2.1 Combinatorics on the lattice of bi-non-crossing partitions

For $n\in \mathbb {N}$ , the collection of all partitions on $\{1,\ldots ,n\}$ is denoted by $\mathcal {P}(n)$ . The elements of any $\pi \in \mathcal {P}(n)$ are called the blocks of $\pi $ . A partial ordered on $\mathcal {P}(n)$ is defined via refinement where for $\pi ,\sigma \in \mathcal {P}(n)$ we write $\pi \leq \sigma $ if every block of $\pi $ is contained in a block of $\sigma $ . The maximal element of $\mathcal {P}(n)$ with respect to this partial order is the partition consisting of one block and is denoted by $1_n$ , whereas the minimal element is the partition consisting of n-blocks and is denoted by $0_n$ . Note that $\mathcal {P}(n)$ becomes a lattice under this partial ordering. For $\pi ,\sigma \in \mathcal {P}(n)$ , the join of $\pi $ and $\sigma $ , denoted $\pi \vee \sigma $ , is the minimum element of the non-empty set $\{\upsilon \in \mathcal {P}(n)\, \mid \, \upsilon \geq \pi ,\sigma \}$ . A partition $\pi $ on $\{1,\ldots , n\}$ is said to be non-crossing if whenever $V,W$ are blocks of $\pi $ and $v_1,v_2\in V , w_1,w_2\in W$ are such that

$$\begin{align*}v_1 < w_1 < v_2 < w_2, \end{align*}$$

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

In the bi-free setting, all operators are implicitly imbued with a direction, either left or right. Given a sequence of n operators, we will use a map $\chi \in {\{\ell ,r\}}^n$ to distinguish whether the $k{\text {th}}$ operator is a left or a right operator. Such a map automatically gives rise to a permutation $s_{\chi }$ on $\{1,\ldots ,n\}$ defined as follows: if ${\chi }^{-1}(\{\ell \}) = \{i_1<\cdots <i_p\}$ and ${\chi }^{-1}(\{r\}) = \{j_1<\cdots <j_{n-p}\}$ , then

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

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

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

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

Definition 2.1 Let $n\in \mathbb {N}$ and $ \chi \in {\{\ell ,r\}}^n$ . A partition $\pi \in \mathcal {P}(n)$ is called bi-non-crossing with respect to $\chi $ if the partition $s_{\chi }^{-1}\circ \pi $ (i.e., the partition obtained by applying the permutation $s_{\chi }^{-1}$ to each entry of every block of $\pi $ ) is non-crossing. Equivalently, $\pi $ is bi-non-crossing with respect to $\chi $ if whenever $V,W$ are blocks of $\pi $ and $v_1,v_2\in V , w_1,w_2\in W$ are such that

$$\begin{align*}v_1 \prec_\chi w_1 \prec_\chi v_2 \prec_\chi w_2, \end{align*}$$

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

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

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

Example 2.1 If $\chi \in {\{\ell ,r\}}^6$ is such that $\chi ^{-1}(\{\ell \}) = \{1,2,3,6\}$ and $\chi ^{-1}(\{r\}) = \{4,5\}$ , then

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

and the partition given by

$$\begin{align*}\pi = \{\{1,4\}, \{2,5\}, \{3,6\}\} \end{align*}$$

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

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

Definition 2.2 The bi-non-crossing Möbius function is the map

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

defined recursively by

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

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

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

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

for all $\pi \leq \sigma \in \operatorname {\mathrm {BNC}}(\chi )$ . Hence, $\mu _{\operatorname {\mathrm {BNC}}}$ inherits permuted analogs of the multiplicative properties of $\mu _{\operatorname {\mathrm {NC}}}$ (see [Reference Charlesworth, Nelson and Skoufranis3, Section 3]). In particular, if $n\in \mathbb {N}$ , if $\chi \in \{\ell ,r\}^n$ , if $\pi ,\sigma \in \operatorname {\mathrm {BNC}}(\chi )$ such that $\pi \leq \sigma $ , and if $V_1,\ldots ,V_m$ are unions of blocks of $\pi $ that partition $\{1,\ldots ,n\}$ , then the natural map

$$ \begin{align*} [\pi, \sigma] & \longrightarrow \prod_{k=1}^m [\pi|_{V_k}, \sigma|_{V_k}] \\ \tau & \mapsto\qquad {({\tau |}_{V_k})}_{k=1}^m \end{align*} $$

is a bijection and

$$\begin{align*}\mu_{\operatorname{\mathrm{BNC}}}(\pi,\sigma) = \mu_{\operatorname{\mathrm{BNC}}}({\pi |}_{V_1}, \sigma|_{V_1}) \cdots \mu_{\operatorname{\mathrm{BNC}}}({\pi |}_{V_m},\sigma|_{V_m}). \end{align*}$$

2.2 B-B-noncommutative probability spaces and bi-freeness

To study operator-valued bi-free independence, certain structures are required. Thus, we shall remind the reader of the general structures as developed in [Reference Charlesworth, Nelson and Skoufranis2] and refer the reader there for more details.

Definition 2.3 Let B be a unital $*$ -algebra, and let $B^{{\text {op}}}$ denote the unital $*$ -algebra with the same elements as B with the opposite multiplication. A B-B-noncommutative probability space consists of a triple $(A,E,\varepsilon )$ , where A is a unital $*$ -algebra, $\varepsilon : B\otimes B^{{\text {op}}}\to A$ is a unital $*$ -homomorphism such that the restrictions ${\varepsilon |}_{B\otimes 1_B}$ and ${\varepsilon |}_{1_B\otimes B^{{\text {op}}}}$ are both injective, and $E:A\to B$ is a unital linear map such that

$$\begin{align*}E(\varepsilon(b_1\otimes b_2) a) = b_1 E(a) b_2 \qquad \text{and}\qquad E(a \varepsilon(b\otimes 1_B)) = E(a \varepsilon(1_B\otimes b)), \end{align*}$$

for all $b,b_1,b_2\in B$ and $a\in A$ . In addition, consider the unital $*$ -subalgebras $A_\ell $ and $A_r$ of A given by

$$\begin{align*}A_\ell = \{a\in A\, \mid \, a \varepsilon (1_B\otimes b) = \varepsilon(1_B\otimes b) a \text{ for all } b\in B \} \end{align*}$$

and

$$\begin{align*}A_r = \{a\in A\, \mid \, a \varepsilon (b\otimes 1_B) = \varepsilon(b\otimes 1_B) a\text{ for all } b\in B \}. \end{align*}$$

We call $A_\ell $ and $A_r$ the left and right algebras of A, respectively.

Note that one can always assume that a B-B-noncommutative probability space is generated as a $*$ -algebra by $A_\ell $ and $A_r$ .

Example 2.2 Let $\mathcal {A}$ and B be unital $*$ -algebras, and let $\varphi : \mathcal {A} \to \mathbb {C}$ be a unital, linear map. If $A = \mathcal {A} \otimes B \otimes B^{{\text {op}}}$ , if $\varepsilon : B \otimes B^{{\text {op}}} \to A$ is defined by $\varepsilon (b_1 \otimes b_2) = 1_{\mathcal {A}} \otimes b_1 \otimes b_2$ for all $b_1, b_2 \in B$ , and $E : A \to B$ is defined by

$$\begin{align*}E(a \otimes b_1 \otimes b_2) = \varphi(a) b_1b_2 \end{align*}$$

for all $a \in A$ and $b_1, b_2 \in B$ , then $(A, E, \varepsilon )$ is a B-B-noncommutative probability space. Indeed, clearly, $\varepsilon $ is a unital injective $*$ -homomorphism. Furthermore, note for all $Z\in \mathcal {A}$ and $b,b_1,b_2,b_3,b_4\in B$ that

$$\begin{align*}E((1_{\mathcal{A}}\otimes b_1\otimes b_2)(Z\otimes b_3\otimes b_4)) = \varphi(Z)b_1b_3b_4b_2 = b_1 E(Z\otimes b_3\otimes b_4)b_2 \end{align*}$$

and

$$\begin{align*}E((Z\otimes b_1\otimes b_2)(1_{\mathcal{A}}\otimes b\otimes 1_B)) = \varphi(Z)b_1bb_2 = E((Z\otimes b_1\otimes b_2)(1_{\mathcal{A}} \otimes 1_B\otimes b)). \end{align*}$$

Hence, E satisfies the required properties.

For future use, notice that

$$\begin{align*}\mathcal{A} \otimes B \otimes 1_B \subseteq \mathcal{A}_\ell \qquad \text{and}\qquad \mathcal{A} \otimes 1_B \otimes B^{{\text{op}}} \subseteq \mathcal{A}_r. \end{align*}$$

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

Example 2.3 Let $\mathfrak {M}$ be a finite von Neumann algebra with a tracial state $\tau : \mathfrak {M} \to \mathbb {C}$ , and let $L_2(\mathfrak {M}, \tau )$ be the Gelfand-Naimark-Segal Hilbert space generated by $(\mathfrak {M}, \tau )$ . For $T \in \mathfrak {M}$ , let $L_T$ denote the left action of T on $L_2(\mathfrak {M}, \tau )$ , and let $R_T$ denote the right action of T on $L_2(\mathfrak {M}, \tau )$ . Furthermore, let A be the algebra generated by $\{L_T, R_T \, \mid \, T \in \mathfrak {M}\}$ .

Let B be a unital von Neumann subalgebra of $\mathfrak {M}$ , and let $E_B : \mathfrak {M} \to B$ be the conditional expectation of $\mathfrak {M}$ onto B. Recall that if $P : L_2(\mathfrak {M}, \tau ) \to L_2(B, \tau )$ is the orthogonal projection of $L_2(\mathfrak {M}, \tau )$ onto $L_2(B, \tau )$ , then $E_B(Z) = P(Z1_{\mathfrak {M}})$ for all $Z \in \mathfrak {M}$ .

Define $\varepsilon : B \otimes B^{{\text {op}}} \to A$ by $\varepsilon (b_1 \otimes b_2) = L_{b_1} R_{b_2}$ and define $E : A \to B$ by

$$\begin{align*}E(Z) = P(Z 1_{\mathfrak{M}}) \end{align*}$$

for all $Z \in A$ . Elementary von Neumann algebra theory implies that the range of E is indeed contained in B. To see that $(A, E, \varepsilon )$ is a B-B-noncommutative probability space, first note that $\varepsilon $ is clearly a unital $*$ -homomorphism that is injective when restricted to $B \otimes 1_B$ and when restricted to $1_B \otimes B^{\text {op}}$ . Moreover, note for all $Z\in A$ and $b,b_1,b_2\in B$ that

$$\begin{align*}E(L_{b_1} R_{b_2}Z ) = P(b_1(Z1_{\mathfrak{M}})b_2) = b_1P(Z1_{\mathfrak{M}})b_2 = b_1 E(Z)b_2 \end{align*}$$

and

$$\begin{align*}E(TL_b) = P(TL_b 1_{\mathfrak{M}})= P(Tb) = P(TR_b 1_{\mathfrak{M}}) = E(TR_b). \end{align*}$$

Hence, E satisfies the required properties.

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

$$\begin{align*}B_\ell = \varepsilon(B\otimes 1_B)=\{L_b \, \mid \, b\in B\} \quad \text{and}\quad B_r = \varepsilon(1_B\otimes B^{{\text{op}}})=\{R_b \, \mid \, b\in B\}. \end{align*}$$

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

Example 2.5 Let $\mathfrak {M}_1$ and $\mathfrak {M}_2$ be finite von Neumann algebras with a common von Neumann subalgebra B and tracial states $\tau _1$ and $\tau _2$ , respectively, such that $\tau _1|_B = \tau _2|_B$ . Let $\mathfrak {M} = \mathfrak {M}_1\ast _B \mathfrak {M}_2$ be the reduced free product von Neumann algebra with amalgamation over B, let $E_B : \mathfrak {M} \to B$ be the conditional expectation of $\mathfrak {M}$ onto B, and let $\tau = \tau _1 \ast \tau _2 = \tau _1|_B \circ E_B$ be the tracial state on $\mathfrak {M}$ . If E and $\varepsilon $ are as in Example 2.3 for $(\mathfrak {M}, \tau )$ , then

$$\begin{align*}\left\{\left(\{L_X \, \mid \, X \in \mathfrak{M}_1\}, \{R_Y \, \mid \, Y \in \mathfrak{M}_1\}\right)\right\}\qquad \text{and}\qquad \left\{\left(\{L_X \, \mid \, X \in \mathfrak{M}_2\}, \{R_Y \, \mid \, Y \in \mathfrak{M}_2\}\right)\right\} \end{align*}$$

are bi-free with amalgamation over B.

In order to study bi-free independence with amalgamation, the operator-valued bi-free moment and cumulant functions are key. These functions have specific properties that are described via the following concept. In what follows and for the remainder of the paper, given an n-tuple $(Z_1, \ldots , Z_n)$ and $V \subseteq \{1,\ldots , n\}$ , we will use $(Z_1, \ldots , Z_n)|_V$ to denote the $|V|$ -tuple where only the entries $Z_k$ where $k \in V$ remain.

Definition 2.5 [Reference Charlesworth, Nelson and Skoufranis2, Definition 4.2.1]

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

$$\begin{align*}\Phi : \bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in\{\ell,r\}^n}\operatorname{\mathrm{BNC}}(\chi)\times A_{\chi(1)}\times A_{\chi(2)}\times\cdots\times A_{\chi(n)}\to B \end{align*}$$

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

1. (1) Let

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

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

   $$\begin{align*}\Phi_{1_{\chi}}(Z_1,\ldots,Z_{n-1},Z_n L_b) = \begin{cases} \Phi_{1_\chi}(Z_1,\ldots,Z_{q-1},Z_q R_b,Z_{q+1},\ldots,Z_n), &\text{if }q\neq -\infty,\\ \Phi_{1_\chi}(Z_1,\ldots,Z_{n-1},Z_n) b, & \text{if } q= - \infty. \\ \end{cases} \end{align*}$$

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

   $$\begin{align*}\Phi_{1_{\chi}}(Z_1,\ldots,Z_{n-1},Z_n R_b) = \begin{cases} \Phi_{1_\chi}(Z_1,\ldots,Z_{q-1},Z_q L_b,Z_{q+1},\ldots,Z_n), &\text{if }q\neq -\infty,\\ b \Phi_{1_\chi}(Z_1,\ldots,Z_{n-1},Z_n), & \text{if } q= - \infty. \\ \end{cases} \end{align*}$$

2. (2) Let $p\in \{1,\ldots ,n\}$ , and let

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

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

   $$\begin{align*}\Phi_{1_\chi}(Z_1,\ldots,Z_{p-1},L_b Z_p,Z_{p+1}\ldots,Z_n)= \begin{cases} \Phi_{1_\chi}(Z_1,\ldots,Z_{q-1},Z_q L_b,Z_{q+1},\ldots,Z_n), &\text{if }q\neq -\infty,\\ b \Phi_{1_\chi}(Z_1,\ldots,Z_{n-1},Z_n), & \text{if }q= - \infty. \\ \end{cases} \end{align*}$$

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

   $$\begin{align*}\Phi_{1_\chi}(Z_1,\ldots,Z_{p-1},R_b Z_p,Z_{p+1}\ldots,Z_n)= \begin{cases} \Phi_{1_\chi}(Z_1,\ldots,Z_{q-1},Z_q R_b,Z_{q+1},\ldots,Z_n), & \text{if }q\neq -\infty,\\ \Phi_{1_\chi}(Z_1,\ldots,Z_{n-1},Z_n) b, & \text{if }q= - \infty. \\ \end{cases} \end{align*}$$

3. (3) Suppose $V_1,\ldots , V_m$ are unions of blocks of $\pi $ that partition $\{1,\ldots ,n\}$ with each being a $\chi $ -interval (i.e., an interval in the $\chi $ -ordering) and the sets $V_1,\ldots ,V_m$ are ordered by $\preceq _{\chi }$ (i.e., $(\min _{\preceq _{\chi }}V_k)\prec _{\chi } (\min _{\preceq _{\chi }}V_{k+1})$ for all k). Then,

   $$\begin{align*}\Phi_\pi(Z_1,\ldots,Z_n) = \Phi_{\pi|_{V_1}}({(Z_1,\ldots,Z_n)|}_{V_1})\cdots \Phi_{\pi|_{V_m}}({(Z_1,\ldots,Z_n)|}_{V_m}). \end{align*}$$

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

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

   Then, we have that

   $$ \begin{align*} \Phi_\pi(Z_1,\ldots,Z_n) &= \begin{cases} \Phi_{\pi|_W}\left({(Z_1,\ldots,Z_{p-1},Z_p L_{\Phi_{\pi|_V}({(Z_1,\ldots,Z_n)|}_V)},Z_{p+1},\ldots,Z_n) |}_W\right), &\text{if }\chi(p)=\ell,\\ \Phi_{\pi|_W}\left({(Z_1,\ldots,Z_{p-1},R_{\Phi_{\pi|_V}({(Z_1,\ldots,Z_n)|}_V)} Z_p ,Z_{p+1},\ldots,Z_n) |}_W\right), & \text{if }\chi(p)=r, \\ \end{cases} \\ &= \begin{cases} \Phi_{\pi|_W}\left({(Z_1,\ldots,Z_{q-1},L_{\Phi_{\pi|_V}({(Z_1,\ldots,Z_n)|}_V)} Z_q, Z_{q+1},\ldots,Z_n) |}_W\right), &\text{if }\chi(q)=\ell,\\ \Phi_{\pi|_W}\left({(Z_1,\ldots,Z_{q-1}, Z_q R_{\Phi_{\pi|_V}({(Z_1,\ldots,Z_n)|}_V)} ,Z_{q+1},\ldots,Z_n) |}_W\right), & \text{if } \chi(q)=r. \\ \end{cases} \end{align*} $$

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

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

4.1 Analytical operator-valued bi-moment function

We will now construct the analytical extension of the operator-valued bi-moment function via recursion and the map $\widetilde {E}:L_2(A,\tau )\to L_2(B,\tau _B)$ from Section 3. Note that the recursive process in the following definition is different than that from [Reference Charlesworth, Nelson and Skoufranis2, Definition 5.1] and [Reference Gu and Skoufranis6, Definition 4.4], in order to facilitate the introduction of the $L_2(A, \tau )$ element. The same recursive process could have been used in [Reference Charlesworth, Nelson and Skoufranis2, Definition 5.1] and [Reference Gu and Skoufranis6, Definition 4.4], as these processes are equivalent in those settings. Note that we use $\Psi $ in the following to avoid confusion with $\widetilde {E}$ in Section 3, although $\Psi $ is a multi-entry extension of $\widetilde {E}$ .

Definition 4.2 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space. The analytical bi-moment function

$$\begin{align*}\Psi:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in \{\ell,r\}^n}\operatorname{\mathrm{BNC}}(\chi)\times A_{\chi(1)}\times\cdots\times A_{\chi(n-1)}\times L_2(A,\tau)\to L_2(B,\tau_B) \end{align*}$$

is defined recursively as follows: let $n\in \mathbb {N}$ , $\chi \in \{\ell ,r\}^n$ , $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ , $\xi \in L_2(A,\tau )$ , and $Z_k \in A_{\chi (k)}$ .

• • If $\pi =1_\chi $ , then

  $$\begin{align*}\Psi_{1_\chi}(Z_1,Z_2,\ldots,Z_{n-1},\xi) = \widetilde{E}(Z_1 Z_2\cdots Z_{n-1}\xi). \end{align*}$$

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

  • – Suppose that $\min _{\preceq _{\chi }}V=s_{\chi }(1)$ and $\max _{\preceq _{\chi }}V=s_{\chi }(n)$ , and let

    $$\begin{align*}p = \min_{\preceq_{\chi}}\{i\in\{1,\ldots,n\} \, \mid \, i\notin V\}, \quad q &= \min_{\preceq_{\chi}}\{j\in V \, \mid \, p \prec_{\chi} j\}, \quad \text{and}\\ m &= \max_{\preceq_{\chi}}\{i \in V \, \mid \, i \prec_\chi p \}. \end{align*}$$
    Set
    $$\begin{align*}W = \{i\in\{1,\ldots,n\} \, \mid \, p \preceq_{\chi} i \prec_{\chi} q\}. \end{align*}$$
    Note by construction and the fact that $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ that W is equal to a union of blocks of $\pi $ and $\chi (p)=\chi (j)$ for all $j \in W$ . Thus, we define
    $$\begin{align*}&\Psi_\pi(Z_1,\ldots,Z_{n-1},\xi)\\&\quad = \begin{cases} \Psi_{\pi|_{W^{\mathsf{c}}}}\left({(Z_1,\ldots,Z_{p-1}, Z_m L_{E_{\pi|_W}({(Z_1,\ldots,Z_{n-1},\xi)|}_W)} ,\ldots,Z_{n-1},\xi)|}_{W^{\mathsf{c}}}\right), &\text{if }\chi(p)=\ell,\\ \Psi_{\pi|_{W^{\mathsf{c}}}}\left({(Z_1,\ldots,Z_{q-1}, Z_q R_{E_{\pi|_W}({(Z_1,\ldots,Z_{n-1},\xi)|}_W)} ,\ldots,Z_{n-1},\xi)|}_{W^{\mathsf{c}}}\right), & \text{if } \chi(p)=r. \\ \end{cases} \end{align*}$$
    (Note that as $n\notin W$ , the quantity $E_{\pi |_W}({(Z_1,\ldots ,Z_{n-1},\xi )|}_W)$ is always a well-defined element of B in this case. Note in the case that $\chi (p) = \ell $ that $m \prec _{\chi } p \prec _{\chi } n$ and thus $Z_m \neq \xi $ , so $\Psi _\pi (Z_1,\ldots ,Z_{n-1},\xi )$ is well defined. Also, in the case when $\chi (p)=r$ , note that $n\prec _{\chi } q$ and thus $Z_q\neq \xi $ , so $\Psi _\pi (Z_1,\ldots ,Z_{n-1},\xi )$ is well defined.)

  • – Otherwise, set

    $$\begin{align*}\widetilde{V} = \left\{i\in\{1,\ldots,n\}\, \left| \, \min_{\preceq_{\chi}}V \preceq_{\chi} i \preceq_{\chi} \max_{\preceq_{\chi}}V \right.\right\}. \end{align*}$$
    Note that $\widetilde {V}$ is a proper subset of $\{1,\ldots ,n\}$ that is a union of blocks of $\pi $ and is such that $n\in V\subseteq \widetilde {V}$ . For $q=\max _\leq {\widetilde {V}^{\mathsf {c}}}$ , define
    $$\begin{align*}\Psi_\pi(Z_1,\ldots,Z_{n-1},\xi) = \Psi_{\pi|_{\widetilde{V}^{\mathsf{c}}}}\left({(Z_1,\ldots,Z_{q-1}, Z_q \Psi_{\pi|_{\widetilde{V}}}{({(Z_1,\ldots,Z_{n-1},\xi)|}_{\widetilde{V}})} ,\ldots,Z_{n-1},\xi)|}_{\widetilde{V}^{\mathsf{c}}}\right). \end{align*}$$
    (Note that the quantity $\Psi _{\pi |_{\widetilde {V}}}{({(Z_1,\ldots ,Z_{n-1},\xi )|}_{\widetilde {V}})}$ is a well-defined element of $L_2(B,\tau _B)$ due to the recursive nature of our definition. Moreover, the last element of the sequence
    $$\begin{align*}{(Z_1,\ldots,Z_{q-1}, Z_q \Psi_{\pi|_{\widetilde{V}}}{({(Z_1,\ldots,Z_{n-1},\xi)|}_{\widetilde{V}})} ,\ldots,Z_{n-1},\xi)|}_{\widetilde{V}^{\mathsf{c}}} \end{align*}$$
    is equal to $Z_q \Psi _{\pi |_{\widetilde {V}}}{({(Z_1,\ldots ,Z_{n-1},\xi )|}_{\widetilde {V}})}$ , which is an element of $L_2(A,\tau )$ , so $\Psi _\pi (Z_1,\ldots ,Z_{n-1},\xi ) $ is well defined.)

To aid in the comprehension of Definition 4.2, we provide an example using bi-non-crossing diagrams to show the recursive construction. We note that $\xi $ will always appear last in a sequence of operators and is an element of $L_2(A,\tau )$ and thus neither a left nor a right operator. As such, we treat it as neither. This is reminiscent of [Reference Skoufranis12, Lemma 2.17], where it does not matter whether we treat the last operator in a list as a left or as a right operator, and of [Reference Charlesworth and Skoufranis4, Lemma 2.29 and Proposition 2.30], where the last entry can be a mixture of left and right operators.

Example 4.5 Let $\chi \in \{\ell ,r\}^{12}$ be such that $\chi ^{-1}(\{\ell \})=\{1,5,8,9,11,12\}$ , let $\xi \in L_2(A,\tau )$ , let $Z_k\in A_{\chi (k)}$ , and let $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ be the partition with blocks

$$\begin{align*}V_1 = \{1,3\}, \quad V_2 = \{2\}, \quad V_3 = \{4,5,11,12\}, \quad V_4 = \{6,10\}, \quad V_5 = \{7\}, \quad \text{and}\quad V_6 = \{8,9\}. \end{align*}$$

To compute $\Psi _\pi (Z_1, \ldots , Z_{11}, \xi )$ , we note that the second part of the second step of the recursive definition from Definition 4.2 applies first. In particular,

$$\begin{align*}\widetilde{V} = \bigcup_{k=3}^6 V_k. \end{align*}$$

Thus, if

$$\begin{align*}X = \Psi_{\pi|_{\widetilde{V}}}(Z_4,Z_5,Z_6,Z_7,Z_8,Z_9,Z_{10},Z_{11},\xi), \end{align*}$$

then

$$\begin{align*}\Psi_\pi(Z_1,\ldots,Z_{11},\xi) = \Psi_{\pi|_{{\widetilde{V}^{\mathsf{c}}}}}(Z_1,Z_2,Z_3 X). \end{align*}$$

Diagrammatically, this first reduction is seen as follows:

Note that

$$\begin{align*}\Psi_{\pi|_{{\widetilde{V}^{\mathsf{c}}}}}(Z_1,Z_2,Z_3 X) = \Psi_{\pi|_{V_2}}(Z_2 \Psi_{\pi|_{V_1}}(Z_1,Z_3X))= \widetilde{E}\left(Z_2 \widetilde{E}(Z_1Z_3 X)\right), \end{align*}$$

where the first equality holds by the same recursive idea, whereas the second equality holds by the first step of Definition 4.2.

When computing the value of X, the minimal and maximal elements of $\{4,5,\ldots ,11,12\}$ in the ${\chi |}_{\widetilde {V}}$ -order are $5$ and $4$ , respectively, and the block that contains the index corresponding to $\xi $ contains both 5 and 4. Thus, the first part of the second step of Definition 4.2 should be used. The algorithm in Definition 4.2 then calculates the value of X by “stripping out” the $\chi $ -intervals $V_6$ and $V_4\cup V_5$ successively and this is seen via the following two diagrammatic reductions:

and

It is readily verified using the fact that the operator-valued bi-free moment function is bi-multiplicative that $E_{\pi |_{V_6}}(Z_8,Z_9)= E(Z_8Z_9)$ and $E_{\pi |_{V_4\cup V_6}}(Z_6,Z_7,Z_{10}) = E(Z_6R_{E(Z_7)}Z_{10})$ . Thus, using the fact that $\pi |_{V_3} = 1_{{\chi |}_{V_3}}$ , the first step in Definition 4.2 yields

$$\begin{align*}X = \widetilde{E}\left(Z_4 R_{E(Z_6R_{E(Z_7)}Z_{10})} Z_5 L_{E(Z_8Z_9)} Z_{11}\xi \right). \end{align*}$$

Hence,

$$ \begin{align*} \Psi_\pi(Z_1,\ldots,Z_{11},\xi) &= \widetilde{E}\left(Z_2 \widetilde{E}\left(Z_1Z_3 \widetilde{E}\left(Z_4 R_{E\left(Z_6R_{E(Z_7)}Z_{10}\right)} Z_5 L_{E(Z_8Z_9)} Z_{11}\xi\right)\right)\right)\\ &= \widetilde{E}\left(Z_1Z_3 \widetilde{E}\left(Z_4 R_{E\left(Z_6R_{E(Z_7)}Z_{10}\right)} Z_5 L_{E(Z_8Z_9)} Z_{11}\xi\right)\right)E(Z_2), \end{align*} $$

with the last equality following from Proposition 3.8.

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

Theorem 4.6 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space. For any $n\in \mathbb {N}$ , $\chi \in \{\ell ,r\}^n$ , $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ , and $Z_k\in A_{\chi (k)}$ ,

$$\begin{align*}\Psi_\pi(Z_1,\ldots,Z_{n-1},Z_n+N_\tau) = E_\pi(Z_1,\ldots,Z_{n-1},Z_n) + N_{\tau_B}. \end{align*}$$

Proof Note that each step in the recursive definition of Definition 4.2 is a step that can be performed to the operator-valued bi-free moment function as the operator-valued bi-free moment function is bi-multiplicative (see Definition 2.5). Therefore, as Proposition 3.6 implies that

$$\begin{align*}\widetilde{E}(a) = E(a) + N_{\tau_B} \end{align*}$$

for all $a \in A$ , by applying the same recursive properties to $E_\pi (Z_1,\ldots ,Z_{n-1},Z_n)$ as used to compute $\Psi _\pi (Z_1,\ldots ,Z_{n-1},Z_n+N_\tau )$ , the result follows.

Like with the construction of the operator-valued bi-free moment function in [Reference Charlesworth, Nelson and Skoufranis2], although the construction of the analytical bi-moment function is done using specific rules from the operator-valued bi-free moment function in a specific order, we desire more flexibility in the reductions that can be done and the order they can be done in. In particular, we desire to show that the analytical bi-moment function is an analytic extension of the operator-valued bi-free moment function.

The main ideas used to prove this are similar to those utilized in the proof of [Reference Charlesworth, Nelson and Skoufranis2, Theorem 5.1.4], and hence we shall be concerned with demonstrating that the inclusion of the $L_2(A,\tau )$ term and the slightly modified recursive definition are not issue and pose next to no changes. In particular, it may appear that $\Psi $ behaves differently than the operator-valued bi-free moment function as entries in $L_2(A, \tau )$ can also act as mixtures of left and right operators, which was not dealt with in [Reference Charlesworth, Nelson and Skoufranis2]. However, using the properties of $\widetilde {E}$ as developed in Proposition 3.8, one familiar with [Reference Charlesworth, Nelson and Skoufranis2] can easily see that the desired results will hold with simple adaptations. We note that similar adaptations were done in [Reference Gu and Skoufranis6] without issue.

When examining the proof of the following, Example 4.5 serves as a good example to keep in mind, just as Example 4.3 aided in comprehending why analytic extensions of bi-multiplicative functions work.

Theorem 4.7 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space, and let

$$\begin{align*}\Psi:\bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in \{\ell,r\}^n}\operatorname{\mathrm{BNC}}(\chi)\times A_{\chi(1)}\times\cdots\times A_{\chi(n-1)}\times L_2(A,\tau)\to L_2(B,\tau_B) \end{align*}$$

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

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

Proof of Theorem 4.7 property (ii)

To see (ii), fix $n\in \mathbb {N}$ , $\chi \in \{\ell ,r\}^n$ , $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ , $\xi \in L_2(A,\tau )$ , $b\in B$ , and $Z_k\in A_{\chi (k)}$ , and let p and q be as in the statement of (ii). In the case that $\chi (p) = \ell $ , note that

$$\begin{align*}\Psi_{1_\chi}(Z_1,\ldots,Z_{p-1},L_b Z_p, Z_{p+1}, \ldots,Z_{n-1},\xi) = \widetilde{E}(Z_1 \cdots Z_{p-1}L_bZ_p Z_{p+1} \cdots Z_{n-1} \xi). \end{align*}$$

If $q \neq \infty $ , then $Z_{q+1},\ldots , Z_{p-1} \in A_r$ and thus commute with $L_b$ . Hence,

$$ \begin{align*} \Psi_{1_\chi}(Z_1,\ldots,Z_{p-1},L_b Z_p, Z_{p+1}, \ldots,Z_{n-1},\xi) &= \widetilde{E}(Z_1 \cdots Z_{q-1}Z_qL_b Z_{q+1} \cdots Z_{n-1} \xi) \\ &= \Psi_{1_\chi}(Z_1,\ldots,Z_{q-1},Z_q L_b, Z_{q+1}, \ldots,Z_{n-1},\xi) \end{align*} $$

(and note $Z_qL_b \in A_\ell $ ).

If $q = \infty $ , then $Z_1, \ldots , Z_{p-1} \in A_r$ and thus commute with $L_b$ . Hence,

$$ \begin{align*} \Psi_{1_\chi}(Z_1,\ldots,Z_{p-1},L_b Z_p, Z_{p+1}, \ldots,Z_{n-1},\xi) &= \widetilde{E}(L_b Z_1 \cdots Z_{n-1} \xi) \\ &= b \widetilde{E}(Z_1 \cdots Z_{n-1} \xi) \\ &= b \Psi_{1_\chi}(Z_1,\ldots,Z_{n-1},\xi) \end{align*} $$

by Proposition 3.8. The case $\chi (p) = r$ is similar.

Proof of Theorem 4.7 property (iii)

To highlight how the proof works, we begin with the case that $\pi $ consists of exactly three blocks that are $\chi $ -intervals and n is contained in the middle block under the $\chi $ -ordering. Suppose that $\pi = \{V_1,V_2,V_3\}$ and thus there exists $i\in \{2,\ldots ,n-1\}$ and $j\in \{0,\ldots ,n-2\}$ such that

$$ \begin{align*} V_1 &= \{s_{\chi}(1),s_{\chi}(2),\ldots,s_{\chi}(i-1)\},\\ V_2 &= \{s_{\chi}(i),s_{\chi}(i+1),\ldots,s_{\chi}(i+j)\}, \text{ and } \\ V_3 &=\{s_{\chi}(i+j+1),s_{\chi}(i+j+2),\ldots,s_{\chi}(n)\}. \end{align*} $$

Thus, $n=s_{\chi }(k)$ for some $i\leq k\leq i+j$ . This implies that $\chi (p)=\ell $ for all $p\in V_1$ and $\chi (p)=r$ for all $p\in V_3$ . If $W=V_1\cup V_3$ , observe that the definition of the permutation $s_{\chi }$ yields that

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

Consider the case $q=s_{\chi }(i-1)$ so that $q \in V_1$ , and let $m= s_{\chi }(i+j+1)$ . Notice that if

$$\begin{align*}X = \Psi_{\pi|_{V_2}}({(Z_1,\ldots,Z_{n-1},\xi)|}_{V_2}), \end{align*}$$

then by Definition 4.2, we obtain that

$$\begin{align*}\Psi_\pi(Z_1,\ldots,Z_{n-1},\xi) = \Psi_{\pi|_W}({(Z_1,\ldots,Z_{q-1},Z_qX,\ldots,Z_{n-1},\xi)|}_W). \end{align*}$$

Thus, if

$$\begin{align*}Y = \Psi_{\pi|_{V_1}}({(Z_1,\ldots,Z_{q-1},Z_qX,\ldots,Z_{n-1},\xi)|}_{V_1}), \end{align*}$$

then again Definition 4.2 implies that

$$ \begin{align*} \Psi_\pi(Z_1,\ldots,Z_{n-1},\xi) &= \Psi_{\pi|_{V_3}}({(Z_1,\ldots,Z_{m-1},Z_mY,\ldots,Z_{n-1},\xi)|}_{V_3})\\ &= \widetilde{E}\left(Z_{s_{\chi}(n)} Z_{s_{\chi}(n-1)}\ldots Z_{s_{\chi}(i+j+1)} Y\right). \end{align*} $$

Therefore, since $Z_{s_{\chi }(n)} Z_{s_{\chi }(n-1)}\ldots Z_{s_{\chi }(i+j+1)} \in A_r$ , Proposition 3.8 implies that

$$\begin{align*}\Psi_\pi(Z_1,\ldots,Z_{n-1},\xi) = YE\left(Z_{s_{\chi}(n)} Z_{s_{\chi}(n-1)}\ldots Z_{s_{\chi}(i+j+1)}\right) = Y E_{\pi|_{V_3}}((Z_1,\ldots,Z_{n-1},\xi)|_{V_3}). \end{align*}$$

Since $Z_{s_{\chi }(1)}, Z_{s_{\chi }(2)},\ldots , Z_{s_{\chi }(i-1)} \in A_r$ , Proposition 3.8 implies that

$$ \begin{align*} Y = E(Z_{s_{\chi}(1)} Z_{s_{\chi}(2)}\ldots Z_{s_{\chi}(i-1)})X = E_{\pi|_{V_1}}((Z_1,\ldots,Z_{n-1},\xi)|_{V_1})X, \end{align*} $$

so the result follows. Note that the case $q=s_{\chi }(i+j+1)$ is handled similarly by interchanging the orders of the $\chi $ - intervals $V_1$ and $V_3$ . This argument can be extended via induction to any bi-non-crossing partition $\pi $ all of whose blocks are $\chi $ -interval.

By the same argument as [Reference Charlesworth, Nelson and Skoufranis2, Lemma 5.2.1], one need only consider the case in property (iii) that for each $\chi $ -interval, the $\chi $ -maximal and $\chi $ -minimal elements belong to the same block. When using the recursive procedure in Definition 4.2 to reduce $\Psi _\pi $ , one of the $\chi $ -intervals (which will either be entirely on the left or entirely on the right) will have the $L_2(A, \tau )$ term added to the last entry as above. This $L_2(A, \tau )$ entry can be pulled out on the appropriate side leaving only the bi-moment function expression for the $\chi $ -interval, which can be undone as usual. By repetition, eventually, all that remains is the expression for the $\chi $ -interval containing n as desired.

4.2 Analytical operator-valued bi-free cumulant function

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

Definition 4.3 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space, and denote by $\Psi $ the analytical bi-moment function. The analytical bi-cumulant function

$$\begin{align*}\widetilde{\kappa}: \bigcup_{n\in\mathbb{N}}\bigcup_{\chi\in \{\ell,r\}^n}\operatorname{\mathrm{BNC}}(\chi)\times A_{\chi(1)}\times\cdots\times A_{\chi(n-1)}\times L_2(A,\tau)\to L_2(B,\tau_B) \end{align*}$$

is defined by

$$\begin{align*}\widetilde{\kappa}_\pi(Z_1,\ldots,Z_{n-1},\xi) = \sum_{\substack{\sigma\in\operatorname{\mathrm{BNC}}(\chi)\\ \sigma\leq\pi\\}}\Psi_\sigma(Z_1,\ldots,Z_{n-1},\xi)\mu_{\operatorname{\mathrm{BNC}}}(\sigma,\pi), \end{align*}$$

for all $n\in \mathbb {N}$ , $\chi \in \{\ell ,r\}^n$ , $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ , $\xi \in L_2(A,\tau )$ , and $Z_k\in A_{\chi (k)}$ .

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

Theorem 4.10 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space. For all $n\in \mathbb {N}$ , $\chi \in \{\ell ,r\}^n$ , $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ , and $Z_k\in A_{\chi (k)}$ , we have that

$$\begin{align*}\widetilde{\kappa}_\pi(Z_1,\ldots,Z_{n-1}, Z_n + N_{\tau}) = \kappa^B_\pi(Z_1,\ldots,Z_{n-1},Z_n) + N_{\tau_B}. \end{align*}$$

Proof By Theorem 4.6, we know that

$$\begin{align*}\widetilde{\Psi}_\pi(Z_1,\ldots,Z_{n-1}, Z_n + N_{\tau}) = E_\pi(Z_1,\ldots,Z_{n-1},Z_n) + N_{\tau_B} \end{align*}$$

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

Of course, as [Reference Charlesworth, Nelson and Skoufranis2, Theorem 8.1.1] demonstrated that bi-freeness with amalgamation over B is equivalent to the mixed operator-valued bi-free cumulants vanishing, Theorem 4.10 immediately implies the following.

4.3 Vanishing analytical cumulants

However, something stronger than Corollary 4.11 holds. Indeed, note that the analytic operator-valued bi-free cumulant function has the added benefit that the last entry can be an element of $L_2(A, \tau )$ and thus the $L_2$ -image of a product of left and right operators. As such, it is possible to verify that additional analytic bi-cumulants vanish.

The desired result is analogous to the scalar-valued result demonstrated in [Reference Charlesworth and Skoufranis4, Proposition 2.30] and proved in a similar manner. Thus, we begin with a generalization of [Reference Charlesworth, Nelson and Skoufranis2, Theorem 9.1.5] where we can expand out a cumulant involving products of operators. In [Reference Charlesworth, Nelson and Skoufranis2, Theorem 9.1.5], only products of left and right operators were considered in the operator-valued setting, whereas [Reference Charlesworth and Skoufranis4, Lemma 2.29] expanded out scalar-valued cumulants involving a product of left and right operator in the last entry.

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

$$\begin{align*}k(0)=0<k(1)< \cdots<k(m)=n, \end{align*}$$

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

$$\begin{align*}\widehat{\chi}(q)=\chi(p_q). \end{align*}$$

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

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

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

of $\widehat {\pi }$ . It is not difficult to see that $\widehat {\pi } \in \operatorname {\mathrm {BNC}}(\widehat {\chi })$ as $\widehat {\chi }$ is constant on $\{k(p - 1) + 1, \ldots ,k(p)\}$ for all $p \in V \setminus \{m\}$ and although the block containing $\{k(n - 1) + 1, \ldots ,k(n)\}$ has both left and right entries, it occurs at the bottom of the bi-non-crossing diagram and thus poses no problem. Alternatively, this map can be viewed as an analog of the map on non-crossing partitions from [Reference Nica and Speicher9, Notation 11.9] after applying $s^{-1}_\chi $ .

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

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

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

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

Remark 4.12 Recall that since $\mu _{\operatorname {\mathrm {BNC}}}$ is the bi-non-crossing Möbius function, we have for each $\sigma ,\pi \in \operatorname {\mathrm {BNC}}(\chi )$ with $\sigma \leq \pi $ that

$$\begin{align*}\sum_{\substack{ \upsilon \in \operatorname{\mathrm{BNC}}(\chi) \\ \sigma \leq \upsilon \leq \pi }} \mu_{\operatorname{\mathrm{BNC}}}(\upsilon, \pi) = \left\{ \begin{array}{ll} 1, & \mbox{if } \sigma = \pi, \\ 0, & \mbox{otherwise. } \end{array} \right. \end{align*}$$

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

It is also easy to see that the partial Möbius inversion from [Reference Nica and Speicher9, Proposition 10.11] holds in the bi-free setting; that is, if $f, g : \operatorname {\mathrm {BNC}}(\chi ) \to \mathbb {C}$ are such that

$$\begin{align*}f(\pi) = \sum_{\substack{\sigma \in \operatorname{\mathrm{BNC}}(\chi) \\ \sigma \leq \pi}} g(\sigma) \end{align*}$$

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

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

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

Thus, by following the proofs of either [Reference Nica and Speicher9, Theorem 11.12], [Reference Charlesworth, Nelson and Skoufranis2, Theorem 9.1.5], or [Reference Charlesworth and Skoufranis4, Lemma 2.29], we arrive at the following.

Proposition 4.13 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space. Under the above notation, if $\pi \in \operatorname {\mathrm {BNC}}(\chi )$ and $Z_k\in A_{\widehat {\chi }(k)}$ , then

$$\begin{align*}&\widetilde{\kappa}_{\pi}(Z_1\cdots Z_{k(1)},\ldots, Z_{k(m-2)+1}\cdots Z_{k(m-1)}, Z_{k(m-1)+1}\cdots Z_{k(m)} + N_\tau)\\ &\quad = \sum_{\substack{\sigma\in \operatorname{\mathrm{BNC}}(\widehat{\chi})\\ \sigma \vee\widehat{0_{\chi}} = \widehat{\pi}}} \widetilde{\kappa}_\sigma(Z_1,\ldots,Z_{n-1},Z_n+ N_\tau). \end{align*}$$

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

$$\begin{align*}&\widetilde{\kappa}_{\chi}(Z_1\cdots Z_{k(1)},\ldots, Z_{k(m-2)+1}\cdots Z_{k(m-1)}, Z_{k(m-1)+1}\cdots Z_{k(m)} + N_\tau)\\&\quad = \sum_{\substack{\sigma\in \operatorname{\mathrm{BNC}}(\widehat{\chi})\\ \sigma\vee\widehat{0_{\chi}} = 1_{\widehat{\chi}}\\}} \widetilde{\kappa}_\sigma(Z_1,\ldots,Z_{n-1},Z_n+ N_\tau), \end{align*}$$

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

$$\begin{align*}&\Psi_{\upsilon}(Z_1\cdots Z_{k(1)},\ldots, Z_{k(m-2)+1}\cdots Z_{k(m-1)}, Z_{k(m-1)+1}\cdots Z_{k(m)} + N_\tau)\\&\quad = \Psi_{\widehat{\upsilon}} (Z_1,\ldots,Z_{n-1},Z_n+ N_\tau) \end{align*}$$

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

$$ \begin{align*} &\widetilde{\kappa}_{\pi}(Z_1\cdots Z_{k(1)},\ldots, Z_{k(m-2)+1}\cdots Z_{k(m-1)}, Z_{k(m-1)+1}\cdots Z_{k(m)} + N_\tau) \\ &= \sum_{\substack{\upsilon \in \operatorname{\mathrm{BNC}}(\chi) \\ \upsilon \leq \pi}} \Psi_\upsilon(Z_1\cdots Z_{k(1)},\ldots, Z_{k(m-2)+1}\cdots Z_{k(m-1)}, Z_{k(m-1)+1}\cdots Z_{k(m)} + N_\tau) \mu_{\operatorname{\mathrm{BNC}}}(\upsilon, \pi) \\ & = \sum_{\substack{\upsilon \in \operatorname{\mathrm{BNC}}(\chi)\\ \upsilon \leq \pi}} \Psi_{\widehat{\upsilon}} (Z_1,\ldots,Z_{n-1},Z_n+ N_\tau) \mu_{\operatorname{\mathrm{BNC}}}(\widehat{\upsilon}, \widehat{\pi}) \\ & = \sum_{\substack{\sigma \in \operatorname{\mathrm{BNC}}(\widehat{\chi}) \\ \widehat{0_\chi} \leq \sigma \leq \widehat{\pi}}} \Psi_{\sigma} (Z_1,\ldots,Z_{n-1},Z_n+ N_\tau) \mu_{\operatorname{\mathrm{BNC}}}(\sigma, \widehat{\pi}) \\ & = \sum_{\substack{\sigma \in \operatorname{\mathrm{BNC}}(\widehat{\chi})\\ \sigma \vee \widehat{0_\chi} = \widehat{\pi}}} \widetilde{\kappa}_\sigma(Z_1,\ldots,Z_{n-1},Z_n+ N_\tau), \end{align*} $$

where the last line follows from Remark 4.12.

Theorem 4.14 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space containing a family of pairs of B-algebras $\{(C_k,D_k)\}_{k\in K}$ that are bi-free with amalgamation over B with respect to E. For each $k \in K$ , let $L_2(A_k, \tau )$ be the closed subspace of $L_2(A, \tau )$ generated by

$$\begin{align*}\operatorname{\mathrm{alg}}(C_k, D_k) + N_\tau. \end{align*}$$

Then, for all $n\in \mathbb {N}$ , $\chi \in \{\ell ,r\}^n$ , nonconstant maps $\gamma :\{1,\ldots ,n\}\to K$ , $\xi \in L_2(A_{\gamma (n)}, \tau )$ , and $Z_1,\ldots ,Z_{n-1}\in A$ such that

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

it follows that

$$\begin{align*}\widetilde{\kappa}_\chi(Z_1,\ldots,Z_{n-1},\xi)=0. \end{align*}$$

Proof Fix an $m\in \mathbb {N}$ , $\chi \in \{\ell ,r\}^m$ , nonconstant map $\gamma :\{1,\ldots ,m\}\to K$ , and $Z_1,\ldots ,Z_{m-1}\in A$ . For any $n \geq m$ , if $Z_m, \ldots , Z_n \in C_{\gamma (m)} \cup D_{\gamma (m)}$ and $\widehat {\chi }$ is defined by

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

then by Proposition 4.13 and Theorem 4.6 imply that

$$ \begin{align*} \widetilde{\kappa}_{\chi}(Z_1,\ldots, Z_{m-1}, Z_{m}\cdots Z_{n} + N_\tau) & = \sum_{\substack{\sigma\in \operatorname{\mathrm{BNC}}(\widehat{\chi})\\ \sigma\vee\widehat{0_{\chi}} = 1_{\widehat{\chi}}\\}} \widetilde{\kappa}_\sigma(Z_1,\ldots,Z_{n-1},Z_n+ N_\tau)\\ & = \sum_{\substack{\sigma\in \operatorname{\mathrm{BNC}}(\widehat{\chi})\\ \sigma\vee\widehat{0_{\chi}} = 1_{\widehat{\chi}}\\}} \kappa^B_\sigma(Z_1,\ldots,Z_{n-1},Z_n) + N_{\tau_B}. \end{align*} $$

As the conditions $\sigma \in \operatorname {\mathrm {BNC}}(\widehat {\chi })$ and $\sigma \vee \widehat {0_{\chi }} = 1_{\widehat {\chi }}$ automatically imply that each block of $\sigma $ containing one of $\{1,2,\ldots , m-1\}$ must also contain an element of $\{m, \ldots , n\}$ , the bi-multiplicative properties of the operator-valued bi-free cumulant function imply that each cumulant $\kappa ^B_\sigma (Z_1,\ldots ,Z_{n-1},Z_n)$ appearing in the sum above can be reduced down to an expression involving a mixed $\kappa ^B$ term which must be 0 by [Reference Charlesworth, Nelson and Skoufranis2, Theorem 8.1.1]. Hence,

$$\begin{align*}\widetilde{\kappa}_{\chi}(Z_1,\ldots, Z_{m-1}, Z_{m}\cdots Z_{n} + N_\tau) = 0. \end{align*}$$

Since $\widetilde {E}$ is a continuous function and left multiplication of A on $L_2(A, \tau )$ yields bounded operators, due to the recursive nature of $\Psi $ , we see that $\Psi $ is continuous in the $L_2(A, \tau )$ entry. Therefore, by Möbius inversion, $\widetilde {\kappa }$ is continuous in the $L_2(A, \tau )$ entry. Hence, the result follows.

Corollary 4.15 Let $(A,E,\varepsilon ,\tau )$ be an analytical B-B-noncommutative probability space, and let $n\geq 2$ , $\chi \in \{\ell ,r\}^n$ , $\xi \in L_2(A,\tau )$ , $b\in B$ , and $Z_k\in A_{\chi (k)}$ . Suppose that either there exists $p\in \{1,\ldots ,n-1\}$ such that

$$\begin{align*}Z_p= \begin{cases} L_b, &\text{if } \chi (p)=\ell\\ R_b, &\text{if } \chi (p) = r \end{cases} \end{align*}$$

or that $\xi \in L_2(B, \tau _B)$ . Then,

$$\begin{align*}\widetilde{\kappa}_\chi(Z_1,\ldots,Z_{n-1},\xi)=0. \end{align*}$$

Proof If $Z_p = L_b$ or $Z_p = R_b$ for some p, then we may proceed as in the proof of Theorem 4.14 by assuming that $\xi $ is an element of A, expanding out the analytic operator-valued bi-free cumulant function with the aid of Proposition 4.13, and using the fact that non-singleton operator-valued bi-free cumulants involving $L_b$ or $R_b$ terms are zero by [Reference Charlesworth, Nelson and Skoufranis2, Proposition 6.4.1] and then taking a limit at the end.

In the case where $\xi \in L_2(B, \tau _B)$ , $\xi $ is a limit of terms of the form $L_b + N_{\tau }$ . As

$$\begin{align*}\widetilde{\kappa}_\chi(Z_1,\ldots,Z_{n-1}, L_b + N_{\tau}) = \kappa_\chi(Z_1,\ldots,Z_{n-1}, L_b ) + N_{\tau_B} = 0 + N_{\tau_B}, \end{align*}$$

by Theorem 4.10 and [Reference Charlesworth, Nelson and Skoufranis2, Proposition 6.4.1], the result follows by taking a limit.