# Conjugate variables approach to mixed $q$-Araki-Woods Algebras: Factoriality
and non-injectivity

Manish Kumar<EMAIL_ADDRESS>Institute of Mathematics of the Polish Academy
of Sciences, Warsaw, Poland

###### Abstract.

We establish factoriality and non-injectivity in full generality for the mixed
$q$-Araki-Woods von Neumann algebra associated to a separable real Hilbert
space $\mathsf{H}_{\mathbb{R}}$ with $\dim\mathsf{H}_{\mathbb{R}}\geq 2$, a
strongly continuous one parameter group of orthogonal transformations on
$\mathsf{H}_{\mathbb{R}}$, a direct sum decomposition
$\mathsf{H}_{\mathbb{R}}=\oplus_{i}\mathsf{H}_{\mathbb{R}}^{(i)}$, and a real
symmetric matrix $(q_{ij})$ with $\sup_{i,j}|q_{ij}|<1$. This is achieved by
first proving the existence of conjugate variables for a finite number of
generators of the algebras (when $\dim\mathsf{H}_{\mathbb{R}}<\infty$),
following the lines of Miyagawa-Speicher and Kumar-Skalski-Wasilewski. The
conjugate variables belong to the factors in question and satisfy certain
Lipschitz conditions.

###### Key words and phrases:

mixed $q$-Araki-Woods von Neumann algebra, factoriality, non-injectivity,
conjugate variables

###### 2020 Mathematics Subject Classification:

Primary: 46L36; Secondary 46L10, 46L53, 46L65

##  1\. Introduction

We study the factoriality and non-injectivity question of mixed $q$-Araki-
Woods von Neumann algebras. These are non-tracial counterpart of mixed
$q$-Gaussian algebras introduced by Speicher [22] and Bożejko-Speicher [3]
which followed their definition of (non-mixed) $q$-Gaussian algebras in [2].
Mixed $q$-Gaussian algebras are associated with two data
$(\mathsf{H}_{\mathbb{R}},(q_{ij}))$ where $\mathsf{H}_{\mathbb{R}}$ is a real
separable Hilbert space and $-1<q_{ij}=q_{ji}<1$ are real numbers for $1\leq
i,j\leq\dim\mathsf{H}_{\mathbb{R}}$ such that $\sup_{i,j}|q_{ij}|<1$. The von
Neumann algebra is then generated by a family of operators
$\\{l_{i}+l_{i}^{*}\\}_{1\leq i\leq\dim\mathsf{H}_{\mathbb{R}}}$ acting on
certain deformed Fock space where $l_{i}$’s obey the mixed $q_{ij}$
commutation relations:

$\displaystyle l_{i}l_{j}^{*}-q_{ij}l_{j}^{*}l_{i}=\delta_{ij}.$

The non-mixed scenario corresponds to the case where $q_{ij}=q$ for all $i,j$.
The structure of these algebras have attracted deep analysis over the decades;
in particular factoriality of (mixed) $q$-Gaussian algebras is work of many
hands. Ricard [20] established the factoriality of $q$-Gaussian algebras in
full generality (for $\dim\mathsf{H}_{\mathbb{R}}\geq 2$) while several
partial results were obtained earlier in [1, 12, 24]. Following the ideas of
[20], Skalski and Wang in [23] solved the factoriality question of mixed
$q$-Gaussian algebras in full generality ($\dim\mathsf{H}_{\mathbb{R}}\geq 2$)
while again some partial results preceded this in [13, 18]. The citations
listed here are by no means complete for the study of various other structural
properties of the Gaussian algebras.

Both $q$-Gaussian and mixed $q$-Gaussian von Neumann algebras are tracial and
are natural deformations of Voiculescu’s free Gaussian functor associated to
real Hilbert spaces. Another natural generalization of the free Gaussian
functor was introduced by Shlyakhtenko [21] in the non-tracial framework which
can also be regarded as analogues of injective factors coming from CAR
relations. The associated algebras depend on one parameter groups of
orthogonal transformations $U_{t}$ on real Hilbert spaces
$\mathsf{H}_{\mathbb{R}}$. They are non-injective factors of type
$\mathrm{III}$ as soon as $\dim\mathsf{H}_{\mathbb{R}}\geq 2$ and $U_{t}$ is
non-trivial, and they are called free Araki-Woods factors.

Hiai in [11] combined the functor of [2] and [21] to produce the so called
$q$-Araki-Woods von Neumann algebras associated to
$(\mathsf{H}_{\mathbb{R}},U_{t},q)$. These algebras have equally received a
lot of attention, particularly because they provide a concrete approach to the
study of certain type $\mathrm{III}$ factors. On one hand factoriality and
non-injectivity was determined completely for the free Araki-Woods algebras
(i.e. when $q=0$) in [21], whereas the same for $q$-deformation counterparts
for $q\neq 0$ appears to be more involved, with only partial results obtained
over the years in [11, 16, 7, 23, 8, 14]. Remark that the factoriality was
established by Bikram-Mukherjee-Ricard-Wang in [8] for
$\dim\mathsf{H}_{\mathbb{R}}\geq 3$, and for $\dim\mathsf{H}_{\mathbb{R}}\geq
2$ with $U_{t}$ ergodic and $q$ large. The complete answer to the factoriality
and non-injectivity question of such algebras have been settled recently by
Skalski, Wasilewski and the author in [14]. Unlike the free case scenario,
various structural properties of $q$-deformed algebras like fullness, absence
of Cartan subalgebras, solidity etc are still far from full understanding.

In [5], Bikram, Kumar R. and Mukherjee further extended Hiai’s construction to
now include the mixed $q$-Gaussian functor of [3] as well as the functor of
[21]. The concerned von Neumann algebras are called mixed $q$-Araki-Woods von
Neumann algebras. Now the algebras are associated with data consisting of
$(U_{t},\mathsf{H}_{\mathbb{R}}^{(i)},(q_{ij})_{i,j\in{N}},T)$ where ${N}$ is
a countable set, $\mathsf{H}_{\mathbb{R}}^{(i)},i\in{N}$ is a real separable
Hilbert space, parameters $q_{ij}\in(-1,1)$ satisfy $\sup_{i,j\in
N}|q_{ij}|<1$, $U_{t}$ is a strongly continuous one parameter group of
orthogonal operators on
$\mathsf{H}_{\mathbb{R}}:=\oplus_{i\in{N}}\mathsf{H}_{\mathbb{R}}^{(i)}$ such
that each $\mathsf{H}_{\mathbb{R}}^{(i)}$ is invariant under $U_{t}$, and
$T\in\mathbf{B}(\mathsf{H}_{\mathbb{R}}\otimes\mathsf{H}_{\mathbb{R}})$ is a
Yang-Baxter operator which interpolates
$\mathsf{H}_{\mathbb{R}}^{(i)}\otimes\mathsf{H}_{\mathbb{R}}^{(j)}$ and
$\mathsf{H}_{\mathbb{R}}^{(j)}\otimes\mathsf{H}_{\mathbb{R}}^{(i)}$ with the
twist $q_{ij}$. The mixed $q$-Araki-Woods algebra
$\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ is then generated by a set of
operators
$\\{a(\xi)+a^{*}(\xi);\xi\in\mathsf{H}_{\mathbb{R}}^{(i)},i\in{N}\\}$ acting
on a deformed Fock-space, satisfying the following commutation relations:

$a(\xi)a^{*}(\eta)-q_{ij}a^{*}(\eta)a(\xi)=\langle\xi,\eta\rangle_{U}$

for $\xi\in\mathsf{H}_{\mathbb{R}}^{(i)}$ and
$\eta\in\mathsf{H}_{\mathbb{R}}^{(j)}$ where $\langle\cdot,\cdot\rangle_{U}$
is a twisted scalar product on the complexification of
$\mathsf{H}_{\mathbb{R}}$ with the twist arising out of the representation
$U_{t}$ (see the formal construction in Section 2).

Previous investigations of mixed $q$-Araki-Woods algebras in [5] established
factoriality and the type classification of
$\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ under different conditions i.e.
whenever $U_{t}$ is weakly mixing or $U_{t}$ admits a fixed non-zero vector or
when $U_{t}$ is almost-periodic on an infinite dimensional space
$\mathsf{H}_{\mathbb{R}}$ with the set of eigenvalues of $A$ having a non-zero
limit point where $A$ is the analytic generator of $U_{t}$. They also prove
non-injectivity of the algebras in certain cases i.e. when the weakly mixing
part of $U_{t}$ is non-zero. Several other nice properties are illustrated in
[5] mostly generalizing those of their sister algebras of non-mixed situation.

However the case when the dimension of the underlying Hilbert space
$\mathsf{H}_{\mathbb{R}}$ is finite could not be fully understood. In this
paper, we finally settle the factoriality and non-injectivity question of
mixed $q$-Araki-Woods algebras in full generality. We summarize our main
results in the following theorem which is a combination of Theorem 4.2 and
Theorem 4.3:

######  Theorem 1.1.

Let
$(\mathsf{H}_{\mathbb{R}}=\oplus_{i\in{N}}\mathsf{H}_{\mathbb{R}}^{(i)},U_{t},(q_{ij})_{i,j\in
N},T)$ be given with $\dim\mathsf{H}_{\mathbb{R}}\geq 2$,
$q=\max_{i,j}|q_{ij}|<1$ and $N$ countable. Then the von Neumann algebra
$\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ is a non-injective factor of type

$\displaystyle\left\\{\begin{array}[]{ll}\mathrm{III}_{1}&\text{ if
}G=\mathbb{R}^{\times}_{\ast},\\\ \mathrm{III}_{\lambda}&\text{ if
}G=\lambda^{\mathbb{Z}},0<\lambda<1,\\\ \mathrm{II}_{1}&\text{ if
}G=\\{1\\},\end{array}\right.$

where $G<\mathbb{R}^{\times}_{\ast}$ is the closed subgroup generated by the
eigenvalues of the generator $A$ of $(U_{t})_{t\in\mathbb{R}}$.

Our approach towards the proof utilises the concepts of conjugate variables
following the lines of arguments by Miyagawa-Speicher [15], Kumar-Skalski-
Wasilewski [14] and Nelson [17]. The key idea of previous attempts towards
factoriality in [5] is the study of mixing properties of certain subalgebras
which allows one to control the relative commutant of the centralizer, whereas
our methods take completely different route.

We begin with the construction and basic properties of mixed $q$-Araki-Woods
von Neumann algebras in Section 2. We recollect the notions of dual/conjugate
variables in Section 3 and then exhibit in Theorem 3.4 the existence of a
system of conjugate variables for a set of generators for the von Neumann
algebra $\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ whenever
$\dim\mathsf{H}_{\mathbb{R}}<\infty$. The conjugate variables are shown to lie
in the ambient von Neumann algebras (Corollary 3.5).

Moreover we prove in Corollary 3.10 that the conjugate variables are Lipschitz
(see Section 3 for the definition of Lipschitz condition). Remark that the
Lipschitz condition was not addressed in [14] even for $q_{ij}=q$ case. In
this paper we do not precisely indicate an application of the Lipschitz
conjugate variables, nevertheless we believe that the Lipschitz condition may
provide pleasant applications similar to the tracial case and we are working
to achieve some in our upcoming work. To mention one striking application from
the tracial settings, in conjugation with abstract results of [10], the
Lipschitz condition allowed [15] to furnish a different proof for the absence
of Cartan subalgebras in $q$-Gaussian factors with finite symbols.

Finally in Section 4, the existence of conjugate variables allows us to invoke
the abstract results of Nelson [17] to prove factoriality and non-injectivity
of $\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ for $\mathsf{H}_{\mathbb{R}}$
finite dimensional. For a general (separable) Hilbert space
$\mathsf{H}_{\mathbb{R}}$ with $U_{t}$ being almost periodic (remaining cases
are dealt in [5]), an inductive limit argument is used to deduce factoriality
while some additional arguments are added to deduce non-injectivity.

After the completion of this article, the author learnt that parts of the
results about non-injectivity of $\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$
have been obtained independently by [6]. I thank Rahul Kumar for making me
aware of their results from [6] and for sending the draft which was not
available online. More precisely they prove non-injectivity of
$\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ under certain conditions on the
dimension of a fixed spectral projection of the analytic generator $A$ of
$U_{t}$ (see [6, Theorem 4.3, Corollary 4.4]). However their investigations
which still leave a large number of cases open are inspired from the ideas of
[19] proving the equivalent conditions of non-semi-discreetness, and they do
not take into account the point of view of conjugate and dual variables as we
have done.

##  2\. Construction of mixed $q$-Araki-Woods von Neumann algebras

We briefly describe the construction of mixed $q$-Araki-Woods von Neumann
algebras and refer the readers to [5] for a detailed treatment. As a
convention, all Hilbert spaces are separable and inner products are linear in
the second variable. For any real or complex Hilbert space $H$,
$\mathbf{B}(H)$ denotes the algebra of bounded operators on $H$.

Let $\mathsf{H}_{\mathbb{R}}$ be a real Hilbert space, and let $U_{t}$ be a
strongly continuous orthogonal representations of $\mathbb{R}$ on
$\mathsf{H}_{\mathbb{R}}$. Consider the complexification
$\mathsf{H}_{\mathbb{C}}=\mathsf{H}_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}$
of $\mathsf{H}_{\mathbb{R}}$, whose inner product is denoted by
$\langle\cdot,\cdot\rangle_{\mathsf{H}_{\mathbb{C}}}$. Identify
$\mathsf{H}_{\mathbb{R}}$ inside $\mathsf{H}_{\mathbb{C}}$ by
$\mathsf{H}_{\mathbb{R}}\otimes 1$, so that
$\mathsf{H}_{\mathbb{C}}=\mathsf{H}_{\mathbb{R}}+i\mathsf{H}_{\mathbb{R}}$.
Given $\xi=\xi_{1}+i\xi_{2}\in\mathsf{H}_{\mathbb{C}}$ for some
$\xi_{1},\xi_{2}\in\mathsf{H}_{\mathbb{R}}$, write the canonical involution
$\bar{\xi}$ as

$\bar{\xi}=\xi_{1}-i\xi_{2}.$

We extend $U_{t}$ to a strongly continuous one parameter group of unitaries on
the complexification $\mathsf{H}_{\mathbb{C}}$, which is still denoted by
$U_{t}$. Let $A$ denote the positive, self-adjoint, and non-singular analytic
(unbounded) operator on $\mathsf{H}_{\mathbb{C}}$ obtained via Stone’s
theorem, which satisfies $U_{t}=A^{it}$ for all $t\in\mathbb{R}$. Note that
$\overline{A\xi}=A^{-1}\bar{\xi}$ for any $\xi$ in the domain of $A$. In
particular, $\sigma(A)\cap(0,\infty)$ is symmetric around $1$ i.e. for
$\lambda\neq 0$, $\lambda\in\sigma(A)$ if and only if $1/\lambda\in\sigma(A)$,
where $\sigma(A)$ is the spectrum of $A$.

Following Shlyakhtenko [21], consider the deformed inner product
$\langle\cdot,\cdot\rangle_{U}$ on $\mathsf{H}_{\mathbb{C}}$ defined by

$\langle\xi,\zeta\rangle_{U}=\langle\xi,2(1+A^{-1})^{-1}\zeta\rangle\quad\forall\xi,\zeta\in\mathsf{H}_{\mathbb{C}}.$

We denote the completion of $\mathsf{H}_{\mathbb{C}}$ with respect to
$\langle\cdot,\cdot\rangle_{U}$ by $\mathsf{H}_{U}$. The space
$(\mathsf{H}_{\mathbb{R}},\|\cdot\|_{\mathsf{H}_{\mathbb{R}}})$ is naturally
identified as a subspace of $\mathsf{H}_{U}$ such that
$\mathsf{H}_{\mathbb{R}}+i\mathsf{H}_{\mathbb{R}}$ is dense in
$\mathsf{H}_{U}$ and $\mathsf{H}_{\mathbb{R}}\cap
i\mathsf{H}_{\mathbb{R}}=\\{0\\}$. Let ${N}$ denote a countable set
$\\{1,\ldots,r\\}$ for $r\in\mathbb{N}$ or $r=\infty$. Fix a decomposition of
$\mathsf{H}_{\mathbb{R}}$

$\mathsf{H}_{\mathbb{R}}=\bigoplus_{i\in{N}}\mathsf{H}_{\mathbb{R}}^{(i)}$

into a direct sum of invariant subspaces of $U_{t}.$ Fix $-1<q_{ij}=q_{ji}<1$
for $i,j\in{N}$ such that $\sup_{i,j\in{N}}|q_{ij}|<1.$ We will often write
$q(i,j)$ instead of $q_{ij}$. For each $i,j\in N$ define the operator
$T_{ij}:\mathsf{H}_{\mathbb{R}}^{(i)}\otimes\mathsf{H}_{\mathbb{R}}^{(j)}\to\mathsf{H}_{\mathbb{R}}^{(j)}\otimes\mathsf{H}_{\mathbb{R}}^{(i)}$
to be the bounded linear extension of the assignment

$\xi\otimes\eta\mapsto q_{ij}\eta\otimes\xi$ (2.1)

for
$\xi\in\mathsf{H}_{\mathbb{R}}^{(i)},\eta\in\mathsf{H}_{\mathbb{R}}^{(j)}$.
Set
$T=\oplus_{i,j\in{N}}T_{ij}\in\mathbf{B}(\mathsf{H}_{\mathbb{R}}\otimes\mathsf{H}_{\mathbb{R}})$.
Naturally $T$ extends linearly to
$\mathsf{H}_{\mathbb{C}}\otimes\mathsf{H}_{\mathbb{C}}$ and then to a bounded
operator on $\mathsf{H}_{U}\otimes\mathsf{H}_{U}$ in such a way that
$T(\mathsf{H}_{U}^{(i)}\otimes\mathsf{H}_{U}^{(j)})\subseteq\mathsf{H}_{U}^{(j)}\otimes\mathsf{H}_{U}^{(i)}$.
Moreover $T$ satisfies the following properties:

$\displaystyle T=T^{*},$ $\displaystyle\|T\|<1,\mbox{ and }$
$\displaystyle(I\otimes T)(T\otimes I)(I\otimes T)=(T\otimes I)(I\otimes
T)(T\otimes I)$

where $I$ denotes the identity on $\mathsf{H}_{U}$. The last of the above
conditions is called Yang-Baxter equation. For each $k\in\mathbb{N}$, define
$T_{k}\in\mathbf{B}(\mathsf{H}_{U}^{\otimes{k+1}})$ by

$T_{k}={\underbrace{I\otimes\cdots\otimes I}_{k-1}}\otimes T$ (2.2)

and by amplification also on $\mathsf{H}_{U}^{\otimes n}$ for all $n\geq k+1$
(which again is denoted by $T_{k}$ by abuse of notation). For
$n\in\mathbb{N}$, let $S_{n}$ denote the symmetric group of $n$ symbols and
let $\tau_{k}$ be the transposition between $k$ and $k+1$ for $1\leq k\leq
n-1.$ Let $\Phi:S_{n}\to\mathbf{B}(\mathsf{H}_{U}^{\otimes n})$ be the unique
(quasi-multiplicative) map satisfying $\Phi(1)=I$ and $\Phi(\tau_{k})=T_{k}$,
and define $P^{(n)}\in\mathbf{B}(\mathsf{H}_{U}^{\otimes n})$ by

$P^{(n)}=\sum_{\sigma\in S_{n}}\Phi(\sigma).$

Then $P^{(n)}$ is a strictly positive operator on
$\mathsf{H}_{U}^{\otimes^{n}}$ ([3]); Consider the inner product on
$\mathsf{H}_{U}^{\otimes n}$ defined by

$\langle\xi,\eta\rangle_{T}=\langle\xi,P^{(n)}\eta\rangle_{\mathsf{H}_{U}^{\otimes
n}}$

and denote by $\mathsf{H}_{U}^{\otimes_{T}^{n}}$ the completion of
$\mathsf{H}_{U}^{\otimes n}$ with respect to $\langle\cdot,\cdot\rangle_{T}$.
Define the $T$-deformed Fock space $\mathcal{F}_{T}(\mathsf{H}_{U})$ to be the
direct sum $\oplus_{n\geq 0}\mathsf{H}_{U}^{\otimes_{T}^{n}}$, where
$\mathsf{H}_{U}^{0}$ is understood to be the one-dimensional space
$\mathbb{C}\Omega$ for a distinguished unit vector $\Omega$. We now consider
the free left annihilation operator $l(\xi)$ for any $\xi\in\mathsf{H}_{U}$
given on a dense subspace of $\mathcal{F}_{T}(\mathsf{H}_{U})$ by the
assignment

$\displaystyle
l(\xi)\Omega=0,\;\;l(\xi)\xi_{1}\otimes\cdots\otimes\xi_{n}=\langle\xi,\xi_{1}\rangle\xi_{2}\otimes\cdots\otimes\xi_{n}$
(2.3)

for $\xi_{i}\in\mathsf{H}_{U},1\leq i\leq n,n\geq 1.$ The following lemma is
crucial for our purposes (see [15, Lemma 2.2] and [4, Theorem 6] for proof):

######  Lemma 2.1.

The free annihilation operator $l(\xi)$ extends to a bounded linear operator
on $\mathcal{F}_{T}(\mathsf{H}_{U})$ with

$\displaystyle\|l(\xi)\|\leq\frac{\|\xi\|_{U}}{\sqrt{w(q)}},\;\;\;w(q)=(1-q^{2})^{-1}\prod_{k=1}^{\infty}\frac{(1-q^{k})}{(1+q^{k})}$

where $q=\max_{i,j\in{N}}|q_{ij}|<1$.

We next consider the $T$-deformed creation and annihilation operators on
$\mathcal{F}_{T}(\mathsf{H}_{U})$, denoted $a^{*}(\xi)$ and $a(\xi)$
respectively for $\xi\in\mathsf{H}$, given by the assignments

$\displaystyle\begin{split}&a^{*}(\xi)\Omega=\xi,\;\;a^{*}(\xi)(\xi_{1}\otimes\ldots\otimes\xi_{n})=\xi\otimes\xi_{1}\otimes\cdots\xi_{n},\;\;\;\text{and}\\\
&a(\xi)\Omega=0,\;\;a(\xi)(\xi_{1}\otimes\cdots\otimes\xi_{n})=l(\xi)(1+T_{1}+T_{1}T_{2}+\ldots+T_{1}\ldots
T_{n-1})\xi_{1}\otimes\ldots\otimes\xi_{n}\end{split}$ (2.4)

for $\xi_{1}\otimes\cdots\otimes\xi_{n}\in\mathsf{H}_{U}^{\otimes n},n\geq 1$,
where $l(\xi)$ is the free annihilation operator as in (2.3) and each
$T_{k}\in\mathbf{B}(\mathsf{H}_{U}^{\otimes n})$ is the operator as in (2.2)
for $1\leq k\leq n$. The operators $a^{*}(\xi)$ and $a(\xi)$ extend to bounded
linear maps on $\mathcal{F}_{T}(\mathsf{H}_{U})$ and they are adjoint to each
other satisfying $\|a(\xi)\|=\|a^{*}(\xi)\|\leq\|\xi\|_{U}(1-q)^{-1/2}$ (see
[3]). One easily verifies that if $\xi_{k}\in\mathsf{H}_{U}^{(i_{k})}$ for
$i_{k}\in{N}$ and $1\leq k\leq n$, then

$\displaystyle
a(\xi)(\xi_{n}\otimes\cdots\otimes\xi_{1})=\sum_{k=1}^{n}\langle\xi,\xi_{k}\rangle_{U}\prod_{t=1}^{n-k}q({i_{k},i_{t+k}})\xi_{n}\otimes\cdots\otimes\widehat{\xi}_{k}\otimes\cdots\otimes\xi_{1}$
(2.5)

where $\hat{\xi}_{k}$ means the absence of the vector $\xi_{k}$. We adopt the
following notations in order to minimize the length of calculations:

  * •

$\xi_{1}\cdots\xi_{n}=\xi_{1}\otimes\cdots\otimes\xi_{n}$ for any
$\xi_{i}\in\mathsf{H}_{U},1\leq i\leq n$.

  * •

$C_{q}=\prod_{n=1}^{\infty}\frac{1}{1-q^{n}}$ for $0<q<1$.

  * •

$[n]_{q}=1+q+\cdots+q^{n-1}$ for $n\geq 1$ and $[0]_{q}=1$.

  * •

$[n]_{q}!=\prod_{k=1}^{n}[k]_{q}$ for $n\geq 1$ and $[0]_{q}!=1$.

We now mention an explicit description for the inner product of two simple
tensors. We could not trace anywhere a formula written in the following form,
so we include the proof for completeness (also see [5, Lemma 2.1]).

######  Lemma 2.2.

For $\xi_{k}\in\mathsf{H}_{U}^{(i_{k})}$ and
$\eta_{k}\in\mathsf{H}_{U}^{(j_{k})}$, $i_{k},j_{k}\in{N},1\leq k\leq n$, we
have

$\displaystyle\begin{split}\langle\xi_{n}\cdots\xi_{1},\eta_{n}\cdots\eta_{1}\rangle_{T}&=\sum_{\sigma\in
S_{n}}\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n}}q(i_{u},i_{v})\prod_{t=1}^{n}\langle\xi_{t},\eta_{\sigma(t)}\rangle_{U}\\\
&=\sum_{\sigma\in S_{n}}\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n}}q(j_{\sigma(u)},j_{\sigma(v)})\prod_{t=1}^{n}\langle\xi_{t},\eta_{\sigma(t)}\rangle_{U}\\\
&=\sum_{\sigma\in S_{n}}\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n}}q(j_{u},j_{v})\prod_{t=1}^{n}\langle\xi_{\sigma(t)},\eta_{t}\rangle_{U}\\\
&=\sum_{\sigma\in S_{n}}\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n}}q(i_{\sigma(v)},i_{\sigma(u)})\prod_{t=1}^{n}\langle\xi_{\sigma{(t)}},\eta_{t}\rangle_{U}.\end{split}$
(2.6)

###### Proof.

Note first that the last four expressions are indeed the same; for example to
check that the first two expressions on the right side are same, observe that
each term in the first sum of these is non-zero only if
$\langle\xi_{t},\eta_{\sigma(t)}\rangle_{U}\neq 0$ for all $1\leq t\leq n$
which is possible only if $i_{t}=j_{\sigma(t)}$, in which case we have
$q(i_{u},i_{v})=q(j_{\sigma(u)},j_{\sigma(v)})$ for $1\leq v,u\leq n$.
Similarly, the first sum and the last sum on the right are equal by noting
that

$\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n}}q(i_{u},i_{v})=\prod_{\underset{\sigma^{-1}(r)>\sigma^{-1}(s)}{1\leq
r<s\leq n}}q(i_{\sigma^{-1}(r)},i_{\sigma^{-1}(s)})$ (2.7)

for any $\sigma\in S_{n}$. We shall prove equality of the left hand side with
the first on the right. The proof is by induction on $n$. For $n=1$ the
equality is obvious. So assume them to be true for some $n\geq 1$; then
calculate

$\displaystyle\langle$
$\displaystyle\xi_{n+1}\cdots\xi_{1},\eta_{n+1}\cdots\eta_{1}\rangle_{T}=\langle\xi_{n+1}\cdots\xi_{1},a^{*}(\eta_{n+1})\eta_{n}\cdots\eta_{1}\rangle_{T}$
$\displaystyle=\langle
a({\eta_{n+1}})(\xi_{n+1}\cdots\xi_{1}),\eta_{n}\cdots\eta_{1}\rangle_{T}$
$\displaystyle=\left\langle\sum_{k=1}^{n+1}\langle\eta_{n+1},\xi_{k}\rangle_{U}\prod_{t=1}^{n+1-k}q(i_{k},i_{k+t})\xi_{n+1}\cdots\widehat{\xi}_{k}\cdots\xi_{1},\eta_{n}\cdots\eta_{1}\right\rangle_{T}\quad\quad\quad(\text{by
eq. }\eqref{eq:annihilation operator acting on simple tensors})$
$\displaystyle=\sum_{k=1}^{n+1}\langle\xi_{k},\eta_{n+1}\rangle_{U}\prod_{t=1}^{n+1-k}q(i_{k},i_{k+t})\left\langle\xi_{n+1}\cdots\widehat{\xi}_{k}\cdots\xi_{1},\eta_{n}\cdots\eta_{1}\right\rangle_{T}$
$\displaystyle=\sum_{k=1}^{n+1}\prod_{t=1}^{n+1-k}q(i_{k},i_{k+t})\sum_{\sigma\in
S_{n}}\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n}}q(i_{u},i_{v})\;\langle\xi_{k},\eta_{n+1}\rangle_{U}\left\langle\xi_{n+1}\cdots\widehat{\xi}_{k}\cdots\xi_{1},\eta_{\sigma(n)}\cdots\eta_{\sigma(1)}\right\rangle_{\mathsf{H}_{U}^{\otimes
n}}$
$\displaystyle=\sum_{k=1}^{n+1}\prod_{t=1}^{n+1-k}q(i_{k},i_{k+t})\sum_{\underset{\sigma(k)=n+1}{\sigma\in
S_{n+1}}}\prod_{\underset{\sigma^{-1}(u)>\sigma^{-1}(v)}{1\leq u<v\leq
n}}q(i_{\sigma^{-1}(u)},i_{\sigma^{-1}(v)})\left\langle\xi_{n+1}\cdots\xi_{1},\eta_{\sigma(n+1)}\cdots\eta_{\sigma(1)}\right\rangle_{\mathsf{H}_{U}^{\otimes{n+1}}}$

where in the last equality we have used (2.7), and for any fixed
$k\in\\{1,\ldots,n\\}$ we have identified any permutation $\sigma\in S_{n}$
with a permutation, say $\sigma^{(k)}$, in $S_{n+1}$ by declaring
$\sigma^{(k)}(k)=n+1$ and keeping the order preserved, that is,

$\displaystyle\sigma^{(k)}(t)=\left\\{\begin{array}[]{ll}\sigma(t)&\mbox{ if
}1\leq t<k,\\\ n+1&\mbox{ if }t=k,\\\ \sigma(t-1)+1&\mbox{ if }k<t\leq
n+1.\end{array}\right.$

Note that the extra crossings that occur in the permutation $\sigma^{(k)}$ as
compared to $\sigma$ are exactly those between the pairs $(k,n+1)$ and
$(t,\sigma^{(k)}(t))$ for $t>k$; in such case we get from (2.7) (and by
setting $\tau=\sigma^{(k)}$) that

$\displaystyle\prod_{\underset{\tau(u)>\tau(v)}{1\leq u<v\leq
n+1}}q(i_{u},i_{v})$
$\displaystyle=\prod_{\underset{\tau^{-1}(u)>\tau^{-1}(v)}{1\leq u<v\leq
n+1}}q(i_{\tau^{-1}(v)},i_{\tau^{-1}(u)})$
$\displaystyle=\prod_{\underset{\tau^{-1}(u)>\tau^{-1}(v)}{1\leq u<v\leq
n}}q(i_{\tau^{-1}(v)},i_{\tau^{-1}(u)})\cdot\prod_{t=k+1}^{n+1}q(i_{k},i_{t}).$

Finally we obtain

$\displaystyle\langle\xi_{n+1}\cdots\xi_{1},\eta_{n+1}\cdots\eta_{1}\rangle_{T}$
$\displaystyle=\sum_{k=1}^{n+1}\sum_{\underset{\sigma(k)=n+1}{\sigma\in
S_{n+1}}}\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n+1}}q(i_{u},i_{v})\langle\xi_{n+1}\cdots\xi_{1},\eta_{\sigma(n+1)}\cdots\eta_{\sigma(1)}\rangle_{\mathsf{H}_{U}^{\otimes{n+1}}}$
$\displaystyle=\sum_{{\sigma\in
S_{n+1}}}\prod_{\underset{\sigma(u)>\sigma(v)}{1\leq u<v\leq
n+1}}q(i_{u},i_{v})\langle\xi_{n+1}\cdots\xi_{1},\eta_{\sigma(n+1)}\cdots\eta_{\sigma(1)}\rangle_{\mathsf{H}_{U}^{\otimes{n+1}}}$

which is exactly what was to be shown. ∎

For each $\xi\in\mathsf{H}_{\mathbb{R}}$, consider the operator
$s(\xi)=a(\xi)+a^{*}(\xi)\in\mathbf{B}(\mathcal{F}_{T}(\mathsf{H}))$ and
denote by $\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ the von Neumann algebra
generated by $\\{s(\xi);\xi\in\mathsf{H}_{\mathbb{R}}\\}$, which we call as
mixed $q$-Araki-Woods von Neumann algebra. We will often denote this von
Neumann algebra by $\mathsf{M}_{T}$ (or simply by $\mathsf{M}$ if $T$ is clear
from the context). The vacuum vector $\Omega$ is a cyclic and separating
vector for $\mathsf{M}_{T}$; so $\mathsf{M}_{T}$ is in standard form as an
algebra acting on $\mathcal{F}_{T}(\mathsf{H}_{U})$. Let $\varphi$ denote the
canonical normal faithful state $\langle\Omega,\cdot\Omega\rangle$ on
$\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$. Denote by $L^{2}(\mathsf{M}_{T})$
(or by $L^{2}(\mathsf{M}_{T},\varphi)$) the GNS Hilbert space of
$\mathsf{M}_{T}$ with respect to the state $\varphi$. As mentioned above
$L^{2}(\mathsf{M}_{T})$ is naturally identified with
$\mathcal{F}_{T}(\mathsf{H}_{U})$. Denote by
$\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$ the subspace spanned by
finite simple tensors. Then $\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$
is contained in $\mathsf{M}_{T}\Omega$ (see [5]); hence for all
$\xi\in\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$, there is a unique
operator $W(\xi)\in\mathsf{M}_{T}$ such that $W(\xi)\Omega=\xi$. We refer the
readers to [5] for modular theory and other basic facts about the algebras
$\mathsf{M}_{T}$.

##  3\. Dual and conjugate variables for mixed $q$-Araki-Woods algebras

In this section we will consider only the case of finite-dimensional
$\mathsf{H}_{\mathbb{R}}$ and an orthogonal representation $U_{t}$ acting on
$\mathsf{H}_{\mathbb{R}}$. Assume then that ${N}$ is a finite set and
$\mathsf{H}_{\mathbb{R}}=\oplus_{i\in{N}}\mathsf{H}_{\mathbb{R}}^{(i)}$ where
each $\mathsf{H}_{\mathbb{R}}^{(i)}$ is a $U_{t}$-invariant finite dimensional
real Hilbert space. Fix $-1<q(i,j)=q(j,i)<1$ for $i,j\in{N}$, let
$q=\max_{i,j\in{N}}|q(i,j)|<1$ and consider the operator $T$ on
$\mathbf{B}(\mathsf{H}_{U}\otimes\mathsf{H}_{U})$ as defined in (2.1). Assume
that $\dim{\mathsf{H}_{\mathbb{R}}}=d$. For this whole Section, we fix an
orthonormal basis $\\{e_{1},\ldots,e_{d}\\}$ of $\mathsf{H}_{U}$ constructed
in such a way that each $e_{k}\in\mathsf{H}_{\mathbb{C}}^{(j_{k})}$ for some
$j_{k}\in{N}$ (the basis can be chosen so as to contain eigenvectors of the
analytic generator $A$ of $U_{t}$; see [21, 11]). For each $1\leq k\leq d$,
set $A_{k}=W(e_{k})$. Note that the set $\\{A_{k}\\}_{1\leq k\leq d}$ is
algebraically free such that the unital $*$-subalgebra
$\mathbb{C}[A_{1},\ldots,A_{d}]$ generated by the set is strongly dense in
$\mathsf{M}_{T}=\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$.

We say that a tuple $(D_{1},\ldots,D_{d})$ of unbounded operators on
$\mathcal{F}_{T}(\mathsf{H}_{U})$ with
$\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$ contained in their domains
and $\Omega$ contained in the domain of their adjoints is a (normalized) dual
system for $(A_{1},\ldots,A_{d})$ if for all $k,\ell\in\\{1,\ldots,d\\}$

${[D_{k},A_{\ell}]}=\langle\bar{e}_{\ell},e_{k}\rangle_{U}P_{\mathbb{C}\Omega}=\varphi(A_{\ell}A_{k})P_{\mathbb{C}\Omega}\;\;\mbox{
and }D_{k}\Omega=0,$

where $P_{\mathbb{C}\Omega}$ is the projection onto the one-dimensional
subspace $\mathbb{C}\Omega$, and as before $\bar{e}_{\ell}$ is the complex
conjugate of $e_{\ell}$. The existence of dual system ensures that each
$D_{k}$ is a closable operator on $\mathcal{F}_{T}(\mathsf{H}_{U})$.

Next recall that if $\mathsf{B}\subseteq\mathsf{M}_{T}$ is a
$\ast$-subalgebra, then a _derivation_ is a $\mathbb{C}$-linear map
$\delta:\mathsf{B}\to\mathsf{M}_{T}\otimes\mathsf{M}_{T}^{\operatorname{op}}$
which satisfies the Leibniz rule: $\delta(ab)=a\delta(b)+b\delta(a)$ for all
$a,b\in\mathsf{B}$. Here $\mathsf{M}_{T}^{\operatorname{op}}$ denotes the
opposite algebra equipped with the natural state
$\varphi^{\operatorname{op}}$. Then the quasi-free difference quotients
$\partial_{k}$ are defined as unique derivations from
$\mathbb{C}[A_{1},\dots,A_{d}]$ into
$\mathsf{M}_{T}\overline{\otimes}\mathsf{M}_{T}^{\operatorname{op}}$ such that

$\partial_{k}(A_{\ell}):=\varphi(A_{\ell}A_{k})\mathds{1}\otimes\mathds{1}$

for all $k,\ell\in\\{1,\ldots,d\\}$, where $\mathds{1}$ is the identity of
$\mathsf{M}_{T}$. The _conjugate variable_ will be a vector $\xi_{k}\in
L^{2}(\mathsf{M}_{T})$ such that

$\langle\xi_{k},x\mathds{1}\rangle_{L^{2}(\mathsf{M}_{T})}=\langle\mathds{1}\otimes\mathds{1},\partial_{k}(x)\mathds{1}\otimes\mathds{1}\rangle_{L^{2}(M_{T}\otimes\mathsf{M}_{T}^{\operatorname{op}})}$

for all $x\in\mathbb{C}[A_{1},\ldots,A_{d}]$. Remark that the existence of the
conjugate variable is equivalent to
$\mathds{1}\otimes\mathds{1}\in\operatorname{Dom}(\partial_{k}^{*})$ when
$\partial_{k}$ is considered an (unbounded) operator from
$L^{2}(\mathsf{M}_{T})$ to
$L^{2}(\mathsf{M}_{T}\overline{\otimes}\mathsf{M}_{T}^{\operatorname{op}})$;
in this case it is given precisely by
$\xi_{k}=\partial_{k}^{*}(\mathds{1}\otimes\mathds{1})$. We also recall that
the existence of dual variables implies existence of conjugate variables:

######  Proposition 3.1 (See [15, Theorem 2.5], [14, Proposition 3.1]).

Suppose that $(D_{1},\dots,D_{d})$ is a normalized dual system for
$(A_{1},\dots,A_{d})$. Then $(D_{1}^{\ast}\Omega,\dots,D_{d}^{\ast}\Omega)$
are conjugate variables for $(A_{1},\dots,A_{d})$.

### Existence of Dual variables

We now show the existence of a system of dual variables for the tuple
$(A_{1},\ldots,A_{d})$ of operators corresponding to a fixed orthonormal basis
$\\{e_{1},\ldots,e_{d}\\}$ of $\mathsf{H}_{U}$. We follow the lines of
investigation as in [15, 14]. In order to keep this article self-contained, we
repeat here some of the notions considered in [15]. Firstly we recall below
the rules of drawing partitions of the set $\\{0,1,\ldots,n\\}$ consisting of
singletons and pairs:

  1. (1)

Consider $n+1$ vertices $n>n-1>\cdots>1>0$.

  2. (2)

$0$ must be connected to some $k\in\\{1,\ldots,n\\}$ with height $1$.

  3. (3)

$\ell\in\\{1,\ldots,k-1\\}$ must be coupled with one of $\\{k+1,\ldots,n\\}$
with height $\ell+1$.

  4. (4)

Vertices which are not coupled with $\\{1,\ldots,k-1\\}$ should be singletons
and are drawn with straight lines to the top.

Define $B(n+1)$ as the set of partitions that satisfy the above rules. For
$\pi\in B(n+1)$, denote by $p(\pi)$ the set of pairings in $\pi$, and by
$s(\pi)$ the set of singletons in $\pi$. Also for any $k\in\\{0,\ldots,n\\}$,
write $\pi(k)=\ell$ if $(k,\ell)\in p(\pi)$ and $\pi(k)=k$ if $k\in s(\pi)$.

We set some notations. Write $[d]=\\{1,\ldots,d\\}$ and let $[d]^{*}$ denote
the set of words in $[d]$. Write

$e_{w}=e_{\alpha_{1}}\cdots e_{\alpha_{n}}=e_{\alpha_{1}}\otimes\ldots\otimes
e_{\alpha_{n}}\;\;\mbox{ and }\;\;e_{0}=e_{()}=\Omega,$

for any word $w=\alpha_{1}\cdots\alpha_{n}\in[d]^{*}$ of length $n\geq 1$ and
empty word $()$. The set $\\{e_{w};w\in[d]^{*}\\}$ is linearly independent and
spans $\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$. Observe that for any
$\alpha\in[d]$ and for any word $\alpha_{n}\cdots\alpha_{1}\in[d]^{*}$ of
length $n\geq 1$, the expression
$A_{\alpha}=W(e_{\alpha})=a^{*}(e_{\alpha})+a(\bar{e}_{\alpha})$ yields

$\displaystyle\begin{split}A_{\alpha}(e_{\alpha_{n}\ldots\alpha_{1}})&=e_{\alpha}e_{\alpha_{n}\cdots\alpha_{1}}+\sum_{k=1}^{n}\langle\bar{e}_{\alpha},e_{\alpha_{k}}\rangle_{U}\prod_{k<t\leq
n}q(i_{k},i_{t})e_{\alpha_{n}\cdots\widehat{\alpha}_{k}\ldots\alpha_{1}}\\\
&=e_{\alpha}e_{\alpha_{n}\cdots\alpha_{1}}+\sum_{k=1}^{n}\langle\bar{e}_{\alpha},e_{\alpha_{k}}\rangle_{U}\prod_{t=1}^{n-k}q(i_{k},i_{n-t+1})e_{\alpha_{n}\cdots\widehat{\alpha}_{k}\ldots\alpha_{1}}\end{split}$
(3.1)

where $i_{k}\in{N}$ is such that
$e_{\alpha_{k}}\in\mathsf{H}_{\mathbb{C}}^{(i_{k})}$ for $1\leq k\leq n$.

Taking cue from the approach of [15] and [14], we now establish the existence
of dual/conjugate variables in the non-tracial context of mixed $q$-Araki-
Woods von Neumann algebras.

######  Proposition 3.2.

The algebraic formula for the tuple $(D_{1},\ldots,D_{d})$ of the dual
variables of $(A_{1},\ldots,A_{d})$ is given as follows; for fixed
$\alpha\in[d]$ and $w=\alpha_{n}\cdots\alpha_{1}\in[d]^{*}$ such that
$e_{\alpha}\in\mathsf{H}_{\mathbb{C}}^{(i_{0})}$ and
$e_{\alpha_{k}}\in\mathsf{H}_{\mathbb{C}}^{(i_{k})}$, for some
$i_{0},i_{k}\in{N}$, $1\leq k\leq n$:

$D_{\alpha}\Omega=0,\;\;\;\;\;D_{\alpha}(e_{\alpha_{n}\cdots\alpha_{1}})=\sum_{\pi\in
B(n+1)}(-1)^{{\pi(0)-1}}Q(\pi)E(\pi)e_{s(\pi)},$ (3.2)

where $s(\pi)$ is the set of singletons of $\pi$, and

$\displaystyle Q(\pi):=\prod_{0\leq
v<u<\pi(0)}q(i_{v},i_{u})\cdot\prod_{\underset{\pi(v)>\pi(u)}{0<v<u<\pi(0)}}q(i_{\pi(v)},i_{\pi(u)})\cdot\prod_{\underset{v\in
s(\pi),u\notin s(\pi)}{\pi(0)<v<u\leq n}}q(i_{v},i_{u})$ $\displaystyle
E(\pi):=\prod_{\underset{{k>\ell}}{(k,\ell)}\in
p(\pi)}\langle\bar{e}_{\alpha_{k}},e_{\alpha_{\ell}}\rangle_{U}$
$\displaystyle
e_{s(\pi)}:=e_{\alpha_{\ell_{s}}\cdots\alpha_{\ell_{1}}}=e_{\alpha_{\ell_{s}}}\otimes\cdots\otimes
e_{\alpha_{\ell_{1}}}\;\mbox{ for }s(\pi)=\\{\ell_{s}>\ldots>\ell_{1}\\}$

with the prescription that $\alpha_{0}=\alpha$.

######  Remark 3.3.

We have divided the terms of $Q(\pi)$ into three products. The first product
(resp. the second product) corresponds to the crossings in the partition
occurring on the line of the vertex $`u$’ (resp. $`\pi(u)^{\prime}$) for
$0<u<\pi(0)$, with the lines of another pair. While the third product
corresponds to the crossings on the lines of the singletons. Note that
$E(\pi)$ is non-zero only if $i_{k}=i_{\pi(k)}$ for any pair $(k,\pi(k))\in
p(\pi)$ (because $\xi\in\mathsf{H}_{\mathbb{C}}^{(i)}$ if and only if
$\bar{\xi}\in\mathsf{H}_{\mathbb{C}}^{(i)}$ and
$\mathsf{H}_{\mathbb{C}}^{(i)}\perp\mathsf{H}_{\mathbb{C}}^{(j)}$ if $i\neq
j$). Thus in such cases, in the expression for $Q(\pi)$, the term
$q(i_{v},i_{u})$ can be replaced by $q(i_{\pi(v)},i_{\pi(u)})$ for any indices
$u,v$ which are part of the pairings of $\pi$ and then we use symmetry of the
matrix $(q(i,j))$ to get the following:

$\displaystyle\begin{split}E(\pi)Q(\pi)&=E(\pi)\prod_{{0\leq
v<u<\pi(0)}}q(i_{\pi(v)},i_{\pi(u)})\cdot\prod_{\underset{\pi(v)>\pi(u)}{0<v<u<\pi(0)}}q(i_{\pi(v)},i_{\pi(u)})\cdot\prod_{\underset{v\in
s(\pi),u\notin s(\pi)}{\pi(0)<v<u\leq n}}q(i_{v},i_{u})\\\
&=E(\pi)\prod_{{0\leq
v<u<\pi(0)}}q(i_{v},i_{u})\cdot\prod_{\underset{\pi(v)>\pi(u)}{0<v<u<\pi(0)}}q(i_{v},i_{u})\cdot\prod_{\underset{v\in
s(\pi),u\notin s(\pi)}{\pi(0)<v<u\leq n}}q(i_{v},i_{u}).\end{split}$ (3.3)

###### Proof.

Consider the (unbounded) operator $D_{\alpha}$ on
$\mathcal{F}_{T}(\mathsf{H}_{U})$ defined as in (3.2) on vectors which span
the subspace $\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$. We will show
that
$[D_{\alpha},A_{\beta}]=\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}P_{\mathbb{C}\Omega}$
on $\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$ for all $\beta\in[d]$. It
is straightforward to check that

$[D_{\alpha},A_{\beta}]\Omega=D_{\alpha}A_{\beta}\Omega=D_{\alpha}e_{\beta}=\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}\Omega.$

Then for $\alpha_{n+1}\in[d]$, use (3.1) to compute

$\displaystyle A_{\alpha_{n+1}}$ $\displaystyle
D_{\alpha}(e_{\alpha_{n}\ldots\alpha_{1}})=\sum_{\sigma\in
B(n+1)}(-1)^{\sigma(0)-1}Q(\sigma)E(\sigma)A_{\alpha_{n+1}}e_{s(\sigma)}$
$\displaystyle=\sum_{\sigma\in
B(n+1)}(-1)^{\sigma(0)-1}Q(\sigma)E(\sigma)\;{e}_{\alpha_{n+1}}e_{s(\sigma)}$
$\displaystyle\quad+\sum_{\sigma\in
B(n+1)}\sum_{v=1}^{|s(\sigma)|}(-1)^{\sigma(0)-1}Q(\sigma)E(\sigma)\langle\bar{e}_{\alpha_{n+1}},{e}_{\alpha_{s(\sigma)_{v}}}\rangle_{U}\prod_{\underset{{t\in
s(\sigma)}}{s(\sigma)_{v}<t\leq
n}}q(i_{s(\sigma)_{v}},i_{t})\;e_{s(\sigma)\setminus s(\sigma)_{v}}$

where $e_{s(\sigma)\setminus s(\sigma)_{v}}$ means that we omit the term
$e_{\alpha_{s(\sigma)_{v}}}$ from $e_{s(\sigma)}$. Next we compute

$\displaystyle D_{\alpha}$ $\displaystyle
A_{\alpha_{n+1}}(e_{\alpha_{n}\ldots\alpha_{1}})$
$\displaystyle=D_{\alpha}\left(e_{\alpha_{n+1}\cdots\alpha_{1}}+\sum_{u=1}^{n}\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}\prod_{u<t\leq
n}q(i_{u},i_{t})e_{\alpha_{n}\cdots\widehat{{\alpha}}_{u}\cdots\alpha_{1}}\right)$
$\displaystyle=D_{\alpha}e_{\alpha_{n+1}\cdots\alpha_{1}}$
$\displaystyle\quad\quad+\sum_{u=1}^{n}\sum_{\pi\in
B(n)}(-1)^{\pi(0)-1}Q(\pi,u)E(\pi,u)\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}\prod_{u<t\leq
n}q(i_{u},i_{t})\;e_{s(\pi,u)}$

where $e_{s(\pi,u)}$, $Q(\pi,u)$ and $E(\pi,u)$ respectively denote
$e_{s(\pi)}$, $Q(\pi)$ and $E(\pi)$ only with the emphasis that $\pi\in B(n)$
acts on the $n-1$ letters
$\alpha_{n},\ldots,\widehat{\alpha}_{u},\ldots,\alpha_{1}$. As in the proof of
[15, Proposition 4.5] we identify all terms in the last sum of
$A_{\alpha_{n+1}}D_{\alpha}(e_{\alpha_{n}\ldots\alpha_{1}})$ with some terms
in the last sum of
$D_{\alpha}A_{\alpha_{n+1}}(e_{\alpha_{n}\cdots\alpha_{1}})$ as follows: take
a pair $(\sigma,v)$ with $\sigma\in B(n+1)$ and
$v\in\\{1,\ldots,|s(\sigma)|\\}$, then get a pair $(\pi,u)$ where
$u=s(\sigma)_{v}$ and $\pi\in B(n)$ is a partition of the set
$\\{0,\ldots,n\\}\setminus\\{u\\}$ obtained from $\sigma$ in an order
preserving fashion by removing the singleton $s(\sigma)_{v}$. Then observe
that $\sigma(0)=\pi(0)$ and the pairings in $p(\sigma)$ and $p(\pi)$ remain
the same while the singleton sets satisfy
$s(\sigma)=s(\pi)\cup\\{s(\sigma)_{v}\\}$. This means that
$e_{s(\pi,u)}=e_{s(\sigma)\setminus s(\sigma)_{v}}$ and $E(\pi,u)=E(\sigma)$
so that

$E(\pi,u)\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}e_{s(\pi,u)}=E(\sigma)\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{s(\sigma)_{v}}}\rangle_{U}e_{s(\sigma)\setminus
s(\sigma)_{v}}.$

Moreover the extra terms which appear in the product of $Q(\sigma)$ as
compared to that of $Q(\pi,u)$ are the ones due to the crossing points on the
line of the singleton $s(\sigma)_{v}$. Such crossings occur with those
pairings in $\sigma$ which have one of the vertices on the left of
$u=s(\sigma)_{v}$. Therefore we get

${Q(\sigma)}={Q(\pi,u)}\cdot\prod_{\underset{t\notin
s(\sigma)}{s(\sigma)_{v}<t\leq
n}}q(i_{s(\sigma)_{v}},i_{t})=Q(\pi,u)\cdot\frac{\prod_{{s(\sigma)_{v}<t\leq
n}}q(i_{s(\sigma)_{v}},i_{t})}{\prod_{\underset{t\in
s(\sigma)}{s(\sigma)_{v}<t\leq n}}q(i_{s(\sigma)_{v}},i_{t})}$

where $\frac{0}{0}$ is understood to be $0$; this implies by using
$u=s(\sigma)_{v}$ that

$\displaystyle Q(\sigma)\cdot\prod_{\underset{t\in
s(\sigma)}{s(\sigma)_{v}<t\leq
n}}q(i_{s(\sigma)_{v}},i_{t})=Q(\pi,u)\cdot\prod_{{u<t\leq n}}q(i_{u},i_{t}).$

This shows that the term corresponding to $\sigma$ and $v$ in the second sum
of $A_{\alpha_{n+1}}D_{\alpha}(e_{\alpha_{n}\ldots\alpha_{1}})$ appears as the
term corresponding to $\pi$ and $u$ in the second sum of
$D_{\alpha}A_{\alpha_{n+1}}(e_{\alpha_{n}\ldots\alpha_{1}})$, and at the same
time note that this correspondence exhausts all the pairs $(\pi,u)$ with
$\pi\in B(n)$ and $u>\pi(0)$. Thus after the subtracting we get the following:

$\displaystyle\begin{split}-[D_{\alpha},A_{\alpha_{n+1}}]&e_{\alpha_{n}\ldots\alpha_{1}}+D_{\alpha}e_{\alpha_{n+1}\cdots\alpha_{1}}\\\
&=\sum_{\sigma\in
B(n+1)}(-1)^{\sigma(0)-1}Q(\sigma)E(\sigma)\;e_{\alpha_{n+1}}e_{s(\sigma)}\\\
&\quad-\sum_{u=1}^{n}\sum_{\underset{\pi(0)\geq u}{\pi\in
B(n)}}(-1)^{\pi(0)-1}Q(\pi,u)E(\pi,u)\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}\prod_{u<t\leq
n}q(i_{u},i_{t})e_{s(\pi,u)}.\end{split}$ (3.4)

Now we try to understand the terms in the last sum above. Recall that each
pair $(\pi,u)$ with $\pi\in B(n)$ and $u\in\\{1,\ldots,n\\}$ acts on the $n-1$
letters $\alpha_{n},\ldots,\widehat{\alpha}_{u},\ldots,\alpha_{1}$. As in the
second step of the proof of [15, Proposition 4.5], to each such pair $(\pi,u)$
with $u\leq\pi(0)$, we associate a new partition $\pi^{\prime}\in B(n+2)$ as
follows: construct the partition $\pi^{\prime}\in B(n+2)$ of
$\\{1,\ldots,n+1\\}$ out of $(\pi,u)$ by inserting a new point at $u$ and
pairing it with $n+1$ with the requirement that
$\pi^{\prime}\setminus(n+1,u)=\pi$. We then have

$\displaystyle\pi^{\prime}(0)=\pi(0)+1.$

Observe that the set of singletons remains the same in both $\pi$ and
$\pi^{\prime}$ while the set of pairings satisfies
$p(\pi^{\prime})=p(\pi)\cup\\{(n+1,u)\\}$; hence we get

$e_{s(\pi^{\prime})}=e_{s(\pi,u)}\;\;\mbox{ and
}\;\;E(\pi^{\prime})=E(\pi,u)\;\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}.$

Further note that there are some extra crossing points for the partition
$\pi^{\prime}$ as compared to $(\pi,u)$ due to the coupling $(n+1,u)$; such
crossings occur on the line of $t$ of the couplings $(t,\pi^{\prime}(t))$ for
$0\leq t<u$, on the lines of both $t$ and $\pi(t)$ of the couplings
$(t,\pi^{\prime}(t))$ for $\pi^{\prime}(0)>t>u$, and with the singletons of
$\pi^{\prime}$. Therefore we obtain (whenever $E(\pi^{\prime})\neq 0$) that

$\displaystyle Q(\pi^{\prime})$ $\displaystyle={Q(\pi,u)}\cdot\prod_{0\leq
t<u}q(i_{t},i_{u})\cdot\prod_{{u<t<\pi^{\prime}(0)}}q(i_{t},i_{u})\cdot\prod_{u<t<\pi^{\prime}(0)}q(i_{\pi^{\prime}(u)},i_{\pi^{\prime}(t)})\cdot\prod_{t\in
s(\pi^{\prime})}q(i_{t},i_{n+1})$ $\displaystyle={Q(\pi,u)}\prod_{0\leq
t<u}q(i_{\pi^{\prime}(t)},i_{u})\cdot\prod_{{u<t<\pi^{\prime}(0)}}q(i_{\pi^{\prime}(t)},i_{u})\cdot\prod_{u<t<\pi^{\prime}(0)}q(i_{u},i_{\pi^{\prime}(t)})\cdot\prod_{t\in
s(\pi^{\prime})}q(i_{t},i_{u})$

where to get the last expression, we have used the symmetry of the matrix
$(q(i,j))$ and the same argument as in Remark 3.3 that whenever
$E(\pi^{\prime})\neq 0$, then $i_{t}=i_{\pi^{\prime}(t)}$ for any coupling
$(t,\pi^{\prime}(t))$ so that we can replace any $q(i_{t},i_{r})$ by
$q(i_{\pi(t)},i_{r})$, $r\in{N}$. On the other hand, if $E(\pi^{\prime})=0$
then $E(\pi,u)=0$, so the corresponding term does not contribute in the sum. A
careful observation of the last product above (and again the symmetry of the
matrix $(q(i,j))$) then yields

$Q(\pi^{\prime})=Q(\pi,u)\cdot\prod_{u<t\leq n}q(i_{u},i_{t}).$ (3.5)

Therefore the expression in (3.4) can finally be rewritten in the following
way:

$\displaystyle-[D_{\alpha},A_{\alpha_{n+1}}]$
$\displaystyle(e_{\alpha_{n}\ldots\alpha_{1}})+D_{\alpha}e_{\alpha_{n+1}}\ldots
e_{\alpha_{1}}$ $\displaystyle=\sum_{\sigma\in
B(n+1)}(-1)^{\sigma(0)-1}Q(\sigma)E(\sigma)\;e_{\alpha_{n+1}}e_{s(\sigma)}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+\sum_{\underset{\pi^{\prime}(n+1)\mbox{ is
not a singleton}}{\pi^{\prime}\in
B(n+2)}}(-1)^{\pi^{\prime}(0)-1}Q(\pi^{\prime})E(\pi^{\prime})\;e_{s(\pi^{\prime})}$
$\displaystyle=\sum_{{\pi^{\prime}\in
B(n+2)}}(-1)^{\pi^{\prime}(0)-1}Q(\pi^{\prime})E(\pi^{\prime})\;e_{s(\pi^{\prime})}$

where the last equality follows because each partition $\sigma\in B(n+1)$ gets
identified exactly to a partition $\sigma^{\prime}\in B(n+2)$ in an order
preserving fashion such that $n+1$ is a singleton in $\sigma^{\prime}$; in
such case
$\sigma(0)=\sigma^{\prime}(0),Q(\sigma)=Q(\sigma^{\prime}),E(\sigma)=E(\sigma^{\prime})$
and $s(\sigma^{\prime})=\\{n+1\\}\cup s(\sigma)$ so that
$e_{s(\sigma^{\prime})}=e_{\alpha_{n+1}}e_{s(\pi)}$. We have thus shown that
$[D_{\alpha},A_{\alpha_{n+1}}]e_{\alpha_{n}\cdots\alpha_{1}}=0$. This
completes the proof. ∎

We now give the main result of this Section in the following theorem. Note
that since $\pi(n+1)$ has at least one singleton whenever $n$ is even,
$\langle\Omega,D_{\alpha}e_{w}\rangle=0$ for any word $w$ of even length and
$\alpha\in[d]$. Also if $n=2m-1$ for $m\geq 1$, then in the summation of
$D_{\alpha}$, only those $\pi\in B(2m)$ contribute a non-zero term in the
inner product $\langle\Omega,D_{\alpha}e_{w}\rangle$ which has no singleton;
this occurs only if $\pi(0)=m$. In this case, we have

$\displaystyle\langle\Omega,D_{\alpha}e_{\omega}\rangle=\sum_{\underset{\pi(0)=m}{\pi\in
B(2m)}}(-1)^{m-1}Q(\pi,\omega)E(\pi,\omega)$ (3.6)

where $Q(\pi,\omega)$ and $E(\pi,\omega)$ are written respectively for
$Q(\pi)$ and $E(\pi)$ in order to make the dependency on $\omega$ explicit.

######  Theorem 3.4.

For each $\alpha\in[d]$ we have $\Omega\in\operatorname{Dom}D_{\alpha}^{*}$.
Thus $(D_{1}^{\ast}\Omega,\dots,D_{d}^{\ast}\Omega)$ forms a set of conjugate
variables for $(A_{1},\dots,A_{d})$.

###### Proof.

Fix $\alpha\in[d]$ and let
$e_{\alpha}\in\mathsf{H}_{\mathbb{C}}^{(i_{\alpha})}$ for some
$i_{\alpha}\in{N}$. As in [15, Theorem 4.6] and [14, Proposition 1.4] we will
prove that the functional $\langle\Omega,D_{\alpha}(\cdot)\rangle_{T}$ on
$\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$ extends to a bounded map on
$\mathcal{F}_{T}(\mathsf{H}_{U})$. Fix then an element
$\sum_{w\in[d]^{*}}\lambda_{w}e_{w}$ in
$\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$ where the scalars
$\lambda_{w}\in\mathbb{C}$ are $0$ except for finitely many, and then
calculate using (3.6):

$\displaystyle\langle\Omega,D_{\alpha}(\sum_{w\in[d]^{*}}\lambda_{w}e_{w})\rangle_{T}=\sum_{m=1}^{\infty}\sum_{\underset{\pi(0)=m}{\pi\in
B(2m)}\;\;}\sum_{|w|=2m-1}\lambda_{w}(-1)^{m-1}Q(\pi,w)E(\pi,w).$ (3.7)

Fix $m\geqslant 1$ and rewrite each word $w$ of length $2m-1$ as $w_{1}\beta
w_{1}^{\prime}$ with $w_{1},w_{1}^{\prime}$ being words of length $m-1$ and
$\beta\in[d]$; then identify each $\pi\in B(2m)$, $\pi(0)=m$ with a
permutation $\rho:=\rho_{\pi}\in S_{m-1}$ requiring that the couplings satisfy
$(t,m+\rho(t))\in p(\pi)$ for $1\leq t\leq m-1$. Under such identification
write $Q(\rho,w_{1},w_{1}^{\prime},\beta)$ and
$E(\rho,w_{1},w_{1}^{\prime},\beta)$ respectively for $Q(\pi,w)$ and
$E(\pi,w)$; which can further be rewritten as follows: if
$w_{1}=\alpha_{m-1}\cdots\alpha_{1}$,
$w_{1}^{\prime}=\alpha_{m-1}^{\prime}\cdots\alpha_{1}^{\prime}$ and
$e_{\alpha_{t}}\in\mathsf{H}_{\mathbb{C}}^{(i_{t})}$,
$e_{\alpha^{\prime}_{t}}=\mathsf{H}_{\mathbb{C}}^{(i_{t}^{\prime})}$,
$e_{\beta}\in\mathsf{H}_{\mathbb{C}}^{(j)}$ for some
$j,i_{t},i^{\prime}_{t}\in{N}$, $1\leq t\leq m-1$, then

$\displaystyle
E(\rho,w_{1},w_{1}^{\prime},\beta)=\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}\prod_{t=1}^{m-1}\langle\bar{e}_{\alpha_{\rho(t)}},e_{\alpha^{\prime}_{t}}\rangle_{U}=\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}\prod_{t=1}^{m-1}\langle\bar{e}_{\alpha_{t}},e_{\alpha^{\prime}_{\rho^{-1}(t)}}\rangle_{U}$
$\displaystyle
Q(\rho,w_{1},w_{1}^{\prime},\beta)=\prod_{u=1}^{m-1}q(i_{\alpha},i_{u}^{\prime})\cdot\prod_{{1\leq
v<u\leq
m-1}}q(i_{v}^{\prime},i_{u}^{\prime})\cdot\prod_{\underset{\rho(v)>\rho(u)}{1\leq
v<u\leq m-1}}q(i_{\rho(v)},i_{\rho(u)}).$

Moreover, whenever $E(\pi,w)\neq 0$, then we use the same arguments as in
Remark 3.3 to get the following expression:

$Q(\rho,w_{1},w_{1}^{\prime},\beta)=\prod_{t=1}^{m-1}q(i_{\alpha},i_{\rho(t)})\cdot\prod_{{1\leq
v<u\leq
m-1}}q(i_{\rho(v)},i_{\rho(u)})\cdot\prod_{\underset{\rho(v)>\rho(u)}{1\leq
v<u\leq m-1}}q(i_{\rho(v)},i_{\rho(u)})$

which can further be written as follows by rearranging the indices of the
first two products and then by using the symmetry of the matrix $(q(i,j))$:

$Q(\rho,w_{1},w_{1}^{\prime},\beta)=\prod_{t=1}^{m-1}q(i_{\alpha},i_{t})\prod_{{1\leq
v<u\leq m-1}}q(i_{v},i_{u})\cdot\prod_{\underset{\rho(v)>\rho(u)}{1\leq
v<u\leq m-1}}q(i_{\rho(v)},i_{\rho(u)}).$

Therefore the sum in (3.7) can be rewritten as follows:

$\displaystyle\langle\Omega,D_{\alpha}$
$\displaystyle(\sum_{w\in[d]^{*}}\lambda_{w}e_{w})\rangle_{T}$
$\displaystyle=\sum_{m=1}^{\infty}(-1)^{m-1}\sum_{\beta\in[d]}\sum_{\rho\in
S_{m-1}\;\;}\sum_{|w|=m-1}\sum_{|w^{\prime}|=m-1}\lambda_{w\beta
w^{\prime}}\;Q(\rho,w,w^{\prime},\beta)\;E(\rho,w,w^{\prime},\beta)$

We set the following notations: For each word $w=\alpha_{n}\cdots\alpha_{1}$
with $e_{\alpha_{t}}=\mathsf{H}_{\mathbb{C}}^{(i_{t})}$ for $i_{t}\in{N},1\leq
t\leq n$, write $i_{t}=i_{t}^{(w)}$ in order to show dependency on $w$. Also
write
$\bar{e}_{w}=\bar{e}_{\alpha_{n}}\otimes\cdots\otimes\bar{e}_{\alpha_{1}}$ and
$\rho(w)=\alpha_{\rho(n)}\cdots\alpha_{\rho(1)}$ for any permutation $\rho\in
S_{n}$. In this notation, we note that
$E(\rho,w_{1},w_{1}^{\prime},\beta)=\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}\langle
e_{\rho(w)},e_{w^{\prime}}\rangle_{\mathsf{H}_{U}^{\otimes^{n}}}$. We now
calculate

$\displaystyle\sum_{\beta\in[d]}\sum_{\rho\in
S_{m-1}}\sum_{|w|=m-1}\sum_{|w^{\prime}|=m-1}\lambda_{w\beta
w^{\prime}}\;Q(\rho,w,w^{\prime},\beta)\;E(\rho,w,w^{\prime},\beta)$
$\displaystyle=\sum_{\beta\in[d]}\sum_{|w|=m-1}\sum_{|w^{\prime}|=m-1}\prod_{{1\leq
v<u\leq
m-1}}q(i^{(w)}_{v},i^{(w)}_{u})\prod_{t=1}^{m-1}q(i_{\alpha},i_{t}^{(w)})\lambda_{w\beta
w^{\prime}}\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}$
$\displaystyle\quad\quad\quad\cdot\sum_{\rho\in
S_{m-1}}\prod_{\underset{\rho(t)>\rho(r)}{1\leq t<r\leq
m-1}}q(i^{(w)}_{\rho(r)},i^{(w)}_{\rho(t)})\prod_{t=1}^{m-1}\langle\bar{e}_{w_{\rho(t)}},e_{w^{\prime}_{t}}\rangle_{U}$
$\displaystyle=\sum_{\beta\in[d]}\sum_{|w|=m-1}\sum_{|w^{\prime}|=m-1}\prod_{{1\leq
v<u\leq
m-1}}q(i^{(w)}_{v},i^{(w)}_{u})\prod_{t=1}^{m-1}q(i_{\alpha},i_{t}^{(w)})\lambda_{w\beta
w^{\prime}}\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}\langle\bar{e}_{w},e_{w^{\prime}}\rangle_{T}$
$\displaystyle=\sum_{\beta\in[d]}\sum_{|w|=m-1}\prod_{{1\leq v<u\leq
m-1}}q(i^{(w)}_{v},i^{(w)}_{u})\prod_{t=1}^{m-1}q(i_{\alpha},i_{t}^{(w)})\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}{\left\langle\bar{e}_{w},\sum_{|w^{\prime}|=m-1}\lambda_{w\beta
w^{\prime}}\;e_{w^{\prime}}\right\rangle}_{T}.$

Now denote by ${L}_{w\beta}$ the composition of the $m$ relevant free
annihilation operators i.e. $L_{w\beta}=l(e_{\beta})l(e_{\alpha_{1}})\cdots
l(e_{\alpha_{n}})$ for any word $w=\alpha_{n}\cdots\alpha_{1}$, where
$l(e_{\gamma})$ is as defined in (2.3) for any $\gamma\in[d]$. Recall our
assumption that the set $\\{e_{\gamma}\\}_{\gamma\in[d]}$ is an orthonormal
basis for $\mathsf{H}_{U}$ so that $L_{w\beta}(e_{w^{\prime\prime}\gamma
w^{\prime}})=0$ for words $w^{\prime},w^{\prime\prime}$ of length $m-1$ and
$\gamma\in[d]$ such that $w^{\prime\prime}\gamma\neq w\beta$. Then for fixed
$w,\beta,\alpha$ as above we obtain

$\displaystyle\sum_{|w^{\prime}|=m-1}\lambda_{w\beta
w^{\prime}}\;e_{w^{\prime}}$
$\displaystyle=\sum_{\gamma\in[d]}\sum_{|w^{\prime\prime}|=m-1}\sum_{|w^{\prime}|=m-1}\lambda_{w^{\prime\prime}\gamma
w^{\prime}}\;{L}_{w\beta}(e_{w^{\prime\prime}\gamma w^{\prime}})$
$\displaystyle={L}_{w\beta}\left(\sum_{\gamma\in[d]}\sum_{|w^{\prime\prime}|=m-1}\sum_{|w^{\prime}|=m-1}\lambda_{w^{\prime\prime}\gamma
w^{\prime}}e_{w^{\prime\prime}\gamma w^{\prime}}\right)$
$\displaystyle={L}_{w\beta}\left(\sum_{|w^{\prime\prime}|=2m-1}\lambda_{w^{\prime\prime}}e_{w^{\prime\prime}}\right)$

so that

$\displaystyle\langle\bar{e}_{w},\sum_{|w^{\prime}|=m-1}\lambda_{w\beta
w^{\prime}}\;e_{w^{\prime}}\rangle_{T}=\langle\bar{e}_{w},{L}_{w\beta}\left(\sum_{w^{\prime\prime}\in[d]^{*}}\lambda_{w^{\prime\prime}}e_{w^{\prime\prime}}\right)\rangle_{T}.$

which follows by noting that the operator ${L}_{w\beta}$ maps
${e}_{w^{\prime}}$ with $|w^{\prime}|=2m^{\prime}-1$ to the subspace spanned
by $\\{e_{v}\\}_{|v|=m-1}$ and hence
$\langle\bar{e}_{w},{L}_{w\beta}(e_{w^{\prime}})\rangle_{T}=0$ if $m\neq
m^{\prime}$ and $|w|=m-1,|w^{\prime}|=2m^{\prime}-1$. Thus we get

$\displaystyle\sum_{\beta\in[d]}\sum_{\rho\in
S_{m-1}}\sum_{|w|=m-1}\sum_{|w^{\prime}|=m-1}\lambda_{w\beta
w^{\prime}}\;Q(\rho,w,w^{\prime},\beta)\;E(\rho,w,w^{\prime},\beta)$
$\displaystyle=\sum_{\beta\in[d]}\sum_{|w|=m-1}\prod_{{1\leq v<u\leq
m-1}}q(i^{(w)}_{v},i^{(w)}_{u})\prod_{t=1}^{m-1}q(i_{\alpha},i_{t}^{(w)})\langle\bar{e}_{\beta},e_{\alpha}\rangle_{U}\left\langle\bar{e}_{w},{L}_{w\beta}\left(\sum_{w^{\prime\prime}\in[d]^{*}}\lambda_{w^{\prime\prime}}e_{w^{\prime\prime}}\right)\right\rangle_{T}$

If we set

$\displaystyle\xi_{m,\alpha}=\sum_{\beta\in[d]}\sum_{|w|=m-1}\prod_{{1\leq
v<u\leq
m-1}}q(i^{(w)}_{v},i^{(w)}_{u})\prod_{t=1}^{m-1}q(i_{\alpha},i_{t}^{(w)})\langle
e_{\alpha},\bar{e}_{\beta}\rangle_{U}{L}_{w\beta}^{*}(\bar{e}_{w})$ (3.8)

then we have

$\displaystyle\langle\Omega,D_{\alpha}(\sum_{w\in[d]^{*}}\lambda_{w}e_{w})\rangle_{T}=\langle\sum_{m=1}^{\infty}(-1)^{m}\xi_{m,\alpha},\sum_{w\in[d]^{*}}\lambda_{w}e_{w}\rangle_{T}$

so long as the sum $\sum_{m=1}^{\infty}(-1)^{m}\xi_{m,\alpha}$ is a well-
defined vector. We show that it indeed is and converges in norm:

$\displaystyle\begin{split}\|\xi_{m,\alpha}\|_{T}&\leq\sum_{\beta\in[d]}\sum_{|w|=m-1}q^{\frac{m(m-1)}{2}}\|{e}_{\alpha}\|_{U}\;\|\bar{e}_{\beta}\|_{U}\;\|{L}_{w\beta}^{*}\|\;\|\bar{e}_{w}\|_{T}\\\
&\leq d\;B\sum_{|w|=m-1}q^{\frac{m(m-1)}{2}}C^{m}\|\bar{e}_{w}\|_{T}\\\
&=d^{m}B^{m}C^{m}\sqrt{[m-1]_{q}!}\;q^{\frac{m(m-1)}{2}}\end{split}$ (3.9)

where $d=|[d]|$, $B=\max_{\gamma\in[d]}\|\bar{e}_{\gamma}\|_{U}$,
$C=\max_{\gamma\in[d]}\|l(e_{\gamma})\|<\infty$ (by Lemma 2.1) and
$q=\max_{i,j}|q(i,j)|<1$; and the last equality follows by noting from Lemma
2.6 that

$\|\bar{{e}}_{w}\|_{T}^{2}\leq\sum_{\sigma\in
S_{m-1}}q^{{\text{inv}}(\sigma)}B^{2(m-1)}=[m-1]_{q}!B^{2(m-1)}$ (3.10)

where ${\text{inv}}(\sigma)$ denotes the number of inversions in the
permutation $\sigma$, so that

$\sum_{|w|=m-1}\|\bar{e}_{w}\|_{T}\leq d^{m-1}B^{m-1}\sqrt{[m-1]_{q}!}.$

Therefore we finally arrive at the claim:

$\displaystyle\sum_{m=1}^{\infty}\|\xi_{m,\alpha}\|\leq\sum_{m=1}^{\infty}q^{\frac{m(m-1)}{2}}d^{m}B^{m}C^{m}\sqrt{[m-1]_{q}!}<\;\infty$

by using ratio test. We have thus shown that

$\xi_{\alpha}:=\sum_{m=1}^{\infty}(-1)^{m}\xi_{m,\alpha}\in\mathcal{F}_{T}(\mathsf{H}_{U})$
(3.11)

and
$\langle\Omega,D_{\alpha}(\cdot)\rangle_{T}=\langle\xi_{\alpha},\cdot\rangle_{T}$
on $\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$; hence the functional
extends to a bounded linear map on $\mathcal{F}_{T}(\mathsf{H}_{U})$. This
proves that $\Omega\in\operatorname{Dom}D_{\alpha}^{*}$ and
$\xi_{\alpha}=D_{\alpha}^{*}(\Omega)$. ∎

In our next result, we use Bożejko-Haagerup type inequality (see [19, Lemma
2]): For any $\xi\in\mathsf{H}_{U}^{\otimes n},n\geq 1$, we have

$\displaystyle\|\xi\|_{T}\leq\|W(\xi)\|\leq C_{q}^{3/2}(n+1)\|\xi\|_{T}$
(3.12)

where as before $q=\sup_{i,j}|q{(i,j)}|$ and
$C_{q}=\prod_{n=1}^{\infty}(1-q^{n})^{-1}$. With the notations as in the proof
of Theorem 3.4, we establish the following:

######  Corollary 3.5.

The conjugate system $(\xi_{1},\ldots,\xi_{d})$ corresponding to
$(D_{1},\ldots,D_{d})$ is given by
$\xi_{\alpha}=D_{\alpha}^{*}\Omega=\sum_{m=1}^{\infty}(-1)^{m-1}\xi_{m,{\alpha}}$
for $1\leq\alpha\leq d:=\dim{\mathsf{H}_{\mathbb{R}}}$ where for $m\geq 1$,

$\xi_{m,\alpha}=\sum_{\beta\in[d]}\sum_{|w|=m-1}\prod_{{1\leq v<u\leq
m-1}}q(i^{(w)}_{v},i^{(w)}_{u})\prod_{t=1}^{m-1}q(i_{\alpha},i^{(w)}_{t})\langle
e_{\alpha},\bar{e}_{\beta}\rangle_{U}{L}_{w\beta}^{*}(\bar{e}_{w}).$

Moreover the sum $\sum_{m=1}^{\infty}(-1)^{m-1}W(\xi_{m,\alpha})$ converges in
operator norm i.e. $\xi_{\alpha}\in\mathsf{M}_{T}\Omega$.

###### Proof.

We have already seen the expression for the conjugate variables $\xi_{\alpha}$
in (3.11) in Theorem 3.4. Note that the vector $L_{w\beta}^{*}(\bar{e}_{w})$
belongs to the particle space $\mathsf{H}_{U}^{\otimes{2m-1}}$ for any word
$w$ of length $m-1$ and $\beta\in[d]$. Arguing as in [15], we invoke Bożejko’s
Haagerup type inequality as mentioned in (3.12) to infer that

$\displaystyle\|W(\xi_{m,\alpha})\|$
$\displaystyle\leq\sum_{\beta\in[d]}\sum_{|w|=m-1}q^{\frac{m(m-1)}{2}}\|{e}_{\alpha}\|_{U}\;\|\bar{e}_{\beta}\|_{U}\;\|W({L}_{w\beta}^{*}\bar{e}_{w})\|$
$\displaystyle\leq
B\;\sum_{\beta\in[d]}\sum_{|w|=m-1}q^{\frac{m(m-1)}{2}}\;C_{q}^{3/2}(2m)\;\|{L}_{w\beta}^{*}(\bar{e}_{w})\|_{T}$
$\displaystyle\leq
C_{q}^{3/2}d^{m}q^{\frac{m(m-1)}{2}}(2m)\;C^{m}B^{m}\sqrt{[m-1]_{q}!}$

where $B,C$ are as written towards the end of proof of Theorem 3.4 and we have
used estimates as in (3.9) and (3.10). Thus, by using the ratio test, it
follows that

$\sum_{m=1}^{\infty}\|W(\xi_{m,\alpha})\|\leq
C_{q}^{3/2}\sum_{m=1}^{\infty}d^{m}\;q^{\frac{m(m-1)}{2}}(2m)\;C^{m}B^{m}\sqrt{[m-1]_{q}!}<\infty$

concluding the proof. ∎

######  Remark 3.6.

(1) In the expression for the conjugate variables given in [15, Corollary
4.7], certain products of free _right_ annihilation operators appear, while in
our result products of _left_ annihilation operators appear. One can however
verify that in the $q$-Gaussian case, the two situations indeed agree. This
follows by noting that if ${R}_{\beta w}$ is a product of appropriate right
free annihilation operators then

$\displaystyle\sum_{|v|=2m-1}\sum_{|w|=m-1}\lambda_{v}\langle\bar{e}_{w},{R}_{\beta
w}e_{v}\rangle_{q}$
$\displaystyle=\sum_{|w^{\prime}|=m-1}\sum_{|w|=m-1}\lambda_{w^{\prime}\beta
w}\langle{e}_{w},e_{w^{\prime}}\rangle_{q}$
$\displaystyle=\sum_{|w|=m-1}\sum_{|w^{\prime}|=m-1}\lambda_{w^{\prime}\beta
w}\langle e_{w^{\prime}},e_{w}\rangle_{q}$
$\displaystyle=\sum_{|v|=2m-1}\sum_{|w^{\prime}|=m-1}\lambda_{v}\langle\bar{e}_{w^{\prime}},{L}_{w^{\prime}\beta}e_{v}\rangle_{q}$

simply because in the $q$-Gaussian case, one chooses an orthonormal basis
$\\{e_{1},\cdots,e_{d}\\}$ with $e_{i}\in\mathsf{H}_{\mathbb{R}}$ so that
$\bar{e}_{w}=e_{w}$ and $\langle e_{w},e_{w^{\prime}}\rangle_{T}=\langle
e_{w^{\prime}},e_{w}\rangle_{T}$ as they are always real for any two words
$w,w^{\prime}$ (summation of the above form appears in the proof of Theorem
3.4). In particular, we can replace $L_{{\beta w}}$ by $R_{\beta w}$ in
Corollary 3.5 which produces exactly the same formula as in [15, Corollary
4.7]. Note that equality of above forms no longer holds in non-tracial
settings, so left free annihilation operators seem to be more convenient
choice.

(2) The orthonormal set $\\{e_{1},\cdots,e_{d}\\}$ can be chosen to be
eigenvectors of the analytic generator $A$ of $U_{t}$ so that each element
$W(e_{k})$ is analytic in $\mathsf{M}_{T}$ (w.r.t the modular automorphism
group $\sigma^{\varphi}$ of the canonical state $\varphi$); consequently the
conjugate variables $\xi_{i}$’s are also analytic as they are norm-limits of
vectors which are linear combinations of tensors of $e_{i}$’s.

(3) When $q_{ij}=q$ for all $i,j$, we get

$\xi_{\alpha}=\sum_{m=1}^{\infty}(-1)^{m-1}\sum_{\beta\in[d]}\sum_{|w|=m-1}q^{\frac{m(m-1)}{2}}\langle
e_{\alpha},\bar{e}_{\beta}\rangle_{U}{L}_{w\beta}^{*}(\bar{e}_{w}),$

which is slightly different that the formula in [14, Remark 1.5] because the
initial assumption on the set
$\\{e_{1},\ldots,e_{d}\\}\subseteq\mathsf{H}_{\mathbb{C}}$ in [14] is that it
is orthonormal in the undeformed inner product of $\mathsf{H}_{\mathbb{C}}$
while our assumption is that the set is orthonormal in the deformed inner
product of $\mathsf{H}_{U}$. In particular, if $q=0$ then $\xi_{m,\alpha}=0$
for all $m\geq 2$, so that

$\xi_{\alpha}=\xi_{1,\alpha}=\sum_{\beta\in[d]}\langle
e_{\alpha},\bar{e}_{\beta}\rangle_{U}l(\beta)^{*}\Omega=\sum_{\beta\in[d]}\langle
e_{\alpha},\bar{e}_{\beta}\rangle_{U}\;e_{\beta}.$

### Lipschitz Conjugate Variables

Our aim now is to prove that the conjugate variables
$(\xi_{1},\ldots,\xi_{d})$ as obtained above are Lipschitz i.e.

$\xi_{\ell}\in\operatorname{Dom}\bar{\partial}_{k}\;\;\text{and}\;\;\bar{\partial}_{k}(\xi_{\ell})\in\mathsf{M}_{T}\otimes\mathsf{M}_{T}^{\operatorname{op}}$

for all $1\leq k,\ell\leq d$, where $\partial_{k}$ is the quasi-free
difference quotients as defined in the beginning of this Section. For this
purpose, we consider a different set of rules for partitions and counting
their crossings exactly as in [15, Section 5]. The partitions again consist of
singletons and pairs, and we draw them in the following ways to locate
crossings:

  1. (1)

Consider $n+1$ vertices $n>\cdots>1>0.$

  2. (2)

The vertex $0$ must be coupled with some $k\in\\{1,\ldots,n\\}$ with height 1.

  3. (3)

Each $\ell\in\\{1,\ldots,k-1\\}$ is a singleton or is coupled with one of
$\\{k+1,\ldots,n\\}$ with height $\ell+1$

  4. (4)

Vertices which are not coupled with one of $\\{1,\ldots,k-1\\}$ should be
singletons and are drawn with straight lines to the top.

Let $C(n+1)$ be the set of partitions defined by the rules above. For each
$\pi\in C(n+1)$, denote by $s_{\ell}(\pi)$ and $s_{r}(\pi)$ respectively the
set of singletons in the left area $n\geq l>\pi(0)$ and the set of singletons
in the right area $\pi(0)>\ell\geq 1$. Let $p(\pi)$ denote the set of
pairings.

We now give a formula for the action of $\partial_{k}$’s on the algebra
$\mathbb{C}[A_{1},\ldots,A_{d}]$. The algebra $\mathbb{C}[A_{1},\ldots,A_{d}]$
is nothing but the space $\operatorname{span}\\{W(e_{w});w\in[d]^{*}\\}$. This
follows by noting the following calculation (see eq. (2.5)):

$\displaystyle
W(\xi_{n+1}\cdots\xi_{1})=W(\xi_{n+1})W(\xi_{n}\cdots\xi_{1})-\sum_{k=1}^{n}\langle\bar{\xi}_{n+1},\xi_{k}\rangle_{U}\prod_{t=1}^{n-k}q(i_{k},i_{k+t})W(\xi_{n}\cdots\widehat{\xi}_{k}\cdots\xi_{1})$
(3.13)

whenever $\xi_{k}\in\mathsf{H}_{U}^{(i_{k})}$ for $i_{k}\in{N}$, $1\leq k\leq
n+1$. Therefore it suffices to understand the action of $\partial_{k}$’s on
the operators $W(e_{w})$ for each word $w$.

######  Proposition 3.7.

For each $\alpha\in[d]$ and $w=\alpha_{n}\cdots\alpha_{1}\in[d]^{*}$, $n\geq
1$, we have

$\partial_{\alpha}W(e_{w})=\sum_{\pi\in
C(n+1)}(-1)^{|p(\pi)|-1}{D}(\pi)\;{E}(\pi)W(e_{s_{\ell}(\pi)})\otimes
W(e_{s_{r}(\pi)})^{\operatorname{op}}$

where

$\displaystyle{D(\pi)}:=\prod_{\underset{0\leq v<u<\pi(0)}{u,v\text{ belong to
a pairing}}}q(i_{v},i_{u})\cdot\prod_{\begin{subarray}{c}u,v\text{ belong to a
pairing}\\\ {0<v<u\leq\pi(0)}\\\
\pi(u)>\pi(v)\end{subarray}}q(i_{\pi(v)},i_{\pi(u)})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\cdot\prod_{\underset{v\notin
s_{r}(\pi),u\in
s_{r}(\pi)}{0<v<u<\pi(0)}}q(i_{v},i_{u})\cdot\prod_{\underset{v\in
s_{\ell}(\pi),u\notin s_{\ell}(\pi)}{\pi(0)<v<u\leq n}}q(i_{v},i_{u})$
$\displaystyle{E}(\pi):=\prod_{\underset{{k>\ell}}{(k,\ell)}\in
p(\pi)}\langle\bar{e}_{\alpha_{k}},e_{\alpha_{\ell}}\rangle_{U}$
$\displaystyle e_{s_{\ell}(\pi)}=e_{\alpha_{j_{r}}}\cdots
e_{\alpha_{j_{1}}}\;\;\text{ for
}\;\;{s_{\ell}(\pi)}=\\{j_{r}>\ldots>j_{1}\\}\;\;\;\text{ and }$
$\displaystyle e_{s_{r}(\pi)}=e_{\alpha_{j^{\prime}_{s}}}\cdots
e_{\alpha_{j^{\prime}_{1}}}\;\;\text{ for
}\;\;s_{r}(\pi)=\\{j^{\prime}_{s}>\ldots>j^{\prime}_{1}\\},$

with the prescription that $\alpha_{0}=\alpha$.

######  Remark 3.8.

As in Proposition 3.2, we have divided the expression for $D(\pi,w)$ into four
products. The terms in the first product (resp. the second product) correspond
to the crossings occurring on the line of the vertex $`u$’ (resp. $`\pi(u)$’)
of any pair $(u,\pi(u))$ for $0<u<\pi(0)$ with the lines of vertices of
another pair. The third product corresponds to the crossings in the right side
of $\pi(0)$ on the lines of singletons with all pairs except $(0,\pi(0))$,
while the fourth product corresponds to the crossings on the lines of
singletons in the left side of $\pi(0)$.

###### Proof.

As in [15, Proposition 5.1], the proof goes by induction on $n$. For $n=1$,
the expression is obvious because

$\partial_{\alpha}(W(e_{\alpha_{1}}))=\partial_{\alpha}(A_{\alpha_{1}})=\langle\bar{e}_{\alpha_{1}},e_{\alpha}\rangle_{U}\mathds{1}\otimes\mathds{1}.$

So assume this to be true for some $n\geq 1$. Use then (3.13) to calculate:

$\displaystyle\partial_{\alpha}$ $\displaystyle
W(e_{\alpha_{n+1}\cdots\alpha_{1}})=\partial_{\alpha}\left(W(e_{\alpha_{n+1}})W(e_{\alpha_{n}\cdots\alpha_{1}})-\sum_{u=1}^{n}\prod_{u<t\leq
n}q(i_{u},i_{t})\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}\;W(e_{\alpha_{n}\ldots\widehat{\alpha}_{u}\ldots\alpha_{1}})\right)$
$\displaystyle=\langle\bar{e}_{\alpha_{n+1}},e_{\alpha}\rangle_{U}\mathds{1}\otimes
W(e_{\alpha_{n}\cdots\alpha_{1}})^{\operatorname{op}}+(W(e_{\alpha_{n+1}})\otimes\mathds{1})\partial_{\alpha}W(e_{\alpha_{n}\cdots\alpha_{1}})$
$\displaystyle\quad\quad\quad\quad\quad-\sum_{v=1}^{n}\prod_{v<t\leq
n}q(i_{v},i_{t})\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{v}}\rangle_{U}\;\partial_{\alpha}W(e_{\alpha_{n}\ldots\widehat{\alpha}_{u}\ldots\alpha_{1}}).$

By using induction argument, we separately compute the last two terms in the
above sum:

$\displaystyle\begin{split}&(W(e_{\alpha_{n+1}})\otimes\mathds{1})\partial_{\alpha}W(e_{\alpha_{n}\cdots\alpha_{1}})=\sum_{\pi\in
C(n+1)}(-1)^{|p(\pi)|-1}{D}(\pi){E}(\pi)W(e_{\alpha_{n+1}})W(e_{s_{\ell}(\pi)})\otimes
W(e_{s_{r}(\pi)})^{\operatorname{op}}\\\ &=\sum_{\pi\in
C(n+1)}(-1)^{|p(\pi)|-1}{D}(\pi){E}(\pi)W(e_{\alpha_{n+1}}e_{s_{\ell}(\pi)})\otimes
W(e_{s_{r}(\pi)})^{\operatorname{op}}\\\ &+\sum_{\pi\in
C(n+1)}(-1)^{|p(\pi)|-1}{D}(\pi){E}(\pi)\sum_{u=1}^{n(\pi)}\prod_{t=u+1}^{n(\pi)}q(i_{j_{u}^{\pi}},i_{j_{t}^{\pi}})\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{j_{u}^{\pi}}}\rangle_{U}W(e_{\alpha_{j_{n(\pi)}^{\pi}}\ldots\hat{\alpha}_{j_{u}^{\pi}}\ldots\alpha_{j_{1}^{\pi}}})\otimes
W(e_{s_{r}(\pi)})^{\operatorname{op}}\end{split}$ (3.14)

where $n(\pi)=|s_{\ell}(\pi)|$ and
$s_{\ell}(\pi)=\\{j_{n(\pi)}^{\pi}>\ldots>j_{1}^{\pi}\\}$; then compute

$\displaystyle\begin{split}\sum_{v=1}^{n}&\prod_{v<t\leq
n}q(i_{v},i_{t})\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{v}}\rangle_{U}\;\partial_{\alpha}W(e_{\alpha_{n}\ldots\widehat{\alpha_{v}}\ldots\alpha_{1}})\\\
&=\sum_{\sigma\in C(n)}\sum_{v=1}^{n}\prod_{v<t\leq
n}q(i_{v},i_{t})\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{v}}\rangle_{U}(-1)^{|p(\sigma)|-1}{D}(\sigma,v){E}(\sigma,v)W(e_{s_{\ell}(\sigma)})\otimes
W(e_{s_{r}(\sigma)})^{\operatorname{op}}\end{split}$ (3.15)

where $D(\sigma,v)$ and $E(\sigma,v)$ are written respectively for $D(\sigma)$
and $E(\sigma)$ only but just to emphasize that each $\sigma\in C(n)$ acts on
the word $\alpha_{n}\cdots\widehat{\alpha}_{v}\cdots\alpha_{1}$. Note that
each term of the last sum of (3.14) appears as one of the terms in the last
sum of (3.15). To see this, we follow the similar argument as in Theorem 3.4.
Let $\pi\in C(n+1)$ and $u\in\\{1,\ldots,n(\pi)\\}$, where
$n(\pi)=|s_{\ell}(\pi)|$. Set $v=s_{\ell}(\pi)_{u}=j_{u}^{\pi}$ and obtain the
partition $\sigma\in C(n)$ of the set $\\{0,\ldots,n\\}\setminus\\{v\\}$ in an
order-preserving fashion by removing the singleton $s_{\ell}(\pi)_{u}=v$. Note
that $p(\pi)=p(\sigma),s_{\ell}(\pi)=s_{\ell}{(\sigma)}\setminus\\{v\\}$ and
$s_{r}(\pi)=s_{r}(\sigma)$. Hence ${E}(\pi)={E}(\sigma)$. Also note that the
extra crossings in $\pi$ as compared to $\sigma$ may be occurring only due to
the straight line at $v=j_{u}^{\pi}$. Therefore we have

$\displaystyle{{D}(\pi)}={{D}(\sigma,v)}\cdot\prod_{\underset{t\notin
s_{\ell}(\pi)}{t>v}}q(i_{v},i_{t})=\frac{\prod_{v<t\leq
n}q(i_{v},i_{t})}{\prod_{\underset{t\in s_{\ell}(\pi)}{v<t\leq
n}}q(i_{v},i_{t})}=\frac{\prod_{v<t\leq n}q(i_{v},i_{t})}{\prod_{{u<r\leq
n(\pi)}}q(i_{j_{u}^{\pi}},i_{j_{r}^{\pi}})}$

where it is to be understood that $\frac{0}{0}$ is $0$. This shows our claim
that all terms of the last sum of (3.14) appear in the last sum of (3.15), and
the same time note that the remaining terms in (3.15) correspond to those
$\sigma$ and $v$ for which $v\leq\sigma(0)$. Again as in Theorem 3.4, we
identify such pair $(\sigma,v)$ i.e. $\sigma\in C(n)$ and $v\leq\sigma(0)$,
with a partition $\sigma^{\prime}\in C(n+2)$ which creates a new pair between
$v$ and $n+1$ with the requirement that
$\sigma^{\prime}\setminus(n+1,v)=\sigma$. Note that
$\sigma^{\prime}(0)=\sigma(0)+1$,
$s_{\ell}(\sigma^{\prime})=s_{\ell}(\sigma),s_{r}(\sigma^{\prime})=s_{r}(\sigma)$
and $|p(\sigma^{\prime})|=|p(\sigma)|+1$. Also note that the extra crossings
in $\sigma^{\prime}$ occur with all the pairs (twice) and singletons (once) to
the left of $v$ or on the line of $v$; so that whenever $E(\sigma)\neq 0$,
then we have

$\displaystyle{D}(\sigma^{\prime})={D}(\sigma)\cdot\prod_{v<t\leq
n}q(i_{v},i_{t})$

which follows by similar arguments as in (3.5) from Proposition 3.2. In
particular, we get

$E(\sigma^{\prime}){D}(\sigma^{\prime})=E(\sigma){D}(\sigma)\cdot\prod_{v<t\leq
n}q(i_{v},i_{t}).$

Finally by subtracting (3.14) from (3.15) yields

$\displaystyle\partial_{\alpha}W(e_{\alpha_{n+1}\cdots\alpha_{1}})$
$\displaystyle=\langle\bar{e}_{\alpha_{n+1}},e_{\alpha}\rangle_{U}\mathds{1}\otimes
W(e_{\alpha_{n}\cdots\alpha_{1}})^{\operatorname{op}}$
$\displaystyle+\sum_{\pi\in
C(n+1)}(-1)^{|p(\pi)|-1}{D}(\pi)\;{E}(\pi)\;W(e_{\alpha_{n+1}}e_{s_{\ell}(\pi)})\otimes
W(e_{s_{r}(\pi)})^{\operatorname{op}}$
$\displaystyle+\sum_{\begin{subarray}{c}{\sigma^{\prime}\in C(n+2)}\\\
\sigma^{\prime}(n+1)\text{ is not a singleton}\\\ \sigma^{\prime}(0)\neq
n+1\end{subarray}}\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{v}}\rangle_{U}(-1)^{|p(\sigma)|-1}{D}(\sigma^{\prime}){E}(\sigma^{\prime})W(e_{s_{\ell}(\sigma)})\otimes
W(e_{s_{r}(\sigma)})^{\operatorname{op}}$
$\displaystyle=\sum_{{\sigma^{\prime}\in
C(n+2)}}(-1)^{|p(\sigma)|-1}{D}(\sigma^{\prime}){E}(\sigma^{\prime})\;W(e_{s_{\ell}(\sigma)})\otimes
W(e_{s_{r}(\sigma)})^{\operatorname{op}}$

where the last equality follows by noting that
$\langle\bar{e}_{{\alpha_{n+1}}},e_{\alpha}\rangle_{U}\mathds{1}\otimes
W(e_{\alpha_{n}\cdots\alpha_{1}})^{\operatorname{op}}$ corresponds to the
partition in $C(n+2)$ which connects $0$ to $n+1$ (in which case all other
points are right singletons); while each $\pi\in C(n+1)$ corresponds to those
partitions $\sigma^{\prime}\in C(n+2)$ such that $n+1$ is a singleton, in
which case we have
$p(\sigma^{\prime})=p(\pi),{D}(\sigma^{\prime})={D(\pi)},s_{r}(\sigma^{\prime})=s_{r}(\pi)$
and $s_{\ell}(\sigma^{\prime})=\\{n+1\\}\cup s_{\ell}(\pi)$ so that
$e_{\alpha_{n+1}}e_{s(\pi)}=e_{s(\sigma^{\prime})}$. This completes the proof
by induction. ∎

######  Remark 3.9.

One can also obtain Theorem 3.4 using the formulas in Proposition 3.7 for the
quasi-free difference quotients as the conjugate variables $\xi_{\alpha}$’s
are nothing but $\partial_{\alpha}^{*}(\mathds{1}\otimes\mathds{1})$ and it is
an easy check that the expression
$\langle\mathds{1}\otimes\mathds{1},\partial_{\alpha}(\sum_{w\in[d]^{*}}\lambda_{w}W(e_{w}))\rangle_{L^{2}(\mathsf{M}_{T}\otimes\mathsf{M}_{T}^{\operatorname{op}})}$
is exactly the one in eq. (3.7). In fact, one can obtain Proposition 3.2 by
checking that
$D_{\alpha}:=({\textrm{id}}\otimes\varphi^{\operatorname{op}})\partial_{\alpha}$
form a normalized dual system for $1\leq\alpha\leq d$.

We keep the notations of Corollary 3.5 and Proposition 3.7 for the following
result.

######  Corollary 3.10.

The system of conjugate variables $(\xi_{1},\ldots,\xi_{d})$ corresponding to
the tuple $(D_{1},\ldots,D_{d})$ is Lipschitz.

###### Proof.

Fix $\alpha,\beta\in[d]$. First note that each
$\xi_{m,\alpha}\in\mathcal{F}_{T}(\mathsf{H}_{U})_{\textup{alg}}$ so that
$W(\xi_{m,\alpha})\in\operatorname{Dom}\partial_{\beta}$; hence by Proposition
3.7,
$\partial_{\beta}W(\xi_{m,\alpha})\in\mathsf{M}_{T}\bar{\otimes}\mathsf{M}_{T}^{\operatorname{op}}$.
We show that
$\sum_{m=1}^{\infty}\|\partial_{\beta}W(\xi_{m,\alpha})\|<\infty$, from which
it will follow that
$\sum_{m=1}^{\infty}(-1)^{m-1}\partial_{\beta}W(\xi_{m,\alpha})$ converges
both in norm and in
$L^{2}(\mathsf{M}_{T}\bar{\otimes}\mathsf{M}_{T}^{\operatorname{op}})$ proving
that $W(\xi_{\alpha})\in\operatorname{Dom}\bar{\partial}_{\beta}$ and
$\bar{\partial}_{\beta}W(\xi_{\alpha})\in\mathsf{M}_{T}\bar{\otimes}\mathsf{M}_{T}^{\operatorname{op}}$.
Assume that $e_{\alpha}\in\mathsf{H}_{\mathbb{C}}^{(i_{\alpha})}$ so that

$\displaystyle\xi_{m,\alpha}=\sum_{\gamma\in[d]}\sum_{|w|=m-1}\prod_{{1\leq
v<u\leq
m-1}}q(i^{(w)}_{v},i^{(w)}_{u})\prod_{t=1}^{m-1}q(i_{\alpha},i^{(w)}_{t})\langle
e_{\alpha},\bar{e}_{\gamma}\rangle_{U}{L}_{w\gamma}^{*}(\bar{e}_{w}).$

By Proposition 3.7, note that for each word $w^{\prime}$ of length $n$

$\displaystyle\|\partial_{\beta}W(e_{w^{\prime}})\|\leq\sum_{\pi\in
C(n+1)}|{D}(\pi)|\;|{E}(\pi)|\;\|W(e_{s_{\ell}(\pi)})\|\;\|W(e_{s_{r}(\pi)})\|.$

But for each $\pi\in C(n+1)$, we have

$\displaystyle|{D}(\pi)|\leq 1,$ $\displaystyle|{E}(\pi)|\leq B^{|p(\pi)|}\leq
B^{n}\;\;\;\mbox{ where
}{B=\max_{\gamma\in[d]}\\{1,\|\bar{e}_{\gamma}\|_{U}\\}},$
$\displaystyle\|W(e_{s_{\ell}(\pi)})\|\leq
C_{q}^{3/2}(|s_{\ell}(\pi)|+1)\|e_{s_{\ell}(\pi)}\|_{T}\leq
C_{q}^{3/2}(|s_{\ell}(\pi)|+1)B^{|s_{\ell}(\pi)|}\sqrt{[|s_{\ell}(\pi)|]_{q}!}$
$\displaystyle\quad\quad\quad\quad\quad\;\leq
C_{q}^{3/2}(n+1)B^{n}\sqrt{[n]_{q}!},$ $\displaystyle\|W(e_{s_{r}(\pi)})\|\leq
C_{q}^{3/2}(n+1)B^{n}\sqrt{[n]_{q}!}$

where the last two inequalities are due to Bożejko-Haagerup type inequality
and the estimate as in (3.10). Consequently we obtain

$\displaystyle\begin{split}\|\partial_{\beta}W(e_{w^{\prime}})\|&\leq|C(n+1)|\;B^{3n}\;C_{q}^{3}\;(n+1)^{2}\;[n]_{q}!\\\
&\leq(n+1)!\;B^{3n}\;C_{q}^{3}\;(n+1)^{2}\;[n]_{q}!,\end{split}$ (3.16)

where we have used the fact that the cardinality $|C(n+1)|$ of $C(n+1)$ is
upper-bounded by $(n+1)!$ as we can think $C(n+1)$ as a subset of the
symmetric group $S_{n+1}$. Note that the upper-bound of each
$\|\partial_{\beta}W(e_{w^{\prime}})\|$ depends only on the length of the word
$w^{\prime}$ and is independent of the choice of the word itself. In
particular, since for each fixed word $w$ of length $m-1$ and $\gamma\in[d]$,
${L}^{*}_{w\gamma}(\bar{e}_{w})$ belongs to the subspace spanned by
$\\{e_{w^{\prime}}\\}_{|w^{\prime}|=2m-1}$, we write
${L}^{*}_{w\gamma}(\bar{e}_{w})=\sum_{|w^{\prime}|=2m-1}\lambda_{w^{\prime}}e_{w^{\prime}}$
for some scalars $\lambda_{w^{\prime}}\in\mathbb{C}$ and then estimate

$\displaystyle\|\partial_{\beta}W({L}^{*}_{w\gamma}\bar{e}_{w})\|$
$\displaystyle\leq\sum_{|w^{\prime}|=2m-1}|\lambda_{w^{\prime}}|\;\|\partial_{\beta}W(e_{w^{\prime}})\|$
$\displaystyle\leq\sum_{|w^{\prime}|=2m-1}|\lambda_{w^{\prime}}|\;(2m)!\;B^{6m-3}\;C_{q}^{3}\;(2m)^{2}\;[2m-1]_{q}!.$

Further observe that for all $|w^{\prime}|=2m-1$, we have
$\lambda_{w^{\prime}}=L_{w^{\prime}}L^{*}_{\gamma w}(\bar{e}_{w})$ so that

$|\lambda_{w^{\prime}}|\leq\|L_{w^{\prime}}\|\;\|{L}^{*}_{w\gamma}\|\;\|\bar{e}_{w}\|_{T}\leq
C^{3m-1}B^{{m-1}}\sqrt{[m-1]_{q}!},$

where as in Theorem 3.4, $C=\max_{\gamma\in[d]}\|{l}(e_{\gamma})\|$, $B$ is as
above and the estimate of (3.10) is used. It follows that

$\displaystyle\|\partial_{\beta}W({L}^{*}_{w\gamma}\bar{e}_{w})\|\leq
d^{2m-1}\;(2m)!\;C^{3m-1}\;B^{7m-4}\;C_{q}^{3}\;(2m)^{2}\;[2m-1]_{q}!\;\sqrt{[m-1]_{q}!}.$

Gathering all the estimates above, we finally arrive at the following:

$\displaystyle\|\partial_{\beta}W(\xi_{m,\alpha})\|$
$\displaystyle\leq\sum_{\gamma\in[d]}\sum_{|w|=m-1}q^{\frac{m(m-1)}{2}}\|e_{\alpha}\|_{U}\;\|\bar{e}_{\gamma}\|_{U}\;\|\partial_{\beta}W({L}^{*}_{w\gamma}\bar{e}_{w})\|$
$\displaystyle\leq
q^{\frac{m(m-1)}{2}}\;d^{3m-1}\;(2m)!\;C^{3m-1}\;B^{7m-4}\;C_{q}^{3}\;(2m)^{2}\;[2m-1]_{q}!\;\sqrt{[m-1]_{q}!}$

therefore we conclude by using the ratio test that

$\displaystyle\sum_{m=1}^{\infty}\|\partial_{\beta}W(\xi_{m,\alpha})\|\leq\sum_{m=1}^{\infty}q^{\frac{m(m-1)}{2}}\;d^{3m-1}(2m)!C^{3m-1}B^{7m-4}C_{q}^{3}\;(2m)^{2}[2m-1]_{q}!\;\sqrt{[m-1]_{q}!}<\infty$

thus completing the proof. ∎

### Power Series Expansion of Conjugate Variables

The fact that the conjugate variables belong to $\mathsf{M}_{T}$ and are
Lipschitz can also be obtained by writing each $W(e_{w})$ (for a word $w$) as
a non-commutative polynomial in $\\{W(e_{1}),\ldots,W(e_{d})\\}$ and then by
estimating their norms. This approach avoids the use of Bożejko-Haagerup
inequality. The non-commutative power series thus obtained will have infinite
radius of convergence. We describe this method below by following the
techniques of [15]. Also see [25, Corllary 5.2].

For the purpose, we denote by $D(n)$ the collection of partitions of
$\\{1,\ldots,n\\}$ consisting of either singleton or pairs for $n\geq 1$. As
before, denote by $p(\pi)$ the set of pairs and by $s(\pi)$ the set of
singletons for any partition $\pi\in D(n)$. Also we set $\pi(u)=v$ if
$(u,v)\in p(\pi)$, and $\pi(u)=u$ if $u\in s(\pi)$.

######  Proposition 3.11.

For $n\geq 1$ and $\alpha_{1},\ldots,\alpha_{n}\in[d]$, we have

$\displaystyle W(e_{\alpha_{n}\cdots\alpha_{1}})=\sum_{\pi\in
D(n)}(-1)^{|p(\pi)|}Q(\pi)E(\pi)W^{s(\pi)}$

where

$\displaystyle Q(\pi)=\prod_{\underset{v\in
s(\pi)}{u<v<\pi(u)}}q(i_{u},i_{v})\cdot\prod_{u<v<\pi(v)<\pi(u)}q(i_{u},i_{v})q(i_{\pi(u)},i_{\pi(v)})\cdot\prod_{u<v<\pi(u)<\pi(v)}q(i_{u},i_{v})$
$\displaystyle E(\pi):=\prod_{\underset{{k>\ell}}{(k,\ell)}\in
p(\pi)}\langle\bar{e}_{\alpha_{k}},e_{\alpha_{\ell}}\rangle_{U}$
$\displaystyle W^{s(\pi)}:=W(e_{\alpha_{\ell_{s}}})\cdots
W(e_{\alpha_{\ell_{1}}})\;\mbox{ for }s(\pi)=\\{\ell_{s}>\ldots>\ell_{1}\\}.$

###### Proof.

Remark that the terms in $Q(\pi)$ correspond to the crossings on the line of
$v$ (and on $\pi(v)$ in the second product) as $v$ varies. Similar to Remark
3.3, the terms $q(i_{u},i_{v})$ in the product $Q(\pi)E(\pi)$ can be replaced
by $q(i_{\pi(u)},i_{\pi(v)}),q(i_{u},i_{\pi(v)})$ etc because $E(\pi)$ is non-
zero only if $i_{u}=i_{\pi(u)}$ for any pair $(u,\pi(u))$.

We sketch the proof below which goes by the induction on $n$. For $n=1$, the
formula is obvious. So assume the expression to be true for some $n\geq 1$.
Then calculate by using eq. (3.13) and by the induction argument:

$\displaystyle W(e_{\alpha_{n+1}\cdots{\alpha_{1}}})$
$\displaystyle=W(e_{\alpha_{n+1}})W(e_{\alpha_{n}\cdots{\alpha_{1}}})-\sum_{u=1}^{n}\prod_{u<t\leq
n}q(i_{u},i_{t})\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}W(e_{\alpha_{n}\cdots\hat{\alpha}_{u}\cdots\alpha_{1}})$
$\displaystyle=\sum_{\pi\in
D(n)}(-1)^{|p(\pi)|}Q(\pi)E(\pi)W(e_{\alpha_{n+1}})W^{s(\pi)}$
$\displaystyle-\sum_{u=1}^{n}\prod_{u<t\leq
n}q(i_{u},i_{t})\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle_{U}\sum_{\sigma\in
D(n-1)}(-1)^{|p(\sigma)|}Q(\sigma,u)E(\sigma,u)W^{s(\sigma,u)}$

where $Q(\sigma,u),E(\sigma,u)$ and $W^{s(\sigma,u)}$ denote
$Q(\sigma),E(\sigma)$ and $W^{s(\sigma)}$ respectively with emphasis that
$\sigma\in D(n-1)$ acts on the word
$\alpha_{n}\cdots\hat{\alpha}_{u}\cdots\alpha_{1}$. The two sums above give
nothing but the expression $\sum_{\tilde{\pi}\in
D(n+1)}(-1)^{|p(\tilde{\pi})|}Q(\tilde{\pi})E(\tilde{\pi})W^{s(\tilde{\pi})}$.
To see this, note that the first sum corresponds to those $\tilde{\pi}\in
D(n+1)$ where $n+1$ is a singleton. Whereas for the terms in the second sum,
for each $u\in\\{1,\ldots,n\\}$ and $\sigma\in D(n-1)$ acting on the word
$\alpha_{n}\cdots\hat{\alpha}_{u}\cdots\alpha_{1}$, one constructs
$\tilde{\pi}_{\sigma,u}\in D(n+1)$ by connecting $u$ to $n+1$ so that
$\tilde{\pi}_{\sigma,u}\setminus(n+1,u)=\sigma$. Then observe that
$|p(\tilde{\pi}_{\sigma,u})|=1+|p(\sigma)|$ and
$s(\tilde{\pi}_{\sigma,u})=s(\sigma,u)$ so that
$(-1)^{|p(\tilde{\pi}_{\sigma,u})|}=-(-1)^{|p(\sigma)|}$,
$W^{s(\tilde{\pi}_{\sigma,u})}=W^{s(\sigma,u)}$ and
$E(\tilde{\pi}_{\sigma,u})=\langle\bar{e}_{\alpha_{n+1}},e_{\alpha_{u}}\rangle
E(\sigma,u)$. Moreover, the extra crossings in $\tilde{\pi}_{\sigma,u}$ as
compared to $\sigma$ occur due to the pair $(n+1,u)$ with all the lines to its
left; hence we have $Q(\tilde{\pi}_{\sigma,u})=Q(\sigma,u)\prod_{u<t\leq
n}q(i_{u},i_{t})$. These observations complete the proof. ∎

######  Remark 3.12.

(1). For any $n\geq 1$ and $\alpha_{1},\ldots,\alpha_{n}\in[d]$, we observe
that

$|W(e_{\alpha_{n}\cdots\alpha_{1}})\|\leq n!B^{n}A^{n}$ (3.17)

where $B=\max_{\gamma\in[d]}\\{1,\|\bar{e}_{\gamma}\|_{U}\\}$ and
$A=\max_{\gamma\in[d]}\|W(e_{\gamma})\|\leq 2C_{q}^{3/2}$. This follows by
noting that $|D(n)|\leq n!$, and for any $\pi\in D(n)$, we have $|Q(\pi)|\leq
1$, $|E(\pi)|\leq B^{|p(\pi)|}\leq B^{n}$ and $\|W^{s(\pi)}\|\leq
A^{|s(\pi)|}\leq A^{n}$.

(2). By proposition 3.11, we get a concrete formula for each conjugate
variable $\xi_{\alpha}$ as a non-commutative power series in
$W(e_{1}),\ldots,W(e_{d})$. Moreover, we can show that the power series has
infinite radius of convergence and reprove Corollary 3.5 as follows: Indeed,
with the notations as in Corollary 3.5, for any $m\geq 1$ and $\alpha\in[d]$,
we have

$\displaystyle\|W(\xi_{m,\alpha})\|\leq\sum_{\beta\in[d]}\sum_{|w|=m-1}q^{\frac{m(m-1)}{2}}B\|W(L_{w\beta}^{*}\bar{e}_{w})\|.$

As in the proof of Corollary 3.10, write
$L_{w\beta}^{*}\bar{e}_{w}=\sum_{|w^{\prime}|=2m-1}\lambda_{w^{\prime}}e_{w^{\prime}}$
where the scalars $\lambda_{w^{\prime}}$ satisfy

$|\lambda_{w^{\prime}}|\leq C^{3m-1}B^{m-1}\sqrt{[m-1]_{q}!}$

for $C=\max_{\gamma\in[d]}\|l(e_{\gamma})\|$. By eq. (3.17) we get

$\|W(L_{w\beta}^{*}\bar{e}_{w})\|\leq\sum_{|w^{\prime}|=2m-1}|\lambda_{w^{\prime}}|\;\|W(e_{w}^{\prime})\|\leq
d^{2m-1}C^{3m-1}B^{m-1}\sqrt{[m-1]_{q}!}(2m-1)!B^{2m-1}A^{2m-1}$

which further yields

$\displaystyle\|W(\xi_{m,\alpha})\|\leq
d^{3m-1}q^{\frac{m(m-1)}{2}}C^{3m-1}B^{3m-1}\sqrt{[m-1]_{q}!}\;(2m-1)!\;A^{2m-1}$

and then ratio test says that
$\sum_{m=1}^{\infty}\|W(\xi_{m,\alpha})\|<\infty$. By a similar argument, one
can show that $\partial_{\beta}(W(\xi_{\alpha}))\in\mathsf{M}_{T}$ for all
$\beta,\alpha\in[d]$ and hence $\xi_{\alpha}$ is Lipschitz.

In terminologies of [17], the above results can be rephrased as follows. Let
$\varphi=\langle\Omega,\cdot\Omega\rangle_{T}$ be the canonical normal
faithful state on $\mathsf{M}_{T}$ and let $\sigma^{\varphi}$ be the
corresponding modular automorphism group. An element $x\in\mathsf{M}_{T}$ is
an eigenoperator (with respect to $\sigma^{\varphi}$) if
$\sigma^{\varphi}_{t}(x)=\lambda^{it}x$ $\forall t\in\mathbb{R}$ for some
$\lambda>0$. We choose the orthonormal set $\\{e_{1},\ldots,e_{d}\\}$ in
$\mathsf{H}_{U}$ at the beginning of this Section in such a way that each
$e_{k}\in\mathsf{H}_{\mathbb{C}}$ is an eigenvector for the analytic generator
$A$ of $U_{t}$, so that $W(e_{k})$ will be an eigenoperator for
$\sigma^{\varphi}$ because $\sigma_{t}(W(e_{k}))=W(A^{-it}e_{k})$ $\forall
t\in\mathbb{R}$, and that the set $\\{W(e_{1}),\ldots,W(e_{k})\\}$ is self-
adjoint (see [11, Proof of Theorem 2.2] for such construction). Since the
tuple $(W(e_{1}),\ldots,W(e_{d}))$ admits a system $(\xi_{1},\ldots,\xi_{d})$
of conjugate variables as shown in Theorem 3.4, it has finite free Fisher
information i.e.

$\displaystyle\sum_{k=1}^{d}\|\xi_{k}\|_{L^{2}(\mathsf{M}_{T})}^{2}<\infty,$

in the terminology of [17, Definition 3.6]. We summarise our discussions in
the following result:

######  Corollary 3.13 (see [14], Corollary 1.5).

Let
$(\mathsf{H}_{\mathbb{R}}=\oplus_{i\in{N}}\mathsf{H}_{\mathbb{R}}^{(i)},U_{t},(q_{ij})_{i,j\in{N}})$
be given with $\mathsf{H}_{\mathbb{R}}$ finite dimensional (and hence ${N}$ a
finite set), the von Neumann algebra $\mathsf{M}_{T}$ equipped with the
canonical state $\varphi$ is generated by a finite set $G=G^{*}$ of
eigenoperators of the modular group of automorphisms $\sigma_{t}^{\varphi}$
with finite free Fisher information.

##  4\. Factoriality and non-injectivity

In this section, we prove that the mixed $q$-Araki-Woods von Neumann algebra
$\mathsf{M}_{T}$ is a factor i.e. its center
$\mathsf{M}_{T}\cap\mathsf{M}_{T}^{\prime}=\mathbb{C}$, and that
$\mathsf{M}_{T}$ is non-injective. Recall that a von Neumann algebra
$M\subseteq\mathbf{B}(H)$ is _injective_ if there exists an idempotent from
$\mathbf{B}(H)$ onto $M$ of norm $1$. Moreover we also discuss the type
classifications of the algebras $\mathsf{M}_{T}$. Connes [9] defined the
$S$-invariant of a factor $M$ with a separable predual to be the intersection
of the spectra of the modular operators $\Delta_{\phi}$ of all normal faithful
states $\phi$. The factor $M$ is of type $\mathrm{III}$ if and only if $0\in
S(M)$. In this case, the type classification of $M$ is given as follows:

$\displaystyle
S(M)=\left\\{\begin{array}[]{ll}\lambda^{\mathbb{Z}}\cup\\{0\\}&\mbox{if
}M\mbox{ is of type }\mathrm{III}_{\lambda},0<\lambda<1,\\\
\left[0,\infty\right)&\mbox{if }M\mbox{ is of type }\mathrm{III}_{1},\\\
\\{0,1\\}&\mbox{if }M\mbox{ is of type }\mathrm{III}_{0}.\end{array}\right.$

Recall that the centralizer of $M$ with respect to a normal faithful state
$\phi$ is defined by $M^{\phi}=\\{x\in M;\phi(xy)=\phi(yx)\;\forall y\in
M\\}$. If the centralizer $M^{\phi}$ is a factor, then $S(M)$ is just the
spectrum of the modular operator $\Delta_{\phi}$ corresponding to $\phi$. We
now state the following abstract results of Nelson [17]:

######  Theorem 4.1 ([17], Theorem A, Theorem B).

Let $M$ be a von Neumann algebra with a faithful normal state $\phi$. Suppose
$M$ is generated by a finite set $G=G^{\ast}$, $|G|\geqslant 2$ of
eigenoperators of the modular group $\sigma^{\phi}$ with finite free Fisher
information. Then $(M^{\phi})^{\prime}\cap M=\mathbb{C}$. In particular,
$M^{\phi}$ is a $\mathrm{II}_{1}$ factor and if $H<\mathbb{R}^{\times}_{\ast}$
is the closed subgroup generated by the eigenvalues of $G$ then $M$ is a
factor of type

$\left\\{\begin{array}[]{ll}\mathrm{III}_{1}&\text{ if
}H=\mathbb{R}^{\times}_{\ast}\\\ \mathrm{III}_{\lambda}&\text{ if
}H=\lambda^{\mathbb{Z}},0<\lambda<1\\\ \mathrm{II}_{1}&\text{ if
}H=\\{1\\}.\end{array}\right.$

Furthermore $M^{\varphi}$ is full, and if $M$ is a type
$\mathrm{III}_{\lambda}$ factor, $0<\lambda<1$, then $M$ is full.

Recall that a factor $M$ is full if for any bounded sequence $x_{n}$ in $M$
with $\|\phi(\cdot x_{n})-\phi(x_{n}\cdot)\|\to 0$ for all normal states
$\phi$ on $M$, there is a bounded sequence $\lambda_{n}\in\mathbb{C}$ such
that $x_{n}-\lambda_{n}\to 0$ $*$\- strongly. Remark that a full factor is
necessarily non-injective.

We mention in the passing that if $U_{t}$ is a strongly continuous orthogonal
representations on a real Hilbert space $\mathsf{H}_{\mathbb{R}}$, then
$\mathsf{H}_{\mathbb{R}}$ decomposes into a direct sum
$\mathsf{K}_{\mathbb{R}}\oplus\mathsf{L}_{\mathbb{R}}$ of invariant subspaces
of $U_{t}$, where $\mathsf{K}_{U}$ is densely spanned by the eigenvectors of
$U_{t}$. We call the part
$(\mathsf{K}_{\mathbb{R}},{U_{t}}_{|_{\mathsf{K}_{\mathbb{R}}}})$ as the
almost periodic and the part
$(\mathsf{L}_{\mathbb{R}},{U_{t}}_{|_{\mathsf{L}_{\mathbb{R}}}})$ as weakly
mixing. Note that if $\mathsf{L}_{\mathbb{R}}$ is non-zero, then it must be
infinite-dimensional. Also the almost periodic part has a nice decomposition
as representations on a direct sum of either one or two dimensional real
spaces (see [21]). In particular if
$\mathsf{H}_{\mathbb{R}}=\oplus_{i\in{N}}\mathsf{H}_{\mathbb{R}}^{(i)}$ is
finite dimensional then each $\mathsf{H}_{\mathbb{R}}^{(i)}$ is almost
periodic, hence decomposes as a direct sum of one or two dimensional invariant
subspaces. We are now ready to give our main results. As before, we denote
below the canonical normal faithful state
$\langle\Omega,\cdot\Omega\rangle_{T}$ on $\mathsf{M}$ by $\varphi$.

######  Theorem 4.2.

Let
$(\mathsf{H}_{\mathbb{R}}=\oplus_{i\in{N}}\mathsf{H}_{\mathbb{R}}^{(i)},U_{t},(q_{ij})_{i,j\in{N}})$
be given for some countable set $N$, with
$\dim(\mathsf{H}_{\mathbb{R}})\geqslant 2$. Then
$\mathsf{M}:=\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ is a factor. Moreover,
if $G<\mathbb{R}^{\times}_{\ast}$ is the closed subgroup generated by the
spectrum of $A$ then $\mathsf{M}$ is a factor of type

$\left\\{\begin{array}[]{ll}\mathrm{III}_{1}&\text{ if
}G=\mathbb{R}^{\times}_{\ast}\\\ \mathrm{III}_{\lambda}&\text{ if
}G=\lambda^{\mathbb{Z}},0<\lambda<1\\\ \mathrm{II}_{1}&\text{ if
}G=\\{1\\}.\end{array}\right.$

###### Proof.

If $\dim\mathsf{H}_{\mathbb{R}}<\infty$ then the factoriality and type
classification of $\mathsf{M}$ follow from Corollary 3.13 and Theorem 4.1.
Therefore assume that $\dim\mathsf{H}_{\mathbb{R}}=\infty$. If the weakly
mixing part of $U_{t}$ is non-trivial, then $\mathsf{M}$ is a type III1 factor
by [5, Theorem 5.9].

So we may assume that $U_{t}$ is almost periodic on an infinite dimensional
space $\mathsf{H}_{\mathbb{R}}$. We use an inductive limit argument to prove
factoriality and type classification; this may be apparent to experts but we
provide details for completeness. As mentioned above, one can find an
increasing sequence $\mathsf{H}_{\mathbb{R}}(n)$ of $U_{t}$-invariant finite-
dimensional subspaces of $\mathsf{H}_{\mathbb{R}}$ such that each
$\mathsf{H}_{\mathbb{R}}(n)$ is compatible with the direct sum i.e.
$\mathsf{H}_{\mathbb{R}}(n)=\oplus_{i\in{N}}(\mathsf{H}_{\mathbb{R}}(n)\cap\mathsf{H}_{\mathbb{R}}^{(i)})$,
and that their union spans a dense subspace of $\mathsf{H}_{\mathbb{R}}$.
Denote the von Neumann algebra
$\Gamma_{T}(\mathsf{H}_{\mathbb{R}}(n),{U_{t}}_{|_{\mathsf{H}_{\mathbb{R}}(n)}})$
by $\mathsf{M}_{n}$ and the restriction of the state $\varphi$ to
$\mathsf{M}_{n}$ by $\varphi_{n}$. Through our assumption, we have implication
that the unital $\ast$-algebra $\cup_{n\geq 1}\mathsf{M}_{n}$ is strongly
dense in $\mathsf{M}$. An appeal to [5, Proposition 6.1] entail that each
$\mathsf{M}_{n}$ is naturally embedded in $\mathsf{M}$ as a subalgebra and
that there is a $\varphi$-preserving conditional expectation $E_{n}$ from
$\mathsf{M}$ onto $\mathsf{M}_{n}$, so that $\mathsf{M}_{n}$ is invariant
under the modular automorphism group $\sigma_{t}^{\varphi}$ of the state
$\varphi$. Observe that $E_{n}(x)\to x$ strongly for all $x\in\mathsf{M}$. One
easily verifies that
${\mathsf{M}_{n}^{\varphi_{n}}}\subseteq\mathsf{M}^{\varphi}$.

We now show that $\mathsf{M}^{\varphi}$ is a factor. Indeed if
$x\in({\mathsf{M}^{\varphi}})^{\prime}\cap\mathsf{M}$, then
$E_{n}(x)\in({\mathsf{M}_{n}^{\varphi_{n}}})^{\prime}\cap\mathsf{M}_{n}$ for
all $n\geq 1$; but we know from Corollary 3.13 and Theorem 4.1 that
$({\mathsf{M}_{n}^{\varphi_{n}}})^{\prime}\cap\mathsf{M}_{n}=\mathbb{C}$
(since $\dim\mathsf{H}_{\mathbb{R}}(n)<\infty$), therefore
$E_{n}(x)=\varphi_{n}(E_{n}(x))=\varphi(x)$ for all $n\geq 1$ which finally
implies that $x=\varphi(x)$. We have thus shown that
$({\mathsf{M}^{\varphi}})^{\prime}\cap\mathsf{M}=\mathbb{C}$ so that
$\mathsf{M}^{\varphi}$ is a factor; hence $\mathsf{M}$ is a factor. Since
$\mathsf{M}^{\varphi}$ is a factor, the type classification of $\mathsf{M}$
follows from the spectral data of the modular operator $\Delta_{\varphi}$ of
the state $\varphi$, which is related to the analytic generator $A$ as given
in [5, Theorem 3.13]. ∎

We now show that the factor $\mathsf{M}_{T}$ is non-injective with the same
data as in Theorem 4.2.

######  Theorem 4.3.

The factor $\mathsf{M}=\Gamma_{T}(\mathsf{H}_{\mathbb{R}},U_{t})$ is not
injective as soon as $\dim(H_{\mathbb{R}})\geqslant 2$.

###### Proof.

If $U_{t}$ has non-trivial weakly mixing part, then $\mathsf{M}$ is non-
injective by [5, Theorem 8.4]. So we may assume that $U_{t}$ is almost
periodic. As in the proof of [14, Theorem 3.5], we will find a non-injective
subalgebra with a conditional expectation from the ambient factor
$\mathsf{M}$. So let $\mathsf{K}_{\mathbb{R}}$ be a two-dimensional invariant
subspace of $U_{t}$ compatible with the direct sum decomposition i.e.
$\mathsf{K}_{\mathbb{R}}=\oplus_{i\in{N}}(\mathsf{K}_{\mathbb{R}}\cap\mathsf{H}_{\mathbb{R}}^{(i)})$.
As in the proof of Theorem 4.2 above, there is a conditional expectation from
$\mathsf{M}$ to the subalgebra
$\mathsf{M}_{1}:=\Gamma_{T}(\mathsf{K}_{\mathbb{R}},{U_{t}}_{|_{\mathsf{K}_{\mathbb{R}}}})$.
Now if ${U_{t}}_{|_{\mathsf{K}_{\mathbb{R}}}}$ is trivial then
$\mathsf{M}_{1}=\mathsf{M}_{1}^{\varphi_{|_{\mathsf{M}_{1}}}}$ which in
conjugation with Corollary 3.13 and Theorem 4.1 yields that $\mathsf{M}_{1}$
is full and hence $\mathsf{M}_{1}$ is non-injective. On the other hand if
${U_{t}}_{|_{\mathsf{K}_{\mathbb{R}}}}$ is ergodic then by Theorem 4.2,
$\mathsf{M}_{1}$ is a type IIIλ factor for some $0<\lambda<1$; the same
combination of Corollary 3.13 and Theorem 4.1 then concludes that
$\mathsf{M}_{1}$ is full and hence non-injective. The proof is now complete. ∎

Acknowledgments. The author is grateful to Adam Skalski and Mateusz Wasilewski
for carefully reading the initial draft and giving valuable comments. He also
thanks the anonymous referee for carefully reading the manuscript and
providing valuable suggestions. The author is partially supported by the
National Science Center (NCN) grant no. 2020/39/I/ST1/01566.

## References

  * [1] M. Bożejko, B. Kümmerer, and R. Speicher, $q$-Gaussian processes: noncommutative and classical aspects, _Comm. Math. Phys._ 185 (1997), no. 1, 129–154.
  * [2] M. Bożejko and R. Speicher, An example of a generalized Brownian motion, Comm. Math. Phys. 137 (1991), no. 3, 519–531.
  * [3] M. Bożejko and R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, _Math. Ann._ 300 (1994), 97–120.
  * [4] M. Bożejko, Completely positive maps on Coxeter groups and the ultracontractivity of the $q$-Ornstein-Uhlenbeck semigroup, _Banach Center Publ._ 43 (1998), 87–93.
  * [5] P. Bikram, R. Kumar R., and K. Mukherjee, Mixed $q$-deformed Araki-Woods von Neumann algebras, _J. Noncommut. Geom._ , to appear.
  * [6] P. Bikram, R. Kumar R., and K. Mukherjee, On non-injectivity of mixed $q$-deformed Araki-Woods von Neumann algebras, _Kyoto J. Math._ , to appear.
  * [7] P. Bikram and K. Mukherjee, Generator masas in $q$-deformed Araki-Woods von Neumann algebras and factoriality, _J. Funct. Anal._ 273 (2017), no. 4, 1443–1478.
  * [8] P. Bikram, K. Mukherjee, É. Ricard and S. Wang, On the factoriality of $q$-deformed Araki-Woods von Neumann algebras, _Comm. Math. Phys._ 398 (2023), no. 2, 797–821.
  * [9] A. Connes, Une classification des facteurs de type III, Ann. Scient. Ec. Norm. Sup. 6 (1973), 133-252.
  * [10] Y. Dabrowski and A. Ioana, Unbounded derivations, free dilations, and indecomposability results for II1 factors, Trans. Amer. Math. Soc. 368 (2016), no. 7, 4525-4560.
  * [11] F. Hiai, $q$-deformed Araki-Woods algebras, _in_ “Operator algebras and mathematical physics (Constanta, 2001),” Theta, Bucharest, 2003, pp. 169–202.
  * [12] I. Królak, Factoriality of von Neumann algebras connected with general commutation relations – finite dimensional case, in “Quantum probability”, _Banach Center Publ._ 73 (2006), pp. 277–284.
  * [13] I. Królak, Wick product for commutation relations connected with Yang-Baxter operators and new constructions of factors, Comm. Math. Phys. 210 (2000), no. 3, 685–701.
  * [14] M. Kumar, A. Skalski and M. Wasilewski, Full solution of the factoriality question for q-Araki-Woods von Neumann algebras via conjugate variables, Comm. Math. Phys. 402 (2023), 157–167.
  * [15] A. Miyagawa and R. Speicher, A dual and conjugate system for $q$-Gaussians for all $q$, _Adv. Math._ 413 (2023), 108834.
  * [16] B. Nelson, Free monotone transport without a trace, _Comm. Math. Phys._ 334 (2015), no. 3, 1245–1298.
  * [17] B. Nelson, On finite free Fisher information for eigenvectors of a modular operator, _J. Funct. Anal._ 273 (2017), no. 7, 2292–2352.
  * [18] B. Nelson and Q. Zeng, An application of free transport to mixed q-Gaussian algebras, Proc. Amer. Math. Soc. 144 (2016), no. 10, 4357–4366,
  * [19] A. Nou, Non injectivity of the $q$-deformed von Neumann algebra, _Math. Ann._ 330 (2004), no. 1, 17–38.
  * [20] É. Ricard, Factoriality of $q$-Gaussian von Neumann algebras, _Comm. Math. Phys._ 257 (2005), no. 3, 659–665.
  * [21] D. Shlyakhtenko, Free quasi-free states, _Pacific J. Math._ 177 (2) (1997), 329–368.
  * [22] R. Speicher, Generalized statistics of macroscopic fields, _Lett. Math. Phys._ 27 (1993), no. 2, 97–104.
  * [23] A. Skalski and S. Wang, Remarks on factoriality and $q$-deformations, _Proc. Amer. Math. Soc._ 146 (2018), no. 9, 3813–3823.
  * [24] P. Śniady, Factoriality of Bożejko-Speicher von Neumann algebras, _Comm. Math. Phys._ 246 (2004), no. 3, 561–567.
  * [25] Z. Yang, A conjugate system for twisted Araki-Woods von Neumann algebras of finite dimensional spaces, _preprint_ , available on arXiv:2304.13856.