# On the von Neumann algebra of class functions on a compact quantum group

Jacek Krajczok  Institute of Mathematics of the Polish Academy of Sciences,
Warsaw, Poland<EMAIL_ADDRESS>and  Mateusz Wasilewski  Department of
Mathematics, KU Leuven, Belgium _Currently_ : Institute of Mathematics of the
Polish Academy of Sciences, Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract.

We study analogues of the radial subalgebras in free group factors (called the
algebras of class functions) in the setting of compact quantum groups. For the
free orthogonal quantum groups we show that they are not MASAs, as soon as we
are in a non-Kac situation. The most important notion to our present work is
that of a (quasi-)split inclusion. We prove that the inclusion of the algebra
of class functions is quasi-split for some unitary quantum groups – in this
case the subalgebra is non-abelian and we also obtain a result concerning its
relative commutant. In the positive direction, we construct certain bicrossed
products from the quantum group $\mathrm{SU}_{q}(2)$ for which the algebra of
class functions is a MASA.

###### Key words and phrases:

Compact quantum group, Quasi-split inclusion, Maximal abelian subalgebra

###### 2020 Mathematics Subject Classification:

Primary 46L67, 20G42

##  1\. Introduction

It is a well established fact that discrete (quantum or classical) groups
provide a source of interesting examples to the theory of von Neumann
algebras. One of the most prominent examples of von Neumann algebras
associated with groups are the free group factors. Let $F_{n}$ the free group
with generators $g_{1},\dotsc,g_{n}\,(n\geq 2)$ and let
$\operatorname{L}(F_{n})=\lambda(F_{n})^{\prime\prime}$ be the corresponding
group von Neumann algebra. Inside $\operatorname{L}(F_{n})$ one finds the so
called radial subalgebra $\mathscr{R}$, the von Neumann algebra generated by
the operator
$(\lambda_{g_{1}}+\lambda_{g_{1}}^{*})+\cdots+(\lambda_{g_{n}}+\lambda_{g_{n}}^{*})$.
Its name stems from the property that if we (informally) write $x=\sum_{w\in
F_{n}}f(w)\lambda_{w}\in\operatorname{L}(F_{n})$ then $x\in\mathscr{R}$ if and
only if $f$ is a radial function on $F_{n}$, i.e. $f(w)$ depends only on the
length $|w|$ of $w$. The radial algebra was intensively studied – let us
mention that it was proved to be maximal abelian (MASA) in
$\operatorname{L}(F_{n})$ [Pyt81] and later to be singular [R9̆1] and even
maximal injective [CFRW10].

One may look at the element
$(\lambda_{g_{1}}+\lambda_{g_{1}}^{*})+\cdots+(\lambda_{g_{n}}+\lambda_{g_{n}}^{*})$
from a different perspective. It is the character of the fundamental
representation of the compact quantum group $\widehat{F_{n}}$, dual to
$F_{n}$. Consequently, the radial algebra $\mathscr{R}$ is the von Neumann
algebra generated by this character. It is therefore natural to wonder whether
similar properties hold for other discrete (or, by duality, compact) quantum
groups. This question was studied in particular in the case of the (Kac type)
free orthogonal quantum group $O_{N}^{+}$. In [FV16] it was shown that
$\mathscr{C}_{O_{N}^{+}}$, the von Neumann algebra generated by the character
of the fundamental representation, is a singular MASA in
$\operatorname{L}^{\infty}(O_{N}^{+})$. Observe that now
$\mathscr{C}_{O_{N}^{+}}$ has also a different description – it coincides with
the von Neumann algebra generated by all characters of irreducible
representations. Let us take this description as a general definition of
$\mathscr{C}_{\mathbb{G}}$ for a compact quantum group $\mathbb{G}$111Note
however that for $\mathbb{G}=\widehat{F_{n}}$ we do not have equality of
$\mathscr{R}$ and $\mathscr{C}_{\mathbb{G}}$. In fact,
$\mathscr{C}_{\mathbb{G}}=\operatorname{L}^{\infty}(\mathbb{G})$ holds for all
abelian compact quantum groups (meaning that the function algebra
$\operatorname{L}^{\infty}(\mathbb{G})$ is cocommutative)..

######  Definition 1.1 (See [Wor87a, after Corollary 5.10]).

For a compact quantum group $\mathbb{G}$ we define the von Neumann algebra of
class functions
$\mathscr{C}_{\mathbb{G}}:=\\{\chi_{\alpha}\,|\,\alpha\in\operatorname{Irr}(\mathbb{G})\\}^{\prime\prime}\subseteq
L^{\infty}(\mathbb{G})$ .

We choose to call $\mathscr{C}_{\mathbb{G}}$ “the von Neumann algebra of class
functions” because the two coincide for classical compact groups (see Lemma
1.2).

In this article we will focus on non-Kac type compact quantum groups, thus in
many cases we enter the realm of type $\operatorname{III}$ von Neumann
algebras. It means that we cannot use one of the most important tools – the
conditional expectation – unless our subalgebra is preserved by the modular
group. In fact, many examples of MASAs in type III factors admit a conditional
expectation, for example the Cartan subalgebras inside crossed products by
non-singular actions or the von Neumann subalgebra generated by the positive
part in the polar decomposition of a generalized circular element in the free
Araki-Woods factors (see [Shl97, Theorem 4.8]). However, the abelian
subalgebras of free Araki-Woods factors that are counterparts of the famous
generator subalgebras of the free group factors, are not preserved by the
modular group and are not MASAs (this is also known in the case of $q$-Araki-
Woods algebras, see [BM20]). The crucial notion in this setting is that of a
(quasi-)split inclusion (see [Buc74] and [DL84]), which will also be central
to our present work.

In Section 2 we will discuss the basics of quasi-split inclusions of von
Neumann algebras. We record there precise statements that will allow us to
prove that the inclusion of the von Neumann algebra of class functions is
quasi-split in some cases. In our first result we obtain a concrete criterion
in terms of the relation between the usual and quantum dimensions of
irreducible representations (this criterion is satisfied for many, but not
all, non-Kac type quantum groups, see Remark 4.15).

######  Theorem A (Theorem 2.7).

Let $\mathbb{G}$ be a compact quantum group. Suppose that
$\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\sqrt{\tfrac{\operatorname{dim}(\alpha)}{\operatorname{dim}_{q}(\alpha)}}$
converges. Then the inclusion of the von Neumann algebra of class functions
into $\operatorname{L}^{\infty}(\mathbb{G})$ is quasi-split.

This result is sufficient to prove that the von Neumann algebra of class
functions is not a MASA in the case that
$\operatorname{L}^{\infty}(\mathbb{G})$ is a type $\operatorname{III}$ factor.
One of the cases of most interest to us is the compact quantum group
$O_{F}^{+}$, for which it is not known in full generality whether
$\operatorname{L}^{\infty}(O_{F}^{+})$ is a factor. Therefore in Section 3 we
develop a new approach, involving the scaling group, a unique symmetry of the
non-Kac type quantum groups. What we show, in broad strokes, is that if being
non-Kac is witnessed by all non-trivial irreducible representations then the
von Neumann algebra of class functions cannot be a MASA as soon as the
inclusion is quasi-split. More precisely, we prove that if it were a MASA then
$\operatorname{L}^{\infty}(\mathbb{G})$ would have to be factor, and the
quasi-split property would force this factor to be of type $\operatorname{I}$.
In this case a non-trivial inner scaling automorphism must exist, which we
show to be excluded by our assumptions.

######  Theorem B (Theorem 3.4).

Let $\mathbb{G}$ be a non-trivial compact quantum group such that the
inclusion of the von Neumann algebra of class functions is quasi-split and
$\uprho_{\alpha}\neq\mathds{1}_{\alpha}$ for all non-trivial representations
$\alpha\in\operatorname{Irr}(\mathbb{G})$. Then the von Neumann algebra of
class functions is not a MASA.

Here $\uprho_{\alpha}\in\operatorname{B}(\mathsf{H}_{\alpha})$ is the unique
positive matrix intertwining the irreducible representation $\alpha$ with its
double conjugate and satisfying
$\operatorname{Tr}(\uprho_{\alpha})=\operatorname{Tr}(\uprho_{\alpha}^{-1})$;
for an arbitrary finite dimensional representation $U$ we can define
$\uprho_{U}$ using the decomposition of $U$ into irreducible representations,
see [NT13, Section 1.4].

In Section 4 we focus on studying concrete examples. More specifically, in
Subsection 4.3 we introduce a class of compact quantum groups for which we can
verify the criterion from Section 2 for the quasi-split property. Combined
with the results from Section 3 we obtain the following.

######  Theorem C (Corollary 4.11).

Let $\mathbb{G}=O_{F}^{+}$ – the free orthogonal quantum group – or
$\mathbb{G}=\mathbb{G}_{Aut}(B,\psi)$ – the quantum automorphism group of a
finite dimensional $C^{\ast}$-algebra equipped with a $\delta$-form. If
$\mathbb{G}$ is not of Kac type then $\mathscr{C}_{\mathbb{G}}$ is not a MASA
in $\operatorname{L}^{\infty}(\mathbb{G})$.

In the same section we study the compact quantum group $\mathrm{SU}_{q}(2)$.
In this case the von Neumann algebra of class functions is not a MASA in
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$, but it is a MASA in the fixed
point subalgebra of the scaling group. From the action by scaling
automorphisms (restricted to the rational numbers, treated as a discrete
group) we build a new compact quantum group using the bicrossed product
construction. It is a non-Kac type compact quantum group for which we can
prove that the von Neumann algebra of class functions is a MASA.

######  Theorem D (Proposition 4.5).

Let $\mathbb{H}$ be the bicrossed product
$\mathbb{Q}\bowtie\mathrm{SU}_{q}(2)$. Then $\mathscr{C}_{\mathbb{H}}$ is a
MASA in $\operatorname{L}^{\infty}(\mathbb{H})$.

The bicrossed product construction above actually depends on the choice of a
non-zero real number and for most choices
$\operatorname{L}^{\infty}(\mathbb{H})$ is the unique injective factor of type
$\operatorname{II}_{\infty}$.

In the last section of the paper we investigate the properties of the free
unitary groups $U_{F}^{+}$. In this case the von Neumann algebra of class
functions is non-commutative, but it remains tracial (see [Wor87a, Equation
(5.33)]). We show that the inclusion is quasi-split as long as the quantum
group is “sufficiently non-Kac”.

######  Theorem E (See Corollary 4.16 for a precise statement).

Let $\mathbb{G}=U_{F}^{+}$. If the ratio
$\frac{\dim(\alpha)}{\dim_{q}(\alpha)}$, where $\alpha$ is the fundamental
representation, is sufficiently small then the inclusion
$\mathscr{C}_{U_{F}^{+}}\subseteq\operatorname{L}^{\infty}(U_{F}^{+})$ is
quasi-split. As a consequence, the relative commutant
$\mathscr{C}_{U_{F}^{+}}^{\prime}\cap\operatorname{L}^{\infty}(U_{F}^{+})$ is
not contained in $\mathscr{C}_{U_{F}^{+}}$.

###  1.1. Preliminaries

We would like to start by motivating our terminology from Definition 1.1.

######  Lemma 1.2.

Let $G$ be a compact group. Then
$\\{\chi_{\alpha}\,|\,\alpha\in\operatorname{Irr}(G)\\}^{\prime\prime}$ equals
the von Neumann algebra of essentially bounded measurable class functions,
i.e. the set of $f\in\operatorname{L}^{\infty}(G)$ satisfying
$f(hgh^{-1})=f(g)\,(g,h\in G)$.

###### Proof.

Since every character $\chi_{\alpha}$ is a class function, one of the
inclusions is clear. Let $\mu_{G}$ be the Haar measure on $G$ and
$\mathbb{E}\colon f\mapsto\int_{G}f(h\cdot
h^{-1})\operatorname{d}\\!\mu_{G}(h)\;(f\in\operatorname{L}^{\infty}(G))$ the
normal conditional expectation onto the von Neumann algebra of essentially
bounded measurable class fuctions. As matrix coefficients of irreducible
representations span a $w^{*}$-dense subspace in
$\operatorname{L}^{\infty}(G)$, it is enough to show that
$\mathbb{E}(u^{\alpha}_{\xi,\eta})=\frac{\langle\xi\,|\,\eta\rangle}{\dim(\alpha)}\chi_{\alpha}$
for all $\alpha\in\operatorname{Irr}(G)$ and vectors
$\xi,\eta\in\mathsf{H}_{\alpha}$. Using orthogonality relations one can check
that
$\langle\Lambda_{h}(\chi_{\alpha})\,|\,\Lambda_{h}(u_{v,w}^{\beta})\rangle=\delta_{\alpha\beta}\frac{\langle
v\,|\,w\rangle}{\dim(\alpha)}$, so it is sufficient to check that
$\langle\Lambda_{h}(\mathbb{E}(u_{\xi,\eta}^{\alpha}))\,|\,\Lambda_{h}(u_{v,w}^{\beta})\rangle=\frac{\delta_{\alpha\beta}}{(\dim(\alpha))^{2}}\langle\eta\,|\,\xi\rangle\langle
v\,|\,w\rangle$. This can, once again, be verified using the orhogonality
relations:

$\displaystyle\langle\Lambda_{h}(\mathbb{E}(u_{\xi,\eta}^{\alpha}))\,|\,\Lambda_{h}(u_{v,w}^{\beta})\rangle$
$\displaystyle=\int_{G}\int_{G}\langle\alpha(hgh^{-1})\eta\,|\,\xi\rangle\langle
v\,|\,\beta(g)w\rangle\operatorname{d}\\!\mu_{G}(h)\operatorname{d}\\!\mu_{G}(g)$
$\displaystyle=\int_{G}\langle\Lambda_{h}(u_{\alpha(h^{-1})\xi,\alpha(h^{-1})\eta}^{\alpha})\,|\,\Lambda_{h}(u_{v,w}^{\beta})\rangle\operatorname{d}\\!\mu_{G}(h)$
$\displaystyle=\frac{\delta_{\alpha\beta}}{\dim(\alpha)}\int_{G}\langle
v\,|\,\alpha(h^{-1})\xi\rangle\langle\alpha(h^{-1})\eta\,|\,w\rangle\operatorname{d}\\!\mu_{G}(h)$
$\displaystyle=\frac{\delta_{\alpha\beta}}{\dim(\alpha)}\langle\Lambda_{h}(u_{\xi,v}^{\alpha})\,|\,\Lambda_{h}(u_{\eta,w}^{\alpha})\rangle$
$\displaystyle=\frac{\delta_{\alpha\beta}}{(\dim(\alpha))^{2}}\langle\eta\,|\,\xi\rangle\langle
v\,|\,w\rangle.$

∎

######  Remark 1.3.

Actually, it has been proved in [AC17, Theorem 3.7] that for any compact
quantum group $\mathbb{G}$ the algebra of class functions
$\mathscr{C}_{\mathbb{G}}$ is equal to
$\\{x\in\operatorname{L}^{\infty}(\mathbb{G}):\Delta(x)=\Sigma\Delta(x)\\}$,
where
$\Delta:\operatorname{L}^{\infty}(\mathbb{G})\to\operatorname{L}^{\infty}(\mathbb{G})\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{G})$
is the comultiplication and
$\Sigma:\operatorname{L}^{\infty}(\mathbb{G})\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{G})\to\operatorname{L}^{\infty}(\mathbb{G})\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{G})$
is the flip; this answered a question left open by Woronowicz, see [Wor87a,
after Proposition 5.11]. In the classical case it corresponds to functions $f$
satisfying $f(gh)=f(hg)$, i.e. precisely the class functions.

######  Lemma 1.4 ([Wor87a, Equation (5.33)]).

Let $\mathbb{G}$ be a compact quantum group. The Haar integral integral $h$
restricted to $\mathscr{C}_{\mathbb{G}}$ is tracial.

In most of the examples $\mathscr{C}_{\mathbb{G}}$ will actually be
commutative (the fusion rules will be commutative), with the important
exception of the free unitary quantum groups $U_{F}^{+}$.

We will work with non-Kac type compact quantum groups and these objects
naturally admit two one-parameter automorphism groups: the modular group
(because the Haar integral is non-tracial) and the scaling group (because the
antipode is not involutive). For more information on compact quantum groups
consult e.g. [NT13].

There will be two main classes of compact quantum groups to which our results
apply: we will start with free orthogonal (unitary) quantum groups.

######  Definition 1.5 ([VDW96]).

Let $N\geqslant 2$ be an integer and let $F$ be an invertible matrix in
$M_{N}(\mathbb{C})$. Let

  1. (i)

$\operatorname{Pol}(U_{F}^{+})$ be the universal $\ast$-algebra generated by
the entries of a unitary matrix $U\in M_{N}(\operatorname{Pol}(U_{F}^{+}))$
subject to the condition that $F\overline{U}F^{-1}$ is also a unitary matrix;

  2. (ii)

$\operatorname{Pol}(O_{F}^{+})$ be the universal $\ast$-algebra generated by
the entries of a unitary matrix $U\in M_{N}(\operatorname{Pol}(O_{F}^{+}))$
subject to the condition that $U=F\overline{U}F^{-1}$; we also assume in this
case that $F\overline{F}=c\mathds{1}$ for some $c\in\mathbb{R}$.

Then the comultiplication $\Delta(u_{ij}):=\sum_{k=1}^{N}u_{ik}\otimes u_{kj}$
gives both $\operatorname{Pol}(U_{F}^{+})$ and $\operatorname{Pol}(O_{F}^{+})$
the structure of a Hopf $\star$-algebra. After the universal
$\mathrm{C}^{*}$-completion, they give rise to compact quantum groups
$U_{F}^{+}$ and $O_{F}^{+}$.

Before introducing the quantum automorphism groups of finite-dimensional
$\mathrm{C}^{\ast}$-algebras, we need to recall the definition of a
$\delta$-form. Let $B$ be a finite-dimensional $C^{\ast}$-algebra, let
$m:B\otimes B\to B$ be the multiplication map, and let $\psi:B\to\mathbb{C}$
be a faithful state. Then $\psi$ defines natural inner products on both $B$
and $B\otimes B$ and we can compute the Hermitian adjoint of $m$, i.e.
$m^{\ast}$. We say that $\psi$ is a $\delta$-form if
$mm^{\ast}=\delta^{2}\operatorname{Id}$.

######  Definition 1.6 ([Wan98]).

Let $B$ be a finite-dimensional $C^{\ast}$-algebra equipped with a
$\delta$-form $\psi$. Then there exists a universal compact quantum group
acting on $B$ in a $\psi$-preserving way – we denote it by
$\mathbb{G}_{Aut}(B,\psi)$ and the corresponding action by $\alpha$.

It means that whenever we have a compact quantum group $\mathbb{G}$ with a
$\psi$-preserving action $\beta:B\to\mathrm{C}_{u}(\mathbb{G})\otimes B$, i.e.
$\beta$ is a $\ast$-homomorphism satsifying
$(\Delta\otimes\operatorname{Id})\circ\beta=(\operatorname{Id}\otimes\beta)\circ\beta$
and $(\operatorname{Id}\otimes\psi)(\beta(f))=\psi(f)\mathds{1}$, then there
exists a unique quantum group morphism
$\Phi:\mathrm{C}_{u}(\mathbb{G}_{Aut}(B,\psi))\to\mathrm{C}_{u}(\mathbb{G})$
such that $(\Phi\otimes\operatorname{Id})\alpha=\beta$.
Let us mention here that in the Kac case, operator algebras associated with
$\mathbb{G}_{Aut}(B,\psi)$ were extensively studied e.g. in [Bra13].

###  1.2. Notation

For a faithful normal state $\omega$ on a von Neumann algebra
$\operatorname{N}$ we will denote by $(\sigma^{\omega}_{t})_{t\in\mathbb{R}}$
the modular group, by $\nabla_{\omega}$ the modular operator, by $J_{\omega}$
the modular conjugation and by $\Lambda_{\omega}$ the canonical map
$\operatorname{N}\rightarrow\operatorname{L}^{2}(\operatorname{N},\omega)$.
When there is no risk of confusion, we will simply write
$(\sigma_{t})_{t\in\mathbb{R}}$ etc. Similarly, for a compact quantum group
$\mathbb{G}$ we will often write $(\tau_{t})_{t\in\mathbb{R}}$ for the scaling
group. We will use von Neumann subalgebras
$\operatorname{L}^{\infty}(\mathbb{G})^{\tau},\operatorname{L}^{\infty}(\mathbb{G})^{\sigma}\subseteq\operatorname{L}^{\infty}(\mathbb{G})$
of those $x\in\operatorname{L}^{\infty}(\mathbb{G})$ which are invariant under
the scaling group (respectively the modular group of the Haar integral $h$).
Algebraic tensor products will be denoted by $\odot$.

##  2\. Quasi-split inclusions

The split property arose in the study of inclusions of von Neumann algebras
appearing in algebraic quantum field theory (see [Buc74]). A more systematic
study of related properties was conducted in [DL84]. The main tool for proving
that a given inclusion is split was developed in [BDL90], albeit only in the
case of factor inclusions, and we need to resort to subsequent work [Fid02],
which deals with the general case. The precise criterion that we will use
comes from [BM20].

######  Definition 2.1.

Let $\mathsf{N}\subseteq\mathsf{M}$ be an inclusion of von Neumann algebras.
We say this inclusion is split if there exists an intermediate type
$\operatorname{I}$ factor $\mathsf{B}$. We say that the inclusion is quasi-
split if the $\ast$-homomorphism $\mathsf{N}\odot\mathsf{M}^{op}\ni x\otimes
y^{op}\mapsto
xJy^{\ast}J\in\operatorname{B}(\operatorname{L}^{2}(\mathsf{M}))$, where
$L^{2}(\mathsf{M})$ is the standard form, extends to a surjective normal
$\ast$-homomorphism from $\mathsf{N}\overline{\otimes}\mathsf{M}^{op}$ onto
$\mathsf{N}\vee\mathsf{M}^{\prime}$.

######  Remark 2.2.

If both $\mathsf{N}$ and $\mathsf{M}$ are factors or one of them is a type
$\operatorname{III}$ algebra then a quasi-split inclusion is automatically
split (see [DL83, Corollary 1, (iv) and (vi)]).

The most important consequence of the quasi-split property for our work is
that it can be used as a tool for proving that some abelian subalgebras are
not MASAs.

######  Proposition 2.3 ([BM20, Corollary 3.11]).

If $\mathsf{M}$ is a type $\operatorname{III}$ von Neumann algebra and
$\mathsf{N}\subseteq\mathsf{M}$ is a quasi-split inclusion then
$\mathsf{N}^{\prime}\cap\mathsf{M}$ is also of type $\operatorname{III}$. In
particular $\mathsf{N}$ cannot be a MASA.

We would like to present now a useful criterion for proving that a given
inclusion is quasi-split, which is a variant of Proposition 2.3 from [BDL90].
Recall first that if a von Neumann algebra $\mathsf{M}$ is represented on a
Hilbert space $\mathsf{H}$ with a cyclic and separating vector $\Omega$ then
$\mathsf{H}$ can be identified with $\operatorname{L}^{2}(\mathsf{M})$ and we
have an inclusion $\Phi_{2}:\mathsf{M}\to\operatorname{L}^{2}(\mathsf{M})$
given by $x\mapsto\nabla^{\frac{1}{4}}x\Omega$.

We will also need the notion of a nuclear map between two Banach spaces $X$
and $Y$. A map $T:X\to Y$ is called nuclear if there are sequences
$(y_{n})_{n\in\mathbb{N}}\subseteq Y$ and
$(x_{n}^{\ast})_{n\in\mathbb{N}}\subseteq X^{\ast}$ such that
$Tx=\sum_{n\in\mathbb{N}}x_{n}^{\ast}(x)y_{n}$ and
$\sum_{n\in\mathbb{N}}\|x_{n}^{\ast}\|\|y_{n}\|<\infty$.

######  Proposition 2.4 ([BM20, Proposition 3.7]).

Let $\mathsf{N}\subseteq\mathsf{M}$ be a pair of von Neumann algebras. If the
map $\Phi_{2|\mathsf{N}}:\mathsf{N}\to\operatorname{L}^{2}(\mathsf{M})$ is
nuclear then the inclusion $\mathsf{N}\subseteq\mathsf{M}$ is quasi-split.

###  2.1. Quasi-split inclusion of the algebra of class functions

In this subsection we will present examples of compact quantum groups for
which the inclusion of the algebra of class functions is quasi-split. Before
doing that we need some preparation regarding the action of the modular group
on characters of unitary representations.

Recall that our aim is to show that the map
$\Phi_{2|\mathscr{C}_{\mathbb{G}}}:\mathscr{C}_{\mathbb{G}}\to\operatorname{L}^{2}(\mathbb{G})$
given by $x\mapsto\nabla^{\frac{1}{4}}\Lambda_{h}(x)$ is nuclear. Note that
$\mathscr{C}_{\mathbb{G}}$ is given by the closed linear span of the
characters of the (finite dimensional) unitary representations and these
characters are analytic elements for the modular group. Therefore
$\nabla^{\frac{1}{4}}\Lambda_{h}(\chi)=\Lambda_{h}(\sigma_{-\frac{i}{4}}(\chi))$
holds for every character $\chi$; we first have to understand the action of
the modular group on the characters.

Recall that (see [NT13, Page 30]) for any representation $U$ on
$\mathsf{H}_{U}$ we have
$(\operatorname{Id}\otimes\sigma_{z})(U)=(\uprho_{U}^{iz}\otimes\mathds{1}_{U})U(\uprho_{U}^{iz}\otimes\mathds{1}_{U})$.
If we choose an orthonormal basis of $\mathsf{H}_{U}$ in which $\uprho_{U}$ is
diagonal then we can write more concretely that
$\sigma_{z}(u_{kl})=\uprho_{U,k}^{iz}\uprho_{U,l}^{iz}u_{kl}$. Therefore for
the character $\chi_{U}:=\sum_{k}u_{kk}$ we have
$\sigma_{z}(\chi_{U})=\sum_{k}\uprho_{U,k}^{2iz}u_{kk}$. We will now compute
the $\operatorname{L}^{2}$-norm of this element.

######  Lemma 2.5.

If $U$ is irreducible then we have
$\|\sigma_{a+ib}(\chi_{U})\|_{2}^{2}=\frac{\operatorname{Tr}(\uprho_{U}^{-4b-1})}{\operatorname{Tr}(\uprho_{U})}$
for all $a,b\in\mathbb{R}$.

###### Proof.

Recall that by definition $\|x\|_{2}^{2}=h(x^{\ast}x)$, so in our case we get

$\|\sigma_{a+ib}(\chi_{U})\|_{2}^{2}=\sum_{k,l}\uprho_{U,k}^{-2ia-2b}\uprho_{U,l}^{2ia-2b}h(u_{kk}^{\ast}u_{ll}).$

Using the orthogonality relations (see [NT13, Theorem 1.4.3]) we get
$h(u_{kk}^{\ast}u_{ll})=\delta_{k,l}\frac{\uprho^{-1}_{U,k}}{\operatorname{dim}_{q}(U)}$.
Therefore we obtain

$\|\sigma_{a+ib}(\chi_{U})\|_{2}^{2}=\sum_{k}\uprho_{U,k}^{-4b}\frac{\uprho_{U,k}^{-1}}{\operatorname{dim}_{q}(U)}=\frac{\operatorname{Tr}(\uprho_{U}^{-4b-1})}{\operatorname{dim}_{q}(U)}.$

To finish the proof we just have to recall that
$\dim_{q}(U)=\operatorname{Tr}(\uprho_{U})$. ∎

######  Corollary 2.6.

We have $\|\chi_{U}\|_{2}=1$ and
$\|\sigma_{-\frac{i}{4}}(\chi_{U})\|_{2}^{2}=\frac{\operatorname{dim}(U)}{\operatorname{dim}_{q}(U)}$.

The relation between quantum dimension and the usual dimension will be crucial
for proving that the inclusion of the algebra of class functions is quasi-
split. We will present here a precise criterion and then describe a class of
compact quantum groups to which it applies.

######  Theorem 2.7.

Let $\mathbb{G}$ be a compact quantum group. Suppose that
$\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\left(\frac{\operatorname{dim}(\alpha)}{\operatorname{dim}_{q}(\alpha)}\right)^{\frac{1}{2}}<\infty$.
Then the inclusion
$\mathscr{C}_{\mathbb{G}}\subseteq\operatorname{L}^{\infty}(\mathbb{G})$ is
quasi-split.

###### Proof.

We want to show that the map
$\Phi_{2|\mathscr{C}_{\mathbb{G}}}:\mathscr{C}_{\mathbb{G}}\to\operatorname{L}^{2}(\mathbb{G})$
is nuclear. Recall that it is a composition of two maps: the inclusion
$\Lambda_{h}:\mathscr{C}_{\mathbb{G}}\to\Lambda_{h}(\mathscr{C}_{\mathbb{G}})$
and
$\nabla^{\frac{1}{4}}:\Lambda_{h}(\mathscr{C}_{\mathbb{G}})\to\operatorname{L}^{2}(\mathbb{G})$.

We will first show that $\nabla^{\frac{1}{4}}$ extends to a contraction from
$\operatorname{L}^{2}(\mathscr{C}_{\mathbb{G}}):=\overline{\Lambda_{h}(\mathscr{C}_{\mathbb{G}})}$
to $\operatorname{L}^{2}(\mathbb{G})$. Note that the characters of irreducible
representations are linearly dense in
$\operatorname{L}^{2}(\mathscr{C}_{\mathbb{G}})$ and the characters form an
orthonormal set, by the orthogonality relations. Let
$x:=\sum_{\alpha}c_{\alpha}\chi_{\alpha}$ be a finite sum of characters of
irreducible representations. Note that
$\|\Lambda_{h}(x^{\ast})\|^{2}=\|\sum_{\alpha}\overline{c_{\alpha}}\Lambda_{h}(\chi_{\overline{\alpha}})\|^{2}=\sum_{\alpha}|c_{\alpha}|^{2}=\|\Lambda_{h}(x)\|^{2}$.
It follows that

$\displaystyle\|\nabla^{\frac{1}{4}}\Lambda_{h}(x)\|^{2}$
$\displaystyle=\langle\Lambda_{h}(x)\,|\,\nabla^{\frac{1}{2}}\Lambda_{h}(x)\rangle$
$\displaystyle=\langle
J\nabla^{\frac{1}{2}}\Lambda_{h}(x)\,|\,J\Lambda_{h}(x)\rangle$
$\displaystyle=\langle\Lambda_{h}(x^{\ast})\,|\,J\Lambda_{h}(x)\rangle$
$\displaystyle\leqslant\|\Lambda_{h}(x^{\ast})\|\cdot\|\Lambda_{h}(x)\|=\|\Lambda_{h}(x)\|^{2},$

so $\nabla^{\frac{1}{4}}$ extends to a contraction from
$\operatorname{L}^{2}(\mathscr{C}_{\mathbb{G}})$ to
$\operatorname{L}^{2}(\mathbb{G})$.

We will now show that $\Phi_{2}$ from $\mathscr{C}_{\mathbb{G}}$ to
$\operatorname{L}^{2}(\mathbb{G})$ is a nuclear map. We have
$\Phi_{2}(x)=\nabla^{\frac{1}{4}}\Lambda_{h}(x)$. As
$\Lambda_{h}(x)\in\operatorname{L}^{2}(\mathscr{C}_{\mathbb{G}})$, we can
write
$\Lambda_{h}(x)=\sum_{\alpha}\langle\Lambda_{h}(\chi_{\alpha})\,|\,\Lambda_{h}(x)\rangle\Lambda_{h}(\chi_{\alpha})$.
As
$\nabla^{\frac{1}{4}}:\operatorname{L}^{2}(\mathscr{C}_{\mathbb{G}})\to\operatorname{L}^{2}(\mathbb{G})$
is bounded, we have

$\Phi_{2}(x)=\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\langle\Lambda_{h}(\chi_{\alpha})\,|\,\Lambda_{h}(x)\rangle\nabla^{\frac{1}{4}}\Lambda_{h}(\chi_{\alpha}).$

If we define functionals
$\omega_{\alpha}(x):=\langle\Lambda_{h}(\chi_{\alpha})\,|\,\Lambda_{h}(x)\rangle$
then it suffices to check that
$\sum_{\alpha}\|\omega_{\alpha}\|\cdot\|\nabla^{\frac{1}{4}}\Lambda_{h}(\chi_{\alpha})\|<\infty$.
We already know that
$\|\nabla^{\frac{1}{4}}\Lambda_{h}(\chi_{\alpha})\|=\left(\frac{\operatorname{dim}(\alpha)}{\operatorname{dim}_{q}(\alpha)}\right)^{\frac{1}{2}}$
and it is clear that $\|\omega_{\alpha}\|\leqslant\|\chi_{\alpha}\|_{2}=1$,
hence

$\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\|\omega_{\alpha}\|\cdot\|\nabla^{\frac{1}{4}}\Lambda_{h}(\chi_{\alpha})\|\leqslant\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\left(\frac{\operatorname{dim}(\alpha)}{\operatorname{dim}_{q}(\alpha)}\right)^{\frac{1}{2}}<\infty.$

By Proposition 2.4 the inclusion
$\mathscr{C}_{\mathbb{G}}\subseteq\operatorname{L}^{\infty}(\mathbb{G})$ is
quasi-split. ∎

This result is already enough to prove that in many cases the radial
subalgebra in $\operatorname{L}^{\infty}(O_{F}^{+})$ is not a MASA; it follows
from [VV07] that $\operatorname{L}^{\infty}(O_{F}^{+})$ is often a type
$\operatorname{III}$ factor and we can use Proposition 2.3. We will generalize
this result in the next section (see Corollary 4.11).

##  3\. Relative commutant of $\mathscr{C}_{\mathbb{G}}$ and inner scaling
automorphisms

We will be interested in the relative commutant
$\mathscr{C}_{\mathbb{G}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{G})$.
If $\mathscr{C}_{\mathbb{G}}$ is commutative, as is the case for example for
the free orthogonal quantum groups, then the condition
$\mathscr{C}_{\mathbb{G}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{G})\subseteq\mathscr{C}_{\mathbb{G}}$
precisely means that $\mathscr{C}_{\mathbb{G}}$ is a MASA in
$\operatorname{L}^{\infty}(\mathbb{G})$.
Our strategy for proving that $\mathscr{C}_{\mathbb{G}}$ cannot be a MASA in
many cases will be the following. We will show that if
$\mathscr{C}_{\mathbb{G}}$ was a MASA then
$\operatorname{L}^{\infty}(\mathbb{G})$ would have to be a factor. Moreover,
if the inclusion
$\mathscr{C}_{\mathbb{G}}\subseteq\operatorname{L}^{\infty}(\mathbb{G})$ is
quasi-split then it would have to be a type $\operatorname{I}$ factor. We will
then use properties of the scaling automorphisms to exclude this case. Let us
now move on to more precise statements.

######  Proposition 3.1.

Let $\mathbb{G}$ be a compact quantum group. Then

$\overline{\operatorname{span}}^{w^{\ast}}\\{\chi_{\alpha}\,|\,\uprho_{\alpha}=\mathds{1}_{\alpha}\\}=\mathscr{C}_{\mathbb{G}}\cap\operatorname{L}^{\infty}(\mathbb{G})^{\sigma}=\bigcap_{t\in\mathbb{R}}\sigma_{t}(\mathscr{C}_{\mathbb{G}}).$

If moreover
$\mathscr{C}_{\mathbb{G}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{G})\subseteq\mathscr{C}_{\mathbb{G}}$
then

$\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{G}))\subseteq\overline{\operatorname{span}}^{w^{\ast}}\\{\chi_{\alpha}\,|\,\uprho_{\alpha}=\mathds{1}_{\alpha}\\}$

In particular, if $\uprho_{\alpha}\neq\mathds{1}_{\alpha}$ for all non-trivial
irreducible representations of $\mathbb{G}$ then
$\operatorname{L}^{\infty}(\mathbb{G})$ is a factor.

###### Proof.

It is clear that
$\overline{\operatorname{span}}^{w^{\ast}}\\{\chi_{\alpha}\,|\,\uprho_{\alpha}=\mathds{1}_{\alpha}\\}\subseteq\mathscr{C}_{\mathbb{G}}\cap\operatorname{L}^{\infty}(\mathbb{G})^{\sigma}\subseteq\bigcap_{t\in\mathbb{R}}\sigma_{t}(\mathscr{C}_{\mathbb{G}})$.
Now let $x\in\bigcap_{t\in\mathbb{R}}\sigma_{t}(\mathscr{C}_{\mathbb{G}})$; we
can write
$\Lambda_{h}(\sigma_{t}(x))=\sum_{\alpha}c_{\alpha}\Lambda_{h}(\sigma_{t}(\chi_{\alpha}))=\sum_{\alpha}c_{\alpha,t}\Lambda_{h}(\chi_{\alpha})$.
It follows from the orthogonality relations that the elements
$\sigma_{t}(\chi_{\alpha})$ and $\chi_{\beta}$ are orthogonal unless
$\alpha=\beta$. Therefore we get
$c_{\alpha}\sigma_{t}(\chi_{\alpha})=c_{\alpha,t}\chi_{\alpha}$ for all $t$,
hence either $c_{\alpha}=c_{\alpha,t}=0$ or $\chi_{\alpha}$ is an eigenvector
of the modular group. If $\alpha$ is an irreducible representation on the
Hilbert space $\mathsf{H}_{\alpha}$ and we pick an orthonormal basis
$(e_{k})_{k}$ in which the matrix $\uprho_{\alpha}$ is diagonal then
$\sigma_{z}(\chi_{\alpha})=\sum_{k}\uprho_{\alpha,k}^{2iz}u^{\alpha}_{k,k}$,
where $\chi_{\alpha}=\sum_{k}u^{\alpha}_{k,k}$. Thus $\chi_{\alpha}$ is an
eigenvector of the modular group iff the matrix $\uprho_{\alpha}$ is a
multiple of the identity, which actually forces $\uprho_{\alpha}=\mathds{1}$
due to the condition
$\operatorname{Tr}(\uprho_{\alpha})=\operatorname{Tr}(\uprho_{\alpha}^{-1})$.
It follows that if a character of an irreducible representation is an
eigenvector of the modular group then it is in fact a fixed point.

Denote
$A:=\overline{\operatorname{span}}^{w^{\ast}}\\{\chi_{\alpha}\,|\,\uprho_{\alpha}=\mathds{1}_{\alpha}\\}$.
We showed that any element
$x\in\bigcap_{t\in\mathbb{R}}\sigma_{t}(\mathscr{C}_{\mathbb{G}})$ satisfies
$\Lambda_{h}(x)\in\overline{\Lambda_{h}(A)}$. Since $A$ is contained in the
centralizer, there exists a normal, faithful, state-preserving conditional
expectation onto it, and it is easy to conclude that it implies $x\in A$.
Alternatively one can invoke Lemma 4.6 because the Haar integral on the
algebra $A$ is tracial.

To finish the proof, note that the condition
$\mathscr{C}_{\mathbb{G}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{G})\subseteq\mathscr{C}_{\mathbb{G}}$
implies that
$\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{G}))\subseteq\mathscr{C}_{\mathbb{G}}$.
Moreover the center is always contained in the centralizer, so we obtain
$\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{G}))\subseteq\mathscr{C}_{\mathbb{G}}\cap\operatorname{L}^{\infty}(\mathbb{G})^{\sigma}$.
∎

The next technical ingredient, featuring the scaling group, is the following
proposition.

######  Proposition 3.2.

Let $\mathbb{G}$ be a compact quantum group and let $t\in\mathbb{R}$. Suppose
that
$\mathscr{C}_{\mathbb{G}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{G})\subseteq\mathscr{C}_{\mathbb{G}}$,
the scaling automorphism $\tau_{t}$ is inner and is implemented by
$v\in\operatorname{L}^{\infty}(\mathbb{G})$. Then
$v\in\overline{\operatorname{span}}^{w^{\ast}}\\{\chi_{\alpha}\,|\,\uprho_{\alpha}=\mathds{1}_{\alpha}\\}$.

###### Proof.

The scaling group acts trivially on characters, so we have
$\chi_{\alpha}=\tau_{t}(\chi_{\alpha})=v\chi_{\alpha}v^{\ast}$ for any
$\alpha\in\operatorname{Irr}(\mathbb{G})$. Therefore
$v\in\mathscr{C}_{\mathbb{G}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{G})\subseteq\mathscr{C}_{\mathbb{G}}$.
Because of that we can write
$\Lambda_{h}(v)=\sum_{\alpha}c_{\alpha}\Lambda_{h}(\chi_{\alpha})$.

We will now use the fact that the scaling group and the modular group commute,
so for any $x\in\operatorname{L}^{\infty}(\mathbb{G})$ we can write

$vxv^{\ast}=\tau_{t}(x)=\sigma_{s}\tau_{t}(\sigma_{-s}(x))=\sigma_{s}(v\sigma_{-s}(x)v^{\ast})=\sigma_{s}(v)x\sigma_{s}(v^{\ast}).$

It follows that $\sigma_{s}(v^{\ast})vx=x\sigma_{s}(v^{\ast})v$, hence
$\sigma_{s}(v^{\ast})v\in\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{G}))$
for every $s\in\mathbb{R}$. We can write $\sigma_{s}(v)=vw_{s}$ with
$w_{s}=v^{\ast}\sigma_{s}(v)\in\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{G}))$.
As
$\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{G}))\subseteq\mathscr{C}_{\mathbb{G}}$,
we have $vw_{s}\in\mathscr{C}_{\mathbb{G}}$. It follows that
$\sigma_{s}(v)\in\mathscr{C}_{\mathbb{G}}$ for any $s\in\mathbb{R}$, i.e.
$v\in\bigcap_{t\in\mathbb{R}}\sigma_{t}(\mathscr{C}_{\mathbb{G}})$;
Proposition 3.1 allows us to conclude. ∎

Before we state and prove the main result of this section, we need to record
another useful consequence of having a quasi-split inclusion.

######  Lemma 3.3 ([BM20, Remark 3.10 (2)]).

Suppose that the inclusion $\mathsf{B}\subseteq\mathsf{M}$ is quasi-split and
$\mathsf{B}$ is a MASA in $\mathsf{M}$. Then $\mathsf{M}$ is a direct sum of
type $\operatorname{I}$ factors.

######  Theorem 3.4.

Suppose that $\mathbb{G}$ is a non-trivial compact quantum group such that the
inclusion
$\mathscr{C}_{\mathbb{G}}\subseteq\operatorname{L}^{\infty}(\mathbb{G})$ is
quasi-split and $\uprho_{\alpha}\neq\mathds{1}_{\alpha}$ for any non-trivial
$\alpha\in\operatorname{Irr}(\mathbb{G})$. Then $\mathscr{C}_{\mathbb{G}}$ is
not a MASA in $\operatorname{L}^{\infty}(\mathbb{G})$.

###### Proof.

Suppose that $\mathscr{C}_{\mathbb{G}}$ is a MASA in
$\operatorname{L}^{\infty}(\mathbb{G})$. It follows from Lemma 3.3 that
$\operatorname{L}^{\infty}(\mathbb{G})$ is a direct sum of type
$\operatorname{I}$ factors. Moreover, it follows from Proposition 3.1 that
$\operatorname{L}^{\infty}(\mathbb{G})$ is a factor, so
$\operatorname{L}^{\infty}(\mathbb{G})\simeq\operatorname{B}(\mathsf{H})$ for
some Hilbert space $\mathsf{H}$.

By our assumptions there exists an irreducible representation $\alpha$ with
$\uprho_{\alpha}\neq\mathds{1}_{\alpha}$, thus $\mathbb{G}$ is not of Kac
type, so there exists a non-trivial scaling automorphism $\tau_{t}$. All
automorphisms of $\operatorname{B}(\mathsf{H})$ are inner, so there exists
$v\in\operatorname{L}^{\infty}(\mathbb{G})$ implementing it. By Proposition
3.2
$v\in\overline{\operatorname{span}}^{w^{\ast}}\\{\chi_{\alpha}\,|\,\uprho_{\alpha}=\mathds{1}_{\alpha}\\}=\mathbb{C}\mathds{1}$.
But that means that $\tau_{t}$ is a trivial automorphism and this leads us to
the desired contradiction. ∎

We will provide examples to which this result can be applied in Subsection 4.3
(see Corollary 4.11).

######  Remark 3.5.

In the proof of Theorem 3.4 we argued using Proposition 3.2 that if
$\mathscr{C}_{\mathbb{G}}$ is a MASA in
$\operatorname{L}^{\infty}(\mathbb{G})$, then
$\operatorname{L}^{\infty}(\mathbb{G})$ is not isomorphic to
$\operatorname{B}(\mathsf{H})$. In fact, one can show that
$\operatorname{L}^{\infty}(\mathbb{G})$ is not isomorphic to
$\operatorname{B}(\mathsf{H})$ for any non-trivial compact quantum group
$\mathbb{G}$, see [KS].

##  4\. Examples

###  4.1. Properties of $\mathrm{SU}_{q}(2)$

Fix $q\in(-1,1)\setminus\\{0\\}$. In this subsection we establish a number of
properties of $\mathscr{C}_{\mathrm{SU}_{q}(2)}$: we show that it is not a
MASA in $\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$ (Proposition 4.2),
however it is a MASA in
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau}$, the fixed point
subalgebra of the scaling group (Proposition 4.3). This property will be used
in the next subsection, where we construct a new compact quantum group out of
$\mathrm{SU}_{q}(2)$ and $\mathbb{Q}$, using a bicrossed product construction.

Let us start with recalling a description of $\mathrm{SU}_{q}(2)$ from the
dual perspective. Since $\mathrm{C}(\mathrm{SU}_{q}(2))$ is a type
$\operatorname{I}$, separable $\mathrm{C}^{*}$-algebra [Wor87b, Theorem
A.2.3], $\widehat{\mathrm{SU}_{q}(2)}$ is a type $\operatorname{I}$ (second
countable) discrete quantum group and we can use Desmedt’s theorem to describe
its structure using decompositions into direct integrals (see [Des03] and
[Kra20, Example 7.1]). For each $\lambda\in\mathbb{T}$ there is an irreducible
representation

$\psi_{\lambda}\colon\mathrm{C}(\mathrm{SU}_{q}(2))\rightarrow\operatorname{B}(\mathsf{H}_{\lambda})=\operatorname{B}(\ell^{2}(\mathbb{Z}_{+})),$

given by

$\psi_{\lambda}(\alpha)\phi_{k}=\sqrt{1-q^{2k}}\phi_{k-1},\quad\psi_{\lambda}(\gamma)\phi_{k}=\lambda
q^{k}\phi_{k}\quad(k\in\mathbb{Z}_{+})$

($\\{\phi_{k}\\}_{k\in\mathbb{Z}_{+}}$ is the standard orthonormal basis in
$\ell^{2}(\mathbb{Z}_{+}),\,\phi_{-1}=0$ and $\alpha,\gamma$ are the standard
generators of $\mathrm{C}(\mathrm{SU}_{q}(2))$). Denote by $\mu$ the
normalized Lebesgue measure on the unit circle $\mathbb{T}$. Desmedt’s theorem
provides us with a unitary operator
$\mathcal{Q}_{L}\colon\operatorname{L}^{2}(\mathrm{SU}_{q}(2))\rightarrow\int_{\mathbb{T}}^{\oplus}\operatorname{HS}(\mathsf{H}_{\lambda})\operatorname{d}\\!\mu(\lambda)$
given by

$\mathcal{Q}_{L}\colon\Lambda_{h}(a)\mapsto\int_{\mathbb{T}}^{\oplus}\psi_{\lambda}(a)D_{\lambda}^{-1}\operatorname{d}\\!\mu(\lambda)\quad(a\in\mathrm{C}(\mathrm{SU}_{q}(2)))$

where $D_{\lambda}^{-1}$ is a positive Hilbert-Schmidt operator

$D_{\lambda}^{-1}=\sqrt{1-q^{2}}\operatorname{diag}(1,|q|,|q|^{2},\dotsc)$

written with respect to the basis $\\{\phi_{k}\\}_{k\in\mathbb{Z}_{+}}$.
Furthermore, we have

$\mathcal{Q}_{L}\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\mathcal{Q}_{L}^{*}=\int_{\mathbb{T}}^{\oplus}\operatorname{B}(\mathsf{H}_{\lambda})\otimes\mathds{1}_{\overline{\mathsf{H}_{\lambda}}}\,\operatorname{d}\\!\mu(\lambda).$
(4.1)

The next lemma says that the von Neumann algebra generated by the real part of
a weighted shift is MASA in $\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$.

######  Lemma 4.1.

Let $s\in\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$ be the shift operator
given by $s\phi_{k}=\phi_{k-1}\,(k\in\mathbb{Z}_{+})$, and let
$\operatorname{M}_{f}\in\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$ be the
multiplication operator associated with a function
$f\in\ell^{\infty}(\mathbb{Z}_{+})$. If
$f(\mathbb{N})\subseteq\mathbb{R}_{>0}$ then the von Neumann algebra
$\mathscr{B}$ generated by $T=s\operatorname{M}_{f}+\operatorname{M}_{f}s^{*}$
is maximal abelian in $\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$.

###### Proof.

According to [Ped89, Theorem 4.7.7], the claim follows once we show that there
exists a cyclic vector for $\mathscr{B}=\\{T\\}^{\prime\prime}$. We claim that
$\phi_{0}$ is such a vector. Indeed, it is clear that $\phi_{0}$ belongs to
$V=\overline{\mathscr{B}\phi_{0}}$. Next, assume that
$\phi_{0},\dotsc,\phi_{n}\in V$ for some $n\in\mathbb{Z}_{+}$. Then
$\phi_{n+1}=\tfrac{1}{f(n+1)}T(\phi_{n})-\tfrac{f(n)}{f(n+1)}\phi_{n-1}$
belongs to $V$ and consequently
$V=\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$. ∎

One easily sees that $\mathscr{C}_{\mathrm{SU}_{q}(2)}$ is abelian and
$\mathscr{C}_{\mathrm{SU}_{q}(2)}=\\{\alpha+\alpha^{*}\\}^{\prime\prime}$.
Indeed, $\alpha+\alpha^{*}$ is the character of the fundamental representation
and the fusion rules of $\mathrm{SU}_{q}(2)$ imply that
$\chi(U^{n})\in\\{\alpha+\alpha^{*}\\}^{\prime\prime}$ for all
$U^{n}\in\operatorname{Irr}(\mathrm{SU}_{q}(2))$. Consequently222By an abuse
of notation, we will write $\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})$ for
the von Neumann algebra generated by $\psi_{1}(\alpha+\alpha^{*})$.,

$\mathcal{Q}_{L}\mathscr{C}_{\mathrm{SU}_{q}(2)}\mathcal{Q}_{L}^{*}=\bigl{\\{}\int_{\mathbb{T}}^{\oplus}T\otimes\mathds{1}_{\overline{\mathsf{H}_{\lambda}}}\operatorname{d}\\!\mu(\lambda)\,|\,T\in\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})\bigr{\\}}\simeq\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})\otimes\mathds{1}_{\operatorname{L}^{\infty}(\mathbb{T})}$
(4.2)

(observe that $\psi_{\lambda}(\alpha+\alpha^{*})=\psi_{1}(\alpha+\alpha^{*})$
for all $\lambda\in\mathbb{T}$). Note, using equation (4.2), that
$\mathscr{C}_{\mathrm{SU}_{q}(2)}$ does not contain the center of
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\simeq\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{T})$,
so it is certainly not a MASA. The next result describes its relative
commutant.

######  Proposition 4.2.

The relative commutant of $\mathscr{C}_{\mathrm{SU}_{q}(2)}$ is given by

$\begin{split}\mathcal{Q}_{L}(\mathscr{C}_{\mathrm{SU}_{q}(2)}^{\prime}\cap\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2)))\mathcal{Q}_{L}^{*}&=\bigl{\\{}\int_{\mathbb{T}}^{\oplus}T_{\lambda}\otimes\mathds{1}_{\overline{\mathsf{H}_{\lambda}}}\operatorname{d}\\!\mu(\lambda)\,|\,T_{\lambda}\in\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})\text{
for a.e. }\lambda\in\mathbb{T}\bigr{\\}}\\\
&\simeq\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{T}).\end{split}$

###### Proof.

Inclusion $\supseteq$ clearly follows from equation (4.2), assume that $T$
belongs to
$\mathcal{Q}_{L}(\mathscr{C}_{\mathrm{SU}_{q}(2)}^{\prime}\cap\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2)))\mathcal{Q}_{L}^{*}$.
Using (4.1) we can write
$T=\int_{\mathbb{T}}^{\oplus}T_{\lambda}\otimes\mathds{1}_{\overline{\mathsf{H}_{\lambda}}}\operatorname{d}\\!\mu(\lambda)$
for some $T_{\lambda}\in\operatorname{B}(\mathsf{H}_{\lambda})$. Our
assumption forces
$T_{\lambda}\in\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})^{\prime}$ for almost
all $\lambda\in\mathbb{T}$. From the definition of $\psi_{1}$ we see that
$\psi_{1}(\alpha)$ is a weighted shift and Lemma 4.1 applies –
$\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})$ is a MASA in
$\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$, hence
$T_{\lambda}\in\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})$ and the claim
follows. ∎

Despite the above negative result, we can nonetheless obtain some examples of
MASAs. First of all, since $\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})$ is a
MASA in $\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$ we immediately get that
the subalgebra generated by $\mathscr{C}_{\mathrm{SU}_{q}(2)}$ and the center
of $\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$ is a MASA in
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$. More nontrivially, we will
prove that $\mathscr{C}_{\mathrm{SU}_{q}(2)}$ is MASA in the smaller von
Neumann algebra of fixed points for the scaling group. Recall that we denote
this algebra by $\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau}$.

######  Proposition 4.3.

The algebra of class functions $\mathscr{C}_{\mathrm{SU}_{q}(2)}$ is a MASA in
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau}$, i.e.

$\mathscr{C}_{\mathrm{SU}_{q}(2)}^{\prime}\cap\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau}=\mathscr{C}_{\mathrm{SU}_{q}(2)}.$

###### Proof.

Observe first that since $\mathscr{C}_{\mathrm{SU}_{q}(2)}$ is generated by
characters, we have
$\mathscr{C}_{\mathrm{SU}_{q}(2)}\subseteq\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau}$.
Take
$T=\int_{\mathbb{T}}^{\oplus}T_{\lambda}\otimes\mathds{1}_{\overline{\mathsf{H}_{\lambda}}}\operatorname{d}\\!\mu(\lambda)$
in
$\mathcal{Q}_{L}(\mathscr{C}_{\mathrm{SU}_{q}(2)}^{\prime}\cap\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau})\mathcal{Q}_{L}^{*}$.
Proposition 4.2 implies that
$T_{\lambda}\in\psi_{1}(\mathscr{C}_{\mathrm{SU}_{q}(2)})$ for almost all
$\lambda\in\mathbb{T}$. Denote by $P$ the operator implementing the scaling
group for $\mathrm{SU}_{q}(2)$ and its dual [KV00, Definition 6.9]. We know
how to express this operator on the level of direct integrals (see [Kra20,
Proposition 7.3]): for all $t\in\mathbb{R}$ we have

$\mathcal{Q}_{L}P^{it}\mathcal{Q}_{L}^{*}\colon\int_{\mathbb{T}}^{\oplus}\operatorname{HS}(\mathsf{H}_{\lambda})\operatorname{d}\\!\mu(\lambda)\ni\int_{\mathbb{T}}^{\oplus}\xi_{\lambda}\operatorname{d}\\!\mu(\lambda)\mapsto\int_{\mathbb{T}}^{\oplus}\xi_{\lambda|q|^{2it}}\operatorname{d}\\!\mu(\lambda)\in\int_{\mathbb{T}}^{\oplus}\operatorname{HS}(\mathsf{H}_{\lambda})\operatorname{d}\\!\mu(\lambda),$
(4.3)

and since $T$ is invariant under the (transformed) scaling group we have

$\int_{\mathbb{T}}^{\oplus}T_{\lambda}\otimes\mathds{1}_{\overline{\mathsf{H}_{\lambda}}}\operatorname{d}\\!\mu(\lambda)=T=(\mathcal{Q}_{L}P^{it}\mathcal{Q}_{L}^{*})T(\mathcal{Q}_{L}P^{-it}\mathcal{Q}_{L}^{*})=\int_{\mathbb{T}}^{\oplus}T_{\lambda|q|^{2it}}\otimes\mathds{1}_{\overline{\mathsf{H}_{\lambda}}}\operatorname{d}\\!\mu(\lambda).$
(4.4)

It follows that $\mathbb{T}\ni\lambda\mapsto
T_{\lambda}\in\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$ is constant almost
everywhere. Indeed, for $\kappa\in\mathbb{T}$ denote by
$f_{\kappa}\in\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{T})$
the function $\lambda\mapsto T_{\lambda\kappa}$. Equation (4.4) implies
$f_{1}=f_{\kappa}\,(\kappa\in\mathbb{T})$. For all
$\theta\in\operatorname{L}^{1}(\mathbb{T}),\omega\in\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))_{*}$
we get333Integrals of $\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$ or
$\operatorname{L}^{\infty}(\mathbb{T})\overline{\otimes}\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$–valued
functions are understood in the sense of Pettis, where the von Neumann
algebras are equipped with the $w^{*}$–topology.

$\begin{split}&\quad\;(\omega\otimes\theta)f_{1}=(\omega\otimes\theta)\bigl{(}\int_{\mathbb{T}}f_{\kappa}\operatorname{d}\\!\mu(\kappa)\bigr{)}=\int_{\mathbb{T}}(\omega\otimes\theta)(f_{\kappa})\operatorname{d}\\!\mu(\kappa)\\\
&=\int_{\mathbb{T}}\int_{\mathbb{T}}\theta(\lambda)\omega(f_{\kappa}(\lambda))\operatorname{d}\\!\mu(\lambda)\operatorname{d}\\!\mu(\kappa)=\int_{\mathbb{T}}\theta(\lambda)\int_{\mathbb{T}}\omega(T_{\lambda\kappa})\operatorname{d}\\!\mu(\kappa)\operatorname{d}\\!\mu(\lambda)\\\
&=\bigl{(}\int_{\mathbb{T}}\theta(\lambda)\operatorname{d}\\!\mu(\lambda)\bigr{)}\int_{\mathbb{T}}\omega(T_{\kappa})\operatorname{d}\\!\mu(\kappa)=(\omega\otimes\theta)\bigl{(}\int_{\mathbb{T}}T_{\kappa}\operatorname{d}\\!\mu(\kappa)\otimes\mathds{1}_{\operatorname{L}^{\infty}(\mathbb{T})}\bigr{)},\end{split}$

hence
$f_{1}=\int_{\mathbb{T}}T_{\kappa}\operatorname{d}\\!\mu(\kappa)\otimes\mathds{1}_{\operatorname{L}^{\infty}(\mathbb{T})}$.
Consequently, $T$ belongs to
$\mathcal{Q}_{L}\mathscr{C}_{\mathrm{SU}_{q}(2)}\mathcal{Q}_{L}^{*}$ (see
equation (4.2)). ∎

###  4.2. A certain bicrossed product construction

In this subsection we present a construction of a class of compact quantum
groups $\mathbb{H}$ given by a bicrossed product of a compact quantum group
$\mathbb{G}$ and the additive group of rational numbers $\mathbb{Q}$ (in this
paper we equip $\mathbb{Q}$ with the discrete topology), where $\mathbb{Q}$
acts on $\operatorname{L}^{\infty}(\mathbb{G})$ using the scaling group of
$\mathbb{G}$. Our construction is a slight variation of a construction
presented in [DDS19, Section 4.1] – the main difference is that we replace
$\mathbb{R}$ with a discrete group $\mathbb{Q}$ in order to get a compact
quantum group as the bicrossed product. The principal reason why we are
interested in this family of quantum groups is the fact that they admit
nontrivial inner scaling automorphisms – a property that appeared in
Proposition 3.2 (see Lemma 4.4. Observe also that equation (4.3) implies that
nontrivial scaling automorphisms of $\mathrm{SU}_{q}(2)$ are never inner.
Another reason is that these bicrossed products provide examples of compact
quantum groups $\mathbb{H}$ with $\operatorname{L}^{\infty}(\mathbb{H})$ being
the type $\operatorname{II}_{\infty}$ injective factor.

Later on we will specify to $\mathbb{G}=\mathrm{SU}_{q}(2)$, for now let
$\mathbb{G}$ be an arbitrary compact quantum group. Fix a nonzero number
$\nu\in\mathbb{R}\setminus\\{0\\}$ and denote by $\alpha$ the action of
$\mathbb{Q}$ on $\operatorname{L}^{\infty}(\mathbb{G})$ given by

$\alpha_{\gamma}(x)=\tau_{\nu\gamma}^{\mathbb{G}}(x)\quad(\gamma\in\mathbb{Q},\,x\in\operatorname{L}^{\infty}(\mathbb{G})).$

Let $\mathbb{H}=\mathbb{Q}\bowtie\mathbb{G}$ be the resulting quantum group
(see [FMP15, Section 6]. One can also define $\mathbb{H}$ as a bicrossed
product of $\mathbb{Q}$ and $\mathbb{G}$, hence the notation [VV03, Definition
2.1]). For the details of this construction and its properties we refer the
reader to [VV03, Wan95, FMP15], here we will recall only some of its aspects.
By abuse of notation, let us also denote by $\alpha$ the map
$\alpha\colon\operatorname{L}^{\infty}(\mathbb{G})\rightarrow\ell^{\infty}(\mathbb{Q})\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{G})$
given by $\alpha(x)(\gamma)=\tau^{\mathbb{G}}_{\nu\gamma}(x)$. We have

$\operatorname{L}^{\infty}(\mathbb{H})=\mathbb{Q}\ltimes_{\alpha}\operatorname{L}^{\infty}(\mathbb{G})=\\{\alpha(x),u_{\gamma}\,|\,x\in\operatorname{L}^{\infty}(\mathbb{G}),\gamma\in\mathbb{Q}\\}^{\prime\prime}$

(where
$\mathbb{Q}\ni\gamma\mapsto\lambda_{\gamma}\in\operatorname{B}(\ell^{2}(\mathbb{Q}))$
is the left regular representation,
$u_{\gamma}=\lambda_{\gamma}\otimes\mathds{1}$) and

$\ell^{\infty}(\widehat{\mathbb{H}})=\ell^{\infty}(\mathbb{Q})\overline{\otimes}\ell^{\infty}(\widehat{\mathbb{G}}).$

These von Neumann algebras are represented on the Hilbert space

$\operatorname{L}^{2}(\mathbb{H})=\ell^{2}(\mathbb{Q})\otimes\operatorname{L}^{2}(\mathbb{G}).$

The bicrossed product $\mathbb{H}$ is compact as it is built from a discrete
and a compact quantum group. In fact, the GNS map for $h_{\mathbb{H}}$ is
given by

$\Lambda_{h_{\mathbb{H}}}(u_{\gamma}\alpha(x))=\Lambda_{h_{\widehat{\mathbb{Q}}}}(\lambda_{\gamma})\otimes\Lambda_{h_{\mathbb{G}}}(x)\quad(x\in\operatorname{L}^{\infty}(\mathbb{G}),\gamma\in\mathbb{Q}).$
(4.5)

We can also identify the (left) Haar integral on $\widehat{\mathbb{H}}$ – it
is equal to $\varphi_{\mathbb{Q}}\otimes\varphi_{\widehat{\mathbb{G}}}$ (where
$\varphi_{\mathbb{Q}},\varphi_{\widehat{\mathbb{G}}}$ are the left Haar
integrals on $\mathbb{Q}$ and $\widehat{\mathbb{G}}$), hence

$\nabla_{\varphi_{\widehat{\mathbb{H}}}}=\mathds{1}\otimes\nabla_{\varphi_{\widehat{\mathbb{G}}}}$

(it is a combination of Proposition 2.9, Theorem 2.13 and Proposition 2.16 in
[VV03]). Let $P_{\mathbb{G}}$ be the positive operator implementing scaling
group, see [VD14, Definition 5.1, Remark 5.2]. Since equality
$\nabla_{\varphi_{\widehat{\mathbb{G}}}}^{it}=P_{\mathbb{G}}^{it}$ holds for
any unimodular locally compact quantum group ([VD14, Proposition 5.6]), we
arrive at

$P_{\mathbb{H}}^{it}=\nabla_{\varphi_{\widehat{\mathbb{H}}}}^{it}=\mathds{1}\otimes\nabla_{\varphi_{\widehat{\mathbb{G}}}}^{it}=\mathds{1}\otimes
P_{\mathbb{G}}^{it}\quad(t\in\mathbb{R}).$ (4.6)

The irreducible representations of $\mathbb{H}$ are indexed by
$\mathbb{Q}\times\operatorname{Irr}(\mathbb{G})$. The corresponding characters
and $\uprho$ – operators are given by

$\chi_{(\gamma,\delta)}=u_{\gamma}\alpha(\chi_{\delta}),\quad\uprho_{(\gamma,\delta)}=\uprho_{\delta}\quad((\gamma,\delta)\in\mathbb{Q}\times\operatorname{Irr}(\mathbb{G}))$

(see [FMP15, Theorem 6.1]). It is a well known property of crossed products
that automorphisms with which $\mathbb{Q}$ acts on
$\operatorname{L}^{\infty}(\mathbb{G})$ become inner after the inclusion of
$\operatorname{L}^{\infty}(\mathbb{G})$ into
$\mathbb{Q}\ltimes_{\alpha}\operatorname{L}^{\infty}(\mathbb{G})$:

$\alpha(\tau^{\mathbb{G}}_{\nu\gamma}(x))=u_{\gamma}\alpha(x)u_{\gamma}^{*}\quad(x\in\operatorname{L}^{\infty}(\mathbb{G}),\gamma\in\mathbb{Q}).$
(4.7)

Let us now record a simple result concerning the scaling group of
$\mathbb{H}$.

######  Lemma 4.4.

  * •

We have $\tau^{\mathbb{H}}_{t}(\alpha(x))=\alpha(\tau^{\mathbb{G}}_{t}(x))$
and $\tau^{\mathbb{H}}_{t}(u_{\gamma})=u_{\gamma}$ for all
$t\in\mathbb{R},x\in\operatorname{L}^{\infty}(\mathbb{G}),\gamma\in\mathbb{Q}$.

  * •

For every $t\in\mathbb{R}$, the scaling automorphism $\tau^{\mathbb{H}}_{t}$
is trivial if and only if $\tau^{\mathbb{G}}_{t}$ is trivial. If
$\gamma\in\mathbb{Q}$, then $\tau^{\mathbb{H}}_{\nu\gamma}$ is inner.

###### Proof.

The first part is a direct consequence of equations (4.5), (4.6):

$\begin{split}&\quad\;\Lambda_{h_{\mathbb{H}}}(\tau_{t}^{\mathbb{H}}(\alpha(x)))=(\mathds{1}\otimes
P_{\mathbb{G}}^{it})(\Lambda_{h_{\widehat{\mathbb{Q}}}}(\mathds{1})\otimes\Lambda_{h_{\mathbb{G}}}(x))=\Lambda_{h_{\widehat{\mathbb{Q}}}}(\mathds{1})\otimes\Lambda_{h_{\mathbb{G}}}(\tau^{\mathbb{G}}_{t}(x))=\Lambda_{h_{\mathbb{H}}}(\alpha(\tau_{t}^{\mathbb{G}}(x)))\end{split}$

and

$\Lambda_{h_{\mathbb{H}}}(\tau^{\mathbb{H}}_{t}(u_{\gamma}))=(\mathds{1}\otimes
P_{\mathbb{G}}^{it})(\Lambda_{h_{\widehat{\mathbb{Q}}}}(\lambda_{\gamma})\otimes\Lambda_{h_{\mathbb{G}}}(\mathds{1}))=\Lambda_{h_{\widehat{\mathbb{Q}}}}(\lambda_{\gamma})\otimes\Lambda_{h_{\mathbb{G}}}(\mathds{1})=\Lambda_{h_{\mathbb{H}}}(u_{\gamma}).$

Since $h_{\mathbb{H}}$ is faithful on $\operatorname{L}^{\infty}(\mathbb{H})$
we get the first claim. As $\alpha$ is a monomorphism, $\tau^{\mathbb{H}}_{t}$
is trivial if and only so is $\tau^{\mathbb{G}}_{t}$. The last claim follows
from equation (4.7). ∎

Let us end these general considerations with an observation that

$u_{\gamma}\in\mathscr{C}_{\mathbb{H}}\quad(\gamma\in\mathbb{Q})\quad\textnormal{and}\quad\alpha(\mathscr{C}_{\mathbb{G}})\subseteq\mathscr{C}_{\mathbb{H}}.$

Indeed, it is a consequence of [Wan95, Theorem 3.7].

Fix $q\in(-1,1)\setminus\\{0\\}$. From now on we consider the special case
$\mathbb{G}=\mathrm{SU}_{q}(2)$ – accordingly $\mathbb{H}$ is given by
$\mathbb{H}=\mathbb{Q}\bowtie\mathrm{SU}_{q}(2)$. Note that this quantum group
depends on two parameters: $q$ and $\nu$. Using Proposition 4.3
($\mathscr{C}_{\mathrm{SU}_{q}(2)}$ is MASA in
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau}$) we are able to deduce
the following interesting property of $\mathbb{H}$:

######  Proposition 4.5.

Let $\mathbb{H}=\mathbb{Q}\bowtie\mathrm{SU}_{q}(2)$. The von Neumann algebra
$\mathscr{C}_{\mathbb{H}}$ is a MASA in
$\operatorname{L}^{\infty}(\mathbb{H})$.

###### Proof.

First, it is clear that $\mathscr{C}_{\mathbb{H}}$ is commutative. Indeed,
since $\mathscr{C}_{\mathrm{SU}_{q}(2)}$ is commutative, commutativity of
$\mathscr{C}_{\mathbb{H}}$ follows from [Wan95, Theorem 3.7]. Take now
$T\in\mathscr{C}_{\mathbb{H}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{H})$
– we want to show $T\in\mathscr{C}_{\mathbb{H}}$. Let
$\mathbb{E}\colon\operatorname{L}^{\infty}(\mathbb{H})=\mathbb{Q}\ltimes_{\alpha}\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\rightarrow\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$
be the canonical faithful normal conditional expectation satisfying
$\mathbb{E}(u_{\gamma}\alpha(x))=\delta_{\gamma,0}x$. Define operators

$T_{\gamma}=\mathbb{E}(u_{\gamma}^{*}T)\in\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\quad(\gamma\in\mathbb{Q}).$

Clearly we have

$\langle\xi\,|\,T_{\gamma}\eta\rangle=\langle\xi\,|\,\mathbb{E}(u_{\gamma}^{*}T)\eta\rangle=\langle\delta_{0}\otimes\xi\,|\,(u_{\gamma}^{*}T)(\delta_{0}\otimes\eta)\rangle$
(4.8)

for all $\xi,\eta\in\operatorname{L}^{2}(\mathrm{SU}_{q}(2))$. Fix
$\gamma\in\mathbb{Q}$. Using the fact that
$T\in\mathscr{C}_{\mathbb{H}}^{\prime}\cap\operatorname{L}^{\infty}(\mathbb{H})$
we will now show $T_{\gamma}\in\mathscr{C}_{\mathrm{SU}_{q}(2)}^{\prime}$.
Since for any $y\in\mathscr{C}_{\mathrm{SU}_{q}(2)}$ operator $\alpha(y)$
belongs to $\mathscr{C}_{\mathbb{H}}$, we get

$\begin{split}&\quad\;\langle\xi\,|\,T_{\gamma}y\eta\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,T(\delta_{0}\otimes
y\eta)\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,T\alpha(y)(\delta_{0}\otimes\eta)\rangle=\langle\alpha(y^{*})(\delta_{\gamma}\otimes\xi)\,|\,T(\delta_{0}\otimes\eta)\rangle\\\
&=\langle\delta_{0}\otimes
y^{*}\xi\,|\,(u_{\gamma}^{*}T)(\delta_{0}\otimes\eta)\rangle=\langle
y^{*}\xi\,|\,T_{\gamma}\eta\rangle=\langle\xi\,|\,yT_{\gamma}\eta\rangle\end{split}$

for all vectors $\xi,\eta\in\operatorname{L}^{2}(\mathrm{SU}_{q}(2))$ and
consequently $T_{\gamma}\in\mathscr{C}_{\mathrm{SU}_{q}(2)}^{\prime}$.
Take $\gamma^{\prime}\in\mathbb{Q}$. Observe that Lemma 4.4 together with
equation (4.7) implies that $\tau^{\mathbb{H}}_{\nu\gamma^{\prime}}$ is
implemeneted by $u_{\gamma^{\prime}}\in\mathscr{C}_{\mathbb{H}}$. Using
equation 4.6 we calculate

$\begin{split}&\quad\;\langle\xi\,|\,\tau^{\mathrm{SU}_{q}(2)}_{\nu\gamma^{\prime}}(T_{\gamma})\eta\rangle=\langle\delta_{0}\otimes
P^{-i\nu\gamma^{\prime}}_{\mathrm{SU}_{q}(2)}\xi\,|\,(u_{\gamma}^{*}T)(\delta_{0}\otimes
P^{-i\nu\gamma^{\prime}}_{\mathrm{SU}_{q}(2)}\eta)\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,P_{\mathbb{H}}^{i\nu\gamma^{\prime}}TP_{\mathbb{H}}^{-i\nu\gamma^{\prime}}(\delta_{0}\otimes\eta)\rangle\\\
&=\langle\delta_{\gamma}\otimes\xi\,|\,\tau^{\mathbb{H}}_{\nu\gamma^{\prime}}(T)(\delta_{0}\otimes\eta)\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,u_{\gamma^{\prime}}Tu_{\gamma^{\prime}}^{*}(\delta_{0}\otimes\eta)\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,T(\delta_{0}\otimes\eta)\rangle=\langle\xi\,|\,T_{\gamma}\eta\rangle\end{split}$

and as before we arrive at
$\tau_{\nu\gamma^{\prime}}^{\mathrm{SU}_{q}(2)}(T_{\gamma})=T_{\gamma}$.
Density of $\nu\mathbb{Q}$ in $\mathbb{R}$ implies
$T_{\gamma}\in\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))^{\tau}$.
Alternatively, one can also obtain this result by showing that $\mathbb{E}$
commutes with the scaling group and
$T\in\operatorname{L}^{\infty}(\mathbb{H})^{\tau}$.
These two properties of $T_{\gamma}$ imply that
$T_{\gamma}\in\mathscr{C}_{\mathrm{SU}_{q}(2)}$ (Proposition 4.3) and
consequently $\alpha(T_{\gamma})\in\mathscr{C}_{\mathbb{H}}$.

Formally we have $T=\sum_{\gamma\in\mathbb{Q}}u_{\gamma}\alpha(T_{\gamma})$.
However, this series does not need to converge in the $w^{*}$-topology (see
[Mer85]), which is why we will argue on the $\operatorname{L}^{2}$-level. Let
us first prove that

$\Lambda_{h_{\mathbb{H}}}(T)=\sum_{\gamma\in\mathbb{Q}}\Lambda_{h_{\mathbb{H}}}(u_{\gamma}\alpha(T_{\gamma}))=\sum_{\gamma\in\mathbb{Q}}\Lambda_{h_{\widehat{\mathbb{Q}}}}(\lambda_{\gamma})\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(T_{\gamma}).$
(4.9)

Since $\\{\delta_{\gamma}\\}_{\gamma\in\mathbb{Q}}$ forms an orthonormal basis
in $\ell^{2}(\mathbb{Q})$, we can write
$\Lambda_{h_{\mathbb{H}}}(T)=\sum_{\gamma\in\mathbb{Q}}\delta_{\gamma}\otimes\tilde{T}_{\gamma}$
for some $\tilde{T}_{\gamma}\in\operatorname{L}^{2}(\mathrm{SU}_{q}(2))$. Then

$\langle\xi\,|\,\tilde{T}_{\gamma}\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,\sum_{\gamma^{\prime}\in\mathbb{Q}}\delta_{\gamma^{\prime}}\otimes\tilde{T}_{\gamma^{\prime}}\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,\Lambda_{h_{\mathbb{H}}}(T)\rangle=\langle\delta_{\gamma}\otimes\xi\,|\,T(\delta_{0}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(\mathds{1})\rangle=\langle\xi\,|\,\Lambda_{h_{\mathrm{SU}_{q}(2)}}(T_{\gamma})\rangle$

for all $\gamma\in\mathbb{Q},\xi\in\operatorname{L}^{2}(\mathrm{SU}_{q}(2))$
which proves (4.9). Recall that $h_{\mathbb{H}}$ is tracial on
$\mathscr{C}_{\mathbb{H}}$, hence the claim follows from equation (4.9) and
the following lemma. ∎

######  Lemma 4.6.

Let $(\operatorname{M},\omega)$ be a von Neumann algebra with a fixed faithful
normal state. Assume that $\operatorname{N}\subseteq\operatorname{M}$ is a von
Neumann subalgebra such that $\omega|_{\operatorname{N}}$ is tracial. If
$x\in\operatorname{M}$ and
$\Lambda_{\omega}(x)\in\overline{\Lambda_{\omega}(N)}$ then
$x\in\operatorname{N}$.

This lemma is well-known to experts but we were not able to locate a precise
reference, so we decided to add a proof for completeness.

###### Proof.

We will show that $x$ commutes with every $y\in\operatorname{N}^{\prime}$.
Take $a,b\in\operatorname{M}$ that are analytic with respect to
$(\sigma_{t})_{t\in\mathbb{R}}$ and fix a net $(\Lambda_{\omega}(x_{i}))_{i\in
I}\,(x_{i}\in\operatorname{N})$ which converges to $\Lambda_{\omega}(x)$.
Observe that since $\omega|_{\operatorname{N}}$ is tracial,
$J\nabla^{\frac{1}{2}}$ is an isometry on
$\Lambda_{\omega}(\operatorname{N})$. As $J\nabla^{\frac{1}{2}}$ is closed, it
follows that $\lim_{i\in
I}\Lambda_{\omega}(x_{i}^{*})=\Lambda_{\omega}(x^{*})$. Consequently

$\begin{split}&\quad\;\langle\Lambda_{\omega}(a)\,|\,yx\Lambda_{\omega}(b)\rangle=\langle\Lambda_{\omega}(a)\,|\,yJ\sigma_{i/2}(b)^{*}J\Lambda_{\omega}(x)\rangle=\lim_{i\in
I}\langle\Lambda_{\omega}(a)\,|\,yJ\sigma_{i/2}(b)^{*}J\Lambda_{\omega}(x_{i})\rangle\\\
&=\lim_{i\in
I}\langle\Lambda_{\omega}(a)\,|\,yx_{i}\Lambda_{\omega}(b)\rangle=\lim_{i\in
I}\langle
J\sigma_{i/2}(a)^{*}J\Lambda_{\omega}(x_{i}^{*})\,|\,y\Lambda_{\omega}(b)\rangle=\langle
J\sigma_{i/2}(a)^{*}J\Lambda_{\omega}(x^{*})\,|\,y\Lambda_{\omega}(b)\rangle\\\
&=\langle
x^{*}\Lambda_{\omega}(a)\,|\,y\Lambda_{\omega}(b)\rangle=\langle\Lambda_{\omega}(a)\,|\,xy\Lambda_{\omega}(b)\rangle.\end{split}$

A standard density argument implies
$x\in\operatorname{N}^{\prime\prime}=\operatorname{N}$. ∎

######  Remark 4.7.

In the proof of Proposition 4.5, we argued on the $\operatorname{L}^{2}$-level
that $\alpha(T_{\gamma})\in\mathscr{C}_{\mathbb{H}}\,(\gamma\in\mathbb{Q})$
implies that $T\in\mathscr{C}_{\mathbb{H}}$. Alternatively, we could use a
Fejér-type theorem for crossed products and arrive at the same conclusion (see
e.g. [CN19, Theorem 4.10] for a general result).

In the penultimate result we prove about
$\mathbb{H}=\mathbb{Q}\bowtie\mathrm{SU}_{q}(2)$, we study its von Neumann
algebra of bounded functions. In particular, we show that for some values of
$\nu,q$, it is a factor of type $\operatorname{II}_{\infty}$ – we are not
aware of another example of a compact quantum group in the literature with
this property.

######  Proposition 4.8.

  * •

$\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{H}))$ is equal to
$\\{u_{\gamma}\,|\,\gamma\in\mathbb{Q}\,\cap\,\tfrac{\pi}{\nu\log(|q|)}\,\mathbb{Z}\\}^{\prime\prime}$.
In particular, it is trivial if $\nu\log(|q|)\notin\pi\mathbb{Q}$ and
isomorphic to $\operatorname{L}^{\infty}(\mathbb{T})$ otherwise.

  * •

Let $t\in\mathbb{R}$. The scaling automorphism $\tau^{\mathbb{H}}_{t}$ is
trivial if and only if $t\in\tfrac{\pi}{\log(|q|)}\mathbb{Z}$. It is inner if
and only if $t\in\nu\mathbb{Q}+\tfrac{\pi}{\log(|q|)}\mathbb{Z}$.

  * •

$\mathbb{H}$ is coamenable and consequently
$\operatorname{L}^{\infty}(\mathbb{H})$ is injective.

  * •

If $\nu\log(|q|)\notin\pi\mathbb{Q}$ then
$\operatorname{L}^{\infty}(\mathbb{H})$ is a factor of type
$\operatorname{II}_{\infty}$.

###### Proof.

Observe first that for all $t\in\mathbb{R}$, the scaling automorphism
$\tau^{\mathbb{H}}_{t}$ is trivial if and only $\tau^{\mathrm{SU}_{q}(2)}_{t}$
is trivial (Lemma 4.4) which happens if and only if
$t\in\tfrac{\pi}{\log(|q|)}\mathbb{Z}$ (equation (4.4)).
Take $x\in\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{H}))$. Since
$\mathscr{C}_{\mathbb{H}}$ is MASA in $\operatorname{L}^{\infty}(\mathbb{H})$,
we know that

$x\in\overline{\operatorname{span}\\{\chi_{\beta}\,|\,\beta\in\operatorname{Irr}(\mathbb{H}):\uprho_{\beta}=\mathds{1}_{\beta}\\}}^{\,\operatorname{sot}}=\overline{\operatorname{span}\\{u_{\gamma}\,|\,\gamma\in\mathbb{Q}\\}}^{\,\operatorname{sot}}$

(Proposition 3.1). Write

$x=\operatorname{sot}-\underset{i\in
I}{\lim}\sum_{\gamma\in\mathbb{Q}}C^{i}_{\gamma}u_{\gamma},\quad\Lambda_{h_{\mathbb{H}}}(x)=\sum_{\gamma\in\mathbb{Q}}C_{\gamma}\Lambda_{h_{\mathbb{H}}}(u_{\gamma})$

for some $C_{\gamma},C^{i}_{\gamma}\in\mathbb{C}$, where
$\sum_{\gamma\in\mathbb{Q}}C^{i}_{\gamma}u_{\gamma}$ belongs to
$\operatorname{span}\\{u_{\gamma}\,|\,\gamma\in\mathbb{Q}\\}$ for each $i\in
I$. Take now $y\in\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$. Since
$x\in\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{H}))$, we have

$\begin{split}&\quad\;\sum_{\gamma\in\mathbb{Q}}C_{\gamma}\delta_{\gamma}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(\tau^{\mathrm{SU}_{q}(2)}_{\nu\gamma}(y))=\alpha(y)\bigl{(}\sum_{\gamma\in\mathbb{Q}}C_{\gamma}\Lambda_{h_{\mathbb{H}}}(u_{\gamma})\bigr{)}=\Lambda_{h_{\mathbb{H}}}(\alpha(y)x)=\Lambda_{h_{\mathbb{H}}}(x\alpha(y))\\\
&=x(\delta_{0}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(y))=\lim_{i\in
I}\sum_{\gamma\in\mathbb{Q}}C^{i}_{\gamma}(\delta_{\gamma}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(y)),\end{split}$

which implies

$C_{\gamma}\tau^{\mathrm{SU}_{q}(2)}_{\nu\gamma}(y)=\lim_{i\in
I}C^{i}_{\gamma}\,y\quad(\gamma\in\mathbb{Q}).$

As this equation holds for every
$y\in\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$, we must have
$C_{\gamma}=0$ whenever $\tau^{\mathrm{SU}_{q}(2)}_{\nu\gamma}$ is nontrivial,
i.e. for $\nu\gamma\notin\tfrac{\pi}{\log(|q|)}\mathbb{Z}$. Lemma 4.6 gives us

$\mathcal{Z}(\operatorname{L}^{\infty}(\mathbb{H}))\subseteq\\{u_{\gamma}\,|\,\gamma\in\mathbb{Q}\cap\tfrac{\pi}{\nu\log(|q|)}\mathbb{Z}\\}^{\prime\prime}.$

Inclusion $\supseteq$ is clear, hence we have identified the center of
$\operatorname{L}^{\infty}(\mathbb{H})$. If $\nu\log(|q|)\notin\pi\mathbb{Q}$
then clearly $\mathbb{Q}\cap\tfrac{\pi}{\nu\log(|q|)}\mathbb{Z}=\\{0\\}$ and
$\operatorname{L}^{\infty}(\mathbb{H})$ is a factor. Otherwise
$\mathbb{Q}\cap\tfrac{\pi}{\nu\log(|q|)}\mathbb{Z}$ is a subgroup of
$\mathbb{Q}$ isomorphic to $\mathbb{Z}$ and
$\\{u_{\gamma}\,|\,\gamma\in\mathbb{Q}\cap\tfrac{\pi}{\nu\log(|q|)}\mathbb{Z}\\}^{\prime\prime}$
is therefore isomorphic to
$\operatorname{L}(\mathbb{Z})\simeq\operatorname{L}^{\infty}(\mathbb{T})$
[Her73, Theorem A]. This proves the first point444We could also argue that
$\operatorname{L}^{\infty}(\mathbb{H})$ is a factor if
$\nu\log(|q|)\notin\pi\mathbb{Q}$ using [Ped18, Theorem 7.11.11]..

Take now $t\in\mathbb{R}$. If
$t=\nu\gamma+\tfrac{\pi}{\log(|q|)}s\in\nu\mathbb{Q}+\tfrac{\pi}{\log(|q|)}\mathbb{Z}$
then
$\tau^{\mathbb{H}}_{t}=\tau^{\mathbb{H}}_{\nu\gamma}\tau^{\mathbb{H}}_{\pi
s/\log(|q|)}=\tau^{\mathbb{H}}_{\nu\gamma}$ is inner by Lemma 4.4. Assume that
$t\notin\nu\mathbb{Q}+\tfrac{\pi}{\log|q|}\mathbb{Z}$ and
$\tau^{\mathbb{H}}_{t}=\operatorname{Ad}_{v}$ for some unitary
$v\in\operatorname{L}^{\infty}(\mathbb{H})$. Proposition 3.2 implies that
$v\in\\{u_{\gamma}\,|\,\gamma\in\mathbb{Q}\\}^{\prime\prime}$, hence we can
write

$\Lambda_{h_{\mathbb{H}}}(v)=\sum_{\gamma\in\mathbb{Q}}D_{\gamma}\Lambda_{h_{\mathbb{H}}}(u_{\gamma})=\sum_{\gamma\in\mathbb{Q}}D_{\gamma}(\delta_{\gamma}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(\mathds{1}))$
(4.10)

for some $D_{\gamma}\in\mathbb{C}$. Since $v$ is unitary, we have
$\sum_{\gamma\in\mathbb{Q}}|D_{\gamma}|^{2}=1$. Let
$f\in\operatorname{L}^{\infty}(\mathbb{T})$ be the characteristic function of
the arc $\\{e^{i\theta}\,|\,\theta\in[0,\pi]\\}\subseteq\mathbb{T}$ and
$F=\mathcal{Q}_{L}^{*}(\int_{\mathbb{T}}^{\oplus}f(\lambda)\mathds{1}_{\operatorname{HS}(\mathsf{H}_{\lambda})}\operatorname{d}\\!\mu(\lambda))\mathcal{Q}_{L}\in\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$.
Equation (4.10) together with Lemma 4.4 gives us

$\begin{split}&\quad\;\sum_{\gamma\in\mathbb{Q}}D_{\gamma}(\delta_{\gamma}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(\tau^{\mathrm{SU}_{q}(2)}_{\nu\gamma}(F)))=\alpha(F)\bigl{(}\sum_{\gamma\in\mathbb{Q}}D_{\gamma}(\delta_{\gamma}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(\mathds{1}))\bigr{)}=\Lambda_{h_{\mathbb{H}}}(\alpha(F)v)\\\
&=v\Lambda_{h_{\mathbb{H}}}(\tau^{\mathbb{H}}_{-t}(\alpha(F)))=v\bigl{(}\delta_{0}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(\tau^{\mathrm{SU}_{q}(2)}_{-t}(F))\bigr{)}.\end{split}$

Since
$v\in\\{\lambda_{\gamma}\otimes\mathds{1}\,|\,\gamma\in\mathbb{Q}\\}^{\prime\prime}$,
the last vector belongs to
$\overline{\operatorname{span}}\\{\delta_{\gamma}\otimes\Lambda_{h_{\mathrm{SU}_{q}(2)}}(\tau^{\mathrm{SU}_{q}(2)}_{-t}(F))\,|\,\gamma\in\mathbb{Q}\\}$.
It follows that there exists $\gamma\in\mathbb{Q}$ such that

$\tau^{\mathrm{SU}_{q}(2)}_{\nu\gamma}(F)=c\tau^{\mathrm{SU}_{q}(2)}_{-t}(F)$

for some $c\in\mathbb{C}$. Each scaling automorphism acts by a rotation
(equation (4.4)), hence $c=1$ and
$\tau^{\mathrm{SU}_{q}(2)}_{t+\nu\gamma}(F)=F$. However,
$\tau^{\mathrm{SU}_{q}(2)}_{t+\nu\gamma}$ is a nontrivial rotation. Indeed,
otherwise $t+\nu\gamma\in\tfrac{\pi}{\log(|q|)}\mathbb{Z}$ and we assume that
it is not the case. It follows that $f$ is equal to its proper rotation, a
contradiction. This ends the proof of the second bullet.

The compact quantum group $\mathbb{H}=\mathbb{Q}\bowtie\mathrm{SU}_{q}(2)$ is
coamenable because $\mathbb{Q}$ is amenable and $\mathrm{SU}_{q}(2)$ is
coamenable [DQV02, Theorem 15]. It follows that
$\operatorname{L}^{\infty}(\mathbb{H})$ is injective [BT03, Theorem 3.3].
Alternatively, to obtain injectivity of
$\operatorname{L}^{\infty}(\mathbb{H})$ one can also use the fact that a
crossed product of an injective von Neumann algebra by an action of an
amenable group is injective [Tak03c, Theorem 3.16].

Assume $\nu\log(|q|)\notin\pi\mathbb{Q}$. We already know that
$\operatorname{L}^{\infty}(\mathbb{H})=\mathbb{Q}\ltimes_{\alpha}\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$
is a factor. Since the n.s.f. tracial weight on
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\simeq\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{T})$
given by $\operatorname{Tr}\otimes h_{\mathbb{T}}$ is invariant under the
action of $\mathbb{Q}$, it gives rise to a n.s.f. tracial weight on
$\operatorname{L}^{\infty}(\mathbb{H})$ ([Tak03b, Theorem 1.17]) and
consequently $\operatorname{L}^{\infty}(\mathbb{H})$ is not of type
$\operatorname{III}$. It follows from the proof of [Wor98, Theorem 1.3] that
if there was a faithful normal tracial state on
$\operatorname{L}^{\infty}(\mathbb{H})$, then $\mathbb{H}$ would be of Kac
type. As this is not the case, $\operatorname{L}^{\infty}(\mathbb{H})$ cannot
be of type $\operatorname{II}_{1}$; we are left with two cases,
$\operatorname{I}_{\infty}$ and $\operatorname{II}_{\infty}$. Clearly
$|\nu\mathbb{Q}+\tfrac{\pi}{\log(|q|)}\mathbb{Z}|=\aleph_{0}<|\mathbb{R}|$
hence there exists a scaling automorphism $\tau^{\mathbb{H}}_{t}$ which is not
inner. It is well known that all automorphisms of
$\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))$ are inner ([Bla06, II.5.5.14]),
hence $\operatorname{L}^{\infty}(\mathbb{H})$ has to be of type
$\operatorname{II}_{\infty}$.
Let us also give an alternative proof of the result that
$\operatorname{L}^{\infty}(\mathbb{H})$ is not of type
$\operatorname{I}_{\infty}$. Let

$\mathbb{E}\colon\operatorname{L}^{\infty}(\mathbb{H})=\mathbb{Q}\ltimes_{\alpha}\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\rightarrow\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))$

be the canonical faithful normal conditional expectation. Assume by
contradiction that $\operatorname{L}^{\infty}(\mathbb{H})$ is of type
$\operatorname{I}_{\infty}$. Then it is purely atomic and it follows that
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\simeq\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{T})$
is purely atomic as well ([Bla06, Theorem IV.2.2.4]), which gives us a
contradiction.

Yet another way to prove that $\operatorname{L}^{\infty}(\mathbb{H})$ is a
factor of type $\operatorname{II}_{\infty}$ is to use properties of crossed
products. Indeed, we know that
$\operatorname{L}^{\infty}(\mathrm{SU}_{q}(2))\simeq\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))\overline{\otimes}\operatorname{L}^{\infty}(\mathbb{T})$
and by formula 4.3 and the scaling group acts trivially on the first factor
and by rotations on the second factor, hence
$\operatorname{L}^{\infty}(\mathbb{H})\simeq\operatorname{B}(\ell^{2}(\mathbb{Z}_{+}))\overline{\otimes}\left(\operatorname{L}^{\infty}(\mathbb{T})\rtimes\mathbb{Q}\right)$.
If $\nu\log(|q|)\notin\pi\mathbb{Q}$ then all these rotations are irrational,
therefore the action is free, ergodic, and it preserves the Lebesgue measure
on the unit circle. By [Tak03a, Corollary 7.8]
$\operatorname{L}^{\infty}(\mathbb{T})\rtimes\mathbb{Q}$ is a factor of type
$\operatorname{II}_{1}$, hence $\operatorname{L}^{\infty}(\mathbb{H})$ is a
factor of type $\operatorname{II}_{\infty}$. ∎

As a corollary, we can show that our family of bicrossed products contains
uncountably many different isomorphism classes of quantum groups. To formulate
this result, let us denote by $\mathbb{H}_{\nu,q}$ the bicrossed product
$\mathbb{Q}\bowtie\mathrm{SU}_{q}(2)$ constructed using parameter $\nu$.

######  Corollary 4.9.

Let
$\nu,\nu^{\prime}\in\mathbb{R}\setminus\\{0\\},q,q^{\prime}\in(-1,1)\setminus\\{0\\}$.
If $\mathbb{H}_{\nu,q}$ and $\mathbb{H}_{\nu^{\prime},q^{\prime}}$ are
isomorphic, then $|q|=|q^{\prime}|$ and
$\nu\mathbb{Q}+\tfrac{\pi}{\log(|q|)}\mathbb{Z}=\nu^{\prime}\mathbb{Q}+\tfrac{\pi}{\log(|q|)}\mathbb{Z}$.
In particular, for each $q\in(-1,1)\setminus\\{0\\}$ the family
$\\{\mathbb{H}_{\nu,q}\,|\,\nu\in\mathbb{R}\setminus\\{0\\}\\}$ consists of
$\mathfrak{c}$ isomorphism classes of compact quantum groups.

###### Proof.

Let
$\phi\colon\mathrm{C}(\mathbb{H}_{\nu,q})\rightarrow\mathrm{C}(\mathbb{H}_{\nu^{\prime},q^{\prime}})$
be a Hopf $\star$-isomorphism implementing the isomorphism between
$\mathbb{H}_{\nu,q}$ and $\mathbb{H}_{\nu^{\prime},q^{\prime}}$ (recall that
$\mathbb{H}_{\nu,q}$ is coamenable). Since $\phi$ intertwines scaling groups
([MRW12, Proposition 3.15]) it follows that for each $t\in\mathbb{R}$,
$\tau^{\mathbb{H}_{\nu,q}}_{t}$ is trivial if and only if
$\tau^{\mathbb{H}_{\nu^{\prime},q^{\prime}}}_{t}$ is trivial and consequently
Proposition 4.8 implies
$\tfrac{\pi}{\log(|q|)}\mathbb{Z}=\tfrac{\pi}{\log(|q^{\prime}|)}\mathbb{Z}\Rightarrow|q|=|q^{\prime}|$.
Next, since inner scaling automorphisms of $\mathbb{H}_{\nu,q}$ are
implemented by elements of $\mathrm{C}(\mathbb{H}_{\nu,q})$ (similarly for
$\mathbb{H}_{\nu^{\prime},q^{\prime}}$) it follows from the same proposition
that
$\nu\mathbb{Q}+\tfrac{\pi}{\log(|q|)}\mathbb{Z}=\nu^{\prime}\mathbb{Q}+\tfrac{\pi}{\log(|q|)}\mathbb{Z}$.
The last claim is a consequence of
$\dim_{\mathbb{Q}}(\mathbb{R}/(\mathbb{Q}\tfrac{\pi}{\log(|q|)}))=\mathfrak{c}$.
∎

###  4.3. Examples with commutative $\mathscr{C}_{\mathbb{G}}$

In this subsection we will prove that the condition
$\sum_{\alpha}\sqrt{\tfrac{\dim(\alpha)}{\dim_{q}(\alpha)}}<+\infty$ from
Theorem 2.7 holds for a certain class of non-Kac type compact quantum groups.
More precisely, in this subsection we consider any compact quantum group
$\mathbb{G}$ with the following properties:

  1. (1)

there exists an irreducible fundamental representation $U$ with
$\dim_{q}(U)>\dim(U)$ and $\overline{U}\simeq U$,

  2. (2)

irreducible representations of $\mathbb{G}$ are labeled by $\mathbb{Z}_{+}$,
so that $\operatorname{Irr}(\mathbb{G})=\\{U^{n}\\}_{n\in\mathbb{Z}_{+}}$,
where $U^{1}=U$ and $U^{0}$ is the trivial representation,

  3. (3)

we have $U^{1}\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{
\raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!U^{n}\simeq\bigoplus_{k=0}^{n+1}C(k,n)U^{k}$, with $C(n+1,n)\geq 1$
and $\sup_{n\in\mathbb{Z}_{+}}C(n+1,n)<+\infty$.

Let us mention two classes of compact quantum groups that fit into the above
description:

  * •

Non-Kac type free orthogonal quantum group $\mathbb{G}=O_{F}^{+}$ satisfies
the above conditions with $U$ being the standard fundamental representation
(see [Ban96]).

  * •

Let $(B,\psi)$ be a finite dimensional $\mathrm{C}^{*}$-algebra with a non-
tracial $\delta$-form. The non-Kac type quantum automorphism group
$\mathbb{G}_{Aut}(B,\psi)$ also satisfies the above conditions (see [Ban02]).

To keep the notation lighter, let us write $\dim(n)=\dim(U^{n})$ and
$\dim_{q}(n)=\dim_{q}(U^{n})$ for all $n\in\mathbb{Z}_{+}$. Using our
assumptions on the representation theory of $\mathbb{G}$ we can show that
$(\tfrac{\dim(n)}{\dim_{q}(n)})_{n\in\mathbb{Z}_{+}}$ decays at an exponential
rate.

######  Lemma 4.10.

  * •

We have $\overline{U^{n}}\simeq U^{n}$ and $U^{n}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ } }}\\!\\!U^{m}\simeq
U^{m}\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝
} }}\\!\\!U^{n}$ for all $n,m\in\mathbb{Z}_{+}$.

  * •

There exists $d>0,c>1$ such that
$\tfrac{\dim(n)}{\dim_{q}(n)}\leq\tfrac{d}{c^{n}}$ for all
$n\in\mathbb{Z}_{+}$.

Let us note that when $\mathbb{G}=O_{F}^{+}$ or
$\mathbb{G}=\mathbb{G}_{Aut}(B,\psi)$ then numbers $\dim(n),\dim_{q}(n)$ are
known, see [Ban96, Ban99].

###### Proof.

Observe that we have $(U^{1})^{\\!\\!\\!{\scriptscriptstyle\text{
\raisebox{0.5pt}{ \raisebox{-1.2pt}{$\,\top$}⃝ }
}}\\!\\!\\!n}\simeq\bigoplus_{k=0}^{n+1}c^{\prime}_{k,n}U^{k}$ for some
$c^{\prime}_{k,n}\in\mathbb{Z}_{+}$. As $\overline{U^{1}}\simeq U^{1}$, it
follows inductively that $\overline{U^{n}}\simeq U^{n}$ for all
$n\in\mathbb{Z}_{+}$. Equivalence $U^{n}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ } }}\\!\\!U^{m}\simeq
U^{m}\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝
} }}\\!\\!U^{n}$ can now be justified with the following calculations

$U^{n}\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝
} }}\\!\\!U^{m}\simeq\overline{U^{n}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!U^{m}}\simeq\overline{U^{m}}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!\overline{U^{n}}\simeq U^{m}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ } }}\\!\\!U^{n}.$

To prove the second bullet, let us introduce positive numbers $A_{n}\geq 1$
via $\dim_{q}(n)=A_{n}\dim(n)\,(n\in\mathbb{Z}_{+})$. Clearly $A_{0}=1$ and we
assume that $A_{1}>1$. The fusion rule $U^{1}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!U^{n}\simeq\bigoplus_{k=0}^{n+1}C(k,n)U^{k}$ implies

$A_{1}A_{n}\dim(1)\dim(n)=\dim_{q}(U^{1}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!U^{n})=\dim_{q}\bigl{(}\bigoplus_{k=0}^{n+1}C(k,n)U^{k}\bigr{)}=\sum_{k=0}^{n+1}C(k,n)A_{k}\dim(k)$

and

$\dim(1)\dim(n)=\dim(U^{1}\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{
\raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!U^{n})=\dim\bigl{(}\bigoplus_{k=0}^{n+1}C(k,n)U^{k}\bigr{)}=\sum_{k=0}^{n+1}C(k,n)\dim(k).$

Combining these equations gives us

$\begin{split}&\quad\;A_{1}A_{n}\dim(1)\dim(n)\leq(\max_{k\in\\{0,\dotsc,n\\}}A_{k})\sum_{k=0}^{n}C(k,n)\dim(k)+C(n+1,n)A_{n+1}\dim(n+1)\\\
&=(\max_{k\in\\{0,\dotsc,n\\}}A_{k})\bigl{(}\dim(1)\dim(n)-C(n+1,n)\dim(n+1)\bigr{)}+C(n+1,n)A_{n+1}\dim(n+1),\end{split}$

hence

$\begin{split}A_{n+1}&\geq\max_{k\in\\{0,\dotsc,n\\}}A_{k}+\bigl{(}A_{1}A_{n}-\max_{k\in\\{0,\dotsc,n\\}}A_{k}\bigr{)}\tfrac{\dim(1)\dim(n)}{C(n+1,n)\dim(n+1)}.\end{split}$

The above inequality implies $A_{n+1}=\max_{p\in\\{0,\dotsc,n+1\\}}A_{p}$.
Consequently, we can further write

$\begin{split}A_{n+1}\geq
A_{n}+A_{n}\tfrac{A_{1}-1}{\sup_{m\in\mathbb{Z}_{+}}C(m+1,m)}\tfrac{\dim(1)\dim(n)}{\dim(n+1)}.\end{split}$

Since $U^{n+1}$ is a subrepresentation of $U^{1}\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ } }}\\!\\!U^{n}$, we have
$\dim(1)\dim(n)\geq\dim(n+1)$ and

$A_{n+1}\geq
A_{n}\bigl{(}1+\tfrac{A_{1}-1}{\sup_{m\in\mathbb{Z}_{+}}C(m+1,m)}\bigr{)}.$

Write $c=1+\tfrac{A_{1}-1}{\sup_{m\in\mathbb{Z}_{+}}C(m+1,m)}>1$. We have
shown $A_{n+1}\geq cA_{n}$. Using $\dim_{q}(n)=A_{n}\dim(n)$ we arrive at

$\dim_{q}(n)=A_{n}\dim(n)\geq c^{n-1}A_{1}\dim(n)\quad(n\in\mathbb{N}).$

∎

In particular, the above lemma implies that $\mathscr{C}_{\mathbb{G}}$ is an
abelian von Neumann algebra. Theorems 2.7 and 3.4 give us the following
corollary (it follows from the fusion rules that the assumptions are
satisfied).

######  Corollary 4.11.

We have $\sum_{n=0}^{\infty}\sqrt{\tfrac{\dim(n)}{\dim_{q}(n)}}<+\infty$,
hence the inclusion
$\mathscr{C}_{\mathbb{G}}\subseteq\operatorname{L}^{\infty}(\mathbb{G})$ is
quasi-split. Furthermore, $\mathscr{C}_{\mathbb{G}}$ is not MASA.

###  4.4. Quantum unitary group $U_{F}^{+}$

Let $F$ be an invertible matrix with complex entries and $U_{F}^{+}$ the
associated compact quantum group. In this subsection we show that the sum
condition

$\sum_{\gamma\in\operatorname{Irr}(U_{F}^{+})}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}<+\infty$
(4.11)

is satisfied provided $U_{F}^{+}$ is “sufficiently non-Kac” (see Proposition
4.14 for a precise result). Consequently, in this case we obtain information
about the inclusion
$\mathscr{C}_{U_{F}^{+}}\subseteq\operatorname{L}^{\infty}(U_{F}^{+})$.
The representation theory of $U_{F}^{+}$ was described by Banica in [Ban97,
Théorème 1], let us recall some of its aspects. We can identify
$\operatorname{Irr}(U_{F}^{+})$ with the free product of monoids
$\mathbb{Z}_{+}\star\mathbb{Z}_{+}$; let $\alpha,\beta$ be the generators and
$e$ the neutral element. Then $\alpha$ corresponds to the fundamental
representation, $\beta$ to its conjugate and $e$ to the trivial
representation. The unique antimultiplicative involution
$\mathbb{Z}_{+}\star\mathbb{Z}_{+}\ni
x\mapsto\overline{x}\in\mathbb{Z}_{+}\star\mathbb{Z}_{+}$ satisfying
$\overline{\alpha}=\beta,\overline{e}=e$ corresponds to the conjugation of
representations. Finally, the tensor product of representations is given by

$x\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!y\simeq\bigoplus_{\overset{a,b,c\in\mathbb{Z}_{+}\star\mathbb{Z}_{+}:}{x=ac,\;y=\overline{c}b}}ab\quad(x,y\in\mathbb{Z}_{+}\star\mathbb{Z}_{+}).$
(4.12)

In order to efficiently calculate the sum (4.11), we need to single out a
family of irreducible representations out of which all of
$\operatorname{Irr}(U_{F}^{+})$ is built. Observe that each nontrivial word
$\gamma\in\operatorname{Irr}(U_{F}^{+})\setminus\\{e\\}$ has a well defined
beginning and an end $s(\gamma),t(\gamma)\in\\{\alpha,\beta\\}$, e.g.
$s(\alpha\beta)=\beta$. Let us define sets

$\begin{split}I_{\alpha}&=\\{\alpha(\beta\alpha)^{n}\,|\,n\in\mathbb{Z}_{+}\\}\cup\\{(\alpha\beta)^{n}\,|\,n\in\mathbb{N}\\},\\\
I_{\beta}&=\\{\beta(\alpha\beta)^{n}\,|\,n\in\mathbb{Z}_{+}\\}\cup\\{(\beta\alpha)^{n}\,|\,n\in\mathbb{N}\\}.\end{split}$

The following observation was already made e.g. in [MN19]:

######  Lemma 4.12.

Every nontrivial word $\gamma\in\operatorname{Irr}(U_{F}^{+})\setminus\\{e\\}$
can be uniquely written as

$\gamma=x_{1}\cdots x_{p}=x_{1}\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{
\raisebox{-1.7pt}{$\,\top$}⃝ } }}\\!\\!\cdots\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ } }}\\!\\!x_{p}$

for some $p\in\mathbb{N}$, $x_{1},\dotsc,x_{p}\in I_{\alpha}\cup I_{\beta}$
such that $s(x_{k})=t(x_{k+1})$ for $1\leq k\leq p-1$.

The above result follows easily from the observation that if
$\delta\alpha^{n}\delta^{\prime}$ for some
$\delta,\delta^{\prime}\in\operatorname{Irr}(U_{F}^{+})$ and $n\geq 2$ then
(4.12) implies

$\delta\alpha^{n}\delta^{\prime}=\delta\alpha\\!\\!{\scriptstyle\text{
\raisebox{0.8pt}{ \raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!\alpha^{n-1}\delta^{\prime}$

(and similarly for $\delta\beta^{n}\delta^{\prime}$). It follows that in order
to calculate the sum (4.11) we need to find a (quantum) dimension of
representations from the sets $I_{\alpha},I_{\beta}$.

Let us introduce the q-numbers
$\left[n\right]_{q}=\tfrac{q^{-n}-q^{n}}{q^{-1}-q}$, where
$n\in\mathbb{Z}_{+}$ and $0<q<1$ [KS97]. Furthermore, for $n\in\mathbb{N}$ we
will write $w_{n}^{\gamma}=\gamma\overline{\gamma}\gamma\cdots$ ($n$ letters),
where $\gamma=\alpha$ or $\gamma=\beta$. Thus $t(w^{\gamma}_{n})=\gamma$ and
$s(w^{\gamma}_{n})$ is equal to $\gamma$ if $n$ is odd and equal to
$\overline{\gamma}$ if $n$ is even. We also define
$w^{\alpha}_{0}=w^{\beta}_{0}=e$.

######  Lemma 4.13.

Let $d$ be the classical or the quantum dimension function,
$\gamma\in\\{\alpha,\beta\\}$ and $n\in\mathbb{Z}_{+}$. If $d(\alpha)=2$, then
$d(w^{\gamma}_{n})=n+1$. If $d(\alpha)>2$, then $d(w^{\gamma}_{n})=[n+1]_{q}$,
where $0<q<1$ is chosen so that $q^{-1}+q=d(\alpha)$.

###### Proof.

Fix $n\in\mathbb{Z}_{+}$. As

$(\alpha\beta)^{n}\alpha\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{
\raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!\beta=(\alpha\beta)^{n+1}\oplus(\alpha\beta)^{n},$

we have

$d(\alpha)d((\alpha\beta)^{n}\alpha)=d((\alpha\beta)^{n+1})+d((\alpha\beta)^{n}).$
(4.13)

Similarly,

$(\alpha\beta)^{n}\alpha\beta\\!\\!{\scriptstyle\text{ \raisebox{0.8pt}{
\raisebox{-1.7pt}{$\,\top$}⃝ }
}}\\!\\!\alpha=(\alpha\beta)^{n+1}\alpha\oplus(\alpha\beta)^{n}\alpha$

and

$d(\alpha)d((\alpha\beta)^{n+1})=d((\alpha\beta)^{n+1}\alpha)+d((\alpha\beta)^{n}\alpha).$
(4.14)

The formula for $d(w^{\alpha}_{n})$ follows from the following elementary
claim which can be proven by induction.

If $(a_{n})_{n\in\mathbb{Z}_{+}}$ is a sequence in $\mathbb{R}_{\geq 0}$ such
that $a_{0}=1,a_{1}\geq 2$ and $a_{1}a_{n}=a_{n-1}+a_{n+1}$ for all
$n\in\mathbb{N}$, then

$a_{n}=\begin{cases}n+1&a_{1}=2\\\ [n+1]_{q}&a_{1}>2\end{cases}$

where $0<q<1$ is such that $q^{-1}+q=a_{1}$.

The formula for $d(w^{\beta}_{n})$ can be proven analogously using
$d(\alpha)=d(\beta)$. ∎

Using the above result we can show that for “sufficiently non-Kac” quantum
unitary groups, the sum condition (4.11) is satisfied.

######  Proposition 4.14.

Let $0<q_{\operatorname{q}}\leq q_{\operatorname{c}}\leq 1$ be given by
$q_{\operatorname{c}}^{-1}+q_{\operatorname{c}}=\dim(\alpha)$ and
$q_{\operatorname{q}}^{-1}+q_{\operatorname{q}}=\dim_{q}(\alpha)$. If
$\dim(\alpha)=2$ and $q_{\operatorname{q}}<0.0861$ or $\dim(\alpha)\geq 3$ and
$\tfrac{q_{\operatorname{q}}}{q_{\operatorname{c}}}<\bigl{(}1+\sqrt{\tfrac{3\sqrt{5}+5}{10}}\,\bigr{)}^{-2}=0.2306\dotsc$,
then
$\sum_{\gamma\in\operatorname{Irr}(U_{F}^{+})}\sqrt{\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}}<+\infty$.

###### Proof.

Lemma 4.12 shows

$\begin{split}&\sum_{\gamma\in\operatorname{Irr}(U_{F}^{+})}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}=1+\sum_{\delta\in\\{\alpha,\beta\\}}\sum_{p=1}^{\infty}\sum_{\gamma_{1}\in
I_{\delta}}\sum_{\gamma_{2}\in I_{s(\gamma_{1})}}\cdots\sum_{\gamma_{p}\in
I_{s(\gamma_{p-1})}}\bigl{(}\tfrac{\dim(\gamma_{1}\\!\\!\\!{\scriptscriptstyle\text{
\raisebox{0.5pt}{ \raisebox{-1.2pt}{$\,\top$}⃝ }
}}\\!\\!\\!\cdots\\!\\!\\!{\scriptscriptstyle\text{ \raisebox{0.5pt}{
\raisebox{-1.2pt}{$\,\top$}⃝ }
}}\\!\\!\\!\gamma_{p})}{\dim_{q}(\gamma_{1}\\!\\!\\!{\scriptscriptstyle\text{
\raisebox{0.5pt}{ \raisebox{-1.2pt}{$\,\top$}⃝ }
}}\\!\\!\\!\cdots\\!\\!\\!{\scriptscriptstyle\text{ \raisebox{0.5pt}{
\raisebox{-1.2pt}{$\,\top$}⃝ }
}}\\!\\!\\!\gamma_{p})}\bigr{)}^{\frac{1}{2}}.\end{split}$ (4.15)

We have

$\sum_{\gamma\in
I_{\alpha}}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}=\sum_{n=1}^{\infty}\bigl{(}\tfrac{\dim(w^{\alpha}_{n})}{\dim_{q}(w^{\alpha}_{n})}\bigr{)}^{\frac{1}{2}},\quad\sum_{\gamma\in
I_{\beta}}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}=\sum_{n=1}^{\infty}\bigl{(}\tfrac{\dim(w^{\beta}_{n})}{\dim_{q}(w^{\beta}_{n})}\bigr{)}^{\frac{1}{2}},$
(4.16)

consequently Lemma 4.13 implies

$S:=\sum_{\gamma\in
I_{\alpha}}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}=\sum_{\gamma\in
I_{\beta}}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}.$

It follows from equation (LABEL:eq4.4.6) that

$\sum_{\gamma\in\operatorname{Irr}(U_{F}^{+})}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}=1+\sum_{\delta\in\\{\alpha,\beta\\}}\sum_{p=1}^{\infty}S^{p}=1+2\sum_{p=1}^{\infty}S^{p}$

thus
$\sum_{\gamma\in\operatorname{Irr}(U_{F}^{+})}\sqrt{\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}}<+\infty$
if, and only if $S<1$.

Let us first consider the case $\dim(\alpha)\geq 3$. Note that in this case
$q_{\operatorname{c}}<1$. Using equation (4.16) and Lemma 4.13 we calculate

$\begin{split}S&=\sum_{n=1}^{\infty}\bigl{(}\tfrac{[n+1]_{q_{\operatorname{c}}}}{[n+1]_{q_{\operatorname{q}}}}\bigr{)}^{\frac{1}{2}}=\sum_{n=2}^{\infty}\bigl{(}\tfrac{q_{\operatorname{c}}^{-n}-q_{\operatorname{c}}^{n}}{q_{\operatorname{q}}^{-n}-q_{\operatorname{q}}^{n}}\,\tfrac{q_{\operatorname{q}}^{-1}-q_{\operatorname{q}}}{q_{\operatorname{c}}^{-1}-q_{\operatorname{c}}}\bigr{)}^{\frac{1}{2}}=\bigl{(}\tfrac{q_{\operatorname{q}}^{-1}-q_{\operatorname{q}}}{q_{\operatorname{c}}^{-1}-q_{\operatorname{c}}}\bigr{)}^{\frac{1}{2}}\sum_{n=2}^{\infty}(\tfrac{q_{\operatorname{c}}}{q_{\operatorname{q}}})^{-\frac{n}{2}}\bigl{(}\tfrac{1-q_{\operatorname{c}}^{2n}}{1-q_{\operatorname{q}}^{2n}}\bigr{)}^{\frac{1}{2}}\\\
&\leq(\tfrac{q_{\operatorname{q}}}{q_{\operatorname{c}}})^{-\frac{1}{2}}\bigl{(}\tfrac{1-q_{\operatorname{q}}^{2}}{1-q_{\operatorname{c}}^{2}}\bigr{)}^{\frac{1}{2}}\tfrac{1}{(1-q_{\operatorname{q}}^{4})^{\frac{1}{2}}}\sum_{n=2}^{\infty}(\tfrac{q_{\operatorname{c}}}{q_{\operatorname{q}}})^{-\frac{n}{2}}=\tfrac{1}{(1-q_{\operatorname{c}}^{2})^{\frac{1}{2}}(1+q_{\operatorname{q}}^{2})^{\frac{1}{2}}}\;(\tfrac{q_{\operatorname{c}}}{q_{\operatorname{q}}})^{\frac{1}{2}}\;\bigl{(}-1-(\tfrac{q_{\operatorname{q}}}{q_{\operatorname{c}}})^{\frac{1}{2}}+\tfrac{1}{1-(\frac{q_{\operatorname{q}}}{q_{\operatorname{c}}})^{\frac{1}{2}}}\bigr{)}\\\
&=\tfrac{1}{(1-q_{\operatorname{c}}^{2})^{\frac{1}{2}}(1+q_{\operatorname{q}}^{2})^{\frac{1}{2}}}\;(\tfrac{q_{\operatorname{c}}}{q_{\operatorname{q}}})^{\frac{1}{2}}\;\tfrac{1-(1-\frac{q_{\operatorname{q}}}{q_{\operatorname{c}}})}{1-(\frac{q_{\operatorname{q}}}{q_{\operatorname{c}}})^{\frac{1}{2}}}=\tfrac{1}{(1-q_{\operatorname{c}}^{2})^{\frac{1}{2}}(1+q_{\operatorname{q}}^{2})^{\frac{1}{2}}}\;\tfrac{(\frac{q_{\operatorname{q}}}{q_{\operatorname{c}}})^{\frac{1}{2}}}{1-(\frac{q_{\operatorname{q}}}{q_{\operatorname{c}}})^{\frac{1}{2}}}.\end{split}$

In the above inequality we have used
$\tfrac{1-q_{\operatorname{c}}^{2n}}{1-q_{\operatorname{q}}^{2n}}\leq\tfrac{1}{1-q_{\operatorname{q}}^{4}}$
for all $n\geq 2$. Now, as $\dim(\alpha)\geq 3$, we have
$q_{\operatorname{c}}\leq(3-\sqrt{5})/2$ and since
$\tfrac{q_{\operatorname{q}}}{q_{\operatorname{c}}}<\bigl{(}1+\sqrt{\tfrac{3\sqrt{5}+5}{10}}\,\bigr{)}^{-2}$,

$S<\tfrac{2}{(4-(3-\sqrt{5})^{2})^{\frac{1}{2}}}\;\tfrac{\bigl{(}1+\sqrt{\frac{3\sqrt{5}+5}{10}}\,\bigr{)}^{-1}}{1-\bigl{(}1+\sqrt{\frac{3\sqrt{5}+5}{10}}\,\bigr{)}^{-1}}=\tfrac{2}{(6\sqrt{5}-10)^{\frac{1}{2}}}\;\tfrac{1}{\bigl{(}1+\sqrt{\frac{3\sqrt{5}+5}{10}}\bigr{)}-1}=\bigl{(}\tfrac{3\sqrt{5}+5}{10}\bigr{)}^{\frac{1}{2}}\tfrac{1}{\sqrt{\frac{3\sqrt{5}+5}{10}}}=1$

which proves the claim.

Let us now consider the case $\dim(\alpha)=2$. We calculate using again Lemma
4.13

$\begin{split}S&=\sum_{n=1}^{\infty}\bigl{(}\tfrac{n+1}{[n+1]_{q_{\operatorname{q}}}}\bigr{)}^{\frac{1}{2}}=\sum_{n=2}^{\infty}\bigl{(}\tfrac{n}{q^{-n}_{\operatorname{q}}-q^{n}_{\operatorname{q}}}\,(q^{-1}_{\operatorname{q}}-q_{\operatorname{q}})\bigr{)}^{\frac{1}{2}}=(q_{\operatorname{q}}^{-1}-q_{\operatorname{q}})^{\frac{1}{2}}\sum_{n=2}^{\infty}(nq_{\operatorname{q}}^{n})^{\frac{1}{2}}\,\tfrac{1}{(1-q_{\operatorname{q}}^{2n})^{\frac{1}{2}}}\\\
&\leq
q_{\operatorname{q}}^{-\frac{1}{2}}\bigl{(}\tfrac{1-q_{\operatorname{q}}^{2}}{1-q_{\operatorname{q}}^{4}}\bigr{)}^{\frac{1}{2}}\bigl{(}-q_{\operatorname{q}}^{\frac{1}{2}}+\sum_{n=1}^{\infty}nq_{\operatorname{q}}^{\frac{n}{2}}\bigr{)}=\tfrac{q_{\operatorname{q}}^{-\frac{1}{2}}}{(1+q_{\operatorname{q}}^{2})^{\frac{1}{2}}}\bigl{(}-q_{\operatorname{q}}^{\frac{1}{2}}+\tfrac{q_{\operatorname{q}}^{\frac{1}{2}}}{(1-q_{\operatorname{q}}^{\frac{1}{2}})^{2}}\bigr{)}=\tfrac{1-(1-q_{\operatorname{q}}^{\frac{1}{2}})^{2}}{(1+q_{\operatorname{q}}^{2})^{\frac{1}{2}}(1-q_{\operatorname{q}}^{\frac{1}{2}})^{2}}=\tfrac{q_{\operatorname{q}}^{\frac{1}{2}}(2-q_{\operatorname{q}}^{\frac{1}{2}})}{(1+q_{\operatorname{q}}^{2})^{\frac{1}{2}}(1-q_{\operatorname{q}}^{\frac{1}{2}})^{2}}.\end{split}$

One can check that the above expression is increasing for
$0<q_{\operatorname{q}}<1$ and smaller than $1$ when
$q_{\operatorname{q}}<0.0861$. ∎

######  Remark 4.15.

Calculations in the proof of Proposition 4.14 are far from optimal, however it
is clear that there are non-Kac type quantum unitary groups $U_{F}^{+}$ with
$\sum_{\gamma\in\operatorname{Irr}(U_{F}^{+})}\bigl{(}\tfrac{\dim(\gamma)}{\dim_{q}(\gamma)}\bigr{)}^{\frac{1}{2}}=+\infty$.
Indeed, for $\dim(\alpha)=2$ we have
$S=\sum_{n=2}^{\infty}(\tfrac{n}{[n]_{q_{\operatorname{q}}}})^{\frac{1}{2}}>(2/[2]_{q_{\operatorname{q}}})^{\frac{1}{2}}+(3/[3]_{q_{\operatorname{q}}})^{\frac{1}{2}}$
hence $S>1$ when $q_{\operatorname{q}}>0.2134$.

It follows from the rule (4.12) and Lemma 4.13 that if $U_{F}^{+}$ is not of
Kac type, the nontrivial irreducible representations $\gamma$ of $U_{F}^{+}$
have $\uprho_{\gamma}\neq\mathds{1}_{\gamma}$. Using Theorem 2.7 we get the
following corollary.

######  Corollary 4.16.

Assume that $\dim(\alpha)=2$ and $q_{\operatorname{q}}<0.0861$ or
$\dim(\alpha)\geq 3$ and
$\tfrac{q_{\operatorname{q}}}{q_{\operatorname{c}}}<\bigl{(}1+\sqrt{\tfrac{3\sqrt{5}+5}{10}}\bigr{)}^{-2}=0.2306\dotsc$.
Then the inclusion
$\mathscr{C}_{U_{F}^{+}}\subseteq\operatorname{L}^{\infty}(U_{F}^{+})$ is
quasi-split and the relative commutant
$\mathscr{C}_{U_{F}^{+}}^{\prime}\cap\operatorname{L}^{\infty}(U_{F}^{+})$ is
not contained in $\mathscr{C}_{U_{F}^{+}}$.

###### Proof.

By [DCFY14, Theorem 33] $\operatorname{L}^{\infty}(U_{F}^{+})$ is a type
$\operatorname{III}$ factor. Therefore Proposition 2.3 applies and we know
that
$\mathscr{C}_{U_{F}^{+}}^{\prime}\cap\operatorname{L}^{\infty}(U_{F}^{+})$ is
a type $\operatorname{III}$ algebra, hence cannot be contained in
$\mathscr{C}_{U_{F}^{+}}$, which is a finite von Neumann algebra.

An alternative argument can go as follows. By [Ban97, Théorème 1 (iii)] the
character of the fundamental representation of $U_{F}^{+}$ has the same
distribution (with respect to the Haar integral) as a circular
variable555Recall that $x$ is circular if $x=s_{1}+is_{2}$, where $s_{1}$ and
$s_{2}$ are freely independent semicircular variables., so we can conclude
that $\mathscr{C}_{U_{F}^{+}}$ is isomorphic to the free group factor
$\operatorname{L}(F_{2})$, in particular it is a factor. If the relative
commutant
$\mathscr{C}_{U_{F}^{+}}^{\prime}\cap\operatorname{L}^{\infty}(U_{F}^{+})$ was
contained in $\mathscr{C}_{U_{F}^{+}}$ then the center of
$\operatorname{L}^{\infty}(U_{F}^{+})$ would be contained in
$\mathscr{C}_{U_{F}^{+}}$, so $\operatorname{L}^{\infty}(U_{F}^{+})$ has to be
a factor. What is more, if
$\mathscr{C}_{U_{F}^{+}}^{\prime}\cap\operatorname{L}^{\infty}(U_{F}^{+})\subseteq\mathscr{C}_{U_{F}^{+}}$
then
$\mathscr{C}_{U_{F}^{+}}^{\prime}\cap\operatorname{L}^{\infty}(U_{F}^{+})=\mathscr{C}_{U_{F}^{+}}\cap\mathscr{C}_{U_{F}^{+}}^{\prime}\cap\operatorname{L}^{\infty}(U_{F}^{+})=\mathbb{C}\mathds{1}$,
i.e. the inclusion is irreducible. By Remark 2.2 a quasi-split inclusion of
factors is actually split and it is easy to check that a proper split
inclusion cannot be irreducible. ∎

### Acknowledgements

MW was supported by the Research Foundation –- Flanders (FWO) through a
Postdoctoral Fellowship, by long term structural funding – Methusalem grant of
the Flemish Government – and by European Research Council Starting Grant
677120 INDEX. JK was partially supported by the NCN (National Centre of
Science) grant 2014/14/E/ST1/00525. Furthermore, JK would like to express his
gratitude towards Adam Skalski and Piotr M. Sołtan for many helpful
discussions. The authors would like to thank the anonymous referees for
several useful remarks.

## References

  * [AC17] M. Alaghmandan and J. Crann.  Character density in central subalgebras of compact quantum groups.  Canad. Math. Bull., 60(3):449–461, 2017.
  * [Ban96] T. Banica.  Théorie des représentations du groupe quantique compact libre ${\rm O}(n)$.  C. R. Acad. Sci. Paris Sér. I Math., 322(3):241–244, 1996.
  * [Ban97] T. Banica.  Le groupe quantique compact libre ${\rm U}(n)$.  Comm. Math. Phys., 190(1):143–172, 1997.
  * [Ban99] T. Banica.  Symmetries of a generic coaction.  Math. Ann. 314, no. 4, 763–780, 1999.
  * [Ban02] T. Banica.  Quantum groups and Fuss-Catalan algebras.  Comm. Math. Phys., 226(1):221–232, 2002.
  * [BT03] E. Bédos and L. Tuset.  Amenability and co-amenability for locally compact quantum groups.  Internat. J. Math., 14(8):865–884, 2003.
  * [BM20] P. Bikram and K. Mukherjee.  On the commutants of generators of q-deformed Araki-Woods von Neumann algebras.  arXiv:2004.10012, 2020.
  * [Bla06] B. Blackadar.  Operator algebras, volume 122 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2006.  Theory of $C^{*}$-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III.
  * [Bra13] M. Brannan.  Reduced operator algebras of trace-preserving quantum automorphism groups.  Doc. Math., 18:1349–1402, 2013.
  * [Buc74] D. Buchholz.  Product states for local algebras.  Comm. Math. Phys., 36:287–304, 1974.
  * [BDL90] D. Buchholz, C. D’Antoni, and R. Longo.  Nuclear maps and modular structures. I. General properties.  J. Funct. Anal., 88(2):233–250, 1990.
  * [CFRW10] J. Cameron, J. Fang, M. Ravichandran, and S. White.  The radial masa in a free group factor is maximal injective.  J. Lond. Math. Soc. (2), 82(3):787–809, 2010.
  * [CN19] J. Crann and M. Neufang.  A non-commutative Fejér theorem for crossed products, the approximation property, and applications.  arXiv:1901.08700, to appear in Int. Math. Res. Notices, January 2019\.
  * [DL83] C. D’Antoni and R. Longo.  Interpolation by type ${\rm I}$ factors and the flip automorphism.  J. Functional Analysis, 51(3):361–371, 1983.
  * [DDS19] B. Das, M. Daws, and P. Salmi.  Admissibility conjecture and Kazhdan’s property (T) for quantum groups.  J. Funct. Anal., 276(11):3484–3510, 2019.
  * [DCFY14] K. De Commer, A. Freslon, and M. Yamashita.  CCAP for universal discrete quantum groups.  Comm. Math. Phys., 331(2):677–701, 2014.  With an appendix by Stefaan Vaes.
  * [Des03] P. Desmedt.  Aspects of the theory of locally compact quantum groups: Amenability - Plancherel measure.  PhD thesis, Katholieke Universiteit Leuven, 2003.
  * [DQV02] P. Desmedt, J. Quaegebeur, and S. Vaes.  Amenability and the bicrossed product construction.  Illinois J. Math., 46(4):1259–1277, 2002.
  * [DL84] S. Doplicher and R. Longo.  Standard and split inclusions of von Neumann algebras.  Invent. Math., 75(3):493–536, 1984.
  * [Fid02] F. Fidaleo.  On the split property for inclusions of $W^{*}$-algebras.  Proc. Amer. Math. Soc., 130(1):121–127, 2002.
  * [FMP15] Pierre Fima, Kunal Mukherjee, and Issan Patri.  On compact bicrossed products.  J Noncommut Geom, 11, 03 2015.
  * [FV16] A. Freslon and R. Vergnioux.  The radial MASA in free orthogonal quantum groups.  J. Funct. Anal., 271(10):2776–2807, 2016.
  * [Her73] C. Herz.  Harmonic synthesis for subgroups.  Ann. Inst. Fourier (Grenoble), 23(3):91–123, 1973.
  * [KS97] A. Klimyk and K. Schmüdgen.  Quantum groups and their representations.  Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997.
  * [Kra20] J. Krajczok.  Modular properties of type I locally compact quantum groups.  arXiv:2006.02746v2, to appear in J. Operator Theory, 2020.
  * [KS] J. Krajczok and P. M. Sołtan.  Examples of compact quantum groups with $\operatorname{L}^{\infty}(\mathbb{G})$ a factor.  In preparation.
  * [KV00] J. Kustermans and S. Vaes.  Locally compact quantum groups.  Ann. Sci. École Norm. Sup. (4), 33(6):837–934, 2000.
  * [MN19] S. Malacarne and S. Neshveyev.  Martin boundaries of the duals of free unitary quantum groups.  Compos. Math., 155(6):1171–1193, 2019.
  * [Mer85] R. Mercer.  Convergence of Fourier series in discrete crossed products of von Neumann algebras.  Proc. Amer. Math. Soc., 94(2):254–258, 1985.
  * [MRW12] R. Meyer, S. Roy, and S. L. Woronowicz.  Homomorphisms of quantum groups.  Münster J. Math., 5:1–24, 2012.
  * [NT13] S. Neshveyev and L. Tuset.  Compact quantum groups and their representation categories, volume 20 of Cours Spécialisés [Specialized Courses].  Société Mathématique de France, Paris, 2013.
  * [Ped89] G. K. Pedersen.  Analysis now, volume 118 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1989.
  * [Ped18] G. K. Pedersen.  $C^{*}$-algebras and their automorphism groups.  Pure and Applied Mathematics (Amsterdam). Academic Press, London, 2018\.  Second edition of [ MR0548006], Edited and with a preface by Søren Eilers and Dorte Olesen.
  * [Pyt81] T. Pytlik.  Radial functions on free groups and a decomposition of the regular representation into irreducible components.  J. Reine Angew. Math., 326:124–135, 1981.
  * [R9̆1] F. Rădulescu.  Singularity of the radial subalgebra of ${\mathscr{L}}(F_{N})$ and the Pukánszky invariant.  Pacific J. Math., 151(2):297–306, 1991.
  * [Shl97] D. Shlyakhtenko.  Free quasi-free states.  Pacific J. Math., 177(2):329–368, 1997.
  * [Tak03a] M. Takesaki.  Theory of operator algebras. I, volume 124 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2003.  Operator Algebras and Non-commutative Geometry, 6.
  * [Tak03b] M. Takesaki.  Theory of operator algebras. II, volume 125 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2003.  Operator Algebras and Non-commutative Geometry, 6.
  * [Tak03c] M. Takesaki.  Theory of operator algebras. III, volume 127 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2003.  Operator Algebras and Non-commutative Geometry, 8.
  * [VV03] S. Vaes and L. Vainerman.  Extensions of locally compact quantum groups and the bicrossed product construction.  Adv. Math., 175(1):1–101, 2003.
  * [VV07] S. Vaes and R. Vergnioux.  The boundary of universal discrete quantum groups, exactness, and factoriality.  Duke Math. J., 140(1):35–84, 2007.
  * [VD14] A. Van Daele.  Locally compact quantum groups. A von Neumann algebra approach.  SIGMA Symmetry Integrability Geom. Methods Appl., 10:Paper 082, 41, 2014.
  * [VDW96] A. Van Daele and S. Wang.  Universal quantum groups.  Internat. J. Math., 7(2):255–263, 1996.
  * [Wan95] S. Wang.  Tensor products and crossed products of compact quantum groups.  Proc. London Math. Soc. (3), 71(3):695–720, 1995.
  * [Wan98] S. Wang.  Quantum symmetry groups of finite spaces.  Comm. Math. Phys., 195(1):195–211, 1998.
  * [Wor87a] S. L. Woronowicz.  Compact matrix pseudogroups.  Comm. Math. Phys., 111(4):613–665, 1987.
  * [Wor87b] S. L. Woronowicz.  Twisted SU(2) group. An example of a noncommutative differential calculus.  Publ. Res. Inst. Math. Sci., 23(1):117–181, 1987.
  * [Wor98] S. L. Woronowicz.  Compact quantum groups.  In Symétries quantiques (Les Houches, 1995), pages 845–884. North-Holland, Amsterdam, 1998.