# The approximation property for locally compact quantum groups

Matthew Daws  Department of Mathematics and Statistics, Lancaster University,
Lancaster, LA1 4YF, United Kingdom<EMAIL_ADDRESS>,  Jacek Krajczok
School of Mathematics and Statistics
University of Glasgow
University Place
Glasgow G12 8QQ
United Kingdom<EMAIL_ADDRESS>and  Christian Voigt  School of
Mathematics and Statistics
University of Glasgow
University Place
Glasgow G12 8QQ
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We study the Haagerup–Kraus approximation property for locally compact quantum
groups, generalising and unifying previous work by Kraus–Ruan and Crann. Along
the way we discuss how multipliers of quantum groups interact with the
$\mathrm{C}^{*}$-algebraic theory of locally compact quantum groups. Several
inheritance properties of the approximation property are established in this
setting, including passage to quantum subgroups, free products of discrete
quantum groups, and duals of double crossed products. We also discuss a
relation to the weak∗ operator approximation property. For discrete quantum
groups, we introduce a central variant of the approximation property, and
relate this to a version of the approximation property for rigid
$\mathrm{C}^{*}$-tensor categories, building on work of Arano–De Laat–Wahl.

###### Key words and phrases:

Locally compact quantum groups, approximation property

###### 2020 Mathematics Subject Classification:

Primary 46L67, Secondary 22D55, 43A30

This work was supported by EPSRC grants EP/T03064X/1 and EP/T030992/1. For the
purpose of open access, the authors have applied a CC BY public copyright
licence to any Author Accepted Manuscript version arising. No data were
created or analysed in this study.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Completely bounded multipliers
    1. 3.1 Definitions and fundamental facts
    2. 3.2 Predual
    3. 3.3 Viewing multipliers as bimodule maps
  4. 4 The approximation property
    1. 4.1 Equivalent characterisations
    2. 4.2 Further general properties
    3. 4.3 Left versus right
  5. 5 Relation to other approximation properties
  6. 6 Discrete quantum groups and operator algebraic approximation properties
  7. 7 Permanence properties
    1. 7.1 Quantum subgroups
    2. 7.2 Direct limits
    3. 7.3 Free products
      1. 7.3.1 Ricard-Xu results
      2. 7.3.2 AP for free products
    4. 7.4 Double crossed products
      1. 7.4.1 Preliminaries
      2. 7.4.2 $\mathbb{G}_{1}^{\operatorname{op}},\mathbb{G}_{2}$ are quantum subgroups of $\mathbb{G}_{\operatorname{m}}$
      3. 7.4.3 AP for $\widehat{\mathbb{G}_{\operatorname{m}}}$
    5. 7.5 Direct products
  8. 8 Categorical AP

##  1\. Introduction

The approximation property (AP) for locally compact groups, introduced by
Haagerup and Kraus [32], can be viewed as an analogue of Grothendieck’s
approximation property for Banach spaces [28]. It belongs to a family of
widely studied analytical properties like amenability, weak amenability, and
the Haagerup property. In fact, AP is a weakening of weak amenability and thus
a very loose form of amenability. It is known that AP passes from locally
compact groups to their lattices and vice versa, and that it has better
permanence properties with respect to standard constructions like extensions
and free products, in comparison to weak amenability.

It was an open problem for a long time to exhibit examples of exact groups
without AP. In the remarkable paper [48], Lafforgue and de la Salle proved
that $\operatorname{SL}(3,\mathbb{R})$ fails to have AP, thus confirming a
conjecture in [32]. Building on this result, it was shown later by Haagerup,
Knudby and De Laat that a connected Lie group has AP if and only if all simple
factors in its Levi decomposition have real rank at most one [29], [30], [31].

The approximation property has a wide range of applications. As shown by
Haagerup-Kraus [32], in the case of discrete groups there is a connection
between AP and the slice map property (or equivalently, the operator
approximation property) of the associated crossed products. It was recently
proven in the full generality of locally compact groups that AP implies
exactness [64] (see also [15]), which makes it relevant to a number of
problems in operator algebras. Let us also mention that AP was shown to be
equivalent to a non-commutative version of Fejér’s theorem [15], and used to
prove results concerning convolution operators on $\operatorname{L}^{p}(G)$
[12, 21].

Amenability, weak amenability and the Haagerup property have also been studied
extensively in the broader setting of locally compact quantum groups, see [9]
for a survey. An interesting new feature in the quantum setting is the
interplay between discrete quantum groups, their Drinfeld doubles, and the
associated $\mathrm{C}^{*}$-tensor categories [22]. In fact, the central
versions of amenability, the Haagerup property, weak amenability and central
property (T) for discrete quantum groups have been recast at the level of
$\mathrm{C}^{*}$-tensor categories [57], thus building a natural bridge to the
study of subfactors.

In the present paper we undertake a systematic study of the approximation
property for locally compact quantum groups. Kraus and Ruan introduced a
version of the approximation property for Kac algebras in [42], requiring the
existence of a net in the Fourier algebra $\operatorname{A}(\mathbb{G})$ such
that the associated net of completely bounded operators on
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ converges to the identity in
the stable point weak∗-topology of
$\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$.
Crann studied this property for general locally compact quantum groups, and
showed for example that in the presence of this property, amenability is
equivalent to coamenability of the dual quantum group [14, Corollary 7.4].

Our starting point is the original work by Haagerup and Kraus. We say that a
locally compact quantum group has AP if it admits a net of elements in the
Fourier algebra $\operatorname{A}(\mathbb{G})$ which converges weak∗ to
$\mathbbm{1}$ in the space of left CB multipliers
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. We show that this
definition is in fact equivalent to the definition of AP used in [42], [14],
thereby verifying a conjecture in [42]. This result was proved by Haagerup and
Kraus in the case of locally compact groups ([32, Theorem 1.9]). Our methods
are necessarily different, as we cannot work with points or compactly
supported functions. Along the way, we obtain a useful alternative description
of the weak∗-topology on the space of left CB multipliers (Theorem 3.8). We
prove that working with left or right multipliers does not change the theory,
and that passing from a quantum group to its opposite or commutant preserves
AP. We also show that if a quantum group $\mathbb{G}$ has the AP exhibited by
a net which is uniformly bounded in the norm of $\operatorname{A}(\mathbb{G})$
(resp. $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$), then the
dual is coamenable (resp. the quantum group is weakly amenable).

We then derive a number of permanence properties of AP in analogy to the
classical setting. In particular, we show that AP passes to closed quantum
subgroups of locally compact quantum groups, and to duals of double crossed
products. This includes the passage to direct products of quantum groups as a
special case. In the setting of discrete quantum groups we verify that AP is
inherited by free products and direct limits of directed systems of discrete
quantum groups with injective connecting maps. We also introduce a central
version of the approximation property for discrete quantum groups and show
that it is related to a natural notion of AP for rigid $\mathrm{C}^{*}$-tensor
categories, building on work of Arano–De Laat–Wahl [2], [3].

Using our results we can provide a range of examples of quantum groups with
and without AP. For instance, the Drinfeld double of
$\operatorname{SL}(3,\mathbb{R})$ does not have AP, and the free product
$\operatorname{SL}(3,\mathbb{Z})\star\widehat{\operatorname{SU}_{q}(2)}$ does
not have AP. Examples with AP can be obtained via free products as well, for
example
$\operatorname{SL}(2,\mathbb{Z})\star\widehat{\operatorname{SU}_{q}(2)}$.
Using infinite products, one may find examples of discrete quantum groups with
AP which are not weakly amenable. We make further remarks at the start of
Section 4.

Let us now briefly describe more of our results and explain how the paper is
organised. In Section 2 we collect some background material on locally compact
quantum groups and fix our notation. In Section 3 we review several
characterisations of the space
$\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))$ of left cb-
multipliers of the Fourier algebra $\operatorname{A}(\mathbb{G})$ and its
natural predual $Q^{l}(\operatorname{A}(\mathbb{G}))$. By definition,
$\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))$ is a (in general not
closed) subalgebra of $\operatorname{L}^{\infty}(\mathbb{G})$, but it is also
isomorphic to
$\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$,
the space of a normal left module maps on
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$. Any map in
$\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
restricts to $\mathrm{C}_{0}(\widehat{\mathbb{G}})$, and we provide a
characterisation of the maps on $\mathrm{C}_{0}(\widehat{\mathbb{G}})$ which
so arise. This leads to a description of $Q^{l}(\operatorname{A}(\mathbb{G}))$
as a quotient of the projective tensor product
$\mathrm{C}_{0}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})$.
We were unable to locate this description in the literature, even for
classical groups.

In Section 4 we define the Haagerup-Kraus approximation property for locally
compact quantum groups, and survey some examples and counter-examples, making
use of later results in the paper. We verify that AP passes to opposite and
commutant quantum groups. We compare our definition to the version of AP given
by Kraus and Ruan, showing that they are equivalent. We show that
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ admits an involution
linked to the antipode of $\mathbb{G}$, and to the fact that elements of
$\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
act boundedly on the Hilbert space $\operatorname{L}^{2}(\mathbb{G})$. We
finish the section by showing that AP is independent of the choice of working
with left or right multipliers.

In Section 5 we discuss the relation of AP with weak amenability and
coamenability.

Section 6 is devoted to the special case of discrete quantum groups. When
$\mathbbold{\Gamma}$ is discrete, we have the notion of a finitely-supported
function leading to the algebra $\mathrm{c}_{00}(\mathbbold{\Gamma})$. It
suffices to work with $\mathrm{c}_{00}(\mathbbold{\Gamma})$ when considering
AP, and we show further that the approximating net can be chosen to satisfy
other properties. We introduce the central approximation property for discrete
quantum groups and prove that central AP is equivalent to AP in the unimodular
case. Building on the work of Kraus-Ruan [42] and Crann [14], we show that if
a locally compact quantum group $\mathbb{G}$ has AP then the von Neumann
algebra $\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ has W∗OAP. We study
the relation between AP of $\mathbbold{\Gamma}$ and (strong) OAP of
$\mathrm{C}(\widehat{\mathbbold{\Gamma}})$ or W∗OAP of
$\operatorname{L}^{\infty}(\widehat{\mathbbold{\Gamma}})$. Finally, we
introduce strengthenings of these concepts which take into account also the
algebra $\ell^{\infty}(\mathbbold{\Gamma})$, and show that these are
equivalent to AP even in the non-unimodular case.

In Section 7 we establish a number of permanence properties. We show that the
AP is inherited by arbitrary closed quantum subgroups and by the duals of
double crossed products. In particular, the direct product of two quantum
groups with AP also has AP. For discrete quantum groups we investigate the
passage to free products and direct unions, again showing that AP is
preserved.

Finally, in Section 8 we define the approximation property for rigid
$\mathrm{C}^{*}$-tensor categories and verify that the categorical AP is
equivalent to the central AP for discrete quantum groups. This implies in
particular that the central AP is invariant under monoidal equivalence. We
also relate these properties to the AP of the Drinfeld double.

We conclude with some general remarks on notation. If $\mathcal{A}$ is a
$\mathrm{C}^{*}$-algebra we write $\operatorname{M}(\mathcal{A})$ for its
multiplier algebra. For a map $\Phi\colon\mathcal{A}\rightarrow\mathcal{A}$,
the symbol $\Phi^{\dagger}$ stands for the map $\mathcal{A}\ni
a\mapsto\Phi(a^{*})^{*}\in\mathcal{A}$. If
$\omega:\mathcal{A}\rightarrow\mathbb{C}$ is a linear functional we write
$\overline{\omega}$ for the linear functional given by
$\overline{\omega}(x)=\overline{\omega(x^{*})}$.

We write $\odot$ for the algebraic tensor product, $\otimes$ for the tensor
product of Hilbert spaces or the minimal tensor product of
$\mathrm{C}^{*}$-algebras, $\check{\otimes}$ for the injective tensor product
of operator spaces and $\bar{\otimes}$ for the spatial tensor product of von
Neumann algebras. We denote by $\chi$ the flip map for tensor products of
algebras, and use the symbol $\Sigma$ for the flip map of Hilbert spaces.

We freely use the basic theory of operator spaces, following [23], see also
[54, 55] for example. When $X,Y$ are operator spaces, $\operatorname{CB}(X,Y)$
denotes the space of completely bounded (CB) linear maps $X\rightarrow Y$. For
dual operator spaces $X,Y$ we write $\operatorname{CB}^{\sigma}(X,Y)$ for the
subset of $\operatorname{CB}(X,Y)$ consisting of all maps which are
weak∗-weak∗-continuous. In the case $X=Y$ we simply write
$\operatorname{CB}(X)=\operatorname{CB}(X,X)$ and
$\operatorname{CB}^{\sigma}(X)=\operatorname{CB}^{\sigma}(X,X)$. If
$\operatorname{M}$ is a von Neumann algebra, then
$\operatorname{CB}^{\sigma}(\operatorname{M})$ can be equipped with the stable
point-weak∗-topology: $T_{i}\xrightarrow[i\in I]{}T$ with respect to this
topology if and only if $(T_{i}\otimes\textnormal{id})x\xrightarrow[i\in
I]{}(T\otimes\textnormal{id})x$ in the weak∗-topology, for any separable
Hilbert space $\mathsf{H}$ and
$x\in\operatorname{M}\bar{\otimes}\operatorname{B}(\mathsf{H})$, see [32].
Whenever we have a left $\operatorname{N}$-module structure on an operator
space $X$, the space of left $\operatorname{N}$-module CB maps is denoted by
$\prescript{}{\operatorname{N}}{\operatorname{CB}}(X)$. Similarly, if $X$ is a
right module or a bimodule, the corresponding spaces are denoted by
$\operatorname{CB}_{\operatorname{M}}(X)$ and
$\prescript{}{\operatorname{N}}{\operatorname{CB}}_{\operatorname{M}}(X)$,
respectively. We denote the operator space projective tensor product by
$\widehat{\otimes}$, and recall that
$(X\widehat{\otimes}Y)^{*}=\operatorname{CB}(X,Y^{*})$ completely
isometrically. The canonical pairing between an operator space $X$ and its
dual $X^{*}$ is denoted by $\langle\omega,x\rangle_{X^{*},X}$ for $\omega\in
X^{*},x\in X$, or simply $\langle\omega,x\rangle$ if there is no risk of
confusion.

For a n.s.f. weight $\theta$ on a von Neumann algebra $\operatorname{M}$, we
denote the GNS Hilbert space by $\mathsf{H}_{\theta}$, and we use the notation
$\mathfrak{N}_{\theta}=\\{x\in\operatorname{M}\,|\,\theta(x^{*}x)<+\infty\\}$.
We write
$\Lambda_{\theta}\colon\mathfrak{N}_{\theta}\rightarrow\mathsf{H}_{\theta}$
for the GNS map. Typically we then represent $\operatorname{M}$ on
$\mathsf{H}_{\theta}$ and identify
$\operatorname{M}\subseteq\operatorname{B}(\mathsf{H}_{\theta})$.

##  2\. Preliminaries

Throughout the paper we will work in the setting of locally compact quantum
groups introduced by Kustermans and Vaes [46]. In this section we recall some
fundamental constructions and results of the theory, more information can be
found in [44, 47, 70]. For background on operator algebras and operator spaces
we refer to [10, 23, 65].

By definition, a _locally compact quantum group_ $\mathbb{G}$ is given by a
von Neumann algebra $\operatorname{L}^{\infty}(\mathbb{G})$ together with a
normal unital $\star$-homomorphism
$\Delta_{\mathbb{G}}\colon\operatorname{L}^{\infty}(\mathbb{G})\rightarrow\operatorname{L}^{\infty}(\mathbb{G})\bar{\otimes}\operatorname{L}^{\infty}(\mathbb{G})$
called _comultiplication_ , satisfying
$(\Delta_{\mathbb{G}}\otimes\textnormal{id})\Delta_{\mathbb{G}}=(\textnormal{id}\otimes\Delta_{\mathbb{G}})\Delta_{\mathbb{G}}$,
and _left_ resp. _right_ _Haar integrals_ $\varphi$ and $\psi$. These are
normal, semifinite, faithful (n.s.f.) weights on
$\operatorname{L}^{\infty}(\mathbb{G})$ satisfying certain invariance
conditions with respect to $\Delta_{\mathbb{G}}$. In general, the von Neumann
algebra $\operatorname{L}^{\infty}(\mathbb{G})$ is non-commutative and will
not be an algebra of functions on a measure space. Following this notational
convention, the predual of $\operatorname{L}^{\infty}(\mathbb{G})$ is denoted
by $\operatorname{L}^{1}(\mathbb{G})$ and the GNS Hilbert space of $\varphi$
is denoted by $\operatorname{L}^{2}(\mathbb{G})$.

Every locally compact group $G$ can be seen as a locally compact quantum group
$\mathbb{G}$ by taking
$\operatorname{L}^{\infty}(\mathbb{G})=\operatorname{L}^{\infty}(G)$, the
algebra of classes of measurable, bounded functions on $G$, and letting
$\Delta_{\mathbb{G}}$ be the pullback of multiplication in $G$. The weights
$\varphi,\psi$ are given by integration with respect to left (right) Haar
measure in this case.

Out of the axioms, one can construct a number of additional objects associated
to a locally compact quantum group $\mathbb{G}$. First of all, there is the
_Kac-Takesaki operator_
$\mathrm{W}^{\mathbb{G}}\in\operatorname{L}^{\infty}(\mathbb{G})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$,
which is a unitary operator on
$\operatorname{L}^{2}(\mathbb{G})\otimes\operatorname{L}^{2}(\mathbb{G})$
defined via

(2.1)
$((\omega\otimes\textnormal{id})\mathrm{W}^{\mathbb{G}*})\Lambda_{\varphi}(x)=\Lambda_{\varphi}((\omega\otimes\textnormal{id})\Delta_{\mathbb{G}}(x))\qquad(\omega\in\operatorname{L}^{1}(\mathbb{G}),x\in\mathfrak{N}_{\varphi}).$

It implements the comultiplication via
$\Delta_{\mathbb{G}}(x)=\mathrm{W}^{\mathbb{G}*}(\mathbbm{1}\otimes
x)\mathrm{W}^{\mathbb{G}}$ for $x\in\operatorname{L}^{\infty}(\mathbb{G})$.
Tomita-Takesaki theory yields two groups of _modular automorphisms_
$(\sigma^{\varphi}_{t})_{t\in\mathbb{R}},(\sigma^{\psi}_{t})_{t\in\mathbb{R}}$
and _modular conjugations_ $J_{\varphi},J_{\psi}$ associated with the weights
$\varphi,\psi$, respectively [65]. The left and right Haar integrals are
linked by the _modular element_ $\delta_{\mathbb{G}}$, which is a strictly
positive, self-adjoint operator affiliated with
$\operatorname{L}^{\infty}(\mathbb{G})$.

In the theory of quantum groups, the role of the inverse operation is played
by two maps: the _antipode_ $S_{\mathbb{G}}$ and the _unitary antipode_
$R_{\mathbb{G}}$. The antipode is in general an unbounded (densely defined)
map on $\operatorname{L}^{\infty}(\mathbb{G})$ such that

$(\textnormal{id}\otimes\omega)\mathrm{W}^{\mathbb{G}}\in\operatorname{Dom}(S_{\mathbb{G}})\;\textnormal{
and
}S_{\mathbb{G}}((\textnormal{id}\otimes\omega)\mathrm{W}^{\mathbb{G}})=(\textnormal{id}\otimes\omega)\mathrm{W}^{\mathbb{G}*}\qquad(\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})).$

The unitary antipode, on the other hand, is a bounded, normal,
$\star$-preserving, antimultiplicative map on
$\operatorname{L}^{\infty}(\mathbb{G})$ satisfying
$\Delta_{\mathbb{G}}R_{\mathbb{G}}=\chi(R_{\mathbb{G}}\otimes
R_{\mathbb{G}})\Delta_{\mathbb{G}}$. These maps are linked via
$S_{\mathbb{G}}=R_{\mathbb{G}}\tau^{\mathbb{G}}_{-i/2}=\tau^{\mathbb{G}}_{-i/2}R_{\mathbb{G}}$,
where $(\tau^{\mathbb{G}}_{t})_{t\in\mathbb{R}}$ is the group of _scaling
automorphisms_ of $\operatorname{L}^{\infty}(\mathbb{G})$. There is a self-
adjoint, strictly positive (in general unbounded) operator $P_{\mathbb{G}}$
acting on $\operatorname{L}^{2}(\mathbb{G})$ which implements scaling
automorphisms via
$\tau^{\mathbb{G}}_{t}(x)=P^{it}_{\mathbb{G}}xP^{-it}_{\mathbb{G}}\,(x\in\operatorname{L}^{\infty}(\mathbb{G}),t\in\mathbb{R})$.
Furthermore, $P_{\mathbb{G}}$ is self-dual in the sense that
$P_{\widehat{\mathbb{G}}}=P_{\mathbb{G}}$. The left and right Haar integrals
are unique up to a scalar, and we shall fix normalisations such that
$\varphi=\psi\circ R_{\mathbb{G}}$. The scaling constant of $\mathbb{G}$ will
be denoted by $\nu_{\mathbb{G}}>0$. It is characterised via for example
$\varphi\circ\tau^{\mathbb{G}}_{t}=\nu_{\mathbb{G}}^{-t}\varphi$ for
$t\in\mathbb{R}$.

With any locally compact quantum group $\mathbb{G}$ one can associate the
_dual_ locally compact quantum group $\widehat{\mathbb{G}}$ in such a way that
the correspondence between $\mathbb{G}$ and $\widehat{\mathbb{G}}$ extends
Pontryagin duality. Furthermore, the Hilbert spaces
$\operatorname{L}^{2}(\mathbb{G}),\operatorname{L}^{2}(\widehat{\mathbb{G}})$
can be identified in a canonical way, and the Kac-Takesaki operators of
$\mathbb{G}$ and $\widehat{\mathbb{G}}$ are linked via
$\mathrm{W}^{\widehat{\mathbb{G}}}=\chi(\mathrm{W}^{\mathbb{G}*})$. If there
is no risk of confusion we will simply write $\Delta$ for
$\Delta_{\mathbb{G}}$, $\widehat{\Delta}$ for $\Delta_{\widehat{\mathbb{G}}}$,
and similarly $R,S,\widehat{R},\widehat{S}$ for the (unitary) antipode of
$\mathbb{G}$ or $\widehat{\mathbb{G}}$. Using the canonical identification of
$\operatorname{L}^{2}(\mathbb{G})$ and
$\operatorname{L}^{2}(\widehat{\mathbb{G}})$ one obtains a number of useful
formulae. Let us mention in particular that the right regular representation
$\mathrm{V}^{\mathbb{G}}\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}})^{\prime}\bar{\otimes}\operatorname{L}^{\infty}(\mathbb{G})$
is given by $\mathrm{V}^{\mathbb{G}}=(J_{\widehat{\varphi}}\otimes
J_{\widehat{\varphi}})\chi(\mathrm{W}^{\mathbb{G}})^{*}(J_{\widehat{\varphi}}\otimes
J_{\widehat{\varphi}})$.

We will also work with the weak∗-dense $\mathrm{C}^{*}$-subalgebra
$\mathrm{C}_{0}(\mathbb{G})\subseteq\operatorname{L}^{\infty}(\mathbb{G})$. It
is defined as the norm-closure of the space
$\\{(\textnormal{id}\otimes\omega)\mathrm{W}^{\mathbb{G}}\,|\,\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\\}$.
After restriction, the comultiplication becomes a non-degenerate
$\star$-homomorphism
$\mathrm{C}_{0}(\mathbb{G})\rightarrow\operatorname{M}(\mathrm{C}_{0}(\mathbb{G})\otimes\mathrm{C}_{0}(\mathbb{G}))$.
Similarly one defines $\mathrm{C}_{0}(\widehat{\mathbb{G}})$, and then
$\mathrm{W}^{\mathbb{G}}\in\operatorname{M}(\mathrm{C}_{0}(\mathbb{G})\otimes\mathrm{C}_{0}(\widehat{\mathbb{G}}))$.
Using the comultiplication of $\operatorname{L}^{\infty}(\mathbb{G})$, we
define a Banach algebra structure on $\operatorname{L}^{1}(\mathbb{G})$ via
$\omega\star\nu=(\omega\otimes\nu)\Delta_{\mathbb{G}}$ for
$\omega,\nu\in\operatorname{L}^{1}(\mathbb{G})$. As
$\operatorname{L}^{\infty}(\mathbb{G})$ is the dual of
$\operatorname{L}^{1}(\mathbb{G})$, we have a canonical
$\operatorname{L}^{1}(\mathbb{G})$-bimodule structure on
$\operatorname{L}^{\infty}(\mathbb{G})$, which is given by $\omega\star
x=(\textnormal{id}\otimes\omega)\Delta_{\mathbb{G}}(x)$ and
$x\star\omega=(\omega\otimes\textnormal{id})\Delta_{\mathbb{G}}(x)$. Treating
$\operatorname{L}^{1}(\mathbb{G})$ as the predual of the von Neumann algebra
$\operatorname{L}^{\infty}(\mathbb{G})$ gives, as usual, an
$\operatorname{L}^{\infty}(\mathbb{G})$-bimodule structure on
$\operatorname{L}^{1}(\mathbb{G})$ defined via
$x\omega=\omega(\cdot\,x),\omega x=\omega(x\,\cdot)$ for
$x\in\operatorname{L}^{\infty}(\mathbb{G}),\omega\in\operatorname{L}^{1}(\mathbb{G})$.

Let us introduce the map
$\lambda_{\mathbb{G}}\colon\operatorname{L}^{1}(\mathbb{G})\rightarrow\mathrm{C}_{0}(\widehat{\mathbb{G}})$
by
$\lambda_{\mathbb{G}}(\omega)=(\omega\otimes\textnormal{id})\mathrm{W}^{\mathbb{G}}$,
and similarly for $\widehat{\mathbb{G}}$. Using this we define the _Fourier
algebra of $\mathbb{G}$_ as
$\operatorname{A}(\mathbb{G})=\lambda_{\widehat{\mathbb{G}}}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$.
One can check that $\lambda_{\widehat{\mathbb{G}}}$ is multiplicative, so that
$\operatorname{A}(\mathbb{G})$ is a dense subalgebra of
$\mathrm{C}_{0}(\widehat{\mathbb{G}})$. As $\lambda_{\widehat{\mathbb{G}}}$ is
also injective, we can define an operator space structure on
$\operatorname{A}(\mathbb{G})$ by imposing the condition that
$\lambda_{\widehat{\mathbb{G}}}\colon\operatorname{L}^{1}(\widehat{\mathbb{G}})\rightarrow\operatorname{A}(\mathbb{G})$
is completely isometric.

In the text, we will use certain subspaces of
$\operatorname{L}^{1}(\mathbb{G})$, which consist of functionals having nice
additional properties. Firstly, let us introduce

(2.2)
$\begin{split}\operatorname{L}^{1}_{\sharp}(\mathbb{G})&=\\{\omega\in\operatorname{L}^{1}(\mathbb{G})\,|\,\exists_{\theta\in\operatorname{L}^{1}(\mathbb{G})}\,\lambda_{\mathbb{G}}(\omega)^{*}=\lambda_{\mathbb{G}}(\theta)\\}\\\
&=\\{\omega\in\operatorname{L}^{1}(\mathbb{G})\,|\,\exists_{\theta\in\operatorname{L}^{1}(\mathbb{G})}\,\forall_{x\in\operatorname{Dom}(S_{\mathbb{G}})}\,\overline{\omega}(S_{\mathbb{G}}(x))=\theta(x)\\}.\end{split}$

For a given $\omega\in\operatorname{L}^{1}_{\sharp}(\mathbb{G})$, the
functional $\theta$ is characterised uniquely by any of the properties in
(2.2), hence we can write $\theta=\omega^{\sharp}$. The mapping
$\omega\mapsto\omega^{\sharp}$, and the restriction of the multiplication from
$\operatorname{L}^{1}(\mathbb{G})$, turn
$\operatorname{L}^{1}_{\sharp}(\mathbb{G})$ into a $\star$-algebra. The second
subspace we will use is

(2.3)
$\mathscr{J}=\\{\omega\in\operatorname{L}^{1}(\mathbb{G})\,|\,\exists_{M>0}\,\forall_{x\in\mathfrak{N}_{\varphi}}\,|\omega(x^{*})|\leq
M\|\Lambda_{\varphi}(x)\|\\}.$

This subspace appears in the construction of the left Haar integral
$\widehat{\varphi}$ for $\widehat{\mathbb{G}}$. Indeed, for
$\omega\in\mathscr{J}$ we have

$\lambda_{\mathbb{G}}(\omega)\in\mathfrak{N}_{\widehat{\varphi}}\quad\textnormal{
and
}\quad\forall_{x\in\mathfrak{N}_{\varphi}}\,\langle\Lambda_{\varphi}(x)\,|\,\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\rangle=\omega(x^{*}).$

In a couple of places we will need the following result, which says that there
are many functionals with desirable properties.

######  Lemma 2.1.

The subspace

$\begin{split}\mathscr{J}_{0}=\\{\omega\in\mathscr{J}\cap\operatorname{L}^{1}_{\sharp}(\mathbb{G})\,|\,&\overline{\omega},\overline{\omega}^{\sharp}\in\mathscr{J}\cap\operatorname{L}^{1}_{\sharp}(\mathbb{G})\textnormal{
and }f\colon\mathbb{R}\ni
t\mapsto(\omega\delta_{\mathbb{G}}^{it})\circ\tau^{\mathbb{G}}_{t}\in\operatorname{L}^{1}(\mathbb{G})\\\
&\textnormal{extends to an entire function such that
}\forall_{z\in\mathbb{C}}\,f(z)\in\mathscr{J}\cap\operatorname{L}^{1}_{\sharp}(\mathbb{G})\\}\end{split}$

is dense in $\operatorname{L}^{1}(\mathbb{G})$, and
$\lambda_{\mathbb{G}}(\mathscr{J}_{0})$ forms a
$\sigma\textsc{-sot}^{*}\times\|\cdot\|$ core for
$\Lambda_{\widehat{\varphi}}$.

###### Proof.

Our approach is standard, compare for example [39, Lemma 14.5] for a similar
result. Therefore we only give a sketch of the argument.

According to [47, Proposition 2.6], the space
$\mathscr{J}^{\sharp}=\\{\omega\in\mathscr{J}\cap\operatorname{L}^{1}_{\sharp}(\mathbb{G})\,|\,\omega^{\sharp}\in\mathscr{J}\\}$
is dense in $\operatorname{L}^{1}(\mathbb{G})$ and
$\lambda_{\mathbb{G}}(\mathscr{J}^{\sharp})$ is a
$\sigma\textsc{-sot}^{*}\times\|\cdot\|$ core of
$\Lambda_{\widehat{\varphi}}$. Let us introduce three mollifier operations:
for $n\in\mathbb{N}$ let

$\begin{split}M^{\varphi}_{n}&\colon\operatorname{L}^{1}(\mathbb{G})\ni\omega\mapsto\sqrt{\tfrac{n}{\pi}}\int_{\mathbb{R}}e^{-nt^{2}}\omega\circ\sigma^{\varphi}_{t}\operatorname{d}\\!t\in\operatorname{L}^{1}(\mathbb{G}),\\\
M^{\tau}_{n}&\colon\operatorname{L}^{1}(\mathbb{G})\ni\omega\mapsto\sqrt{\tfrac{n}{\pi}}\int_{\mathbb{R}}e^{-ns^{2}}\omega\circ\tau^{\mathbb{G}}_{s}\operatorname{d}\\!s\in\operatorname{L}^{1}(\mathbb{G}),\\\
M^{\delta}_{n}&\colon\operatorname{L}^{1}(\mathbb{G})\ni\omega\mapsto\sqrt{\tfrac{n}{\pi}}\int_{\mathbb{R}}e^{-np^{2}}\omega\delta_{\mathbb{G}}^{ip}\operatorname{d}\\!p\in\operatorname{L}^{1}(\mathbb{G}).\end{split}$

Next, let $\omega_{n}=M^{\tau}_{n}\circ M^{\varphi}_{n}\circ
M^{\delta}_{n}(\omega)$ and set
$\mathscr{J}_{00}=\operatorname{span}\\{\omega_{n}\,|\,n\in\mathbb{N},\omega\in\mathscr{J}^{\sharp}\\}$.
It suffices to show that $\mathscr{J}_{00}$ is dense in
$\operatorname{L}^{1}(\mathbb{G})$, that
$\mathscr{J}_{00}\subseteq\mathscr{J}_{0}$, and that
$\lambda_{\mathbb{G}}(\mathscr{J}_{00})$ forms a
$\sigma\textsc{-sot}^{*}\times\|\cdot\|$ core for
$\Lambda_{\widehat{\varphi}}$.

Choose
$n\in\mathbb{N},\omega\in\mathscr{J}^{\sharp},x\in\mathfrak{N}_{\varphi},y\in\operatorname{Dom}(S_{\mathbb{G}})$.
It is elementary to check that
$\omega_{n}\in\mathscr{J}\cap\operatorname{L}^{1}_{\sharp}(\mathbb{G})$ and
$\mathbb{R}\ni
t\mapsto(\omega_{n}\delta_{\mathbb{G}}^{it})\circ\tau_{t}^{\mathbb{G}}\in\operatorname{L}^{1}(\mathbb{G})$
extends to an entire function with the desired property. Since

$\begin{split}&\quad\;|\overline{\omega_{n}}(x^{*})|=|\omega_{n}(x)|=(\tfrac{n}{\pi})^{3/2}\bigl{|}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+s^{2}+p^{2})}\omega(\delta_{\mathbb{G}}^{ip}\sigma^{\varphi}_{t}\circ\tau^{\mathbb{G}}_{s}(x))\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{|}\\\
&=(\tfrac{n}{\pi})^{3/2}\bigl{|}\omega\bigl{(}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+s^{2}+p^{2})}\delta_{\mathbb{G}}^{ip}\sigma^{\varphi}_{t}\circ\tau^{\mathbb{G}}_{s}(x)\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{|}\\\
&\leq(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}\Lambda_{\varphi}\bigl{(}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+s^{2}+p^{2})}\sigma^{\varphi}_{t}\circ\tau^{\mathbb{G}}_{s}(x^{*})\delta_{\mathbb{G}}^{-ip}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&=(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}J_{\varphi}\nabla_{\varphi}^{1/2}\Lambda_{\varphi}\bigl{(}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+s^{2}+p^{2})}\delta_{\mathbb{G}}^{ip}\sigma^{\varphi}_{t}\circ\tau^{\mathbb{G}}_{s}(x)\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&=(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}\Lambda_{\varphi}\bigl{(}\int_{\mathbb{R}^{3}}e^{-n((t+i/2)^{2}+s^{2}+p^{2})}\nu_{\mathbb{G}}^{p/2}\delta_{\mathbb{G}}^{ip}\sigma^{\varphi}_{t}\circ\tau^{\mathbb{G}}_{s}(x)\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&\leq(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{(}\int_{\mathbb{R}^{3}}|e^{-n((t+i/2)^{2}+s^{2}+p^{2})}|\nu_{\mathbb{G}}^{p/2-s/2}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\|\Lambda_{\varphi}(x)\|,\end{split}$

where $\nu_{\mathbb{G}}$ is the scaling constant of $\mathbb{G}$, we have
$\overline{\omega_{n}}\in\mathscr{J}$. Here we used
$\sigma^{\varphi}_{t}(\delta_{\mathbb{G}}^{ip})=\nu_{\mathbb{G}}^{itp}\delta^{ip}$
and $\varphi\circ\tau_{s}=\nu_{\mathbb{G}}^{-s}\varphi$. It is automatic that
$\overline{\omega_{n}}\in\operatorname{L}^{1}_{\sharp}(\mathbb{G})$. Indeed,

$\begin{split}\overline{\overline{\omega_{n}}}(S_{\mathbb{G}}(y))=\omega_{n}(S_{\mathbb{G}}(y))&=\sqrt{\tfrac{n}{\pi}}\int_{\mathbb{R}}e^{-ns^{2}}M^{\varphi}_{n}\circ
M^{\delta}_{n}(\omega)(\tau^{\mathbb{G}}_{s}\circ
R_{\mathbb{G}}\circ\tau^{\mathbb{G}}_{-i/2}(y))\operatorname{d}\\!s\\\
&=\bigl{(}\sqrt{\tfrac{n}{\pi}}\int_{\mathbb{R}}e^{-n(s+i/2)^{2}}M^{\varphi}_{n}\circ
M^{\delta}_{n}(\omega)\circ\tau^{\mathbb{G}}_{s}\circ
R_{\mathbb{G}}\operatorname{d}\\!s\bigr{)}(y).\end{split}$

The above calculation shows also that
$\overline{\omega_{n}}^{\sharp}=\sqrt{\tfrac{n}{\pi}}\int_{\mathbb{R}}e^{-n(s+i/2)^{2}}M^{\varphi}_{n}\circ
M^{\delta}_{n}(\omega)\circ\tau^{\mathbb{G}}_{s}\circ
R_{\mathbb{G}}\operatorname{d}\\!s$. Moreover we have
$\overline{\omega_{n}}^{\sharp}\in\mathscr{J}$, which is a consequence of the
following calculation:

$\begin{split}&\quad\;|\overline{\omega_{n}}^{\sharp}(x^{*})|=(\tfrac{n}{\pi})^{3/2}\bigl{|}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+(s+i/2)^{2}+p^{2})}\omega\bigl{(}\delta_{\mathbb{G}}^{ip}\sigma^{\varphi}_{t}\circ\tau_{s}^{\mathbb{G}}\circ
R_{\mathbb{G}}(x^{*})\bigr{)}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{|}\\\
&\leq(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}\Lambda_{\varphi}\bigl{(}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+(s-i/2)^{2}+p^{2})}\sigma^{\varphi}_{t}\circ\tau^{\mathbb{G}}_{s}\circ
R_{\mathbb{G}}(x)\delta_{\mathbb{G}}^{-ip}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&=(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}\Lambda_{\psi}\bigl{(}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+(s-i/2)^{2}+(p+i/2)^{2})}\sigma^{\varphi}_{t}\circ\tau^{\mathbb{G}}_{s}\circ
R_{\mathbb{G}}(x)\delta_{\mathbb{G}}^{ip}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&=(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}\Lambda_{\psi}\bigl{(}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+(s-i/2)^{2}+(p+i/2)^{2})}\delta_{\mathbb{G}}^{-it}\sigma^{\psi}_{t}\circ\tau^{\mathbb{G}}_{s}\circ
R_{\mathbb{G}}(x)\delta_{\mathbb{G}}^{i(t+p)}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&=\\!(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}J_{\psi}\nabla_{\psi}^{1/2}\Lambda_{\psi}\bigl{(}\\!\int_{\mathbb{R}^{3}}\\!\\!e^{-n(t^{2}+(s+i/2)^{2}+(p-i/2)^{2})}\delta_{\mathbb{G}}^{-i(t+p)}\sigma^{\psi}_{t}\circ\tau^{\mathbb{G}}_{s}\circ
R_{\mathbb{G}}(x^{*})\delta_{\mathbb{G}}^{it}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&=\\!(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{\|}\Lambda_{\psi}\\!\bigl{(}\\!\int_{\mathbb{R}^{3}}\\!\\!e^{-n((t+i/2)^{2}\\!+(s+i/2)^{2}\\!+(p-i/2)^{2})}\\!\nu_{\mathbb{G}}^{-p/2}\\!\delta_{\mathbb{G}}^{-i(t+p)}\sigma^{\psi}_{t}\\!\circ\\!\tau^{\mathbb{G}}_{s}\\!\circ\\!R_{\mathbb{G}}(x^{*})\delta_{\mathbb{G}}^{it}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\bigr{\|}\\\
&\leq(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\int_{\mathbb{R}^{3}}|e^{-n((t+i/2)^{2}+(s+i/2)^{2}+(p-i/2)^{2})}|\nu_{\mathbb{G}}^{s/2-t/2-p/2}\|\Lambda_{\psi}(R_{\mathbb{G}}(x^{*}))\|\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\\\
&=(\tfrac{n}{\pi})^{3/2}\|\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\|\bigl{(}\int_{\mathbb{R}^{3}}|e^{-n((t+i/2)^{2}+(s+i/2)^{2}+(p-i/2)^{2})}|\nu_{\mathbb{G}}^{s/2-t/2-p/2}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\bigr{)}\|\Lambda_{\varphi}(x)\|,\end{split}$

where we used
$\Lambda_{\varphi}(z)=\Lambda_{\psi}(z\delta_{\mathbb{G}}^{1/2})$ for
sufficiently good operators $z$,
$\sigma^{\psi}_{t}=\operatorname{Ad}(\delta^{it}_{\mathbb{G}})\circ\sigma^{\varphi}_{t}$,
$\psi\circ\tau^{\mathbb{G}}_{s}=\nu_{\mathbb{G}}^{-s}\psi$ and
$\sigma^{\psi}_{t}(\delta_{\mathbb{G}}^{ip})=\nu_{\mathbb{G}}^{itp}\delta_{\mathbb{G}}^{ip}$.
We are left to argue that $\mathscr{J}_{00}$ is dense and that
$\lambda_{\mathbb{G}}(\mathscr{J}_{00})$ is a
$\textsc{sot}^{*}\times\|\cdot\|$ core for $\Lambda_{\widehat{\varphi}}$. It
follows from

$\omega_{n}\xrightarrow[n\to\infty]{}\omega,\quad\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega_{n}))\xrightarrow[n\to\infty]{}\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))$

for $\omega\in\mathscr{J}^{\sharp}$. The first claim is standard (see e.g.
[43, Proposition 2.25] where a similar method is used), the second one can be
seen as follows: first, observe that for $t,s,p\in\mathbb{R}$ we have using
$(\sigma^{\varphi}_{t}\otimes\textnormal{id})\mathrm{W}^{\mathbb{G}}=(\textnormal{id}\otimes\tau^{\widehat{\mathbb{G}}}_{-t})(\mathrm{W}^{\mathbb{G}})(\mathbbm{1}\otimes\delta_{\widehat{\mathbb{G}}}^{it})$
(see equation (4.5)) and
$(\tau^{\mathbb{G}}_{s}\otimes\tau^{\widehat{\mathbb{G}}}_{s})\mathrm{W}^{\mathbb{G}}=\mathrm{W}^{\mathbb{G}}$
that
$(\tau^{\mathbb{G}}_{-t}\otimes\sigma^{\widehat{\varphi}}_{-t})\mathrm{W}^{\mathbb{G}}=(\delta_{\mathbb{G}}^{it}\otimes\mathbbm{1})\mathrm{W}^{\mathbb{G}}$
and

$\begin{split}&\quad\;\lambda_{\mathbb{G}}((\omega\delta_{\mathbb{G}}^{ip})\circ\tau^{\mathbb{G}}_{s}\circ\sigma^{\varphi}_{t})=((\omega\delta_{\mathbb{G}}^{ip})\circ\tau^{\mathbb{G}}_{s}\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\tau^{\widehat{\mathbb{G}}}_{-t})(\mathrm{W}^{\mathbb{G}})(\mathbbm{1}\otimes\delta_{\widehat{\mathbb{G}}}^{it})\bigr{)}\\\
&=(\omega\otimes\tau^{\widehat{\mathbb{G}}}_{-t-s})\bigl{(}(\delta^{ip}_{\mathbb{G}}\otimes\mathbbm{1})\mathrm{W}^{\mathbb{G}}\bigr{)}\delta_{\widehat{\mathbb{G}}}^{it}=(\omega\circ\tau^{\mathbb{G}}_{-p}\otimes\tau^{\widehat{\mathbb{G}}}_{-t-s}\circ\sigma^{\widehat{\varphi}}_{-p})(\mathrm{W}^{\mathbb{G}}\bigr{)}\delta_{\widehat{\mathbb{G}}}^{it}\\\
&=(\omega\otimes\tau^{\widehat{\mathbb{G}}}_{p-t-s}\circ\sigma^{\widehat{\varphi}}_{-p})(\mathrm{W}^{\mathbb{G}}\bigr{)}\delta_{\widehat{\mathbb{G}}}^{it}=\tau^{\widehat{\mathbb{G}}}_{p-t-s}\circ\sigma^{\widehat{\varphi}}_{-p}(\lambda_{\mathbb{G}}(\omega))\delta_{\widehat{\mathbb{G}}}^{it}.\end{split}$

Hence

$\begin{split}\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega_{n}))&=(\tfrac{n}{\pi})^{3/2}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+s^{2}+p^{2})}\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}((\omega\delta_{\mathbb{G}}^{ip})\circ\tau^{\mathbb{G}}_{s}\circ\sigma^{\varphi}_{t}))\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\\\
&=(\tfrac{n}{\pi})^{3/2}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+s^{2}+p^{2})}\Lambda_{\widehat{\varphi}}\bigl{(}\tau^{\widehat{\mathbb{G}}}_{p-t-s}\circ\sigma^{\widehat{\varphi}}_{-p}(\lambda_{\mathbb{G}}(\omega))\delta_{\widehat{\mathbb{G}}}^{it}\bigr{)}\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p\\\
&=(\tfrac{n}{\pi})^{3/2}\int_{\mathbb{R}^{3}}e^{-n(t^{2}+s^{2}+p^{2})}\nu_{\widehat{\mathbb{G}}}^{s/2-p/2}J_{\widehat{\varphi}}\delta_{\widehat{\mathbb{G}}}^{-it}J_{\widehat{\varphi}}P^{i(p-t-s)}\nabla_{\widehat{\varphi}}^{-ip}\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))\operatorname{d}\\!t\operatorname{d}\\!s\operatorname{d}\\!p,\end{split}$

where $P$ is the operator implementing the scaling group via
$P^{it}\Lambda_{\varphi}(x)=\nu_{\mathbb{G}}^{t/2}\Lambda_{\varphi}(\tau^{\mathbb{G}}_{t}(x))$.
Convergence
$\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega_{n}))\xrightarrow[n\to\infty]{}\Lambda_{\widehat{\varphi}}(\lambda_{\mathbb{G}}(\omega))$
follows now using a standard argument. ∎

##  3\. Completely bounded multipliers

Unless stated otherwise, in this section $\mathbb{G}$ is an arbitrary locally
compact quantum group.

###  3.1. Definitions and fundamental facts

We start by discussing the notions of (left/right) centralisers and
multipliers. In the main part of the text we will focus on the left version of
these objects – this is simply a matter of choice, compare Proposition 4.14.

Following [34], we say that a linear map
$T\colon\operatorname{L}^{1}(\widehat{\mathbb{G}})\rightarrow\operatorname{L}^{1}(\widehat{\mathbb{G}})$
is a _left (respectively, right) centraliser_ if

$T(\omega\star\omega^{\prime})=T(\omega)\star\omega^{\prime}\quad\bigl{(}\textnormal{respectively}\;T(\omega\star\omega^{\prime})=\omega\star
T(\omega^{\prime})\bigr{)}\qquad(\omega,\omega^{\prime}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})).$

We denote by $C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ the
space of completely bounded left centralisers. Together with the completely
bounded norm and composition as product,
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ becomes a Banach
algebra. Similarly, $C^{r}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$
stands for the space of completely bounded right centralisers, where now it is
natural to use the opposite composition as product. We equip these spaces with
an operator space structure by requiring that the embeddings
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}})),C^{r}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\hookrightarrow\operatorname{CB}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$
are completely isometric; both then become completely contractive Banach
algebras.

An operator $b\in\operatorname{L}^{\infty}(\mathbb{G})$ is said to be a
_completely bounded left multiplier_ if
$b\operatorname{A}(\mathbb{G})\subseteq\operatorname{A}(\mathbb{G})$ and the
associated map

$\Theta^{l}(b)_{*}\colon\operatorname{L}^{1}(\widehat{\mathbb{G}})\rightarrow\operatorname{L}^{1}(\widehat{\mathbb{G}})\quad\text{satisfying}\quad
b\widehat{\lambda}(\omega)=\widehat{\lambda}(\Theta^{l}(b)_{*}(\omega))\qquad(\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}}))$

is completely bounded. As $\widehat{\lambda}$ is injective, this definition
makes sense. We follow here the notation of [13]; sometimes the notation
$m^{l}_{b}=\Theta^{l}(b)_{*}$ is used instead. As $\widehat{\lambda}$ is
multiplicative, for any completely bounded left multiplier $b$ we have that
$\Theta^{l}(b)_{*}\in C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$.
We write $\Theta^{l}(b)=(\Theta^{l}(b)_{*})^{*}$, and denote the space of CB
left multipliers by $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$.
Any Fourier algebra element
$\widehat{\lambda}(\omega)\in\operatorname{A}(\mathbb{G})$ is a CB left
multiplier with
$\Theta^{l}(\widehat{\lambda}(\omega))_{*}\in\operatorname{CB}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$
being the left multiplication by $\omega$ and
$\Theta^{l}(\widehat{\lambda}(\omega))=(\omega\otimes\textnormal{id})\widehat{\Delta}$.
Moreover, it holds that
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\subseteq\operatorname{M}(\mathrm{C}_{0}(\mathbb{G}))$,
see [17, Theorem 4.2].

Conversely, if $T\in C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$
is a left centraliser, then its Banach space dual $T^{*}$ is a normal CB map
on $\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ which is a left
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$-module homomorphism, i.e.
$T^{*}\in\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))}$.
Then, by [35, Corollary 4.4], there exists a unique CB left multiplier
$b\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ satisfying
$\Theta^{l}(b)=T^{*}$, that is, $\Theta^{l}(b)_{*}=T$. These constructions are
mutually inverse, and so the map
$\Theta^{l}(\cdot)_{*}:\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\rightarrow
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ is bijective. We
define the operator space structure on
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ so that these spaces
become completely isometric.

The above notions have right counterparts. Recalling that
$\mathrm{V}^{\widehat{\mathbb{G}}}\in\operatorname{L}^{\infty}(\mathbb{G})^{\prime}\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$
is the right Kac-Takesaki operator, let us introduce the map
$\widehat{\rho}\colon\operatorname{L}^{1}(\widehat{\mathbb{G}})\ni\omega\mapsto(\textnormal{id}\otimes\omega)\mathrm{V}^{\widehat{\mathbb{G}}}\in\operatorname{L}^{\infty}(\mathbb{G})^{\prime}$.
Its image $\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ should
be thought of as a right analogue of the Fourier algebra
$\operatorname{A}(\mathbb{G})=\widehat{\lambda}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$.
An operator $b^{\prime}\in\operatorname{L}^{\infty}(\mathbb{G})^{\prime}$ is
called a _completely bounded right multiplier_ if
$\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))b^{\prime}\in\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$
and the associated map
$\Theta^{r}(b^{\prime})_{*}\colon\operatorname{L}^{1}(\widehat{\mathbb{G}})\rightarrow\operatorname{L}^{1}(\widehat{\mathbb{G}})$
is CB. Similarly as in the left version, we write
$\Theta^{r}(b^{\prime})\in\operatorname{CB}^{\sigma}_{\operatorname{L}^{1}(\widehat{\mathbb{G}})}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
for $(\Theta^{r}(b^{\prime})_{*})^{*}$ and
$\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))$
for the space of CB right multipliers. Any CB right centraliser $T\in
C^{r}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ is associated to a
unique CB right multiplier
$b^{\prime}\in\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))$
via $T=\Theta^{r}(b^{\prime})_{*}$ and this assignment is bijective. We
similarly define an operator space structure on
$\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))$
to make it completely isometric with
$C^{r}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$.

We will write e.g. $\|b\|_{cb}=\|\Theta^{l}(b)\|_{cb}$ for
$b\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. Observe that
$b\widehat{\lambda}(\omega)=\widehat{\lambda}(\Theta^{l}(b)_{*}(\omega))$ for
each $\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ if and only if
$(\mathbbm{1}\otimes
b)\mathrm{W}^{\widehat{\mathbb{G}}}=(\Theta^{l}(b)\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})$,
and from this, it follows that $\|b\|\leq\|b\|_{cb}$. Similarly we have
$\|b^{\prime}\|\leq\|b^{\prime}\|_{cb}$ for
$b^{\prime}\in\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))$.

As a consequence of the above discussion, we have a commutative diagram

(3.1)
---
$\textstyle{C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{L}^{\infty}(\mathbb{G})}$$\textstyle{\operatorname{L}^{1}(\widehat{\mathbb{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{A}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The two diagonal maps to $\operatorname{L}^{\infty}(\mathbb{G})$ are the
canonical inclusions, as is the map
$\operatorname{A}(\mathbb{G})\rightarrow\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$,
while the vertical map $\operatorname{L}^{1}(\widehat{\mathbb{G}})\rightarrow
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ is given by left
multiplication. A simple calculation shows that this diagram indeed commutes.
We obtain an immediate corollary: the map
$\operatorname{L}^{1}(\widehat{\mathbb{G}})\rightarrow
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ is injective,
equivalently, if $\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ with
$\omega\star\omega^{\prime}=0$ for all
$\omega^{\prime}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$, then
$\omega=0$.

There is a canonical way of moving between left and right CB multipliers using
the extension of the unitary antipode $\widehat{R}$ of $\widehat{\mathbb{G}}$.
Recall that it is implemented via
$\widehat{R}=J_{\varphi}(\cdot)^{*}J_{\varphi}$, and let us denote its
canonical extension to a bounded linear map on
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$ by
${\widehat{R}}^{\sim}=J_{\varphi}(\cdot)^{*}J_{\varphi}$. The following result
will be used in Proposition 4.14 to show that it does not matter if we use
left CB multipliers, or right CB multipliers, when we introduce the
approximation property (AP), see Definition 4.1 below.

######  Lemma 3.1.

For $\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ we have
$\widehat{\rho}(\omega)=\widehat{R}^{\sim}(\widehat{\lambda}(\omega\circ\widehat{R}))$.
Furthermore,
$\widehat{R}^{\sim}(\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G})))=\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))$
and for $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ we have
$\Theta^{r}(\widehat{R}^{\sim}(a))=\widehat{R}\circ\Theta^{l}(a)\circ\widehat{R}$.

###### Proof.

Recall that $\mathrm{V}^{\widehat{\mathbb{G}}}=(J_{\varphi}\otimes
J_{\varphi})\chi(\mathrm{W}^{\widehat{\mathbb{G}}})^{*}(J_{\varphi}\otimes
J_{\varphi})$, see [47, Proposition 2.15]. It follows that

(3.2)
$\widehat{\rho}(\omega)=(\textnormal{id}\otimes\omega)\mathrm{V}^{\widehat{\mathbb{G}}}=\widehat{R}^{\sim}\bigl{(}(\textnormal{id}\otimes\omega\circ\widehat{R})(\chi(\mathrm{W}^{\widehat{\mathbb{G}}}))\bigr{)}=\widehat{R}^{\sim}\bigl{(}(\omega\circ\widehat{R}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})\bigr{)}=\widehat{R}^{\sim}(\widehat{\lambda}(\omega\circ\widehat{R})).$

Next, take $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ and
$\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$. We have
$\widehat{R}^{\sim}(a)=J_{\varphi}a^{*}J_{\varphi}\in\operatorname{L}^{\infty}(\mathbb{G})^{\prime}$,
so by (3.2),

$\displaystyle\widehat{\rho}(\omega)\widehat{R}^{\sim}(a)$
$\displaystyle=\widehat{R}^{\sim}(\widehat{\lambda}(\omega\circ\widehat{R}))\widehat{R}^{\sim}(a)=\widehat{R}^{\sim}(a\widehat{\lambda}(\omega\circ\widehat{R}))$
$\displaystyle=\widehat{R}^{\sim}\bigl{(}\widehat{\lambda}(\Theta^{l}(a)_{*}(\omega\circ\widehat{R}))\bigr{)}=\widehat{\rho}\bigl{(}\Theta^{l}(a)_{*}(\omega\circ\widehat{R})\circ\widehat{R}\bigr{)}.$

Hence $\widehat{R}^{\sim}(a)\in\operatorname{M}^{r}_{cb}(a)$ with
$\Theta^{r}(\widehat{R}^{\sim}(a))=\widehat{R}\circ\Theta^{l}(a)\circ\widehat{R}$;
this map is indeed CB, compare Lemma 4.8. We have shown that
$\widehat{R}^{\sim}(\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G})))\subseteq\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))$;
the converse inclusion is analogous. ∎

We finish by recording a known result for which we have not found a convenient
reference.

######  Lemma 3.2.

Let $b\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. There is
$\beta\in\mathbb{C}$ with $\Theta^{l}(b)(\mathbbm{1})=\beta\mathbbm{1}$.

###### Proof.

It suffices to show that for
$T\in\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))}$
there is $\beta$ with $T(\mathbbm{1})=\beta\mathbbm{1}$. By definition,
$\Delta\circ T=(T\otimes\textnormal{id})\Delta$ and so
$\Delta(T(\mathbbm{1}))=T(\mathbbm{1})\otimes\mathbbm{1}$. By [18, Theorem
2.1] (and references therein) it follows that
$T(\mathbbm{1})\in\mathbb{C}\mathbbm{1}$, as required. ∎

###  3.2. Predual

Since the inclusion
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\hookrightarrow\operatorname{L}^{\infty}(\mathbb{G})$
is bounded (actually, contractive), we can consider the restriction of the
Banach space adjoint of this map, giving a map
$\alpha^{l}\colon\operatorname{L}^{1}(\mathbb{G})\rightarrow\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))^{*}$.
Let us define the space $Q^{l}(\operatorname{A}(\mathbb{G}))$ as the closure
of the image of $\alpha^{l}$, so that

$Q^{l}(\operatorname{A}(\mathbb{G}))=\overline{\alpha^{l}(\operatorname{L}^{1}(\mathbb{G}))}\subseteq\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))^{*}.$

According to [34, Theorem 3.4], the space
$Q^{l}(\operatorname{A}(\mathbb{G}))$ is a predual of
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$, i.e. we have

$Q^{l}(\operatorname{A}(\mathbb{G}))^{*}\cong\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$

completely isometrically. Whenever we speak about the weak∗-topology on
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ we will have in mind
this particular choice of predual – to the best of our knowledge, uniqueness
of predual of $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ is
unknown.

Similarly, we can restrict functionals in
$\operatorname{L}^{1}(\mathbb{G}^{\prime})$ to
$\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))$
via map $\alpha^{r}$, and after taking the closure obtain the predual
$Q^{r}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))\subseteq\operatorname{M}^{r}_{cb}(\widehat{\rho}(\operatorname{L}^{1}(\widehat{\mathbb{G}})))^{*}$.
From now on, we will restrict our discussion to the “left” setting.

######  Proposition 3.3.

$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ is a dual Banach
algebra, that is, the multiplication of
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ is separately
weak∗-continuous.

###### Proof.

We turn $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))^{*}$ into a
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$-bimodule in the
usual way. Let
$a,b\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\subseteq\operatorname{L}^{\infty}(\mathbb{G})$
and $f\in\operatorname{L}^{1}(\mathbb{G})$. Writing
$\langle\cdot,\cdot\rangle$ for the pairing between
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G})$ and
$Q^{l}(\operatorname{A}(\mathbb{G}))$, or between
$\operatorname{L}^{\infty}(\mathbb{G})$ and
$\operatorname{L}^{1}(\mathbb{G})$, we have

$\displaystyle\langle ab,\alpha^{l}(f)\rangle=\langle ab,f\rangle$
$\displaystyle=\langle a,bf\rangle=\langle a,\alpha^{l}(bf)\rangle$
$\displaystyle=\langle b,fa\rangle=\langle b,\alpha^{l}(fa)\rangle.$

This calculation shows that $b\cdot\alpha^{l}(f)=\alpha^{l}(bf)$ and
$\alpha^{l}(f)\cdot a=\alpha^{l}(fa)$, so by continuity, it follows that
$Q^{l}(\operatorname{A}(\mathbb{G}))$ is a closed submodule of
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))^{*}$. It is now
standard, see [60, Proposition 1.2] for example, that the product is
separately weak∗-continuous in
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. ∎

Our next goal is to obtain a characterisation of functionals in
$Q^{l}(\operatorname{A}(\mathbb{G}))$. We will do this by obtaining an
alternative description of the weak∗-topology on
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. In the process, we
also discuss CB maps on the $\mathrm{C}^{*}$-algebra
$\mathrm{C}_{0}(\widehat{\mathbb{G}})$ which are associated to left
centralisers.

To start, we observe that the adjoint $T^{*}$ of a CB left centraliser $T\in
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ restricts to a CB map
on $\mathrm{C}_{0}(\widehat{\mathbb{G}})$. Indeed, we can write
$T^{*}=\Theta^{l}(a)$ for some
$a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ and then the
claim follows from the equality $(\mathbbm{1}\otimes
a)\mathrm{W}^{\widehat{\mathbb{G}}}=(T^{*}\otimes\textnormal{id})\mathrm{W}^{\widehat{\mathbb{G}}}$
and density of $\operatorname{A}(\widehat{\mathbb{G}})$ in
$\mathrm{C}_{0}(\widehat{\mathbb{G}})$. We seek a characterisation of which CB
maps on $\mathrm{C}_{0}(\widehat{\mathbb{G}})$ occur in this way as
restrictions of duals to left CB centralisers, in terms of a property similar
to the characterisation
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\cong\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$.

In the following statement, recall that
$(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\widehat{\Delta})$ is bisimplifiable,
and so elements of the form $\widehat{\Delta}(a)(\mathbbm{1}\otimes b)$, for
$a,b\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$, form a linearly dense subset of
$\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathrm{C}_{0}(\widehat{\mathbb{G}})$.
Hence the left-hand-side of (3.3) is contained in
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\otimes\mathrm{C}_{0}(\widehat{\mathbb{G}})\subseteq\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$,
while the right-hand-side is in
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$.
We also recall that, by Kaplansky density,
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$ is (completely) isometrically a
subspace of $\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*}$.

######  Lemma 3.4.

Let
$L\in\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
be such that

(3.3) $(L\otimes\textnormal{id})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b))=\widehat{\Delta}(L(a))(\mathbbm{1}\otimes
b)\qquad(a,b\in\mathrm{C}_{0}(\widehat{\mathbb{G}})).$

Embedding $\operatorname{L}^{1}(\widehat{\mathbb{G}})$ into the duals of
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ and
$\mathrm{C}_{0}(\widehat{\mathbb{G}})$ in the usual way, we have that $L^{*}$
maps $\operatorname{L}^{1}(\widehat{\mathbb{G}})$ to itself, and the resulting
restriction
$T\in\operatorname{CB}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ is a left
centraliser. Furthermore
$T^{*}\in\operatorname{CB}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
restricts to $L$, so consequently
$L\in\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}))$.

###### Proof.

As $\operatorname{L}^{1}(\widehat{\mathbb{G}})$ is an essential
$\mathrm{C}_{0}(\widehat{\mathbb{G}})$-module, by Cohen–Hewitt factorisation,
given $\omega_{2}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ there are
$\omega_{3}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ and
$b\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$ with $\omega_{2}=b\omega_{3}$.
Then, for $\omega_{1}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ and
$a\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$ we have

$\begin{split}&\quad\;\langle
L^{*}(\omega_{1}\star\omega_{2}),a\rangle_{\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*},\mathrm{C}_{0}(\widehat{\mathbb{G}})}=\langle\widehat{\Delta}(L(a)),\omega_{1}\otimes\omega_{2}\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})}\\\
&=\langle\widehat{\Delta}(L(a))(\mathbbm{1}\otimes
b),\omega_{1}\otimes\omega_{3}\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})}\\\
&=\langle(L\otimes\textnormal{id})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b)),\omega_{1}\otimes\omega_{3}\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})}\\\
&=\langle
L^{*}(\omega_{1})\otimes\omega_{3},\widehat{\Delta}(a)(\mathbbm{1}\otimes
b)\rangle_{(\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathrm{C}_{0}(\widehat{\mathbb{G}}))^{*},\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathrm{C}_{0}(\widehat{\mathbb{G}})}\\\
&=\langle
L^{*}(\omega_{1}),(\textnormal{id}\otimes\omega_{3})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b))\rangle_{\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*},\mathrm{C}_{0}(\widehat{\mathbb{G}})}\\\
&=\langle
L^{*}(\omega_{1}),(\textnormal{id}\otimes\omega_{2})\widehat{\Delta}(a)\rangle_{\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*},\mathrm{C}_{0}(\widehat{\mathbb{G}})}\\\
&=\langle
L^{*}(\omega_{1})\star\omega_{2},a\rangle_{\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*},\mathrm{C}_{0}(\widehat{\mathbb{G}})}.\end{split}$

It follows that
$L^{*}(\omega_{1}\star\omega_{2})=L^{*}(\omega_{1})\star\omega_{2}$ in
$\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*}$. As
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$ is an ideal in
$\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*}$ ([46, Proof of Proposition 8.3]),
this shows that
$L^{*}(\omega_{1}\star\omega_{2})\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$,
and as products have dense linear span in
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$ ([13, Section 3]), we conclude
that $L^{*}$ restricts to a map on
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$, say
$T\in\operatorname{CB}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$. Then
$T(\omega_{1}\star\omega_{2})=T(\omega_{1})\star\omega_{2}$ for all
$\omega_{1},\omega_{2}$, and so $T\in
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$. We finally calculate
that, for
$a\in\mathrm{C}_{0}(\widehat{\mathbb{G}}),\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$,

$\displaystyle\langle
T^{*}(a),\omega\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})}$
$\displaystyle=\langle
a,T(\omega)\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})}=\langle
L^{*}(\omega),a\rangle_{\mathrm{C}_{0}(\widehat{\mathbb{G}})^{*},\mathrm{C}_{0}(\widehat{\mathbb{G}})}=\langle
L(a),\omega\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})},$

and so $T^{*}$ restricts to $L$, as required. ∎

We can now characterise what it means for an operator in
$\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}))$ to be a centraliser.
Condition (2) in the following proposition should be thought of as a
$\mathrm{C}_{0}(\widehat{\mathbb{G}})$ variant of what it means to be a left
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$-module homomorphism.

######  Proposition 3.5.

For $L\in\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}))$ the
following are equivalent:

  1. (1)

there is $T\in C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ such
that $T^{*}$ restricts to $L$;

  2. (2)

$(L\otimes\textnormal{id})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b))=\widehat{\Delta}(L(a))(\mathbbm{1}\otimes b)$ for each
$a,b\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$;

Furthermore, the restriction map
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\cong\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))\rightarrow\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}))$
is a complete isometry; in particular, there is a bijection between
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ and the space of all
maps $L\in\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}))$ satisfying
(2).

###### Proof.

If (1) holds then
$\widehat{\Delta}T^{*}=(T^{*}\otimes\textnormal{id})\widehat{\Delta}$ and so
certainly the condition in (2) will hold for $T^{*}$ and hence also for $L$.
Conversely, suppose that (2) holds. Then due to Lemma 3.4 we know that $L^{*}$
restricts to a map $T\in
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ such that $T^{*}$
restricts to $L$, showing (1).

The restriction map
$\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))\rightarrow\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}))$
is clearly a complete contraction. With $T,L$ as above, this restriction map
is given by $T^{*}\mapsto L$, and as $T$ is the restriction of $L^{*}$ and
$L\mapsto L^{*},T\mapsto T^{*}$ are completely isometric, it follows that
$\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))\rightarrow\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}))$
is a complete isometry. ∎

######  Proposition 3.6.

Equip the space
$\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
with the weak∗-topology arising from the canonical predual
$\mathrm{C}_{0}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})$.
The restriction map

$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\cong\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))\rightarrow\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$

is a complete isometry which has weak∗-closed image.

###### Proof.

Proposition 3.5 shows that the restriction map
$\prescript{}{\operatorname{L}^{1}(\widehat{\mathbb{G}})}{\operatorname{CB}}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))\rightarrow\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
is a complete isometry. Let $(T_{i})_{i\in I}$ be a net in
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ such that the image
of the net $(T_{i}^{*})_{i\in I}$ in
$\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
converges weak∗ to
$L\in\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$.
Let $a,b\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$ and
$\omega_{1},\omega_{2}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$, and note
that
$(\textnormal{id}\otimes\omega_{2})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b))\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$. We now calculate that

$\displaystyle\quad\;\langle\widehat{\Delta}(L(a))(\mathbbm{1}\otimes
b),\omega_{1}\otimes\omega_{2}\rangle=\lim_{i\in I}\langle
T_{i}^{*}(a),\omega_{1}\star(b\omega_{2})\rangle=\lim_{i\in I}\langle
a,T_{i}(\omega_{1})\star(b\omega_{2})\rangle$ $\displaystyle=\lim_{i\in
I}\langle\widehat{\Delta}(a)(\mathbbm{1}\otimes
b),T_{i}(\omega_{1})\otimes\omega_{2}\rangle=\lim_{i\in I}\langle
T_{i}^{*}\big{(}(\textnormal{id}\otimes\omega_{2})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b))\big{)},\omega_{1}\rangle$ $\displaystyle=\langle
L\big{(}(\textnormal{id}\otimes\omega_{2})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b))\big{)},\omega_{1}\rangle=\langle(L\otimes\textnormal{id})(\widehat{\Delta}(a)(\mathbbm{1}\otimes
b)),\omega_{1}\otimes\omega_{2}\rangle.$

All the above pairings are between a von Neumann algebra and its predual. It
follows that we have $\widehat{\Delta}(L(a))(\mathbbm{1}\otimes
b)=(L\otimes\textnormal{id})(\widehat{\Delta}(a)(\mathbbm{1}\otimes b))$ in
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$.
By Lemma 3.4, $L^{*}$ restricts to $T\in
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ such that $T^{*}$
restricts back to give $L$. That is, $T_{i}^{*}\xrightarrow[i\in I]{}T^{*}$
weak∗ in
$\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$,
as required. ∎

We now wish to show that the resulting weak∗-topology on
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ given by Proposition
3.6 agrees with the weak∗-topology on
$C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\cong\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$
given by the predual $Q^{l}(\operatorname{A}(\mathbb{G}))$. In the following,
for a Banach space $E$, we denote by $\kappa_{E}\colon E\rightarrow E^{**}$
the canonical map to the bidual.

######  Lemma 3.7.

Let $E,F$ be Banach spaces, and let $\alpha:E^{*}\rightarrow F^{*}$ be a
bounded linear map. Let $D\subseteq F$ be a subset with dense linear span.
Then $\alpha$ is weak∗-weak∗-continuous if and only if
$\alpha^{*}\kappa_{F}(D)\subseteq\kappa_{E}(E)$. In this case, and when
further $\alpha$ is a bijection, the resulting preadjoint
$\alpha_{*}:F\rightarrow E$ is also an isomorphism of Banach spaces and
$\alpha$ is a weak∗-weak∗-homeomorphism.

###### Proof.

If $\alpha$ is weak∗-continuous, then there is a preadjoint operator
$\alpha_{*}:F\rightarrow E$ with $(\alpha_{*})^{*}=\alpha$, and so
$\alpha^{*}\kappa_{F}(D)=(\alpha_{*})^{**}\kappa_{F}(D)=\kappa_{E}\alpha_{*}(D)\subseteq\kappa_{E}(E)$,
as claimed. Conversely, if $\alpha^{*}\kappa_{F}(D)\subseteq\kappa_{E}(E)$
then by norm density of $\operatorname{span}D$ in $F$, and norm continuity of
$\alpha^{*}$, we have that $\alpha^{*}\kappa_{F}(F)\subseteq\kappa_{E}(E)$. We
could now directly apply [16, Lemma 10.1], but let us give the argument. There
is a linear map $T:F\rightarrow E$ with
$\alpha^{*}\kappa_{F}(x)=\kappa_{E}(T(x))$ for each $x\in F$. As
$\kappa_{E},\kappa_{F}$ are isometries, $T$ is bounded with
$\|T\|\leq\|\alpha^{*}\|=\|\alpha\|$. Then for $x\in F,\mu\in E^{*}$,

$\langle
T^{*}(\mu),x\rangle=\langle\mu,T(x)\rangle=\langle\kappa_{E}(T(x)),\mu\rangle=\langle\alpha^{*}\kappa_{F}(x),\mu\rangle=\langle\alpha(\mu),x\rangle.$

Hence $T^{*}=\alpha$ and so $\alpha$ is weak∗-continuous, with preadjoint $T$.

When $\alpha$ is a bijection, by the Open Mapping Theorem, it is an
isomorphism. Thus also $\alpha^{*}$ is an isomorphism, and so as
$\alpha^{*}\kappa_{F}=\kappa_{E}\alpha_{*}$ it follows that $\alpha_{*}$ is
bounded below and so has closed image. If $\mu\in(\alpha_{*}(F))^{\perp}$ then
$0=\langle\mu,\alpha_{*}(x)\rangle=\langle\alpha(\mu),x\rangle$ for all
$x,\mu$ and so $\alpha(\mu)=0$ so $\mu=0$. Hence $\alpha_{*}$ is a surjection,
and so an isomorphism. ∎

######  Theorem 3.8.

The weak∗-topology on $C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$
given by the embedding into
$\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
agrees with the weak∗-topology on
$\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))$ given by
$Q^{l}(\operatorname{A}(\mathbb{G}))$.

###### Proof.

We use Lemma 3.7. Set $E=Q^{l}(\operatorname{A}(\mathbb{G}))$. To avoid
confusion, for this proof only, we shall write
$\theta:C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\rightarrow\operatorname{CB}(\mathrm{C}_{0}(\widehat{\mathbb{G}}),\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
for the complete isometry $T\mapsto
T^{*}|_{\mathrm{C}_{0}(\widehat{\mathbb{G}})}$, given by Proposition 3.6. As
the image of $\theta$ is weak∗-closed, it has canonical predual $F$ which is a
quotient of
$\mathrm{C}_{0}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})$.
Let
$\pi:\mathrm{C}_{0}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})\rightarrow
F$ be the quotient map. We hence corestrict $\theta$ to give an isomorphism
$\theta:C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))\rightarrow
F^{*}$. Let
$\alpha_{0}:E^{*}=\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))\rightarrow
C^{l}_{cb}(\operatorname{L}^{1}(\widehat{\mathbb{G}}))$ be the canonical
bijection, and set $\alpha=\theta\circ\alpha_{0}:E^{*}\rightarrow F^{*}$.

Given $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ set
$T=\alpha_{0}(a)$, so by definition,
$a\widehat{\lambda}(\omega)=\widehat{\lambda}(T(\omega))$ for each
$\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$. Equivalently,
$(\mathbbm{1}\otimes
a){\mathrm{W}}^{\widehat{\mathbb{G}}}=(T^{*}\otimes\textnormal{id})({\mathrm{W}}^{\widehat{\mathbb{G}}})$.
Given
$\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}}),f\in\operatorname{L}^{1}(\mathbb{G})$,
set $u=\pi((\textnormal{id}\otimes
f)({\mathrm{W}}^{\widehat{\mathbb{G}}})\otimes\omega)\in F$, and calculate
that

$\displaystyle\
\langle\kappa_{E}(\widehat{\lambda}(\omega)f),a\rangle_{E^{**},E^{*}}=\langle
a,\widehat{\lambda}(\omega)f\rangle_{E^{*},E}=\langle
a\widehat{\lambda}(\omega),f\rangle_{\operatorname{L}^{\infty}(\mathbb{G}),\operatorname{L}^{1}(\mathbb{G})}=\langle(T^{*}\otimes\textnormal{id})({\mathrm{W}}^{\widehat{\mathbb{G}}}),\omega\otimes
f\rangle$ $\displaystyle=\langle T^{*}\big{(}(\textnormal{id}\otimes
f)({\mathrm{W}}^{\widehat{\mathbb{G}}})\big{)},\omega\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})}=\langle\theta(T)\big{(}(\textnormal{id}\otimes
f)({\mathrm{W}}^{\widehat{\mathbb{G}}})\big{)},\omega\rangle_{\operatorname{L}^{\infty}(\widehat{\mathbb{G}}),\operatorname{L}^{1}(\widehat{\mathbb{G}})}=\langle\alpha(a),u\rangle_{F^{*},F}.$

It follows that
$\alpha^{*}(\kappa_{F}(u))=\kappa_{E}(\widehat{\lambda}(\omega)f)\in\kappa_{E}(E)$.
As the linear span of such elements $u$ is dense in $F$, the conditions of the
lemma are verified, and the result follows. ∎

Using this result we can characterise functionals in the predual space
$Q^{l}(\operatorname{A}(\mathbb{G}))$. In the following, we work with infinite
matrices with entries in operator spaces, see [23, Chapter 10].

######  Proposition 3.9.

For any Hilbert space $\mathsf{H}$ and
$x\in\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathcal{K}(\mathsf{H}),\,\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}$,
the bounded linear functional

$\Omega_{x,\omega}\colon\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\ni
a\mapsto\langle(\Theta^{l}(a)\otimes\textnormal{id})x,\omega\rangle\in\mathbb{C}.$

belongs to $Q^{l}(\operatorname{A}(\mathbb{G}))$. Furthermore, all functionals
in $Q^{l}(\operatorname{A}(\mathbb{G}))$ are of this form for some separable
Hilbert space.

This result was recorded without proof in [13, Proposition 3.2]. In the
classical context of locally compact groups, an analogous result was proved by
Haagerup and Kraus in [32, Proposition 1.5]. For the convenience of the
reader, we give a proof using Theorem 3.8.

###### Proof.

We first show that $\Omega_{x,\omega}$ is a member of
$Q^{l}(\operatorname{A}(\mathbb{G}))$. As
$Q^{l}(\operatorname{A}(\mathbb{G}))\subseteq\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))^{*}$
is norm closed, by first approximating $x,\omega$ by sums of elementary
tensors, and then collapsing the pairing between $\mathcal{K}(\mathsf{H})$ and
$\operatorname{B}(\mathsf{H})_{*}$, we reduce the problem to the case when
$\mathsf{H}=\mathbb{C}$, when
$x=(\textnormal{id}\otimes\theta){\mathrm{W}}^{\widehat{\mathbb{G}}}$ for some
$\theta\in\operatorname{L}^{1}(\mathbb{G})$, and when
$\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$. We then calculate

$\langle\Theta^{l}(a)\bigl{(}(\textnormal{id}\otimes\theta){\mathrm{W}}^{\widehat{\mathbb{G}}}\bigr{)},\omega\rangle=\langle\widehat{\lambda}(\omega\circ\Theta^{l}(a)),\theta\rangle=\langle
a\widehat{\lambda}(\omega),\theta\rangle=\langle
a,\widehat{\lambda}(\omega)\theta\rangle\quad(a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))),$

which shows that
$\Omega_{x,\omega}=\alpha^{l}(\widehat{\lambda}(\omega)\theta)\in
Q^{l}(\operatorname{A}(\mathbb{G}))$, as required.

Now, take any functional in $Q^{l}(\operatorname{A}(\mathbb{G}))$, which by
Theorem 3.8 is represented by some element
$\rho\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\mathrm{C}_{0}(\widehat{\mathbb{G}})$
(note that the projective operator space tensor product is symmetric). By [23,
Theorem 10.2.1], we can find infinite matrices

$\alpha\in\operatorname{M}_{1,\infty\times\infty},\;\beta\in\mathsf{K}_{\infty}(\operatorname{L}^{1}(\widehat{\mathbb{G}})),\;\gamma\in\mathsf{K}_{\infty}(\mathrm{C}_{0}(\widehat{\mathbb{G}})),\;\alpha^{\prime}\in\operatorname{M}_{\infty\times\infty,1}$

such that $\rho=\alpha(\beta\otimes\gamma)\alpha^{\prime}$ (for the
introduction to infinite matrices with entries in an operator space, see [23,
Sections 10.1, 10.2]). Writing
$\alpha=[\alpha_{1,(i,j)}]_{(i,j)\in\mathbb{N}^{2}}$ etc., this means that

(3.4)
$\langle\Theta^{l}(a),\rho\rangle=\sum_{i,j,k,l=1}^{\infty}\alpha_{1,(i,j)}\langle\Theta^{l}(a)(\gamma_{j,l}),\beta_{i,k}\rangle\alpha^{\prime}_{(k,l),1}\qquad(a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))).$

Let $\mathsf{H}$ be an infinite dimensional, separable Hilbert space with
orthonormal basis $\\{e_{n}\\}_{n=1}^{\infty}$ and let
$e_{i,j}\,(i,j\in\mathbb{N})$ be the corresponding rank one operators. Write
$T(\mathsf{H})$ for the operator space of trace class operators, identified in
a completely isometric way with $\operatorname{B}(\mathsf{H})_{*}$. For any
$n\in\mathbb{N}$ we have
$\|[e_{j,i}]_{i,j=1}^{n}\|_{\operatorname{M}_{n}(T(\mathsf{H}))}=1$. Indeed,
the matrix $[e_{j,i}]_{i,j=1}^{n}$ corresponds to the map
$E_{n}\in\operatorname{CB}(\operatorname{B}(\mathsf{H}),\operatorname{M}_{n})$
given by $E_{n}(x)=[\operatorname{Tr}(e_{j,i}x)]_{i,j=1}^{n}$. If we denote by
$V_{n}\colon\mathbb{C}^{n}\rightarrow\mathsf{H}$ the canonical inclusion
associated with the choice of basis, one easily sees that
$E_{n}(x)=V_{n}^{*}xV_{n}\,(x\in\operatorname{B}(\mathsf{H}))$ and
$\|E_{n}\|_{cb}=1$ follows. Consequently $[e_{j,i}]_{i,j=1}^{\infty}$ is a
well defined matrix in $\operatorname{M}_{\infty}(T(\mathsf{H}))$. Finally,
define

$\omega=\alpha(\beta\otimes[e_{j,i}]_{i,j=1}^{\infty})\alpha^{\prime}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}T(\mathsf{H})=\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}.$

A choice of basis gives us an isomorphism $\mathsf{H}\cong\ell^{2}$ and
consequently we can consider $\gamma$ as an element of
$\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathcal{K}(\mathsf{H})$ ([23,
Equation 10.1.2]). Finally, using equation (3.4) we can show that the
functional associated to $\rho$ is of the form $\Omega_{\gamma,\omega}$.
Indeed, we have

$\begin{split}\langle
a,\Omega_{\gamma,\omega}\rangle=\langle(\Theta^{l}(a)\otimes\textnormal{id})\gamma,\omega\rangle&=\sum_{i,j,k,l=1}^{\infty}\alpha_{1,(i,j)}\langle(\Theta^{l}(a)\otimes\textnormal{id})\gamma,\beta_{i,k}\otimes
e_{l,j}\rangle\alpha^{\prime}_{(k,l),1}\\\
&=\sum_{i,j,k,l=1}^{\infty}\alpha_{1,(i,j)}\langle\Theta^{l}(a)(\gamma_{j,l}),\beta_{i,k}\rangle\alpha^{\prime}_{(k,l),1}\end{split}$

for any $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. ∎

###  3.3. Viewing multipliers as bimodule maps

In this section we provide another way of looking at CB multipliers and the
associated weak∗-topology which will be useful in later considerations.

Let us first introduce some terminology. As usual, let
$\operatorname{CB}^{\sigma}(\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))$
be the space of normal CB maps on
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$. Observe
$\operatorname{CB}^{\sigma}(\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))\cong\operatorname{CB}(\mathcal{K}(\operatorname{L}^{2}(\mathbb{G})),\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))$.
Indeed, any normal CB map on
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$ can be restricted to
$\mathcal{K}(\operatorname{L}^{2}(\mathbb{G}))$ without changing its norm, and
conversely, since
$\mathcal{K}(\operatorname{L}^{2}(\mathbb{G}))^{**}\simeq\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$,
any CB map
$\mathcal{K}(\operatorname{L}^{2}(\mathbb{G}))\rightarrow\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$
uniquely extends to a normal CB map on
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$.
$\operatorname{CB}^{\sigma}(\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))$
is an operator space which is equipped with the weak∗-topology given by the
predual
$\mathcal{K}(\operatorname{L}^{2}(\mathbb{G}))\widehat{\otimes}\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))_{*}$.
Via left and right multiplication,
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$ becomes a
$\operatorname{L}^{\infty}(\mathbb{G})^{\prime}$-bimodule, hence we can
consider normal CB bimodule maps on
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$. We can also look at
those maps which leave
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\subseteq\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$
globally invariant. We will denote the set of CB normal
$\operatorname{L}^{\infty}(\mathbb{G})^{\prime}$-bimodule maps on
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$ which leave
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ globally invariant by
$\prescript{}{\operatorname{L}^{\infty}(\mathbb{G})^{\prime}}{\operatorname{CB}}_{\operatorname{L}^{\infty}(\mathbb{G})^{\prime}}^{\sigma,\operatorname{L}^{\infty}(\widehat{\mathbb{G}})}(\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))$.
One easily checks that this space is weak∗-closed in
$\operatorname{CB}^{\sigma}(\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))$,
hence it naturally inherits an operator space structure and a weak∗-topology.

According to [35, Theorem 4.5] (and [18, Proposition 3.3] for the left
version), for any
$a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ there exists a
unique map
$\Phi(a)\in\prescript{}{\operatorname{L}^{\infty}(\mathbb{G})^{\prime}}{\operatorname{CB}}_{\operatorname{L}^{\infty}(\mathbb{G})^{\prime}}^{\sigma,\operatorname{L}^{\infty}(\widehat{\mathbb{G}})}(\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))$
which extends
$\Theta^{l}(a)\in\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$.
This map satisfies

$\mathbbm{1}\otimes\Phi(a)(x)={\mathrm{W}}^{\widehat{\mathbb{G}}}\bigl{(}(\Theta^{l}(a)\otimes\textnormal{id})({\mathrm{W}}^{\widehat{\mathbb{G}}*}(\mathbbm{1}\otimes
x){\mathrm{W}}^{\widehat{\mathbb{G}}})\bigr{)}{\mathrm{W}}^{\widehat{\mathbb{G}}*}\qquad(x\in\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))).$

Furthermore, the resulting map

(3.5) $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\ni
a\mapsto\Phi(a)\in\prescript{}{\operatorname{L}^{\infty}(\mathbb{G})^{\prime}}{\operatorname{CB}}_{\operatorname{L}^{\infty}(\mathbb{G})^{\prime}}^{\sigma,\operatorname{L}^{\infty}(\widehat{\mathbb{G}})}(\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})))$

is a completely isometric isomorphism which is additionally a
weak∗-homeomorphism ([18, Theorem 6.2]).

When $a$ arises from an element of the Fourier algebra, $\Phi(a)$ takes a
special form.

######  Lemma 3.10.

For $\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ let
$a=\widehat{\lambda}(\omega)\in\operatorname{A}(\mathbb{G})$, so that
$\Theta^{l}(a)=(\omega\otimes\textnormal{id})\widehat{\Delta}$. The associated
map $\Phi(a)$ is

$\Phi(a)\colon\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))\ni
x\mapsto(\omega\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}*}(\mathbbm{1}\otimes
x)\mathrm{W}^{\widehat{\mathbb{G}}})\in\operatorname{B}(\operatorname{L}^{2}(\mathbb{G})).$

A similar formula holds for an arbitrary element of the Fourier-Stieltjes
algebra, see the discussion after Proposition 4.1 in [18].

###### Proof.

As before, the left centraliser associated with $a$ is simply left
multiplication by $\omega$, and so $\Theta^{l}(a)$ has the given form. Then
for $x\in\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$, using that
$(\widehat{\Delta}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})=\mathrm{W}^{\widehat{\mathbb{G}}}_{13}\mathrm{W}^{\widehat{\mathbb{G}}}_{23}$,

$\displaystyle\mathbbm{1}\otimes\Phi(a)(x)$
$\displaystyle=\mathrm{W}^{\widehat{\mathbb{G}}}\bigl{(}((\omega\otimes\textnormal{id})\widehat{\Delta}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}*}(\mathbbm{1}\otimes
x)\mathrm{W}^{\widehat{\mathbb{G}}})\bigr{)}\mathrm{W}^{\widehat{\mathbb{G}}*}$
$\displaystyle=\mathrm{W}^{\widehat{\mathbb{G}}}\bigl{(}(\omega\otimes\textnormal{id}\otimes\textnormal{id})(\mathrm{W}_{23}^{\widehat{\mathbb{G}}*}\mathrm{W}_{13}^{\widehat{\mathbb{G}}*}(\mathbbm{1}\otimes\mathbbm{1}\otimes
x)\mathrm{W}^{\widehat{\mathbb{G}}}_{13}\mathrm{W}^{\widehat{\mathbb{G}}}_{23})\bigr{)}\mathrm{W}^{\widehat{\mathbb{G}}*}$
$\displaystyle=\mathrm{W}^{\widehat{\mathbb{G}}}\mathrm{W}^{\widehat{\mathbb{G}}*}(\omega\otimes\textnormal{id}\otimes\textnormal{id})(\mathrm{W}_{13}^{\widehat{\mathbb{G}}*}(\mathbbm{1}\otimes\mathbbm{1}\otimes
x)\mathrm{W}_{13}^{\widehat{\mathbb{G}}})\mathrm{W}^{\widehat{\mathbb{G}}}\mathrm{W}^{\widehat{\mathbb{G}}*}$
$\displaystyle=\mathbbm{1}\otimes(\omega\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}*}(\mathbbm{1}\otimes
x)\mathrm{W}^{\widehat{\mathbb{G}}}),$

and so $\Phi(a)$ has indeed the claimed form. ∎

##  4\. The approximation property

We define the approximation property for a locally compact quantum group
$\mathbb{G}$ (abbreviated AP) in a way completely analogous to the definition
of AP for locally compact groups by Haagerup and Kraus in [32]. Recall that we
fix the predual $Q^{l}(\operatorname{A}(\mathbb{G}))$ of
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$, and we always refer
to the corresponding weak∗-topology on
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$.

######  Definition 4.1.

We say that a locally compact quantum group $\mathbb{G}$ has the
_approximation property (AP)_ if there is a net $(a_{i})_{i\in I}$ in
$\operatorname{A}(\mathbb{G})$ which converges to $\mathbbm{1}$ in the
weak∗-topology of $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$.

######  Remark 4.2.

  * •

We could call the above property “left AP” and introduce also a right variant
of AP. However, in Proposition 4.14 we will show that these properties are
equivalent, so that there is no need to distinguish between them.

  * •

A variant of AP was considered by Kraus-Ruan [42] (for Kac algebras) and Crann
in [14]. Their property is a priori stronger, but in Theorem 4.4 we show that
this variant is in fact equivalent to Definition 4.1. This proves a conjecture
by Kraus-Ruan [42, Remark 4.2].

Let us list some examples and counter-examples:

  * •

In Section 5 we show weak amenability implies AP, therefore all compact
quantum groups and the discrete quantum groups
$\widehat{O_{F}^{+}},\widehat{U_{F}^{+}}$ have AP ([26, 22]). Furthermore, the
locally compact quantum group $\operatorname{SU}_{q}(1,1)_{ext}$ has AP, see
[11].

  * •

Permanence properties of AP with respect to quantum subgroups (Theorem 7.1),
direct products (Proposition 7.22), free products (Theorem 7.8) and the
Drinfeld double construction (Theorem 7.16) allow us to construct examples
with and without AP. For instance, the Drinfeld double of
$\operatorname{SL}(3,\mathbb{R})$, or of any classical locally compact group
without AP, see [48], gives rise to non-classical quantum groups without AP.
Similarly,
$\operatorname{SL}(3,\mathbb{Z})\star\widehat{\operatorname{SU}_{q}(2)}$ is a
quantum group without AP, as AP passes to quantum subgroups.

  * •

Examples of quantum groups with AP similarly arise. As AP passes to free
products,
$\operatorname{SL}(2,\mathbb{Z})\star\widehat{\operatorname{SU}_{q}(2)}$ is a
quantum group with AP. Similarly,
$D(\operatorname{SL}(2,\mathbb{R}))^{\wedge}$, the dual of the quantum double
of $\operatorname{SL}(2,\mathbb{R})$, has AP (Theorem 7.16).

  * •

The paper [24] by Fima, Mukherjee and Patri makes a careful study of compact
bicrossed products, and in particular, [24, Theorem 6.7] provides estimates on
the Cowling–Haagerup constant (see Definition 5.1) of the resulting discrete
quantum groups. Using this, if one starts with a discrete group $\Gamma$ with
AP and with $\Lambda_{cb}(\Gamma)\geq n$ (see for example [10, Theorem 12.3.8]
and references therein) then one can construct a non-classical discrete
quantum group with AP but with Cowling–Haagerup constant $\geq n$. Then by
taking infinite products we obtain a discrete quantum group which has AP but
which is not weakly amenable.

  * •

We leave as a question whether the Drinfeld double of
$\operatorname{SU}_{q}(3)$ might be an example of a locally compact quantum
group which does not have AP; for more see Remark 7.15.

###  4.1. Equivalent characterisations

We check first that the approximation property is preserved under taking the
_commutant_ quantum group $\mathbb{G}^{\prime}$, or the _opposite_ quantum
group $\mathbb{G}^{\operatorname{op}}$, for definitions see [47, Section 4].

######  Proposition 4.3.

The following conditions are equivalent:

  1. (1)

$\mathbb{G}$ has AP,

  2. (2)

$\mathbb{G}^{\prime}$ has AP.

  3. (3)

$\mathbb{G}^{\operatorname{op}}$ has AP,

###### Proof.

Assume that $\mathbb{G}$ has AP, i.e. there is a net $(\omega_{i})_{i\in I}$
in $\operatorname{L}^{1}(\widehat{\mathbb{G}})$ such that
$\lambda_{\widehat{\mathbb{G}}}(\omega_{i})\xrightarrow[i\in I]{}\mathbbm{1}$
weak∗ in $\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))$.

First we will prove that $\mathbb{G}^{\prime}$ has AP. Recall that
$\widehat{\mathbb{G}^{\prime}}=\widehat{\mathbb{G}}^{\operatorname{op}}$ ([47,
Proposition 4.2]). Using Proposition 3.9, choose an arbitrary functional
$\Omega_{y,\nu}\in Q^{l}(\operatorname{A}(\mathbb{G}^{\prime}))$ where
$\mathsf{H}$ is a separable Hilbert space,
$y\in\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathcal{K}(\mathsf{H})$ and
$\nu\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}$.
Pick some selfadjoint antiunitary $\mathcal{J}$ on $\mathsf{H}$ (for example,
pick an orthonormal basis and let $\mathcal{J}$ be coordinate-wise complex
conjugation) and define
$j:\mathcal{K}(\mathsf{H})\rightarrow\mathcal{K}(\mathsf{H})$ by
$j(x)=\mathcal{J}x^{*}\mathcal{J}$, an $\star$-antihomomorphism with
$j^{2}=\textnormal{id}$. Then $\widehat{R}\otimes j$ is a well-defined bounded
map on the spatial tensor product
$\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathcal{K}(\mathsf{H})$, and
$\widehat{R}_{*}\otimes j^{*}$ is well-defined on
$\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}$.
Indeed, $\widehat{R}\otimes j$ acts via $(\widehat{R}\otimes
j)(X)=(J_{\varphi}\otimes\mathcal{J})X^{*}(J_{\varphi}\otimes\mathcal{J})$ for
$X\in\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathcal{K}(\mathsf{H})$ and
then we can define $\widehat{R}_{*}\otimes j^{*}$ as the restriction of
$(\widehat{R}\otimes j)^{*}$ to
$\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}$:
$(\widehat{R}\otimes j)^{*}$ preserves this subspace as $(\widehat{R}\otimes
j)^{*}(\omega_{\xi\otimes\eta})=\omega_{J_{\varphi}\xi\otimes\mathcal{J}\eta}$
for $\xi\in\operatorname{L}^{2}(\mathbb{G}),\eta\in\mathsf{H}$. Since
$\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\mathcal{K}(\mathsf{H})^{*}\subseteq(\mathrm{C}_{0}(\widehat{\mathbb{G}})\otimes\mathcal{K}(\mathsf{H}))^{*}$
is closed, the claim follows. Furthermore, both these maps are isometric
bijections.

Set $x=(\widehat{R}\otimes j)(y),\omega=(\widehat{R}_{*}\otimes j^{*})(\nu)$.
Then using Lemma 3.10

$\displaystyle\langle\lambda_{\widehat{\mathbb{G}}}(\omega_{i}),\Omega_{x,\omega}\rangle$
$\displaystyle=\langle((\omega_{i}\otimes\textnormal{id})\Delta_{\widehat{\mathbb{G}}}\otimes\textnormal{id})(\widehat{R}\otimes
j)(y),(\widehat{R}_{*}\otimes j^{*})(\nu)\rangle$
$\displaystyle=\langle(\widehat{R}(\omega_{i}\otimes\textnormal{id})\Delta_{\widehat{\mathbb{G}}}\widehat{R}\otimes\textnormal{id})(y),\nu\rangle=\langle((\textnormal{id}\otimes\widehat{R}_{*}(\omega_{i}))\Delta_{\widehat{\mathbb{G}}}\otimes\textnormal{id})(y),\nu\rangle$
$\displaystyle=\langle((\widehat{R}_{*}(\omega_{i})\otimes\textnormal{id})\Delta_{\widehat{\mathbb{G}}^{\operatorname{op}}}\otimes\textnormal{id})(y),\nu\rangle=\langle\lambda_{\widehat{\mathbb{G}}^{\operatorname{op}}}(\widehat{R}_{*}(\omega_{i})),\Omega_{y,\nu}\rangle$

and since $\lambda_{\widehat{\mathbb{G}}}(\omega_{i})\xrightarrow[i\in
I]{}\mathbbm{1}$ weak∗, we conclude
$\lambda_{\widehat{\mathbb{G}}^{\operatorname{op}}}(\widehat{R}_{*}(\omega_{i}))\xrightarrow[i\in
I]{}\mathbbm{1}$ weak∗. This shows that $\mathbb{G}^{\prime}$ has AP.

Next we prove that $\mathbb{G}^{\operatorname{op}}$ has AP. By [47,
Proposition 4.2] we have
$\widehat{\mathbb{G}^{\operatorname{op}}}=\widehat{\mathbb{G}}^{\prime}$.
Write $R^{\sim}$ for the extension of the unitary antipode on $\mathbb{G}$,
$\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))\ni x\mapsto
J_{\widehat{\varphi}}x^{*}J_{\widehat{\varphi}}\in\operatorname{B}(\operatorname{L}^{2}(\mathbb{G}))$,
so that $\tilde{\omega}_{i}=\omega_{i}\circ
R^{\sim}\in\operatorname{L}^{1}({\widehat{\mathbb{G}}}^{\prime})$. We claim
that the net $(\lambda_{\widehat{\mathbb{G}}^{\prime}}(\omega_{i}\circ
R^{\sim}))_{i\in I}$ converges weak∗ to $\mathbbm{1}$ in
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}^{\operatorname{op}}))$.
Take
$z\in\mathrm{C}_{0}(\widehat{\mathbb{G}}^{\prime})\otimes\mathcal{K}(\mathsf{H}),\theta\in\operatorname{L}^{1}(\widehat{\mathbb{G}}^{\prime})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}$.
Recall that
$\mathrm{C}_{0}(\widehat{\mathbb{G}}^{\prime})=J_{\widehat{\varphi}}\mathrm{C}_{0}(\widehat{\mathbb{G}})J_{\widehat{\varphi}}$
and

$\begin{split}\Delta_{\widehat{\mathbb{G}}^{\prime}}\colon\operatorname{L}^{\infty}(\widehat{\mathbb{G}})^{\prime}\ni
x\mapsto&(J_{\widehat{\varphi}}\otimes
J_{\widehat{\varphi}})\Delta_{\widehat{\mathbb{G}}}(J_{\widehat{\varphi}}xJ_{\widehat{\varphi}})(J_{\widehat{\varphi}}\otimes
J_{\widehat{\varphi}})\\\ =&(R^{\sim}\otimes
R^{\sim})\Delta_{\widehat{\mathbb{G}}}(R^{\sim}(x))\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}})^{\prime}\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})^{\prime}.\end{split}$

Using this, we obtain

$\begin{split}&\quad\;\langle\lambda_{\widehat{\mathbb{G}}^{\prime}}(\omega_{i}\circ
R^{\sim}),\Omega_{z,\theta}\rangle=\langle(\Theta^{l}(\lambda_{\widehat{\mathbb{G}}^{\prime}}(\omega_{i}\circ
R^{\sim}))\otimes\textnormal{id})z,\theta\rangle=\langle((\omega_{i}\circ
R^{\sim}\otimes\textnormal{id})\Delta_{\widehat{\mathbb{G}}^{\prime}}\otimes\textnormal{id})z,\theta\rangle\\\
&=\langle(R^{\sim}\otimes
j)\bigl{(}(\omega_{i}\otimes\textnormal{id})\Delta_{\widehat{\mathbb{G}}}\otimes\textnormal{id}\bigr{)}(R^{\sim}\otimes
j)z,\theta\rangle=\langle\lambda_{\widehat{\mathbb{G}}}(\omega_{i}),\Omega_{(R^{\sim}\otimes
j)z,\theta\circ(R^{\sim}\otimes j)}\rangle\\\ &\xrightarrow[i\in
I]{}\langle\mathbbm{1},\Omega_{(R^{\sim}\otimes
j)z,\theta\circ(R^{\sim}\otimes j)}\rangle=\langle
z,\theta\rangle=\langle\mathbbm{1},\Omega_{z,\theta}\rangle\end{split}$

and thus $\mathbb{G}^{\operatorname{op}}$ has AP. The converse implications
follow since $(\mathbb{G}^{\prime})^{\prime}=\mathbb{G}$ and
$(\mathbb{G}^{\operatorname{op}})^{\operatorname{op}}=\mathbb{G}$. ∎

The next result shows that the version of the approximation property
considered in [42] and [14] is equivalent to AP as defined in Definition 4.1.
Both [42, Definition 4.1] and [14, Page 1728] take condition (2) of the
following theorem as their definition of AP.

######  Theorem 4.4.

The following conditions are equivalent:

  1. (1)

$\mathbb{G}$ has AP,

  2. (2)

there is a net $(a_{i})_{i\in I}$ in the Fourier algebra
$\operatorname{A}(\mathbb{G})$, such that the corresponding net
$(\Theta^{l}(a_{i}))_{i\in I}$ converges to the identity in the stable point-
weak∗-topology of
$\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$.

In order to prove Theorem 4.4 we need to establish some preliminary results.
Recall that $\operatorname{L}^{\infty}(\mathbb{G})$ is a right
$\operatorname{L}^{1}(\mathbb{G})$-module via
$x\star\omega=(\omega\otimes\textnormal{id})\Delta(x)$ for
$x\in\operatorname{L}^{\infty}(\mathbb{G}),\omega\in\operatorname{L}^{1}(\mathbb{G})$.

######  Proposition 4.5.

Let $a\in\operatorname{L}^{\infty}(\mathbb{G})$ and
$\omega\in\operatorname{L}^{1}(\mathbb{G})$.

  1. (1)

If $a\in\operatorname{A}(\mathbb{G})$ then
$a\star\omega\in\operatorname{A}(\mathbb{G})$.

  2. (2)

If $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ then
$a\star\omega\in\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))$ with
$\|a\star\omega\|_{cb}\leq\|a\|_{cb}\|\omega\|$ and

$\Theta^{l}(a\star\omega)(\widehat{x})=(\omega\otimes\textnormal{id})\big{(}(\textnormal{id}\otimes\Theta^{l}(a))((\mathbbm{1}\otimes\widehat{x}){\mathrm{W}}^{\mathbb{G}*})\mathrm{W}^{\mathbb{G}}\big{)}\qquad(\widehat{x}\in
L^{\infty}(\widehat{\mathbb{G}})).$

###### Proof.

(1) Write $a=\widehat{\lambda}(\widehat{\omega})$ for
$\widehat{\omega}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$. Then

$\begin{split}&\quad\;a\star\omega=(\omega\otimes\textnormal{id})\Delta\bigl{(}(\widehat{\omega}\otimes\textnormal{id}){\mathrm{W}}^{\widehat{\mathbb{G}}}\bigr{)}=(\widehat{\omega}\otimes\omega\otimes\textnormal{id})\bigl{(}{\mathrm{W}}^{\widehat{\mathbb{G}}}_{13}{\mathrm{W}}^{\widehat{\mathbb{G}}}_{12}\bigr{)}\\\
&=(\widehat{\omega}\otimes\textnormal{id})\bigl{(}{\mathrm{W}}^{\widehat{\mathbb{G}}}((\textnormal{id}\otimes\omega)\mathrm{W}^{\widehat{\mathbb{G}}}\otimes\mathbbm{1})\bigr{)}=\widehat{\lambda}\bigl{(}\widehat{\omega}\bigl{(}\cdot\;(\textnormal{id}\otimes\omega)\mathrm{W}^{\widehat{\mathbb{G}}}\bigr{)}\bigr{)}\in\operatorname{A}(\mathbb{G})\end{split}$

as required.
(2) As
$\mathrm{W}^{\mathbb{G}}\in\operatorname{L}^{\infty}(\mathbb{G})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$,
there is a well-defined linear map $T$ on
$\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ given by

$T(\widehat{x})=(\omega\otimes\textnormal{id})\big{(}(\textnormal{id}\otimes\Theta^{l}(a))((\mathbbm{1}\otimes\widehat{x})\mathrm{W}^{\mathbb{G}*})\mathrm{W}^{\mathbb{G}}\big{)}\qquad(\widehat{x}\in
L^{\infty}(\widehat{\mathbb{G}})).$

Clearly $T$ is completely bounded with $\|T\|_{cb}\leq\|a\|_{cb}\|\omega\|$
and weak∗-continuous. We first show that $T$ is the adjoint of a centraliser,
equivalently, that
$\widehat{\Delta}T=(T\otimes\textnormal{id})\widehat{\Delta}$. If
$\widehat{x}\in L^{\infty}(\widehat{\mathbb{G}})$ then using
$\widehat{\Delta}\Theta^{l}(a)=(\Theta^{l}(a)\otimes\textnormal{id})\widehat{\Delta}$
gives

$\displaystyle\widehat{\Delta}T(\widehat{x})$
$\displaystyle=(\omega\otimes\textnormal{id}\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\widehat{\Delta}\Theta^{l}(a))((\mathbbm{1}\otimes\widehat{x})\mathrm{W}^{\mathbb{G}*})\mathrm{W}^{\mathbb{G}}_{13}\mathrm{W}^{\mathbb{G}}_{12}\bigr{)}$
$\displaystyle=(\omega\otimes\textnormal{id}\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\Theta^{l}(a)\otimes\textnormal{id})\bigl{(}(\mathbbm{1}\otimes\widehat{\Delta}(\widehat{x}))\mathrm{W}^{\mathbb{G}*}_{12}\mathrm{W}^{\mathbb{G}*}_{13}\bigr{)}\mathrm{W}^{\mathbb{G}}_{13}\mathrm{W}^{\mathbb{G}}_{12}\bigr{)}$
$\displaystyle=(\omega\otimes\textnormal{id}\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\Theta^{l}(a)\otimes\textnormal{id})\bigl{(}(\mathbbm{1}\otimes\widehat{\Delta}(\widehat{x}))\mathrm{W}^{\mathbb{G}*}_{12}\bigr{)}\mathrm{W}^{\mathbb{G}}_{12}\bigr{)}$
$\displaystyle=(T\otimes\textnormal{id})\widehat{\Delta}(\widehat{x}).$

Consequently $T$ is the adjoint of a centraliser, and so there exists
$b\in\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))$ with
$T=\Theta^{l}(b)$. Then
$b\widehat{\lambda}(\widehat{\omega})=\widehat{\lambda}(\widehat{\omega}\circ
T)$ for each $\widehat{\omega}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$,
equivalently, $(\mathbbm{1}\otimes
b)\mathrm{W}^{\widehat{\mathbb{G}}}=(T\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})$.
In other words, we have

$(b\otimes\mathbbm{1})\mathrm{W}^{\mathbb{G}*}=(\textnormal{id}\otimes
T)(\mathrm{W}^{\mathbb{G}*})=(\textnormal{id}\otimes\omega\otimes\textnormal{id})\big{(}(\textnormal{id}\otimes\textnormal{id}\otimes\Theta^{l}(a))(\mathrm{W}^{\mathbb{G}*}_{13}\mathrm{W}^{\mathbb{G}*}_{23})\mathrm{W}^{\mathbb{G}}_{23}\big{)},$

or equivalently

$\displaystyle b\otimes\mathbbm{1}$
$\displaystyle=(\textnormal{id}\otimes\omega\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\textnormal{id}\otimes\Theta^{l}(a))(\mathrm{W}^{\mathbb{G}*}_{13}\mathrm{W}^{\mathbb{G}*}_{23})\mathrm{W}^{\mathbb{G}}_{23}\mathrm{W}^{\mathbb{G}}_{13}\bigr{)}$
(4.1)
$\displaystyle=(\omega\otimes\textnormal{id}\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\textnormal{id}\otimes\Theta^{l}(a))(\mathrm{W}^{\mathbb{G}*}_{23}\mathrm{W}^{\mathbb{G}*}_{13})\mathrm{W}^{\mathbb{G}}_{13}\mathrm{W}^{\mathbb{G}}_{23}\bigr{)}.$

Now, $(\mathbbm{1}\otimes
a)\mathrm{W}^{\widehat{\mathbb{G}}}=(\Theta^{l}(a)\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})$
so
$a\otimes\mathbbm{1}=(\textnormal{id}\otimes\Theta^{l}(a))(\mathrm{W}^{\mathbb{G}*})\mathrm{W}^{\mathbb{G}}$
and hence

$\displaystyle a\star\omega\otimes\mathbbm{1}$
$\displaystyle=(\omega\otimes\textnormal{id})\Delta(a)\otimes\mathbbm{1}=((\omega\otimes\textnormal{id})\Delta\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\Theta^{l}(a))(\mathrm{W}^{\mathbb{G}*})\mathrm{W}^{\mathbb{G}}\bigr{)}$
(4.2)
$\displaystyle=(\omega\otimes\textnormal{id}\otimes\textnormal{id})\bigl{(}(\textnormal{id}\otimes\textnormal{id}\otimes\Theta^{l}(a))(\mathrm{W}^{\mathbb{G}*}_{23}\mathrm{W}^{\mathbb{G}*}_{13})\mathrm{W}^{\mathbb{G}}_{13}\mathrm{W}^{\mathbb{G}}_{23}\bigr{)}.$

As (4.1) and (4.2) agree, we conclude that $b=a\star\omega$. Thus
$a\star\omega\in\operatorname{M}_{cb}^{l}(\operatorname{A}(\mathbb{G}))$ with
$\Theta^{l}(a\star\omega)=T$ as required. ∎

######  Lemma 4.6.

For any locally compact quantum group $\mathbb{G}$ and
$x\in\operatorname{L}^{\infty}(\mathbb{G}),a\in\mathrm{C}_{0}(\mathbb{G}),\omega\in\operatorname{L}^{1}(\mathbb{G})$
we have $\omega\star(xa)\in\mathrm{C}_{0}(\mathbb{G})$.

###### Proof.

As $\operatorname{L}^{1}(\mathbb{G})$ is a closed
$\mathrm{C}_{0}(\mathbb{G})$-submodule of $\mathrm{C}_{0}(\mathbb{G})^{*}$, by
[61, Lemma 2.1] we know that $\omega=b\omega_{1}$ for some
$b\in\mathrm{C}_{0}(\mathbb{G})$ and
$\omega_{1}\in\operatorname{L}^{1}(\mathbb{G})$. Then

$\omega\star(xa)=(\textnormal{id}\otimes\omega)\Delta(xa)=(\textnormal{id}\otimes\omega_{1})\big{(}\Delta(x)\Delta(a)(\mathbbm{1}\otimes
b)\big{)}.$

As $a,b\in\mathrm{C}_{0}(\mathbb{G})$ we know that
$\Delta(a)(\mathbbm{1}\otimes
b)\in\mathrm{C}_{0}(\mathbb{G})\otimes\mathrm{C}_{0}(\mathbb{G})$, the minimal
$C^{*}$-algebraic tensor product. By continuity, it hence suffices to prove
that

$(\textnormal{id}\otimes\omega_{1})\big{(}\Delta(x)(c\otimes
d)\big{)}\in\mathrm{C}_{0}(\mathbb{G})$

for $c,d\in\mathrm{C}_{0}(\mathbb{G})$. However, this equals
$\bigl{(}(\textnormal{id}\otimes d\omega_{1})\Delta(x)\bigr{)}c$ and by [61,
Theorem 2.4] we know that $(\textnormal{id}\otimes
d\omega_{1})\Delta(x)\in\operatorname{M}(\mathrm{C}_{0}(\mathbb{G}))$, and so
the result follows. ∎

Next we introduce certain functionals in $Q^{l}(\operatorname{A}(\mathbb{G}))$
in analogy to [32, Proposition 1.3]. For a Hilbert space $\mathsf{H}$,
$x\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{B}(\mathsf{H}),\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}$
and $f\in\operatorname{L}^{1}(\mathbb{G})$ define

(4.3)
$\Omega_{x,\omega,f}\colon\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\ni
a\mapsto\langle(\Theta^{l}(a\star
f)\otimes\textnormal{id})x,\omega\rangle\in\mathbb{C}.$

Note that $\Omega_{x,\omega,f}$ is well-defined and bounded by Proposition
4.5.

######  Proposition 4.7.

The linear functional $\Omega_{x,\omega,f}$ is weak∗-continuous, hence is
contained in $Q^{l}(\operatorname{A}(\mathbb{G}))$.

###### Proof.

Clearly $\Omega_{x,\omega,f}$ is bounded with
$\|\Omega_{x,\omega,f}\|\leq\|x\|\|f\|\|\omega\|$, so it suffices to prove the
result when $\omega$ is in the algebraic tensor product of
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$ with
$\operatorname{B}(\mathsf{H})_{*}$, and hence by linearity, we may suppose
that $\omega=\widehat{\omega}\otimes u$. Then

$\Omega_{x,\omega,f}(a)=\langle(\Theta^{l}(a\star
f)\otimes\textnormal{id})(x),\widehat{\omega}\otimes
u\rangle=\langle\Theta^{l}(a\star f)((\textnormal{id}\otimes
u)(x)),\widehat{\omega}\rangle\qquad(a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))).$

Thus, it suffices to show that for
$\widehat{\omega}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$ and
$\widehat{x}\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$

$\mu\colon\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\ni
a\mapsto\langle\Theta^{l}(a\star
f)(\widehat{x}),\widehat{\omega}\rangle\in\mathbb{C}$

is weak∗-continuous. By Proposition 4.5, given the form of $\Theta^{l}(a\star
f)$,

$\mu(a)=\langle(\textnormal{id}\otimes\Theta^{l}(a))((\mathbbm{1}\otimes\widehat{x})\mathrm{W}^{\mathbb{G}*})\mathrm{W}^{\mathbb{G}},f\otimes\widehat{\omega}\rangle\qquad(a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))).$

As
$\mathrm{W}^{\mathbb{G}}\in\operatorname{L}^{\infty}(\mathbb{G})\bar{\otimes}\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$,
also
$\mathrm{W}^{\mathbb{G}}(f\otimes\widehat{\omega})\in\operatorname{L}^{1}(\mathbb{G})\widehat{\otimes}\operatorname{L}^{1}(\widehat{\mathbb{G}})$,
so again by approximation, it suffices to show that for
$f^{\prime}\in\operatorname{L}^{1}(\mathbb{G}),\widehat{\omega}^{\prime}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$
the map

$\mu^{\prime}\colon\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))\ni
a\mapsto\langle(\textnormal{id}\otimes\Theta^{l}(a))((\mathbbm{1}\otimes\widehat{x})\mathrm{W}^{\mathbb{G}*}),f^{\prime}\otimes\widehat{\omega}^{\prime}\rangle=\langle\Theta^{l}(a)(\widehat{x}\widehat{y}\big{)},\widehat{\omega}^{\prime}\rangle\in\mathbb{C}$

is weak∗-continuous, where
$\widehat{y}=(f^{\prime}\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$.

By linear density of products, compare Lemma 5.3, it suffices to consider the
case when
$\widehat{\omega}^{\prime}=\widehat{\omega}_{1}\star\widehat{\omega}_{2}$ for
$\widehat{\omega}_{1},\widehat{\omega}_{2}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$.
As
$\Theta^{l}(a)_{*}(\widehat{\omega}_{1}\star\widehat{\omega}_{2})=\Theta^{l}(a)_{*}(\widehat{\omega}_{1})\star\widehat{\omega}_{2}$
by the left centraliser property, we see that

$\mu^{\prime}(a)=\langle\Theta^{l}(a)(\widehat{x}\widehat{y}),\widehat{\omega}_{1}\star\widehat{\omega}_{2}\rangle=\langle\widehat{\omega}_{2}\star(\widehat{x}\widehat{y}),\Theta^{l}(a)_{*}(\widehat{\omega}_{1})\rangle\qquad(a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))).$

As $\widehat{y}\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$, by Lemma 4.6 applied
to $\widehat{\mathbb{G}}$, we know that
$\widehat{\omega}_{2}\star(\widehat{x}\widehat{y})\in\mathrm{C}_{0}(\widehat{\mathbb{G}})$.
Thus $\mu^{\prime}\in Q^{l}(\operatorname{A}(\mathbb{G}))$ by Proposition 3.9.
∎

We are now ready to prove Theorem 4.4.

###### Proof of Theorem 4.4.

$(2)\Rightarrow(1)$ follows directly from the characterisation of functionals
in the predual $Q^{l}(\operatorname{A}(\mathbb{G}))$ of
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ in Proposition 3.9.

$(1)\Rightarrow(2)$ Assume that $(a_{i})_{i\in I}$ is a net in
$\operatorname{A}(\mathbb{G})$ which converges weak∗ to $\mathbbm{1}$ in
$\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. Pick a state
$f\in\operatorname{L}^{1}(\mathbb{G})$, and for each $i\in I$ set
$b_{i}=a_{i}\star f$. By Proposition 4.5 we have
$b_{i}\in\operatorname{A}(\mathbb{G})$. We now show that
$(\Theta^{l}(b_{i}))_{i\in I}$ converges to the identity in the stable point-
weak∗-topology.

Given a separable Hilbert space $\mathsf{H}$,
$x\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}})\bar{\otimes}\operatorname{B}(\mathsf{H})$,
and
$\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\widehat{\otimes}\operatorname{B}(\mathsf{H})_{*}$,
using Proposition 4.7 we see that

$\begin{split}&\langle(\Theta^{l}(b_{i})\otimes\textnormal{id})x,\omega\rangle=\langle(\Theta^{l}(a_{i}\star
f)\otimes\textnormal{id})x,\omega\rangle=\langle
a_{i},\Omega_{x,\omega,f}\rangle\\\ \xrightarrow[i\in
I]{}&\langle\mathbbm{1},\Omega_{x,\omega,f}\rangle=\langle(\Theta^{l}(\mathbbm{1}\star
f)\otimes\textnormal{id})(x),\omega\rangle,\end{split}$

Since $f$ is a state, we have $\mathbbm{1}\star
f=(f\otimes\textnormal{id})\Delta(\mathbbm{1})=\mathbbm{1}$, hence
$\Theta^{l}(\mathbbm{1}\star f)=\textnormal{id}$, and so
$(\Theta^{l}(b_{i})\otimes\textnormal{id})(x)\xrightarrow[i\in I]{}x$ weak∗ as
required. ∎

###  4.2. Further general properties

Let us start with an auxiliary technical result. Recall that for a von Neumann
algebra $\operatorname{M}$ and a linear map
$T\colon\operatorname{M}\rightarrow\operatorname{M}$ we define
$T^{\dagger}\colon\operatorname{M}\ni x\mapsto
T(x^{*})^{*}\in\operatorname{M}$.

######  Lemma 4.8.

If $T\in\operatorname{CB}^{\sigma}(\operatorname{M})$, then
$T^{\dagger}\in\operatorname{CB}^{\sigma}(\operatorname{M})$ and
$\|T^{\dagger}\|_{cb}=\|T\|_{cb}$. If
$\operatorname{M}=\operatorname{L}^{\infty}(\mathbb{G})$ for a locally compact
quantum group $\mathbb{G}$ then $R\circ T\circ
R\in\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\mathbb{G}))$ and
$\|R\circ T\circ R\|_{cb}=\|T\|_{cb}$. Both operations $T\mapsto
T^{\dagger},T\mapsto R\circ T\circ R$ are continuous with respect to the
stable point-weak∗-topology.

###### Proof.

Let $\operatorname{M}\subseteq\operatorname{B}(\mathsf{H})$. Using the normal
version of Wittstock’s Theorem (compare with the start of the proof of [33,
Theorem 2.5] for example), we can find a Hilbert space $\mathsf{K}$, bounded
linear maps $V,W\colon\mathsf{H}\rightarrow\mathsf{K}$ and a normal
representation
$\pi\colon\operatorname{M}\rightarrow\operatorname{B}(\mathsf{K})$ such that
$T=W^{*}\pi(\cdot)V$ and $\|T\|_{cb}=\|V\|\|W\|$. Now proving the claimed
properties of $T^{\dagger}$ is elementary.

Assume next that $\operatorname{M}=\operatorname{L}^{\infty}(\mathbb{G})$. If
$T\in\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\mathbb{G}))$, then
clearly $R\circ T\circ R$ is normal since the unitary antipode $R$ is normal.
Using again Wittstock’s Theorem, write $T^{\dagger}=W^{*}\pi(\cdot)V$ and
choose an antiunitary $\mathcal{J}$ on $\mathsf{H}$ which satisfies
$\mathcal{J}^{*}=\mathcal{J}$. Then, for
$x\in\operatorname{L}^{\infty}(\mathbb{G})$,

$\begin{split}&\quad\;R\circ T\circ
R(x)=J_{\widehat{\varphi}}T(J_{\widehat{\varphi}}x^{*}J_{\widehat{\varphi}})^{*}J_{\widehat{\varphi}}=J_{\widehat{\varphi}}T^{\dagger}(J_{\widehat{\varphi}}xJ_{\widehat{\varphi}})J_{\widehat{\varphi}}\\\
&=J_{\widehat{\varphi}}W^{*}\pi(J_{\widehat{\varphi}}xJ_{\widehat{\varphi}})VJ_{\widehat{\varphi}}=(\mathcal{J}WJ_{\widehat{\varphi}})^{*}\mathcal{J}\pi(J_{\widehat{\varphi}}xJ_{\widehat{\varphi}})\mathcal{J}(\mathcal{J}VJ_{\widehat{\varphi}}).\end{split}$

As $\operatorname{L}^{\infty}(\mathbb{G})\ni
x\mapsto\mathcal{J}\pi(J_{\widehat{\varphi}}xJ_{\widehat{\varphi}})\mathcal{J}\in\operatorname{B}(\mathsf{H})$
is a $\star$-homomorphism, it follows that $R\circ T\circ R$ is CB with
$\|R\circ T\circ R\|_{cb}\leq\|T\|_{cb}$. Again, as $R\circ(R\circ T\circ
R)\circ R=T$, we have in fact $\|R\circ T\circ R\|_{cb}=\|T\|_{cb}$.

Let $(T_{i})_{i\in I}$ be a net in
$\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\mathbb{G}))$ converging
to $T$ in the stable point-weak∗-topology. Choose a self-adjoint antiunitary
$\mathcal{J}^{\prime}$ on $\ell^{2}$ and define
$j=\mathcal{J}^{\prime}(\cdot)^{*}\mathcal{J}^{\prime},$ a normal
$\star$-antiautomorphism of $\operatorname{B}(\ell^{2})$. Then $R\otimes j$
extends to a well-defined normal bounded linear map on
$\operatorname{L}^{\infty}(\mathbb{G})\bar{\otimes}\operatorname{B}(\ell^{2})$.
Indeed, in the proof of Proposition 4.3 we argued that
$R_{*}\otimes(j|_{\mathcal{K}(\ell^{2})})^{*}$ is a bounded linear map on
$\operatorname{L}^{1}(\mathbb{G})\widehat{\otimes}\operatorname{B}(\ell^{2})_{*}$,
and we just need to take the dual map $R\otimes
j=(R_{*}\otimes(j|_{\mathcal{K}(\ell^{2})})^{*})^{*}$. For
$x\in\operatorname{L}^{\infty}(\mathbb{G})\bar{\otimes}\operatorname{B}(\ell^{2}),\omega\in\operatorname{L}^{1}(\mathbb{G})\widehat{\otimes}\operatorname{B}(\ell^{2})_{*}$
we have

$\begin{split}&\langle(R\circ T_{i}\circ
R\otimes\textnormal{id})x,\omega\rangle=\langle(R\circ T_{i}\circ R\otimes
j^{2})x,\omega\rangle=\langle(T_{i}\otimes\textnormal{id})\bigl{(}(R\otimes
j)(x)\bigr{)},\omega\circ(R\otimes j)\rangle\\\ \xrightarrow[i\in
I]{}&\langle(T\otimes\textnormal{id})\bigl{(}(R\otimes
j)(x)\bigr{)},\omega\circ(R\otimes j)\rangle=\langle(R\circ T\circ
R\otimes\textnormal{id})x,\omega\rangle,\end{split}$

which concludes the proof. ∎

We now show that $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$
admits an interesting involution. Recall that $S$ denotes the antipode of
$\mathbb{G}$.

######  Proposition 4.9.

Let $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. Then
$a^{*}\in\operatorname{Dom}(S)$ and
$S(a^{*})\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ with
$\Theta^{l}(S(a^{*}))=\Theta^{l}(a)^{\dagger}$.

###### Proof.

By Lemma 4.8, we know that
$\Theta^{l}(a)^{\dagger}\in\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\mathbb{G}))$.
One easily checks that $\Theta^{l}(a)^{\dagger}$ is a left
$\operatorname{L}^{1}(\widehat{\mathbb{G}})$-module map, hence
$\Theta^{l}(a)^{\dagger}=\Theta^{l}(b)$ for some
$b\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. The claim
follows now from [17, Theorem 5.9].

For the convenience of the reader let us also indicate a direct argument. As
$\Theta^{l}(a)^{\dagger}=\Theta^{l}(b)$ we have that

$(\mathbbm{1}\otimes
b)\mathrm{W}^{\widehat{\mathbb{G}}}=(\Theta^{l}(b)\otimes\textnormal{id})\mathrm{W}^{\widehat{\mathbb{G}}}=(\Theta^{l}(a)^{\dagger}\otimes\textnormal{id})\mathrm{W}^{\widehat{\mathbb{G}}}=((\Theta^{l}(a)\otimes\textnormal{id})\mathrm{W}^{\widehat{\mathbb{G}}*})^{*},$

and hence

$\mathrm{W}^{\widehat{\mathbb{G}}*}(\mathbbm{1}\otimes
b^{*})=(\Theta^{l}(a)\otimes\textnormal{id})\mathrm{W}^{\widehat{\mathbb{G}}*}\quad\Rightarrow\quad(\textnormal{id}\otimes\widehat{\omega})(\mathrm{W}^{\mathbb{G}})b^{*}=(\textnormal{id}\otimes\widehat{\omega}\circ\Theta^{l}(a))\mathrm{W}^{\mathbb{G}}$

for each $\widehat{\omega}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$. In
what follows, we treat $S$ as a densely-defined, closed operator on
$\operatorname{L}^{\infty}(\mathbb{G})$ equipped with the weak∗-topology. From
[70, Proposition 2.24], compare [46, Proposition 8.3], we know that for any
$\widehat{\omega}$ we get
$(\textnormal{id}\otimes\widehat{\omega})\mathrm{W}^{\mathbb{G}}\in D(S)$ and
$S((\textnormal{id}\otimes\widehat{\omega})\mathrm{W}^{\mathbb{G}})=(\textnormal{id}\otimes\widehat{\omega})\mathrm{W}^{\mathbb{G}*}$.
It follows that
$(\textnormal{id}\otimes\widehat{\omega})(\mathrm{W}^{\mathbb{G}})b^{*}\in
D(S)$ with

$\displaystyle
S\bigl{(}(\textnormal{id}\otimes\widehat{\omega})(\mathrm{W}^{\mathbb{G}})b^{*}\bigr{)}$
$\displaystyle=(\textnormal{id}\otimes\widehat{\omega}\circ\Theta^{l}(a))\mathrm{W}^{\mathbb{G}*}=(\widehat{\omega}\circ\Theta^{l}(a)\otimes\textnormal{id})\mathrm{W}^{\widehat{\mathbb{G}}}=a\,(\widehat{\omega}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})$
(4.4)
$\displaystyle=a\,(\textnormal{id}\otimes\widehat{\omega})\mathrm{W}^{\mathbb{G}*}=a\,S\big{(}(\textnormal{id}\otimes\widehat{\omega})\mathrm{W}^{\mathbb{G}}\big{)}.$

Let
$C=\\{(\textnormal{id}\otimes\widehat{\omega})\mathrm{W}^{\mathbb{G}}\,|\,\widehat{\omega}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})\\}\subseteq
D(S)$. We shall show that $C$ contains a net $(a_{i})_{i\in I}$ such that both
$a_{i}\xrightarrow[i\in I]{}\mathbbm{1}$ and $S(a_{i})\xrightarrow[i\in
I]{}\mathbbm{1}$ weak∗ in $\operatorname{L}^{\infty}(\mathbb{G})$. It follows
that $a_{i}b^{*}\xrightarrow[i\in I]{}b^{*}$ and $aS(a_{i})\xrightarrow[i\in
I]{}a$ weak∗, and so as $S$ is weak∗-closed, it follows from (4.4) that
$b^{*}\in D(S)$ with $S(b^{*})=a$. Hence
$a^{*}=S(b^{*})^{*}\in\operatorname{Dom}(S)$ and $S(a^{*})=S(S(b^{*})^{*})=b$,
as claimed.

We now show the claim about $C$, using some standard “smearing” techniques,
compare [43]. For $a\in\mathrm{C}_{0}(\mathbb{G})$ and $r>0,z\in\mathbb{C}$,
define

$a(r,z)=\frac{r}{\sqrt{\pi}}\int_{\mathbb{R}}\exp(-r^{2}(t-z)^{2})\tau_{t}(a)\operatorname{d}\\!t.$

Then $a(r,z)$ is analytic for the one-parameter automorphism group
$(\tau_{t})_{t\in\mathbb{R}}$ with $\tau_{w}(a(r,z))=a(r,z+w)$ for
$w\in\mathbb{C}$. Similarly, for
$\widehat{\omega}\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$, define

$\widehat{\omega}(r,z)=\frac{r}{\sqrt{\pi}}\int_{\mathbb{R}}\exp(-r^{2}(t-z)^{2})\widehat{\omega}\circ\widehat{\tau}_{t}\operatorname{d}\\!t.$

Given $\widehat{\omega}$, let
$a=(\textnormal{id}\otimes\widehat{\omega})(\mathrm{W}^{\mathbb{G}})$. As
$(\tau_{t}\otimes\widehat{\tau}_{t})(\mathrm{W}^{\mathbb{G}})=\mathrm{W}^{\mathbb{G}}$,
it follows that
$\tau_{t}(a)=(\textnormal{id}\otimes\widehat{\omega}\circ\widehat{\tau}_{-t})(\mathrm{W}^{\mathbb{G}})$
and hence
$a(r,z)=(\textnormal{id}\otimes\widehat{\omega}(r,-z))(\mathrm{W}^{\mathbb{G}})$.
Finally, as $S=R\tau_{-i/2}$, it follows that
$S(a(r,z))=R(a(r,z-i/2))=(\textnormal{id}\otimes(\widehat{\omega}\circ\widehat{R})(r,-z+i/2))(\mathrm{W}^{\mathbb{G}})$.

As $C$ is norm dense in $\mathrm{C}_{0}(\mathbb{G})$, we can find a net
$(\widehat{\omega}_{i})_{i\in I}$ with
$a_{i}=(\textnormal{id}\otimes\widehat{\omega}_{i})(\mathrm{W}^{\mathbb{G}})\xrightarrow[i\in
I]{}\mathbbm{1}$ strictly. By [43, Proposition 2.25], the net
$(a_{i}(r,z))_{i\in I}$ converges strictly to $\mathbbm{1}(r,z)=\mathbbm{1}$,
for any choice of $r,z$. By the above discussion, $a_{i}(r,z)\in C$. Then also
$S(a_{i}(r,z))=R(a_{i}(r,z-i/2))\xrightarrow[i\in
I]{}R(\mathbbm{1})=\mathbbm{1}$ strictly. Cohen–Hewitt’s Factorisation Theorem
shows that strict convergence implies weak∗-convergence in
$\operatorname{L}^{\infty}(\mathbb{G})$, hence we have constructed the
required net. ∎

######  Corollary 4.10.

Let $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. Then
$\Theta^{l}(a)\in\operatorname{CB}^{\sigma}(\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))$
preserves the adjoint if and only if $S(a^{*})=a$.

Next we check that $\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$
is globally invariant under the scaling and modular automorphism groups.

######  Lemma 4.11.

Let $\mathbb{G}$ be a locally compact quantum group and let
$a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. For any
$t\in\mathbb{R}$,

  * •

$\tau_{t}(a)\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ with
$\Theta^{l}(\tau_{t}(a))=\hat{\tau}_{t}\circ\Theta^{l}(a)\circ\hat{\tau}_{-t}$.

  * •

$\sigma^{\varphi}_{t}(a)\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$
with $\Theta^{l}(\sigma^{\varphi}_{t}(a))\colon
x\mapsto\hat{\delta}^{it}\,\hat{\tau}_{t}\circ\Theta^{l}(a)\bigl{(}\hat{\delta}^{-it}\hat{\tau}_{-t}(x)\bigr{)}$.

  * •

$\sigma^{\psi}_{t}(a)\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$
with $\Theta^{l}(\sigma^{\psi}_{t}(a))\colon
x\mapsto\hat{\tau}_{-t}\circ\Theta^{l}(a)\bigl{(}\hat{\tau}_{t}(x)\hat{\delta}^{-it}\bigr{)}\hat{\delta}^{it}$.

###### Proof.

Take $\omega\in\operatorname{L}^{1}(\widehat{\mathbb{G}})$. Using
$(\hat{\tau}_{t}\otimes\tau_{t})\mathrm{W}^{\widehat{\mathbb{G}}}=\mathrm{W}^{\widehat{\mathbb{G}}}$
(see proof of [46, Proposition 6.10]) we obtain

$\begin{split}&\quad\;\tau_{t}(a)\widehat{\lambda}(\omega)=\tau_{t}\bigl{(}a\,\tau_{-t}(\widehat{\lambda}(\omega))\bigr{)}=\tau_{t}\bigl{(}a\widehat{\lambda}(\omega\circ\hat{\tau}_{t})\bigr{)}=\widehat{\lambda}\bigl{(}\Theta^{l}(a)_{*}(\omega\circ\hat{\tau}_{t})\circ\hat{\tau}_{-t}\bigr{)}\end{split}$

which implies
$\tau_{t}(a)\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ and
$\Theta^{l}(\tau_{t}(a))=\hat{\tau}_{t}\circ\Theta^{l}(a)\circ\hat{\tau}_{-t}$;
notice that clearly the right-hand-side of this final expression gives a
completely bounded map.

We now use the following facts. Firstly, by the definition of
$\mathrm{W}^{\mathbb{G}}$ we have
$(\rho\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})\Lambda_{\varphi}(x)=\Lambda_{\varphi}((\rho\otimes\textnormal{id})\Delta(x))$
for $\rho\in\operatorname{L}^{1}(\mathbb{G}),x\in\mathfrak{N}_{\varphi}$.
Secondly,
$(\sigma^{\varphi}_{t}\otimes\sigma^{\psi}_{-t})\circ\Delta=\Delta\circ\tau_{t}$,
see [46, Proposition 6.8]. Thus

$\begin{split}&\quad\;(\rho\circ\sigma^{\varphi}_{t}\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})\Lambda_{\varphi}(x)=\Lambda_{\varphi}((\rho\circ\sigma^{\varphi}_{t}\otimes\textnormal{id})\Delta(x))=\Lambda_{\varphi}\bigl{(}(\rho\otimes\sigma^{\psi}_{t})\Delta(\tau_{t}(x))\bigr{)}\\\
&=\nu^{\frac{t}{2}}\nabla_{\psi}^{it}\Lambda_{\varphi}((\rho\otimes\textnormal{id})\Delta(\tau_{t}(x)))=\nabla_{\psi}^{it}(\rho\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})P^{it}\Lambda_{\varphi}(x),\end{split}$

which implies that
$(\sigma^{\varphi}_{t}\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})=(\mathbbm{1}\otimes\nabla_{\psi}^{it})\mathrm{W}^{\mathbb{G}*}(\mathbbm{1}\otimes
P^{it})$; here we have also used [70, Definition 5.1, Remark 5.2]. Next, since
$\mathrm{W}^{\widehat{\mathbb{G}}}=\chi(\mathrm{W}^{\mathbb{G}*})$ and
$\hat{\delta}^{-it}=\nabla_{\psi}^{it}P^{it}$ ([70, Theorem 5.17]), and using
that $\hat{\tau}_{-t}(y)=P^{-it}yP^{it}$ for
$y\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ see [46, Proposition
8.23], we arrive at

(4.5)
$(\textnormal{id}\otimes\sigma^{\varphi}_{t})(\mathrm{W}^{\widehat{\mathbb{G}}})=(\nabla_{\psi}^{it}\otimes\mathbbm{1})\mathrm{W}^{\widehat{\mathbb{G}}}(P^{it}\otimes\mathbbm{1})=(\hat{\delta}^{-it}\otimes\mathbbm{1})(\hat{\tau}_{-t}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}}).$

Using this formula and $\hat{\tau}_{t}(\hat{\delta})=\hat{\delta}$ ([70,
Theorem 3.11]) we calculate

$\begin{split}&\quad\;\sigma^{\varphi}_{t}(a)\widehat{\lambda}(\omega)=\sigma^{\varphi}_{t}\bigl{(}a\sigma^{\varphi}_{-t}(\widehat{\lambda}(\omega))\bigr{)}=\sigma^{\varphi}_{t}\bigl{(}a(\omega\otimes\textnormal{id})\bigl{(}(\hat{\delta}^{it}\otimes\mathbbm{1})(\hat{\tau}_{t}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})\bigr{)}\bigr{)}\\\
&=\sigma^{\varphi}_{t}\bigl{(}a\widehat{\lambda}((\omega\hat{\delta}^{it})\circ\hat{\tau}_{t})\bigr{)}=\sigma^{\varphi}_{t}\bigl{(}\widehat{\lambda}(\Theta^{l}(a)_{*}((\omega\hat{\delta}^{it})\circ\hat{\tau}_{t}))\bigr{)}\\\
&=\widehat{\lambda}\bigl{(}\bigl{(}\bigl{(}\Theta^{l}(a)_{*}((\omega\hat{\delta}^{it})\circ\hat{\tau}_{t})\bigr{)}\hat{\delta}^{-it}\bigr{)}\circ\hat{\tau}_{-t}\bigr{)}.\end{split}$

The above shows that
$\sigma^{\varphi}_{t}(a)\in\operatorname{M}^{l}(\operatorname{A}(\mathbb{G}))$,
and for $x\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}})$ we get

$\begin{split}&\quad\;\langle\Theta^{l}(\sigma^{\varphi}_{t}(a))(x),\omega\rangle=\langle
x,\Theta^{l}(\sigma^{\varphi}_{t}(a))_{*}(\omega)\rangle=\langle
x,\bigl{(}\Theta^{l}(a)_{*}((\omega\hat{\delta}^{it})\circ\hat{\tau}_{t})\hat{\delta}^{-it}\bigr{)}\circ\hat{\tau}_{-t}\rangle\\\
&=\langle\hat{\delta}^{-it}\hat{\tau}_{-t}(x),\Theta^{l}(a)_{*}((\omega\hat{\delta}^{it})\circ\hat{\tau}_{t})\rangle=\langle\hat{\delta}^{it}\hat{\tau}_{t}\circ\Theta^{l}(a)\bigl{(}\hat{\delta}^{-it}\hat{\tau}_{-t}(x)\bigr{)},\omega\rangle.\end{split}$

The final claim is shown in a similar way. We have
$(\sigma^{\psi}_{t}\otimes\tau_{-t})\circ\Delta=\Delta\circ\sigma^{\psi}_{t}$
([46, Proposition 8.23]) and hence for $x\in\mathfrak{N}_{\varphi}$,

$\begin{split}&\quad\;(\rho\circ\sigma^{\psi}_{t}\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})\Lambda_{\varphi}(x)=\Lambda_{\varphi}((\rho\circ\sigma^{\psi}_{t}\otimes\textnormal{id})\Delta(x))=\Lambda_{\varphi}\bigl{(}(\rho\otimes\tau_{t})\Delta(\sigma^{\psi}_{t}(x))\bigr{)}\\\
&=\nu^{-\frac{t}{2}}P^{it}\Lambda_{\varphi}((\rho\otimes\textnormal{id})\Delta(\sigma^{\psi}_{t}(x)))=P^{it}(\rho\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})\nabla_{\psi}^{it}\Lambda_{\varphi}(x),\end{split}$

consequently
$(\sigma^{\psi}_{t}\otimes\textnormal{id})(\mathrm{W}^{\mathbb{G}*})=(\mathbbm{1}\otimes
P^{it})\mathrm{W}^{\mathbb{G}*}(\mathbbm{1}\otimes\nabla_{\psi}^{it})$ and so

$(\textnormal{id}\otimes\sigma^{\psi}_{t})(\mathrm{W}^{\widehat{\mathbb{G}}})=(P^{it}\otimes\mathbbm{1})\mathrm{W}^{\widehat{\mathbb{G}}}(\nabla_{\psi}^{it}\otimes\mathbbm{1})=(\hat{\tau}_{t}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})(\hat{\delta}^{-it}\otimes\mathbbm{1}).$

Calculating as before,

$\begin{split}&\quad\;\sigma^{\psi}_{t}(a)\widehat{\lambda}(\omega)=\sigma^{\psi}_{t}\bigl{(}a(\omega\otimes\textnormal{id})\bigl{(}(\hat{\tau}_{-t}\otimes\textnormal{id})(\mathrm{W}^{\widehat{\mathbb{G}}})(\hat{\delta}^{it}\otimes\mathbbm{1})\bigr{)}\bigr{)}\\\
&=\sigma^{\psi}_{t}\bigl{(}a\widehat{\lambda}((\hat{\delta}^{it}\omega)\circ\hat{\tau}_{-t})\bigr{)}=\widehat{\lambda}\bigl{(}\bigl{(}\bigl{(}\hat{\delta}^{-it}\,\Theta^{l}(a)_{*}((\hat{\delta}^{it}\omega)\circ\hat{\tau}_{-t})\bigr{)}\bigr{)}\circ\hat{\tau}_{t}\bigr{)}\end{split}$

and so

$\begin{split}&\quad\;\langle\Theta^{l}(\sigma^{\psi}_{t}(a))(x),\omega\rangle=\langle\hat{\tau}_{-t}\circ\Theta^{l}(a)\bigl{(}\hat{\tau}_{t}(x)\hat{\delta}^{-it}\bigr{)}\hat{\delta}^{it},\omega\rangle\qquad(x\in\operatorname{L}^{\infty}(\widehat{\mathbb{G}}))\end{split}$

as desired. ∎

In the next proposition we show that for any
$a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$ the map
$\Theta^{l}(a)$ is bounded on the Hilbert space level; the final claim should
be compared with Proposition 4.9.

######  Proposition 4.12.

Let $a\in\operatorname{M}^{l}_{cb}(\operatorname{A}(\mathbb{G}))$. Then for
$b\in\mathfrak{N}_{\widehat{\varphi}}$ we have
$\Theta^{l}(a)(b)\in\mathfrak{N}_{\widehat{\varphi}}$, and the densely defined
operator

$\operatorname{L}^{2}(\mathbb{G})\supseteq\Lambda_{\widehat{\varphi}}(\mathfrak{N}_{\widehat{\varphi}})\ni\Lambda_{\widehat{\varphi}}(b)\mapsto\Lambda_{\widehat{\varphi}}(\Theta^{l}(a)(b))\in\operatorname{L}^{2}(\mathbb{G})$

is bounded. In fact, we have

$\Lambda_{\widehat{\varphi}}(\Theta^{l}(a)(b))=S^{-1}(a)\Lambda_{\widehat{\varphi}}(b)=S(a^{*})^{*}\Lambda_{\widehat{\varphi}}(b)\qquad(b\in\mathfrak{N}_{\widehat{\varphi}}).$

Here and in the sequel we use the GNS implementation of map $\Theta^{l}(a)$ –
in the literature people have considered also different embeddings of (a
subspace of) $\operatorname{L}^{\infty}(\mathbb{G})$ into
$\operatorname{L}^{2}(\mathbb{G})$, c.f. [62, Section 2].

In order to prove Proposition 4.12 we start with a general lemma; recall (2.3)
for the definition of $\mathscr{J}$.

######  Lemma 4.13.

Let $\omega\in\mathscr{J}\subseteq\operatorname{L}^{1}(\mathbb{G})$, let
$a\in\operatorname{L}^{\infty}(\mathbb{G})$, and let
$b\in\operatorname{Dom}(\sigma^{\varphi}_{-i/2})\subseteq\operatorname{L}^{\infty}(\mathbb{G})$.
Then $a\omega b\in\mathscr{J}$ with
$\Lambda_{\widehat{\varphi}}(\lambda(a\omega
b))=aJ_{\varphi}\sigma^{\varphi}_{-i/2}(b)^{*}J_{\varphi}\Lambda_{\widehat{\varphi}}(\lambda(\omega))$.

###### Proof.

Take $x\in\mathfrak{N}_{\varphi}$. We have

$\begin{split}&\quad\;\langle x^{*},a\omega
b\rangle=\langle(a^{*}xb^{*})^{*},\omega\rangle=\langle\Lambda_{\varphi}(a^{*}xb^{*})\,|\,\Lambda_{\widehat{\varphi}}(\lambda(\omega))\rangle\\\
&=\langle
a^{*}J_{\varphi}\sigma^{\varphi}_{i/2}(b^{*})^{*}J_{\varphi}\Lambda_{\varphi}(x)\,|\,\Lambda_{\widehat{\varphi}}(\lambda(\omega))\rangle=\langle\Lambda_{\varphi}(x)\,|\,aJ_{\varphi}\sigma^{\varphi}_{i/2}(b^{*})J_{\varphi}\Lambda_{\widehat{\varphi}}(\lambda(\omega))\rangle\end{split}$

which proves the claim. ∎

###### Proof of Proposition 4.12.

Take $b=(\textnormal{id}\otimes\omega)\mathrm{W}^{\widehat{\mathbb{G}}}$ for
$\omega\in\operatorname{L}^{1}(\mathbb{G})$ such that