Multilinear function series in conditionally free probability with amalgamation

TL;DR

This paper extends the concept of conditional freeness in non-commutative probability by developing positivity results, a central limit theorem, and an analogue of the R-transform using multilinear function series.

Contribution

It introduces a multilinear function series approach to conditionally free probability with amalgamation, providing new tools and results in the field.

Findings

Positivity results for conditional freeness

A version of the central limit theorem

An analogue of the conditionally free R-transform

Abstract

As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.

Equations

$$\Psi (a_1{a}_2\dots a_n)$$

$$\Phi (a_1{a}_2\dots a_n)$$

$$A_{i,j}=\{\begin{array}{ccc}{\Phi }_1(a_i^{\ast }{a}_j)& \text{if}& i,j\le n\\ {\Phi }_1(a_i^{\ast }){\Phi }_2(a_j)& \text{if}& i\le n,j>n\\ {\Phi }_2(a_i^{\ast }){\Phi }_1(a_j)& \text{if}& i>n,j\le n\\ {\Phi }_2(a_i^{\ast }{a}_j)& \text{if}& i,j>n\end{array}$$

$$⟨\cdot ,\cdot ⟩:\mathfrak{E}\times \mathfrak{E}⟶\mathfrak{B}$$

$$⟨b_1,b_2⟩$$

$$⟨u_j,v_j⟩$$

$$⟨u_1,u_2⟩$$

$$⟨b,u_j⟩$$

$$⟨a,a⟩$$

$$a$$

$$s_{(j,k)}$$

$$a=d+\sum _{k=1}^N{a}_{1,k}\dots a_{n(k),k}$$

$$\Phi (a^{\ast }a)$$

$$S=T^{\ast }T,\text{for some}T\in M_{N+1}(\mathfrak{B}).$$

$$\Phi (a^{\ast }a)$$

$${\text{∗}}_{j\in \mathfrak{J}}\left({\text{∗}}_{i\in {\mathcal{I}}_j}({\mathfrak{A}}_i,{\Phi }_i,{\Psi }_i)\right)={\text{∗}}_{i\in \mathcal{I}}({\mathfrak{A}}_i,{\Phi }_i,{\Psi }_i)$$

$$\left({\text{∗}}_{j\in {\mathfrak{J}}_l}{\Psi }_j\right)((a_{i_l-1}\cdots a_{i_l}))=0,$$

$$\Phi (a_1\cdots a_m)=\prod _{l=1}^k\left[({\text{∗}}_{({\Psi }_j),j\in {\mathfrak{J}}_l}{\Phi }_j)(a_{i_l-1}\cdots a_{i_l})\right].$$

$$\Phi (a_1\cdots a_m)={\Phi }_{\epsilon (1)}(a_1)\cdots {\Phi }_{\epsilon (m)}(a_m)$$

$$({\text{∗}}_{({\Psi }_j),j\in {\mathfrak{J}}_l}{\Phi }_j)(a_{i_l-1}\cdots a_{i_l})={\Phi }_{i_l-1}(a_{i_l-1})\cdots {\Phi }_{i_l}(a_{i_l})$$

$${\Phi }_X,{\Psi }_X:\mathfrak{B}⟨\xi ⟩⟶\mathfrak{B}$$

$${\Phi }_x{\ast }_{({\Psi }_X,{\Psi }_Y)}{\Phi }_Y:\mathfrak{B}⟨{\xi }_1⟩{\ast }_{\mathfrak{B}}\mathfrak{B}⟨{\xi }_2⟩=\mathfrak{B}⟨{\xi }_1,{\xi }_2⟩⟶\mathfrak{B}$$

$$i_Z:\mathfrak{B}⟨\xi ⟩\ni f(\xi )\mapsto f(X+Y)\in \mathfrak{B}⟨{\xi }_1⟩{\ast }_{\mathfrak{B}}\mathfrak{B}⟨{\xi }_2⟩$$

$${\Phi }_{X+Y}=({\Phi }_X{\ast }_{({\Psi }_X,{\Psi }_Y)}{\Phi }_Y)\circ i_{X+Y}.$$

$$m_X^{(n)}$$

$$m_X^{(n)}(b_1,\dots ,b_{n-1})$$

$$m_X(z)=\sum _{n=0}^{\infty }\Psi (X^n)z^n$$

$${\overline{m}}_X(z)=\sum _{n=0}^{\infty }\Psi (X^n)z^n$$

$${\omega }_n:\underset{n\text{times}}{\underbrace{\mathfrak{B}\times \cdots \times \mathfrak{B}}}⟶\mathfrak{B}$$

$$(\alpha +\beta {)}_n(b_1,\dots ,b_n)$$

Full text

1. Introduction

The paper addresses a topic related to conditionally free (or, shortly, using the term from [2], c-free) probability. This notion was developed in the ’90’s (see [1], [2]) as an extension of freeness within the framework of $\ast$-algebras endowed with not one, but two states. Namely, given a family of unital algebras $\{\mathfrak{A}\}_{i\in\mathcal{I}}$, each $\mathfrak{A}_{i}$ endowed with two expectations $\varphi_{i},\psi_{i}:\mathfrak{A}_{i}\longrightarrow\mathbb{C}$, their c-free product is the triple $(\mathfrak{A},\varphi,\psi)$, where:

1. (i)

\mathfrak{A}=\text{\huge{\ast}}_{i\in I}\mathfrak{A}_{i} is the free product of the algebras $\mathfrak{A}_{i}$. 2. (ii)

\psi=\text{\huge{\ast}}_{i\in\mathcal{I}}\psi_{i} and \varphi=\text{\huge{\ast}}_{(\psi_{i})_{i\in\mathcal{I}}}\varphi_{i} are expectations given by the relations

1. (a)

$\psi(a_{1}\cdots a_{n})=0$ 2. (b)

$\varphi(a_{1}\cdots a_{n})=\varphi_{\varepsilon(1)}(a_{1})\cdots\varphi_{\varepsilon(n)}(a_{n})$

for all $a_{j}\in\mathfrak{A}_{\varepsilon(j)},j=1,\dots,n$ such that $\psi_{\varepsilon(j)}(a_{j})=0$ and $\varepsilon(1)\neq\dots\neq\varepsilon(n)$.

A key result is that if the $\mathfrak{A}_{i}$ are $\ast$-algebras and $\varphi_{i},\psi_{i}$ are positive functionals, then $\varphi$ and $\psi$ are also positive.

In [6], the positivity of the free product maps $\varphi,\psi$ is proved for the case when $\varphi_{1},\varphi_{2}$ are positive conditional expectations in a common $C*$-subalgebra, but $\psi_{1},\psi_{2}$ remain positive $\mathbb{C}$-valued maps. A more general situation is indeed discussed (see Theorem 3, Section 6, from [6]), but the question if $\varphi,\psi$ re positive for $\varphi_{1,2},\psi_{1,2}$ arbitrary positive conditional expectations is left unanswered.

A first answer was given in [8], where we showed that for $\mathfrak{A}$ a $\ast$-algebra, the analogous construction with both $\varphi$ and $\psi$ valued in a $C^{*}$-subalgebra $\mathfrak{B}$ of $\mathfrak{A}$ still retains the positivity. The present paper further develops this result (see Theorem 2.3) and also demonstrates the use of multilinear function series in c-free setting.

In [2] is constructed a c-free version of Voiculescu’s $R$-transform, which we will call the ${}^{c}R$-transform, with the property that ${}^{c}R_{X+Y}={}^{c}R_{X}+{}^{c}R_{Y}$ if $X$ and $Y$ are c-free elements from the algebra $\mathfrak{A}$ relative to $\varphi$ and $\psi$ (i.e. the relations (a) and (b) from the definition of the c-free product hold true for the subalgebras generated by $X$ and $Y$.)

The apparatus of multilinear function series is used in recent work of K. Dykema ([3] and [4]) to construct suitable analogues for the $R$ and $S$-transforms in the framework of freeness with amalgamation. We will show that this construction is also appropriate for the ${}^{c}R$-transform mentioned above. The techniques used differ from the ones of [3], the Fock space type construction being substituted by combinatorial techniques similar to [2] and [7]. Particularly, Theorems 3.3 and 3.6 contain new (shorter) proves of the results 6.1–6.13 from [3].

The paper is structured in four sections. In Section 2 are stated the basic definitions and are proved the main positivity results. Section 3 describes the construction and the basic property of the multilinear function series ${}^{c}R$-transform and Section 4 treats the central limit theorem and a related positivity property.

2. Definitions and positivity results

Definition 2.1**.**

Let $\{\mathfrak{A}_{i}\}_{i\in{\mathcal{I}}}$ be a family of algebras, all containing the subalgebra $\mathfrak{B}$. Suppose $\mathfrak{D}$ is a subalgebra of $\mathfrak{B}$ and $\Psi_{i}:\mathfrak{A}_{i}\longrightarrow\mathfrak{D}$ and $\Phi_{i}:\mathfrak{A}_{i}\longrightarrow\mathfrak{B}$ are conditional expectations, $i\in\mathcal{I}$. We say that the triple (\mathfrak{A},\Phi,\Psi)=\text{\huge{\ast}}_{i\in{I}}(\mathfrak{A}_{i},\Phi_{i},\Psi_{i}) is the conditionally free product with amalgamation over ($\mathfrak{B}$,$\mathfrak{D}$), or shortly, the c-free product, of the triples $(\mathfrak{A}_{i},\Phi_{i},\Psi_{i})_{{i\in{I}}}$ if

1. (1)

$\mathfrak{A}$ is the free product with amalgamation over $\mathfrak{B}$ of the family $(\mathfrak{A}_{i})_{i\in\mathcal{I}}$ 2. (2)

\Psi=\text{\huge{\ast}}_{i\in\mathcal{I}}\Psi_{i} and \Phi=\text{\huge{\ast}}_{(\Psi_{i})_{i\in\mathcal{I}}}\Phi_{i} are determined by the relations

[TABLE]

for all $a_{i}\in\mathfrak{A}_{\varepsilon(i)},\varepsilon(i)\in\mathcal{I}$ such that $\varepsilon(1)\neq\varepsilon(2)\neq\dots\neq\varepsilon(n)$ and $\Psi_{\varepsilon(i)}(a_{i})=0$.

When $\mathfrak{D}=\mathbb{C}$, this definition reduces to the one given in [6]. When both $\mathfrak{B}$ and $\mathfrak{D}$ are equal to $\mathbb{C}$, this definition was given in [2].

When discussing positivity, we need a $\ast$-structure on our algebras. We will demand that $\mathfrak{B}$ and $\mathfrak{D}$ be C∗-algebras, while $\mathfrak{A}_{i}$ and $\mathfrak{A}$ are only required to be $\ast$-algebras.

The following results are slightly modified versions of Lemma 6.4 and Theorem 6.5 from [8].

Lemma 2.2**.**

Let $\mathfrak{B}$ be a $C^{*}$-algebra and $\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$ be two $\ast$-algebras containing $\mathfrak{B}$ as a $\ast$-subalgebra, endowed with positive conditional expectations $\Phi_{j}:\mathfrak{A}_{j}\longrightarrow\mathfrak{B}$, $j=1,2$. If $a_{1},\dots,a_{n}\in\mathfrak{A}_{1}$, $a_{n+1},\dots,a_{n+m}\in\mathfrak{A}_{2}$ and $A=(A_{i,j})_{i,j}\in M_{n+m}(\mathfrak{B})$ is the matrix with the entries

[TABLE]

then $A$ is positive.

Proof.

The vector space $\mathfrak{E}=\mathfrak{B}\oplus\ker(\Phi_{1})\oplus\ker(\Phi_{2})$ has a $\mathfrak{B}$-bimodule structure given by the algebraic operations on $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$. Consider the $\mathfrak{B}$-sesquilinear pairing

[TABLE]

determined by the relations:

[TABLE]

With this notation, we have that $A_{i,j}=\langle a_{i},a_{j}\rangle$, hence it suffices to show that $\langle a,a\rangle\geq 0$ for all $a\in\mathfrak{E}$.

Indeed, for an element $a=b+u_{1}+u_{2}$ with $b\in\mathfrak{B},u_{j}\in\ker(\Phi_{j}),j=1,2$, we have:

[TABLE]

∎

Theorem 2.3**.**

Let $\mathfrak{B}$ be a $C^{*}$-algebra and $\mathfrak{D}$ a $C^{*}$-subalgebra of $\mathfrak{B}$. Suppose that $\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$ are $\ast$-algebras containing $\mathfrak{B}$, each endowed with two positive conditional expectations $\Phi_{j}:\mathfrak{A}_{j}\longrightarrow\mathfrak{B}$, and $\Psi_{j}:\mathfrak{A}_{j}\longrightarrow\mathfrak{D}$, $j=1,2$ and consider the c-free product (\mathfrak{A},\Phi,\Psi)=\text{\huge{\ast}}_{i=1,2}(\mathfrak{A}_{i},\Phi_{i},\Psi_{i}).

Then the maps $\Phi$ and $\Psi$ are positive.

Proof.

The positivity of $\Psi$ is by now a classical result in the theory of free probability with amalgamation over a $C^{*}$-algebra (for example, see [9], Theorem 3.5.6). For the positivity of $\Phi$ we have to show that $\Phi(a^{*}a)\geq 0$ for any $a\in\mathfrak{A}$.

Any element of $\mathfrak{A}$ can be written as

[TABLE]

where $s_{j,k}\in\mathfrak{A}_{\varepsilon(j,k)}\ \varepsilon(1,k)\neq\varepsilon(2,k)\neq\dots\neq\varepsilon(n(k),k).$

Writing

[TABLE]

and expanding the product, we can consider $a$ of the form

[TABLE]

with $d\in\mathfrak{D}\subset\mathfrak{B}$ and $a_{j,k}\in\mathfrak{A}_{\varepsilon(j,k)}$ such that $\Psi_{\varepsilon(j,k)}(a_{j,k})=0$ and $\varepsilon(1,k)\neq\varepsilon(2,k)\neq\dots\neq\varepsilon(n(k),k)$.

Therefore

[TABLE]

Since $\Phi$ is a conditional expectation and $d\in\mathfrak{D}\subset\mathfrak{B}$, the above equality becomes

[TABLE]

Using the definition of the conditionally free product with amalgamation over $\mathfrak{B}$ and that $\Psi_{\varepsilon(j,k)}(a_{j,k})=0$ for all $j,k$, one further has

[TABLE]

that is

[TABLE]

From Lemma 2.2, the matrix $S=\left(\Phi(a_{1,i}^{*}a_{1,j})_{i,j=1}^{N+1}\right)$ is positive in $M_{N+1}(\mathfrak{B})$, therefore

[TABLE]

Denote now $a_{1,N+1}=d$ and $v_{k}=\Phi(a_{2,k})\dots\Phi(a_{n(k),k})$.The identity for $\Phi(a^{*}a)$ becomes:

[TABLE]

∎

Theorem 2.4**.**

Assume that $\mathcal{I}=\bigcup_{j\in\mathfrak{J}}\mathcal{I}_{j}$ is a partition of $\mathcal{I}$. Then:

[TABLE]

Proof.

The proof is identical to the proofs of similar results in [6] and [2]. Consider $a_{i}\in\mathfrak{A}_{\varepsilon(i)},1\leq i\leq m$ such that $\varepsilon(1)\neq\varepsilon(2\neq\dots\neq\varepsilon(m)$ and $\Psi_{\varepsilon(i)}(a_{i})=0$. Let $1=i_{0}<i_{1}<\dots<i_{k}=m$ and $\mathfrak{J}_{l}=\{\varepsilon(i),i_{l-1}\leq i<i_{l}\}$.

Since

[TABLE]

it suffices to show that

[TABLE]

But

[TABLE]

while, since $\Psi_{\varepsilon(i)}(a_{i})=0$,

[TABLE]

and the conclusion follows. ∎

Definition 2.5**.**

Let $\mathfrak{A}$ be an algebra (respectively a $\ast$-algebra), $\mathfrak{B}$ a subalgebra ($\ast$-subalgebra) of $\mathfrak{A}$ and $\mathfrak{D}$ a subalgebra ($\ast$-subalgebra) of $\mathfrak{B}$. Suppose $\mathfrak{A}$ is endowed with the conditional expectations $\Psi:\mathfrak{A}\longrightarrow\mathfrak{D}$ and $\Phi:\mathfrak{A}\longrightarrow\mathfrak{D}$.

1. (i)

The subalgebras ($\ast$-subalgebras) $(\mathfrak{A}_{i})_{i\in\mathcal{I}}$ of $\mathfrak{A}$ are said to be c-free with respect to $(\Phi,\Psi)$ if:

1. (a)

$(\mathfrak{A}_{i})_{i\in\mathcal{I}}$ are free with respect to $\Psi$. 2. (b)

if $a_{i}\in\mathfrak{A}_{\varepsilon(i)},1\leq i\leq m$, are such that $\varepsilon(1)\neq\dots\neq\varepsilon(m)$ and $\Psi(a_{i})=0$, then $\Phi(a_{1}\cdots a_{m})=\Phi(a_{1})\cdots\Phi(a_{m}).$ 2. (ii)

The elements $(X_{i})_{i\in\mathcal{I}}$ of $\mathfrak{A}$ are said to be c-free with respect to $(\Phi,\Psi)$ if the subalgebras ($\ast$-subalgebras) generated by $\mathfrak{B}$ and $X_{i}$ are c-free with respect to $(\Phi,\Psi)$.

We will denote by $\mathfrak{B}\langle\xi\rangle$ the non-commutative algebra of polynomials in the symbol $\xi$ and with coefficients from $\mathfrak{B}$ (the coefficients do not commute with the symbol $\xi$). If $\mathcal{I}$ is a family of indices, $\mathfrak{B}\langle\{\xi_{i}\}_{i\in\mathcal{I}}\rangle$ will denote the algebra of polynomials in the non-commuting variables $\{\xi\}_{i\in\mathcal{I}}$ and with coefficients from $\mathfrak{B}$. We will identify $\mathfrak{B}\langle\{\xi_{i}\}_{i\in\mathcal{I}}\rangle$ with the free product with amalgamation over $\mathfrak{B}$ of the family $\{\mathfrak{B}\langle\xi_{i}\rangle\}_{i\in\mathcal{I}}$.

If $\mathfrak{A}$ is a $\ast$-algebra and $\mathfrak{B}$ is with the $C^{*}$-algebra, $\mathfrak{B}\langle\xi\rangle$ will also be considered with a $\ast$-algebra structure, by taking $\xi^{*}=\xi$. If $X$ is a selfadjoint element from $\mathfrak{A}$, we define the conditional expectations

[TABLE]

given by $\Phi_{X}(f(\xi))=\Phi(f(X))$ and $\Psi_{X}(f(\xi))=\Psi(f(X))$ , for any $f(\xi)\in\mathfrak{B}\langle\xi\rangle$.

Corollary 2.6**.**

Suppose that $\mathfrak{A}$ is a $\ast$-algebra and $X$ and $Y$ are c-free selfadjoint elements of $\mathfrak{A}$ such that the maps $\Phi_{X},\Psi_{X}$ and $\Phi_{Y},\Psi_{Y}$ are positive. Then the maps $\Phi_{X+Y}$ and $\Psi_{X+Y}$ are also positive.

Proof.

The positivity of $\Psi_{X+Y}$ is an immediate consequence of the fact that $X$ and $Y$ are free with amalgamation over $\mathfrak{B}$ with respect to $\Psi$. It remains to prove the positivity of $\Phi_{X+Y}$.

Since the maps $\Phi_{X}:\mathfrak{B}\langle\xi_{1}\rangle\longrightarrow\mathfrak{B}$ and $\Phi_{Y}:\mathfrak{B}\langle\xi_{2}\rangle\longrightarrow\mathfrak{B}$ are positive, from Theorem 2.3 so is

[TABLE]

Remark also that

[TABLE]

is a positive $\mathfrak{B}$-functional.

The conclusion follows from the fact that the c-freeness of $X$ and $Y$ is equivalent to

[TABLE]

∎

3. Multilinear function series and the ${}^{c}R$-transform

Let $\mathfrak{A}$ be a $\ast$-algebra containing the $C^{*}$-algebra $\mathfrak{B}$, endowed with a conditional expectation $\Psi:\mathfrak{A}\longrightarrow\mathfrak{B}$. If $X$ is a selfadjoint element of $\mathfrak{A}$, then by the moment of order $n$ of $X$ we will understand the map

[TABLE]

If $\mathfrak{B}=\mathbb{C}$, then the moment-generating series of $X$

[TABLE]

encodes all the information about the moments of $X$. For $\mathfrak{B}\neq\mathbb{C}$, the straightforward generalization

[TABLE]

generally fails to keep track of all the possible moments of $X$. A solution to this inconvenience was proposed in [3], namely the moment-generating multilinear function series of $X$. Before defining this notion, we will briefly recall the construction and several results on multilinear function series.

Let $\mathfrak{B}$ be an algebra over a field $K$. We set $\widetilde{\mathfrak{B}}$ equal to $\mathfrak{B}$ if $\mathfrak{B}$ is unital and to the unitization of $\mathfrak{B}$ otherwise. For $n\geq 1$, we denote by $\mathcal{L}_{n}(\mathfrak{B})$ the set of all $K$-multilinear maps

[TABLE]

A formal multilinear function series over $\mathfrak{B}$ is a sequence $\omega=(\omega_{0},\omega_{1},\dots)$, where $\omega_{0}\in\widetilde{\mathfrak{B}}$ and $\omega_{n}\in\mathcal{L}_{n}(\mathfrak{B})$ for $n\geq 1$. According to [3], the set of all multilinear function series over $\mathfrak{B}$ will de denoted by $Mul[[\mathfrak{B}]]$.

For $\alpha,\beta\in Mul[[\mathfrak{B}]]$, the * formal sum* $\alpha+\beta$ and the formal product $\alpha\beta$ are the elements from $Mul[[\mathfrak{B}]]$ defined by:

[TABLE]

for any $b_{1},\dots,b_{n}\in\mathfrak{B}$.

If $\beta_{0}=0$, then the formal composition $\alpha\circ\beta\in Mul[[\mathfrak{B}]]$ is defined by

[TABLE]

and, for $n\geq 1$, by

[TABLE]

where the second summation is done over all $k$-tuples $p_{1},\dots,p_{k}\geq 1$ such that $p_{1}+\dots+p_{k}=n$ and $q_{j}=p_{1}+\dots+p_{j-1}$.

One can work with elements of $Mul[[\mathfrak{B}]]$ as if they were formal power series. The relevant properties are described in [3], Proposition 2.3 and Proposition 2.6. As in [3], we use $1$, respectively $I$, to denote the identity elements of $Mul[[\mathfrak{B}]]$ relative to multiplication, respectively composition. In other words, $1=(1,0,0,\dots)$ and $I=(0,id_{\mathfrak{B}},0,0,\dots)$. We will also use the fact that an element $\alpha\in Mul[[\mathfrak{B}]]$ has an inverse with respect to formal composition, denoted $\alpha^{{\langle-1\rangle}}$, if and only if $\alpha$ has the form $(0,\alpha_{1},\alpha_{2},\dots)$ with $\alpha_{1}$ an invertible element of $\mathcal{L}_{1}(\mathfrak{B})$.

Definition 3.1**.**

With the above notation, the moment-generating multilinear function series $\mathcal{M}_{X}$ of $X$ is the element of $Mul[[\mathfrak{B}]]$ such that:

[TABLE]

Given an element $\alpha\in Mul[[\mathfrak{B}]]$, the multilinear function series $R_{\alpha}$ is defined by the following equation (see [3], Def 6.1):

[TABLE]

A key property of $R$ is that for any $X,Y\in\mathfrak{A}$ free over $\mathfrak{B}$, we have

[TABLE]

These relations were proved earlier in the particular case $\mathfrak{B}=\mathbb{C}$. One can also describe $R_{\alpha}$ by combinatorial means, via the recurrence relation

[TABLE]

where the second summation is done over all $1=i(0)<i(1)<\dots<i(k)\leq n$ and $p(k)=i(k)-i(k-1)-2$.

Following an idea from [2], the above equation can be graphically illustrated by the picture:

[TABLE]

In the case of scalar c-free probability, an analogue of the Voiculescu’s $R$-transform is developed in [2]. In order to avoid confusions, we will denote it by ${}^{c}R$.

The ${}^{c}R$-transform has the property that it linearizes the c-free convolution of pairs of compactly supported measures. In particular, if $X$ and $Y$ are c-free elements from some algebra $\mathfrak{A}$, then

[TABLE]

If the $\ast$-algebra $\mathfrak{A}$ is endowed with the $\mathbb{C}$-valued states $\varphi,\psi$ and $X$ is a selfadjoint element of $\mathcal{A}$, then (see [2]), the coefficients $\{{}^{c}R_{m}\}_{m}\geq 0$ of ${}^{c}R_{X}$ are defined by the recurrence:

[TABLE]

equation that can be graphically illustrated by the picture, were the dark boxes stand for the application of $\varphi$ and the light ones for the application of $\psi$:

[TABLE]

The above considerations lead to the following definition:

Definition 3.2**.**

Let $\beta,\gamma\in Mul[[\mathfrak{B}]]$. The multilinear function series ${}^{c}R_{\beta,\gamma}$ is the element of $Mul[[\mathfrak{B}]]$ defined by the recurrence relation

[TABLE]

where the second summation is done over all $1=i(0)<i(1)<\dots<i(k)\leq n$ and $p(k)=i(k)-i(k-1)-2$.

The following analytical description of ${}^{c}R_{\beta,\gamma}$ also shows that it is unique and well-defined:

Theorem 3.3**.**

For any $\beta,\gamma\in Mul[[\mathfrak{B}]]$,

[TABLE]

Before proving the theorem, remark that the right-hand side of (3.3) is well-defined and unique, since $1+I\gamma$ is invertible with respect to the formal multiplication, $I+I\beta I$ is invertible with respect to formal composition and its inverse has 0 as first component (see [3]). We will need the following auxiliary result:

Lemma 3.4**.**

Let $\beta$ be an element of $Mul[[\mathfrak{B}]]$ and $I$ the identity element with respect to formal composition, $I=(0,id_{\mathfrak{B}},0,0\dots)$.

1. (i)

the multilinear function series $I\beta$ is given by:

[TABLE] 2. (ii)

the multilinear function series $I\beta I$ is given by

[TABLE]

Proof.

Since $I=(0,id_{\mathfrak{B}},0,\dots)$, one has:

[TABLE]

If $n\geq 1$,

[TABLE]

For $I\beta I$, the same computations give:

[TABLE]

If $n\geq 2$, one has:

[TABLE]

∎

Proof of the Theorem 3.3:.

Set $\sigma=I+I\beta I$. Then

[TABLE]

where $q_{i}=p_{1}+\dots+p_{i-1}$.

From Lemma (3.4)(ii), we have that

[TABLE]

therefore Definition 3.2 is equivalent to

[TABLE]

Considering now Lemma 3.4(i), the above relation becomes

[TABLE]

therefore

[TABLE]

which is equivalent to (3.3). ∎

Remark 3.5*.*

Up to a shift in the coefficients, equation (3.3) is similar to the result in the case $\mathfrak{B}=\mathbb{C}$ from [2], Theorem 5.1.

Let $X$ be a selfadjoint element of $\mathfrak{A}$. If $\mathfrak{A}$ is endowed with two $\mathfrak{B}$-valued conditional expectations $\Phi,\Psi$, the element $X$ will have two moment-generating multilinear function series, one with respect to $\Psi$, that we will denote by $\mathcal{M}_{X}$, and one with respect to $\Phi$, denoted $\mathfrak{M}_{X}$. For brevity, we will use the notation ${}^{c}R_{X}$ for the multilinear function series ${}^{c}R_{\mathcal{M}_{X},\mathfrak{M}_{X}}$.

Theorem 3.6**.**

Let $X$ and $Y$ be two elements of $\mathfrak{A}$ that are c-free with respect to the pair of conditional expectations $(\Phi,\Psi)$. Then

[TABLE]

Proof.

Let $\mathcal{A}$ be an algebra containing $\mathfrak{B}$ as a subalgebra and endowed with the conditional expectations $\Phi,\Psi:\mathcal{A}\longrightarrow\mathfrak{B}$. Consider the set $\mathcal{A}_{0}=\mathcal{A}\setminus\mathfrak{B}$ (set difference). For $n\geq 1$ define the maps

[TABLE]

given by the recurrence formula:

[TABLE]

Note that ${}^{c}r_{n}$ is well defined, and that, for any $b_{1},\dots,b_{n}\in\mathfrak{B}$,

[TABLE]

As in Section 2, consider $\mathfrak{B}\langle\xi_{i}\rangle$, the noncommutative algebras of polynomials in the symbols $\xi_{i},i=1,2$ and with coefficients from $\mathfrak{B}$ and the conditional expectations

[TABLE]

given by

[TABLE]

and their analogues $\Phi_{Y},\Psi_{Y}$ for $\mathfrak{B}\langle\xi_{2}\rangle$.

On $\mathfrak{B}\langle\xi_{1},\xi_{2}\rangle$, identified to $\mathfrak{B}\langle\xi_{1}\rangle\ast_{\mathfrak{B}}\mathfrak{B}\langle\xi_{2}\rangle$, consider the conditional expectations $\Psi_{0},\Phi_{0},\varphi$ given by:

[TABLE]

where $a_{1},\dots,a_{n}$ are elements of the set $\mathfrak{B}\langle\xi_{1},\xi_{2}\rangle_{0}=\mathfrak{B}\langle\xi_{1}\rangle\cup\mathfrak{B}\langle\xi_{2}\rangle\setminus\mathfrak{B},$ and the maps

[TABLE]

are given by:

[TABLE]

We will show that $\varphi=\Phi_{0}$, in particular $\varphi$ is also well-defined. Consider the element $a\in\mathfrak{B}\langle\xi_{1},\xi_{2}\rangle$ of the form $a=a_{1}\cdots a_{n}$ with $a_{j}\in\mathfrak{B}\langle\xi_{\varepsilon(j)}\rangle,$ such that $\varepsilon(1)\neq\varepsilon(2)\neq\dots\neq\varepsilon(n)$ and $\Psi_{0}(a_{j})=0$. The computation of $\varphi(a_{1}\cdots a_{n})$ is done via the recurrence relation above. Because of the definition of $\rho$ and the fact that $\Psi_{0}=\Psi_{X}\ast\Psi_{Y}$, only the term with $k=1$ contribute at the sum, i.e.

[TABLE]

and the identity between $\varphi$ and $\Phi_{0}$ follows by induction over $n$.

Since $\varphi=\Phi_{0}$, the maps $\rho_{n}$ and ${}^{c}r_{n}$ are satisfying the same recurrence relation, hence

[TABLE]

In particular

[TABLE]

∎

4. Central limit theorem

Consider the ordered set $\langle n\rangle=\{1,2,\dots,n\}$ and $\pi$ a partition of $\langle n\rangle$ with blocks $B_{1},\dots,B_{m}$:

[TABLE]

The blocks $B_{p}$ and $B_{q}$ of $\pi$ are said to be crossing if there exist $i<j<k<l$ in $\langle n\rangle$ such that $i,k\in B_{p}$ and $j,l\in B_{q}$.

The partition $\pi$ is said to be non-crossing if all pairs of distinct blocks of $\pi$ are not crossing. We will denote by $NC_{2}(n)$ the set of all non-crossing partitions of $\langle n\rangle$ whose blocks contain exactly 2 elements and by $NC_{\leq s}(n)$ the set of all non-crossing partitions of $\langle n\rangle$ whose blocks contain at most $s$ elements.

Let now $\gamma$ be a non-crossing partition of $\langle n\rangle$ and $B$ and $C$ be two blocks of $\pi$. We say that $B$ is interior to $C$ if there exist two indices $i<j$ in $\langle n\rangle$ such that $i,j\in C$ and $B\subset\{i+1,\dots,j-1\}$. The block $B$ is said to be outer if it is not interior to any other block of $\gamma$. In a non-crossing partition of $\langle n\rangle$, the block containing $1$ is always outer.

Consider now an element $X$ of $\mathfrak{A}$. Let $\pi$ be a partition from $NC_{2}(n+1)$ ($n$ = odd) and $B_{1}=(1,k)$ be the block of $\pi$ containing $1$. We define, by recurrence, the following expressions:

[TABLE]

Theorem 4.1**.**

(Central Limit Theorem) Let $(X_{n})_{n\geq 1}$ be a sequence of c-free elements of $\mathfrak{A}$ such that:

1. (1)

all $X_{n}$ have the same moment-generating multilinear function series, $\mathfrak{M}$ with respect to $\Phi$ and $M$ with respect to $\Psi$. 2. (2)

$\Psi(X_{n})=\Phi(X_{n})=0.$**

Set

[TABLE]

Then:

1. (i)

$\displaystyle\lim_{N\rightarrow\infty}{}^{c}R_{S_{N}}=(0,\mathfrak{M}_{1}(\cdot),0,\dots)$** 2. (ii)

$\displaystyle\lim_{N\rightarrow\infty}R_{S_{N}}=(0,M_{1}(\cdot),0,\dots)$** 3. (iii)

there exist two conditional expectations $\nu:\mathfrak{B}\langle\xi\rangle\longrightarrow\mathfrak{B}$, depending only on $M_{1}(\cdot)$, and $\mu:\mathfrak{B}\langle\xi\rangle\longrightarrow\mathfrak{B}$, depending only on $M_{1}(\cdot)$ and $\mathfrak{M}_{1}(\cdot)$, such that

[TABLE]

in the weak sense; in particular,

[TABLE]

Proof.

Let $X$ be an element of $\mathfrak{A}$ with the same moment generating series as $X_{j},$  $j\geq 1$. As shown in [3],

[TABLE]

Also, from Theorem 2.4 and Theorem 3.6, it follows that

[TABLE]

Since $R$ and ${}^{c}R$ are multilinear and $M_{0}=\mathfrak{M}_{0}=0$, we have that

[TABLE]

and the similar relations for $R_{S_{N},n}$, hence (i) and (ii) are proved.

For (iii) it suffices to check the relations for $\nu(\xi b_{1}\xi\dots b_{n}\xi)$ and $\mu(\xi b_{1}\xi\dots b_{n}\xi)$, which are a trivial corollary of (i), (ii), and the recurrence formulas that define $R$ and ${}^{c}R$. ∎

Remark 4.2*.*

For $\mathfrak{B}=\mathbb{C}$, the theorem is a weaker version of Theorem 4.3 from [2]. If $\Psi$ is $\mathbb{C}$-valued, then the result is similar to Corollary 5.1 from [6]. Also, under the assumptions that for some $a,b\in\mathfrak{B}$ we have that:

[TABLE]

the same techniques lead to a Poisson-type limit Theorem, similar to Corollary 2, Section 5 of [6].

In the following remaining pages we will describe the positivity of the limit functionals $\mu$ and $\nu$ in terms of $\Phi$ and $\Psi$. The central result is Corollary 4.4.

For simplicity, suppose that $\mathfrak{B}$ is a unital $\ast$-algebra (otherwise, we can replace $\mathfrak{B}$ by its unitisation). Consider the symbol $\xi$, the $\ast$-algebra $\mathfrak{B}\langle\xi\rangle$ of polynomials in $\xi$ with coefficients from $\mathfrak{B}$, as defined before, and consider also the linear space $\mathfrak{B}\xi\mathfrak{B}$ generated by the set $\{b_{1}\xi b_{2};\ b_{1},b_{2}\in\mathfrak{B}\}$ with the $\mathfrak{B}$-bimodule structure given by

[TABLE]

for all $a_{1},a_{2},b_{1},b_{2}\in\mathfrak{B}$.

Lemma 4.3**.**

For any positive $\mathfrak{B}$-sesquilinear pairing $\langle\cdot,\cdot\rangle$ on $\mathfrak{B}\xi\mathfrak{B}$ there exists a positive conditional expectation $\varphi:\mathfrak{B}\langle\xi\rangle\longrightarrow\mathfrak{B}$ such that for any $b_{1},b_{2}\in\mathfrak{B}$ one has that

[TABLE]

Proof.

Without loss of generality, we can suppose that $\mathfrak{B}$ is unital (otherwise we can replace $\mathfrak{B}$ by its unitization).

Consider the Full Fock bimodule over $\mathfrak{B}\xi\mathfrak{B}$

[TABLE]

with the pairing given by

[TABLE]

($a,a_{j},b,b_{j}\in\mathfrak{B},j=1,\dots,n$)

Note that the $\mathfrak{B}$-linear operators $A_{1},A_{2}:\mathcal{F}\langle\xi\rangle\longrightarrow\mathcal{F}\langle\xi\rangle$ described by the relations

[TABLE]

are self-adjoint to each other, in the sense that

[TABLE]

for any $\widetilde{\zeta_{1}},\widetilde{\zeta_{2}}\in\mathcal{F}\langle\xi\rangle$, therefore $S=A_{1}+A_{2}$ is selfadjoint.

Moreover, for any $a,b\in\mathfrak{B}$,

[TABLE]

and the conclusion follows by setting $\varphi(p(\xi))=\langle 1,p(S)1\rangle$ for all $p\in\mathfrak{B}\langle\xi\rangle$. ∎

Corollary 4.4**.**

The maps $\mu$ and $\nu$ from Theorem 4.1 are positive if and only if for any $b\in\mathfrak{B}$ one has that $\Phi(Xb^{*}bX)\geq 0$ and $\Psi(Xb^{*}bX)\geq 0$.

Proof.

One implication is trivial, since, if $\nu$ and $\mu$ are positive, then

[TABLE]

and

[TABLE]

Suppose now that $\Phi(Xb^{*}bX)\geq 0$ and $\Psi(Xb^{*}bX)\geq 0$ for all $b\in\mathfrak{B}$. We will use the same argument as in [9] and [8].

Consider the set of selfadjoint symbols $\{\xi_{i}\}_{i\geq 1}$. On each $\mathfrak{B}$-bimodule $\mathfrak{B}\xi_{i}\mathfrak{B}$ we have the positive $\mathfrak{B}$-sesquilinear pairings $\langle\cdot,\cdot\rangle_{\Phi}$ and $\langle\cdot,\cdot\rangle_{\Psi}$ determined by

[TABLE]

As shown in Lemma 4.3, the above $\mathfrak{B}$-sesquilinear pairings determine positive conditional expectations $\varphi_{1},\psi_{i}:\mathfrak{A}_{i}\longrightarrow\mathfrak{B}$, where $\mathfrak{A}_{i}=\mathfrak{B}\langle\xi_{i}\rangle$ be the $\ast$-algebras of polynomials in $\xi$ with coefficients from $\mathfrak{B}$, $i\geq 1$.

For $\tau:\mathfrak{B}\langle\xi\rangle\longrightarrow\mathfrak{B}$ a conditional expectation, and $\lambda\geq 0$, note with $D_{\lambda}\tau$ the dilation with $\lambda$ of $\tau$, i.e.

[TABLE]

Remark that if $\tau$ is positive, then $D_{\lambda}\tau$ is also positive.

With the notations above, consider, as in Definition 2.1, the conditionally free product (\mathfrak{A},\Phi,\Psi)=\text{\huge{\ast}}_{i\in\mathcal{I}}(\mathfrak{A}_{i},\Phi_{i},\Psi_{i}). The elements $\{\xi_{i}\}_{i\geq 1}$ are conditionally free in $\mathfrak{A}$, so Theorem 4.1 implies that:

[TABLE]

We have that \text{\huge{\ast}}_{i=1}^{N}\Psi_{\xi_{i}}\geq 0 since it is the free product of states (see, for example [9]), hence the positivity of $\nu$.

Also, Theorem 2.4 and Corollary 2.6 imply that $\Phi_{\xi_{1}+\dots+\xi_{N}}\geq 0$, therefore $\mu\geq 0$.

∎

Acknowledgment. This research was partially supported by the Grant 2-CEx06-11-34 of the Romanian Government. I am thankful to Marek Bożejko for presenting me the basics of c-freeness and bringing to my attention the references [2] and [6]. I thank also Hari Bercovici for his constant support and his many advices during the work on this paper.