\chi}$| be the partition of |$\{1, \dots, n\}$| with blocks |$\{\{k(p - 1) + 1, \dots, k(p)\}\}_{p = 1}^m$|⁠. Recursively applying the previous lemma along with [4, Theorem 9.1.5] yields the following operator-valued analogue of [6, Theorem 4.22].

 

Theorem 7.4.

Let |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| be a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. Under the above notation, we have

$${\mathcal{K}}_{1_{\chi }}\left(Z_1\cdots Z_{k(1)},Z_{k(1)+1}\cdots Z_{k(2)},\dots ,Z_{k(m-1)+1}\cdots Z_{k(m)}\right)=\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\hat{\chi })\\ \sigma \vee \widehat{0_{\chi }}=1_{\hat{\chi }}\end{array}}{\mathcal{K}}_{\sigma }(Z_1,\dots ,Z_n)$$

for all |$\chi: \{1, \dots, m\} \to \{\ell, r\}$| and |$Z_k \in {\mathcal{A}}_{\widehat{\chi}(k)}$|⁠. □

7.3 Operator-valued conditionally bi-moment and bi-cumulant pairs

In [12, Subsection 3.2], the classes of operator-valued moment and cumulant functions were introduced as a tool to calculate moment expressions of elements in amalgamated free products. The c-free extension (in the special case |${\mathcal{B}} = {\mathbb{C}}$|⁠) was achieved in [8, Section 3] and the bi-free analogue was obtained in [4, Subsection 6.3]. In this subsection, we extend the notions of operator-valued bi-moment and bi-cumulant functions from [4, Definition 6.3.2] to pairs of functions.

 

Definition 7.5.

Let |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| be a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, and let

$$\varphi :\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{B}$$

and

$$\mathrm{\Phi }:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{D}$$

be an operator-valued conditionally bi-multiplicative pair.

• (1) We say that |$(\phi, \Phi)$| is an operator-valued conditionally bi-moment pair if whenever |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| is such that there exists a |$q \in \{1, \dots, n - 1\}$| with |$\chi(q) = \chi(q + 1)$|⁠, then

  $${\varphi }_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\varphi }_{1_{\chi }}(Z_1,\dots ,Z_n)$$

  and

  $${\varphi }_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\varphi }_{1_{\chi }}(Z_1,\dots ,Z_n)$$

  for all |$Z_k \in \mathcal{A}_{\chi(k)}$|⁠.
• (2) We say that |$(\phi, \Phi)$| is an operator-valued conditionally bi-cumulant pair if whenever |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| is such that there exists a |$q \in \{1, \dots, n - 1\}$| with |$\chi(q) = \chi(q + 1)$|⁠, then

  $${\varphi }_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\varphi }_{1_{\chi }}(Z_1,\dots ,Z_n)+\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ |\pi |=2,q{\nsim }_{\pi }q+1\end{array}}{\varphi }_{\pi }(Z_1,\dots ,Z_n)$$

  and

  $${\mathrm{\Phi }}_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\mathrm{\Phi }}_{1_{\chi }}(Z_1,\dots ,Z_n)+\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ |\pi |=2,q{\nsim }_{\pi }q+1\end{array}}{\mathrm{\Phi }}_{\pi }(Z_1,\dots ,Z_n)$$

  for all |$Z_k \in \mathcal{A}_{\chi(k)}$|⁠, where |$\Phi_\pi(Z_1, \dots, Z_n)$| is defined by the operator-valued conditionally bi-multiplicativity of |$(\phi, \Phi)$| using |$\phi$| for an interior block and |$\Phi$| for an exterior block.

□

The following demonstrates that the two notions of pairs of functions are naturally related by summing over bi-non-crossing partitions.

 

Theorem 7.6.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. If

$$\varphi ,\psi :\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{B}$$

and

$$\mathrm{\Phi },\mathrm{\Psi }:\bigcup _{n\ge 1}\bigcup _{\chi :\{1,\dots ,n\}\to \{\ell ,r\}}\mathcal{B}\mathcal{N}\mathcal{C}(\chi )\times {\mathcal{A}}_{\chi (1)}\times \cdots \times {\mathcal{A}}_{\chi (n)}\to \mathcal{D}$$

are such that |$(\phi, \Phi)$| and |$(\psi, \Psi)$| are operator-valued conditionally bi-multiplicative related by the formulae

$${\varphi }_{\pi }(Z_1,\dots ,Z_n)=\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma \le \pi \end{array}}{\psi }_{\sigma }(Z_1,\dots ,Z_n)$$

and

$${\mathrm{\Phi }}_{\pi }(Z_1,\dots ,Z_n)=\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma \le \pi \end{array}}{\mathrm{\Psi }}_{\sigma }(Z_1,\dots ,Z_n)$$

for all |$\chi: \{1, \dots, n\} \to \{\ell, r\}$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$|⁠, and |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠, then |$(\phi, \Phi)$| is an operator-valued conditionally bi-moment pair if and only if |$(\psi, \Psi)$| is an operator-valued conditionally bi-cumulant pair. □

 

Proof.

Let |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| be such that there exists a |$q \in \{1, \dots, n - 1\}$| with |$\chi(q) = \chi(q + 1)$|⁠. If |$(\psi, \Psi)$| is an operator-valued conditionally bi-cumulant pair, then

$${\varphi }_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\varphi }_{1_{\chi }}(Z_1,\dots ,Z_n)$$

for all |$Z_k \in {\mathcal{A}}_{\chi(k)}$| by [4, Theorem 6.3.5]. On the other hand, using the operator-valued conditionally bi-multiplicativity of |$(\psi, \Psi)$| and part |$(2)$| of Definition 7.5, we have

$${\mathrm{\Psi }}_{\pi }(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},\dots ,Z_n)=\sum _{\begin{array}{c}\sigma \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ \sigma {|}_{q=q+1}=\pi \end{array}}{\mathrm{\Psi }}_{\sigma }(Z_1,\dots ,Z_n)$$

for all |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi|_{\setminus q})$|⁠, and it follows from the same calculations as in the first part of the proof of [4, Theorem 6.3.5] that

$${\mathrm{\Phi }}_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\mathrm{\Phi }}_{1_{\chi }}(Z_1,\dots ,Z_n)$$

for all |$Z_k \in \mathcal{A}_{\chi(k)}$|⁠.

Conversely, if |$(\phi, \Phi)$| is an operator-valued conditionally bi-moment pair, then

$${\psi }_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\psi }_{1_{\chi }}(Z_1,\dots ,Z_n)+\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ |\pi |=2,q{\nsim }_{\pi }q+1\end{array}}{\psi }_{\pi }(Z_1,\dots ,Z_n)$$

for all |$Z_k \in {\mathcal{A}}_{\chi(k)}$| by [4, Theorem 6.3.5], and it follows from the same induction arguments as in the second part of the proof of [4, Theorem 6.3.5] that

$${\mathrm{\Psi }}_{1_{\chi {|}_{\setminus q}}}(Z_1,\dots ,Z_{q-1},Z_q{Z}_{q+1},Z_{q+2},\dots ,Z_n)={\mathrm{\Psi }}_{1_{\chi }}(Z_1,\dots ,Z_n)+\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}(\chi )\\ |\pi |=2,q{\nsim }_{\pi }q+1\end{array}}{\mathrm{\Psi }}_{\pi }(Z_1,\dots ,Z_n)$$

for all |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. ■

As an immediate corollary, we have the following expected result.

 

Corollary 7.7.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. The operator-valued c-bi-free moment pair |$({\mathcal{E}}, {\mathcal{F}})$| is an operator-valued conditionally bi-moment pair and the operator-valued c-bi-free cumulant pair |$(\kappa, {\mathcal{K}})$| is an operator-valued conditionally bi-cumulant pair. □

7.4 Operations on operator-valued cumulants

The following two results demonstrate how certain operations affect operator-valued c-bi-free cumulants under certain conditions. The same effects in the scalar-valued setting were observed in [6, Lemmata 4.17 and 4.18].

 

Lemma 7.8.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. Let |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| be such that |$\chi(k_0) = \ell$| and |$\chi(k_0 + 1) = r$| for some |$k_0 \in \{1, \dots, n - 1\}$|⁠, and let |$X \in {\mathcal{A}}_\ell$| and |$Y \in {\mathcal{A}}_r$| be such that |${\mathbb{E}}(ZXYZ') = {\mathbb{E}}(ZYXZ')$| and |${\mathbb{F}}(ZXYZ') = {\mathbb{F}}(ZYXZ')$| for all |$Z, Z' \in {\mathcal{A}}$|⁠. Define |$\chi': \{1, \dots, n\} \to \{\ell, r\}$| by

$${\chi }^{\prime }(k)=\{\begin{array}{ll}r\hfill & \text{if }k=k_0\hfill \\ \ell \hfill & \text{if }k=k_0+1\hfill \\ \chi (k)\hfill & \text{otherwise }\hfill \end{array}.$$

Then

$${\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_{k_0-1},X,Y,Z_{k_0+2},\dots ,Z_n)={\mathcal{K}}_{1_{{\chi }^{\prime }}}(Z_1,\dots ,Z_{k_0-1},Y,X,Z_{k_0+2},\dots ,Z_n)$$

for all |$Z_1, \dots, Z_{k_0 - 1}, Z_{k_0 + 2}, \dots, Z_n \in {\mathcal{A}}$| with |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. □

 

Proof.

By repeatedly applying Definition 4.7 and using Definition 2.7 for interior blocks, we have

$${\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_{k_0-1},X,Y,Z_{k_0+2},\dots ,Z_n)=\sum _{(\pi ,\iota )\in \mathcal{B}\mathcal{N}\mathcal{C}(\chi ,ie)}d(\chi ;\pi ,\iota ){\mathrm{\Theta }}_{(\pi ,\iota )}(Z_1,\dots ,Z_{k_0-1},X,Y,Z_{k_0+2},\dots ,Z_n)$$

for some integer coefficients such that |$d(\chi; \pi, \iota) = 0$| if there is an interior block |$V$| of |$\pi$| with |$i(V) = e$|⁠, and |$\Theta_{(\pi, \iota)}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n)$| for non-zero |$d(\chi; \pi, \iota)$| is defined as in the proof of Theorem 6.4. Similarly, we have

$$\begin{array}{cc}\hfill {\mathcal{K}}_{1_{{\chi }^{\prime }}}(Z_1,\dots ,Z_{k_0-1},Y,& X,Z_{k_0+2},\dots ,Z_n)\hfill \\ & \hspace{0.17em}=\sum _{({\pi }^{\prime },{\iota }^{\prime })\in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }^{\prime },ie)}d({\chi }^{\prime };{\pi }^{\prime },{\iota }^{\prime }){\mathrm{\Theta }}_{({\pi }^{\prime },{\iota }^{\prime })}(Z_1,\dots ,Z_{k_0-1},Y,X,Z_{k_0+2},\dots ,Z_n).\hfill \end{array}$$

Note that there is a bijection from |${\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi, ie)$| to |${\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi', ie)$| which sends a pair |$(\pi, \iota)$| to the pair |$(\pi', \iota')$| obtained by swapping |$k_0$| and |$k_0 + 1$|⁠. Furthermore, as only the lattice structure affects the expansions of the above formulae (alternatively, by appealing to the scalar-valued case in [6, Subsection 4.2]), |$d(\chi; \pi, \iota) = d(\chi'; \pi', \iota')$| under this bijection.

To complete the proof, it suffices to show that

$${\mathrm{\Theta }}_{(\pi ,\iota )}(Z_1,\dots ,Z_{k_0-1},X,Y,Z_{k_0+2},\dots ,Z_n)={\mathrm{\Theta }}_{({\pi }^{\prime },{\iota }^{\prime })}(Z_1,\dots ,Z_{k_0-1},Y,X,Z_{k_0+2},\dots ,Z_n)$$

for all |$(\pi, \iota) \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi, ie)$|⁠. If |$k_0$| and |$k_0 + 1$| are in the same block of |$\pi$|⁠, then one may reduce

$${\mathrm{\Theta }}_{(\pi ,\iota )}(Z_1,\dots ,Z_{k_0-1},X,Y,Z_{k_0+2},\dots ,Z_n)$$

to an expression involving |${\mathbb{E}}(ZXYZ')$| or |${\mathbb{F}}(ZXYZ')$| for some |$Z, Z' \in {\mathcal{A}}$|⁠, commute |$X$| and |$Y$| to get |${\mathbb{E}}(ZYXZ')$| or |${\mathbb{F}}(ZYXZ')$|⁠, and undo the reduction to obtain

$${\mathrm{\Theta }}_{({\pi }^{\prime },{\iota }^{\prime })}(Z_1,\dots ,Z_{k_0-1},Y,X,Z_{k_0+2},\dots ,Z_n).$$

On the other hand, if |$k_0$| and |$k_0 + 1$| are in different blocks of |$\pi$|⁠, then the reductions of

$${\mathrm{\Theta }}_{(\pi ,\iota )}(Z_1,\dots ,Z_{k_0-1},X,Y,Z_{k_0+2},\dots ,Z_n)\text{and}{\mathrm{\Theta }}_{({\pi }^{\prime },{\iota }^{\prime })}(Z_1,\dots ,Z_{k_0-1},Y,X,Z_{k_0+2},\dots ,Z_n)$$

agree. Consequently, the proof is complete. ■

 

Lemma 7.9.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. Let |$\chi: \{1, \dots, n\} \to \{\ell, r\}$| be such that |$\chi(n) = \ell$|⁠, and let |$X \in {\mathcal{A}}_\ell$| and |$Y \in {\mathcal{A}}_r$| be such that |${\mathbb{E}}(ZX) = {\mathbb{E}}(ZY)$| and |${\mathbb{F}}(ZX) = {\mathbb{F}}(ZY)$| for all |$Z \in {\mathcal{A}}$|⁠. Define |$\chi': \{1, \dots, n\} \to \{\ell, r\}$| by

$${\chi }^{\prime }(k)=\{\begin{array}{ll}r\hfill & \text{if }k=n\hfill \\ \chi (k)\hfill & \text{otherwise }\hfill \end{array}.$$

Then

$${\mathcal{K}}_{1_{\chi }}(Z_1,\dots ,Z_{n-1},X)={\mathcal{K}}_{1_{{\chi }^{\prime }}}(Z_1,\dots ,Z_{n-1},Y)$$

for all |$Z_1, \dots, Z_{n - 1} \in {\mathcal{A}}$| with |$Z_k \in {\mathcal{A}}_{\chi(k)}$|⁠. □

 

Proof.

By the same arguments as the previous lemma, we have

$$d(\chi ;\pi ,\iota ){\mathrm{\Theta }}_{(\pi ,\iota )}(Z_1,\dots ,Z_{n-1},X)=d({\chi }^{\prime };{\pi }^{\prime },{\iota }^{\prime }){\mathrm{\Theta }}_{({\pi }^{\prime },{\iota }^{\prime })}(Z_1,\dots ,Z_{n-1},Y)$$

for all |$(\pi, \iota) \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi, ie)$|⁠, where |$(\pi', \iota')$| is obtained from |$(\pi, \iota)$| by changing the last node from a left node to a right node. Consequently, the proof is complete. ■

In [4, Theorem 10.2.1], it was demonstrated that for a family of |${\mathcal{B}}$|-algebras with certain conditions, bi-free independence over |${\mathcal{B}}$| can be deduced from free independence over |${\mathcal{B}}$| of either the left |${\mathcal{B}}$|-algebras or the right |${\mathcal{B}}$|-algebras. The c-bi-free analogue in the scalar-valued setting was proved in [6, Theorem 4.20].

 

Theorem 7.10.

Let |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| be a |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations. If |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is a family of pairs of |${\mathcal{B}}$|-algebras in |$({\mathcal{A}}, \mathbb{E}, \mathbb{F}, \varepsilon)$| such that

• (1) |${\mathcal{A}}_{m, \ell}$| and |${\mathcal{A}}_{n, r}$| commute for all |$m, n \in K$|⁠,

• (2) for every |$Y \in {\mathcal{A}}_{k, r}$|⁠, there exists an |$X \in {\mathcal{A}}_{k, \ell}$| such that |${\mathbb{E}}(ZY) = {\mathbb{E}}(ZX)$| and |${\mathbb{F}}(ZY) = {\mathbb{F}}(ZX)$| for all |$Z \in {\mathcal{A}}$|⁠,

then |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| if and only if |$\{{\mathcal{A}}_{k, \ell}\}_{k \in K}$| is c-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠. Consequently, if |$\{{\mathcal{A}}_{k, \ell}\}_{k \in K}$| is c-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠, then |$\{{\mathcal{A}}_{k, r}\}_{k \in K}$| is c-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠. □

 

Proof.

If |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$| is c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠, then it is clear that |$\{{\mathcal{A}}_{k, \ell}\}_{k \in K}$| is c-free over |$({\mathcal{B}}, {\mathcal{D}})$| and |$\{{\mathcal{A}}_{k, r}\}_{k \in K}$| is c-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠.

Suppose |$\{{\mathcal{A}}_{k, \ell}\}_{k \in K}$| is c-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠. Given a mixed operator-valued bi-free or c-bi-free cumulant from |$\{({\mathcal{A}}_{k, \ell}, {\mathcal{A}}_{k, r})\}_{k \in K}$|⁠, assumptions |$(1)$| and |$(2)$| imply that we may apply the previous two lemmata (or [11, Lemmata 2.16 and 2.17]) and reduce it to a mixed operator-valued free or c-free cumulant from |$\{{\mathcal{A}}_{k, \ell}\}_{k \in K}$|⁠, which vanishes by c-free independence over |$({\mathcal{B}}, {\mathcal{D}})$|⁠. Thus the result follows from Theorem 6.4. ■

8 The Operator-Valued C-Bi-Free Partial |$\mathcal{R}$|-Transform

In this section, we construct an operator-valued c-bi-free partial |${\mathcal{R}}$|-transform generalizing [6, Definition 5.3] and relate it to certain operator-valued moment transforms. As we will see in the proof, such transform is a function of three |${\mathcal{B}}$|-variables instead of two by a similar reason as the operator-valued bi-free partial |${\mathcal{R}}$|-transform developed in [11, Section 5]. As in [11, Section 5], our proof will follow the combinatorial techniques used in [10, Section 7]. In that which follows, all algebras are assumed to be Banach algebras.

 

Definition 8.1.

A Banach |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations is a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| such that |${\mathcal{A}}$|⁠, |${\mathcal{B}}$|⁠, and |${\mathcal{D}}$| are Banach algebras, and |$\varepsilon|_{{\mathcal{B}} \otimes 1}$|⁠, |$\varepsilon_{1 \otimes {\mathcal{B}}^{\mathrm{op}}}$|⁠, |${\mathbb{E}}$|⁠, and |${\mathbb{F}}$| are bounded. □

Let |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| be a Banach |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, let |$Z_\ell \in {\mathcal{A}}_\ell$|⁠, |$Z_r \in {\mathcal{A}}_r$|⁠, and let |$b, d \in {\mathcal{B}}$|⁠. Consider the following series:

$$\begin{array}{cc}\hfill M_{Z_{\ell }}^{\ell }(b)& \hspace{0.17em}=1+\sum _{m\ge 1}\mathbb{E}((L_b{Z}_{\ell }{)}^m),\hfill \\ \hfill {\mathbb{M}}_{Z_{\ell }}^{\ell }(b)& \hspace{0.17em}=1+\sum _{m\ge 1}\mathbb{F}((L_b{Z}_{\ell }{)}^m),\hfill \\ \hfill {\mathcal{C}}_{Z_{\ell }}^{\ell }(b)& \hspace{0.17em}=1+\sum _{m\ge 1}{\mathcal{K}}_{1_{{\chi }_{m,0}}}(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}}),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill M_{Z_r}^r(d)& \hspace{0.17em}=1+\sum _{n\ge 1}\mathbb{E}((R_d{Z}_r{)}^n),\hfill \\ \hfill {\mathbb{M}}_{Z_r}^r(d)& \hspace{0.17em}=1+\sum _{n\ge 1}\mathbb{F}((R_d{Z}_r{)}^n),\hfill \\ \hfill {\mathcal{C}}_{Z_r}^r(d)& \hspace{0.17em}=1+\sum _{n\ge 1}{\mathcal{K}}_{1_{{\chi }_{0,n}}}(\underset{n\text{entries}}{\underbrace{R_d{Z}_r,\dots ,R_d{Z}_r}}).\hfill \end{array}$$

By similar arguments as in [11, Remark 5.2], all of the series above converge absolutely for |$b, d$| sufficiently small.

In the proof of Theorem 8.3 below, the following relations will be used. Since the statements are slightly different than the ones in the literature (see, e.g., [1, equation (15)]), we will provide a proof.

 

Lemma 8.2.

Under the above assumptions and notation, we have

$$\begin{array}{cc}& {\mathcal{C}}_{Z_{\ell }}^{\ell }\left(M_{Z_{\ell }}^{\ell }(b)b\right)=1+M_{Z_{\ell }}^{\ell }(b)-M_{Z_{\ell }}^{\ell }(b){\mathbb{M}}_{Z_{\ell }}^{\ell }(b{)}^{-1}\text{and}\hfill \\ & {\mathcal{C}}_{Z_r}^r\left(d{M}_{Z_r}^r(d)\right)=1+M_{Z_r}^r(d)-{\mathbb{M}}_{Z_r}^r(d{)}^{-1}{M}_{Z_r}^r(d)\hfill \end{array}$$

for |$b, d$| sufficiently small. □

 

Proof.

For |$m \geq 1$|⁠, we have

$$\mathbb{F}((L_b{Z}_{\ell }{)}^m)=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{m,0})}{\mathcal{K}}_{\pi }(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}}).$$

For every partition |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_{m, 0})$|⁠, let |$W_\pi$| denote the block of |$\pi$| containing |$1$|⁠, which is necessarily an exterior block. Rearrange the above sum (which may be done as it converges absolutely) by first choosing |$s \in \{1, \dots, m\}$|⁠, |$W = \{1 = w_1 < \cdots < w_s\} \subset \{1, \dots, m\}$|⁠, and then summing over all |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_{m, 0})$| such that |$W_\pi = W$|⁠, i.e.,

$$\mathbb{F}((L_b{Z}_{\ell }{)}^m)=\sum _{s=1}^m\sum _{\begin{array}{c}W=\{1=w_1<\cdots <w_s\}\\ W\subset \{1,\dots ,m\}\end{array}}\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{m,0})\\ W_{\pi }=W\end{array}}{\mathcal{K}}_{\pi }(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}}).$$

Furthermore, using operator-valued conditionally bi-multiplicative properties, the right-most sum in the above expression is

$$b{\mathcal{K}}_{1_{{\chi }_{s,0}}}(Z_{\ell },L_{b_2}{Z}_{\ell },\dots ,L_{b_s}{Z}_{\ell })\mathbb{F}((L_b{Z}_{\ell }{)}^{m-w_s}),$$

where |$b_k = {\mathbb{E}}((L_bZ_\ell)^{w_k - w_{k - 1} - 1})b$|⁠. Thus

$$\mathbb{F}((L_b{Z}_{\ell }{)}^m)=\sum _{\begin{array}{c}1\le s\le m\\ 0\le i_1,\dots ,i_s\le m\\ i_1+\cdots +i_s=m-s\end{array}}b{\mathcal{K}}_{1_{{\chi }_{s,0}}}(Z_{\ell },L_{f(i_1)}{Z}_{\ell },\dots ,L_{f(i_{s-1})}{Z}_{\ell })\mathbb{F}((L_b{Z}_{\ell }{)}^{i_s}),$$

where |$f(k) = {\mathbb{E}}((L_bZ_\ell)^k)b$|⁠. Note that

$$\sum _{k\ge 0}f(k)=M_{Z_{\ell }}^{\ell }(b)b\text{and}\sum _{k\ge 0}\mathbb{F}((L_b{Z}_{\ell }{)}^k)={\mathbb{M}}_{Z_{\ell }}^{\ell }(b).$$

Consequently, we obtain

$$\sum _{m\ge 1}\mathbb{F}((L_b{Z}_{\ell }{)}^m)=\sum _{s\ge 1}b{\mathcal{K}}_{1_{{\chi }_{s,0}}}(Z_{\ell },L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell },\dots ,L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell }){\mathbb{M}}_{Z_{\ell }}^{\ell }(b),$$

therefore

$$M_{Z_{\ell }}^{\ell }(b){\mathbb{M}}_{Z_{\ell }}^{\ell }(b)=M_{Z_{\ell }}^{\ell }(b)+\sum _{s\ge 1}{\mathcal{K}}_{1_{{\chi }_{s,0}}}(L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell },L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell },\dots ,L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell }){\mathbb{M}}_{Z_{\ell }}^{\ell }(b),$$

and hence

$$M_{Z_{\ell }}^{\ell }(b)-M_{Z_{\ell }}^{\ell }(b){\mathbb{M}}_{Z_{\ell }}^{\ell }(b{)}^{-1}={\mathcal{C}}_{Z_{\ell }}^{\ell }\left(M_{Z_{\ell }}^{\ell }(b)b\right)-1,$$

which proves the first equation. The proof for the second equation is nearly identical once one uses the fact that |$d \mapsto R_d$| is an anti-homomorphism. ■

For |$b, c, d \in {\mathcal{B}}$|⁠, |$Z_\ell \in {\mathcal{A}}_\ell$|⁠, and |$Z_r \in {\mathcal{A}}_r$|⁠, consider the following series of the pair |$(Z_\ell, Z_r)$|⁠:

$$\begin{array}{cc}\hfill M_{(Z_{\ell },Z_r)}(b,c,d)& \hspace{0.17em}=\sum _{m,n\ge 0}\mathbb{E}((L_b{Z}_{\ell }{)}^m(R_d{Z}_r{)}^n{R}_c),\hfill \\ \hfill {\mathbb{M}}_{(Z_{\ell },Z_r)}(b,c,d)& \hspace{0.17em}=\sum _{m,n\ge 0}\mathbb{F}((L_b{Z}_{\ell }{)}^m(R_d{Z}_r{)}^n{R}_c),\hfill \\ \hfill {\mathcal{C}}_{(Z_{\ell },Z_r)}(b,c,d)& \hspace{0.17em}=c+\sum _{m\ge 1}{\mathcal{K}}_{1_{{\chi }_{m,0}}}(\underset{m-1\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}},L_b{Z}_{\ell }{L}_c)\hfill \\ & \hspace{0.17em}+\sum _{\begin{array}{c}\hfill m\ge 0\\ \hfill n\ge 1\end{array}}{\mathcal{K}}_{1_{{\chi }_{m,n}}}(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}},\underset{n-1\text{entries}}{\underbrace{R_d{Z}_r,\dots ,R_d{Z}_r}},R_d{Z}_r{R}_c),\hfill \end{array}$$

which converge absolutely for |$b, c, d$| sufficiently small by similar arguments as in [11, Remarks 5.2 and 5.5].

Notice if |$(Z_{1, \ell}, Z_{1, r})$| and |$(Z_{2, \ell}, Z_{2, r})$| are c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠, then

$${\mathcal{C}}_{(Z_{1,\ell }+Z_{2,\ell },Z_{1,r}+Z_{2,r})}(b,c,d)-c=({\mathcal{C}}_{(Z_{1,\ell },Z_{1,r})}(b,c,d)-c)+({\mathcal{C}}_{(Z_{2,\ell },Z_{2,r})}(b,c,d)-c)$$

by Theorem 6.4; that is, |${\mathcal{C}}_{(Z_\ell, Z_r)}(b, c, d) - c$| is an operator-valued c-bi-free partial |${\mathcal{R}}$|-transform.

 

Theorem 8.3.

Let |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| be a Banach |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations, let |$Z_\ell \in {\mathcal{A}}_\ell$|⁠, and let |$Z_r \in {\mathcal{A}}_r$|⁠. Then

$$\begin{array}{cc}& {\mathcal{C}}_{(Z_{\ell },Z_r)}(M_{Z_{\ell }}^{\ell }(b)b,M_{(Z_{\ell },Z_r)}(b,c,d),d{M}_{Z_r}^r(d))\hfill \\ & \hspace{0.17em}=M_{Z_{\ell }}^{\ell }(b){\mathbb{M}}_{Z_{\ell }}^{\ell }(b{)}^{-1}{\mathbb{M}}_{(Z_{\ell },Z_r)}(b,c,d){\mathbb{M}}_{Z_r}^r(d{)}^{-1}{M}_{Z_r}^r(d)+M_{(Z_{\ell },Z_r)}(b,c,d)\hfill \\ & \hspace{0.17em}+M_{Z_{\ell }}^{\ell }(b)(1-{\mathbb{M}}_{Z_{\ell }}^{\ell }(b{)}^{-1})M_{(Z_{\ell },Z_r)}(b,c,d)+M_{(Z_{\ell },Z_r)}(b,c,d)(1-{\mathbb{M}}_{Z_r}^r(d{)}^{-1})M_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}-M_{Z_{\ell }}^{\ell }(b)c{M}_{Z_r}^r(d)\hfill \end{array}$$

for |$b, c, d \in {\mathcal{B}}$| sufficiently small. □

 

Remark 8.4.

Note that if |${\mathcal{B}} = {\mathcal{D}} = {\mathbb{C}}$|⁠, |$b = z$|⁠, |$d = w$|⁠, and |$c = 1$|⁠, then Theorem 8.3 produces exactly equation |$(9)$| in [6, Theorem 5.6] for the scalar-valued setting. On the other hand, if |${\mathcal{B}} = {\mathcal{D}}$| and |${\mathbb{E}} = {\mathbb{F}}$|⁠, then Theorem 8.3 produces exactly equation |$(10)$| in [11, Theorem 5.6] for the operator-valued bi-free setting. □

 

Proof of Theorem 8.3.

For |$m, n \geq 1$|⁠, let |${\mathcal{B}}{\mathcal{N}}{\mathcal{C}}_{\mathrm{vs}}(\chi_{m, n})$| denote the set of bi-non-crossing partitions where no block contains both left and right nodes. Using operator-valued conditionally bi-multiplicativity, we obtain

$$\begin{array}{cc}\hfill \mathbb{F}& \hspace{0.17em}((L_b{Z}_{\ell }{)}^m(R_d{Z}_r{)}^n{R}_c)\hfill \\ & \hspace{0.17em}=\sum _{\pi \in \mathcal{B}\mathcal{N}{\mathcal{C}}_{\mathrm{v}\mathrm{s}}({\chi }_{m,n})}{\mathcal{K}}_{\pi }(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}},\underset{n-1\text{entries}}{\underbrace{R_d{Z}_r,\dots ,R_d{Z}_r}},R_d{Z}_r{R}_c)\hfill \\ & \hspace{0.17em}+\sum _{\begin{array}{c}\hfill \pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{m,n})\\ \hfill \pi \notin \mathcal{B}\mathcal{N}{\mathcal{C}}_{\mathrm{v}\mathrm{s}}({\chi }_{m,n})\end{array}}{\mathcal{K}}_{\pi }(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}},\underset{n-1\text{entries}}{\underbrace{R_d{Z}_r,\dots ,R_d{Z}_r}},R_d{Z}_r{R}_c)\hfill \\ & \hspace{0.17em}=\mathbb{F}((L_b{Z}_{\ell }{)}^m)c\mathbb{F}((R_d{Z}_r{)}^n)+{\mathrm{\Theta }}_{m,n}(b,c,d),\hfill \end{array}$$

where |$\Theta_{m, n}(b, c, d)$| denotes the sum

$$\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{m,n})\\ \pi \notin \mathcal{B}\mathcal{N}{\mathcal{C}}_{\mathrm{v}\mathrm{s}}({\chi }_{m,n})\end{array}}{\mathcal{K}}_{\pi }(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}},\underset{n-1\text{entries}}{\underbrace{R_d{Z}_r,\dots ,R_d{Z}_r}},R_d{Z}_r{R}_c).$$

For every partition |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_{m, n}) \setminus {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}_{\mathrm{vs}}(\chi_{m, n})$|⁠, let |$V_\pi$| denote the block of |$\pi$| with both left and right indices such that |$\min(V_\pi)$| is the smallest among all blocks of |$\pi$| with this property. Note that |$V_\pi$| is necessarily an exterior block. Rearrange the sum in |$\Theta_{m, n}(b, c, d)$| (which may be done as it converges absolutely) by first choosing |$s \in \{1, \dots, m\}$|⁠, |$t \in \{1, \dots, n\}$|⁠, |$V \subset \{1, \dots, m + n\}$| such that

$$V_{\ell }:=V\cap \{1,\dots ,m\}=\{u_1<\cdots <u_s\}\text{and}{V}_r:=V\cap \{m+1,\dots ,m+n\}=\{v_1<\cdots <v_t\}$$

and then summing over all |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_{m, n}) \setminus {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}_{\mathrm{vs}}(\chi_{m, n})$| such that |$V_\pi = V$|⁠. The result is

$${\mathrm{\Theta }}_{m,n}(b,c,d)=\sum _{s=1}^m\sum _{t=1}^n\sum _{\begin{array}{c}V\\ V_{\ell }=\{u_1<\cdots <u_s\}\\ V_r=\{v_1<\cdots <v_t\}\end{array}}\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{m,n})\\ \pi \notin \mathcal{B}\mathcal{N}{\mathcal{C}}_{\mathrm{v}\mathrm{s}}({\chi }_{m,n})\\ V_{\pi }=V\end{array}}{\mathcal{K}}_{\pi }(\underset{m\text{entries}}{\underbrace{L_b{Z}_{\ell },\dots ,L_b{Z}_{\ell }}},\underset{n-1\text{entries}}{\underbrace{R_d{Z}_r,\dots ,R_d{Z}_r}},R_d{Z}_r{R}_c).$$

Using operator-valued conditionally bi-multiplicative properties, the right-most sum in the above expression is

$$F_b{\mathcal{K}}_{1_{{\chi }_{s,t}}}(Z_{\ell },L_{b_2}{Z}_{\ell },\dots ,L_{b_s}{Z}_{\ell },Z_r,R_{d_2}{Z}_r,\dots ,R_{d_{t-1}}{Z}_r,R_{d_t}{Z}_r{R}_{\mathbb{E}((L_b{Z}_{\ell }{)}^{m-u_s}(R_d{Z}_r{)}^{n-v_t}{R}_c)})G_d,$$

where

$$\begin{array}{c}\hfill b_k=\mathbb{E}((L_b{Z}_{\ell }{)}^{u_k-u_{k-1}-1})b,d_k=d\mathbb{E}((R_d{Z}_r{)}^{v_k-v_{k-1}-1}),\\ \hfill F_b=\mathbb{F}((L_b{Z}_{\ell }{)}^{u_1-1})b,\text{and}{G}_d=d\mathbb{F}((R_d{Z}_r{)}^{v_1-1}).\end{array}$$

Consequently, we obtain that |$\Theta_{m, n}(b, c, d)$| equals

$$\begin{array}{c}\sum _{\begin{array}{c}1\le s\le m\\ 0\le i_0,i_1,\dots ,i_s\le m\\ i_0+i_1+\cdots +i_s=m-s\end{array}}\sum _{\begin{array}{c}1\le t\le n\\ 0\le j_0,j_1,\dots ,j_t\le n\\ j_0+j_1+\cdots +j_t=n-t\end{array}}\hfill \\ F(i_1){\mathcal{K}}_{1_{{\chi }_{s,t}}}(Z_{\ell },L_{f(i_2)}{Z}_{\ell },\dots ,L_{f(i_s)}{Z}_{\ell },Z_r,R_{g(j_2)}{Z}_r,\dots ,R_{g(j_t-1)}{Z}_r,R_{g(j_t)}{Z}_r{R}_{\mathbb{E}((L_b{Z}_{\ell }{)}^{i_0}(R_d{Z}_r{)}^{j_0}{R}_c)})G(j_1),\end{array}$$

(8)

where

$$f(k)=\mathbb{E}((L_b{Z}_{\ell }{)}^k)b,g(k)=d\mathbb{E}((R_d{Z}_r{)}^k),F(k)=\mathbb{F}((L_b{Z}_{\ell }{)}^k)b,\text{and}G(k)=d\mathbb{F}((R_d{Z}_r{)}^k).$$

Note that

$$\sum _{k\ge 0}f(k)=M_{Z_{\ell }}^{\ell }(b)b,\sum _{k\ge 0}g(k)=d{M}_{Z_r}^r(d),\sum _{k\ge 0}F(k)={\mathbb{M}}_{Z_{\ell }}^{\ell }(b)b,\text{and}\sum _{k\ge 0}G(k)=d{\mathbb{M}}_{Z_r}^r(d).$$

On the other hand, expanding |$\mathbb{M}_{(Z_\ell, Z_r)}(b, c, d)$| using the fact everything converges absolutely produces

$$\begin{array}{cc}& {\mathbb{M}}_{(Z_{\ell },Z_r)}(b,c,d)\hfill \\ & \hspace{0.17em}=c+\sum _{m\ge 1}\mathbb{F}((L_b{Z}_{\ell }{)}^m{R}_c)+\sum _{n\ge 1}\mathbb{F}((R_d{Z}_r{)}^n{R}_c)+\sum _{m,n\ge 1}\mathbb{F}((L_b{Z}_{\ell }{)}^m(R_d{Z}_r{)}^n{R}_c)\hfill \\ & \hspace{0.17em}=c+\sum _{m\ge 1}\mathbb{F}((L_b{Z}_{\ell }{)}^m{R}_c)+\sum _{n\ge 1}\mathbb{F}((R_d{Z}_r{)}^n{R}_c)+\sum _{m,n\ge 1}\mathbb{F}((L_b{Z}_{\ell }{)}^m)c\mathbb{F}((R_d{Z}_r{)}^n)+\sum _{m,n\ge 1}{\mathrm{\Theta }}_{m,n}(b,c,d)\hfill \\ & \hspace{0.17em}=\sum _{m,n\ge 0}\mathbb{F}((L_b{Z}_{\ell }{)}^m)c\mathbb{F}((R_d{Z}_r{)}^n)+\sum _{m,n\ge 1}{\mathrm{\Theta }}_{m,n}(b,c,d)\hfill \\ & \hspace{0.17em}={\mathbb{M}}_{Z_{\ell }}^{\ell }(b)c{\mathbb{M}}_{Z_r}^r(d)+\sum _{m,n\ge 1}{\mathrm{\Theta }}_{m,n}(b,c,d).\hfill \end{array}$$

By rearranging the remaining sum involving |$\Theta_{m, n}(b, c, d)$| to sum over all fixed |$s, t$| in equation (8), and by choosing |$b, d$| sufficiently small so that |$M_{Z_\ell}^\ell(b)$|⁠, |$M_{Z_r}^r(d)$|⁠, |$\mathbb{M}_{Z_\ell}^\ell(b)$|⁠, and |$\mathbb{M}_{Z_r}^r(d)$| are invertible, we obtain

$$\begin{array}{cc}& \hspace{0.17em}\sum _{m,n\ge 1}{\mathrm{\Theta }}_{m,n}(b,c,d)\hfill \\ & \hspace{0.17em}=\sum _{s,t\ge 1}{\mathbb{M}}_{Z_{\ell }}^{\ell }(b)b\hfill \\ & \hspace{0.17em}\times {\mathcal{K}}_{1_{{\chi }_{s,t}}}(\underset{s\text{entries}}{\underbrace{Z_{\ell },L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell },\dots ,L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell }}},\underset{t-1\text{entries}}{\underbrace{Z_r,R_{d{M}_{Z_r}^r(d)}{Z}_r,\dots ,R_{d{M}_{Z_r}^r(d)}{Z}_r}},R_{d{M}_{Z_r}^r(d)}{Z}_r{R}_{M_{(Z_{\ell },Z_r)}(b,c,d)})\hfill \\ & \hspace{0.17em}\times d{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}={\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}\hfill \\ & \hspace{0.17em}\times \sum _{s,t\ge 1}\left({\mathcal{K}}_{1_{{\chi }_{s,t}}}(\underset{s\text{entries}}{\underbrace{L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell },\dots ,L_{M_{Z_{\ell }}^{\ell }(b)b}{Z}_{\ell }}},\underset{t-1\text{entries}}{\underbrace{R_{d{M}_{Z_r}^r(d)}{Z}_r,\dots ,R_{d{M}_{Z_r}^r(d)}{Z}_r}},R_{d{M}_{Z_r}^r(d)}{Z}_r{R}_{M_{(Z_{\ell },Z_r)}(b,c,d)})\right)\hfill \\ & \hspace{0.17em}\times M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}={\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}[{\mathcal{C}}_{(Z_{\ell },Z_r)}(M_{Z_{\ell }}^{\ell }(b)b,M_{(Z_{\ell },Z_r)}(b,c,d),d{M}_{Z_r}^r(d))-{\mathcal{C}}_{Z_{\ell }}^{\ell }(M_{Z_{\ell }}^{\ell }(b)b)M_{(Z_{\ell },Z_r)}(b,c,d)\hfill \\ & \hspace{0.17em}-M_{(Z_{\ell },Z_r)}(b,c,d){\mathcal{C}}_{Z_r}^r(d{M}_{Z_r}^r(d))+M_{(Z_{\ell },Z_r)}(b,c,d)]M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}={\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}{\mathcal{C}}_{(Z_{\ell },Z_r)}(M_{Z_{\ell }}^{\ell }(b)b,M_{(Z_{\ell },Z_r)}(b,c,d),d{M}_{Z_r}^r(d))M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}-{\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}(1+M_{Z_{\ell }}^{\ell }(b)-M_{Z_{\ell }}^{\ell }(b){\mathbb{M}}_{Z_{\ell }}^{\ell }(b{)}^{-1})M_{(Z_{\ell },Z_r)}(b,c,d)M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}-{\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}{M}_{(Z_{\ell },Z_r)}(b,c,d)(1+M_{Z_r}^r(d)-{\mathbb{M}}_{Z_r}^r(d{)}^{-1}{M}_{Z_r}^r(d))M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}+{\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}{M}_{(Z_{\ell },Z_r)}(b,c,d)M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}={\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}\left({\mathcal{C}}_{(Z_{\ell },Z_r)}(M_{Z_{\ell }}^{\ell }(b)b,M_{(Z_{\ell },Z_r)}(b,c,d),d{M}_{Z_r}^r(d))-M_{(Z_{\ell },Z_r)}(b,c,d)\right)\hfill \\ & \hspace{0.17em}\times M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)\hfill \\ & \hspace{0.17em}-({\mathbb{M}}_{Z_{\ell }}^{\ell }(b)-1)M_{(Z_{\ell },Z_r)}(b,c,d)M_{Z_r}^r(d{)}^{-1}{\mathbb{M}}_{Z_r}^r(d)-{\mathbb{M}}_{Z_{\ell }}^{\ell }(b)M_{Z_{\ell }}^{\ell }(b{)}^{-1}{M}_{(Z_{\ell },Z_r)}(b,c,d)({\mathbb{M}}_{Z_r}^r(d)-1),\hfill \end{array}$$

where the fourth equality follows from Lemma 8.2. The result now follows by combining these equations. ■

9 Operator-Valued C-Bi-Free Limit Theorems

In this section, operator-valued conditionally bi-free limit theorems are studied. Recall first from Definition 6.2 that if |${\mathcal{Z}} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$| is a two-faced family in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$|⁠, then the moment and cumulant series |$(\nu^{\mathcal{Z}}, \mu^{\mathcal{Z}})$| and |$(\rho^{\mathcal{Z}}, \eta^{\mathcal{Z}})$| completely describe the joint distribution of |${\mathcal{Z}}$| with respect to |$({\mathbb{E}}, {\mathbb{F}})$|⁠. In that which follows, given a bi-non-crossing partition |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi)$| it is often convenient to define

$$({\nu }_{\omega }^{\mathcal{Z}}{)}_{\pi }(b_1,\dots ,b_{n-1}),({\mu }_{\omega }^{\mathcal{Z}}{)}_{\pi }(b_1,\dots ,b_{n-1}),({\rho }_{\omega }^{\mathcal{Z}}{)}_{\pi }(b_1,\dots ,b_{n-1}),\text{and}({\eta }_{\omega }^{\mathcal{Z}}{)}_{\pi }(b_1,\dots ,b_{n-1})$$

by using operator-valued conditionally bi-multiplicativity and replacing |$1_{\chi_\omega}$| with |$\pi$| in Notation 6.1.

9.1 The operator-valued c-bi-free central limit theorem

Like any non-commutative probability theory, the first result is a central limit theorem in the operator-valued c-bi-free setting.

 

Definition 9.1.

A two-faced family |${\mathcal{Z}} = ((Z_i)_{i \in I}, (Z_j)_{j \in J})$| in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| is said to have a centred |$({\mathcal{B}}, {\mathcal{D}})$|-valued c-bi-free Gaussian distribution if

$${\rho }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})={\eta }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})=0$$

for all |$n \geq 1$| with |$n \neq 2$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in \mathcal{B}$|⁠. □

In view of the definition above and the moment-cumulant formulae, it is enough to specify |$\nu_\omega^{\mathcal{Z}}(b)$| and |$\mu_\omega^{\mathcal{Z}}(b)$| for |$\omega: \{1, 2\} \to I \sqcup J$| and |$b \in {\mathcal{B}}$|⁠.

 

Definition 9.2.

Let |$I$| and |$J$| be non-empty disjoint finite sets, let |$M_{|I \sqcup J|}({\mathcal{B}})$| and |$M_{|I \sqcup J|}({\mathcal{D}})$| denote the |$|I \sqcup J|$| by |$|I \sqcup J|$| matrices with entries in |${\mathcal{B}}$| and |${\mathcal{D}}$| respectively, and let

$$\sigma :\mathcal{B}\to M_{|I\bigsqcup J|}(\mathcal{B}),b\mapsto ({\sigma }_{k,\ell }(b){)}_{k,\ell \in I\bigsqcup J}\text{and}\tau :\mathcal{B}\to M_{|I\bigsqcup J|}(\mathcal{D}),b\mapsto ({\tau }_{k,\ell }(b){)}_{k,\ell \in I\bigsqcup J}$$

be linear maps. A two-faced family |${\mathcal{Z}} = ((Z_i)_{i \in I}, (Z_j)_{j \in J})$| in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| is said to have a centred |$({\mathcal{B}}, {\mathcal{D}})$|-valued c-bi-free Gaussian distribution with covariance matrices |$(\sigma, \tau)$| if, in addition to having a centred |$({\mathcal{B}}, {\mathcal{D}})$|-valued c-bi-free Gaussian distribution,

$${\nu }_{\omega }^{\mathcal{Z}}(b)={\sigma }_{\omega (1),\omega (2)}(b)\in \mathcal{B}\text{and}{\mu }_{\omega }^{\mathcal{Z}}(b)={\tau }_{\omega (1),\omega (2)}(b)\in \mathcal{D}$$

for all |$\omega: \{1, 2\} \to I \sqcup J$| and |$b \in {\mathcal{B}}$|⁠. □

 

Theorem 9.3.

Let |$\{{\mathcal{Z}}_m = ((Z_{m; i})_{i \in I}, (Z_{m; j})_{j \in J})\}_{m = 1}^\infty$| be a sequence of two-faced families in a Banach |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| which are c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠. Moreover assume

• (1) |${\mathbb{E}}(Z_{m; k}) = {\mathbb{F}}(Z_{m; k}) = 0$| for all |$m \geq 1$| and |$k \in I \sqcup J$|⁠;

• (2) |$\sup_{m \geq 1}\|\nu_\omega^{{\mathcal{Z}}_m}(b_1, \dots, b_{n - 1})\| < \infty$| and |$\sup_{m \geq 1}\|\mu_\omega^{{\mathcal{Z}}_m}(b_1, \dots, b_{n - 1})\| < \infty$| for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠;

• (3) there are linear maps |$\sigma: {\mathcal{B}} \to M_{|I \sqcup J|}({\mathcal{B}})$| and |$\tau: {\mathcal{B}} \to M_{|I \sqcup J|}({\mathcal{D}})$| such that

  $$\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{m=1}^N{\nu }_{\omega }^{{\mathcal{Z}}_m}(b)={\sigma }_{\omega (1),\omega (2)}(b)\text{and}\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{m=1}^N{\mu }_{\omega }^{{\mathcal{Z}}_m}(b)={\tau }_{\omega (1),\omega (2)}(b)$$

  for all |$\omega: \{1, 2\} \to I \sqcup J$| and |$b \in {\mathcal{B}}$|⁠.

Then the two-faced families |$\{\S_N = ((S_{N ; i})_{i \in I}, (S_{N ; j})_{j \in J})\}_{N = 1}^\infty$|⁠, defined by

$$S_{N;k}=\frac{1}{\sqrt{N}}\sum _{m=1}^N{Z}_{m;k},k\in I\bigsqcup J$$

converges in distribution to a two-faced family |${\mathcal{Y}} = ((Y_i)_{i \in I}, (Y_j)_{j \in J})$| which has a centred |$({\mathcal{B}}, {\mathcal{D}})$|-valued c-bi-free Gaussian distribution with covariance matrices |$(\sigma, \tau)$|⁠. □

 

Proof.

Since the cumulant series uniquely determine the joint distributions, it suffices to show that

$$\underset{N\to \mathrm{\infty }}{lim}{\rho }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})={\rho }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})\text{and}\underset{N\to \mathrm{\infty }}{lim}{\eta }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})={\eta }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. By definitions, this means

$$\underset{N\to \mathrm{\infty }}{lim}{\rho }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}{\eta }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})=0$$

for all |$\omega: \{1, \dots, n\} \to I \sqcup J$| such that |$n \neq 2$|⁠,

$$\underset{N\to \mathrm{\infty }}{lim}{\rho }_{\omega }^{{\mathrm{\S }}_N}(b)={\sigma }_{\omega (1),\omega (2)}(b)\text{and}\underset{N\to \mathrm{\infty }}{lim}{\eta }_{\omega }^{{\mathrm{\S }}_N}(b)={\tau }_{\omega (1),\omega (2)}(b)$$

for all |$\omega: \{1, 2\} \to I \sqcup J$| and |$b \in {\mathcal{B}}$|⁠.

For fixed |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠, by the additive and multilinear properties of cumulants, we have

$$\begin{array}{cc}\hfill E& \hspace{0.17em}:=\underset{N\to \mathrm{\infty }}{lim}{\rho }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N^{n/2}}\sum _{m=1}^N{\rho }_{\omega }^{{\mathcal{Z}}_m}(b_1,\dots ,b_{n-1}),\hfill \\ \hfill F& \hspace{0.17em}:=\underset{N\to \mathrm{\infty }}{lim}{\eta }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N^{n/2}}\sum _{m=1}^N{\eta }_{\omega }^{{\mathcal{Z}}_m}(b_1,\dots ,b_{n-1}).\hfill \end{array}$$

If |$n = 1$|⁠, then

$$\begin{array}{cc}\hfill E& \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{\sqrt{N}}\sum _{m=1}^N\mathbb{E}(Z_{m;\omega (1)})=0,\hfill \\ \hfill F& \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{\sqrt{N}}\sum _{m=1}^N\mathbb{F}(Z_{m;\omega (1)})=0\hfill \end{array}$$

by assumption |$(1)$|⁠. If |$n \geq 3$|⁠, then assumption |$(2)$| and operator-valued conditionally bi-multiplicativity imply

$$\underset{m\ge 1}{sup}\Vert {\rho }_{\omega }^{{\mathcal{Z}}_m}(b_1,\dots ,b_{n-1})\Vert :=B<\mathrm{\infty }\text{and}\underset{m\ge 1}{sup}\Vert {\eta }_{\omega }^{{\mathcal{Z}}_m}(b_1,\dots ,b_{n-1})\Vert :=D<\mathrm{\infty },$$

hence

$$\Vert E\Vert \le \underset{N\to \mathrm{\infty }}{lim}\frac{B}{N^{(n-2)/2}}=0\text{and}\Vert F\Vert \le \underset{N\to \mathrm{\infty }}{lim}\frac{D}{N^{(n-2)/2}}=0.$$

Otherwise |$n = 2$| and

$$E=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{m=1}^N{\rho }_{\omega }^{{\mathcal{Z}}_m}(b)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{m=1}^N{\nu }_{\omega }^{{\mathcal{Z}}_m}(b)={\sigma }_{\omega (1),\omega (2)}(b)$$

and similarly |$F = \tau_{\omega(1), \omega(2)}(b)$|⁠, for all |$\omega: \{1, 2\} \to I \sqcup J$| and |$b \in {\mathcal{B}}$| by assumptions |$(1)$| and |$(3)$|⁠. ■

9.2 The operator-valued compound c-bi-free Poisson limit theorem

The next result is a Poisson type limit theorem in the operator-valued c-bi-free setting. In what follows, all two-faced families are assumed to have non-empty disjoint left and right index sets |$I$| and |$J$|⁠, respectively. To formulate the statement, we introduce the following notation.

Let |$(\nu_1, \mu_1)$| and |$(\nu_2, \mu_2)$| be the moment series of two-faced families. For |$\lambda \in {\mathbb{R}}$|⁠, denote by

$$(\lambda {\nu }_1+(1-\lambda ){\nu }_2,\lambda {\mu }_1+(1-\lambda ){\mu }_2)$$

the moment series of some two-faced family such that

$$(\lambda {\nu }_1+(1-\lambda ){\nu }_2{)}_{\omega }(b_1,\dots ,b_{n-1})=\lambda ({\nu }_1{)}_{\omega }(b_1,\dots ,b_{n-1})+(1-\lambda )({\nu }_2{)}_{\omega }(b_1,\dots ,b_{n-1})$$

and

$$(\lambda {\mu }_1+(1-\lambda ){\mu }_2{)}_{\omega }(b_1,\dots ,b_{n-1})=\lambda ({\mu }_1{)}_{\omega }(b_1,\dots ,b_{n-1})+(1-\lambda )({\mu }_2{)}_{\omega }(b_1,\dots ,b_{n-1})$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. Such a realization always exists by similar (and simpler) constructions as in the proofs of [11, Lemma 3.8] and Lemma 6.3. Moreover, let |$(\nu^\delta, \mu^\delta)$| be the special moment series such that

$${\nu }_{\omega }^{\delta }(b_1,\dots ,b_{n-1})={\mu }_{\omega }^{\delta }(b_1,\dots ,b_{n-1})=0$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠.

 

Definition 9.4.

Let |$(\nu, \mu)$| be the moment series of some two-faced family and let |$\lambda \in {\mathbb{R}}$|⁠. A two-faced family |${\mathcal{Z}} = ((Z_i)_{i \in I}, (Z_j)_{j \in J})$| in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| is said to have a |$({\mathcal{B}}, {\mathcal{D}})$|-valued compound c-bi-free Poisson distribution with rate |$\lambda$| and jump distribution |$(\nu, \mu)$| if

$${\rho }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})=\lambda {\nu }_{\omega }(b_1,\dots ,b_{n-1})\text{and}{\eta }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})=\lambda {\mu }_{\omega }(b_1,\dots ,b_{n-1})$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. □

 

Theorem 9.5.

Let |$(\nu, \mu)$| be the moment series of some two-faced family, let |$\lambda \in {\mathbb{R}}$|⁠, and consider the sequence |$\{(\nu_N, \mu_N)\}_{N = 1}^\infty$| of moment series defined by

$${\nu }_N=\left(1-\frac{\lambda }{N}\right){\nu }^{\delta }+\frac{\lambda }{N}\nu \text{and}{\mu }_N=\left(1-\frac{\lambda }{N}\right){\mu }^{\delta }+\frac{\lambda }{N}\mu .$$

If |$\{{\mathcal{Z}}_{N; m} = ((Z_{N; m; i})_{i \in I}, (Z_{N; m; j})_{j \in J})\}_{m = 1}^N$| is a sequence of identically distributed two-faced families in a |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| which are c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$| with moment series |$(\nu_N, \mu_N)$|⁠, then the two-faced families |$\{\S_N = ((S_{N ; i})_{i \in I}, (S_{N ; j})_{j \in J})\}_{N = 1}^\infty$|⁠, defined by

$$S_{N;k}=\sum _{m=1}^N{Z}_{N;m;k},k\in I\bigsqcup J$$

converges in distribution to a two-faced family |${\mathcal{Z}} = ((Z_i)_{i \in I}, (Z_j)_{j \in J})$| which has a |$({\mathcal{B}}, {\mathcal{D}})$|-valued compound c-bi-free Poisson distribution with rate |$\lambda$| and jump distribution |$(\nu, \mu)$|⁠. □

 

Proof.

For each |$N \geq 1$|⁠, let |$(\rho_N, \eta_N)$| be the cumulant series corresponding to |$(\nu_N, \mu_N)$|⁠. For |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠, we have

$$\begin{array}{cc}\hfill ({\rho }_N{)}_{\omega }(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}({\nu }_N{)}_{\pi }(b_1,\dots ,b_{n-1}){\mu }_{\mathcal{B}\mathcal{N}\mathcal{C}}(\pi ,1_{{\chi }_{\omega }})\hfill \\ & \hspace{0.17em}=({\nu }_N{)}_{\omega }(b_1,\dots ,b_{n-1})+O(1/N^2)\hfill \\ & \hspace{0.17em}=\frac{\lambda }{N}{\nu }_{\omega }(b_1,\dots ,b_{n-1})+O(1/N^2)\hfill \end{array}$$

and thus

$$\begin{array}{cc}\hfill {\rho }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}{\rho }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\left(\lambda {\nu }_{\omega }(b_1,\dots ,b_{n-1})+O(1/N)\right)\hfill \\ & \hspace{0.17em}=\lambda {\nu }_{\omega }(b_1,\dots ,b_{n-1}).\hfill \end{array}$$

Similarly, we have |$(\eta_N)_\pi(b_1, \dots, b_{n - 1}) = O(1/N^2)$| for |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_\omega)$| with at least two blocks, therefore

$$({\eta }_N{)}_{\omega }(b_1,\dots ,b_{n-1})=({\mu }_N{)}_{\omega }(b_1,\dots ,b_{n-1})+O(1/N^2)=\frac{\lambda }{N}{\mu }_{\omega }(b_1,\dots ,b_{n-1})+O(1/N^2)$$

and thus

$${\eta }_{\omega }^{\mathcal{Z}}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}{\eta }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})=\lambda {\mu }_{\omega }(b_1,\dots ,b_{n-1})$$

as required. ■

9.3 A general operator-valued c-bi-free limit theorem

We finish this section with an operator-valued analogue of [6, Theorem 6.8].

 

Lemma 9.6.

For every |$N \in \mathbb{N}$|⁠, let |${\mathcal{Z}}_N = ((Z_{N; i})_{i \in I}, (Z_{N; j})_{j \in J})$| be a two-faced family in a Banach |$\mathcal{B}$|-|$\mathcal{B}$|-non-commutative-probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$(\mathcal{A}_N, {\mathbb{E}}_{{\mathcal{A}}_N}, {\mathbb{F}}_{{\mathcal{A}}_N}, \varepsilon_N)$|⁠. The following assertions are equivalent.

• (1) For all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in \mathcal{B}$|⁠, the limits

  $$\underset{N\to \mathrm{\infty }}{lim}N{\nu }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})\in \mathcal{B}\text{and}\underset{N\to \mathrm{\infty }}{lim}N{\mu }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})\in \mathcal{D}$$

  exist.
• (2) For all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in \mathcal{B}$|⁠, the limits

  $$\underset{N\to \mathrm{\infty }}{lim}N{\rho }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})\in \mathcal{B}\text{and}\underset{N\to \mathrm{\infty }}{lim}N{\eta }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})\in \mathcal{D}$$

  exist.

Moreover, if these assertions hold, then

$$\begin{array}{cc}& \hspace{0.17em}\underset{N\to \mathrm{\infty }}{lim}N{\nu }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}N{\rho }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})\text{and}\hfill \\ & \hspace{0.17em}\underset{N\to \mathrm{\infty }}{lim}N{\mu }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}N{\eta }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})\hfill \end{array}$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in \mathcal{B}$|⁠. □

 

Proof.

Suppose assertion |$(2)$| holds. Since |$(\kappa_{{\mathcal{A}}_N}, {\mathcal{K}}_{{\mathcal{A}}_N})$| is operator-valued conditionally bi-multiplicative, we have

$$({\rho }_{\omega }^{{\mathcal{Z}}_n}{)}_{\pi }(b_1,\dots ,b_{n-1})=O\left(1/N^2\right)\text{and}({\eta }_{\omega }^{{\mathcal{Z}}_n}{)}_{\pi }(b_1,\dots ,b_{n-1})=O\left(1/N^2\right)$$

for |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_\omega)$| with at least two blocks. Hence

$$\begin{array}{cc}\hfill \underset{N\to \mathrm{\infty }}{lim}N{\nu }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}N\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}({\rho }_{\omega }^{{\mathcal{Z}}_n}{)}_{\pi }(b_1,\dots ,b_{n-1})\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\left(N{\rho }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})+O\left(1/N\right)\right)\hfill \end{array}$$

and similarly

$$\underset{N\to \mathrm{\infty }}{lim}N{\mu }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}\left(N{\eta }_{\omega }^{{\mathcal{Z}}_n}(b_1,\dots ,b_{n-1})+O\left(1/N\right)\right)$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in \mathcal{B}$|⁠.

The proof for the other direction is analogous by the operator-valued conditionally bi-multiplicativity of |$({\mathcal{E}}_{{\mathcal{A}}_N}, {\mathcal{F}}_{{\mathcal{A}}_N})$| and the moment-cumulant formulae from Definitions 2.7 and 4.7. ■

 

Theorem 9.7.

For every |$N \in {\mathbb{N}}$|⁠, let |$({\mathcal{A}}_N, {\mathbb{E}}_N, {\mathbb{F}}_N, \varepsilon_N)$| be a Banach |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative-probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations and let |$\{{\mathcal{Z}}_{N; m} = ((Z_{N; m; i})_{i \in I}, (Z_{N; m; j})_{j \in J})\}_{m = 1}^N$| be a sequence of identically distributed two-faced families in |$({\mathcal{A}}_N, {\mathbb{E}}_N, {\mathbb{F}}_N, \varepsilon_N)$| which are c-bi-free over |$({\mathcal{B}}, {\mathcal{D}})$|⁠. Furthermore, let |$\S_N = ((S_{N; i})_{i \in I}, (S_{N; j})_{j \in J})$| be the two-faced family in |$({\mathcal{A}}_N, {\mathbb{E}}_N, {\mathbb{F}}_N, \varepsilon_N)$| defined by

$$S_{N;k}=\sum _{m=1}^N{Z}_{N;m;k},k\in I\bigsqcup J.$$

The following assertions are equivalent.

• (1) There exists a two-faced family |${\mathcal{Y}} = ((Y_i)_{i \in I}, (Y_j)_{j \in J})$| in a Banach |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative-probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| such that |$\S_N$| converges in distribution to |${\mathcal{Y}}$| as |$N \to \infty$|⁠.

• (2) For all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠, the limits

  $$\underset{N\to \mathrm{\infty }}{lim}N{\nu }_{\omega }^{{\mathcal{Z}}_{N;m}}(b_1,\dots ,b_{n-1})\text{and}\underset{N\to \mathrm{\infty }}{lim}N{\mu }_{\omega }^{{\mathcal{Z}}_{N;m}}(b_1,\dots ,b_{n-1})$$

  exist and are independent of |$m$|⁠.

Moreover, if these assertions hold, then the operator-valued bi-free and c-bi-free cumulants of |${\mathcal{Y}}$| are given by

$${\rho }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}N{\nu }_{\omega }^{{\mathcal{Z}}_{N;m}}(b_1,\dots ,b_{n-1})\text{and}{\eta }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}N{\mu }_{\omega }^{{\mathcal{Z}}_{N;m}}(b_1,\dots ,b_{n-1})$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. □

 

Proof.

Suppose assertion |$(1)$| holds. For |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠, we have

$$\begin{array}{cc}& {\nu }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}{\nu }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}({\mathcal{E}}_{{\mathcal{A}}_N}{)}_{1_{{\chi }_{\omega }}}\left(S_{N;\omega (1)},C_{b_1}^{\omega (2)}{S}_{N;\omega (2)},\dots ,C_{b_{n-2}}^{\omega (n)}{S}_{N;\omega (n)}{C}_{b_{n-1}}^{\omega (n)}\right)\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\sum _{t(1),\dots ,t(n)=1}^N({\mathcal{E}}_{{\mathcal{A}}_N}{)}_{1_{{\chi }_{\omega }}}\left(Z_{N;t(1);\omega (1)},C_{b_1}^{\omega (2)}{Z}_{N;t(2);\omega (2)},\dots ,C_{b_{n-2}}^{\omega (n)}{Z}_{N;t(n);\omega (n)}{C}_{b_{n-1}}^{\omega (n)}\right)\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\sum _{t(1),\dots ,t(n)=1}^N\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}({\kappa }_{{\mathcal{A}}_N}{)}_{\pi }\left(Z_{N;t(1);\omega (1)},C_{b_1}^{\omega (2)}{Z}_{N;t(2);\omega (2)},\dots ,C_{b_{n-2}}^{\omega (n)}{Z}_{N;t(n);\omega (n)}{C}_{b_{n-1}}^{\omega (n)}\right)\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}{N}^{|\pi |}({\kappa }_{{\mathcal{A}}_N}{)}_{\pi }\left(Z_{N;m;\omega (1)},C_{b_1}^{\omega (2)}{Z}_{N;m;\omega (2)},\dots ,C_{b_{n-2}}^{\omega (n)}{Z}_{N;m;\omega (n)}{C}_{b_{n-1}}^{\omega (n)}\right)\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}{N}^{|\pi |}({\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1}),\hfill \end{array}$$

where the second equality follows from Notation 6.1 by assuming |$\{\omega(k)\}_{k = 1}^n$| intersects both |$I$| and |$J$| (the special cases that |$\omega(k) \in I$| or |$\omega(k) \in J$| for all |$k$| can be checked similarly), and the fifth equality, which is independent of |$m$|⁠, follows from the assumptions of c-bi-free independence over |$({\mathcal{B}}, {\mathcal{D}})$| and identical distribution. Since |$\nu_\omega^{{\mathcal{Y}}}(b_1, \dots, b_{n - 1})$| exist for all |$n \geq 1$| and |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, it can be shown by induction on |$n$| that the limits

$$\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})$$

exist for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_\omega)$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. Indeed, the base case |$n = 1$| follows from the assumption that

$$\underset{N\to \mathrm{\infty }}{lim}N({\kappa }_{{\mathcal{A}}_n}{)}_{1_{{\chi }_{\omega }}}\left(Z_{N;m;\omega (1)}\right)=\underset{N\to \mathrm{\infty }}{lim}({\kappa }_{{\mathcal{A}}_N}{)}_{1_{{\chi }_{\omega }}}\left(S_{N;\omega (1)}\right)=\underset{N\to \mathrm{\infty }}{lim}({\mathcal{E}}_{{\mathcal{A}}_N}{)}_{1_{{\chi }_{\omega }}}\left(S_{N;\omega (1)}\right)={\mathcal{E}}_{1_{{\chi }_{\omega }}}(Y_{\omega (1)})$$

exist for all |$\omega: \{1\} \to I \sqcup J$|⁠. For the inductive step, the limit

$${\nu }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})$$

exists by assumption, and the limit

$$\underset{N\to \mathrm{\infty }}{lim}\sum _{\begin{array}{c}\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })\\ \pi \ne 1_{{\chi }_{\omega }}\end{array}}{N}^{|\pi |}({\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})$$

exists by induction hypothesis with operator-valued conditionally bi-multiplicativity, thus the limit

$$\underset{N\to \mathrm{\infty }}{lim}N{\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}(b_1,\dots ,b_{n-1})$$

exists, and equals

$$\underset{N\to \mathrm{\infty }}{lim}N{\nu }_{\omega }^{{\mathcal{Z}}_{N;m}}(b_1,\dots ,b_{n-1})$$

by Lemma 9.6. On the other hand, it follows from a similar calculation as above that

$$\begin{array}{cc}\hfill {\mu }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})& \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}{\mu }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}{N}^{|\pi |}({\eta }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})\hfill \end{array}$$

for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠, and a similar induction argument on |$n$| shows that the limits

$$\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\eta }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})$$

exist for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_\omega)$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. In particular, choose |$\pi = 1_{\chi_\omega}$| and apply Lemma 9.6, we obtain the existence of the limit

$$\underset{N\to \mathrm{\infty }}{lim}N{\mu }_{\omega }^{{\mathcal{Z}}_{N;m}}(b_1,\dots ,b_{n-1}).$$

Conversely, suppose assertion |$(2)$| holds. By Lemma 9.6 and operator-valued conditionally bi-multiplicativity, the limits

$$\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})\text{and}\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\eta }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})$$

exist for all |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_\omega)$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠. Therefore, by the calculations above,

$$\underset{N\to \mathrm{\infty }}{lim}{\nu }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})$$

(9)

and

$$\underset{N\to \mathrm{\infty }}{lim}{\mu }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\eta }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})$$

(10)

and hence these limits exist. One concludes assertion |$(1)$| by using Lemma 6.3 to construct a two-faced family |${\mathcal{Y}} = ((Y_i)_{i \in I}, (Y_j)_{j \in J})$| in a Banach |${\mathcal{B}}$|-|${\mathcal{B}}$|-non-commutative-probability space with a pair of |$({\mathcal{B}}, {\mathcal{D}})$|-valued expectations |$({\mathcal{A}}, {\mathbb{E}}, {\mathbb{F}}, \varepsilon)$| and define |$\nu_\omega^{{\mathcal{Y}}}(b_1, \dots, b_{n - 1})$| and |$\mu_\omega^{{\mathcal{Y}}}(b_1, \dots, b_{n - 1})$| to be the corresponding limit in equations (9) and (10), respectively.

Finally, for |$n \geq 1$|⁠, |$\omega: \{1, \dots, n\} \to I \sqcup J$|⁠, and |$b_1, \dots, b_{n - 1} \in {\mathcal{B}}$|⁠, we have

$$\begin{array}{cc}\hfill \sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}({\rho }_{\omega }^{\mathcal{Y}}{)}_{\pi }(b_1,\dots ,b_{n-1})& \hspace{0.17em}={\nu }_{\omega }^{\mathcal{Y}}(b_1,\dots ,b_{n-1})\hfill \\ & \hspace{0.17em}=\underset{N\to \mathrm{\infty }}{lim}{\nu }_{\omega }^{{\mathrm{\S }}_N}(b_1,\dots ,b_{n-1})\hfill \\ & \hspace{0.17em}=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})\hfill \end{array}$$

and similarly

$$\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}({\eta }_{\omega }^{\mathcal{Y}}{)}_{\pi }(b_1,\dots ,b_{n-1})=\sum _{\pi \in \mathcal{B}\mathcal{N}\mathcal{C}({\chi }_{\omega })}\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\eta }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1}).$$

A similar induction argument on |$n$| shows that

$$\begin{array}{cc}& \hspace{0.17em}({\rho }_{\omega }^{\mathcal{Y}}{)}_{\pi }(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\rho }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})\text{and}\hfill \\ & \hspace{0.17em}({\eta }_{\omega }^{\mathcal{Y}}{)}_{\pi }(b_1,\dots ,b_{n-1})=\underset{N\to \mathrm{\infty }}{lim}{N}^{|\pi |}({\eta }_{\omega }^{{\mathcal{Z}}_{N;m}}{)}_{\pi }(b_1,\dots ,b_{n-1})\hfill \end{array}$$

for all |$\pi \in {\mathcal{B}}{\mathcal{N}}{\mathcal{C}}(\chi_\omega)$|⁠, from which the last claims follow from Lemma 9.6 applied to |$\pi = 1_{\chi_\omega}$|⁠. ■

Communicated by Prof. Dan Virgil Voiculescu