# On the factoriality of $q$-deformed Araki-Woods von Neumann algebras

Panchugopal Bikram  ,  Kunal Mukherjee  ,  Éric Ricard  and  Simeng Wang
School of Mathematical Sciences, National Institute of Science Education and
Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, Jatni-
752050, India  Department of Mathematics, IIT Madras, Chennai - 600036, India.
Normandie Univ, UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme,
14000 Caen, France  Institute for Advanced Study in Mathematics, Harbin
Institute of Technology, Harbin, 150001, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The $q$-deformed Araki-Woods von Neumann algebras
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ are factors for
all $q\in(-1,1)$ whenever $dim(\mathcal{H}_{\mathbb{R}})\geqslant 3$. When
$dim(\mathcal{H}_{\mathbb{R}})=2$ they are factors as well for all $q$ so long
as the parameter defining $(U_{t})$ is ‘small’ or $1$ $($trivial$)$ as the
case may be.

###### Key words and phrases:

$q$-commutation relations, von Neumann algebras

###### 2010 Mathematics Subject Classification:

Primary 46L10; Secondary 46L65, 46L55.

##  1\. Introduction

To any strongly continuous orthogonal representation $(U_{t})$ of $\mathbb{R}$
on a real Hilbert space $\mathcal{H}_{\mathbb{R}}$ with
$dim(\mathcal{H}_{\mathbb{R}})\geqslant 2$, Hiai in [Hi03] constructed the
$q$-deformed Araki-Woods von Neumann algebras $($hereafter abbreviated as
$q$-Araki-Woods algebras$)$ for $-1<q<1$. These are $W^{\ast}$-algebras
usually arising from non-tracial representations of the $q$-commutation
relations (a Yang-Baxter deformation of the canonical commutation relations),
thereby yielding an interpolation between the Bosonic and Fermionic
statistics. Hiai’s functor is a fusion of Shlyakhtenko’s free CAR functor
[S97] $($associated $W^{*}$-algebras are called free Araki-Woods factors$)$
and the $q$-Gaussian functor of Bo$\overset{.}{\text{z}}$ejko-Speicher
$($associated $W^{*}$-algebras are called Bo$\overset{.}{\text{z}}$ejko-
Speicher factors$)$ $($see [BS91]$)$. All of these constructions are
generalizations of Voiculescu’s $C^{*}$-free Gaussian functor, which is the
central object of study in free probability [VDN92]. Note that when $q=0$,
i.e. the Yang-Baxter deformation is trivial $($free case$)$, Hiai’s functor
reduces to Shlyakhtenko’s functor. However, when $(U_{t})$ is trivial, Hiai’s
functor reduces to the $q$-Gaussian functor. Further, when $q=0$ and $(U_{t})$
is trivial, one obtains Voiculescu’s functor.

The $q$-Araki-Woods algebras are quite complicated objects. Structural
properties of the Bo$\overset{.}{\text{z}}$ejko-Speicher factors have been
studied in [A11, BS91, BKS97, D14, R05, N04, Sn04, S04, S09], and those of the
free Araki-Woods factors have been studied in [H09, HR11, BHV15]; though, this
list is by no means complete. On the contrary, very little is known about the
$q$-Araki-Woods algebras $($see [N04, N06, W21]$)$. In fact, the simplest
question regarding its factoriality is not known in full generality. This is
because, unlike the case when $q=0$, there is very little room to perform
meaningful calculations with the standard generators of these algebras owing
to the complicated nature of the scalar product $($of the GNS space$)$ and the
interference of the modular group. The case when
$dim(\mathcal{H}_{\mathbb{R}})=2$ is the hardest and most notorious.

This paper attempts the factoriality problem of the $q$-Araki-Woods algebras.
Before describing our results, though, we note the past efforts in this
direction.

The factoriality of the Bo$\overset{.}{\text{z}}$ejko-Speicher algebras was
not a single-handed attempt. In [BKS97], the factoriality was established by
Bo$\overset{.}{\text{z}}$ejko, Kümmerer and Speicher when
$\dim(\mathcal{H}_{\mathbb{R}})$ is infinite. By making careful estimates of
norms of certain operators on the $q$-deformed full Fock space, Śniady
established the factoriality when $dim(\mathcal{H}_{\mathbb{R}})$ is finite
but greater than a constant depending on $q$. It was finally settled in [R05]
by showing that any standard generator of the Bo$\overset{.}{\text{z}}$ejko-
Speicher algebra generates a strongly mixing $($see [CFM] for defn.$)$ MASA in
the ambient algebra. Therefore, the center of the algebra gets arrested in two
orthogonal $($with respect to the vacuum state$)$ MASAs and is thus reduced to
scalars. By using freeness and modular data, Shlyakhtenko established the
factoriality of the free Araki-Woods algebras in [S97].

Hiai established the factoriality of the $q$-Araki-Woods algebras in [Hi03] in
the case when the dimension of the almost periodic part of $(U_{t})$ is
infinite or $(U_{t})$ is weakly mixing, by showing that the centralizer of the
vacuum state has trivial relative commutant $($irreducible inclusion$)$.
Unfortunately, there is a gap in the proof of [Hi03, Thm. 3.2]. To be precise,
Hiai’s proof holds only in the case when the set of eigenvalues of the
analytic generator of $(U_{t})$ has a limit point in $\mathbb{R}$ other than
$0$. Without this assumption, the conclusion ‘$\varphi(y^{*}x)=0$’ in the last
equation in [Hi03, Thm. 3.2] would fail, and hence, the final statement cannot
be concluded. Using the theory of free monotone transport $($see [Ne15, Thm.
4.5]$)$, Nelson proved that when $\mathcal{H}_{\mathbb{R}}$ is finite-
dimensional, the $q$-Araki-Woods algebras are isomorphic to the free Araki-
Woods factors for sufficiently small values of $\left|q\right|$; in
particular, they are factors, in this case.

[BM] closely followed [R05]. In [BM], the authors observe that a non-zero
vector $\xi\in\mathcal{H}_{\mathbb{R}}$ fixed by $(U_{t})$ enables the
construction of an orthonormal basis of analytic vectors $($of the GNS
space$)$ which behaves well as long as one only considers its interaction with
elements of the algebra $vN(s_{q}(\xi))$, where $s_{q}(\xi)$ is the standard
self-adjoint generator of the $q$-Araki-Woods algebra corresponding to $\xi$.
This allows exploiting the ideas in [R05] to show that $vN(s_{q}(\xi))$ is a
strongly mixing MASA living inside the centralizer of the vacuum state. Using
this, the factoriality of the $q$-Araki-Woods algebras was shown to be true if
$(U_{t})$ is not ergodic or has a non-zero weakly mixing component.
Irreducibility of the centralizer was also obtained when $(U_{t})$ is not
ergodic, and the almost periodic component is at least two-dimensional. At the
same time, a proof of the same statement on factoriality was obtained
independently by Skalski and Wang [SW]; note that the phenomenon of mixing is
implicit in this proof too.

When $(U_{t})$ is ergodic and almost periodic, the above ideas involving MASAs
freeze. This is because the standard self-adjoint generators no longer
generate abelian von Neumann subalgebras with appropriate conditional
expectations or operator-valued weights. In fact, it is far worse that these
algebras are quasi split [BM2] and therefore admit very large relative
commutants.

Given the previous attempts, we are left to deal with the case when
$\mathcal{H}_{\mathbb{R}}=\mathbb{R}^{2}\oplus\mathcal{K}_{\mathbb{R}}$, where
$\mathcal{K}_{\mathbb{R}}$ is a real Hilbert space $($could be $0)$, and
$\mathbb{R}^{2}$ is reducing subspace for $(U_{t})$ with associated sub
representation being ergodic. As discussed before, we cannot rely on MASAs
anymore, but our strategy is to leverage with a bit of ‘mixing’. The analysis
is split into two cases.

When $\mathcal{K}_{\mathbb{R}}=0$, denote $\mathcal{A}$ to be the algebra
generated by the centralizer and the commutant of the ambient algebra. The
novelty of this approach is to identify a suitable subspace inside the GNS
space of the centralizer which possesses ‘sufficient mixing’ $($the associated
subspace in [BM, R05] was the GNS space of a MASA$)$, and use an appropriate
orthonormal basis of that subspace to track a vector $\xi$, so that the cyclic
subspace $\mathcal{A}\xi$ fully captures the ‘size’ of the relative commutant
of the centralizer. However, there is a payoff. Since we are working with a
‘mixing subspace’, we lose algebraic techniques (most importantly cannot
locate unitaries) and depend on norm estimates of operators involving creation
and annihilation operators. This forces us a bargain with the characteristic
parameter that defines the two-dimensional representation, but our results
remain valid for all $q\neq 0$.

When $\mathcal{K}_{\mathbb{R}}\neq 0$, a similar idea with slight modification
works. This time, the increment in dimension allows us to choose unitaries
from an orthogonal subalgebra and frees us from any bargain with the
characteristic parameter as was in the previous case, and the conclusions are
no longer subject to any constraints.

The key to the factoriality is Lemma 3.12 in which ‘some mixing’ is analyzed
to control the relative commutant of the centralizer. The main results of this
paper are summarized as follows:

Theorem: Let
$\mathcal{H}_{\mathbb{R}}=\mathbb{R}^{2}\oplus\mathcal{K}_{\mathbb{R}}$, where
$\mathcal{K}_{\mathbb{R}}$ is a real Hilbert space $($could be $0)$, and let
$\mathbb{R}^{2}$ be reducing subspace for $(U_{t})$ with associated sub
representation being ergodic. Let $\lambda\in(0,1)$ be the parameter that
defines $(U_{t})$ on $\mathbb{R}^{2}$. Then, the following holds:

  1. (1)

if $dim(\mathcal{H}_{\mathbb{R}})=2$ and $\lambda$ is small (depending on
$q$), then the centralizer of the $q$-quasi free $($vacuum$)$ state is
irreducible if $q\neq 0$;

  2. (2)

if $dim(\mathcal{H}_{\mathbb{R}})\geqslant 3$, the $q$-Araki-Woods algebras
are factors for all $-1<q<1$;

  3. (3)

if the almost periodic part of $(U_{t})$ is sufficiently large, then the
centralizer of the $q$-quasi free state is irreducible for all $-1<q<1$.

This paper is organized as follows. In §2, we lay out all the technical
prerequisites that are needed to address the problem. The technical lemmas
that will be used to deal with factoriality is divided into two groups under
§3; the case when $dim(\mathcal{H}_{\mathbb{R}})=2$ and $(U_{t})$ is ergodic
appears in §3.1 and that for all cases
$(dim(\mathcal{H}_{\mathbb{R}})\geqslant 2)$ appears in §3.2. The main
theorems on factoriality appear in §4. Irreducibility of the centralizer is
discussed in §5.

##  2\. Preliminaries

In this section, we accumulate some well known facts about $q$-deformed Araki-
Woods von Neumann algebras constructed by Hiai in [Hi03] that will be
indispensable for our purpose. As a convention, all Hilbert spaces in this
paper are separable, all von Neumann algebras have separable preduals,
inclusions of von Neumann algebras are unital and inner products are linear in
the _second variable_. This section has overlap with [BM, §2].

###  2.1. Hiai’s Construction

Let $\mathcal{H}_{\mathbb{R}}$ be a real Hilbert space with
$\dim(\mathcal{H}_{\mathbb{R}})\geqslant 2$ and let $(U_{t})_{t\in\mathbb{R}}$
be a strongly continuous orthogonal representation of $\mathbb{R}$ on
$\mathcal{H}_{\mathbb{R}}$. Let
$\mathcal{H}_{\mathbb{C}}=\mathcal{H}_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}$
denote the complexification of $\mathcal{H}_{\mathbb{R}}$. Denote the inner
product and norm on $\mathcal{H}_{\mathbb{C}}$ by
$\langle\cdot,\cdot\rangle_{\mathcal{H}_{\mathbb{C}}}$ and
$\left\|\cdot\right\|_{\mathcal{H}_{\mathbb{C}}}$ respectively. Identify
$\mathcal{H}_{\mathbb{R}}$ in $\mathcal{H}_{\mathbb{C}}$ by
$\mathcal{H}_{\mathbb{R}}\otimes 1$. Thus,
$\mathcal{H}_{\mathbb{C}}=\mathcal{H}_{\mathbb{R}}+i\mathcal{H}_{\mathbb{R}}$,
and as a real Hilbert space the inner product of $\mathcal{H}_{\mathbb{R}}$ in
$\mathcal{H}_{\mathbb{C}}$ is given by
$\Re\langle\cdot,\cdot\rangle_{\mathcal{H}_{\mathbb{C}}}$. Consider the
bounded anti-linear $($complex $($left$)$ conjugation$)$ operator
$\mathcal{J}:\mathcal{H}_{\mathbb{C}}\rightarrow\mathcal{H}_{\mathbb{C}}$
given by $\mathcal{J}(\xi+i\eta)=\xi-i\eta$,
$\xi,\eta\in\mathcal{H}_{\mathbb{R}}$, and note that $\mathcal{J}\xi=\xi$ for
$\xi\in\mathcal{H}_{\mathbb{R}}$. Moreover,

$\displaystyle\langle\xi,\eta\rangle_{\mathcal{H}_{\mathbb{C}}}=\overline{\langle\eta,\xi\rangle}_{\mathcal{H}_{\mathbb{C}}}=\langle\eta,\mathcal{J}\xi\rangle_{\mathcal{H}_{\mathbb{C}}},\text{
for all }\xi\in\mathcal{H}_{\mathbb{C}},\eta\in\mathcal{H}_{\mathbb{R}}.$

By abuse of notation we denote the linear extension of $(U_{t})$ on
$\mathcal{H}_{\mathbb{C}}$ by the same notation, which is again a strongly
continuous one-parameter group of unitaries in $\mathcal{H}_{\mathbb{C}}$. Let
$A$ denote the analytic generator of $(U_{t})$. Then $A$ is positive,
nonsingular and self-adjoint. Note that $\mathcal{J}A=A^{-1}\mathcal{J}$.

Introduce a new inner product on $\mathcal{H}_{\mathbb{C}}$ by
$\langle\xi,\eta\rangle_{U}=\langle\frac{2}{1+A^{-1}}\xi,\eta\rangle_{\mathcal{H}_{\mathbb{C}}}$,
$\xi,\eta\in\mathcal{H}_{\mathbb{C}}$, and let $\left\|\cdot\right\|_{U}$
denote the associated norm on $\mathcal{H}_{\mathbb{C}}$. Let $\mathcal{H}$
denote the complex Hilbert space obtained by completing
$(\mathcal{H}_{\mathbb{C}},\left\|\cdot\right\|_{U})$. The inner product and
norm of $\mathcal{H}$ will respectively be denoted by
$\langle\cdot,\cdot\rangle_{U}$ and $\left\|\cdot\right\|_{U}$ as well. Then,
$(\mathcal{H}_{\mathbb{R}},\left\|\cdot\right\|_{\mathcal{H}_{\mathbb{C}}})\ni\xi\overset{\imath}{\mapsto}\xi\in(\mathcal{H}_{\mathbb{C}},\left\|\cdot\right\|_{U})\subseteq(\mathcal{H},\left\|\cdot\right\|_{U})$,
is an isometric embedding of the real Hilbert space $\mathcal{H}_{\mathbb{R}}$
in $\mathcal{H}$ $($in the sense of [S97]$)$. With abuse of notation, we will
identify $\mathcal{H}_{\mathbb{R}}$ with its image
$\iota(\mathcal{H}_{\mathbb{R}})$. Then, $\mathcal{H}_{\mathbb{R}}\cap
i\mathcal{H}_{\mathbb{R}}=\\{0\\}$ and
$\mathcal{H}_{\mathbb{R}}+i\mathcal{H}_{\mathbb{R}}$ is dense in $\mathcal{H}$
$($see pp. 332 [S97]$)$.

Note that $(U_{t})$ extends to a strongly continuous unitary representation
$(\widetilde{U}_{t})$ of $\mathbb{R}$ on $\mathcal{H}$. Let $\widetilde{A}$ be
the analytic generator associated to $(\widetilde{U}_{t})$, which is obviously
an extension of $A$. Any eigenvector of $\widetilde{A}$ is an eigenvector of
$A$ corresponding to the same eigenvalue [BM, Prop. 2.1]. Since the spectral
data of $A$ and $\widetilde{A}$ $($and hence of $(U_{t})$ and
$(\widetilde{U}_{t}))$ are essentially the same $($see [BM, §2] for
details$)$, and $\widetilde{U}_{t},\widetilde{A}$ are respectively extensions
of $U_{t},A$ for all $t\in\mathbb{R}$, so we would now write $\widetilde{A}=A$
and $\widetilde{U}_{t}=U_{t}$ for all $t\in\mathbb{R}$.

For $q\in(-1,1)$ and for the Hilbert space $\mathcal{H}$, consider the
associated $q$-Fock space $\mathcal{F}_{q}(\mathcal{H})$ introduced in [BS91].
$\mathcal{F}_{q}(\mathcal{H})$ is constructed as follows. Let $\Omega$ be a
distinguished unit vector in $\mathbb{C}$ usually referred to as the vacuum
vector. Denote $\mathcal{H}^{\otimes 0}=\mathbb{C}\Omega$, and, for
$n\geqslant 1$, let $\mathcal{H}^{\otimes n}=\text{
span}_{\mathbb{C}}\\{\xi_{1}\otimes\cdots\otimes\xi_{n}:\xi_{i}\in\mathcal{H}\text{
for }1\leqslant i\leqslant n\\}$ denote the algebraic tensor products. Let
$\mathcal{F}_{fin}(\mathcal{H})=\text{
span}_{\mathbb{C}}\\{\mathcal{H}^{\otimes n}:n\geqslant 0\\}$. For
$n,m\geqslant 0$ and
$f=\xi_{1}\otimes\cdots\otimes\xi_{n}\in\mathcal{H}^{\otimes n}$,
$g=\zeta_{1}\otimes\cdots\otimes\zeta_{m}\in\mathcal{H}^{\otimes m}$, the
association

$\displaystyle\langle f,g\rangle_{q}=\delta_{m,n}\sum_{\pi\in
S_{n}}q^{i(\pi)}\langle\xi_{1},\zeta_{\pi(1)}\rangle_{U}\cdots\langle\xi_{n},\zeta_{\pi(n)}\rangle_{U},$
(1)

where $i(\pi)$ denotes the number of inversions of the permutation $\pi\in
S_{n}$, defines a positive definite sesquilinear form on
$\mathcal{F}_{fin}(\mathcal{H})$ and the $q$-Fock space
$\mathcal{F}_{q}(\mathcal{H})$ is the completion of
$\mathcal{F}_{fin}(\mathcal{H})$ with respect to the norm
$\left\|\cdot\right\|_{q}$ induced by $\langle\cdot,\cdot\rangle_{q}$. Denote
$\mathcal{H}^{\otimes_{q}n}:=\overline{\mathcal{H}^{\otimes
n}}^{\left\|\cdot\right\|_{q}}$, $n\geqslant 0$. Note that
$\left\|\cdot\right\|_{q}=\left\|\cdot\right\|_{U}$ on
$\mathcal{H}^{\otimes_{q}1}=\mathcal{H}$.

For $\xi\in\mathcal{H}$, the left $q$-creation and $q$-annihilation operators
on $\mathcal{F}_{q}(\mathcal{H})$ are respectively defined by:

$\displaystyle c_{q}(\xi)\Omega=\xi,$ (2) $\displaystyle
c_{q}(\xi)(\xi_{1}\otimes\cdots\otimes\xi_{n})=\xi\otimes\xi_{1}\otimes\cdots\otimes\xi_{n},$
$\displaystyle\text{and},$ $\displaystyle c_{q}(\xi)^{*}\Omega=0,$
$\displaystyle
c_{q}(\xi)^{*}(\xi_{1}\otimes\cdots\otimes\xi_{n})=\sum_{i=1}^{n}{q^{i-1}}\langle\xi,\xi_{i}\rangle_{U}\xi_{1}\otimes\cdots\otimes\xi_{i-1}\otimes\xi_{i+1}\otimes\cdots\otimes\xi_{n},$

where $\xi_{1}\otimes\cdots\otimes\xi_{n}\in\mathcal{H}^{\otimes_{q}n}$ for
$n\geqslant 1$. The operators $c_{q}(\xi)$ and $c_{q}(\xi)^{*}$ are bounded on
$\mathcal{F}_{q}(\mathcal{H})$ and they are adjoints of each other. Moreover,

$\left\|c_{q}(\xi)\right\|=\begin{cases}\frac{1}{\sqrt{1-q}}\left\|\xi\right\|_{U},\quad&\text{
if }0\leqslant q<1;\\\ \left\|\xi\right\|_{U},\quad&\text{ if }-1<q\leqslant
0.\end{cases}$ (3)

Moreover, they satisfy the following $q$-commutation relation:

$\displaystyle
c_{q}(\xi)^{*}c_{q}(\zeta)-qc_{q}(\zeta)c_{q}(\xi)^{*}=\langle\xi,\zeta\rangle_{U}1,\text{
for all }\xi,\zeta\in\mathcal{H}.$ (4)

The following Lemma from [BM] will be crucial for our purpose.

######  Lemma 2.1.

[BM, Lemma 2.3] Let $\xi,\xi_{i},\eta_{j}\in\mathcal{H}$, for $1\leqslant
i\leqslant n$, $1\leqslant j\leqslant m$. Then,

$\displaystyle c_{q}$
$\displaystyle(\xi)^{*}\bigg{(}(\xi_{1}\otimes\cdots\otimes\xi_{n})\otimes(\eta_{1}\otimes\cdots\otimes\eta_{m})\bigg{)}$
$\displaystyle=\bigg{(}c_{q}(\xi)^{*}(\xi_{1}\otimes\cdots\otimes\xi_{n})\bigg{)}\otimes(\eta_{1}\otimes\cdots\otimes\eta_{m})$
$\displaystyle\indent\indent\indent\indent\indent+q^{n}(\xi_{1}\otimes\cdots\otimes\xi_{n})\otimes\bigg{(}c_{q}(\xi)^{*}(\eta_{1}\otimes\cdots\otimes\eta_{m})\bigg{)}.$

Following [Hi03, S97], consider the $C^{*}$-algebra
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t}):=C^{*}\\{s_{q}(\xi):\xi\in\mathcal{H}_{\mathbb{R}}\\}$
and the von Neumann algebra
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$, where

$\displaystyle s_{q}(\xi)=c_{q}(\xi)+c_{q}(\xi)^{*},\text{
}\xi\in\mathcal{H}_{\mathbb{R}}.$

$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ is known as the
$q$-deformed Araki-Woods von Neumann algebra (see [Hi03, §3]). The vacuum
state $\varphi_{q,U}:=\langle\Omega,\cdot\text{ }\Omega\rangle_{q}$ $($also
called the $q$-quasi free state$)$, is a faithful normal state of
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ and
$\mathcal{F}_{q}(\mathcal{H})$ is the GNS Hilbert space of
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ associated to
$\varphi_{q,U}$. Thus,
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ acting on
$\mathcal{F}_{q}(\mathcal{H})$ is in standard form [H75].

Making slight violation of the traditional notations, we will use the symbols
$\langle\cdot,\cdot\rangle_{q}$ and $\left\|\cdot\right\|_{q}$ respectively to
denote the inner product and two-norm of elements of the GNS Hilbert space.

###  2.2. Modular Theory

Most of what follows in §2.2 and §2.3 is taken from [S97, Hi03]. We need to
have a convenient description of the commutant and centralizer of
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ $($which has been
recorded in the case $q=0$ in [S97] and a similar collection of operators in
the commutant has been identified in [Hi03]$)$. Thus, we need to record some
facts related to the modular theory of the $q$-quasi free state
$\varphi_{q,U}$. Let $J_{\varphi_{q,U}}$ and $\Delta_{\varphi_{q,U}}$
respectively denote the modular conjugation and modular operator associated to
$\varphi_{q,U}$ and let
$S_{\varphi_{q,U}}=J_{\varphi_{q,U}}\Delta_{\varphi_{q,U}}^{\frac{1}{2}}$.
Then, for $n\in\mathbb{N}$,

$\displaystyle
J_{\varphi_{q,U}}(\xi_{1}\otimes\cdots\otimes\xi_{n})=A^{-1/2}\xi_{n}\otimes\cdots\otimes
A^{-1/2}\xi_{1},\text{ }\forall\text{
}\xi_{i}\in\mathcal{H}_{\mathbb{R}}\cap\mathfrak{D}(A^{-{\frac{1}{2}}});$ (5)
$\displaystyle\Delta_{\varphi_{q,U}}(\xi_{1}\otimes\cdots\otimes\xi_{n})=A^{-1}\xi_{1}\otimes\cdots\otimes
A^{-1}\xi_{n},\text{ }\forall\text{
}\xi_{i}\in\mathcal{H}_{\mathbb{R}}\cap\mathfrak{D}(A^{-1});$ $\displaystyle
S_{\varphi_{q,U}}(\xi_{1}\otimes\cdots\otimes\xi_{n})=\xi_{n}\otimes\cdots\otimes\xi_{1},\text{
}\forall\text{ }\xi_{i}\in\mathcal{H}_{\mathbb{R}}.$

The modular automorphism group $(\sigma_{t}^{\varphi_{q,U}})$ of
$\varphi_{q,U}$ is given by
$\sigma_{-t}^{\varphi_{q,U}}=\text{Ad}(\mathcal{F}(U_{t}))$, where
$\mathcal{F}(U_{t})=id\oplus\oplus_{n\geqslant 1}U_{t}^{\otimes_{q}n}$, for
all $t\in\mathbb{R}$. In particular,

$\displaystyle\sigma^{\varphi_{q,U}}_{-t}(s_{q}(\xi))=s_{q}(U_{t}\xi),\text{
for all }\xi\in\mathcal{H}_{\mathbb{R}}.$ (6)

To reduce notation, the complex $($left$)$ conjugation $\mathcal{J}$
associated to $\mathcal{H}_{\mathbb{R}}\subset\mathcal{H}$ will be denoted by
$\overline{\xi+i\eta}=\xi-i\eta$ for $\xi,\eta\in\mathcal{H}_{\mathbb{R}}$. It
corresponds with $S_{\varphi_{q},U}$.

###  2.3. Commutant

Now we proceed to describe the commutant of
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$. Consider the set

$\displaystyle\mathcal{H}_{\mathbb{R}}^{\prime}=\\{\xi\in\mathcal{H}:\langle\xi,\eta\rangle_{U}\in\mathbb{R}\text{
for all }\eta\in\mathcal{H}_{\mathbb{R}}\\}.$

Then $\mathcal{H}_{\mathbb{R}}^{\prime}$ is a real subspace. Note that
$\overline{\mathcal{H}_{\mathbb{R}}^{\prime}+i\mathcal{H}_{\mathbb{R}}^{\prime}}=\mathcal{H}$
and $\mathcal{H}_{\mathbb{R}}^{\prime}\cap
i\mathcal{H}^{\prime}_{\mathbb{R}}=\\{0\\}$. It is easy to check that
$A^{-1/2}\zeta\in\mathcal{H}_{\mathbb{R}}^{\prime}$ for all
$\zeta\in\mathfrak{D}(A^{-{\frac{1}{2}}})\cap\mathcal{H}_{\mathbb{R}}$.

Now for $\xi\in\mathcal{H}$, define the right creation operator $c_{q,r}(\xi)$
on $\mathcal{F}_{q}(\mathcal{H})$ by

$\displaystyle c_{q,r}(\xi)\Omega=\xi,$ (7) $\displaystyle
c_{q,r}(\xi)(\xi_{1}\otimes\cdots\otimes\xi_{n})=\xi_{1}\otimes\cdots\otimes\xi_{n}\otimes\xi,\text{
}\xi_{i}\in\mathcal{H},n\geqslant 1.$

Clearly, $c_{q,r}(\xi)=\jmath c_{q}(\xi){\jmath}^{*}$, where
$\jmath:\mathcal{F}_{q}(\mathcal{H})\rightarrow\mathcal{F}_{q}(\mathcal{H})$
is the unitary defined by

$\displaystyle\jmath(\xi_{1}\otimes\cdots\otimes\xi_{n})=\xi_{n}\otimes\cdots\otimes\xi_{1},\text{
where }\xi_{i}\in\mathcal{H}\text{ for all }1\leqslant i\leqslant n,n\geqslant
1,$ (8) $\displaystyle\jmath(\Omega)=\Omega.$

Therefore, $c_{q,r}(\xi)\in\mathbf{B}(\mathcal{F}_{q}(\mathcal{H}))$ and its
adjoint $c_{q,r}(\xi)^{*}$ is given by

$\displaystyle c_{q,r}(\xi)^{*}\Omega=0,$ (9) $\displaystyle
c_{q,r}(\xi)^{*}(\xi_{1}\otimes\cdots\otimes\xi_{n})=\sum_{i=1}^{n}q^{n-i}\langle\xi,\xi_{i}\rangle_{U}\xi_{1}\otimes\cdots\otimes\xi_{i-1}\otimes\xi_{i+1}\otimes\cdots\otimes\xi_{n},\text{
}\xi_{i}\in\mathcal{H},n\geqslant 1.$

Write $s_{q,r}(\xi)=c_{q,r}(\xi)+c_{q,r}(\xi)^{*}$, $\xi\in\mathcal{H}$. The
following result describes the commutant of
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$.

######  Theorem 2.2.

[BM, Thm. 2.4] Suppose
$\xi\in\operatorname{\mathfrak{D}}(A^{-1})\cap\mathcal{H}_{\mathbb{R}}$. Then
$J_{\varphi_{q,U}}s_{q}(\xi)J_{\varphi_{q,U}}=s_{q,r}(A^{-\operatorname{\frac{1}{2}}}\xi)$.
Moreover,
$\Gamma(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime}=\\{s_{q,r}(\xi):\xi\in\mathcal{H}_{\mathbb{R}}^{\prime}\\}^{\prime\prime}$.

The complex $($right$)$ conjugation associated to
$\mathcal{H}_{\mathbb{R}}^{\prime}\subset\mathcal{H}$ will be denoted by
$\overline{\xi+i\eta}^{\,r}=\xi-i\eta$ for
$\xi,\eta\in\mathcal{H}_{\mathbb{R}}^{\prime}$. It corresponds with $\jmath
S_{\varphi_{q},U}\jmath^{*}$.

###  2.4. Notations and some technical facts

In this paper, we are interested in the factoriality of
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ and the orthogonal
representation remains arbitrary but fixed. Thus, to reduce notation, we will
write $M_{q}=\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ and
$\varphi=\varphi_{q,U}$. We will also denote $J_{\varphi_{q,U}}$ by $J$ and
$\Delta_{\varphi_{q,U}}$ by $\Delta$. As $\Omega$ is separating for both
$M_{q}$ and $M_{q}^{\prime}$, for $\zeta\in M_{q}\Omega$ and $\eta\in
M_{q}^{\prime}\Omega$ there exist unique $x_{\zeta}\in M_{q}$ and
$x^{\prime}_{\eta}\in M_{q}^{\prime}$ such that $\zeta=x_{\zeta}\Omega$ and
$\eta=x_{\eta}^{\prime}\Omega$. In this case, we will write

$\displaystyle W(\zeta)=x_{\zeta}\text{ and }W_{r}(\eta)=x_{\eta}^{\prime}.$

Note that $W_{r}(\eta)=JW(J\eta)J$, as $J\eta\in M_{q}\Omega$ from Tomita’s
fundamental theorem. Thus, for example, as $\xi\in M_{q}\Omega$ for every
$\xi\in\mathcal{H}_{\mathbb{R}}$, so $W(\xi+i\eta)=s_{q}(\xi)+is_{q}(\eta)$
for all $\xi,\eta\in\mathcal{H}_{\mathbb{R}}$.

Write $\mathcal{Z}(M_{q})=M_{q}\cap M_{q}^{\prime}$. Let
$M_{q}^{\varphi}=\\{x\in M_{q}:\sigma_{t}^{\varphi}(x)=x\text{ for all
}t\in\mathbb{R}\\}$ denote the centralizer of $M_{q}$ associated to the state
$\varphi$. Recall that $x\in M_{q}$ is analytic with respect to
$(\sigma_{t}^{\varphi})$ if and only if the function $\mathbb{R}\ni
t\mapsto\sigma_{t}^{\varphi}(x)\in M_{q}$ extends to a weakly entire function.
We say that a vector $\xi\in M_{q}\Omega$ is analytic, if $W(\xi)$ is analytic
for $(\sigma_{t}^{\varphi})$.

In order to control calculations within the page limit, we adopt the following
notations for convenience.

  1. (1)

$\xi_{1}\cdots\xi_{n}:=\xi_{1}\otimes\cdots\otimes\xi_{n}\in\mathcal{H}^{\otimes_{q}n}$
for $\xi_{i}\in\mathcal{H}$, $1\leqslant i\leqslant n$;

  2. (2)

$c_{q}(\xi)=c_{q}(\xi_{1})\cdots c_{q}(\xi_{n})$ for
$\xi=\xi_{1}\cdots\xi_{n}\in\mathcal{H}^{\otimes_{q}n}$;

  3. (3)

$c_{q,r}(\xi)=c_{q,r}(\xi_{n})\cdots c_{q,r}q(\xi_{1})$ for
$\xi=\xi_{1}\cdots\xi_{n}\in\mathcal{H}^{\otimes_{q}n}$;

  4. (4)

$C_{q}=\prod_{i=1}^{\infty}\frac{1}{1-|q|^{i}}$;

  5. (5)

$d_{0}=1$, $d_{j}=\prod_{i=1}^{j}(1-q^{i})$, $j\in\mathbb{N}$, and
$d_{\infty}=\prod_{i=1}^{\infty}(1-q^{i})$;

  6. (6)

$[n]_{q}:=1+q+\cdots+q^{(n-1)}$, $[n]_{q}!:=\prod_{j=1}^{n}[j]_{q},\text{ for
}n\geqslant 1$, and $[0]_{q}:=0$, $[0]_{q}!:=1$ by convention.

Note that $d_{k},\frac{1}{d_{k}}\leqslant C_{q}$ for all $k\geqslant 0$ and
$\left|q\right|<1$.

The following norm estimates will be crucial (see [BKS97, BS91, B99, R05]).
$\bullet$ For all $\xi\in\mathcal{H}^{\otimes_{q}n}$,

$\displaystyle\|c_{q}(\xi)\|\leqslant\sqrt{C_{q}}\|\xi\|_{q}.$ (10)

$\bullet$ If $\xi\in\mathcal{H}$ and
${\left\|\xi\right\|}_{U}=1(=\left\|\xi\right\|_{q})$, then

$\displaystyle{\left\|\xi^{n}\right\|}_{q}^{2}=[n]_{q}!=d_{n}(1-q)^{-n}.$ (11)

$\bullet$ If $\xi_{1},\cdots,\xi_{n},\xi\in\mathcal{H}$ with
${\left\|\xi_{j}\right\|}_{U}={\left\|\xi\right\|}_{U}=1$ for all $1\leqslant
j\leqslant n$, then

$\displaystyle{\left\|\xi_{1}\cdots\xi_{n}\xi^{m}\right\|}_{q}={\left\|\xi^{m}\xi_{n}\cdots\xi_{1}\right\|}_{q}\leqslant
C_{q}^{\frac{n}{2}}\sqrt{[m]_{q}!},\text{ }m\geqslant 0.$ (12)

$\bullet$ We recall the following $q$-analogue of the Pascal’s identity for
$q$-binomial coefficients (cf. [BKS97, Prop. 1.8]):

$\displaystyle q^{k}\left(\begin{matrix}n\\\
k\end{matrix}\right)_{q}+\left(\begin{matrix}n\\\
k-1\end{matrix}\right)_{q}=\left(\begin{matrix}n+1\\\
k\end{matrix}\right)_{q},\quad k\leqslant n.$ (13)

We also recall the Wick formula from [B99, N04], and its right version which
can be obtained using the $($right$)$ complex conjugation.

######  Proposition 2.3.

Let $\xi_{1},\cdots,\xi_{n}$ be in $\mathcal{H}_{\mathbb{C}}$. Then,

$W(\xi_{1}\cdots\xi_{n})=\sum_{i=0}^{n}\sum_{\sigma\in
S_{n,i}}q^{\left|\sigma\right|}c_{q}(\xi_{\sigma(1)})\cdots
c_{q}(\xi_{\sigma(i)})c_{q}(\overline{\xi_{\sigma(i+1)}})^{*}\cdots
c_{q}(\overline{\xi_{\sigma(n)}})^{*},$

where $S_{n,i}$ is the set of permutations of $\\{1,\ldots,n\\}$ that are
increasing on $\\{1,\ldots,i\\}$ and $\\{i+1,\ldots,n\\}$ and $|\sigma|$ is
the number of inversions of $\sigma$. Further, if
$\xi_{1},\cdots,\xi_{n}\in\mathcal{H}_{\mathbb{R}}^{\prime}+i\mathcal{H}_{\mathbb{R}}^{\prime}$,
then

$W_{r}(\xi_{1}\cdots\xi_{n})=\sum_{i=0}^{n}\sum_{\sigma\in
S_{n,i}}q^{\left|{\rm
flip\,}\circ\,\sigma\right|}c_{q,r}(\xi_{\sigma(1)})\cdots
c_{q,r}(\xi_{\sigma(i)})c_{q,r}(\overline{\xi_{\sigma(i+1)}}^{\,r})^{*}\cdots
c_{q,r}(\overline{\xi_{\sigma(n)}}^{\,r})^{*},$

where ${\rm flip}$ is the permutation ${\rm flip}(k)=n-k$, $1\leqslant
k\leqslant n$.

There is also another convenient way to write the Wick formula using crossings
of partitions.

Any $\sigma\in S_{n,i}$ is completely determined by a subset
$\mathfrak{J}=\\{j_{1}<\cdots<j_{i}\\}$ with complement
$\mathfrak{J}^{c}=\\{k_{i+1}<\cdots<k_{n}\\}$. Then $\left|\sigma\right|$ is
the number of crossings of the partition $\mathfrak{J}\cup\mathfrak{J}^{c}$,
i.e.,

$\left|\sigma\right|=c(\mathfrak{J},\mathfrak{J}^{c})=\\#\\{(a,b)\;|\;j_{a}>k_{b}\\}.$

Thus,

$W(\xi_{1}\cdots\xi_{n})=\sum_{i=0}^{n}\sum_{\begin{subarray}{c}\mathfrak{J}=\\{j_{1}<\cdots<j_{i}\\}\\\
\mathfrak{J}^{c}=\\{k_{i+1}<\cdots<k_{n}\\}\\\
\mathfrak{J}\cup\mathfrak{J}^{c}=\\{1,...,n\\}\end{subarray}}q^{c(\mathfrak{J},\mathfrak{J}^{c})}c_{q}(\xi_{j_{1}})\cdots
c_{q}(\xi_{j_{i}})c_{q}(\overline{\xi_{k_{i+1}}})^{*}\cdots
c_{q}(\overline{\xi_{k_{n}}})^{*}.$ (14)

###  2.5. Centralizer

We need a convenient description of the centralizer $M_{q}^{\varphi}$ which
depends on the almost periodic part of the orthogonal representation
$(U_{t})$. We need some preparation. For details check [BM].

Recall that for a strongly continuous orthogonal representation $t\mapsto
U_{t}$, $t\in\mathbb{R}$, on the real Hilbert space
$\mathcal{H}_{\mathbb{R}}$, there is a unique decomposition $($see [S97]$)$,

$\displaystyle(\mathcal{H}_{\mathbb{R}},U_{t})=\left(\bigoplus_{j=1}^{N_{1}}(\mathbb{R},\text{id})\right)\oplus\left(\bigoplus_{k=1}^{N_{2}}(\mathcal{H}_{\mathbb{R}}(k),U_{t}(k))\right)\oplus(\widetilde{\mathcal{H}}_{\mathbb{R}},\widetilde{U}_{t}),$

where $0\leqslant N_{1},N_{2}\leqslant\aleph_{0}$,

$\displaystyle\mathcal{H}_{\mathbb{R}}(k)=\mathbb{R}^{2},\quad
U_{t}(k)=\left(\begin{matrix}\cos(t\log\lambda_{k})&-\sin(t\log\lambda_{k})\\\
\sin(t\log\lambda_{k})&\cos(t\log\lambda_{k})\end{matrix}\right),\text{
}0<\lambda_{k}<1,$

and $(\widetilde{\mathcal{H}}_{\mathbb{R}},\widetilde{U}_{t})$ corresponds to
the weakly mixing component of the orthogonal representation; thus
$\widetilde{\mathcal{H}}_{\mathbb{R}}$ is either $0$ or infinite-dimensional.

If $N_{1}\neq 0$, let $e_{j}=0\oplus\cdots\oplus 0\oplus 1\oplus
0\oplus\cdots\oplus 0\in\bigoplus_{j=1}^{N_{1}}\mathbb{R}$, where $1$ appears
at the $j$-th place for $1\leqslant j\leqslant N_{1}$. Similarly, if
$N_{2}\neq 0$, let $f_{k}^{1}=0\oplus\cdots\oplus
0\oplus\left(\begin{matrix}1\\\ 0\end{matrix}\right)\oplus 0\oplus\cdots\oplus
0\in\bigoplus_{k=1}^{N_{2}}\mathcal{H}_{\mathbb{R}}(k)$ and
$f_{k}^{2}=0\oplus\cdots\oplus 0\oplus\left(\begin{matrix}0\\\
1\end{matrix}\right)\oplus 0\oplus\cdots\oplus
0\in\bigoplus_{k=1}^{N_{2}}\mathcal{H}_{\mathbb{R}}(k)$ be vectors with
nonzero entries in the $k$-th position for $1\leqslant k\leqslant N_{2}$.
Denote

$\displaystyle
e^{1}_{k}=\frac{\sqrt{\lambda_{k}+1}}{2}(f_{k}^{1}+if_{k}^{2})\text{ and
}e^{2}_{k}=\frac{\sqrt{{\lambda}^{-1}_{k}+1}}{2}(f_{k}^{1}-if_{k}^{2});$

thus
$e_{k}^{1},e_{k}^{2}\in\mathcal{H}_{\mathbb{R}}(k)+i\mathcal{H}_{\mathbb{R}}(k)$
are orthonormal basis of
$(\mathcal{H}_{\mathbb{R}}(k)+i\mathcal{H}_{\mathbb{R}}(k),\langle\cdot,\cdot\rangle_{U})$
for $1\leqslant k\leqslant N_{2}$. Fix $1\leqslant k\leqslant N_{2}$. The
analytic generator $A(k)$ of $(U_{t}(k))$ is given by

$\displaystyle
A(k)=\frac{1}{2}\left(\begin{matrix}\lambda_{k}+\frac{1}{\lambda_{k}}&i(\lambda_{k}-\frac{1}{\lambda_{k}})\\\
-i(\lambda_{k}-\frac{1}{\lambda_{k}})&\lambda_{k}+\frac{1}{\lambda_{k}}\end{matrix}\right).$

Moreover,

$\displaystyle A(k)e_{k}^{1}=\frac{1}{\lambda_{k}}e_{k}^{1}\text{ and
}A(k)e_{k}^{2}=\lambda_{k}e_{k}^{2}.$

Write $\mathcal{S}=\\{e_{j}:1\leqslant j\leqslant
N_{1}\\}\cup\\{e_{k}^{1},e_{k}^{2}:1\leqslant k\leqslant N_{2}\\}$ if
$N_{1}\neq 0$ or $N_{2}\neq 0$, else set $\mathcal{S}=\\{0\\}$. If
$\mathcal{S}\neq\\{0\\}$, then $\mathcal{S}$ is an orthogonal set in
$(\mathcal{H}_{\mathbb{C}},\langle\cdot,\cdot\rangle_{U})$ and the collection
of eigenvectors of the analytic generator $A$ of $(U_{t})$ is contained in
$\text{span }\mathcal{S}$. In the event $\mathcal{S}\neq\\{0\\}$, rename the
elements of the set $\mathcal{S}$ as $\zeta_{1},\zeta_{2},\cdots$, i.e.,
$\mathcal{S}=\\{\zeta_{i}:1\leqslant i\leqslant N_{1}+2N_{2}\\}$, whence
$A\zeta_{l}=\beta_{l}\zeta_{l}$ with $\beta_{l}\in\mathcal{E}_{A}$ for all
$l$, where $\mathcal{E}_{A}=\\{1\\}\cup\\{\lambda_{k}:1\leqslant k\leqslant
N_{2}\\}\cup\\{\frac{1}{\lambda_{k}}:1\leqslant k\leqslant N_{2}\\}$. It is to
be understood that when $N_{1}=\aleph_{0}$ $($resp. $N_{2}=\aleph_{0})$, the
constraints $j\leqslant N_{1}$ and $i\leqslant N_{1}+2N_{2}$ $($resp.
$k\leqslant N_{2}$ and $i\leqslant N_{1}+2N_{2})$ $($in defining $\mathcal{S}$
and $\mathcal{E}_{A})$ is replaced by $j<N_{1}$ and $i<N_{1}+2N_{2}$ $($resp.
$k<N_{2}$ and $i<N_{1}+2N_{2})$.

Now we are ready to write down the description of $M_{q}^{\varphi}$.

######  Theorem 2.4.

[BM, Thm. 3.4] Let

$\displaystyle\mathcal{W}_{0}=\begin{cases}&\\{\zeta_{i_{1}}\cdots\zeta_{i_{n}}:\zeta_{i_{j}}\in\mathcal{S},1\leqslant
i_{j}\leqslant
N_{1}+2N_{2},\prod_{j=1}^{n}\beta_{i_{j}}=1,n\in\mathbb{N}\\},\\\
&\indent\indent\indent\indent\indent\indent\indent\indent\indent\text{ if
}\max(N_{1},N_{2})<\infty;\\\
&\\{\zeta_{i_{1}}\cdots\zeta_{i_{n}}:\zeta_{i_{j}}\in\mathcal{S},1\leqslant
i_{j}<N_{1}+2N_{2},\prod_{j=1}^{n}\beta_{i_{j}}=1,n\in\mathbb{N}\\},\\\
&\indent\indent\indent\indent\indent\indent\indent\indent\indent\text{ if
}\max(N_{1},N_{2})=\infty.\\\ \end{cases}$

Let $\mathcal{W}=\mathbb{C}\Omega\oplus\overline{\text{span
}\mathcal{W}_{0}}^{\left\|\cdot\right\|_{q}}$. Then,
$M_{q}^{\varphi}\Omega=\mathcal{W}\cap M_{q}\Omega$.

######  Remark 2.5.

If $\mathcal{E}_{A}=\\{1\\}$, then $M_{q}^{\varphi}$ is isomorphic to the
$q$-Gaussian von Neumann algebra of Bo$\overset{.}{\text{z}}$ejko and Speicher
[BS91, BKS97]. Thus, in this case, if $1$ is eigenvalue of multiplicity more
than or equal to $2$, then $M_{q}^{\varphi}$ is a factor by [R05].

##  3\. Technical Analysis

The factoriality of $M_{q}$ is decided in several steps. When
$\dim(\mathcal{H}_{\mathbb{R}})=2$, then $(U_{t})$ is not ergodic if and only
if it is trivial and in that case $M_{q}^{\varphi}=M_{q}$ is a $\rm{II}_{1}$
factor $($see [BM, R05]$)$. The most important and difficult case is the one
when $\dim(\mathcal{H}_{\mathbb{R}})=2$ and $(U_{t})$ is ergodic. The
difficulty arises in lack of room to perform meaningful calculations.

In this section, we lay out the technical analysis that will lead to the
factoriality of $M_{q}$. This section is divided into two subsections. In §3.1
we deal with the case when $\dim(\mathcal{H}_{\mathbb{R}})=2$ and $(U_{t})$ is
ergodic and in §3.2 we prepare the machinery that will help deal with all the
cases.

###  3.1. $\mathbf{\Gamma(\mathbb{R}^{2},U_{t})^{\prime\prime}}$ with small
$\lambda$ and $(U_{t})$ ergodic

Following the discussion in §2.5, it follows that $N_{1}=0$ and $N_{2}=1$.
Further, there exists a $\lambda\in(0,1)$ such that

$\displaystyle
U_{t}=\left(\begin{matrix}\cos(t\log\lambda)&-\sin(t\log\lambda)\\\
\sin(t\log\lambda)&\cos(t\log\lambda)\end{matrix}\right).$

Reducing notation, write
$e_{1}:=e_{1}^{1}=\frac{\sqrt{\lambda+1}}{2}\begin{pmatrix}1\\\
i\end{pmatrix}$ and
$e_{2}:=e_{1}^{2}=\frac{\sqrt{\lambda^{-1}+1}}{2}\begin{pmatrix}1\\\
-i\end{pmatrix}$ so that $Ae_{1}=\frac{1}{\lambda}e_{1}\text{ and
}Ae_{2}=\lambda e_{2}$ and $\\{e_{1},e_{2}\\}$ forms a orthonormal basis of
$(\mathbb{C}^{2},\langle\cdot,\cdot\rangle_{U})$.

Now we set $e=\lambda^{-\frac{1}{4}}e_{1}$. Then,
$\bar{e}=\lambda^{\frac{1}{4}}e_{2}$. Note that
$\left\|e\right\|_{U}=\lambda^{-\frac{1}{4}}$ and
$\left\|\bar{e}\right\|_{U}=\lambda^{\frac{1}{4}}$. Note that in this context
$\mathcal{H}=\mathbb{C}^{2}$ considered with respect to the inner product
$\langle\cdot,\cdot\rangle_{U}$. We also have the formulas
$\bar{e}^{\,r}=\lambda^{-1}\bar{e}$ and $\overline{\,\bar{e}\,}^{\,r}=\lambda
e$.

We begin with some useful lemmas.

######  Lemma 3.1.

For $m\geqslant 0$ and $\xi\in\mathcal{H}$, we have

$c_{q}(\xi)^{*n}(\xi^{m})=\begin{cases}\frac{[m]_{q}!}{[m-n]_{q}!}\left\|\xi\right\|_{q}^{2n}\xi^{m-n},\quad&n\leqslant
m;\\\ 0,\quad&n>m.\end{cases}$

###### Proof.

The proof follows directly from Eq. (2). $\Box$

######  Lemma 3.2.

$W(e^{n})=W(e)^{n}=\displaystyle\sum_{k=0}^{n}\left(\begin{matrix}n\\\
k\end{matrix}\right)_{q}c_{q}(e)^{n-k}c_{q}(\bar{e})^{*k}$ for $n\geqslant 1$.

###### Proof.

The proof is straightforward and follows by using Eq. (13) and induction. So
we omit the proof. $\Box$

The next lemma initiates the interplay between the parameters $\lambda$ and
$q$ in the case $\dim(\mathcal{H}_{\mathbb{R}})=2$, which will force
constraints to the factoriality of $M_{q}$.

######  Lemma 3.3.

Let $D(q)=\sup_{n}d_{n}$. With the above notations, the following are true:

  1. $(i)$

$\\{\lambda^{\frac{n}{4}}(1-q)^{\frac{n}{2}}c_{q}(e)^{n}\\}_{n\geqslant 1}$,
$\\{\lambda^{-\frac{n}{4}}(1-q)^{\frac{n}{2}}c_{q}(\bar{e})^{n}\\}_{n\geqslant
1}$ are bounded $($actually by $1$ if $q\geqslant 0$ and by $\sqrt{C_{q}D(q)}$
if $q<0)$.

  2. $(ii)$

$\\{\lambda^{\frac{n}{4}}(1-q)^{\frac{n}{2}}W(e^{n})\\}_{n\geqslant 1}$ is
bounded.

###### Proof.

$(i)$. First assume $0\leqslant q<1$. Then, from Eq. (3) it follows that

$\displaystyle\left\|c_{q}(e)^{n}\right\|\leqslant\left\|c_{q}(e)\right\|^{n}=\left\|e\right\|_{q}^{n}(1-q)^{-\frac{n}{2}}=\lambda^{-\frac{n}{4}}(1-q)^{-\frac{n}{2}}.$

If $-1<q<0$, from Eq. (10) and Eq. (11) it follows that

$\displaystyle\left\|c_{q}(e)^{n}\right\|$
$\displaystyle=\left\|c_{q}(e^{n})\right\|\leqslant\sqrt{C_{q}}\left\|e^{n}\right\|_{q}$
$\displaystyle=\sqrt{C_{q}}\left\|e\right\|_{q}^{n}\sqrt{[n]_{q}!}$
$\displaystyle=\sqrt{C_{q}}\lambda^{-\frac{n}{4}}\sqrt{[n]_{q}!}$
$\displaystyle=\sqrt{C_{q}d_{n}}\lambda^{-\frac{n}{4}}(1-q)^{-\frac{n}{2}}\leqslant\sqrt{C_{q}D(q)}\lambda^{-\frac{n}{4}}(1-q)^{-\frac{n}{2}}.$

The argument for $\bar{e}$ is similar. Thus $(i)$ follows.

$(ii)$. We just use Lemma 3.2 and the triangle inequality. Note that

$\displaystyle\|W(e^{n})\|$ $\displaystyle\leqslant$
$\displaystyle\sum_{k=0}^{n}\frac{d_{n}}{d_{n-k}d_{k}}\|c_{q}(e)^{n-k}\|.\|c_{q}(\bar{e})^{*k}\|\quad(\text{use
Eq. \eqref{Normelt2}})$ $\displaystyle\leqslant$ $\displaystyle
C_{q}\sum_{k=0}^{n}\frac{d_{n}}{d_{n-k}d_{k}}\|e^{n-k}\|_{q}.\|\bar{e}^{k}\|_{q}\quad(\text{use
Eq. \eqref{Normelt}})$ $\displaystyle=$ $\displaystyle
C_{q}\sum_{k=0}^{n}\frac{d_{n}}{d_{n-k}d_{k}}\|e\|_{q}^{n-k}\sqrt{[n-k]_{q}!}\|\bar{e}\|_{q}^{k}\sqrt{[k]_{q}!}\quad(\text{use
Eq. \eqref{Normelt2}})$ $\displaystyle=$ $\displaystyle
C_{q}\sum_{k=0}^{n}\frac{d_{n}}{\sqrt{d_{n-k}d_{k}}}\lambda^{(2k-n)/4}(1-q)^{-n/2}\quad(\text{use
Eq. \eqref{Normelt2}}).$

The result follows as $(d_{i})$ is bounded from above and below and
$\lambda<1$. $\Box$

######  Lemma 3.4.

Let $T_{n}=(1-q)^{n}\lambda^{n/2}c_{q}(e)^{*n}c_{q}(e)^{n}$ for $n\geqslant
1$, then $T_{n}$ is norm convergent to some $T\in C^{*}\langle
c_{q}(e)\rangle$.

Moreover, if $q\geqslant 0$ then $d_{\infty}\leqslant T\leqslant 1$ and if
$q\leqslant 0$ then $d_{\infty}/(1-q)\leqslant T\leqslant 1-q$.

###### Proof.

Let $\mathcal{K}=\mathbb{C}e$. Then by [W17, Thm. 3.6] (see also [W18],
[JSW96] Example 2), the $C^{*}$-algebras generated by the creation operators
associated to $e$ in $\mathcal{F}_{q}(\mathcal{H})$ and in
$\mathcal{F}_{q}(\mathcal{K})$ are isomorphic. Thus, we only need to prove the
result in $\mathbf{B}(\mathcal{F}_{q}(\mathcal{K}))$; but
$\mathcal{F}_{q}(\mathcal{K})=\oplus_{k=0}^{\infty}\mathbb{C}e^{k}$. With
respect to this decomposition, $T_{n}$ is a diagonal operator with

$\displaystyle T_{n}(e^{k})$
$\displaystyle=(1-q)^{n}\lambda^{n/2}c_{q}(e)^{*n}(e^{(n+k)})=(1-q)^{n}\frac{[n+k]_{q}!}{[k]_{q}!}e^{k}$
$\displaystyle=\Big{(}\prod_{j=1}^{n}(1-q^{k+j})\Big{)}e^{k},\quad\forall
k\geqslant 0.$

Let $T\in\mathbf{B}(\mathcal{F}_{q}(\mathcal{K}))$ be the diagonal operator
with eigenvalues $\frac{d_{\infty}}{d_{k}}=\prod_{j=1}^{\infty}(1-q^{k+j})$
and associated eigenvectors $e^{k}$ for all $k\geqslant 0$. Then,

$\displaystyle\|T_{n}-T\|\leqslant
K\sup_{k}|1-\prod_{j=n+1}^{\infty}(1-q^{k+j})|\to 0\text{ as }n\to\infty,$

where $K$ is a constant (independent of $n,k$). Consequently, $0\leqslant T\in
C^{*}\langle c_{q}(e)\rangle$.

The estimations for $T$ are clear from its spectrum if $q\geqslant 0$. When
$q<0$, we have to estimate
$\alpha_{q}=\inf_{k}\prod_{j=1}^{\infty}(1-q^{k+j})$ and
$\beta_{q}=\sup_{k}\prod_{j=1}^{\infty}(1-q^{k+j})$. Note that for all
$m\geqslant 0$, we have that $(1-q^{2m})(1-q^{2m+1})\leqslant 1$. Thus, it
follows that $\alpha_{q}=\prod_{j=2}^{\infty}(1-q^{j})=d_{\infty}/(1-q)$. In
the same way since $\prod_{j=2m}^{\infty}(1-q^{k+j})\leqslant 1$, we must have
$\beta_{q}\leqslant 1-q$. $\Box$

######  Remark 3.5.

When $q<0$, if moreover $|q|(1+|q|)\leqslant 1$, we have that for all
$m\geqslant 0$, $(1-q^{2m+1})(1-q^{2m+2})\geqslant 1$. Thus, it follows that
$\beta_{q}=\sup_{k}d_{\infty}/d_{k}=d_{\infty}$ in that case.

Now, consider the following operator

$S_{n}=(1-q)^{n}\lambda^{\frac{n}{2}}{c_{q}(e)}^{*n}W(e^{n}).$

Also note that, for each fixed $k$, as $n$ goes to infinity,
$\left(\begin{matrix}n\\\ k\end{matrix}\right)_{q}=\frac{d_{n}}{d_{n-k}d_{k}}$
converges to $c_{k}=d_{k}^{-1}\leqslant C_{q}$.

######  Lemma 3.6.

The sequence $S_{n}$ converges in norm to

$\displaystyle{S_{\infty}=\sum_{k=0}^{\infty}c_{k}(1-q)^{k}\lambda^{k/2}c_{q}(e)^{*k}Tc_{q}(\overline{e})^{k*}}.$

Moreover, if $q>0$ and $\lambda<(1+\frac{C_{q}}{d_{\infty}})^{-2}$ or if $q<0$
and $\lambda<(1+\frac{C_{q}^{2}D(q)(1-q)^{2}}{d_{\infty}})^{-2}$, the operator
$S_{\infty}$ is invertible.

###### Proof.

Let $B(q)=\sqrt{C_{q}D(q)}$. First note that the series defining $S_{\infty}$
converges absolutely. Indeed by Lemma 3.3 $(i)$, we have

$\displaystyle\|(1-q)^{k/2}c_{q}(e)^{k}\|\leqslant\lambda^{-k/4},\quad\|(1-q)^{k/2}c_{q}(\overline{e})^{k}\|\leqslant\lambda^{k/4},\quad\text{
when }0\leqslant q<1;$ $\displaystyle\|(1-q)^{k/2}c_{q}(e)^{k}\|\leqslant
B(q)\lambda^{-k/4},\,\,\|(1-q)^{k/2}c_{q}(\overline{e})^{k}\|\leqslant
B(q)\lambda^{k/4},\text{ when }-1<q<0;$

Thus, using Lemma 3.4 we have

$\sum_{k=0}^{\infty}\|c_{k}(1-q)^{k}\lambda^{k/2}c_{q}(e)^{*k}Tc_{q}(\overline{e})^{k*}\|\leqslant\begin{cases}C_{q}\sum_{k=0}^{\infty}{\lambda}^{k/2}<\infty,&\text{
if }q\geqslant 0;\\\
C_{q}^{2}D(q)(1-q)\sum_{k=0}^{\infty}{\lambda}^{k/2}<\infty,&\text{ if
}q<0.\end{cases}$

We note the following:

$\displaystyle S_{n}$
$\displaystyle=(1-q)^{n}\lambda^{\frac{n}{2}}{c_{q}(e)}^{*n}W(e^{n})$
$\displaystyle=(1-q)^{n}\lambda^{\frac{n}{2}}{c_{q}(e)}^{*n}\displaystyle\sum_{k=0}^{n}\left(\begin{matrix}n\\\
k\end{matrix}\right)_{q}c_{q}(e)^{n-k}c_{q}(\bar{e})^{*k}\quad(\text{Lemma
\ref{wen}})$
$\displaystyle=\displaystyle\sum_{k=0}^{n}\left(\begin{matrix}n\\\
k\end{matrix}\right)_{q}(1-q)^{k}\lambda^{\frac{k}{2}}{c_{q}(e)}^{*k}(1-q)^{n-k}\lambda^{\frac{n-k}{2}}c_{q}(e)^{*(n-k)}c_{q}(e)^{n-k}c_{q}(\bar{e})^{*k}$
$\displaystyle=\displaystyle\sum_{k=0}^{n}\left(\begin{matrix}n\\\
k\end{matrix}\right)_{q}(1-q)^{k}\lambda^{\frac{k}{2}}{c_{q}(e)}^{*k}T_{n-k}c_{q}(\bar{e})^{*k}\quad(\text{Lemma
\ref{Tn}}).$

We have just seen in Lemma 3.4 that $T_{n-k}$ is norm convergent to $T$ (and
bounded by $1$ if $q\geqslant 0$ or $1-q$ if $q<0$). Thus, the general term in
$S_{n}$ converges to the general term of $S_{\infty}$ in norm and one
concludes the convergence in the statement with the dominated convergence
theorem as $\sum_{k=0}^{\infty}\lambda^{k/2}<\infty$.

We also have $S_{\infty}=T(1+T^{-1}V)$ $($use Lemma 3.4$)$, where

$\|V\|\leqslant\begin{cases}C_{q}\sum_{k=1}^{\infty}{\lambda}^{k/2},\quad&\text{
if }q>0;\\\ C_{q}^{2}D(q)(1-q)\sum_{k=1}^{\infty}{\lambda}^{k/2},\quad&\text{
if }q<0.\end{cases}$

The rest follows by ensuring that $\left\|T^{-1}V\right\|<1$ using Lemma 3.4.
We know that $\|T^{-1}\|\leqslant\frac{1}{d_{\infty}}$ if $q>0$ and
$\|T^{-1}\|\leqslant\frac{1-q}{d_{\infty}}$ if $q<0$ by Lemma 3.4. Thus,
$S_{\infty}$ is invertible if
$\frac{C_{q}}{d_{\infty}}\frac{\sqrt{\lambda}}{1-\sqrt{\lambda}}<1$ when $q>0$
and
$C_{q}^{2}D(q)\frac{(1-q)^{2}}{d_{\infty}}\frac{\sqrt{\lambda}}{1-\sqrt{\lambda}}<1$
when $q<0$. $\Box$

######  Remark 3.7.

  1. (1)

When $q=0$,
$S_{\infty}=\sum_{k=0}^{\infty}\lambda^{k/2}c_{0}(e)^{*k}c_{0}(\bar{e})^{*k}$.
The operators $l_{k}=c_{0}(\bar{e})^{k}c_{0}(e)^{k}$ are partial isometries
with orthogonal ranges when $k\geqslant 1$. On the smallest subspace containg
$\Omega$ and invariant by $l_{k}$’s, $S_{\infty}$ acts like
$1+\sqrt{\frac{\lambda}{1-\lambda}}l^{*}$ where $l$ is a unilateral shift.
Hence, $S_{\infty}$ has a kernel if $\frac{\lambda}{1-\lambda}>1$, that is
$\lambda>\frac{1}{2}$.

  2. (2)

As a result of Lemma 3.6, we have

$\displaystyle S_{\infty}^{*}\Omega$
$\displaystyle=\sum_{k=0}^{\infty}c_{k}(1-q)^{k}\lambda^{k/2}c_{q}(\bar{e})^{k}Tc_{q}(e)^{k}\Omega=\sum_{k=0}^{\infty}c_{k}(1-q)^{k}\lambda^{k/2}c_{q}(\bar{e})^{k}T(e^{k})$
$\displaystyle=\sum_{k=0}^{\infty}c_{k}(1-q)^{k}\lambda^{k/2}c_{q}(\bar{e})^{k}\frac{d_{\infty}}{d_{k}}e^{k}\quad(\text{Lemma
\ref{Tn}})$
$\displaystyle=d_{\infty}\sum_{k=0}^{\infty}c_{k}^{2}(1-q)^{k}\lambda^{k/2}{\bar{e}}^{k}e^{k}.$

The vector $S_{\infty}^{*}\Omega$ will be useful later in the proof.

As a consequence of the Wick formula, we explicitly get:

######  Lemma 3.8.

For every $n\geqslant 0$, there are reals $q_{k,l}$, $0\leqslant k,l\leqslant
n$ with $|q_{k,l}|\leqslant C_{q}^{2}|q|^{(n-k)l}$ such that

$W(\bar{e}^{n}e^{n})=\sum_{k,l=0}^{n}q_{k,l}c_{q}(\bar{e})^{k}c_{q}(e)^{l}c_{q}(e)^{*{(n-k)}}c_{q}(\bar{e})^{*(n-l)}.$

###### Proof.

We use the Wick formula in Eq. (14). Terms of the form
$c_{q}(\bar{e})^{k}c_{q}(e)^{l}c_{q}(e)^{*{(n-k)}}c_{q}(\bar{e})^{*(n-l)}$
occur when we choose $\mathfrak{J}=J_{1}\cup J_{2}\subset\\{1,\ldots,2n\\}$
with $J_{1}\subset\\{1,\ldots,n\\}$ of cardinal $k$ and
$J_{2}\subset\\{n+1,\ldots,2n\\}$ of cardinal $l$. Thus, the formula holds
with $q_{k,l}=\sum_{J_{1},J_{2}}q^{c(\mathfrak{J},\mathfrak{J}^{c})}$, where
the sum runs over all possibilities as mentioned.

Writing $\mathfrak{J}^{c}=K_{1}\cup K_{2}$ with
$K_{1}=\mathfrak{J}^{c}\cap\\{1,\ldots,n\\}$ and
$K_{2}=\mathfrak{J}^{c}\cap\\{n+1,\ldots,2n\\}$, we have
$c(\mathfrak{J},\mathfrak{J}^{c})=c(J_{1},K_{1})+c(J_{2},K_{2})+(n-k)l$, has
all elements in $K_{1}$ cross those of $J_{2}$. Hence,

$q_{k,l}=q^{(n-k)l}.\sum_{J_{1}}q^{c(J_{1},K_{1})}.\sum_{J_{2}}q^{c(J_{2},K_{2})}.$

It follows from [B99] that $|q_{k,l}|\leqslant|q|^{(n-k)l}C_{q}^{2}$. $\Box$

As a result of Lemma 3.8 we have:

######  Lemma 3.9.

The sequence $((1-q)^{n}W({\bar{e}}^{n}e^{n}))$ is bounded.

###### Proof.

We rely on Lemma 3.8 and the triangle inequality with the help of Eq. (10) and
Eq. (11). Note that

$\displaystyle\|(1-q)^{n}W({\bar{e}}^{n}e^{n})\|$ $\displaystyle\leqslant$
$\displaystyle\sum_{k,l=0}^{n}C_{q}^{4}|q|^{(n-k)l}\sqrt{d_{l}d_{n-l}d_{k}d_{n-k}}\lambda^{(k-l)/2}$
$\displaystyle\leqslant$
$\displaystyle\sum_{k,l=0}^{n}C_{q}^{6}|q|^{(n-k)l}\lambda^{(k-l)/2}$
$\displaystyle=$ $\displaystyle
C_{q}^{6}\sum_{j=-n}^{n}\lambda^{j/2}\sum_{\begin{subarray}{c}0\leqslant
k,l\leqslant n\\\ k-l=j\end{subarray}}|q|^{(n-k)l}.$

If we denote by $c_{j,n}$ the last quantity, we have for $j<0$

$c_{j,n}=\sum_{s=0}^{n+j}|q|^{(n-s)(s-j)}\leqslant(n+j+1)|q|^{-nj}.$

If $j\geqslant 0$, then

$c_{j,n}=\sum_{k=j}^{n}|q|^{(n-k)(k-j)}\leqslant\frac{1}{1-|q|}.$

Choose $n_{0}$ large so that $\frac{\left|q\right|^{n}}{\sqrt{\lambda}}<1$ for
all $n\geqslant n_{0}$. Combining the estimates

$\displaystyle\|(1-q)^{n}W({\bar{e}}^{n}e^{n})\|$ $\displaystyle\leqslant$
$\displaystyle
C_{q}^{6}\Big{(}\frac{1}{1-|q|}\sum_{j=0}^{n}\lambda^{j/2}+\sum_{j=1}^{n}\lambda^{-j/2}(n+1)\left|q\right|^{nj}\Big{)}$
$\displaystyle\leqslant$ $\displaystyle
C_{q}^{6}\Big{(}\frac{1}{(1-|q|)(1-\sqrt{\lambda})}+\frac{(n+1)\frac{|q|^{n}}{\sqrt{\lambda}}}{1-\frac{|q|^{n}}{\sqrt{\lambda}}}\Big{)},\quad\forall
n\geqslant n_{0}.$

This allows to conclude as $(n+1)\left|q\right|^{n}\rightarrow 0$ as
$n\rightarrow\infty$. $\Box$

We end this section with the following lemma.

######  Lemma 3.10.

Let $n_{1},n_{2}\in\mathbb{N}\cup\\{0\\}$ and
$\Psi\in\mathcal{H}^{\otimes_{q}l}$ for some $l\geqslant 0$. Then,

$\displaystyle(1-q)^{n}c_{q}(e)^{*}(\bar{e}^{(n+n_{1})}e^{(n+n_{2})}\Psi)\rightarrow
0,$
$\displaystyle(1-q)^{n/2}\lambda^{-n/4}c_{q}(e)^{*}(\bar{e}^{(n+n_{1})}\Psi)\rightarrow
0,\text{ as }n\rightarrow\infty.$

###### Proof.

We only prove the first one, the second one is similar. Using Lemma 2.1 and
Eq. (2), it follows that

$\displaystyle\left\|(1-q)^{n}c_{q}(e)^{*}\bar{e}^{(n+n_{1})}e^{(n+n_{2})}\Psi\right\|_{q}$
$\displaystyle=$
$\displaystyle(1-q)^{n}\left|q\right|^{(n_{1}+n)}\left\|\bar{e}^{(n+n_{1})}\big{(}c_{q}(e)^{*}e^{(n+n_{2})}\Psi\big{)}\right\|_{q}$
$\displaystyle=$
$\displaystyle(1-q)^{n}\left|q\right|^{(n_{1}+n)}\left\|c_{q}(\bar{e})^{(n+n_{1})}c_{q}(e)^{*}c_{q}(e^{n+n_{2}})\Psi\right\|_{q}$
$\displaystyle\leqslant$ $\displaystyle
C_{q}^{\frac{3}{2}}(1-q)^{n}\left|q\right|^{(n_{1}+n)}\left\|\bar{e}^{(n+n_{1})}\right\|_{q}\left\|e^{(n+n_{2})}\right\|_{q}\left\|e\right\|_{q}\left\|\Psi\right\|_{q}\quad(\text{by
\eqref{Normelt}})$ $\displaystyle=$ $\displaystyle
C_{q}^{\frac{3}{2}}(1-q)^{n}\left|q\right|^{(n_{1}+n)}\lambda^{\frac{(n+n_{1}-n-n_{2}-1)}{4}}\sqrt{d_{n+n_{1}}d_{n+n_{2}}d_{1}}(1-q)^{-\frac{2n+n_{1}+n_{2}+1}{2}}\left\|\Psi\right\|_{q}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(\text{by
Eq. \eqref{Normelt2}})$ $\displaystyle\leqslant$ $\displaystyle
C_{q}^{3}(1-q)^{-\frac{n_{1}+n_{2}+1}{2}}\lambda^{\frac{(n_{1}-n_{2}-1)}{4}}\left|q\right|^{(n_{1}+n)}\left\|\Psi\right\|_{q}\xrightarrow{n\rightarrow\infty}0.$

$\Box$

###  3.2. Technicalities to treat all cases

The results in this section are crucial for tracking the relative commutant of
$M_{q}^{\varphi}$ and will be used even in the case when
$dim(\mathcal{H}_{\mathbb{R}})\geqslant 3$. Thus, we adjust the set up of this
section in such a way that it applies to all cases.

For the next three results, we assume
$\mathcal{H}_{\mathbb{R}}=\mathbb{R}^{2}\oplus\mathcal{K}_{\mathbb{R}}$, where
$\mathcal{K}_{\mathbb{R}}$ is a real Hilbert space $($could be $0)$, and
$\mathbb{R}^{2}$ is reducing subspace for $(U_{t})$ with associated sub
representation being ergodic i.e., $(\mathbb{R}^{2},U_{t})$ is as before.
Define
$\mathcal{O}=\\{e,\bar{e},\Omega\\}\cup\\{\xi\in\mathcal{K}_{\mathbb{C}}:W(\xi)\text{
is analytic for }(\sigma_{t}^{\varphi})\\}$.

The next lemma tracks $w^{*}$-limits of certain sequences of operators, which
in turn tracks the relative commutant of the centralizers.

######  Lemma 3.11.

For any $\chi\in\mathcal{H}^{\otimes_{q}i}$ and
$\eta\in\mathcal{H}^{\otimes_{q}j}$, $a,\,b,\,\alpha,\,\beta\in\mathbb{Z}$, we
have

$\displaystyle\lim_{n\to\infty}(1-q)^{2n}\langle\eta\bar{e}^{n+b}e^{n+\beta},\,\bar{e}^{n+a}e^{n+\alpha}\chi\rangle_{q}$
$\displaystyle=$ $\displaystyle\,\delta_{\begin{subarray}{c}a=b+j\\\
\alpha+i=\beta\end{subarray}}{d_{\infty}^{2}}\,\frac{\lambda^{(a-\beta)/2}}{(1-q)^{a+\beta}}\langle\eta,\,\frac{\bar{e}^{j}}{\|\bar{e}^{j}\|_{q}^{2}}\rangle_{q}\langle\frac{{e}^{i}}{\|e^{i}\|_{q}^{2}},\,\chi\rangle_{q}.$

###### Proof.

As one is taking limit as $n\rightarrow\infty$, one can assume that $n+a$,
$n+\alpha$, $n+b$ and $n+\beta$ are all positive and large. By linearity, we
can assume that $\eta$ and $\chi$ are elementary tensors in the letters from
$\mathcal{O}$. If $n+a\geqslant n+\alpha$, then $a-\alpha\geqslant 0$.
Consequently,
$(1-q)^{n}\bar{e}^{n+a}e^{n+\alpha}\chi=c_{q,r}(\chi)c_{q}(\bar{e}^{a-\alpha})(1-q)^{n}W(\bar{e}^{n+\alpha}e^{n+\alpha})\Omega$
is a bounded sequence from Eq. (11) and Lemma 3.9. Dealing with the case
$n+a<n+\alpha$ similarly (replacing $\chi$ by $e^{\alpha-a}\chi$), it follows
that $(1-q)^{n}\bar{e}^{n+a}e^{n+\alpha}\chi$ is bounded. Similarly,
$(1-q)^{n}\eta\bar{e}^{n+b}e^{n+\beta}$ is bounded.

First, assume that $\eta$ or $\chi$ contains a letter different from
$e,\bar{e},\Omega$. For simplicity assume $\eta=\eta_{1}\cdots\eta_{j}$ is
such that at least one letter in $\eta\not\in\\{e,\bar{e},\Omega\\}$. Let
$t=\min\\{l:\eta_{l}\not\in\\{e,\bar{e},\Omega\\},\,1\leqslant l\,\leqslant
j\\}$. Let $T=\prod_{l=1}^{t-1}c_{q}(\eta_{l})$. By Lemma 2.1, it follows that
there exists a finite set $F$ $($depending on $l)$ and scalars $c_{f,n}$ with
$\sup_{n}\left|c_{f,n}\right|<\infty$ for all $f\in F$,
$a_{f},\alpha_{f}\in\mathbb{Z}$ and vectors
$\chi_{f}\in\mathcal{F}_{q}(\mathcal{H})$ for all $f\in F$ such that

$\displaystyle T^{*}\bar{e}^{n+a}e^{n+\alpha}\chi=\sum_{f\in
F}c_{f,n}\bar{e}^{n+a_{f}}e^{n+\alpha_{f}}\chi_{f}.$

Fix $f\in F$. From Lemma 2.1 again, it follows that as $n\rightarrow\infty$,

$\displaystyle(1-q)^{n}c_{q}(\eta_{t})^{*}\bar{e}^{n+a_{f}}e^{n+\alpha_{f}}\chi_{f}=q^{2n+a_{f}+\alpha_{f}}(1-q)^{n}\bar{e}^{n+a_{f}}e^{n+\alpha_{f}}c_{q}(\eta_{t})^{*}\chi_{f}\rightarrow
0.$

Summing over $f\in F$, it follows that the limit in the statement is $0$ and
matches with the right hand side of the statement. Arguing similarly with
$\chi$ and using right creation operators, it follows that it is sufficient to
assume that the letters in $\eta,\chi$ are all in $\\{e,\bar{e},\Omega\\}$,
otherwise both sides in the statement are $0$.

We do the proof by induction on $i+j$. Note that
$\langle\bar{e}^{j},\,\frac{\bar{e}^{j}}{\|\bar{e}^{j}\|_{q}^{2}}\rangle_{q}=1$
and similarly for $e$. So, if $\eta$ is an elementary tensor in the letter $e$
and $\bar{e}$, the quantity
$\langle\eta,\,\frac{\bar{e}^{j}}{\|\bar{e}^{j}\|_{q}^{2}}\rangle_{q}$ is 1 if
$\eta=\bar{e}^{j}$ and 0 otherwise. Thus, counting the number of letters $e$
and $\bar{e}$ gives the Dirac condition.

We start with $i+j=0$. The Dirac condition is clear and from Eq. (11) we have

$\displaystyle\|\bar{e}^{n+a}e^{n+\beta}\|_{q}^{2}=\lambda^{(a-\beta)/2}[n+a]_{q}![n+\beta]_{q}!=\lambda^{(a-\beta)/2}(1-q)^{-2n-a-\beta}d_{n+a}d_{n+\beta}.$
(15)

Thus, the limit in the statement is
$\lambda^{(a-\beta)/2}(1-q)^{-(a+\beta)}d_{\infty}^{2}$.

Assume for instance $j>0$. Let $\eta=e_{1}\cdots e_{j}$ with
$e_{k}\in\\{e,\bar{e}\\}$. We have

$\displaystyle(1-q)^{2n}\langle\eta\bar{e}^{n+b}e^{n+\beta},\,\bar{e}^{n+a}e^{n+\alpha}\chi\rangle_{q}=(1-q)^{2n}\langle\eta^{\prime}\bar{e}^{n+b}e^{n+\beta},c_{q}(e_{1})^{*}\bar{e}^{n+a}e^{n+\alpha}\chi\rangle_{q},$
(16)

with $\eta^{\prime}=e_{2}\cdots e_{j}$. By making arguments as in the first
paragraph of the proof, it follows that
$(1-q)^{n}\eta^{\prime}\bar{e}^{n+b}e^{n+\beta}$ is bounded. Thus, using Lemma
3.10, we get that the limit in Eq. (16) is zero unless $e_{1}=\bar{e}$. If
this is so, then from Lemma 2.1 we have

$\displaystyle
c_{q}(e_{1})^{*}\bar{e}^{n+a}e^{n+\alpha}\chi=\lambda^{1/2}[n+a]_{q}\bar{e}^{n+a-1}e^{n+\alpha}\chi+q^{2n+a+\alpha}\bar{e}^{n+a}e^{n+\alpha}\big{(}c_{q}(e_{1})^{*}\chi\big{)}.$

Thus, by induction the limit exists and

$\lim_{n}(1-q)^{2n}\langle\eta\bar{e}^{n+b}e^{n+\beta},\bar{e}^{n+a}e^{n+\alpha}\chi\rangle_{q}=\frac{\lambda^{1/2}}{1-q}\lim_{n}(1-q)^{2n}\langle\eta^{\prime}\bar{e}^{n+b}e^{n+\beta},\bar{e}^{n+a-1}e^{n+\alpha}\chi\rangle_{q}.$

One can argue in the same way if $i>0$ using right creation operators and Eq.
(9) when dealing with $\chi$. $\Box$

The next lemma is the key to factoriality and irreducible centralizers.

######  Lemma 3.12.

The $w^{*}$-limit of $(1-q)^{2n}W_{r}(\bar{e}^{n}e^{n})W(\bar{e}^{n}e^{n})$
exists and is the positive rank-one operator $T_{\xi}$ where

$\displaystyle\xi=S_{\infty}^{*}(\Omega)=\|\cdot\|_{q}\text{-}\lim_{n\to\infty}(1-q)^{n}\lambda^{n/2}W(\bar{e}^{n})e^{n}=d_{\infty}\sum_{k=0}^{\infty}c_{k}^{2}(1-q)^{k}\lambda^{k/2}{\bar{e}}^{k}e^{k}.$

Needless to say, $T_{\xi}$ is a scalar multiple of the rank-one projection
$P_{\frac{\xi}{\left\|\xi\right\|_{q}}}$.

###### Proof.

First note that $e,\bar{e}$ are analytic vectors for
$(\Delta^{it}_{\varphi})$, thus words in them belong to the Tomita algebra
associated to $\varphi$. Further, $\bar{e}^{n}e^{n}\in M_{q}^{\varphi}\Omega$
and $J(\bar{e}^{n}e^{n})=\bar{e}^{n}e^{n}$ for all $n$ $($see Thm. 2.4$)$.
Consequently,
$z_{n}=(1-q)^{2n}W_{r}(\bar{e}^{n}e^{n})W(\bar{e}^{n}e^{n})\in\mathbf{B}(\mathcal{F}_{q}(\mathcal{H}))$
is a bounded sequence; also $\xi_{n}=(1-q)^{n}\bar{e}^{n}e^{n}$ is bounded in
$n$ $($by Lemma 3.9$)$.

To show the $w^{*}$-convergence, we just need to find the limit of $\langle
z_{n}\Phi,\Psi\rangle_{q}$ where $\Psi\in\mathcal{H}^{\otimes_{q}l}$ and
$\Phi\in\mathcal{H}^{\otimes_{q}k}$, $k,l\geqslant 0$. By density, we can also
assume that $\Phi=e_{1}e_{2}\cdots e_{k}$ and $\Psi=f_{1}f_{2}\cdots f_{l}$
where $e_{i},f_{j}\in\mathcal{O}$. Then, we have

$\displaystyle\langle\Psi,\,z_{n}\Phi\rangle_{q}$
$\displaystyle=\langle\Psi,~{}~{}(1-q)^{2n}W_{r}(\bar{e}^{n}e^{n})W(\bar{e}^{n}e^{n})\Phi\rangle_{q}$
$\displaystyle=(1-q)^{2n}\langle
W(\Psi)\bar{e}^{n}e^{n},~{}~{}W_{r}(\Phi)\bar{e}^{n}e^{n}\rangle_{q}.$

By the Wick formulas $($Prop. 2.3$)$,

$W(\Psi)=\sum_{0\leqslant j\leqslant l,\,\sigma\in
S_{l,\,j}}q^{|\sigma|}u_{\sigma,\,j}\,\text{ if }l>0,\quad
W_{r}(\Phi)=\sum_{0\leqslant i\leqslant k,\,\rho\in S_{k,\,i}}q^{|{\rm
flip}\,\circ\,\rho|}v_{\rho,\,i}\text{ if }k>0,$

where

$\displaystyle u_{\sigma,\,j}=c_{q}(f_{\sigma(1)})\cdots
c_{q}({f_{\sigma(j)}})c_{q}(\overline{f_{\sigma(j+1)}})^{*}c_{q}(\overline{f_{\sigma(l)}})^{*}$
and $\displaystyle v_{\rho,\,i}=c_{q,r}(e_{\rho(1)})\cdots
c_{q,r}({e_{\rho(i)}})c_{q,r}(\overline{g_{\rho(i+1)}}^{r})^{*}c_{q,r}(\overline{g_{\rho(k)}}^{r})^{*},$

where $g_{\rho(i+1)},\cdots,g_{\rho(k)}\in\mathcal{H}_{\mathbb{C}}$ are
vectors that correct the difference between the left and right conjugations
$($by $e_{\rho(i+1)},\cdots,e_{\rho(k)}$ respectively$)$. Note that
$W(\Psi)=1$ if $l=0$ and $W_{r}(\Phi)=1$ if $k=0$.

Let $\min(k,l)>0$. One observes that the left or right annhilation operators
in a symbol other than $e,\bar{e}$ does not contribute in the inner product
$\langle\Psi,\,z_{n}\Phi\rangle_{q}$. Thus, the contributing factor in
$\langle\Psi,\,z_{n}\Phi\rangle_{q}$ comprises of two scenarios: $(i)$ when a
generic term in the Wick expansion formula of $W_{r}(\Phi)$ and $W(\Psi)$ both
consists of creation operators only, $(ii)$ when the annihilation operators in
a generic term in the Wick expansion formula of $W_{r}(\Phi)$ or $W(\Psi)$
consists only of the symbols $e$ or $\bar{e}$.

For both cases, if the letters in the creation operators $($either left or
right or both, as the case may be$)$ consist of a symbol different from $e$ or
$\bar{e}$, then by Lemma 3.11, the associated limit contributing to
$\langle\Psi,\,z_{n}\Phi\rangle_{q}$ goes to $0$ as $n\rightarrow\infty$.
Consequently, we can assume that $e_{i},f_{j}\in\\{e,\bar{e}\\}$. The same
conclusion holds when $\min(k,l)=0$ and $\max(k,l)>0$. It is clear that, this
reduction is nothing but compressing $M_{q}$ by the Jones’ projection onto its
subalgebra $\Gamma_{q}(\mathbb{R}^{2},U_{t})^{\prime\prime}$ $($which possess
$\varphi$-preserving conditional expectation$)$ to reduce to the set up when
$\dim(\mathcal{H}_{\mathbb{R}})=2$.

Therefore, we can now assume that $W_{r}(\Phi)=\sum_{0\leqslant i\leqslant
k,\,\rho\in S_{k,\,i}}q^{|{\rm
flip}\,\circ\,\rho|}\lambda^{n_{k,i}}v_{\rho,\,i}$ if $k>0$, where $n_{k,i}$
is an integer to correct the difference between the left and right
conjugations and the right annihilation operators in $v_{\rho,\,i}$ consists
of symbols from $\\{e,\bar{e}\\}$.

If $l>0$, then using Lemma 3.10 $(l-j)$ times and Lemma 2.1, we get that
$u_{\sigma,\,j}(\xi_{n})$ goes to $0$ in $\left\|\cdot\right\|_{q}$ unless
$f_{\sigma(j+1)}=\cdots=f_{\sigma(l)}=e$. If this is so, then using Lemma 2.1
and Lemma 3.1 we have:
$u_{\sigma,\,j}(\xi_{n})=\frac{[n]_{q}!}{[n-l+j]_{q}!}\lambda^{(l-j)/2}(1-q)^{n}\Psi_{j}\bar{e}^{n-l+j}e^{n}$,
where $\Psi_{j}=f_{\sigma(1)}f_{\sigma(2)}\cdots f_{\sigma(j)}$. Setting
$\Psi^{c}_{j}=f_{\sigma(j+1)}\cdots f_{\sigma(l)}$, we may rewrite
$u_{\sigma,\,j}(\xi_{n})$ in full generality as

$u_{\sigma,\,j}(\xi_{n})=\frac{[n]_{q}!}{[n-l+j]_{q}!}\lambda^{(l-j)/2}(1-q)^{n}\langle\Psi_{j}^{c},\,\frac{e^{l-j}}{\|{e}^{l-j}\|_{q}^{2}}\rangle_{q}\Psi_{j}\bar{e}^{n-l+j}e^{n}.$

We can do the same for $\Phi$ to get
$v_{\rho,\,i}(\xi_{n})=\frac{[n]_{q}!}{[n-k+i]_{q}!}\lambda^{(i-k)/2}(1-q)^{n}\bar{e}^{n}e^{n-k+i}e_{\rho(i)}\cdots
e_{\rho(1)}$ provided that $e_{\rho(i+1)}=\cdots=e_{\rho(k)}=\bar{e}$ if
$k>0$. Set $\Phi_{i}=e_{\rho(i)}\cdots e_{\rho(1)}$ and
$\Phi_{i}^{c}=e_{\rho(k)}\cdots e_{\rho(i+1)}$. Then,

$v_{\rho,\,i}(\xi_{n})=\frac{[n]_{q}!}{[n-k+i]_{q}!}\lambda^{(i-k)/2}(1-q)^{n}\langle\frac{\bar{e}^{k-i}}{\left\|\bar{e}^{k-i}\right\|_{q}^{2}},\,\Phi_{i}^{c}\rangle_{q}\,\bar{e}^{n}e^{n-k+i}\Phi_{i}.$

We are in position of using Lemma 3.11 with $a=\beta=0$ to get the existence
of

$\displaystyle\lim_{n}\langle
u_{\sigma,\,j}(\xi_{n}),\,v_{\rho,\,i}(\xi_{n})\rangle_{q}$ (17)
$\displaystyle=\,$ $\displaystyle\delta_{\begin{subarray}{c}2j=l\\\
2i=k\end{subarray}}\frac{d_{\infty}^{2}\lambda^{(i+j)/2}}{(1-q)^{i+j}}\langle\Psi_{j},\,\frac{\bar{e}^{j}}{\|\bar{e}^{j}\|_{q}^{2}}\rangle_{q}\langle\Psi_{j}^{c},\,\frac{e^{j}}{\|{e}^{j}\|_{q}^{2}}\rangle_{q}\langle\frac{{e}^{i}}{\|e^{i}\|_{q}^{2}},\,\Phi_{i}\rangle_{q}\langle{\frac{{\bar{e}}^{i}}{\|{\bar{e}}^{i}\|_{q}^{2}}},\,\Phi_{i}^{c}\rangle_{q},$

when $k,l>0$. This can be interpreted as decoupled scalar products. When $k$
or $l$ is $0$, there is no Wick product expansion in terms of creation and
annihilation operators, but the obvious modifications justify the existence of
the desired limit(s) using Lemma 3.11 and matches Eq. (17).

Thus, summing all the terms we get that for some
$\chi_{k}\in\mathcal{H}^{\otimes_{q}k}$ and
$\eta_{l}\in\mathcal{H}^{\otimes_{q}l}$,

$\lim_{n}\langle\Psi,\,z_{n}\Phi\rangle_{q}=\langle\chi_{k},\,\Phi\rangle_{q}.\langle\Psi,\,\eta_{l}\rangle_{q},\quad\forall\,k,l\geqslant
0.$

Now we will use some arguments to avoid the use of $q$-symetrization operators
to identify the vectors. Since $z_{n}$ is bounded, this justifies that the
$w^{*}$-limit exists and has rank 1. As $z_{n}=z_{n}^{*}$, we have
$\chi_{l}=\eta_{l}$ for all $l\geqslant 0$. Taking $\Phi=\Omega$, we have that
$z_{n}\Omega$ is weakly converging to
$\zeta=\langle\chi_{0},\Omega\rangle_{q}\big{(}\oplus_{m=0}^{\infty}\chi_{m}\big{)}$.

We want to identify $\xi=\oplus_{m=0}^{\infty}\chi_{m}$ as
$z_{n}\overset{w.o.t.}{\rightarrow}T_{\xi}$ $($the rank-one limit$)$, and to
do so we identify $\zeta$. To find $\zeta$, we want to consider
$\lambda^{-n/2}(1-q)^{n}\langle W(\Psi)\bar{e}^{n},\,\bar{e}^{n}\rangle_{q}$.
This is a bounded sequence $($see Lemma 3.3$)$.

Note that if $l=0$ (i.e., $\Psi=\Omega$), then
$\lim_{n}\lambda^{-n/2}(1-q)^{n}\langle
W(\Psi)\bar{e}^{n},\,\bar{e}^{n}\rangle_{q}=d_{\infty}$. Let $l>0$. As above,
we can use the Wick formula for $\Psi$ and the situation as in Lemma 3.11. We
get that $u_{\sigma,\,j}(\lambda^{-n/4}\bar{e}^{n})$ is $0$ unless
$f_{\sigma(j+1)}=\cdots=f_{\sigma(l)}=e$. If this is so,
$u_{\sigma,\,j}(\lambda^{-n/4}\bar{e}^{n})=\frac{[n]_{q}!}{[n-l+j]_{q}!}\lambda^{(l-j)/2-n/4}\Psi_{j}\bar{e}^{n-l+j}$
$($Lemma 3.1$)$. Taking scalar product with $\lambda^{-n/4}\bar{e}^{n}$,
recalling that $\|\bar{e}^{n}\|_{q}^{2}=[n]_{q}!\lambda^{n/2}$ and computing
again as in Lemma 3.11, we have

$\lim_{n}\lambda^{-n/2}(1-q)^{n}\langle
u_{\sigma,\,j}(\bar{e}^{n}),\,\bar{e}^{n}\rangle_{q}=\delta_{2j=l}\,d_{\infty}\lambda^{j/2}(1-q)^{-j}\langle\Psi_{j},\,\frac{\bar{e}^{j}}{\|\bar{e}^{j}\|_{q}^{2}}\rangle_{q}\langle\Psi_{j}^{c},\,\frac{e^{j}}{\|{e}^{j}\|_{q}^{2}}\rangle_{q}.$

Thus, comparing with Eq. (17) $($setting $k=0)$ we have

$\lim_{n}\langle\Psi,\,z_{n}\Omega\rangle_{q}=d_{\infty}\lim_{n}\lambda^{-n/2}(1-q)^{n}\langle
W(\Psi)\bar{e}^{n},\,\bar{e}^{n}\rangle_{q}.$

But using Eq. (5), we have

$\langle
W(\Psi)\bar{e}^{n},\,\bar{e}^{n}\rangle_{q}=\langle\Psi,\,W(\bar{e}^{n})W_{r}(\bar{e}^{n})^{*}\Omega\rangle_{q}=\lambda^{n}\langle\Psi,\,W(\bar{e}^{n})W(e^{n})\Omega\rangle_{q}.$

Since $S_{n}^{*}(\Omega)=\lambda^{n/2}(1-q)^{n}W(\bar{e}^{n})W(e^{n})\Omega$
$($use Lemma 3.2 or Eq. (5)$)$, we get that $\zeta=d_{\infty}S^{*}(\Omega)$.
To compute its value, one can use Lemma 3.1 and Lemma 3.2 to derive

$\displaystyle\lambda^{n/2}(1-q)^{n}W(\bar{e}^{n})e^{n}=\sum_{k=0}^{n}(1-q)^{k}\frac{d_{n}^{2}}{d_{n-k}d_{k}^{2}}\lambda^{k/2}\bar{e}^{k}e^{k}.$

By taking the limit in $n$ and using the dominated convergence theorem, we
have

$\zeta=d_{\infty}^{2}\sum_{k=0}^{\infty}c_{k}^{2}(1-q)^{k}\lambda^{k/2}{\bar{e}}^{k}e^{k},$

and this expression tallies with the expression in $(2)$ of Rem. 3.7.

Comparing the two expressions for $\zeta$ we have
$\langle\chi_{0},\Omega\rangle_{q}\chi_{0}=d_{\infty}^{2}\Omega$. Since
$\chi_{0}=\kappa\Omega$ where $\kappa\in\mathbb{C}$, it follows that
$\left|\kappa\right|^{2}=d_{\infty}^{2}$ $($scalar product is linear on the
right$)$, i.e., $\left|\kappa\right|=d_{\infty}$. Consequently,

$\displaystyle\xi=\frac{1}{\langle\chi_{0},\Omega\rangle_{q}}\zeta=\frac{1}{\bar{\kappa}}d_{\infty}^{2}\sum_{k=0}^{\infty}c_{k}^{2}(1-q)^{k}\lambda^{k/2}{\bar{e}}^{k}e^{k}=e^{-i\arg{(\kappa)}}d_{\infty}\sum_{k=0}^{\infty}c_{k}^{2}(1-q)^{k}\lambda^{k/2}{\bar{e}}^{k}e^{k}.$

Finally, note that a rank-one self-adjoint operator is uniquely determined
upto a phase factor of the associated vector. Also note that $\xi$ is not an
unit vector, thus $T_{\xi}$ is a positive scalar multiple of
$P_{\frac{\xi}{\left\|\xi\right\|_{q}}}$. $\Box$

The next lemma will be used particularly when
$\dim(\mathcal{H}_{\mathbb{R}})\geqslant 3$. It describes some amount of
mixing available when $\dim(\mathcal{H}_{\mathbb{R}})$ is ‘large’, which frees
one from any bargain with the parameters $\lambda$ and $q$ to decide the
factoriality unlike the case when $\dim(\mathcal{H}_{\mathbb{R}})=2$.

######  Lemma 3.13.

Suppose $\text{dim}(\mathcal{H}_{\mathbb{R}})\geqslant 3$ with $(U_{t})$ as in
the set up. Let $0\neq\eta\in\mathcal{H}_{\mathbb{R}}$ be such that
$\eta\perp\bar{e},e$ in $\langle\cdot,\cdot\rangle_{q}$. Further, let
$\\{u_{n}\\}$ be a sequence of unitaries in $vN(W(\eta))$ converging to $0$ in
the w.o.t. Then, $u_{n}W(\xi)u_{n}^{*}\rightarrow c1$ in the $w.o.t$ for some
$c>0$, where
$\xi=d_{\infty}\sum_{k=0}^{\infty}c_{k}^{2}(1-q)^{k}\lambda^{k/2}{\bar{e}}^{k}e^{k}$.

###### Proof.

Firstly, note that $vN(W(\eta))$ is a diffuse abelian von Neumann algebra [BM,
§4]. Thus, the desired sequence $\\{u_{n}\\}$ exists as in the statement.

By Lemma 3.12, the series defining $\xi$ is convergent in
$\mathcal{F}_{q}(\mathcal{H})$ in $\left\|\cdot\right\|_{q}$. Further,
$\\{(1-q)^{k}W(\bar{e}^{k}e^{k})\\}$ is a bounded sequence by Lemma 3.9. Since
$c_{k}$ is a bounded sequence, so
$d_{\infty}\sum_{k=0}^{N}c_{k}^{2}(1-q)^{k}\lambda^{k/2}W(\bar{e}^{k}e^{k})$
is convergent in norm as $N\rightarrow\infty$ in $M_{q}^{\varphi}$. Using Thm.
2.4, it follows that

$\displaystyle
W(\xi)=d_{\infty}\sum_{k=0}^{\infty}c_{k}^{2}(1-q)^{k}\lambda^{k/2}W(\bar{e}^{k}e^{k})\in
M_{q}^{\varphi}.$

Therefore, it is sufficient to show that
$(1-q)^{k}u_{n}W(\bar{e}^{k}e^{k})u_{n}^{*}$ converges to $0$ in the $w.o.t$
as $n\rightarrow\infty$ for each $k\in\mathbb{N}$. Further, the boundedness of
the sequence entails that it is sufficient to verify the convergence with
vectors of the form $\Psi=e_{1}e_{2}\cdots e_{l}\in\mathcal{H}^{\otimes_{q}l}$
and $\Phi=f_{1}f_{2}\cdots f_{m}\in\mathcal{H}^{\otimes_{q}l^{\prime}}$ where
$e_{i},f_{j}\in\mathcal{O}$ and $l,l^{\prime}\geqslant 0$.

Now, note that

$\displaystyle\langle\Phi,\,u_{n}W(\bar{e}^{k}e^{k})u_{n}^{*}\Psi\rangle_{q}$
$\displaystyle=\langle
u_{n}^{*}\Phi,\,W(\bar{e}^{k}e^{k})u_{n}^{*}\Psi\rangle_{q}$
$\displaystyle=\langle
W_{r}(\Phi)u_{n}^{*}\Omega,\,W(\bar{e}^{k}e^{k})W_{r}(\Psi)u_{n}^{*}\Omega\rangle_{q}$
$\displaystyle=\langle
W_{r}(\Psi)^{*}W_{r}(\Phi)u_{n}^{*}\Omega,\,W(\bar{e}^{k}e^{k})u_{n}^{*}\Omega\rangle_{q}.$

When $k>0$, this last quantity goes to 0 as $n\to\infty$ as
$\bar{e}^{k}e^{k}\bot\,span\\{\eta^{l}:\ l\geqslant 0\\}$ $($see [BM, §4]$)$.
This a consequence of the Wick formula and Lemma 1 in [SW] as explained around
(4.2) there.

As $\text{span}_{\mathbb{C}}\mathcal{O}$ is dense in
$\mathcal{F}_{q}(\mathcal{H})$ and $(1-q)^{k}W({\bar{e}}^{k}e^{k})$ is
uniformly bounded $($Lemma 3.9$)$, it follows that
$(1-q)^{k}u_{n}W(\bar{e}^{k}e^{k})u_{n}^{*}\rightarrow 0$ as
$n\rightarrow\infty$ in the $w.o.t$ for each $k\geqslant 1$. Clearly
$c=d_{\infty}$ and the proof is complete. $\Box$

##  4\. Factoriality

This is the main section of this paper. Here we establish the factoriality of
$M_{q}$. The set up of §3.2 will be in force in this section. As informed
earlier, we will have to compromise with the parameter $\lambda$ to assert the
factoriality of $M_{q}$ when $\dim(\mathcal{H}_{\mathbb{R}})=2$. In this case
though, we will directly deduce that $M_{q}^{\varphi}$ is irreducible. When
$\dim(\mathcal{H}_{\mathbb{R}})\geqslant 3$, $M_{q}$ is a factor regardless of
the value of the parameter $\lambda$ defining $(U_{t})$ on $\mathbb{R}^{2}$
and $-1<q<1$. However, deducing that $M_{q}^{\varphi}$ is irreducible will not
be direct and will be constrained. When $q=0$, $M_{0}$ is the free Araki-Woods
factor [S97] and there is nothing to prove in this case.

As an outcome of the results in §3, we have the following:

######  Lemma 4.1.

With $\xi$ as in Lemma 3.12, let $P$ be the orthogonal projection onto
$[M_{q}^{\varphi}M_{q}^{\prime}\xi]^{\perp}$. Then,
$(M_{q}^{\varphi})^{\prime}\cap M_{q}$ is trivial if and only if $P=0$. In
particular, $M_{q}^{\varphi}$ and $M_{q}$ are factors if and only if $P=0$.

###### Proof.

It is clear that $P\in\mathcal{(}M_{q}^{\varphi})^{\prime}\cap M_{q}$. Suppose
that $(M_{q}^{\varphi})^{\prime}\cap M_{q}=\mathbb{C}1$. Then, $P=0$ as
$\xi\neq 0$ and $M_{q}^{\varphi}\text{ and }M_{q}$ are factors.

Conversely, by Lemma 3.12 it follows that $T_{\xi}\in M_{q}^{\varphi}\vee
M_{q}^{\prime}$. Let $x\in(M_{q}^{\varphi})^{\prime}\cap M_{q}$ so that
$xT_{\xi}=T_{\xi}x$. In particular, $x(\xi)=\lambda\xi$ for some
$\lambda\in\mathbb{C}$. Then, for any $m\in M_{q}^{\varphi}$ and $n\in
M_{q}^{\prime}$, $x(mn(\xi))=\lambda mn(\xi)$. If $P=0$, this means that
$x(\eta)=\lambda\eta$ for all $\eta\in\mathcal{F}_{q}(\mathcal{H})$ and thus
$x=\lambda 1$. $\Box$

Now we are in position to state the first theorem on factoriality.

######  Theorem 4.2.

Let $q\neq 0$. If $\dim(\mathcal{H}_{\mathbb{R}})=2$ and $(U_{t})$ is ergodic
then $(M_{q}^{\varphi})^{\prime}\cap M_{q}=\mathbb{C}1$ $($and hence
$M_{q}^{\varphi}$ and $M_{q}$ are factors$)$ when

$\lambda<\begin{cases}(1+\frac{C_{q}}{d_{\infty}})^{-2},\quad&\text{if
}q>0;\\\ (1+\frac{(C_{q}(1-q))^{2}D(q)}{d_{\infty}})^{-2},\quad&\text{if
}q<0.\end{cases}$

###### Proof.

Consider $P$ from Lemma 4.1. Then, we have $P(\xi)=0$. Thanks to Lemma 3.12
and Eq. (5), we have that

$\displaystyle 0=\langle P(\xi),\,\Omega\rangle_{q}$ $\displaystyle=$
$\displaystyle\lim_{n}\lambda^{n/2}(1-q)^{n}\langle
PW(e^{n})^{*}W(e^{n})\Omega,\,\Omega\rangle_{q}$ $\displaystyle=$
$\displaystyle\lim_{n}\lambda^{n/2}(1-q)^{n}\langle
W(e^{n})^{*}W(e^{n})P\Omega,\,P\Omega\rangle_{q}$ $\displaystyle=$
$\displaystyle\lim_{n}\lambda^{n/2}(1-q)^{n}\|W(e^{n})P(\Omega)\|_{q}^{2}.$

Thus, $\eta_{n}=\lambda^{n/4}(1-q)^{n/2}W(e^{n})P(\Omega)$ goes to 0 in
$\left\|\cdot\right\|_{q}$. Thanks to Lemma 3.3 $(i)$, we still get that
$\lambda^{n/4}(1-q)^{n/2}c_{q}(e)^{*n}\eta_{n}=S_{n}(P(\Omega))$ goes to 0.
Consequently, $S_{\infty}(P(\Omega))=0$. But $S_{\infty}$ is invertible by
Lemma 3.6 and the hypothesis, and hence $P(\Omega)=0$ forcing $P=0$. Now use
Lemma 4.1. $\Box$

Now, we assume that
$\mathcal{H}_{\mathbb{R}}=\mathbb{R}^{2}\oplus\mathcal{K}_{\mathbb{R}}$ where
$\mathcal{K}_{\mathbb{R}}$ is a non-zero real Hilbert space, and
$\mathbb{R}^{2}$ is reducing subspace for $(U_{t})$ with associated sub
representation being ergodic i.e., $(\mathbb{R}^{2},U_{t})$ is as in §3.1. It
should be noted that unlike Thm. 4.2, the invertibility of the operator
$S_{\infty}$ has no role to play in deciding the factoriality in this case.

######  Theorem 4.3.

Suppose $\text{dim}(\mathcal{H}_{\mathbb{R}})\geqslant 3$. Then $M_{q}$ is a
factor.

###### Proof.

Let $P$ be the orthogonal projection onto $[M_{q}M_{q}^{\prime}\xi]^{\perp}$,
where $\xi$ is as in Lemma 3.13. Since $dim(\mathcal{H}_{\mathbb{R}})\geqslant
3$, choose $0\neq\eta\in\mathcal{H}_{\mathbb{R}}$ such that $s_{q}(\eta)$ is
analytic for $(\sigma_{t}^{\varphi})$ and $\eta\perp\\{e,\bar{e}\\}$ in
$\langle\cdot,\rangle_{q}$. Then, $u_{n}=e^{-ins_{q}(\eta)}$ is an analytic
sequence of unitaries such that $u_{n}\rightarrow 0$ in the $w^{*}$-topology.
The $w^{*}$-convergence holds as the spectral measure of $s_{q}(\eta)$ is
Lebesgue absolutely continuous $($see [BM, §4]$)$. By Lemma 3.13, it follows
that

$\displaystyle
J\big{(}\sigma_{\frac{i}{2}}^{\varphi}(u_{n})\big{)}^{*}Ju_{n}^{*}\xi=u_{n}^{*}W(\xi)u_{n}\Omega\rightarrow
c\Omega\text{ weakly as }n\rightarrow\infty.$

It follows that $[M_{q}M_{q}^{\prime}\xi]$ is weakly dense in
$\mathcal{F}_{q}(\mathcal{H})$, and hence norm dense in
$\mathcal{F}_{q}(\mathcal{H})$ by a theorem of Mazur. It follows that $P=0$.

From Lemma 3.12, we have $T_{\xi}\in M_{q}^{\varphi}\vee M_{q}^{\prime}$. Let
$x\in\mathcal{Z}(M_{q})$. Then, $xT_{\xi}=T_{\xi}x$. Replacing the role of
$M_{q}^{\varphi}$ by $M_{q}$ in the proof of Lemma 4.1 $($second paragraph$)$,
the argument is verbatim. $\Box$

######  Remark 4.4.

The following comments are in order.

  1. (1)

In view of the main results in [BM, R05] and the results in this section, the
factoriality problem for
$\Gamma_{q}(\mathcal{H}_{\mathbb{R}},U_{t})^{\prime\prime}$ remains open only
in the case when $dim(\mathcal{H}_{\mathbb{R}})=2$, $(U_{t})$ is ergodic and
$\lambda$ is not small in the sense of Thm. 4.2.

  2. (2)

When $dim(\mathcal{H}_{\mathbb{R}})=2$, our proof is different than the
existing proofs of the factoriality of $M_{q}$ under various assumptions $($in
[BM, Hi03, R05, S97, Ne15]$)$ because of the role of the operator $S_{\infty}$
in the proof, and this process will not work when $q=0$ and
$\lambda>\frac{1}{2}$ (see Rem. 3.7).

  3. (3)

Let $\eta\in\mathcal{H}_{\mathbb{R}}$ be such that
$\left\|\eta\right\|_{U}=1\,(=\left\|\eta\right\|_{q})$ and $\eta$ is not
fixed by $(U_{t})$. Then, either $(U_{t})$ has a non-zero weakly mixing
component or has a two-dimensional ergodic sub representation characterized by
$\lambda\in(0,1)$. If $dim(\mathcal{H}_{\mathbb{R}})\geqslant 3$ or
$dim(\mathcal{H}_{\mathbb{R}})=2$, $q\neq 0$ and $\lambda$ is ‘small’ in the
sense of Thm. 4.2 or $dim(\mathcal{H}_{\mathbb{R}})=2$ and $q=0$, then
$vN(W(\eta))\subseteq M_{q}$ is a split inclusion $($see [BM2, Thm. 4.6]$)$.

##  5\. Revisiting Centralizer

The first result on $M_{q}^{\varphi}$ is Thm. 3.2 of [Hi03]. In §2.5 and in
[BM, §7] we have discussed on the structure of $M_{q}^{\varphi}$. In this
section, we continue our discussion on $M_{q}^{\varphi}$. Note that
$M_{q}^{\varphi}$ depends on the almost periodic component of $(U_{t})$ $($see
Thm. 2.4$)$. Thm. 7.1 of [BM] says that if there is a non-trivial fixed point
of $(U_{t})$ and the almost periodic component of $(U_{t})$ is at least two-
dimensional then $(M_{q}^{\varphi})^{\prime}\cap M_{q}=\mathbb{C}1$.

Consequently, when $\dim(\mathcal{H}_{\mathbb{R}})=2$ the last statement
together with Thm. 4.2 provide all that is known about $M_{q}^{\varphi}$. When
the almost periodic part of $(U_{t})$ is three-dimensional or five-dimensional
one can apply [BM, Thm. 7.1]. Now we proceed to describe the structure of
$M_{q}^{\varphi}$ when the almost periodic part of $(U_{t})$ is sufficiently
large.

######  Theorem 5.1.

Let $\mathcal{H}_{\mathbb{R}}^{ap}\subseteq\mathcal{H}_{\mathbb{R}}$ denote
the almost periodic part of $(U_{t})$. If
$dim(\mathcal{H}_{\mathbb{R}}^{ap})\geqslant 5$, then
$(M_{q}^{\varphi})^{\prime}\cap M_{q}=\mathbb{C}1$. In particular,
$M_{q}^{\varphi}$ is a factor.

###### Proof.

If $(U_{t})$ has a non-zero fixed point then there is nothing to prove. So we
only have to prove in the case when
$\dim(\mathcal{H}_{\mathbb{R}}^{ap})\geqslant 6$ and $(U_{t})$ is ergodic. In
this case,
$\mathcal{H}_{\mathbb{R}}=\oplus_{i=1}^{3}\mathbb{R}^{2}_{i}\oplus\mathcal{K}_{\mathbb{R}}$,
where $\mathbb{R}^{2}_{i}:=\mathbb{R}^{2}$ for $1\leqslant i\leqslant 3$ and
$\mathcal{K}_{\mathbb{R}}$ are invariant subspaces of $(U_{t})$.

Denote
$M_{q}^{(1)}=\Gamma_{q}(\mathbb{R}^{2}_{1},{U_{t}}_{\upharpoonleft\mathbb{R}^{2}_{1}})^{\prime\prime}$,
$M_{q}^{(2)}=\Gamma_{q}(\mathbb{R}^{2}_{2}\oplus\mathbb{R}_{3}^{2},{U_{t}}_{\upharpoonleft\mathbb{R}^{2}_{2}\oplus\mathbb{R}_{3}^{2}})^{\prime\prime}$.
Then $M_{q}^{(1)},M_{q}^{(2)}$ are unital subalgebras of $M_{q}$ via the
$q$-Gaussian functoriality [Hi03]. Note that $M_{q}^{(1)}$ and $M_{q}^{(2)}$
posses $\varphi$-preserving conditional expectations by Takesaki’s theorem
[T72].

By Thm. 4.3, it follows that $M_{q}$ and $M_{q}^{(2)}$ are both factors. We
claim that both $M_{q}$ and $M_{q}^{(2)}$ cannot be of type $\rm{I}$ and hence
they are diffuse. Let us first assume this claim and finish the proof. Since
$({U_{t}}_{\upharpoonleft\mathbb{R}^{2}_{2}\oplus\mathbb{R}_{3}^{2}})$ is
almost periodic, $(M_{q}^{(2)})^{\varphi_{\upharpoonleft M_{q}^{(2)}}}$ is
also diffuse [DM, Thm. 7.9]. Let
$u_{n}\in(M_{q}^{(2)})^{\varphi_{\upharpoonleft M_{q}^{(2)}}}$ be a sequence
of unitaries that go to $0$ in the $w.o.t$. By Thm. 2.4, it follows that
$u_{n}\in M_{q}^{\varphi}$ as well.

Let $\xi$ be as in Lemma 3.13 obtained by regarding $M_{q}^{(1)}$ as the von
Neumann algebra in §3.1. From Lemma 3.13, it follows that
$u_{n}^{*}W(\xi)u_{n}\overset{w.o.t}{\rightarrow}c1$, where $c>0$ is a scalar.
It follows that
$[M_{q}^{\varphi}M_{q}^{\prime}\xi]=\mathcal{F}_{q}(\mathcal{H})$. Thanks to
Lemma 4.1, the argument is complete.

Now it remains to establish the claim. The proof of the claim is identical for
both $M_{q}^{(2)}$ and $M_{q}$. So we prove it for $M_{q}$ and work with the
sub representation
$(\mathbb{R}_{1}^{2},{U_{t}}_{\upharpoonleft\mathbb{R}_{1}^{2}})$ as in §3.2.

Recall that a factor $M$ equipped with a faithful normal state $\psi$ is of
type $\rm{I}$ if and only if, $M\ni
x\mapsto\Delta_{\psi}^{\frac{1}{4}}x\Omega_{\psi}\in L^{2}(M,\psi)$ is a
compact embedding $($see [BDL, Cor. 2.9] or [DM, Thm. 7.13]$)$. Note that
$\\{(1-q)^{n}W({\bar{e}}^{n}e^{n})\\}_{n\geqslant 1}\subseteq M_{q}^{\varphi}$
is bounded from Lemma 3.9. However,
$\Delta^{\frac{1}{4}}(1-q)^{n}W({\bar{e}}^{n}e^{n})\Omega=(1-q)^{n}{\bar{e}}^{n}e^{n}$,
and this sequence has no converging subsequence in
$\mathcal{F}_{q}(\mathcal{H})$. Therefore, the symmetric embedding of $M_{q}$
in $\mathcal{F}_{q}(\mathcal{H})$ is not compact proving the claim. $\Box$

######  Remark 5.2.

  1. (1)

If $dim(\mathcal{H}_{\mathbb{R}}^{ap})=4$ and $(U_{t})$ is ergodic, then upon
assuming that the parameter $\lambda\in(0,1)$ which defines a two-dimensional
sub representation is small in the sense of Thm. 4.2, one can still conclude
that $(M_{q}^{\varphi})^{\prime}\cap M_{q}=\mathbb{C}1$ $($when $q\neq 0)$.

  2. (2)

In all cases where $(M_{q}^{\varphi})^{\prime}\cap M_{q}=\mathbb{C}1$, the
$S$-invariant of Connes can be directly deduced from the modular theory of the
vacuum state and is exactly as in [BM, Thm. 8.2].

  3. (3)

Note that the last part of the proof of Thm. 5.1 along with Thm. 5.2 of [DM2]
says that if $\mathcal{H}^{ap}\neq 0$, then $M_{q}^{\varphi}$ cannot be
discrete $($direct sum of matrix algebras$)$.

  4. (4)

Suppose that $dim(\mathcal{H}_{\mathbb{R}})=4$ and $(U_{t})$ is ergodic. In
this case if $\zeta\in\mathcal{H}_{\mathbb{R}}$ be such that
$\left\|\zeta\right\|_{q}=1$, then $M_{\zeta}=vN(s_{q}(\zeta))\subseteq M_{q}$
is a quasi-split inclusion $($see [BM2, Thm. 4.6]$)$. Since $M_{q}$ is a
factor $($Thm. 4.3$)$, $M_{\zeta}\otimes 1\subseteq
M_{q}\otimes\mathbf{B}(\mathcal{K})$ is a split inclusion $($see [BM2, Thm.
3.8]$)$, where $\dim(\mathcal{K})=\aleph_{0}$. It follows that there is no
normal conditional expectation from $M_{q}$ onto $M_{\zeta}$. This forces that
$M_{q}$ cannot be a $\rm{II}_{1}$ factor, for if it were there would have been
a normal conditional expectation from $M_{q}$ onto $M_{\zeta}$ preserving the
canonical trace.

In view of Rem. 5.2, we are still not in position to compute the $S$-invariant
when $\dim(\mathcal{H}_{\mathbb{R}})=4$ and $(U_{t})$ is ergodic. Therefore,
we conclude this section by proving that $M_{q}$ is never semifinite.

######  Theorem 5.3.

$M_{q}$ is not semifinite. In particular, if $dim(\mathcal{H}_{\mathbb{R}})=4$
then $M_{q}$ is a type $\rm{III}$ factor.

###### Proof.

If $(U_{t})$ has a fixed (non-zero) vector or a non-trivial weakly mixing
component, the statement follows from [BM]. Therefore, we assume that
$(\mathcal{H}_{\mathbb{R}},U_{t})$ is as in the set up of §3.1.

We claim that $(\sigma_{t}^{\varphi})$ cannot be inner. Suppose there exists a
one-parameter group of unitaries $(u_{t})\subseteq M_{q}^{\varphi}$ such that
$\sigma_{t}^{\varphi}(x)=u_{t}xu_{t}^{*}$ for all $x\in M_{q}$ and
$t\in\mathbb{R}$. Then, by Lemma 3.12, we have $u_{t}T_{\xi}=T_{\xi}u_{t}$ for
all $t\in\mathbb{R}$. Therefore, there exists scalars $z_{t}$ with
$\left|z_{t}\right|=1$ such that $u_{t}\xi=z_{t}\xi$ for all $t$. Replacing
$u_{t}$ by $\overline{z_{t}}u_{t}$, we may assume that $z_{t}=1$ for all $t$.
Then, $u_{t}W(\xi)=W(\xi)$ for all $t$.

Let $\mathcal{K}=Ker(W(\xi))$ and $\mathcal{L}=\mathcal{K}^{\perp}$. Then,
${u_{t}}_{\upharpoonleft\mathcal{L}}=1_{\mathcal{L}}$. Put
$w_{n}=\left|W\big{(}(\lambda^{\frac{1}{4}}(1-q)^{\frac{1}{2}}e)^{n}\big{)}\right|^{2}$.
By the proof of Thm. 3.12, we have

$\displaystyle W(\xi)=w.o.t.\text{-}\lim_{n}w_{n}.$

Hence,
$\mathcal{K}=\\{\zeta\in\mathcal{F}_{q}(\mathcal{H}):w_{n}\zeta\overset{w}{\rightarrow}0\\}$
and it follows that $W(e)\mathcal{K}\subseteq\mathcal{K}$ $($use Lemma 3.2$)$.
Thus, decomposing $\mathbf{B}(\mathcal{F}_{q}(\mathcal{H}))$ with respect
$\mathcal{K}\oplus\mathcal{L}$, one has the form

$\displaystyle W(e)=\begin{pmatrix}x&y\\\ 0&w\end{pmatrix}\text{ and
}u_{t}=\begin{pmatrix}v_{t}&0\\\ 0&1\end{pmatrix}\quad\forall\,t.$

Since $u_{t}W(e)u_{t}^{*}=\sigma_{t}^{\varphi}(W(e))=e^{it\log\lambda}W(e)$
for all $t$ $($use Eq. (5)$)$, so choosing $t$ such that $t\log\lambda\not\in
2\pi\mathbb{Z}$ it follows that $w=0$. Thus, $Ran(W(e))\subseteq\mathcal{K}$.

This is false as $e\not\in\mathcal{K}$. Indeed,

$\displaystyle W(\xi)$
$\displaystyle=w.o.t.\text{-}\lim_{n}\lambda^{\frac{n}{2}}(1-q)^{n}W(\bar{e})^{n}W(e)^{n}$
$\displaystyle=\lambda^{\frac{1}{2}}(1-q)W(\bar{e})W(\xi)W(e).$

Consequently, $\xi=\lambda^{\frac{1}{2}}(1-q)W(\bar{e})W(\xi)e\neq 0$. This
completes the proof. $\Box$

Acknowledgements: P. Bikram, K. Mukherjee and É. Ricard acknowledge the
support of the grant 6101-1 from the CEFIPRA. É. Ricard is also supported by
ANR-19-CE40-0002. S. Wang would like to thank C. Houdayer and M. Wasilewski
for fruitful discussions at the early stage of this project.

## References

  * [A11] S. Avesec, _Strong solidity of the q-Gaussian algebras for all $-1<q<1$_, preprint, arXiv arXiv:1110.4918, (2011).
  * [BM] 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.
  * [BM2] P. Bikram and K. Mukherjee, _On the commutants of generators of $q$-deformed Araki-Woods von Neumann algebras_, to appear in Publ. Res. Inst. Math. Sci., arXiv:2004.10012, (2020).
  * [BHV15] R. Boutonnet, C. Houdayer and S. Vaes, _Strong solidity of free Araki-Woods factors_ , Amer. J. Math., 140 (2018), no. 5, 1231–1252.
  * [B99] M. Bo$\overset{.}{\text{z}}$ejko, _Ultracontractivity and strong Sobolev inequality for $q$-Ornstein-Uhlenbeck semigroup $(-1<q<1)$_, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (1999), no. 2, 203–220.
  * [BKS97] M. Bo$\overset{.}{\text{z}}$ejko, B. Kümmerer and R. Speicher, _$q$ -Gaussian processes: non-commutative and classical aspects_, Comm. Math. Phys., 185 (1997), no. 1, 129–154.
  * [BS91] M. Bo$\overset{.}{\text{z}}$ejko and R. Speicher, _An example of a generalized Brownian motion_ , Comm. Math. Phys., 137 (1991), no. 3, 519–531.
  * [BDL] D. Buchholz, C. D’Antoni and R. Longo, _Nuclear maps and modular structures. $\rm{I}$. General Properties_, J. Funct. Anal., 88 (1990), no. 2, 233–250.
  * [CFM] J. Cameron, J. Fang and K. Mukherjee, _Mixing and weak mixing abelian subalgebras of type $\rm{II}_{1}$ factors_, J. Funct. Anal., 272 (2017), no. 7, 2697–2725.
  * [D14] Y. Dabrowski, _A free stochastic partial differential equation_ , Ann. Inst. H. Poincaré Probab. Statist., 50 (2014), no. 4, 1404–1455.
  * [DM] D. De and K. Mukherjee, _On bounded coordinates in GNS spaces_ , preprint, 2020.
  * [DM2] D. De and K. Mukherjee, _On symmetric embedding of type $\rm{I}$ factors_, J. Math. Phys., 62 (2021), no. 1, 013511, 28 pp.
  * [H75] U. Haagerup, _The standard form of von Neumann algebras_ , Math. Scand., 37 (1975), no. 2, 271–283.
  * [Hi03] F. Hiai, _$q$ -deformed Araki-Woods algebras_, Operator algebras and Mathematical Physics (Constanţa, 2001), Theta, Bucharest, (2003), 169–202.
  * [H09] C. Houdayer, _Free Araki-Woods factors and Connes’ bicentralizer problem_ , Proc. Amer. Math. Soc., 137 (2009), no. 11, 3749–3755.
  * [HR11] C. Houdayer and É. Ricard, _Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors_ , Adv. Math., 228 (2011), no. 2, 764–802.
  * [JSW96] P. E. T. Jorgensen, L. M. Schmitt and R. F. Werner, _q-canonical commutation relations and stability of the Cuntz algebra_ , Pacific J. Math., 165 (1994), no. 1, 131–151.
  * [Ne15] B. Nelson, _Free monotone transport without a trace_ , Comm. Math. Phys., 334 (2015), no. 3, 1245–1298.
  * [N04] A. Nou, _Non injectivity of the $q$-deformed von Neumann algebra_, Math. Ann., 330 (2004), no. 1, 17–38.
  * [N06] A. Nou, _Asymptotic matricial models and QWEP property for $q$-Araki–Woods algebras_, J. Funct. Anal., 232 (2006), no. 2, 295–327.
  * [R05] É. Ricard, _Factoriality of $q$-Gaussian von Neumann algebras_, Comm. Math. Phys., 257 (2005), no. 3, 659–665.
  * [S97] D. Shlyakhtenko, _Free quasi-free states_ , Pacific J. Math., 177 (1997), no. 2, 329–368.
  * [S04] D. Shlyakhtenko, _Some estimates for non-microstates free entropy dimension with applications to q-semicircular families_ , Int. Math. Res. Not., 51, 2757–2772, (2004).
  * [S09] D. Shlyakhtenko, _Lower estimates on microstates free entropy dimension_ , Anal. PDE, 2 (2009), no. 2, 119–146.
  * [SW] A. Skalski and S. Wang, _Remarks on factoriality and $q$-deformations_, Proc. Amer. Math. Soc., 146 (2018), no. 9, 3813–3823.
  * [Sn04] P. Śniady, _Factoriality of Bożejko-Speicher von Neumann algebras_ , Comm. Math. Phys., 246 (2004), no. 3, 561–567.
  * [T72] M. Takesaki, _Conditional expectations in von Neumann algebras_ , J. Funct. Anal., 9 (1972), 306–321.
  * [VDN92] D. V. Voiculescu, K. J. Dykema and A. Nica, _Free random variables_ , CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, (1992).
  * [W17] M. Wasilewski, _$q$ -Araki-Woods algebras: extension of second quantisation and Haagerup approximation property_, Proc. Amer. Math. Soc., 145 (2017), no. 12, 5287–5298.
  * [W18] M. Wasilewski, _On $q$-deformed von Neumann algebras_. PhD thesis 2018.
  * [W21] M. Wasilewski, _A simple proof of the complete metric approximation property for $q$-Gaussian algebras_, Colloq. Math., 163 (2021), no. 1, 1–14.