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# Approximation properties for dynamical W∗-correspondences

K. De Commer and J. De Ro  Vrije Universiteit Brussel<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $\mathbb{G}$ be a locally compact quantum group, and $A,B$ von Neumann
algebras with an action by $\mathbb{G}$. We refer to these as _$\mathbb{G}$
-dynamical W∗-algebras_. We make a study of $\mathbb{G}$-equivariant
$A$-$B$-correspondences, that is, Hilbert spaces $\mathcal{H}$ with an
$A$-$B$-bimodule structure by $*$-preserving normal maps, and equipped with a
unitary representation of $\mathbb{G}$ which is equivariant with respect to
the above bimodule structure. Such structures are a Hilbert space version of
the theory of $\mathbb{G}$-equivariant Hilbert C∗-bimodules. We show that
there is a well-defined Fell topology on equivariant correspondences, and use
this to formulate approximation properties for them. Within this formalism, we
then characterize amenability of the action of a locally compact group on a
von Neumann algebra, using recent results due to Bearden and Crann. We further
consider natural operations on equivariant correspondences such as taking
opposites, composites and crossed products, and examine the continuity of
these operations with respect to the Fell topology.

## Introduction

In the setting of representation theory and abstract harmonic analysis,
amenability properties can often be characterized by the possibility to
approximate a pre-determined restricted class of ‘trivial representations’
within a pre-determined restricted class of ‘regular representations’. We then
say that the former are ‘weakly contained’ in the latter. The above classes
consist in many cases of just one object. Two examples stand out:

  * •

If $G$ is a locally compact group, then $G$ is amenable if and only if the
trivial representation is weakly contained in the regular representation on
$L^{2}(G)$.

  * •

If $A$ is a von Neumann algebra, then $A$ is amenable if and only if the
trivial $A$-bimodule $L^{2}(A)$ is weakly contained in the regular (or
_coarse_) $A$-bimodule $L^{2}(A)\otimes L^{2}(A)$.

In this last example, we mean bimodules in the sense of _correspondences_
[Con80+] (see also [Pop86], [Con94, Appendix B]): we are given two von Neumann
algebras $A,B$ and a Hilbert space $\mathcal{H}$ equipped with a normal unital
$*$-representation $\pi$ of $A$ and a normal unital anti-$*$-representation
$\rho$ of $B$ for which $\mathcal{H}$ forms an $A$-$B$-bimodule.
Characterizing amenability in this setting has led to some early breakthrough
results in the modern theory of von Neumann algebras by its relation to
_injectivity_ [Con76, CE77, EL77], and continues to play a fundamental role in
many aspects of operator theory. For example, the notion of _relative
amenability_ , introduced in [Pop86] and further explored in the general
context of correspondences in [A-D90, A-D95], was only recently shown to be
equivalent with the corresponding notion of relative injectivity [BMO20].

By means of the crossed product construction, close ties can be made between
amenability of _actions_ of locally compact groups on von Neumann algebras
[A-D79] and approximation properties related to correspondences constructed
from the von Neumann algebra and its crossed product. In the setting of
dynamical systems, such approximation properties are connected to amenability
properties for certain associated groupoids [A-DR00]. We also mention the
recent key results relating amenability of C∗-dynamical systems to amenability
of W∗-dynamical systems [BEW21, OS21].

In [KV00, KV03], a generalization was made of the theory of locally compact
groups to that of _locally compact quantum groups_ (we refer to these articles
for more on the history of the subject). It is based on the notion of a Hopf-
von Neumann algebra [Ern67], which we now recall.

######  Definition 0.1.

A Hopf-von Neumann algebra consists of a von Neumann algebra $M$ with a unital
normal faithful $*$-homomorphism $\Delta:M\rightarrow M\bar{\otimes}M$
satisfying the _coassociativity_ condition

$(\Delta\otimes\operatorname{\mathrm{id}})\circ\Delta=(\operatorname{\mathrm{id}}\otimes\Delta)\circ\Delta.$

It is called _involutive_ if it is equipped with an involutive
anti-$*$-automorphism $R:M\rightarrow M$ satisfying

$\Delta\circ R=(R\otimes R)\circ\Delta^{\mathrm{op}},$

where $\Delta^{\mathrm{op}}$ is the composition of $\Delta$ with the flip map
$\varsigma:x\otimes y\mapsto y\otimes x$.

If $(M,\Delta)$ is a Hopf-von Neumann algebra, we sometimes write
$\mathbb{G}=(M,\Delta)$ and $M=L^{\infty}(\mathbb{G})$. Then we say that
$\mathbb{G}$ is a _locally compact quantum group_ if moreover $(M,\Delta)$
possesses invariant weights (see Definition 1.1). Such Hopf-von Neumann
algebras are automatically involutive with respect to a canonical involution
$R$.

The von Neumann algebraic framework for locally compact quantum groups is very
flexible and high-powered. For example, consider the following notions for a
Hopf-von Neumann algebra $\mathbb{G}=(M,\Delta)$.

######  Definition 0.2.

A _unitary representation_ of $\mathbb{G}$ on a Hilbert space $\mathcal{H}$ is
a unitary $U\in B(\mathcal{H})\bar{\otimes}M$ such that

$(\operatorname{\mathrm{id}}\otimes\Delta)U=U_{12}U_{13}.$ (0.1)

Here $U=U\otimes 1$, while $U_{13}$ is
$(\operatorname{\mathrm{id}}\otimes\varsigma)(U\otimes 1)$.

######  Definition 0.3.

If $A$ is a von Neumann algebra, an action of $\mathbb{G}$ on $A$ is a unital
normal faithful $*$-homomorphism

$\alpha:A\rightarrow A\bar{\otimes}M,$

called the associated _coaction_ , and satisfying the _coaction property_

$(\alpha\otimes\operatorname{\mathrm{id}})\alpha=(\operatorname{\mathrm{id}}\otimes\Delta)\alpha.$

We also say that $(A,\alpha)$ is a $\mathbb{G}$-dynamical von Neumann algebra,
or simply a $\mathbb{G}$-W∗-algebra.

In the general context of Hopf-von Neumann algebras, these two notions have no
apparent relation, but when $\mathbb{G}=(M,\Delta)$ is a locally compact
quantum group, there is a beautiful generalisation, due to S. Vaes [Vae01], of
Haagerup’s canonical implementation of a group action on a von Neumann algebra
[Haa78]: there exists a canonical unitary representation $U_{\alpha}$ of
$\mathbb{G}$ on the standard Hilbert space $L^{2}(A)$ such that

$\alpha(a)=U_{\alpha}(a\otimes 1)U_{\alpha}^{*},\qquad\forall a\in A,$ (0.2)

where we view $A\subseteq B(L^{2}(A))$ in standard position.

From an abstract and even an aesthetic point of view, one of the main salient
features of locally compact quantum groups is that they allow an extension of
the Pontryagin duality for arbitrary _abelian_ locally compact groups to a
Pontryagin duality for arbitrary locally compact _quantum_ groups. When
restricting to locally compact groups $G$, this is the correspondence sending
the function algebra $L^{\infty}(G)$ (with its natural coproduct encoding the
group structure) into the group von Neumann algebra $\mathscr{R}(G)$ generated
by right translation operators (with its natural diagonal coproduct), and thus
not an operation which gives a duality on a single category in this restricted
setting. Various approximation properties such as amenability, Haagerup
property, property (T),… can also be formulated naturally within the much more
general setting of locally compact quantum groups, see e.g. [Voi77, Rua96,
DQV02, Tom06, Fim10, KS12, CN15, DFSW16].

In this paper, we combine the above two settings by studying _equivariant
correspondences_. This notion is a verbatim generalisation of the
corresponding notion for $\mathbb{G}$-C∗-algebras with $\mathbb{G}$ a compact
quantum group, introduced in [AS21].

######  Definition 0.4.

Let $\mathbb{G}=(M,\Delta,R)$ be an involutive Hopf-von Neumann algebra, and
let $(A,\alpha),(B,\beta)$ be $\mathbb{G}$-W∗-algebras. A _$\mathbb{G}$
-equivariant $A$-$B$-correspondence_, or simply a
$\mathbb{G}$-$A$-$B$-correspondence, consists of a Hilbert space $\mathcal{H}$
with

  * •

a normal unital $*$-representation $\pi$ of $A$ on $\mathcal{H}$,

  * •

a normal unital anti-$*$-representation $\rho$ of $B$ on $\mathcal{H}$, and

  * •

a unitary representation $U$ of $\mathbb{G}$ on $\mathcal{H}$,

satisfying for all $a\in A$ and $b\in B$ the compatibility relations
$\pi(a)\rho(b)=\rho(b)\pi(a)$ and

$U(\pi(a)\otimes
1)U^{*}=(\pi\otimes\operatorname{\mathrm{id}})\alpha(a),\qquad
U^{*}(\rho(b)\otimes 1)U=(\rho\otimes R)(\beta(b)).$ (0.3)

C∗-algebraic analogues of such equivariant correspondences, in the context of
equivariant Hilbert C∗-modules, have already appeared in the original works
[BS89, BS93] in relation to equivariant KK-theory, and have remained important
in follow-up works (see e.g. [BC17]). In [AS21], concrete use was made rather
of the above Hilbert space picture of equivariant correspondences, however
within the context of C∗-algebras and _compact_ quantum groups. On the other
hand, in [Vae05] there already appear specific equivariant correspondences,
albeit in the particular case arising from inclusions of quantum groups, and
using instead the (equivalent) notion of equivariant von Neumann bimodules.

More particular cases of interest have turned up recently within the setting
of _quantum automorphism groups_ [RV22] (for related results in a more
classical setting, see [AV16]), and _quantizations of semisimple Lie groups_
[DCDT21]. It is our hope that the framework of this paper will be a convenient
one to study analytic approximation properties for these important classes of
examples.

Let us now come to the main results of this paper.

######  Theorem (Theorem 2.15).

Let $\mathbb{G}$ be a locally compact quantum group, and $A,B$ two
$\mathbb{G}$-W∗-algebras. Then there exists a universal C∗-algebra
$C^{\mathbb{G}}(A,B)$ such that any $\mathbb{G}$-equivariant
$A$-$B$-correspondence $\mathcal{H}$ uniquely lifts to a non-degenerate
$*$-representation $\theta_{\mathcal{H}}$ of $C^{\mathbb{G}}(A,B)$ on
$\mathcal{H}$.

The C∗-algebra $C^{\mathbb{G}}(A,B)$ is a C∗/W∗-intermediate of the double
cross product C∗-algebra appearing in [AS21] (see also [NV02, Section 2.6] for
a description in a purely algebraic context). Due to the above theorem, we can
introduce a _Fell topology_ on the class of $\mathbb{G}$-equivariant
$A$-$B$-correspondences, as well as a notion of weak containment: we say
$\mathcal{G}$ is weakly contained in $\mathcal{H}$ if the $*$-representation
$\theta_{\mathcal{G}}$ of $C^{\mathbb{G}}(A,B)$ factors through
$\theta_{\mathcal{H}}(C^{\mathbb{G}}(A,B))$. When we want to emphasize that we
are working in the equivariant setting, we will sometimes use the term
_equivariantly weakly contained_.

Our next main result offers another characterisation of equivariant weak
containment. If $(\mathcal{H},U_{\mathcal{H}}),(\mathcal{G},U_{\mathcal{G}})$
are $\mathbb{G}$-$A$-$B$-correspondences, we use the notation

$X_{B}(\mathcal{G},\mathcal{H})=\\{x:\mathcal{G}\rightarrow\mathcal{H}\mid
x\rho_{\mathcal{G}}(b)=\rho_{\mathcal{H}}(b)x,\forall b\in B\\},$ (0.4)

and

$\alpha_{X_{B}(\mathcal{G},\mathcal{H})}:X_{B}(\mathcal{G},\mathcal{H})\rightarrow
X_{B}(\mathcal{G},\mathcal{H})\bar{\otimes}M,\qquad x\mapsto
U_{\mathcal{H}}(x\otimes 1)U_{\mathcal{G}}^{*}.$

######  Theorem (Theorem 3.6).

Let $\mathbb{G}$ be a locally compact quantum group, and let $A,B$ be
$\mathbb{G}$-W∗-algebras. Let $\mathcal{H},\mathcal{G}$ be
$\mathbb{G}$-$A$-$B$-correspondences, and put
$\mathcal{H}^{\infty}=\oplus_{1}^{\infty}\mathcal{H}$.

Then $\mathcal{G}$ is equivariantly weakly contained in $\mathcal{H}$ if and
only if there exists a $1$-bounded net $y_{i}\in
X_{B}(\mathcal{G},\mathcal{H}^{\infty})$ such that for all $a\in A$

$(y_{i}^{*}\pi_{\mathcal{H}^{\infty}}(a)\otimes
1)\alpha_{X_{B}(\mathcal{G},\mathcal{H})}(y_{i})\rightarrow\pi_{\mathcal{G}}(a)\otimes
1\quad\sigma\textrm{-weakly in }B(\mathcal{G})\bar{\otimes}M.$ (0.5)

The above results naturally lead to a notion of amenability for inclusions of
$\mathbb{G}$-W∗-algebras: if $A\subseteq B$ is $\mathbb{G}$-equivariant, then
we can say that this inclusion is _strongly equivariantly amenable_ if
${}_{A}L^{2}(A)_{A}$ is equivariantly weakly contained in ${}_{A}L^{2}(B)_{A}$
(these correspondences can easily be made equivariant using the results of
[Vae01], see Example 2.2). The following theorem then shows that equivariant
strong amenability implies _equivariant amenability_ (=equivariant relative
injectivity).

######  Theorem (Theorem 4.3).

Let $\mathbb{G}$ be a locally compact quantum group, and let
$(A,\alpha)\subseteq(B,\beta)$ be an equivariant inclusion of
$\mathbb{G}$-W∗-algebras. If ${}_{A}L^{2}(A)_{A}$ is equivariantly weakly
contained in ${}_{A}L^{2}(B)_{A}$, then there exists a _$\mathbb{G}$
-equivariant_ conditional expectation $E:B\rightarrow A$, i.e.

$(E\otimes\operatorname{\mathrm{id}})\circ\beta=\alpha\circ E.$ (0.6)

As a particular case in point, one can consider for a $\mathbb{G}$-W∗-algebra
$(A,\alpha)$ the equivariant inclusion

$A\cong\alpha(A)\subseteq A\bar{\otimes}M,$

where $A\bar{\otimes}M$ is endowed with the $\mathbb{G}$-action
$\operatorname{\mathrm{id}}_{A}\otimes\Delta$. Our next result is then a
converse of the last result above, in the case of a usual locally compact
group $G$. It heavily relies on [BC22].

######  Theorem (Theorem 4.9).

Let $G$ be a locally compact group, and let $A$ be a $G$-W∗-algebra via

$\alpha:A\rightarrow A\bar{\otimes}L^{\infty}(G).$

Then ${}_{A}L^{2}(A)_{A}$ is weakly contained in
${}_{\alpha(A)}(L^{2}(A)\otimes L^{2}(G))_{\alpha(A)}$ if and only if there
exists a $G$-equivariant conditional expectation
$E:A\bar{\otimes}L^{\infty}(G)\rightarrow A$.

As mentioned, an important aspect of locally compact quantum groups concerns
their Pontryagin duality theory. We recall (see Section 1 for details) that if
$A$ is a $\mathbb{G}$-dynamical von Neumann algebra, and $\check{\mathbb{G}}$
the Pontryagin dual, one can construct the _crossed product von Neumann
algebra_ $A\rtimes\mathbb{G}$, which becomes a $\check{\mathbb{G}}$-dynamical
von Neumann algebra. We will show that this duality extends to the level of
$\mathbb{G}$-equivariant correspondences: to any $\mathbb{G}$-equivariant
$A$-$B$-correspondence $\mathcal{H}$ is associated a
$\check{\mathbb{G}}$-equivariant
$A\rtimes\mathbb{G}$-$B\rtimes\mathbb{G}$-correspondence
$\mathcal{H}\rtimes\mathbb{G}$.

Our final main result is then the following topological refinement of the
well-known Takesaki-Takai duality.

######  Theorem 0.5 (Theorem 5.24 and Theorem 6.4).

The assignment

$\mathcal{H}\mapsto\mathcal{H}\rtimes\mathbb{G}$

is an equivalence between the W∗-category of $\mathbb{G}$-equivariant
$A$-$B$-correspondences and the W∗-category of
$\check{\mathbb{G}}$-equivariant
$A\rtimes\mathbb{G}$-$B\rtimes\mathbb{G}$-correspondences. Moreover, this
correspondence preserves the Fell topology.

The precise contents of this paper are as follows.

In the _first section_ , we provide some preliminaries on locally compact
quantum groups and their actions. In the _second section_ , we introduce
equivariant correspondences, provide some elementary examples, and show that
the category of equivariant correspondences admits a well-defined and well-
behaved Fell topology. In the _third section_ , we introduce the notion of
weak containment between equivariant correspondences, and provide an
equivalent formulation for weak containment in terms of one-sided
intertwiners. In the _fourth section_ , we introduce the notion of amenability
and strong amenability for equivariant correspondences, and show that these
notions can be used to introduce corresponding notions for actions of locally
compact quantum groups on von Neumann algebras. We show that in the setting of
locally compact groups, an amenable action is automatically strongly amenable,
using the results of [BC22]. In the _fifth section_ , we consider various
operations on equivariant correspondences such as opposites, composites and
crossed products. The main result here is a generalized Takesaki-Takai-
duality, providing an equivalence of W∗-bicategories between
$\mathbb{G}$-equivariant correspondences and $\check{\mathbb{G}}$-equivariant
correspondences. In the final _sixth section_ , we consider the interplay
between the above constructions and the Fell topology, and consider some
direct corollaries for the associated approximation properties.

Notations and conventions

All our vector spaces are complex. If $V$ is a vector space and $S\subseteq
V$, we denote $\mathrm{Span}(S)$ for the linear span of $S$. If $V,W$ are
subspaces of an algebra $A$, we denote

$VW=\mathrm{Span}\\{vw\mid v\in V,w\in W\\}\subseteq A.$

If $V$ is a normed vector space and $S\subseteq V$, we denote $[S]$ for the
norm-closure of the linear span of $S$. We sometimes use the shorthand

$[S]=[s\mid s\in S].$

We denote $\odot$ for algebraic tensor products (over $\mathbb{C}$). We use
$\otimes$ for the tensor product of Hilbert spaces, or the minimal tensor
product of C∗-algebras or Hilbert C∗-modules. We use $\bar{\otimes}$ for the
spatial tensor product of von Neumann algebras. The inner product of Hilbert
spaces is assumed anti-linear in the first variable.

We will use the terms W∗-algebra and von Neumann algebra interchangeably. When
$M$ is a von Neumann algebra, we denote $M_{*}$ for the predual and $M_{+}$
for the positive cone. If $\Phi$ is an nsf (= normal, semifinite, faithful)
weight on $M$, we use the standard notation

$\mathscr{N}_{\Phi}=\\{x\in M\mid\Phi(x^{*}x)<\infty\\}.$

We let $L^{2}(M)$ be the standard Hilbert space of $M$, with modular
conjugation $J=J_{M}$ and $\Lambda_{\Phi}:\mathscr{N}_{\Phi}\rightarrow
L^{2}(M)$ the canonical GNS-map of $\Phi$. We denote $\pi_{M}$ for the
associated left representation of $M$ on $L^{2}(M)$, and
$\rho_{M}(x)=J\pi_{M}(x)^{*}J$ for the associated anti-$*$-representation of
$M$ on $L^{2}(M)$. When there is no ambiguity, we drop the symbol $\pi_{M}$,
and identify $M\subseteq B(L^{2}(M))$. For the basics of weight theory on von
Neumann algebras, we refer to [Tak03].

If $\mathcal{H}$ is a Hilbert space, we denote $\mathcal{K}(\mathcal{H})$ for
the C∗-algebra of compact operators on $\mathcal{H}$. If $u\in B(\mathcal{H})$
is a unitary, we write
$\operatorname{\mathrm{Ad}}(u)(x)=\operatorname{\mathrm{Ad}}_{u}(x)=uxu^{*}$
for $x\in B(\mathcal{H})$. More generally, when $\mathcal{H},\mathcal{G}$ are
Hilbert spaces, and $x\in B(\mathcal{G},\mathcal{H})$, we will denote by
$\operatorname{\mathrm{Ad}}_{x}$ the completely positive map

$\operatorname{\mathrm{Ad}}_{x}:B(\mathcal{G})\rightarrow
B(\mathcal{H}),\qquad T\mapsto xTx^{*}.$

If $\xi,\eta\in\mathcal{H}$, we write

$\omega_{\xi,\eta}\in B(\mathcal{H})_{*},\qquad B(\mathcal{H})\ni
x\mapsto\langle\xi,x\eta\rangle\in\mathbb{C}.$

If $\mathcal{H},\mathcal{G}$ are Hilbert spaces, we denote the _flip map_ by

$\Sigma:\mathcal{H}\otimes\mathcal{G}\rightarrow\mathcal{G}\otimes\mathcal{H},\qquad\xi\otimes\eta\mapsto\eta\otimes\xi.$

We use the _leg numbering notation_ : if $x$ is an operator on a tensor
product of Hilbert spaces, we write

$x_{12}=x\otimes 1,\qquad x_{23}=1\otimes x,\qquad
x_{13}=(\operatorname{\mathrm{id}}\otimes\operatorname{\mathrm{Ad}}\Sigma)x_{12}=(\operatorname{\mathrm{Ad}}\Sigma\otimes\operatorname{\mathrm{id}})x_{23}.$

In practice, there should be no ambiguity on where the unit acts.

For $C$ a C∗-algebra, we write $M(C)$ for the multiplier C∗-algebra of $C$. We
write $C^{**}$ for the universal W∗-envelope of $C$. The set of _quasi-
multipliers_ of $C$ is defined as the norm-closed subspace

$QM(C)=\\{x\in C^{**}\mid CxC\subseteq C\\}\subseteq C^{**}.$

We recall from [Ped79, Section 3.12] that whenever $C\subseteq B(\mathcal{H})$
is a faithful, non-degenerate $*$-representation, the corresponding normal
unital $*$-homomorphism $C^{**}\rightarrow B(\mathcal{H})$ is faithful on
$QM(C)$, and we may isometrically identify through this map

$QM(C)=\\{x\in B(\mathcal{H})\mid CxC\subseteq C\\}.$

##  1\. Locally compact quantum groups and their actions

We recall some basic results from the theory of locally compact quantum groups
[KV00, KV03, VV03], and their representations and actions [Kus01, Vae01]. The
following is [KV03, Definition 1.1], enhanced by [KV03, Proposition 3.1].

######  Definition 1.1.

A _locally compact quantum group $\mathbb{G}$_ consists of a Hopf-von Neumann
algebra $\mathbb{G}=(M,\Delta)$ possessing nsf weights
$\Phi,\Psi:M_{+}\rightarrow[0,+\infty]$ satisfying the _left_ and the _right_
invariance conditions

$(\operatorname{\mathrm{id}}\otimes\Phi)\Delta(x)=\Phi(x)1,\qquad(\Psi\otimes\operatorname{\mathrm{id}})\Delta(x)=\Psi(x)1,\qquad
x\in M_{+}.$ (1.1)

These weights are then necessarily unique up to multiplication by a positive
scalar.

Whenever convenient, we write
$M=L^{\infty}(\mathbb{G}),M_{*}=L^{1}(\mathbb{G})$ and
$L^{2}(M)=L^{2}(\mathbb{G})$. We identify $M\subseteq B(L^{2}(M))$ by the
standard representation, which we write explicitly as $\pi_{M}$ if we want to
emphasize it. We note that a locally compact quantum group automatically
satisfies the density conditions

$[(M\otimes 1)\Delta(M)]^{\sigma\textrm{-weak}}=[\Delta(M)(1\otimes
M)]^{\sigma-\textrm{weak}}=M\bar{\otimes}M.$ (1.2)

We write the induced _convolution product_ on $M_{*}$ as

$\omega*\omega^{\prime}:=(\omega\otimes\omega^{\prime})\Delta,\qquad\omega,\omega^{\prime}\in
M_{*}.$

Fundamental to the theory of locally compact quantum groups are the unitaries

$V\in B(L^{2}(M))\bar{\otimes}M,\qquad W\in M\bar{\otimes}B(L^{2}(M)),$

called respectively _right_ and _left_ regular unitary representation. They
are uniquely characterized by the identities

$(\operatorname{\mathrm{id}}\otimes\omega)(V)\Lambda_{\Psi}(x)=\Lambda_{\Psi}((\operatorname{\mathrm{id}}\otimes\omega)\Delta(x)),\qquad\omega\in
M_{*},x\in\mathscr{N}_{\Psi},$
$(\omega\otimes\operatorname{\mathrm{id}})(W^{*})\Lambda_{\Phi}(x)=\Lambda_{\Phi}((\omega\otimes\operatorname{\mathrm{id}})\Delta(x)),\qquad\omega\in
M_{*},x\in\mathscr{N}_{\Phi}.$

They are _multiplicative unitaries_ [BS93] in the sense that as operators on
$L^{2}(M)\otimes L^{2}(M)$, they satisfy

$V_{12}V_{13}V_{23}=V_{23}V_{12},\qquad W_{12}W_{13}W_{23}=W_{23}W_{12}.$

These multiplicative unitaries implement the coproduct of $M$:

$W^{*}(1\otimes x)W=\Delta(x)=V(x\otimes 1)V^{*},\qquad x\in M.$ (1.3)

Moreover, we have the equality

$[(\omega\otimes\operatorname{\mathrm{id}})V\mid\omega\in
B(L^{2}(M))_{*}]=[(\operatorname{\mathrm{id}}\otimes\omega)W\mid\omega\in
B(L^{2}(M))_{*}],$

and this defines a $\sigma$-weakly dense C∗-subalgebra of $M$ that we will
write as

$C_{0}^{\mathrm{red}}(\mathbb{G})\subseteq M.$

We recall some aspects of the duality theory of locally compact quantum groups
[KV03, Section 2]. We write

$\hat{M}=[(\omega\otimes\operatorname{\mathrm{id}})W\mid\omega\in
M_{*}]^{\sigma\textrm{-weak}},$

which can be shown to be a von Neumann algebra. It becomes a locally compact
quantum group for the coproduct

$\hat{\Delta}(x)=\Sigma W(x\otimes 1)W^{*}\Sigma,\qquad x\in\hat{M}.$

When useful for the sake of intuition or terminology, we write
$\hat{M}=\mathscr{L}(\mathbb{G})=\mathscr{L}^{\infty}(\hat{\mathbb{G}})$, and
view it as either the left group von Neumann algebra of $\mathbb{G}$, or a
function algebra on what is to be considered the Pontryagin dual
$\hat{\mathbb{G}}$ of $\mathbb{G}$.

As we will need to compare left and right coactions later on, it will also be
convenient to introduce independently the dual
$\check{M}=M^{\mathrm{cop}\wedge}$ of
$M^{\mathrm{cop}}=(M,\Delta^{\mathrm{op}})$, where
$\Delta^{\mathrm{op}}=\operatorname{\mathrm{Ad}}\Sigma\circ\Delta$. Noting
that $(M,\Delta^{\mathrm{op}})$ has the left and right regular unitary
representation equal to respectively $V_{21}^{*}$ and $W_{21}^{*}$, we have

$\check{M}=[(\operatorname{\mathrm{id}}\otimes\omega)V\mid\omega\in
M_{*}]^{\sigma\textrm{-weak}}$

with coproduct

$\check{\Delta}(x)=V^{*}(1\otimes x)V,\qquad x\in\check{M}.$

It can be shown that $\check{M}=\hat{M}^{\prime}$. We then also occasionally
write
$\check{M}=\mathscr{R}(\mathbb{G})=\mathscr{L}^{\infty}(\check{\mathbb{G}})$,
and view it as either the right group von Neumann algebra of $\mathbb{G}$, or
a function algebra on an ‘opposite’ Pontryagin dual $\check{\mathbb{G}}$ of
$\mathbb{G}$. We further note that

$C_{0}^{\mathrm{red}}(\hat{\mathbb{G}})=[(\omega\otimes\operatorname{\mathrm{id}})W\mid\omega\in
M_{*}],\qquad
C_{0}^{\mathrm{red}}(\check{\mathbb{G}})=[(\operatorname{\mathrm{id}}\otimes\omega)V\mid\omega\in
M_{*}].$ (1.4)

There is a canonical identification of $L^{2}(M)$ and $L^{2}(\check{M})$, in
such a way that

$\check{W}=V$

is the left regular representation of $\check{\mathbb{G}}$. With
$\check{J}=J_{\check{M}}$ the resulting associated modular conjugation for
$\check{M}$ on $L^{2}(M)$, we have

$\check{J}M\check{J}=M.$

One calls the induced anti-multiplicative involution

$R:M\rightarrow M,\quad x\mapsto\check{J}x^{*}\check{J}$

the _unitary antipode_. It turns $(M,\Delta)$ into an involutive Hopf-von
Neumann algebra. The following identities involving the modular conjugations
will be useful:

$(\check{J}\otimes J)W(\check{J}\otimes
J)=W^{*},\qquad(J\otimes\check{J})V(J\otimes\check{J})=V^{*}.$ (1.5)

We also recall from [KV03, Corollary 2.12] that one can pick a canonical
unimodular number $c\in\mathbb{C}$ such that

$c\check{J}J=\overline{c}J\check{J}.$

The resulting self-adjoint unitary $u_{\mathbb{G}}:=c\check{J}J$ is called the
_fundamental symmetry_. This unitary allows us to relate $W$ and $V$ directly,
for we have

$(u_{\mathbb{G}}\otimes 1)V(u_{\mathbb{G}}\otimes 1)=W_{21}.$ (1.6)

It also allows us to express the right regular unitary of $\check{M}$ as

$\check{V}=\widetilde{W}:=(u_{\mathbb{G}}\otimes
u_{\mathbb{G}})W(u_{\mathbb{G}}\otimes u_{\mathbb{G}}).$

We similarly write $\widetilde{V}=(u_{\mathbb{G}}\otimes
u_{\mathbb{G}})V(u_{\mathbb{G}}\otimes u_{\mathbb{G}})$. The operation
$M\mapsto\check{M}$ is not on the nose involutive, but we have that

$M^{\vee\vee}=u_{\mathbb{G}}Mu_{\mathbb{G}},\qquad\Delta^{\vee\vee}=\operatorname{\mathrm{Ad}}(u_{\mathbb{G}}\otimes
u_{\mathbb{G}})\circ\Delta\circ\operatorname{\mathrm{Ad}}(u_{\mathbb{G}}).$

Recall from Definition 0.2 the notion of a _unitary representation_ of
$\mathbb{G}$ on a Hilbert space $\mathcal{H}$. If $\omega\in M_{*}$, we write

$U(\omega)=(\operatorname{\mathrm{id}}\otimes\omega)U\in B(\mathcal{H}).$

Clearly

$U(\omega*\omega^{\prime})=U(\omega)U(\omega^{\prime}),\qquad\omega,\omega^{\prime}\in
M_{*}.$

For later use, we recall the following facts, see [Wor96, Theorem 1.6],
[Kus01] and [BDS13, Theorem 4.12].

######  Proposition 1.2.

If $U$ is a unitary $\mathbb{G}$-representation on a Hilbert space
$\mathcal{H}$, then the norm-closure

$C_{U}=[U(\omega)\mid\omega\in M_{*}]$

is a C∗-algebra acting non-degenerately on $\mathcal{H}$, and

$U\in M(C_{U}\otimes C_{0}^{\mathrm{red}}(\mathbb{G}))\subseteq
M(\mathcal{K}(\mathcal{H})\otimes C_{0}^{\mathrm{red}}(\mathbb{G}))\cap
M(C_{U}\otimes\mathcal{K}(L^{2}(\mathbb{G}))).$

The following results from [Vae01] will be crucial. Assume that $A$ is a
$\mathbb{G}$-W∗-algebra. The _crossed product_ von Neumann algebra is defined
as

$A\rtimes\mathbb{G}=\\{\alpha(a)(1\otimes x)\mid a\in
A,x\in\check{M}\\}^{\prime\prime}=[\alpha(a)(1\otimes x)\mid a\in
A,x\in\check{M}]^{\sigma\textrm{-weak}}\subseteq
A\bar{\otimes}B(L^{2}(\mathbb{G})).$

In the following, we write

$A^{\rtimes}:=A\rtimes\mathbb{G}$

as a shorthand. We note that $A^{\rtimes}$ carries the dual right
$\check{\mathbb{G}}$-action

$\alpha^{\rtimes}:A^{\rtimes}\rightarrow
A^{\rtimes}\bar{\otimes}\check{M},\quad\alpha^{\rtimes}(z)=\widetilde{W}_{23}z_{12}\widetilde{W}_{23}^{*},\qquad
z\in A^{\rtimes}.$

In particular, one has

$\alpha^{\rtimes}(\alpha(a))=\alpha(a)\otimes 1,\quad\alpha^{\rtimes}(1\otimes
x)=1\otimes\check{\Delta}(x),\qquad a\in A,x\in\check{M}.$

######  Theorem 1.3.

Let $A$ be a $\mathbb{G}$-W∗-algebra. Then there is a canonical identification

$L^{2}(A^{\rtimes})\cong L^{2}(A)\otimes L^{2}(\mathbb{G}),$

in such a way that

$U_{\alpha}=J_{A^{\rtimes}}(J_{A}\otimes\check{J})\in B(L^{2}(A)\otimes
L^{2}(\mathbb{G}))$ (1.7)

is a unitary $\mathbb{G}$-representation implementing $\alpha$ (in the sense
of (0.2)). Moreover, the associated GNS-representation $\pi_{A^{\rtimes}}$ of
$A^{\rtimes}$ becomes the identity representation of $A^{\rtimes}$ on
$L^{2}(A)\otimes L^{2}(\mathbb{G})$, while

$\rho_{A^{\rtimes}}(\alpha(a))=J_{A^{\rtimes}}\pi_{A^{\rtimes}}(a)^{*}J_{A^{\rtimes}}=\rho_{A}(a)\otimes
1,\qquad a\in A,$ $\rho_{A^{\rtimes}}(1\otimes
x)=J_{A^{\rtimes}}(1\otimes\pi_{\check{M}}(x)^{*})J_{A^{\rtimes}}=U_{\alpha}(1\otimes\rho_{\check{M}}(x))U_{\alpha}^{*},\qquad
x\in\check{M}.$

The dual $\check{\mathbb{G}}$-action has $\widetilde{W}_{23}$ as its canonical
unitary implementation.

Note that we make here the switch from the left coactions of [Vae01] to right
coactions by replacing $(M,\Delta)$ with $(M,\Delta^{\mathrm{op}})$.

###### Proof.

See [Vae01, Lemma 3.3, Definition 3.4, Definition 3.6 and Proposition 3.7].
The last claim follows from [Vae01, Proposition 4.3]. ∎

######  Remark 1.4.

From (1.7), we obtain the identity

$(J_{A}\otimes\check{J})U_{\alpha}(J_{A}\otimes\check{J})=U_{\alpha}^{*}.$
(1.8)

######  Remark 1.5.

When $A=M$ with $\alpha=\Delta$, we have $U_{\Delta}=V$.

We end this section with a concrete computation of a canonical implementing
unitary.

######  Example 1.6.

Let $A$ be a $\mathbb{G}$-W∗-algebra. We can endow
$A^{\rtimes}=A\rtimes\mathbb{G}$ with the _adjoint $\mathbb{G}$-action_

$\alpha_{\operatorname{\mathrm{ad}}}:A^{\rtimes}\rightarrow
A^{\rtimes}\bar{\otimes}M,\quad z\mapsto V_{23}z_{12}V_{23}^{*},\qquad z\in
A^{\rtimes},$ (1.9)

so

$\alpha_{\operatorname{\mathrm{ad}}}(\alpha(a))=(\operatorname{\mathrm{id}}\otimes\Delta)\alpha(a)=(\alpha\otimes\operatorname{\mathrm{id}})\alpha(a),\qquad\alpha_{\operatorname{\mathrm{ad}}}(1\otimes
x)=1\otimes V(x\otimes 1)V^{*},\qquad a\in A,x\in\check{M}.$

We claim that the canonical unitary implementation
$U_{\operatorname{\mathrm{ad}}}$ of $\alpha_{ad}$ is given by

$U_{\operatorname{\mathrm{ad}}}=V_{23}W_{32}U_{\alpha,13}.$ (1.10)

Indeed, note first that the crossed product von Neumann algebra
$A^{\rtimes}\rtimes\mathbb{G}$ is given by

$\displaystyle A^{\rtimes}\rtimes\mathbb{G}$ $\displaystyle=$
$\displaystyle[\alpha_{\operatorname{\mathrm{ad}}}(\alpha(a)(1\otimes
x))(1\otimes 1\otimes y)\mid a\in A,x,y\in\check{M}]^{\sigma\textrm{-weak}}$
$\displaystyle=$ $\displaystyle[V_{23}\alpha(a)_{12}x_{2}V_{23}^{*}y_{3}\mid
a\in A,x,y\in\check{M}]^{\sigma\textrm{-weak}}$ $\displaystyle=$
$\displaystyle[V_{23}\alpha(a)_{12}x_{2}\check{\Delta}(y)_{23}V_{23}^{*}\mid
a\in A,x,y\in\check{M}]^{\sigma\textrm{-weak}}$
$\displaystyle\underset{\eqref{EqDensGal}}{=}$
$\displaystyle[V_{23}\alpha(a)_{12}(1\otimes x\otimes y)V_{23}^{*}\mid a\in
A,x,y\in\check{M}]^{\sigma\textrm{-weak}}$ $\displaystyle=$ $\displaystyle
V_{23}(A^{\rtimes}\bar{\otimes}\check{M})V_{23}^{*},$

Since the dual coaction of $\check{M}$ on $A^{\rtimes}\rtimes\mathbb{G}$ is
given by conjugation with $\widetilde{W}_{34}$, we see that
$A^{\rtimes}\rtimes\mathbb{G}$ is just a conjugated copy of
$A^{\rtimes}\bar{\otimes}\check{M}$ with
$\operatorname{\mathrm{id}}_{A^{\rtimes}}\otimes\check{\Delta}$ as the
coaction. In particular, under the canonical identification

$L^{2}(A^{\rtimes}\rtimes\mathbb{G})=L^{2}(A^{\rtimes})\otimes
L^{2}(\mathbb{G})=L^{2}(A)\otimes L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}),$

the associated modular conjugation of $A^{\rtimes}\rtimes\mathbb{G}$ is given
by

$J_{A^{\rtimes}\rtimes\mathbb{G}}=V_{23}(J_{A^{\rtimes}}\otimes\check{J})V_{23}^{*},$

and so the canonical unitary implementation of
$\alpha_{\operatorname{\mathrm{ad}}}$ is by Theorem 1.3 given by

$\displaystyle U_{\alpha_{\operatorname{\mathrm{ad}}}}$ $\displaystyle=$
$\displaystyle
J_{A^{\rtimes}\rtimes\mathbb{G}}(J_{A^{\rtimes}}\otimes\check{J})$
$\displaystyle=$ $\displaystyle
V_{23}(J_{A^{\rtimes}}\otimes\check{J})V_{23}^{*}(J_{A^{\rtimes}}\otimes\check{J})$
$\displaystyle=$ $\displaystyle
V_{23}U_{\alpha,12}(J_{A}\otimes\check{J}\otimes\check{J})V_{23}^{*}(J_{A}\otimes\check{J}\otimes\check{J})U_{\alpha,12}^{*}$
$\displaystyle\underset{\eqref{EqMultUnJhatJ},\eqref{EqIdVW}}{=}$
$\displaystyle V_{23}U_{\alpha,12}W_{32}U_{\alpha,12}^{*}$
$\displaystyle\underset{\eqref{EqUnitRepDef}}{=}$ $\displaystyle
V_{23}W_{32}U_{\alpha,13},$

proving (1.10).

##  2\. Dynamical W∗-correspondences

Let $\mathbb{G}$ be a locally compact quantum group, and let
$(A,\alpha),(B,\beta)$ be $\mathbb{G}$-dynamical von Neumann algebras. In
Definition 0.4, we introduced the notion of a $\mathbb{G}$-equivariant
$A$-$B$-correspondence $\mathcal{H}=(\mathcal{H},U,\pi,\rho)$. Note that if
$\mathbb{G}$ is the trivial group, we find back the usual notion of an
$A$-$B$-correspondence. On the other hand, when $A=B=\mathbb{C}$, we find back
the notion of a unitary $\mathbb{G}$-representation.

######  Definition 2.1.

We denote by $\mathrm{Corr}^{\mathbb{G}}(A,B)$ the category of
$\mathbb{G}$-$A$-$B$-correspondences, with morphisms from $\mathcal{H}$ to
$\mathcal{H}^{\prime}$ consisting of bounded operators
$T:\mathcal{H}\rightarrow\mathcal{H}^{\prime}$ intertwining the
(anti-)representations of $A,B$ and $M_{*}$.

Clearly, $\mathrm{Corr}^{\mathbb{G}}(A,B)$ is a W∗-category [GLR85], and we
will always understand it as such.

Here are some examples of $\mathbb{G}$-equivariant correspondences.

######  Example 2.2.

Let $A$ be a $\mathbb{G}$-dynamical von Neumann algebra. The _trivial_ or
_identity_ $\mathbb{G}$-$A$-$A$-correspondence $E_{A}^{\mathbb{G}}$ is
$L^{2}(A)$ with its canonical representation $\pi_{A}$ and anti-representation
$\rho_{A}$, and the canonical unitary $\mathbb{G}$-representation $U_{\alpha}$
implementing $\alpha$.

The compatibility relations (0.3) are satisfied by Theorem 1.3 and the
identity (1.8), together with the fact that $\check{J}$ implements the unitary
antipode.

Let $(A,\alpha)$ and $(B,\beta)$ be $\mathbb{G}$-dynamical von Neumann
algebras. We say that $A\subseteq B$ is an _equivariant inclusion_ if
$\beta_{\mid A}=\alpha$.

######  Example 2.3.

Let $A\subseteq C$ and $B\subseteq D$ be $\mathbb{G}$-equivariant inclusions.
Then for any $\mathbb{G}$-$C$-$D$ correspondence $\mathcal{H}$ we obtain the
restriction

${}_{A}\mathrm{Res}_{B}(\mathcal{H})={}_{A}\mathcal{H}_{B}.$

In particular, if $A\subseteq B$ is an equivariant inclusion, we have
equivariant correspondences ${}_{A}L^{2}(B)_{B}$, ${}_{B}L^{2}(B)_{A}$ and
${}_{A}L^{2}(B)_{A}$, each with $U_{\beta}$ as unitary representation for
$\beta$ the given coaction on $B$.

######  Example 2.4.

Let $A,B$ be $\mathbb{G}$-dynamical von Neumann algebras, and let
$(\mathcal{H},U,\pi,\rho)$ be a $\mathbb{G}$-equivariant correspondence.
Consider $C:=\rho(B)^{\prime}$. Then

$\gamma:C\rightarrow C\bar{\otimes}M,\qquad c\mapsto U(c\otimes 1)U^{*}$

is a well-defined coaction, and $({}_{C}\mathcal{H}_{B},U)$ is a
$\mathbb{G}$-equivariant $C$-$B$-correspondence. Moreover, $\pi(A)\subseteq C$
is a $\mathbb{G}$-equivariant inclusion.

As a particular case, let $\pi:A\rightarrow B$ be a normal unital
$\mathbb{G}$-equivariant $*$-homomorphism. Consider
$C:=\rho_{B}(\pi(A))^{\prime}$, so that
$\pi_{B}(\pi(A))\subseteq\pi_{B}(B)\subseteq C$ is the basic construction.
Then

$\gamma:C\rightarrow C\bar{\otimes}M,\qquad c\mapsto U_{\beta}(c\otimes
1)U_{\beta}^{*}$

is a well-defined coaction, and $B\cong\pi_{B}(B)\subseteq C$ becomes a
$\mathbb{G}$-equivariant inclusion of von Neumann algebras.

######  Example 2.5.

The _coarse_ (or regular) $\mathbb{G}$-$A$-$B$-correspondence
$C_{A,B}^{\mathbb{G}}$ is $L^{2}(A)\otimes L^{2}(\mathbb{G})\otimes L^{2}(B)$
endowed with

$\pi(a)=(\pi_{A}\otimes\pi_{M})(\alpha(a))_{12},\qquad\rho(b)=(\rho_{M}\otimes\rho_{B})(\beta^{\mathrm{op}}(b))_{23},$
$U=V_{24}\in B(L^{2}(A)\otimes L^{2}(\mathbb{G})\otimes L^{2}(B)\otimes
L^{2}(\mathbb{G})).$

The necessary compatibilities are satisfied by using the coaction property
together with the identities (1.3) and (1.5). The same reasoning works for the
next examples.

######  Example 2.6.

Let $(\mathcal{H},U,\pi,\rho)$ be a $\mathbb{G}$-$A$-$B$-correspondence. We
define the _regular amplification_ to be the
$\mathbb{G}$-$A$-$B$-correspondence
$S^{\mathbb{G}}(\mathcal{H})=\mathcal{H}\otimes L^{2}(\mathbb{G})$ endowed
with

$\pi_{S}(a)=(\pi\otimes\pi_{M})(\alpha(a)),\qquad\rho_{S}(b)=(\rho\otimes\rho_{M})(\beta(b)),$
$U_{S}=V_{23}\in B(L^{2}(A)\otimes L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G})).$

It is a restriction of ${}_{A\bar{\otimes}M}(\mathcal{H}\otimes
L^{2}(\mathbb{G}))_{B\bar{\otimes}M}$ where $A\bar{\otimes}M,B\bar{\otimes}M$
are endowed with the coaction $\operatorname{\mathrm{id}}\otimes\Delta$, and
where we view $L^{2}(\mathbb{G})=E_{M}$.

In particular, we call $S_{A}^{\mathbb{G}}:=S^{\mathbb{G}}(E_{A})$ the
_$\mathbb{G}$ -semi-coarse_ (or $\mathbb{G}$-semi-regular)
$\mathbb{G}$-$A$-$A$-correspondence.

Note that $S_{A}^{\mathbb{G}}$ becomes the trivial $A$-$A$-correspondence if
$\mathbb{G}$ is trivial, and the regular $\mathbb{G}$-representation if $A$ is
trivial. The following example reverses this situation.

######  Example 2.7.

The _W ∗-semi-coarse_ (or W∗-semi-regular) $\mathbb{G}$-$A$-$B$-correspondence
$S_{A,B}^{\mathbb{G}}$ is $L^{2}(A)\otimes L^{2}(B)$ endowed with

$\pi(a)=\pi_{A}(a)\otimes 1,\qquad\rho(b)=1\otimes\rho_{B}(b),$
$U=U_{\alpha,13}U_{\beta,23}\in B(L^{2}(A)\otimes L^{2}(B)\otimes
L^{2}(\mathbb{G})).$

######  Remark 2.8.

We can make the coarse $\mathbb{G}$-$A$-$B$-correspondence
$C_{A,B}^{\mathbb{G}}$ resemble more the W∗-semi-coarse
$\mathbb{G}$-$A$-$B$-correspondence $S^{\mathbb{G}}_{A,B}$ as follows: write

$\widetilde{U}_{B}=(u_{\mathbb{G}}\otimes
1)U_{\beta,21}^{*}(u_{\mathbb{G}}\otimes
1)\in\rho_{M}(M)\bar{\otimes}B(L^{2}(B)),$ (2.1)

then from the identities (1.5) and (1.6) we find that

$\widetilde{U}_{B,12}^{*}V_{13}\widetilde{U}_{B,12}=V_{13}U_{\beta,23}.$

So, conjugating the coarse $\mathbb{G}$-$A$-$B$-correspondence with the
unitary $\widetilde{U}_{B,23}^{*}U_{\alpha,12}^{*}$, we find that
$C_{A,B}^{\mathbb{G}}$ is equivalent to the
$\mathbb{G}$-$A$-$B$-correspondence on $L^{2}(A)\otimes
L^{2}(\mathbb{G})\otimes L^{2}(B)$ given by

$\pi(a)=\pi_{A}(a)\otimes 1\otimes 1,\qquad\rho(b)=1\otimes
1\otimes\rho_{B}(b),$ $U=U_{\alpha,14}V_{24}U_{\beta,34}.$

As a final example, we consider the following.

######  Example 2.9.

Let $(\mathcal{H},U,\pi,\rho)$ be a $\mathbb{G}$-$A$-$B$-correspondence. The
_adjoint amplification_ is the $\mathbb{G}$-$A$-$B$-correspondence
$D^{\mathbb{G}}(\mathcal{H})=\mathcal{H}\otimes L^{2}(\mathbb{G})$ endowed
with

$\pi_{D}(a)=(\pi\otimes\pi_{M})(\alpha(a)),\qquad\rho_{D}(b)=\rho(b)\otimes
1,$ $U_{D}=V_{23}W_{32}U_{13}.$

Note that $U_{D}$ is indeed a unitary $\mathbb{G}$-representation: it is the
tensor product of $U_{\operatorname{\mathrm{ad}}}$ with $U$, where
$U_{\operatorname{\mathrm{ad}}}=W_{21}V$ is the adjoint representation of
$\mathbb{G}$ on $L^{2}(\mathbb{G})$. The compatibility of $U_{D}$ and
$\pi_{D}$ follows from the fact that $U$ implements $\alpha$, together with
the easily computed commutation relation

$V_{23}W_{32}U_{13}U_{12}=U_{12}U_{13}V_{23}W_{32}.$ (2.2)

The compatibility of $U_{D}$ and $\rho_{D}$ is immediate.

In particular, we call $D_{A}^{\mathbb{G}}:=D^{\mathbb{G}}(E_{A})$ the
_adjoint_ $\mathbb{G}$-$A$-$A$-correspondence. Using Example 1.6, we see that
$D_{A}^{\mathbb{G}}$ is simply the two-sided restriction to $A$ of the
identity correspondence $E_{A\rtimes\mathbb{G}}$ for $A\rtimes\mathbb{G}$ with
the adjoint $\mathbb{G}$-action.

There are some obvious constructions one can perform on
$\mathbb{G}$-equivariant correspondences, such as

  * •

taking (infinite) direct sums, and

  * •

taking $\mathbb{G}$-equivariant subcorrespondences.

Let us make the latter more precise: if $(\mathcal{H},U,\pi,\rho)$ is a
$\mathbb{G}$-$A$-$B$-correspondence, we define
$\mathcal{G}\subseteq\mathcal{H}$ to be a $\mathbb{G}$-equivariant
subcorrespondence if $\mathcal{G}$ is stable under $\pi(A)$, $\rho(B)$ and all
$U(\omega)$ for $\omega\in M_{*}$. It follows from Proposition 1.2 that
$\mathcal{G}$ becomes a $\mathbb{G}$-$A$-$B$-correspondence
$(\mathcal{G},U^{\prime})$ in a unique way such that $U^{\prime}(\omega)$ is
the restriction of $U(\omega)$ to $\mathcal{G}$. Moreover, this description
also makes it clear that $\mathcal{G}^{\perp}$ will be a
$\mathbb{G}$-equivariant subcorrespondence, with
$\mathcal{H}=\mathcal{G}\oplus\mathcal{G}^{\perp}$ as
$\mathbb{G}$-$A$-$B$-correspondences.

######  Definition 2.10.

We call a $\mathbb{G}$-$A$-$B$-correspondence $(\mathcal{H},U,\pi,\rho)$
_cyclic_ if there exists $\xi\in\mathcal{H}$ such that

$\mathcal{H}=[\pi(a)U(\omega)\rho(b)\xi\mid a\in A,\omega\in M_{*},b\in B].$

######  Proposition 2.11.

Any $\mathbb{G}$-$A$-$B$-correspondence $(\mathcal{H},U,\pi,\rho)$ is a direct
sum of cyclic $\mathbb{G}$-$A$-$B$-correspondences.

###### Proof.

Take $\xi\in\mathcal{H}$. It is sufficient to show that

$\mathcal{G}:=[\pi(a)U(\omega)\rho(b)\xi\mid a\in A,\omega\in M_{*},b\in B]$

is a $\mathbb{G}$-equivariant subcorrespondence: since the C∗-algebra
$C_{U}=[\\{U(\omega)\mid\omega\in M_{*}\\}]$ acts non-degenerately on
$\mathcal{H}$, we have $\xi\in\mathcal{G}$, so $\mathcal{G}$ will indeed be
cyclic, and an appeal to Zorn’s lemma then lets us conclude.

Clearly $\mathcal{G}$ is invariant under $\pi(A)$. If $e_{i}$ is an
orthonormal basis of $L^{2}(\mathbb{G})$, then for $\xi,\eta\in
L^{2}(\mathbb{G})$ and $\omega=\omega_{\xi,\eta}\in B(L^{2}(\mathbb{G}))_{*}$,
we have

$U(\omega)\pi(a)=(\operatorname{\mathrm{id}}\otimes\omega)((\pi\otimes\operatorname{\mathrm{id}})\alpha(a)U)=\sum_{i}\pi((\operatorname{\mathrm{id}}\otimes\omega_{\xi,e_{i}})(\alpha(a)))U(\omega_{e_{i},\eta}),$

the latter sum converging e.g. $\sigma$-strongly. Hence $\mathcal{G}$ is
invariant under the $U(\omega)$.

Similarly,

$\rho(b)U(\omega)=(\operatorname{\mathrm{id}}\otimes\omega)(U(\rho\otimes
R)(\beta(b)))=\sum_{i}U(\omega_{\xi,e_{i}})\rho((\operatorname{\mathrm{id}}\otimes\omega_{e_{i},\eta}R)(\beta(b))),$

making it clear that $\mathcal{G}$ is invariant under $\rho(B)$ as well. ∎

It follows easily from the above that we can make a _universal_
$\mathbb{G}$-$A$-$B$-correspondence
$(\mathcal{H}_{u},U_{u},\pi_{u},\rho_{u})$, in the sense that any cyclic
$\mathbb{G}$-$A$-$B$-correspondence is unitarily equivalent to a direct
summand of $\mathcal{H}_{u}$. In the following, we fix such a chosen universal
model.

######  Definition 2.12.

Let $A$ be a $\mathbb{G}$-W∗-algebra. We define

$A_{\mathrm{s}}=[(\operatorname{\mathrm{id}}\otimes\omega)\alpha(a)\mid\omega\in
M_{*},a\in A]\subseteq A.$

It is not clear if $A_{\mathrm{s}}$ will always be a C∗-subalgebra of $A$,
although it is clearly an operator subsystem.111In case $\mathbb{G}$ is semi-
regular, the proof of [BSV03, Proposition 5.7] shows that $A_{s}$ is indeed a
C∗-algebra, but it is not clear if the converse in that Proposition holds in
the von Neumann algebraic setting. In any case, we will not need any
multiplicative structure of $A_{\mathrm{s}}$. Rather the following observation
will be fundamental.

######  Theorem 2.13.

If $(\mathcal{H},U,\pi,\rho)$ is a $\mathbb{G}$-$A$-$B$-correspondence, then

$C^{\mathbb{G}}_{\mathcal{H}}(A):=[C_{U}\pi(A)C_{U}],\qquad
C^{\mathbb{G}}_{\mathcal{H}}(B):=[C_{U}\rho(B)C_{U}],\qquad
C^{\mathbb{G}}_{\mathcal{H}}(A,B):=[C^{\mathbb{G}}_{\mathcal{H}}(A)C^{\mathbb{G}}_{\mathcal{H}}(B)]$
(2.3)

are C∗-algebras, and

$C^{\mathbb{G}}_{\mathcal{H}}(A)=[\pi(A_{\mathrm{s}})C_{U}]=[C_{U}\pi(A_{\mathrm{s}})],\quad
C^{\mathbb{G}}_{\mathcal{H}}(B)=[\rho(B_{\mathrm{s}})C_{U}]=[C_{U}\rho(B_{\mathrm{s}})].$
(2.4)

###### Proof.

Clearly $C^{\mathbb{G}}_{\mathcal{H}}(A)$ and
$C^{\mathbb{G}}_{\mathcal{H}}(B)$ are closed under the $*$-operation.

We now first prove (2.4). Observe that by (1.4), we have

$[(\operatorname{\mathrm{id}}\otimes\omega)(V)\mid\omega\in
B(L^{2}(\mathbb{G}))_{*}]=[(\operatorname{\mathrm{id}}\otimes\omega)(V^{*})\mid\omega\in
B(L^{2}(\mathbb{G}))_{*}].$

We then compute (dropping the notation $\pi$)

$\displaystyle C^{\mathbb{G}}_{\mathcal{H}}(A)$ $\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(U_{12}a_{1}U_{13})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(\alpha(a)_{12}U_{12}U_{13})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(\alpha(a)_{12}(\operatorname{\mathrm{id}}\otimes\Delta)(U))\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(\alpha(a)_{12}V_{23}U_{12}V_{23}^{*})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(\alpha(a)_{12}V_{23}U_{12})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(\alpha(a)_{12}V_{23}^{*}U_{12})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(V_{23}^{*}(\operatorname{\mathrm{id}}\otimes\Delta)\alpha(a)U_{12})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)((\alpha\otimes\operatorname{\mathrm{id}})\alpha(a)U_{12})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$
$\displaystyle[(\operatorname{\mathrm{id}}\otimes\omega)(U_{12}\alpha(a)_{13})\mid
a\in A,\omega\in B(L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G}))_{*}]$
$\displaystyle=$ $\displaystyle[C_{U}A_{\mathrm{s}}].$

Since $C_{\mathcal{H}}^{\mathbb{G}}(A)$ is closed under the $*$-operation, we
obtain (2.4) for $A$. It is now clear that $C_{\mathcal{H}}^{\mathbb{G}}(A)$
is a C∗-algebra:

$[C_{\mathcal{H}}^{\mathbb{G}}(A)C_{\mathcal{H}}^{\mathbb{G}}(A)]=[C_{U}A_{\mathrm{s}}^{2}C_{U}]\subseteq
C_{\mathcal{H}}^{\mathbb{G}}(A).$

For $B$, we can simply observe that $B^{\mathrm{op}}$ has a right coaction by
$(M,\Delta^{\mathrm{op}})$ via

$b^{\mathrm{op}}\mapsto\beta(b)^{\mathrm{op}\otimes R},$

and that $U^{*}$ is a unitary representation for $(M,\Delta^{\mathrm{op}})$
with

$U^{*}(\rho(b)\otimes 1)U=(\rho\otimes R)\beta(b),\qquad b\in B.$

Hence the result for $B$ follows from that for $A$ (since $C_{U}=C_{U^{*}}$).

Finally, $C^{\mathbb{G}}_{\mathcal{H}}(A,B)$ is closed under the $*$-structure
since

$C^{\mathbb{G}}_{\mathcal{H}}(A,B)^{*}=[C_{U}\rho(B_{\mathrm{s}})\pi(A_{\mathrm{s}})C_{U}]=[C_{U}\pi(A_{\mathrm{s}})\rho(B_{\mathrm{s}})C_{U}]=[C_{U}\pi(A)C_{U}C_{U}\rho(B)C_{U}]=C^{\mathbb{G}}_{\mathcal{H}}(A,B).$

A similar computation shows that $C^{\mathbb{G}}_{\mathcal{H}}(A,B)$ is closed
under multiplication. ∎

######  Remark 2.14.

It also follows immediately from (2.3) and (2.4) that for example

$C^{\mathbb{G}}_{\mathcal{H}}(A)=[C_{U}\pi(A_{\mathrm{s}})C_{U}],\qquad
C^{\mathbb{G}}_{\mathcal{H}}(A,B)=[\pi(A_{\mathrm{s}})C_{U}\rho(B_{\mathrm{s}})],$

and that

$\pi(A)\subseteq QM(C_{\mathcal{H}}^{\mathbb{G}}(A,B)),\qquad\rho(B)\subseteq
QM(C_{\mathcal{H}}^{\mathbb{G}}(A,B)).$

######  Theorem 2.15.

There is a well-defined semi-norm on $A_{\mathrm{s}}\odot M_{*}\odot
B_{\mathrm{s}}$ through

$\|\sum_{i=1}^{n}a_{i}\otimes\omega_{i}\otimes
b_{i}\|_{u}=\sup\\{\|\sum_{i=1}^{n}\pi(a_{i})U(\omega_{i})\rho(b_{i})\|\mid(\mathcal{H},U)\textrm{
a }\mathbb{G}\textrm{-}A\textrm{-}B\textrm{-correspondence}\\}.$ (2.5)

Moreover, the associated separation-completion $C^{\mathbb{G}}(A,B)$ has the
unique structure of a C∗-algebra such that, for each
$\mathbb{G}$-$A$-$B$-correspondence $(\mathcal{H},U,\pi,\rho)$, the map

$\theta=\theta_{\mathcal{H}}:C^{\mathbb{G}}(A,B)\rightarrow
B(\mathcal{H}),\quad a\otimes\omega\otimes b\mapsto\pi(a)U(\omega)\rho(b)$

is a well-defined non-degenerate $*$-representation.

###### Proof.

Clearly $\|\textrm{---}\|_{u}$ is well-defined as a semi-norm, with

$\|\sum_{i=1}^{n}a_{i}\otimes\omega_{i}\otimes
b_{i}\|_{u}\leq\sum_{i=1}^{n}\|a_{i}\|\|\omega_{i}\|\|b_{i}\|.$

Now by Proposition 2.11, the supremum in (2.5) does not change if we consider
only cyclic $\mathbb{G}$-equivariant $A$-$B$-correspondences. Hence we see
that $\|-\|_{u}$ is in fact realized by the universal $\mathbb{G}$-equivariant
$A$-$B$-correspondence $(\mathcal{H}_{u},U_{u},\pi_{u},\rho_{u})$, i.e.
$C^{\mathbb{G}}(A,B):=C_{\mathcal{H}_{u}}^{\mathbb{G}}(A,B)$ satisfies the
requirements. ∎

In the following, we identify
$C^{\mathbb{G}}(A,B)=C_{\mathcal{H}_{u}}^{\mathbb{G}}(A,B)$ whenever
convenient. We do have to be a bit careful when using this identification, as
we want to make sure that constructions do not actually depend on the precise
model chosen. Here is a case in point: By means of Remark 2.14, we have a
natural map $\pi_{\mathcal{H}_{u}}:A\rightarrow
QM(C_{\mathcal{H}_{u}}^{\mathbb{G}}(A,B))$, which through the isomorphism

$\theta_{\mathcal{H}_{u}}:C^{\mathbb{G}}(A,B)\cong
C^{\mathbb{G}}_{\mathcal{H}_{u}}(A,B)$

we can pull back to a map

$\pi_{u}:A\rightarrow QM(C^{\mathbb{G}}(A,B)).$ (2.6)

We want to make sure for example that this map does not depend on the chosen
model.

######  Lemma 2.16.

The map (2.6) is independent of the chosen model.

###### Proof.

It is enough to show the following: if $(\mathcal{H},U,\pi,\rho)$ is any
$\mathbb{G}$-equivariant $A$-$B$-correspondence, then (using the same symbol
to denote extensions of non-degenerate $*$-homomorphisms to their quasi-
multipliers)

$\theta_{\mathcal{H}}(\theta_{\mathcal{H}_{u}}^{-1}(\pi_{u}(a)))=\pi_{\mathcal{H}}(a),\qquad\forall
a\in A.$

Since any $\mathcal{H}$ is a direct sum of cyclic ones, it is enough to assume
that $\mathcal{H}$ is cyclic as a $\mathbb{G}$-$A$-$B$-correspondence. But
then since $\mathcal{H}_{u}$ contains any cyclic
$\mathbb{G}$-$A$-$B$-correspondence up to unitary equivalence, we may as well
assume that $\mathcal{H}\subseteq\mathcal{H}_{u}$. It is now easily seen that
if $x\in QM(C^{\mathbb{G}}(A,B))$, then

$\theta_{\mathcal{H}}(x)=\theta_{\mathcal{H}_{u}}(x)_{\mid\mathcal{H}}.$

In particular, taking $x=\pi_{u}(a)$ for $a\in A$, we see that

$\theta_{\mathcal{H}}(\theta_{\mathcal{H}_{u}}^{-1}(\pi_{u}(a)))=\pi_{\mathcal{H}_{u}}(a)_{\mid\mathcal{H}}=\pi_{\mathcal{H}}(a).$

∎

Similarly, one shows that there is a canonical map $B\to QM(C{G}(A,B))$.

Also the following lemma is convenient to have around: it states that the
functor
$\mathrm{Corr}^{\mathbb{G}}(A,B)\rightarrow\operatorname{\mathrm{Rep}}_{*}(C^{\mathbb{G}}(A,B))$
is full, where $\operatorname{\mathrm{Rep}}_{*}(C)$ denotes the W∗-category of
non-degenerate representations of a C∗-algebra $C$.

######  Lemma 2.17.

If
$(\mathcal{H},U,\pi,\rho),(\widetilde{\mathcal{H}},\widetilde{U},\widetilde{\pi},\widetilde{\rho})$
are $\mathbb{G}$-$A$-$B$-correspondences, then an element $x\in
B(\mathcal{H},\widetilde{\mathcal{H}})$ is an intertwiner of
$\mathbb{G}$-$A$-$B$-correspondences if and only if it is an intertwiner of
$C^{\mathbb{G}}(A,B)$-correspondences.

###### Proof.

Clearly an intertwiner of $\mathbb{G}$-$A$-$B$-correspondences is also an
intertwiner of $C^{\mathbb{G}}(A,B)$-representations, by construction of
$\theta_{\mathcal{H}}$. Conversely, if $x$ intertwines $\theta_{\mathcal{H}}$
and $\theta_{\widetilde{\mathcal{H}}}$, we have (again by construction) that

$x\pi(a)U(\omega)\rho(b)=\widetilde{\pi}(a)\widetilde{U}(\omega)\widetilde{\rho}(b)x,\qquad\forall
a\in A_{s},\omega\in M_{*},b\in B_{s}.$

Taking $a=1$ and $b=1$, we find $xU(\omega)=\widetilde{U}(\omega)x$ for all
$\omega\in M_{*}$. But then also (taking $b=1$)

$x\pi(a)U(\omega)=\widetilde{\pi}(a)\widetilde{U}(\omega)x=\widetilde{\pi}(a)xU(\omega),\qquad\forall
a\in A_{s},\omega\in M_{*}$

so that by non-degeneracy of $C_{U}$ we find $x\pi(a)=\widetilde{\pi}(a)x$ for
$a\in A_{s}$. Since $A_{s}$ is $\sigma$-weakly dense in $A$, we find
$x\pi(a)=\widetilde{\pi}(a)x$ for $a\in A$. Similarly
$x\rho(b)=\widetilde{\rho}(b)x$ for $b\in B$. Thus, $x$ is an intertwiner of
$\mathbb{G}$-$A$-$B$-correspondences. ∎

We define a topology on the objects of $\mathrm{Corr}^{\mathbb{G}}(A,B)$ as
follows.

######  Definition 2.18.

Let $(\mathcal{H},U,\pi,\rho)$ be a $\mathbb{G}$-$A$-$B$-correspondence. Let
$F$ be a collection of a scalar $\varepsilon>0$ and finite subsets
$F_{A},F_{B},F_{\mathbb{G}}$ and
$F_{\mathcal{H}}=\\{\xi_{1},\ldots,\xi_{n}\\}$ in resp. $A,B,M_{*}$ and
$\mathcal{H}$.

We say that a $\mathbb{G}$-$A$-$B$-correspondence
$(\mathcal{H}^{\prime},U^{\prime},\pi^{\prime},\rho^{\prime})$ lies in the
_$F$ -neighborhood_ $U(\mathcal{H};F)$ of $(\mathcal{H},U,\pi,\rho)$ if there
exist $\xi_{1}^{\prime},\ldots,\xi_{n}^{\prime}$ in $\mathcal{H}^{\prime}$
with $\|\xi_{i}^{\prime}\|\leq\|\xi_{i}\|$ for all $i$, such that, for all
$a\in F_{A},b\in F_{B},\omega\in F_{\mathbb{G}}$ and $1\leq i,j\leq n$,

$|\langle\xi_{i},\pi(a)U(\omega)\rho(b)\xi_{j}\rangle-\langle\xi_{i}^{\prime},\pi^{\prime}(a)U^{\prime}(\omega)\rho^{\prime}(b)\xi_{j}^{\prime}\rangle|<\varepsilon.$
(2.7)

The $F$ form a directed set in a natural way. Since for any
$\mathbb{G}$-equivariant correspondence $\mathcal{H}$ and its finite
collection of neighborhoods $U(\mathcal{H};F_{i})$, we have

$U(\mathcal{H};F)\subseteq\cap_{i=1}^{n}U(\mathcal{H};F_{i})$ (2.8)

for suitable choice of $F$, and since for each $\mathcal{H}^{\prime}\in
U(\mathcal{H};F)$ we can find $F^{\prime}$ such that

$U(\mathcal{H}^{\prime};F^{\prime})\subseteq U(\mathcal{H};F),$ (2.9)

it is clear that the above $U(\mathcal{H};F)$ form a local basis of open
neighborhoods for a unique topology $\tau$ on
$\mathrm{Corr}^{\mathbb{G}}(A,B)$.222Of course, the complete family of objects
of $\mathrm{Corr}^{\mathbb{G}}(A,B)$ does not form a set, so it should be
explained what is actually meant by this. But since we will really only use
the local properties (2.9) and (2.10), we will gloss over this point.
Moreover, we have the following additional property: for any finite collection
of $\mathbb{G}$-equivariant correspondences
$\mathcal{H}_{1},\ldots,\mathcal{H}_{n}$ and neighborhoods
$U(\mathcal{H}_{i};F_{i})$, we have

$U(\oplus_{i=1}^{n}\mathcal{H}_{i};F)\subseteq\cap_{i=1}^{n}U(\mathcal{H}_{i};F_{i})$
(2.10)

for suitable $F$.

######  Remark 2.19.

The ordering of the factors in (2.7) is important: in general, one seems to
get a different (and less natural) notion if for example the positions of
$U(\omega)$ and $\pi(a)$ are interchanged. However, one may interchange the
positions of $\pi(a)$ and $\rho(b)$ at no cost, as will follow easily from
Theorem 2.23.

Recall now from Theorem 2.15 that any $\mathbb{G}$-$A$-$B$-correspondence
$\mathcal{H}$ leads to a (unique) non-degenerate $*$-representation
$(\mathcal{H},\theta_{\mathcal{H}})$ of $C^{\mathbb{G}}(A,B)$. We will show
that the $\tau$-topology on $\mathrm{Corr}^{\mathbb{G}}(A,B)$ is simply the
restriction of the corresponding ‘Zimmer topology’ on the W∗-category
$\operatorname{\mathrm{Rep}}_{*}(C^{\mathbb{G}}(A,B))$ [Zim84] (see also
[BdlHV08, Appendix F]), passing from the group case to the C∗-algebra setting.

######  Definition 2.20.

Let $C$ be a C∗-algebra, and let $(\mathcal{H},\theta)$ be a non-degenerate
$*$-representation of $C$. Let $F$ consist of $\varepsilon>0$ and finite sets
$F_{C}\subseteq C$ and
$F_{\mathcal{H}}=\\{\xi_{1},\ldots,\xi_{n}\\}\subseteq\mathcal{H}$.

If $(\mathcal{H}^{\prime},\theta^{\prime})$ is a non-degenerate
$*$-representation of $C$, we say that
$(\mathcal{H}^{\prime},\theta^{\prime})$ lies in the $F$-neighborhood
$V(\mathcal{H};F)$ of $(\mathcal{H},\theta)$ if there exist
$\xi_{1}^{\prime},\ldots,\xi_{n}^{\prime}\in\mathcal{H}^{\prime}$ such that
$\|\xi_{i}^{\prime}\|\leq\|\xi_{i}\|$ for all $i$, and, for all $c\in F_{C}$
and $1\leq i,j\leq n$,

$|\langle\xi_{i},\theta(c)\xi_{j}\rangle-\langle\xi_{i}^{\prime},\theta^{\prime}(c)\xi_{j}^{\prime}\rangle|<\varepsilon.$

We denote the resulting topology on $\operatorname{\mathrm{Rep}}_{*}(C)$ by
$\sigma$.

######  Remark 2.21.

By means of a contractive approximate unit in $C$, it is easily verified that
if in Definition 2.20 we drop the requirement that
$\|\xi_{i}^{\prime}\|\leq\|\xi_{i}\|$ for all $i$, the resulting topology
coincides with $\sigma$. We do not know if this still holds for Definition
2.18.

Note now that the definition of $V(\mathcal{H};F)$ makes sense if
$F_{C}\subseteq C^{**}$, using the unique normal extension of $\theta$ to
$C^{**}$. We will then use the following general fact.

######  Lemma 2.22.

Let $S\subseteq QM(C)$ be a subset such that $C\subseteq[S]$. Let $\sigma_{S}$
be the topology on $\operatorname{\mathrm{Rep}}_{*}(C)$ obtained by using as
local neighborhoods the $V(\mathcal{H};F)$ for finite $F_{C}\subseteq S$. Then

$\sigma_{S}=\sigma.$

###### Proof.

Let $(\mathcal{H},\theta)\in\operatorname{\mathrm{Rep}}_{*}(C)$ and
$F=(\varepsilon,F_{C},F_{\mathcal{H}}=\\{\xi_{1},\ldots,\xi_{n}\\})$ with
$F_{C}\subseteq S$. Choose $d\in C$ with $\|d\|\leq 1$ and

$\max_{i}\\{\|\xi_{i}-\theta(d)\xi_{i}\|\\}\max_{c\in
F_{C}}\\{\|c\|\\}\max_{j}\\{\|\xi_{j}\|\\}<\varepsilon/4.$

Then for all $1\leq i,j\leq n$ and all $c\in F_{C}$, we have

$|\langle\xi_{i},\theta(c)\xi_{j}\rangle-\langle\theta(d)\xi_{i},\theta(c)\theta(d)\xi_{j}\rangle|<\varepsilon/2.$

If now $\widetilde{F}=(\varepsilon/2,\widetilde{F}_{C},F_{\mathcal{H}})$ with
$\widetilde{F}_{C}=\\{d^{*}cd\mid c\in F_{C}\\}\subseteq C$, then
$V(\mathcal{H};\widetilde{F})\subseteq V(\mathcal{H};F)$. Indeed, if
$(\mathcal{H}^{\prime},\theta^{\prime})\in V(\mathcal{H};\widetilde{F})$, we
can choose $\xi_{1}^{\prime},\ldots,\xi_{n}^{\prime}\in\mathcal{H}^{\prime}$
with $\|\xi_{i}^{\prime}\|\leq\|\xi_{i}\|$ such that for all $1\leq i,j\leq n$
and $c\in F_{C}$,

$|\langle\theta(d)\xi_{i},\theta(c)\theta(d)\xi_{j}\rangle-\langle\theta^{\prime}(d)\xi_{i}^{\prime},\theta^{\prime}(c)\theta^{\prime}(d)\xi_{j}^{\prime}\rangle|<\varepsilon/2,$

and then clearly

$|\langle\xi_{i},\theta(c)\xi_{j}\rangle-\langle\theta^{\prime}(d)\xi_{i}^{\prime},\theta^{\prime}(c)\theta^{\prime}(d)\xi_{j}^{\prime}\rangle|<\varepsilon.$

In particular, this shows that

$\sigma_{S}\subseteq\sigma.$

We now claim conversely that $\sigma_{S}$ is finer than $\sigma$. In fact, we
have the following easy observations: the topology $\sigma_{S}$ does not
change if we add to $S$

  * •

scalar multiples of elements of $S$,

  * •

sums of elements of $S$, or

  * •

normlimits of elements of $S$.

In particular, we may replace $S$ by $[S]$, and so the assumption
$[S]\supseteq C$ immediately gives the claim. ∎

######  Theorem 2.23.

Let $\mathbb{G}$ be a locally compact quantum group. The topology $\tau$ on
$\mathrm{Corr}^{\mathbb{G}}(A,B)$ is induced by the corresponding topology
$\sigma$ on $\operatorname{\mathrm{Rep}}_{*}(C^{\mathbb{G}}(A,B))$ under the
correspondence
$(\mathcal{H},U,\pi,\rho)\mapsto(\mathcal{H},\theta_{\mathcal{H}})$.

###### Proof.

Observing that

$C^{\mathbb{G}}(A,B)=[\pi_{u}(A_{s})C_{U_{u}}\rho_{u}(B_{s})]\subseteq[\pi_{u}(A)\\{(\operatorname{\mathrm{id}}\otimes\omega)U_{u}\mid\omega\in
M_{*}\\}\rho_{u}(B)]\subseteq QM(C^{\mathbb{G}}(A,B)),$

this follows immediately from Lemma 2.22. ∎

The topology $\tau$ itself is not the most useful: rather, we want to consider
the _Fell topology_ 333It is an instructive exercise to check that this indeed
determines a well-defined topology, in the sense that (2.8) and (2.9) hold.
$\tau^{\infty}$ which has local neighorhoods

$U^{\infty}(\mathcal{H};F)=\\{\mathcal{G}\mid\mathcal{G}^{\infty}\in
U(\mathcal{H};F)\\}$

where here and in the following we use the shorthand notation

$\mathcal{G}^{\infty}:=\oplus_{1}^{\infty}\mathcal{G}.$

The corresponding topology $\sigma^{\infty}$ on
$\operatorname{\mathrm{Rep}}_{*}(C^{\mathbb{G}}(A,B))$ indeed coincides with
the usual Fell (=inner hull-kernel) topology [Fel62].

##  3\. Weak containment

In this section, we fix again a locally compact quantum group $\mathbb{G}$ and
$\mathbb{G}$-W∗-algebras $A,B$.

######  Definition 3.1.

Let $\mathcal{H},\mathcal{H}^{\prime}$ be a
$\mathbb{G}$-$A$-$B$-correspondences. We say that $\mathcal{H}^{\prime}$ is
weakly contained in $\mathcal{H}$ iff $\mathcal{H}^{\prime}$ lies in the
$\tau^{\infty}$-closure of $\\{\mathcal{H}\\}$.

We then write

$\mathcal{H}^{\prime}\preccurlyeq\mathcal{H}.$

In other words, $\mathcal{H}^{\prime}\preccurlyeq\mathcal{H}$ if and only if
for any $F$ as in Definition 2.18, we have $\oplus_{1}^{\infty}\mathcal{H}\in
U(\mathcal{H}^{\prime};F)$.

Clearly, weak containment is a transitive and reflexive relation. We also note
that in terms of the C∗-algebra $C^{\mathbb{G}}(A,B)$, we have
$\mathcal{G}\preccurlyeq\mathcal{H}$ if and only if the representation
$\theta_{\mathcal{G}}$ of $C^{\mathbb{G}}(A,B)$ factors through
$\theta_{\mathcal{H}}(C^{\mathbb{G}}(A,B))$ (see e.g. [Dix77, Section 3.4]).

In this section, we will provide a different characterization of weak
containment. This will generalize to the equivariant setting some of the
techniques and ideas of [A-D90, A-DH90, A-D95].

Before we continue, we prove a small preparatory lemma on von Neumann algebras
and nsf weights. Although it must be well-known, we do not know a convenient
reference, so we briefly provide a proof.

Recall that if $B$ is a von Neumann algebra and $\varphi$ an nsf weight on
$B$, we can endow the opposite von Neumann algebra $B^{\mathrm{op}}$ with the
nsf weight

$\varphi^{\mathrm{op}}:B^{\mathrm{op}}_{+}\rightarrow[0,+\infty],\quad
b^{\mathrm{op}}\mapsto\varphi(b).$

Then
$\mathscr{N}_{\varphi^{\mathrm{op}}}=(\mathscr{N}_{\varphi}^{*})^{\mathrm{op}}$,
and we can identify $L^{2}(B)$ with $L^{2}(B^{\mathrm{op}})$ via

$\Lambda_{\varphi^{\mathrm{op}}}(b^{\mathrm{op}})=J\Lambda_{\varphi}(b^{*}),\qquad
b\in\mathscr{N}_{\varphi}^{*}.$

######  Lemma 3.2.

Let $B$ be a von Neumann algebra. If $\xi_{1},\ldots,\xi_{n}\in L^{2}(B)$,
there exists an nsf weight $\varphi$ on $B$, a vector $\xi\in L^{2}(B)$ and
$b_{1},\ldots,b_{n}\in B$ such that
$b_{i}^{\mathrm{op}}\in\mathscr{N}_{\varphi^{\mathrm{op}}}$ for all $i$ and

$\xi_{i}=\rho_{B}(b_{i})\xi=\Lambda_{\varphi^{\mathrm{op}}}(b_{i}^{\mathrm{op}}),\qquad\forall
i.$ (3.1)

###### Proof.

Let $\mathcal{H}=[\rho_{B}(B)\xi_{1}+\ldots+\rho_{B}(B)\xi_{n}]$. Then
$\mathcal{H}=\pi_{B}(p)L^{2}(B)$ for some selfadjoint projection $p\in B$. Put
$q=1-p$, and let $\psi$ be an nsf weight on $qBq$. Let

$\omega:B\rightarrow\mathbb{C},\qquad
b\mapsto\sum_{i=1}^{n}\langle\xi_{i},\pi_{B}(b)\xi_{i}\rangle,$

and put

$\varphi(b)=\omega(pbp)+\psi(qbq),\qquad b\in B_{+}.$

Then clearly $\varphi$ is an nsf weight on $B$, with $p$ integrable. Moreover,
$p$ is invariant under the modular automorphism group of $\varphi$, so we can
define

$\xi:=\Lambda_{\varphi}(p)=\Lambda_{\varphi^{\mathrm{op}}}(p^{\mathrm{op}}).$

Now an easy calculation gives that

$\langle\xi,\pi_{B}(b)\xi\rangle=\omega(b),\qquad b\in B.$

It follows that there exists $b_{i}\in B$ such that
$\rho_{B}(b_{i})\pi_{B}(b)\xi=\pi_{B}(b)\xi_{i}$. In particular,
$\rho_{B}(pb_{i})\xi=\xi_{i}$, and clearly
$\rho_{B}(pb_{i})\xi=\Lambda_{\varphi^{\mathrm{op}}}((pb_{i})^{\mathrm{op}})$.
∎

Assume now that $\mathcal{H},\mathcal{G}$ are Hilbert spaces with a normal
anti-$*$-representation $\rho_{\mathcal{H}},\rho_{\mathcal{G}}$ by the von
Neumann algebra $B$. Recall the notation (0.4). In particular, we write

$X_{B}(\mathcal{H}):=X_{B}(L^{2}(B),\mathcal{H}).$ (3.2)

We can view $X_{B}(\mathcal{H})$ as a self-dual (right) Hilbert $B$-module
[Pas73, Rie74], and more generally $X_{B}(\mathcal{G},\mathcal{H})$ as a
W∗-TRO [Zet93], but we will not really need this more abstract viewpoint for
now.

Also the following lemma is elementary.

######  Lemma 3.3.

Let $\mathcal{G}$ be a Hilbert space with a normal anti-$*$-representation
$\rho_{\mathcal{G}}$ by $B$. Then there exists a net of contractions
$x_{i}\in\cup_{n=1}^{\infty}X_{B}(L^{2}(B)^{n},\mathcal{G})$ such that the
maps

$B(\mathcal{G})\rightarrow B(\mathcal{G}),\qquad y\mapsto
x_{i}x_{i}^{*}yx_{i}x_{i}^{*}$

converge pointwise $\sigma$-weakly to the identity map.

###### Proof.

It follows immediately from the fact that we can identify

$(\mathcal{G},\rho_{\mathcal{G}})\cong((\pi_{B}\otimes\operatorname{\mathrm{id}})(p)(L^{2}(B)\otimes
l^{2}(I)),\rho_{B}\otimes\operatorname{\mathrm{id}})$

for some set $I$ and some $p\in B\bar{\otimes}B(l^{2}(I))$. ∎

We will also need a slight extension/variation of [A-DH90, Lemma 2.2].
Although the proof almost applies ad verbatim, we prefer to include it for the
convenience of the reader.

######  Lemma 3.4.

Let $B$ be a von Neumann algebra, and let $\mathcal{H},\mathcal{G}$ be Hilbert
spaces with a normal anti-$*$-representation
$\rho_{\mathcal{H}},\rho_{\mathcal{G}}$ of $B$. Let $F\subseteq
X_{B}(\mathcal{G},\mathcal{H})$ be a subset which is closed under right
multiplication with $\rho_{\mathcal{G}}(B)^{\prime}$. Let $S\subseteq
B(\mathcal{H})$ be a subset with $1\in S$, and assume that
$\\{\operatorname{\mathrm{Ad}}_{y^{*}}:S\rightarrow B(\mathcal{G})\mid y\in
F\\}$ is convex. Let $x\in X_{B}(\mathcal{G},\mathcal{H})$ with $\|x\|\leq 1$,
and assume that there exists a net of $x_{i}\in F$ such that for each $s\in S$

$x_{i}^{*}sx_{i}\rightarrow x^{*}sx\quad\textrm{weakly in }B(\mathcal{G}).$
(3.3)

Then there exists a net of _contractive_ $x_{i}\in F$ such that (3.3) holds
for the $\sigma$-weak topology.

###### Proof.

As in [A-DH90, Lemma 2.2], we begin with proving the lemma under the extra
condition that $x^{*}x=1$.

Assume that a net of $x_{i}\in F$ is given such that (3.3) holds for each
$s\in S$. By the assumption that
$\\{(\operatorname{\mathrm{Ad}}_{y^{*}})_{\mid S}\mid y\in F\\}$ is convex, we
may choose this net so that (3.3) in fact holds with respect to the _strong_
topology.

Put

$y_{i}=2x_{i}(1+x_{i}^{*}x_{i})^{-1}\in F.$

Following further the computation as in [A-DH90, Lemma 2.2], we find

$(x_{i}-y_{i})^{*}(x_{i}-y_{i})\leq(x_{i}^{*}x_{i}-1)^{2},$

and for each vector state $\omega=\omega_{\xi}$ on $B(\mathcal{G})$ and $s\in
S$ we have

$|\omega(x_{i}^{*}sx_{i}-y_{i}^{*}sy_{i})|\leq\|s\|\omega((x_{i}^{*}x_{i}-1)^{2})^{1/2}(\omega(x_{i}^{*}x_{i})^{1/2}+\omega(y_{i}^{*}y_{i})^{1/2}).$
(3.4)

Since $\|y_{i}\|\leq 1$ and since
$\operatorname{\mathrm{Ad}}_{x_{i}^{*}}(1)=x_{i}^{*}x_{i}$ converges strongly
to $x^{*}x=1$ by assumption, we find that (3.4) goes to zero in the limit. It
follows that $y_{i}$ is a contractive net in $F$ such that for each $s\in S$
the net $\operatorname{\mathrm{Ad}}_{y_{i}^{*}}(s)$ converges weakly (and
hence $\sigma$-weakly) to $x^{*}sx$, proving the lemma in this special case.

Let us now consider the case where $x\in X_{B}(\mathcal{G},\mathcal{H})$ is an
arbitrary contraction. Let $e\in\rho_{\mathcal{G}}(B)^{\prime}\subseteq
B(\mathcal{G})$ be the support projection for $x^{*}x$. Then we may in fact
view $\operatorname{\mathrm{Ad}}_{x^{*}}$ as a map $B(\mathcal{H})\rightarrow
B(e\mathcal{G})$. By multiplying a net $x_{i}$ as in (3.3) with $e$ on the
right, we see that without loss of generality we may in fact assume that
$e=1$.

Let then $x=u|x|$ be the polar decomposition of $x$, where $u\in
X_{B}(\mathcal{G},\mathcal{H})$ is isometric by the above assumption. Then $u$
lies in the strong closure of the unit ball of
$x\rho_{\mathcal{G}}(B)^{\prime}$. It follows from this last property,
together with the fact that $F$ is closed under right multiplication with
$\rho_{\mathcal{G}}(B)^{\prime}$, that $\operatorname{\mathrm{Ad}}_{u^{*}}$
can be approximated pointwise weakly on $S$ with maps
$\operatorname{\mathrm{Ad}}_{w_{i}^{*}}$ for $w_{i}\in F$. By the first part
of the proof, we may assume the $w_{i}$ to be contractive. But then the
$y_{i}=w_{i}|x|$ will also be contractive, with
$\operatorname{\mathrm{Ad}}_{y_{i}^{*}}$ converging to
$\operatorname{\mathrm{Ad}}_{x^{*}}$ pointwise weakly and hence
$\sigma$-weakly on $S$. ∎

Assume now again that $(A,\alpha_{A})$ and $(B,\alpha_{B})$ are
$\mathbb{G}$-W∗-algebras.

######  Lemma 3.5.

Let $\mathcal{H},\mathcal{G}$ be two $\mathbb{G}$-equivariant
$A$-$B$-correspondences. Assume that $\mathcal{G}$ is weakly contained in
$\mathcal{H}$. Then there exists a contractive net $y_{i}\in
X_{B}(\mathcal{G},\mathcal{H}^{\infty})$ such that for all $c\in
C^{\mathbb{G}}(A,B)$

$y_{i}^{*}\theta_{\mathcal{H}^{\infty}}(c)y_{i}\rightarrow\theta_{\mathcal{G}}(c)\quad\sigma\textrm{-weakly
in }B(\mathcal{G}).$ (3.5)

###### Proof.

We first prove the following: if, for some $n$, we are given a contractive map
$T=(T_{1},\ldots,T_{n})\in X_{B}(L^{2}(B)^{n},\mathcal{G})$, there exists a
$1$-bounded net $y_{i}\in X_{B}(L^{2}(B)^{n},\mathcal{H}^{\infty})$ such that
for all $c\in C^{\mathbb{G}}(A,B)$

$y_{i}^{*}\theta_{\mathcal{H}^{\infty}}(c)y_{i}\rightarrow
T^{*}\theta_{\mathcal{G}}(c)T\quad\sigma\textrm{-weakly in }B(L^{2}(B)^{n}).$
(3.6)

Indeed, pick a finite set $F_{L^{2}(B)}\subseteq L^{2}(B)$ and a finite subset
$F_{C}\subseteq C^{\mathbb{G}}(A,B)\cup\\{1\\}\subseteq
QM(C^{\mathbb{G}}(A,B))$. Take $\varepsilon>0$. We first look for elements
$x_{1},\ldots,x_{n}\in X_{B}(\mathcal{H}^{\infty})$ such that

$|\langle T_{i}\xi,\theta_{\mathcal{G}}(z)T_{j}\eta\rangle-\langle
x_{i}\xi,\theta_{\mathcal{H}^{\infty}}(z)x_{j}\eta\rangle|<\varepsilon,\qquad\forall
z\in F_{C},\forall\xi,\eta\in F_{L^{2}(B)},\forall i,j:1\rightarrow n.$ (3.7)

By Lemma 3.2, we can choose $\zeta\in L^{2}(B)$ and an nsf weight $\varphi$ on
$B$ together with a finite set $F_{B}\subseteq B$ such that each
$b^{\mathrm{op}}\in\mathscr{N}_{\varphi^{\mathrm{op}}}$ for $b\in F_{B}$ and
for each $\xi\in F_{L^{2}(B)}$ there exists $b\in F_{B}$ with

$\xi=\rho_{B}(b)\zeta=\Lambda_{\varphi^{\mathrm{op}}}(b^{\mathrm{op}}).$

Since $(\mathcal{H},\theta_{\mathcal{H}})$ lies in an arbitrary
$\tau^{\infty}$-neighborhood of $(\mathcal{G},\theta_{\mathcal{G}})$, and
since $\rho_{u}(B)\subseteq QM(C^{\mathbb{G}}(A,B))$ implies that also
$\rho_{u}(B)(C^{\mathbb{G}}(A,B)\cup\\{1\\})\rho_{u}(B)\subseteq
QM(C^{\mathbb{G}}(A,B))$, Lemma 2.22 allows us to pick
$\zeta_{i}^{\prime}\in\mathcal{H}^{\infty}$ with
$\|\zeta_{i}^{\prime}\|\leq\|T_{i}\zeta\|\leq\|\zeta\|$ and, for all $b,c\in
F_{B},z\in F_{C}$,

$|\langle
T_{i}\rho_{B}(b)\zeta,\theta_{\mathcal{G}}(z)T_{j}\rho_{B}(c)\zeta\rangle-\langle\rho_{\mathcal{H}^{\infty}}(b)\zeta_{i}^{\prime},\theta_{\mathcal{H}^{\infty}}(z)\rho_{\mathcal{H}^{\infty}}(c)\zeta_{j}^{\prime}\rangle|\\\
=|\langle
T_{i}\zeta,\theta_{\mathcal{G}}(\rho_{u}(b)^{*}z\rho_{u}(c))T_{j}\zeta\rangle-\langle\zeta_{i}^{\prime},\theta_{\mathcal{H}^{\infty}}(\rho_{u}(b)^{*}z\rho_{u}(c))\zeta_{j}^{\prime}\rangle|<\varepsilon.$

By norm-perturbing the $\zeta_{i}^{\prime}$ if necessary, we may assume that
they are left bounded for $\varphi$ (in the sense that
$\zeta_{i}^{\prime}\in\mathfrak{D}(\mathcal{H}^{\infty},\varphi)$, see [Tak03,
Section IX.3]). But then with $x_{i}=L_{\zeta_{i}^{\prime}}\in
X_{B}(\mathcal{H}^{\infty})$, we find that
$\rho_{\mathcal{H}^{\infty}}(b)\zeta_{i}^{\prime}=x_{i}\Lambda_{\varphi^{\mathrm{op}}}(b^{\mathrm{op}})=x_{i}\rho_{B}(b)\zeta$,
so that (3.7) is satisfied.

It follows that we can find a net $x_{i}\in
X_{B}(L^{2}(B)^{n},\mathcal{H}^{\infty})$ such that
$\operatorname{\mathrm{Ad}}_{x_{i}^{*}}\circ\theta_{\mathcal{H}^{\infty}}$
converges weakly to
$\operatorname{\mathrm{Ad}}_{T^{*}}\circ\theta_{\mathcal{G}}$ pointwise on
$C^{\mathbb{G}}(A,B)\cup\\{1\\}$. Now the set

$F=X_{B}(L^{2}(B)^{n},\mathcal{H}^{\infty})\subseteq
X_{B}(L^{2}(B)^{n},\mathcal{G}\oplus\mathcal{H}^{\infty})$

is stable under right multiplication with $\rho_{L^{2}(B)^{n}}(B)^{\prime}$,
and with $\mathrm{diag}_{\infty}:B(\mathcal{H})\rightarrow
B(\mathcal{H}^{\infty})$ the diagonal inclusion, we have that

$\\{\operatorname{\mathrm{Ad}}_{x^{*}}:B(\mathcal{G})\oplus\mathrm{diag}_{\infty}(B(\mathcal{H}))\rightarrow
B(L^{2}(B)^{n})\mid x\in F\\}$

is convex. By viewing also $T\in X_{B}(L^{2}(B)^{n},\mathcal{G})\subseteq
X_{B}(L^{2}(B)^{n},\mathcal{G}\oplus\mathcal{H}^{\infty})$ and applying Lemma
3.4 to $F\subseteq X_{B}(L^{2}(B)^{n},\mathcal{G}\oplus\mathcal{H}^{\infty})$
and
$S=(\theta_{\mathcal{G}}\oplus\theta_{\mathcal{H}^{\infty}})(C^{\mathbb{G}}(A,B)\cup\\{1\\})$,
we can conclude that we can also find a net $y_{i}\in
X_{B}(L^{2}(B)^{n},\mathcal{H}^{\infty})$ with $\|y_{i}\|\leq 1$ such that
$\operatorname{\mathrm{Ad}}_{y_{i}^{*}}\circ\theta_{\mathcal{H}^{\infty}}$
converges $\sigma$-weakly to
$\operatorname{\mathrm{Ad}}_{T^{*}}\circ\theta_{\mathcal{G}}$ pointwise on
$C^{\mathbb{G}}(A,B)\cup\\{1\\}$. This proves in particular (3.6).

We can now prove the lemma. Pick $T\in X_{B}(L^{2}(B)^{n},\mathcal{G})$
contractive for some $n$, and choose a contractive net $y_{i}\in
X_{B}(L^{2}(B)^{n},\mathcal{H}^{\infty})$ such that
$\operatorname{\mathrm{Ad}}_{y_{i}^{*}}\circ\theta_{\mathcal{H}^{\infty}}$
converges weakly to
$\operatorname{\mathrm{Ad}}_{T^{*}}\circ\theta_{\mathcal{G}}$ pointwise on
$C^{\mathbb{G}}(A,B)$. Then $y_{i}T^{*}\in
X_{B}(\mathcal{G},\mathcal{H}^{\infty})$ is still contractive, and for all
$c\in C^{\mathbb{G}}(A,B)$ we have

$(y_{i}T^{*})^{*}\theta_{\mathcal{H}^{\infty}}(c)(y_{i}T^{*})\rightarrow
TT^{*}\theta_{\mathcal{G}}(c)TT^{*}\quad\sigma\textrm{-weakly in
}B(\mathcal{G}).$

By Lemma 3.3, the pointwise $\sigma$-weak closure of the maps
$B(\mathcal{G})\ni x\mapsto TT^{*}xTT^{*}\in B(\mathcal{G})$ contains the
identity map, proving (3.5). ∎

Given $\mathbb{G}$-equivariant $A$-$B$-correspondences
$\mathcal{H},\mathcal{G}$, we obtain a coaction

$\alpha_{X_{B}(\mathcal{G},\mathcal{H})}:X_{B}(\mathcal{G},\mathcal{H})\rightarrow
X_{B}(\mathcal{G},\mathcal{H})\bar{\otimes}M,\quad x\mapsto
U_{\mathcal{H}}(x\otimes 1)U_{\mathcal{G}}^{*}.$ (3.8)

######  Theorem 3.6.

Let $A,B$ be $\mathbb{G}$-W∗-algebras, and let $\mathcal{H},\mathcal{G}$ be
$\mathbb{G}$-$A$-$B$-correspondences. Then $\mathcal{G}$ is equivariantly
weakly contained in $\mathcal{H}$ if and only if there exists a $1$-bounded
net $y_{i}\in X_{B}(\mathcal{G},\mathcal{H}^{\infty})$ such that for all $a\in
A$

$(y_{i}^{*}\pi_{\mathcal{H}^{\infty}}(a)\otimes
1)\alpha_{X_{B}(\mathcal{G},\mathcal{H}^{\infty})}(y_{i})\rightarrow\pi_{\mathcal{G}}(a)\otimes
1\quad\sigma\textrm{-weakly in }B(\mathcal{G}\otimes L^{2}(\mathbb{G})).$
(3.9)

###### Proof.

Assume first that $\mathcal{G}$ is weakly contained in $\mathcal{H}$. By Lemma
3.5, we can find a contractive net $y_{i}\in
X_{B}(\mathcal{G},\mathcal{H}^{\infty})$ such that for all $c\in
C^{\mathbb{G}}(A,B)$ we have

$y_{i}^{*}\theta_{\mathcal{H}^{\infty}}(c)y_{i}\rightarrow\theta_{\mathcal{G}}(c)\quad\sigma\textrm{-weakly
in }B(\mathcal{G}).$ (3.10)

By passing to a subnet, we may assume that

$\operatorname{\mathrm{Ad}}_{y_{i}^{*}}:B(\mathcal{H}^{\infty})\rightarrow
B(\mathcal{G}),\qquad x\mapsto y_{i}^{*}xy_{i}.$ (3.11)

converges pointwise $\sigma$-weakly to a ccp map

$\widetilde{E}:B(\mathcal{H}^{\infty})\rightarrow B(\mathcal{G}).$

Put

$E=\widetilde{E}\circ\mathrm{diag}_{\infty}:B(\mathcal{H})\rightarrow
B(\mathcal{G}),$ (3.12)

where $\mathrm{diag}_{\infty}:B(\mathcal{H})\rightarrow
B(\mathcal{H}^{\infty})$ is the diagonal inclusion. Then clearly $E$ is a ccp
map with

$E(\theta_{\mathcal{H}}(z))=\theta_{\mathcal{G}}(z),\qquad\forall z\in
C^{\mathbb{G}}(A,B).$

In particular, $\theta_{\mathcal{H}}(C^{\mathbb{G}}(A,B))$ belongs to the
multiplicative domain of $E$. But then also

$\theta_{\mathcal{G}}(x)E(\theta_{\mathcal{H}}(z))\theta_{\mathcal{G}}(y)=E(\theta_{\mathcal{H}}(xzy))=\theta_{\mathcal{G}}(x)\theta_{\mathcal{G}}(z)\theta_{\mathcal{G}}(y),\qquad
x,y\in C^{\mathbb{G}}(A,B),z\in QM(C^{\mathbb{G}}(A,B)).$

It follows that

$E(\theta_{\mathcal{H}}(z))=\theta_{\mathcal{G}}(z),\qquad\forall z\in
QM(C^{\mathbb{G}}(A,B)).$ (3.13)

Since $\pi_{u}(A)C_{U_{u}}\subseteq QM(C^{\mathbb{G}}(A,B))$, we find

$y_{i}^{*}\pi_{\mathcal{H}^{\infty}}(a)U_{\mathcal{H}^{\infty}}(\omega)y_{i}\overset{\sigma\textrm{-weakly}}{\underset{i\rightarrow\infty}{\rightarrow}}\pi_{\mathcal{G}}(a)U_{\mathcal{G}}(\omega),\qquad\forall
a\in A,\omega\in M_{*}.$ (3.14)

Hence for all $\xi,\eta\in\mathcal{G}$ and all $f,g\in L^{2}(\mathbb{G})$ and
$a\in A$, we have

$\langle y_{i}\xi\otimes f,(\pi_{\mathcal{H}^{\infty}}(a)\otimes
1)U_{\mathcal{H}^{\infty}}(y_{i}\eta\otimes
g)\rangle\rightarrow\langle\xi\otimes f,(\pi_{\mathcal{G}}(a)\otimes
1)U_{\mathcal{G}}(\eta\otimes g)\rangle.$

By boundedness of the $y_{i}$, this means that for all $a\in A$ we have

$(y_{i}^{*}\pi_{\mathcal{H}^{\infty}}(a)\otimes
1)U_{\mathcal{H}^{\infty}}(y_{i}\otimes
1)\underset{i\rightarrow\infty}{\overset{\sigma\textrm{-weakly}}{\longrightarrow}}(\pi_{\mathcal{G}}(a)\otimes
1)U_{\mathcal{G}}.$ (3.15)

Multiplying to the right with $U_{\mathcal{G}}^{*}$, we find (3.9).

Let us now go back. Assume that we have chosen contractive $y_{k}\in
X_{B}(\mathcal{G},\mathcal{H}^{\infty})$ satisfying (3.9). Choose finitely
many $\xi_{i}\in\mathcal{G}$ as well as a finite set $F_{C}\subseteq
C^{\mathbb{G}}(A,B)$ and $\varepsilon>0$. By Theorem 2.23, it is enough find
$\xi_{i}^{\prime}\in\mathcal{H}^{\infty}$ such that
$\|\xi_{i}^{\prime}\|\leq\|\xi_{i}\|$ and

$|\langle\xi_{i},\theta_{\mathcal{G}}(z)\xi_{j}\rangle-\langle\xi_{i}^{\prime},\theta_{\mathcal{H}^{\infty}}(z)\xi_{j}^{\prime}\rangle|<\varepsilon,\qquad\forall
z\in F_{C},i,j\in\\{1,\ldots,n\\}.$

Now by assumption, we have for all $\omega\in M_{*}$ and $a\in A$ that

$y_{k}^{*}\pi_{\mathcal{H}^{\infty}}(a)U_{\mathcal{H}^{\infty}}(\omega)y_{k}\underset{k\rightarrow\infty}{\overset{\sigma-\textrm{weakly}}{\longrightarrow}}\pi_{\mathcal{G}}(a)U_{\mathcal{G}}(\omega).$

Then clearly also for all $\omega\in M_{*},a\in A$ and $b\in B$ we have

$y_{k}^{*}\pi_{\mathcal{H}^{\infty}}(a)U_{\mathcal{H}^{\infty}}(\omega)\rho_{\mathcal{H}^{\infty}}(b)y_{k}=y_{k}^{*}\pi_{\mathcal{H}^{\infty}}(a)U_{\mathcal{H}^{\infty}}(\omega)y_{k}\rho_{\mathcal{G}}(b)\underset{k\rightarrow\infty}{\overset{\sigma-\textrm{weakly}}{\longrightarrow}}\pi_{\mathcal{G}}(a)U_{\mathcal{G}}(\omega)\rho_{\mathcal{G}}(b).$

Since
$[\pi_{u}(A_{\mathrm{s}})C_{U_{u}}\rho_{u}(B_{\mathrm{s}})]=C^{\mathbb{G}}(A,B)$,
the uniform boundedness of the $y_{k}$ ensures that for all $z\in
C^{\mathbb{G}}(A,B)$ we have

$y_{k}^{*}\theta_{\mathcal{H}^{\infty}}(z)y_{k}\underset{k\rightarrow\infty}{\overset{\sigma-\textrm{weakly}}{\longrightarrow}}\theta_{\mathcal{G}}(z).$

Hence for our finite collection of $\xi_{i}$ and $z\in F_{C}$, we have

$|\langle\xi_{i},\theta_{\mathcal{G}}(z)\xi_{j}\rangle-\langle
y_{k}\xi_{i},\theta_{\mathcal{H}^{\infty}}(z)y_{k}\xi_{j}\rangle|\underset{k\rightarrow\infty}{\longrightarrow}0,$

so we can take $\xi_{i}^{\prime}=y_{k}\xi_{i}$ for $k$ large enough. ∎

Let us recuperate part of the proof of Theorem 3.6 as a separate corollary.

######  Corollary 3.7.

If $\mathcal{G}$ is equivariantly weakly contained in $\mathcal{H}$, there
exists a ucp map $E:B(\mathcal{H})\rightarrow B(\mathcal{G})$ such that

$E(\pi_{\mathcal{H}}(a))=\pi_{\mathcal{G}}(a),\qquad
E(\rho_{\mathcal{H}}(b))=\rho_{\mathcal{G}}(b),\qquad
E(U_{\mathcal{H}}(\omega))=U_{\mathcal{G}}(\omega)\qquad\forall a\in A,b\in
B,\omega\in M_{*}.$ (3.16)

###### Proof.

The map $E$ of (3.12) satisfies

$E(\theta_{\mathcal{H}}(z))=\theta_{\mathcal{G}}(z),\qquad\forall z\in
QM(C^{\mathbb{G}}(A,B)).$ (3.17)

Since $\pi_{u}(A),\rho_{u}(A)\subseteq QM(C^{\mathbb{G}}(A,B))$, we
immediately find that (3.16) is satisfied. ∎

##  4\. Amenability for equivariant correspondences and $\mathbb{G}$-actions

######  Definition 4.1.

Let $\pi:A\rightarrow B$ be a $\mathbb{G}$-equivariant normal unital
$*$-homomorphism.

We say that $\pi:A\rightarrow B$ is _strongly equivariantly amenable_ if
$E_{A}={}_{A}L^{2}(A)_{A}$ is equivariantly weakly contained in
${}_{\pi(A)}L^{2}(B)_{\pi(A)}$, which we view as
$\mathbb{G}$-$A$-$A$-correspondence.444To be very precise, it is the
$\mathbb{G}$-$A$-$A$-correspondence $(L^{2}(B),\pi_{B}\circ\pi:A\to
B(L^{2}(B)),\rho_{B}\circ\pi:A\to B(L^{2}(B)),U_{\beta})$.

We say that $\pi:A\rightarrow B$ is _equivariantly amenable_ if there exists a
$\mathbb{G}$-equivariant ucp map

$P:B\rightarrow A$

which is left inverse to $\pi$.

We may view equivariant amenability as a relative version of
$\mathbb{G}$-equivariant injectivity. Note also that if $\pi$ is (strongly)
equivariantly amenable, then $\pi$ is automatically faithful, so we are
actually dealing with a $\mathbb{G}$-equivariant inclusion $A\subseteq B$.

######  Remark 4.2.

The notion of equivariant amenability was already introduced in [Moa18,
Definition 4.4] in the setting of discrete quantum group actions. Note that we
follow here the terminology of [BMO20], which is dual to the one followed for
example in [A-D90].

Given an ucp map $E:B(\mathcal{H})\to B(\mathcal{G})$, we use the notation

$E\otimes\operatorname{\mathrm{id}}:B(\mathcal{H})\bar{\otimes}B(L^{2}(\mathbb{G}))\rightarrow
B(\mathcal{G})\bar{\otimes}B(L^{2}(\mathbb{G}))$

to denote the unique ucp map such that

$E((\operatorname{\mathrm{id}}\otimes\omega)z)=(\operatorname{\mathrm{id}}\otimes\omega)((E\otimes\operatorname{\mathrm{id}})z),\qquad\forall
z\in B(\mathcal{H})\bar{\otimes}B(L^{2}(\mathbb{G})),\omega\in
B(L^{2}(\mathbb{G}))_{*},$

using for example the identification

$B(\mathcal{G})\bar{\otimes}B(L^{2}(\mathbb{G}))=CB(B(L^{2}(\mathbb{G}))_{*},B(\mathcal{G})),$

where the right hand side means the set of all completely bounded maps [ER00,
Proposition 8.1.2].

######  Theorem 4.3.

If $\pi:A\rightarrow B$ is a strongly $\mathbb{G}$-amenable
$\mathbb{G}$-equivariant unital normal $*$-homomorphism between
$\mathbb{G}$-W∗-algebras, then $\pi:A\rightarrow B$ is $\mathbb{G}$-amenable.

###### Proof.

By assumption, the trivial equivariant $A$-$A$-correspondence $L^{2}(A)$ is
equivariantly weakly contained in the equivariant $A$-$A$-correspondence
${}_{\pi(A)}L^{2}(B)_{\pi(A)}$. Consider the ucp map

$E:B(L^{2}(B))\rightarrow B(L^{2}(A))$

as in Corollary 3.7. Then $E(\pi_{B}(B))$ will lie in $\pi_{A}(A)$, as these
elements commute pointwise with $\rho_{A}(A)$ by a multiplicative domain
argument. We can hence uniquely define a ccp map $P:B\rightarrow A$ by

$\pi_{A}(P(z)):=E(\pi_{B}(z)),\qquad z\in B.$

It is then clear that $P(\pi(a))=a$ for all $a\in A$. Moreover, we have by
(3.16) that $U_{\beta}$, the unitary $\mathbb{G}$-representation for the
$A$-$A$-correspondence ${}_{A}L^{2}(B)_{A}$, lies in the multiplicative domain
of $E\otimes\operatorname{\mathrm{id}}_{M}$, hence for $z\in B$ and $\omega\in
M_{*}$ we get

$\displaystyle\pi_{A}(P((\operatorname{\mathrm{id}}\otimes\omega)\beta(z)))$
$\displaystyle=$
$\displaystyle(\operatorname{\mathrm{id}}\otimes\omega)(E\otimes\operatorname{\mathrm{id}})(U_{\beta}(\pi_{B}(z)\otimes
1)U_{\beta}^{*})$ $\displaystyle=$
$\displaystyle(\operatorname{\mathrm{id}}\otimes\omega)(U_{\alpha}(E(\pi_{B}(z))\otimes
1)U_{\alpha}^{*})$ $\displaystyle=$
$\displaystyle\pi_{A}((\operatorname{\mathrm{id}}\otimes\omega)\alpha(P(z))),$

so that $P$ is $\mathbb{G}$-equivariant. ∎

We can use the definition of (strong) amenability for equivariant normal
unital $*$-homomorphisms to define a notion of left (strong) amenability for
equivariant correspondences. Recall the constructions made in Example 2.4.

######  Definition 4.4.

We say that a $\mathbb{G}$-equivariant $A$-$B$-correspondence
$(\mathcal{H},U,\pi,\rho)$ is _(strongly) left $\mathbb{G}$-amenable_ if
$\pi:A\rightarrow C=\rho(B)^{\prime}$ is (strongly) $\mathbb{G}$-amenable.

######  Remark 4.5.

In [A-D95, Definition 2.1], strong left amenability (called left amenability
loc. cit.) is rather introduced in terms of the Connes fusion product. We use
for now the shortcut provided by the remarks following [A-D95, Defintion 2.1],
and will return to this point in Remark 5.12.

######  Theorem 4.6.

If $\mathcal{H}$ is strongly left equivariantly amenable, then it is left
equivariantly amenable.

###### Proof.

This is a direct consequence of Theorem 4.3. ∎

In [BMO20, Corollary A.2], it was proven that in the non-equivariant setting,
the above can be strengthened: strong left amenability of $\mathcal{H}$ is in
fact _equivalent_ with left amenability. In the equivariant setting, this is
currently yet unknown, as the following special situation demonstrates.

######  Definition 4.7.

Let $(A,\alpha)$ be a $\mathbb{G}$-dynamical von Neumann algebra. We say that
the action of $\mathbb{G}$ on $A$ is _(strongly) $\mathbb{G}$-amenable_ if the
inclusion $\alpha:A\rightarrow A\bar{\otimes}M$ is (strongly)
$\mathbb{G}$-amenable.

Concretely, strong $\mathbb{G}$-amenability of $\alpha$ means that $E_{A}$ is
$\mathbb{G}$-equivariantly weakly contained in $S_{A}^{\mathbb{G}}$ and
$\mathbb{G}$-amenability of $\alpha$ means that there exists a ucp map

$P:A\bar{\otimes}M\rightarrow A$ (4.1)

such that $P(\alpha(a))=a$ for all $a\in A$ and

$P((\operatorname{\mathrm{id}}\otimes(\operatorname{\mathrm{id}}\otimes\omega)\Delta)z)=(\operatorname{\mathrm{id}}\otimes\omega)\alpha(P(z)),\qquad\forall
z\in A\bar{\otimes}M,\omega\in M_{*}.$ (4.2)

It is not clear to us if the definition of $\mathbb{G}$-amenability has
explicitly appeared in print before in the general setting of locally compact
quantum groups, but it is the obvious one, see e.g. [Moa18, Definition 4.1]
for the setting with $\mathbb{G}$ discrete, or [Voi77, Rua96, DQV02] when
$A=\mathbb{C}$ . On the other hand, for $\mathbb{G}=G$ an ordinary locally
compact group, the above is equivalent with the existence of a ucp map

$P:A\bar{\otimes}L^{\infty}(G)\rightarrow A$ (4.3)

such that $P(a\otimes 1)=a$ for all $a\in A$ and which is equivariant with
respect to the diagonal action of $G$ on $A\bar{\otimes}L^{\infty}(G)$, using
the action $\rho$ of $G$ on $L^{\infty}(G)$ by right shifts. This is the
notion of amenability of the action $G\curvearrowright A$ as it appears in
[A-D79, Definition 3.4].555In [A-D79] one uses left shifts, but this is of
course inconsequential.

Specializing Theorem 4.3, we obtain:

######  Theorem 4.8.

Assume $A$ is a von Neumann algebra with a strongly $\mathbb{G}$-amenable
$\mathbb{G}$-action. Then the action of $\mathbb{G}$ on $A$ is
$\mathbb{G}$-amenable.

###### Proof.

This is just a special case of Theorem 4.3. ∎

If we take $A=\mathbb{C}$, then the above simply states that if the trivial
$\mathbb{G}$-representation is weakly contained in the regular
$\mathbb{G}$-representation (= co-amenability of $\hat{\mathbb{G}}$), there
exists a $\mathbb{G}$-invariant state $M\rightarrow\mathbb{C}$ (= amenability
of $\mathbb{G}$). The converse is a well-known open problem, making it hence
difficult to ascertain if the converse of Theorem 4.6 or even Theorem 4.8 will
hold. Let us now show that the converse of the result in Theorem 4.8 holds for
locally compact groups.

######  Theorem 4.9.

If $G$ is a locally compact group, and $A$ a $G$-W∗-algebra, then the action
of $G$ on $A$ is strongly $G$-amenable if and only if it is $G$-amenable.

###### Proof.

Note first that an action of $G$ on $A$ is the same thing as a coaction

$\alpha:A\rightarrow A\bar{\otimes}L^{\infty}(G),\qquad a\mapsto\alpha(a),$

where we view $\alpha(a)\in B(L^{2}(A)\otimes L^{2}(G))=B(L^{2}(G,L^{2}(A)))$
as multiplication by the function $g\mapsto\pi_{A}(\alpha_{g}(a))$. Note that
$M=L^{\infty}(G)$ is endowed with the structure of a locally compact quantum
group by putting $\Delta(f)(g,h)=f(gh)$ for $f\in L^{\infty}(G)$ (this
definition makes sense locally almost everywhere).

By Theorem 4.8, it is enough to prove that $G$-amenability of the action
$G\curvearrowright A$ implies strong amenability, i.e. the weak containment of
$E_{A}$ in the $G$-semi-coarse correspondence $S^{G}_{A}$.

Let $V_{g}$ for $g\in G$ denote the right shift of $g$ on $L^{2}(G)$ (so we
work with respect to $dg$ the right invariant Haar measure). Note that this is
simply the $G$-representation corresponding to the right regular
representation $V$ of $L^{\infty}(G)$ as a locally compact quantum group.
Similarly, we write $g\mapsto U_{g}$ for the unitary $G$-representation
associated to $U_{\alpha}$. Let $Z(A)$ be the center of $A$, and let
$Z(A)_{\mathrm{s}}$ be as in Definition 2.12. In this case,
$Z(A)_{\mathrm{s}}$ is actually the C∗-algebra of elements which are norm-
continuous for the $G$-action [Ped79, Lemma 7.5.1]. Let then
$C_{c}(G,Z(A)_{\mathrm{s}})$ be the $*$-algebra of norm-continuous maps
$G\rightarrow Z(A)_{\mathrm{s}}$ with compact support. By [BC22, Theorem
1.1.(1) $\Rightarrow$ (2)], we can choose a net

$x_{i}\in C_{c}(G,Z(A)_{\mathrm{s}})\subseteq
Z(A)\bar{\otimes}L^{2}(G)\subseteq B(L^{2}(A),L^{2}(A)\otimes L^{2}(G))$

which is contractive for the $B(L^{2}(A),L^{2}(A)\otimes L^{2}(G))$-norm and
such that

$x_{i}^{*}(\alpha_{g}\otimes
V_{g})(x_{i})\overset{\sigma\textrm{-weakly}}{\underset{i\rightarrow\infty}{\longrightarrow}}1$
(4.4)

uniformly on compacts. If we define $z_{i}(g)=x_{i}^{*}(\alpha_{g}\otimes
V_{g})(x_{i})$, it is elementary to verify that $z_{i}\in
C_{c}(G,Z(A)_{\mathrm{s}})$, and that hence

$\int_{G}f(g)z_{i}(g)\operatorname{\mathrm{d}\\!}g\overset{\sigma\textrm{-weakly}}{\underset{i\rightarrow\infty}{\longrightarrow}}\left(\int_{G}f(g)\operatorname{\mathrm{d}\\!}g\right)1$

for each $f\in C_{c}(G)$. A density argument now allows to conclude that

$(x_{i}^{*}\otimes
1)V_{23}(\alpha\otimes\operatorname{\mathrm{id}})(x_{i})_{132}\overset{\sigma\textrm{-weakly}}{\underset{i\rightarrow\infty}{\longrightarrow}}1.$
(4.5)

Put $y_{i}=U_{\alpha}x_{i}$. Then $y_{i}\in X:=X_{A}(L^{2}(A)\otimes
L^{2}(G))$ (using the right $A$-representation as in Example 2.6). Moreover,
the identity $U_{\alpha,12}^{*}V_{23}U_{\alpha,12}=U_{\alpha,12}V_{23}$
transforms (4.5) into

$(y_{i}^{*}\otimes
1)\alpha_{X}(y_{i})\overset{\sigma\textrm{-weakly}}{\underset{i\rightarrow\infty}{\longrightarrow}}1.$
(4.6)

But since the first leg of $x_{i}$ lies in $Z(A)$, multiplying (4.6) to the
left with $\pi_{A}(a)\otimes 1$ for $a\in A$ entails that

$(y_{i}^{*}(\pi_{A}\otimes\pi_{M})\alpha(a)\otimes
1)\alpha_{X}(y_{i})\overset{\sigma\textrm{-weakly}}{\underset{i\rightarrow\infty}{\longrightarrow}}(\pi_{A}\otimes\pi_{M})\alpha(a)\otimes
1,\qquad\forall a\in A.$ (4.7)

It now follows from Theorem 3.6 that $E_{A}$ is weakly contained in
$S_{A}^{G}$. ∎

To end this section, we note that Theorem 3.6 seems to allow for a stronger
and more symmetric condition of weak containment. Namely, let $A,B$ be
$\mathbb{G}$-W∗-algebras, and let $\mathcal{H},\mathcal{G}$ be
$\mathbb{G}$-$A$-$B$-correspondences. Denote

${}_{A}X_{B}(\mathcal{G},\mathcal{H})=\\{x\in
X_{B}(\mathcal{G},\mathcal{H})\mid
x\pi_{\mathcal{G}}(a)=\pi_{\mathcal{H}}(a)x\textrm{ for all }a\in A\\}.$

Let us then say that $\mathcal{H}$ _symmetrically weakly_ contains
$\mathcal{G}$ if there exists a contractive net
$y_{i}\in{}_{A}X_{B}(\mathcal{G},\mathcal{H}^{\infty})$ such that

$(y_{i}^{*}\otimes 1)\alpha_{X}(y_{i})\rightarrow 1\quad\sigma\textrm{-weakly
in }B(\mathcal{G}\otimes L^{2}(\mathbb{G})).$ (4.8)

Clearly, if $\mathcal{H}$ symmetrically weakly contains $\mathcal{G}$, then it
weakly contains $\mathcal{G}$, but in the characterisation of weak containment
in Theorem 3.6, the $x_{i}$ only satisfy
$x_{i}\pi_{\mathcal{G}}(a)=\pi_{\mathcal{H}}(a)x_{i}$ _approximately_.

If we can find $y_{i}\in{}_{A}X_{B}(\mathcal{G},\mathcal{H}^{\infty})$
satisfying (4.8), then it is easily seen that ${}_{A}\mathcal{H}_{B}^{\infty}$
must contain a (not necessarily equivariant) isometric copy of
${}_{A}\mathcal{G}_{B}$ as an $A$-$B$-correspondence. It is hence clear that
symmetric weak containment is in general much stronger than ordinary weak
containment, and already puts a very strong condition on the underlying
correspondences. However, Theorem 4.9 shows that these notions _do coincide_
when considering the weak inclusion of $E_{A}$ inside the $\mathbb{G}$-semi-
coarse correspondence $S_{A}^{G}$ for $G$ a locally compact group. We do not
know if, for a general locally compact quantum group $\mathbb{G}$, the
equivariant weak containment $E_{A}\preccurlyeq S_{A}^{\mathbb{G}}$ will imply
the symmetric equivariant weak containment. Note that the latter, when framed
appropriately in the C∗-context, coincides with how amenability is defined for
actions of discrete quantum groups on unital C∗-algebras in [VV07, Definition
4.1].

##  5\. Operations on equivariant correspondences and the Takesaki-Takai
duality

In this section, we consider particular constructions which can be performed
on equivariant correspondences.

###  5.1. Opposites of $\mathbb{G}$-equivariant correspondences

Fix a locally compact quantum group $\mathbb{G}=(M,\Delta)$ and
$\mathbb{G}$-W∗-algebras $A,B$.

######  Definition 5.1.

Let $\mathcal{H}$ be a $\mathbb{G}$-equivariant $A$-$B$-correspondence. Let
$\overline{\mathcal{H}}$ be the complex conjugate Hilbert space, and let
$C_{\mathcal{H}}:\mathcal{H}\rightarrow\overline{\mathcal{H}}$ be the
canonical anti-unitary conjugation map $\xi\mapsto\overline{\xi}$. The
_opposite_ $B$-$A$-correspondence is obtained by endowing
$\overline{\mathcal{H}}$ with the bimodule structure

$\overline{\pi}(b)=C_{\mathcal{H}}\rho(b^{*})C_{\overline{\mathcal{H}}},\qquad\overline{\rho}(a)=C_{\mathcal{H}}\pi(a^{*})C_{\overline{\mathcal{H}}}$

and the unitary $\mathbb{G}$-representation

$\overline{U}=(C_{\mathcal{H}}\otimes\check{J})U^{*}(C_{\overline{\mathcal{H}}}\otimes\check{J}).$

We can easily see that $\overline{\mathcal{H}}$ is again a
$\mathbb{G}$-equivariant correspondence, using the defining relations of the
$\mathbb{G}$-equivariant correspondence $\mathcal{H}$ and the fact that
$\check{J}$ implements the unitary antipode.

Let us consider this operation for the examples appearing in Section 2. Of
course, we may in Definition 5.1 use _any_ anti-unitary operator
$C_{\mathcal{H}}$ from $\mathcal{H}$ to (another or the same) Hilbert space,
as this will clearly give an equivariant copy.

######  Example 5.2.

We have $\overline{E_{A}}\cong E_{A}$ by taking $C_{\mathcal{H}}=J_{A}$ and
using

$(J_{A}\otimes\check{J})U_{\alpha}(J_{A}\otimes\check{J})=U_{\alpha}^{*}.$

######  Example 5.3.

Let $A\subseteq C$ and $B\subseteq D$ be $\mathbb{G}$-equivariant inclusions
of $\mathbb{G}$-W∗-algebras, and ${}_{C}\mathcal{H}_{D}$ a
$\mathbb{G}$-equivariant $C$-$D$-correspondence. Then

${}_{B}\mathrm{Res}_{A}(\overline{\mathcal{H}})=\overline{{}_{A}\mathrm{Res}_{B}(\mathcal{H})}.$

######  Example 5.4.

With $D^{\mathbb{G}}(-)$ the adjoint amplification, we see that
$D^{\mathbb{G}}(\overline{\mathcal{H}})\cong\overline{D^{\mathbb{G}}(\mathcal{H})}$.
Indeed, implementing the latter by means of the anti-unitary map

$C_{D^{\mathbb{G}}(\mathcal{H})}=C_{\mathcal{H}}\otimes\hat{J}:\mathcal{H}\otimes
L^{2}(\mathbb{G})\rightarrow\overline{\mathcal{H}}\otimes L^{2}(\mathbb{G}),$

we find the unitary intertwiner

$(C_{\mathcal{H}}\otimes\hat{J})U(C_{\overline{\mathcal{H}}}\otimes\hat{J}):D^{\mathbb{G}}(\overline{\mathcal{H}})\rightarrow\overline{D^{\mathbb{G}}(\mathcal{H})},$

using relation (2.2) for the intertwining properties of the unitary
$\mathbb{G}$-representations.

It is easy to see that the statement in the previous example also holds for
the regular amplification.

###  5.2. Composition of $\mathbb{G}$-equivariant correspondences

Let $A,B$ be von Neumann algebras. Recall that $X_{B}(\mathcal{H})$ was
introduced in (3.2) as the self-dual Hilbert $A$-$B$-module of right
$B$-intertwiners from $L^{2}(B)$ to $\mathcal{H}$. If then $\mathcal{G}$ is a
$B$-$C$-correspondence, we can compose $\mathcal{H}$ and $\mathcal{G}$ into an
$A$-$C$-correspondence by the _Connes fusion product_

${}_{A}(\mathcal{H}\boxtimes\mathcal{G})_{C}:={}_{A}(X_{B}(\mathcal{H})\underset{B}{\otimes}\mathcal{G})_{C},$

which is the separation-completion of $X(\mathcal{H})\odot_{B}\mathcal{G}$
with respect to the inner product

$\langle
x\otimes_{B}\xi,y\otimes_{B}\eta\rangle=\langle\xi,x^{*}y\eta\rangle,\qquad
x,y\in X_{B}(\mathcal{H}),\xi,\eta\in\mathcal{G},$

(where we identified $B\cong\pi_{B}(B)$). The $A$-$C$-bimodule structure is
determined by

$\pi_{\boxtimes}(a)(x\otimes_{B}\xi)=\pi(a)x\otimes_{B}\xi,\qquad\rho_{\boxtimes}(c)(x\otimes_{B}\xi)=x\otimes_{B}\rho(c)\xi,\qquad
a\in A,c\in C,x\in X_{B}(\mathcal{H}),\xi\in\mathcal{G}.$

We write $\mathcal{H}\boxtimes_{B}\mathcal{G}$ if we want to emphasize that we
take the fusion product over $B$.

If $\mathcal{K}$ is a $C$-$D$-correspondence, we have a natural intertwining
associativity unitary

$(\mathcal{H}\boxtimes\mathcal{G})\boxtimes\mathcal{K}\cong\mathcal{H}\boxtimes(\mathcal{G}\boxtimes\mathcal{K}),$
(5.1)

using e.g. that $X_{C}(\mathcal{H}\boxtimes\mathcal{G})$ is the $\sigma$-weak
closure of the image of $X_{B}(\mathcal{H})\odot_{B}X_{C}(\mathcal{G})$ in
$B(L^{2}(C),\mathcal{H}\boxtimes\mathcal{G})$ by

$(x\otimes_{B}y)\eta=x\otimes_{B}y\eta,\qquad x\in X_{B}(\mathcal{H}),y\in
X_{C}(\mathcal{G}),\eta\in L^{2}(C).$

The identity correspondence $E_{A}={}_{A}L^{2}(A)_{A}$ acts as the identity
for this composition. See e.g. [Lan01, Bro03] for a detailed treatment.

We can upgrade the above construction to the $\mathbb{G}$-equivariant setting.
Recall that if $(A,\alpha)$ and $(B,\beta)$ are $\mathbb{G}$-W∗-algebras and
$\mathcal{H}$ a $\mathbb{G}$-equivariant $A$-$B$-correspondence, then
$X=X_{B}(\mathcal{H})$ comes equipped with the coaction

$\alpha_{X}=\alpha_{X_{B}(\mathcal{H})}:X\rightarrow X\bar{\otimes}M,\qquad
x\mapsto U(x\otimes 1)U_{\beta}^{*}.$

######  Proposition 5.5.

Assume $(A,\alpha),(B,\beta),(C,\gamma)$ are $\mathbb{G}$-dynamical von
Neumann algebras. If $\mathcal{H}$ is a $\mathbb{G}$-equivariant
$A$-$B$-correspondence and $\mathcal{G}$ a $\mathbb{G}$-equivariant
$B$-$C$-correspondence, we can endow $\mathcal{H}\boxtimes\mathcal{G}$ with a
unique unitary $\mathbb{G}$-representation $U_{\boxtimes}$ such that

$U_{\boxtimes}((x\otimes_{B}\xi)\otimes\eta):=\alpha_{X}(x)\otimes_{B\bar{\otimes}M}U_{\mathcal{G}}(\xi\otimes\eta),\qquad
x\in X_{B}(\mathcal{H}),\xi\in\mathcal{H},\eta\in\mathcal{G}.$ (5.2)

Then $\mathcal{H}\boxtimes\mathcal{G}$ becomes a $\mathbb{G}$-equivariant
$A$-$C$-correspondence.

###### Proof.

First of all, note that in (5.2) one rather defines $U_{\boxtimes}$ as a map

$(X_{B}(\mathcal{H})\odot_{B}\mathcal{G})\odot
L^{2}(M)\rightarrow(X_{B}(\mathcal{H})\bar{\otimes}M)\underset{B\bar{\otimes}M}{\otimes}(\mathcal{G}\otimes
L^{2}(M))\cong(X_{B}(\mathcal{H})\underset{B}{\otimes}\mathcal{G})\otimes
L^{2}(M)=(\mathcal{H}\boxtimes\mathcal{G})\otimes L^{2}(M).$

A direct computation shows that this indeed descends to an isometry

$U_{\boxtimes}:(\mathcal{H}\boxtimes\mathcal{G})\otimes
L^{2}(M)\rightarrow(\mathcal{H}\boxtimes\mathcal{G})\otimes L^{2}(M).$

By the bicommutant theorem, it follows straightforwardly that
$U_{\boxtimes}\in B(\mathcal{H}\boxtimes\mathcal{G})\bar{\otimes}M$.

To check the corepresentation property, various identifications of Hilbert
spaces have to be made, which we will not comment on explicitly. The main goal
is to prove the identity

$U_{\boxtimes,13}U_{\boxtimes,12}=W_{32}^{*}U_{\boxtimes,12}W_{32},$ (5.3)

which we will show by applying both sides to a vector
$(x\otimes_{B}\xi)\otimes\eta\otimes\zeta$ with $x\in
X_{B}(\mathcal{H}),\xi\in\mathcal{G}$ and $\eta,\eta\in L^{2}(\mathbb{G})$.
Then on the one hand, the left hand side gives

$\displaystyle
U_{\boxtimes,13}U_{\boxtimes,12}((x\otimes_{B}\xi)\otimes\eta\otimes\zeta)$
$\displaystyle\cong$ $\displaystyle
U_{\boxtimes,13}((\alpha_{X}(x)\otimes_{B\bar{\otimes}M}U_{\mathcal{G}}(\xi\otimes\eta))\otimes\zeta)$
$\displaystyle\cong$
$\displaystyle(\alpha_{X}\otimes\operatorname{\mathrm{id}})\alpha_{X}(x)_{132}\otimes_{B\bar{\otimes}M\bar{\otimes}M}(U_{\mathcal{G},13}U_{\mathcal{G},12}(\xi\otimes\eta\otimes\zeta)),$

while for the right hand side we get

$\displaystyle
W_{32}^{*}U_{\boxtimes,12}W_{32}((x\otimes_{B}\xi)\otimes\eta\otimes\zeta)$
$\displaystyle=$ $\displaystyle
W_{32}^{*}U_{\boxtimes,12}((x\otimes_{B}\xi)\otimes W_{21}(\eta\otimes\zeta))$
$\displaystyle\cong$ $\displaystyle W_{32}^{*}(\alpha_{X}(x)\otimes
1)\otimes_{B\bar{\otimes}M\bar{\otimes}M}U_{\mathcal{G},12}W_{32}(\xi\otimes\eta\otimes\zeta))$
$\displaystyle=$
$\displaystyle((\operatorname{\mathrm{id}}\otimes\Delta^{\mathrm{op}})\alpha_{X}(x)\otimes
1)\otimes_{B\bar{\otimes}M\bar{\otimes}M}W_{32}^{*}U_{\mathcal{G},12}W_{32}(\xi\otimes\eta\otimes\zeta))$
$\displaystyle=$
$\displaystyle((\operatorname{\mathrm{id}}\otimes\Delta^{\mathrm{op}})\alpha_{X}(x)\otimes
1)\otimes_{B\bar{\otimes}M\bar{\otimes}M}(\operatorname{\mathrm{id}}\otimes\Delta^{\mathrm{op}})(U_{\mathcal{G}})(\xi\otimes\eta\otimes\zeta)),$

so that (5.3) now follows from the coaction property for $\alpha_{X}$ and the
corepresentation property of $U_{\mathcal{G}}$.

One can now see that $U_{\boxtimes}$ must be surjective by invoking [BDS13,
Corollary 4.16]. Alternatively, one could more directly observe that the range
of $U_{\boxtimes}$ will contain all vectors of the form
$\alpha_{X}(x)(1\otimes m)\otimes_{B\bar{\otimes}M}(\xi\otimes\eta)$ for $x\in
X_{B}(\mathcal{H}),m\in M$ and $\xi\in\mathcal{H},\eta\in\mathcal{G}$, and
then conclude from the fact that the linear span of the
$\alpha_{X}(x)(1\otimes m)$ is $\sigma$-strongly dense in
$X_{B}(\mathcal{H})\bar{\otimes}M$.

The equivariance conditions

$U_{\boxtimes}(\pi_{\boxtimes}(a)\otimes
1)=(\pi_{\boxtimes}\otimes\pi_{M})\alpha(a)U_{\boxtimes},\qquad(\rho_{\boxtimes}(c)\otimes
1)U_{\boxtimes}=U_{\boxtimes}(\rho_{\boxtimes}\otimes\pi_{M}R)\gamma(c),\qquad
a\in A,c\in C$

are easily checked. ∎

By using similar identifications as in the proof of Proposition 5.5, one can
check that the associativity intertwiner (5.1) is an intertwiner of
$\mathbb{G}$-equivariant correspondences. We refrain from spelling out the
details here.

The following lemma is convenient for computations.

######  Lemma 5.6.

Assume $(\mathcal{H}\otimes L^{2}(B),U,\pi,\rho)$ is endowed with the
structure of a $\mathbb{G}$-$A$-$B$-correspondence such that

$\rho(b)(\xi\otimes\eta)=\xi\otimes\rho_{B}(b)\eta,\qquad
U=U_{\mathcal{H},13}U_{\beta,23}$

for some unitary $\mathbb{G}$-representation $U_{\mathcal{H}}$ on
$\mathcal{H}$. If
$(\mathcal{G},U_{\mathcal{G}},\pi_{\mathcal{G}},\rho_{\mathcal{G}})$ is a
$\mathbb{G}$-equivariant $B$-$C$-correspondence, we have a unitary intertwiner

$(\mathcal{H}\otimes
L^{2}(B))\boxtimes\mathcal{G}\cong\mathcal{H}\otimes\mathcal{G},$

where the right hand side is endowed with the
$\mathbb{G}$-$A$-$C$-correspondence

$\pi_{\boxtimes}(a)=(\operatorname{\mathrm{id}}\otimes\pi_{\mathcal{G}})\pi(a),\qquad\rho_{\boxtimes}(c)=1\otimes\rho_{\mathcal{G}}(c),\qquad
a\in A,c\in C,$ (5.4) $U_{\boxtimes}=U_{\mathcal{H},13}U_{\mathcal{G},23}.$

###### Proof.

Note that the assumption on $\rho$ implies that $\pi(a)\in
B(\mathcal{H})\bar{\otimes}\pi_{B}(B)$ for $a\in A$, so under the
identification $\pi_{B}(B)\cong B$ we see that $\pi_{\boxtimes}$ in (5.4) is
at least a well-defined unital normal $*$-homomorphism $A\rightarrow
B(\mathcal{H}\otimes\mathcal{G})$.

Now it is immediate that $X_{B}(\mathcal{H}\otimes
L^{2}(B))=\mathcal{H}\overline{\otimes}B$ with its obvious self-dual Hilbert
$A$-$B$-bimodule structure, and with the coaction given by

$\alpha_{X}(\xi\otimes b)=U_{\mathcal{H},13}(\xi\otimes\beta(b)).$

We hence obtain an isometric map with dense image

$(\mathcal{H}\odot
B)\odot_{B}\mathcal{G}\rightarrow\mathcal{H}\otimes\mathcal{G},\qquad(\xi\otimes
b)\otimes\zeta\mapsto\xi\otimes b\zeta.$

It is clear that this induces (5.4) on $\mathcal{H}\otimes\mathcal{G}$. Also
the form for $U_{\boxtimes}$ is immediate. ∎

######  Definition 5.7.

Let $A,B$ be $\mathbb{G}$-W∗-algebras. We say that $A$ is _$\mathbb{G}$
-equivariantly W∗-Morita equivalent with $B$_ if there exists a
$\mathbb{G}$-equivariant $A$-$B$-correspondence $(\mathcal{H},U,\pi,\rho)$
which is full (and hence faithful) as a right $B$-module and faithful as a
left $A$-module, and such that

$\pi(A)=\rho(B)^{\prime}.$

In this case, we call $\mathcal{H}$ a $\mathbb{G}$-W∗-Morita correspondence
(between $A$ and $B$).

If $\mathcal{H}$ is a $\mathbb{G}$-W∗-Morita correspondence between $A$ and
$B$, we put

$Q_{\mathcal{H}}:=X_{B}(L^{2}(B)\oplus\mathcal{H}).$

It follows from basic theory of correspondences [Tak03] that we have a
decomposition

$Q_{\mathcal{H}}\cong\begin{pmatrix}A&X_{B}(\mathcal{H})\\\
X_{B}(\mathcal{H})^{*}&B\end{pmatrix},$

and that we can then decompose

$L^{2}(Q_{\mathcal{H}})=\begin{pmatrix}L^{2}(A)&L^{2}(X_{B}(\mathcal{H}))\\\
L^{2}(X_{B}(\mathcal{H})^{*})&L^{2}(B)\end{pmatrix}\cong\begin{pmatrix}L^{2}(A)&\mathcal{H}\\\
\overline{\mathcal{H}}&L^{2}(B)\end{pmatrix}$ (5.5)

as an $L^{2}$-direct sum. The isomorphism
$L^{2}(X_{B}(\mathcal{H}))\cong\mathcal{H}$ is uniquely determined by
$\pi_{Q}(x)\xi\cong x\xi$ if $x\in X_{B}(\mathcal{H})$ and $\xi\in L^{2}(B)$.
The isomorphism $L^{2}(X_{B}(\mathcal{H})^{*})\cong\overline{\mathcal{H}}$ is
uniquely determined by $J_{Q}\xi\cong\overline{\xi}$ for $\xi\in\mathcal{H}$.

From the fact that we are in the equivariant setting, $Q_{\mathcal{H}}$
obtains a natural $\mathbb{G}$-action $\alpha_{Q}$ on $Q$ by conjugating with
the unitary representation $U_{\beta}\oplus U$ on $L^{2}(B)\oplus\mathcal{H}$.
Let $U_{Q}$ be the associated canonical unitary implementing $\alpha_{Q}$.

######  Lemma 5.8.

Under the identification (5.5), we have

$U_{Q}\cong\begin{pmatrix}U_{\alpha}&U\\\
\overline{U}&U_{\beta}\end{pmatrix}.$ (5.6)

###### Proof.

It is clear that there will be a decomposition

$U_{Q}\cong\begin{pmatrix}U_{11}&U_{12}\\\ U_{21}&U_{22}\end{pmatrix},$

as $\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ and $\begin{pmatrix}0&0\\\
0&1\end{pmatrix}$ are invariant under $\alpha_{Q}$. The identifications on the
diagonal of (5.6) then follow by noticing that the construction of the unitary
$\mathbb{G}$-representation in [Vae05] is compatible with the
($\mathbb{G}$-invariant) projection maps $L^{2}(Q)\rightarrow L^{2}(B)$ and
$L^{2}(Q)\rightarrow L^{2}(A)$.

As $U_{Q}$ implements $\alpha_{Q}$, we must have for $x\in X_{B}(\mathcal{H})$
and $\xi\in L^{2}(B),\eta\in L^{2}(\mathbb{G})$ that

$U_{Q}(x\xi\otimes\eta)=\alpha_{X}(x)U_{\beta}(\xi\otimes\eta)\cong
U(x\xi\otimes\eta),$

showing the isomorphism in the top right corner of (5.6). The isomorphism on
the lower left corner is deduced from the observation that

$U_{Q}(J_{Q}\xi\otimes\hat{J}\eta)=(J_{Q}\otimes\hat{J})U^{*}(\xi\otimes\eta)\cong(C_{\mathcal{H}}\otimes\hat{J})U^{*}(\xi\otimes\eta)=\overline{U}(C_{\mathcal{H}}\xi\otimes\hat{J}\eta),\qquad\xi\in\mathcal{H},\eta\in
L^{2}(\mathbb{G}).$

∎

######  Proposition 5.9.

If $\mathcal{H}$ is a $\mathbb{G}$-W∗-Morita correspondence between $A$ and
$B$, then

$\mathcal{H}\boxtimes\overline{\mathcal{H}}\cong
L^{2}(A),\qquad\overline{\mathcal{H}}\boxtimes\mathcal{H}\cong L^{2}(B).$
(5.7)

###### Proof.

We can construct the maps in (5.7) as equivariant isometries simply by
restriction of the natural unitary $\mathbb{G}$-equivariant intertwiners

$L^{2}(Q)\boxtimes_{Q}L^{2}(Q)\cong L^{2}(Q)\cong
L^{2}(Q)\boxtimes_{Q}L^{2}(Q).$

It is then well-known from the theory of (non-equivariant) correspondences
that these restrictions are still unitary. ∎

######  Corollary 5.10.

If $A,B,C$ are $\mathbb{G}$-W∗-algebras, $\mathcal{H}$ is a
$\mathbb{G}$-equivariant $A$-$B$-correspondence and $\mathcal{G}$ is a
$\mathbb{G}$-equivariant $B$-$C$-correspondence, then

$\overline{\mathcal{H}\boxtimes\mathcal{G}}\cong\overline{\mathcal{G}}\boxtimes\overline{\mathcal{H}},$
(5.8)

###### Proof.

We may replace $\mathcal{G}$ by $\overline{\mathcal{G}}$, for $\mathcal{G}$ a
$C$-$B$-correspondence. By considering $A\oplus C$ and
$\mathcal{H}\oplus\mathcal{G}$, we may moreover assume that $A=C$ and
$\mathcal{H}=\mathcal{G}$. Furthermore, we can then as well assume that
$\rho_{\mathcal{H}}$ is faithful, and $A=\rho_{\mathcal{H}}(B)^{\prime}$. In
other words, $\mathcal{H}$ is now a $\mathbb{G}$-W∗-Morita correspondence
between $A$ and $B$, and the result follows from Proposition 5.9. ∎

######  Corollary 5.11.

Let $A\subseteq B$ be an equivariant inclusion, and put
$C=\rho_{B}(A)^{\prime}$. Then $C$ is $\mathbb{G}$-equivariant by

$\gamma(c):=U_{\beta}(c\otimes 1)U_{\beta}^{*},\qquad c\in C,$

and we obtain a $\mathbb{G}$-W∗-Morita $C$-$A$-correspondence
${}_{C}L^{2}(B)_{A}$. We then have

${}_{B}L^{2}(B)_{A}\boxtimes{}_{A}L^{2}(B)_{B}\cong{}_{B}L^{2}(C)_{B}.$

###### Proof.

The result is just a particular case of Proposition 5.9. ∎

######  Remark 5.12.

More generally, we have that if $\mathcal{H}$ is a $\mathbb{G}$-equivariant
$A$-$B$-correspondence and $C=\rho(B)^{\prime}$, we can identify
${}_{A}\mathcal{H}\boxtimes\overline{\mathcal{H}}_{A}\cong{}_{A}L^{2}(C)_{A}$.
The proof easily reduces to the above case.

Let us look at some more particular examples.

######  Example 5.13.

Using Lemma 5.6 and Corollary 5.10, it is easily seen that
$E_{A}^{\mathbb{G}}$ acts as an identity under $\boxtimes$: for any
$\mathcal{H}\in\mathrm{Corr}^{\mathbb{G}}(A,B)$ we obtain canonical
isomorphisms

$E_{A}^{\mathbb{G}}\boxtimes\mathcal{H}\cong\mathcal{H}\cong\mathcal{H}\boxtimes
E_{B}^{\mathbb{G}}.$

######  Example 5.14.

If $A\subseteq B$ is an equivariant inclusion and $\mathcal{H}$ is a
$\mathbb{G}$-equivariant $B$-$C$-correspondence, we have

${}_{A}L^{2}(B)_{B}\boxtimes{}_{B}\mathcal{H}_{C}\cong{}_{A}\mathcal{H}_{C}.$

Similarly, if $\mathcal{H}$ is a $\mathbb{G}$-equivariant
$C$-$B$-correspondence, then

${}_{C}\mathcal{H}_{B}\boxtimes{}_{B}L^{2}(B)_{A}\cong{}_{C}\mathcal{H}_{A}.$

######  Example 5.15.

By Lemma 5.6, we see that we can identify

$S_{A,B}^{\mathbb{G}}\boxtimes S_{B}^{\mathbb{G}}\cong L^{2}(A)\otimes
L^{2}(B)\otimes L^{2}(\mathbb{G}),$

where the latter is endowed with the normal (anti-)representations

$\pi(a)=\pi_{A}(a)\otimes 1\otimes
1,\qquad\rho(b)=1\otimes(\rho_{B}\otimes\rho_{M})\beta(b),\qquad a\in A,b\in
B,$

and the unitary $\mathbb{G}$-representation $U_{\alpha,14}V_{34}$. Conjugating
with $\Sigma_{23}U_{\alpha,13}$, we find that

$S_{A,B}^{\mathbb{G}}\boxtimes S_{B}^{\mathbb{G}}\cong C_{A,B}^{\mathbb{G}}.$

It follows from the computations in Example 5.2 and Corollary 5.10 that also

$S_{A}^{\mathbb{G}}\boxtimes S_{A,B}^{\mathbb{G}}\cong C_{A,B}^{\mathbb{G}}.$

######  Example 5.16.

Using Lemma 5.6, we immediately obtain that

$S_{A,B}^{\mathbb{G}}\boxtimes S_{B,C}^{\mathbb{G}}\cong L^{2}(A)\otimes
L^{2}(B)\otimes L^{2}(C)$

endowed with the $A$-$C$-correspondence by

$\pi(a)=\pi_{A}(a)\otimes 1\otimes 1,\qquad\rho(c)=1\otimes
1\otimes\rho_{C}(c),\qquad a\in A,c\in C,$

and the unitary $\mathbb{G}$-representation
$U_{\alpha,14}U_{\beta,24}U_{\gamma,34}$. In particular, if we take $B=M$ with
$\beta=\Delta$, we see that

$S_{A,M}^{\mathbb{G}}\boxtimes S_{M,C}^{\mathbb{G}}\cong
C_{A,C}^{\mathbb{G}}.$

Combining this with Example 5.3, we see in particular that

$S_{A,M}^{\mathbb{G}}\boxtimes\overline{S_{A,M}^{\mathbb{G}}}\cong
C_{A,C}^{\mathbb{G}}.$

######  Example 5.17.

Consider the $\mathbb{G}$-regular $\mathbb{G}$-$A$-$A$-correspondence
$S^{A}_{\mathbb{G}}$. Since $S_{A}^{\mathbb{G}}$ is the restriction of the
identity correspondence $E_{A\bar{\otimes}M}$ to $A\cong\alpha(A)$, we see
from Example 5.14 and Example 5.3 that

$S_{A}^{\mathbb{G}}\cong\mathcal{H}\boxtimes\overline{\mathcal{H}},\qquad\mathcal{H}={}_{\alpha(A)}(L^{2}(A)\otimes
L^{2}(\mathbb{G}))_{A\bar{\otimes}M}.$

A similar reasoning applies to the adjoint $\mathbb{G}$-$A$-$A$-correspondence
$D^{A}_{\mathbb{G}}$ as introduced in Example 2.9: since it is the restriction
of the identity correspondence $E_{A\rtimes\mathbb{G}}$ to $A\cong\alpha(A)$,
where $A\rtimes\mathbb{G}$ is endowed with the adjoint $\mathbb{G}$-action, we
find that

$D_{A}^{\mathbb{G}}\cong\mathcal{H}\boxtimes\overline{\mathcal{H}},\qquad\mathcal{H}={}_{\alpha(A)}(L^{2}(A)\otimes
L^{2}(\mathbb{G}))_{A\rtimes\mathbb{G}}.$

######  Example 5.18.

It is easily computed that the regular amplification in Example 2.6 is
obtained by tensoring with the $\mathbb{G}$-semi-coarse correspondence: if
$\mathcal{H}$ is a $\mathbb{G}$-equivariant $A$-$B$-correspondence, then

$S^{\mathbb{G}}(\mathcal{H})\cong\mathcal{H}\boxtimes S_{B}^{\mathbb{G}}\cong
S_{A}^{\mathbb{G}}\boxtimes\mathcal{H}.$

Similarly, the adjoint amplification of $\mathcal{H}$ is gotten by tensoring
with the adjoint $\mathbb{G}$-correspondence,

$D^{\mathbb{G}}(\mathcal{H})\cong\mathcal{H}\boxtimes D_{B}^{\mathbb{G}}\cong
D_{A}^{\mathbb{G}}\boxtimes\mathcal{H}.$

###  5.3. Crossed products

We lift the crossed product construction from actions to the level of
equivariant correspondences. This is an operation which is non-existent (or
more precisely trivial) in the case of non-equivariant correspondences.

######  Proposition 5.19.

Let $(\mathcal{H},U,\pi,\rho)$ be a $\mathbb{G}$-$A$-$B$-correspondence. Then

$\mathcal{H}^{\rtimes}=\mathcal{H}\rtimes\mathbb{G}:=\mathcal{H}\otimes
L^{2}(\mathbb{G})$

becomes a $\check{\mathbb{G}}$-$A^{\rtimes}$-$B^{\rtimes}$-correspondence for
the following structures:

  * •

the normal $A^{\rtimes}$-representation $\pi^{\rtimes}$ determined by

$\pi^{\rtimes}(z)=(\pi\otimes\operatorname{\mathrm{id}})(z),\qquad z\in
A^{\rtimes},$

  * •

the normal anti-$*$-representation $\rho^{\rtimes}$ of $B^{\rtimes}$ given by

$\rho^{\rtimes}(z)=U(\rho\otimes\check{J}(-)^{*}\check{J})(z)U^{*},\qquad z\in
B^{\rtimes},$

  * •

the unitary $\check{\mathbb{G}}$-representation
$U_{\rtimes}:=\widetilde{W}_{23}$.

###### Proof.

It is easily seen that $\pi^{\rtimes}$ and $\rho^{\rtimes}$ indeed define
respectively a normal $*$-representation and a normal anti-$*$-representation
of $A^{\rtimes}$ and $B^{\rtimes}$. Note that the ranges of $\pi^{\rtimes}$
and $\rho^{\rtimes}$ commute: indeed, we have

$\pi^{\rtimes}(\alpha(a))=(\pi\otimes\pi_{M})\alpha(a),\qquad\pi^{\rtimes}(1\otimes
x)=1\otimes\pi_{\check{M}}(x),\qquad a\in A,x\in\check{M},$ (5.9)
$\rho^{\rtimes}(\beta(b))=\rho(b)\otimes 1,\qquad\rho^{\rtimes}(1\otimes
x)=U(1\otimes\rho_{\check{M}}(x))U^{*},\qquad b\in B,x\in\check{M},$ (5.10)

so the only non-trivial commutation to check is the one between
$U(1\otimes\rho_{\check{M}}(\check{M}))U^{*}$ and
$1\otimes\pi_{\check{M}}(\check{M})$. But this follows easily from the
identity $U_{12}^{*}V_{23}U_{12}=U_{13}V_{23}$.

It is clear that $U_{\rtimes}$ interacts in the correct way with
$\pi^{\rtimes}$. To see its $\rho^{\rtimes}$-compatibility, simply note that

$\widetilde{W}_{23}^{*}U_{12}=U_{12}\widetilde{W}_{23}^{*}.$

∎

We now determine how the above crossed product constructions interact with
opposites and composition.

######  Proposition 5.20.

Let $\mathcal{H}$ be a $\mathbb{G}$-$A$-$B$-correspondence. We have

$\overline{\mathcal{H}}^{\rtimes}\cong\overline{\mathcal{H}^{\rtimes}}.$

###### Proof.

The unitary

$\overline{\mathcal{H}}\otimes
L^{2}(\mathbb{G})\rightarrow\overline{\mathcal{H}\otimes
L^{2}(\mathbb{G})},\qquad\overline{\xi}\otimes\eta\mapsto\overline{U(\xi\otimes\check{J}\eta)}$

does the job. ∎

In the proof of the following theorem, it will be convenient to have a
different model for the crossed product correspondence. If $\mathcal{H}$ is a
$\mathbb{G}$-$A$-$B$-correspondence, we will denote by
$\mathcal{H}^{\blacktriangleleft}$ the
$\check{G}$-$A^{\rtimes}$-$B^{\rtimes}$-correspondence $\mathcal{H}\otimes
L^{2}(\mathbb{G})$ obtained by conjugating $\mathcal{H}^{\rtimes}$ with
$U^{*}$, so in particular

$\pi^{\blacktriangleleft}(\alpha(a))=\pi(a)\otimes
1,\quad\pi^{\blacktriangleleft}(1\otimes
x)=U^{*}(1\otimes\pi_{\check{M}}(x))U,\qquad a\in A,x\in\check{M},$
$\rho^{\blacktriangleleft}(\beta(b))=(\rho\otimes\pi_{M}R)\beta(b),\quad\rho^{\blacktriangleleft}(1\otimes
x)=1\otimes\rho_{\check{M}}(x),\qquad b\in B,x\in\check{M},$

while $U_{\blacktriangleleft}=\widetilde{W}_{23}$ is still the associated
unitary $\check{\mathbb{G}}$-representation. Note that under the above
representation, we have for example that

$(\pi^{\blacktriangleleft}\otimes\operatorname{\mathrm{id}})(V_{23})=U_{12}^{*}V_{23}U_{12}=U_{13}V_{23}.$
(5.11)

######  Theorem 5.21.

Let $(\mathcal{H},U_{\mathcal{H}},\pi_{\mathcal{H}},\rho_{\mathcal{H}})$ be a
$\mathbb{G}$-$A$-$B$-correspondence and
$(\mathcal{G},U_{\mathcal{G}},\pi_{\mathcal{G}},\rho_{\mathcal{G}})$ a
$\mathbb{G}$-$B$-$C$-correspondence. Then we have

$(\mathcal{H}\boxtimes\mathcal{G})^{\rtimes}\cong\mathcal{H}^{\rtimes}\boxtimes\mathcal{G}^{\rtimes}.$

###### Proof.

We will show rather that

$(\mathcal{H}\boxtimes\mathcal{G})^{\blacktriangleleft}\cong\mathcal{H}^{\rtimes}\boxtimes\mathcal{G}^{\rtimes}$

Indeed, write $X=X_{B}(\mathcal{H})$ and consider the map

$(X\odot_{B}\mathcal{G})\odot
L^{2}(\mathbb{G})\rightarrow\mathcal{H}^{\rtimes}\boxtimes\mathcal{G}^{\rtimes},\quad(x\otimes_{B}\xi)\otimes\eta\mapsto\alpha_{X}(x)\otimes_{B^{\rtimes}}U_{\mathcal{G}}(\xi\otimes\eta),$
(5.12)

where we have used that $\alpha_{X}(X)\subseteq
X_{B^{\rtimes}}(\mathcal{H}^{\rtimes})$. A direct calculation shows that this
is a well-defined isometric map, which then extends to an isometry

$T:(\mathcal{H}\boxtimes\mathcal{G})\otimes
L^{2}(\mathbb{G})\rightarrow\mathcal{H}^{\rtimes}\boxtimes\mathcal{G}^{\rtimes}.$
(5.13)

In fact, it is an easy observation that
$X_{B^{\rtimes}}(\mathcal{H}^{\rtimes})$ equals the $\sigma$-strong closure of
$\alpha_{X}(X)(1\otimes\check{M})$, hence (5.13) will be unitary.

It is immediate that $T$ is an intertwiner of $A$-representations and of right
$\check{M}$-representations.

To see that $T$ is an intertwiner of $\check{\mathbb{G}}$-representations, we
compute

$\displaystyle
U_{\rtimes\boxtimes}((\alpha_{X}(x)\otimes_{B^{\rtimes}}U_{\mathcal{G}}(\xi\otimes\eta))\otimes\zeta)$
$\displaystyle=$ $\displaystyle(\alpha_{X}(x)\otimes
1)\otimes_{B^{\rtimes}\bar{\otimes}\check{M}}\widetilde{W}_{23}U_{\mathcal{G},12}(\xi\otimes\eta\otimes\zeta)$
$\displaystyle=$ $\displaystyle(\alpha_{X}(x)\otimes
1)\otimes_{B^{\rtimes}\bar{\otimes}\check{M}}U_{\mathcal{G},12}(\xi\otimes\widetilde{W}(\eta\otimes\zeta)),$

from which we see that

$U_{\rtimes\boxtimes}(\omega)(\alpha_{X}(x)\otimes_{B^{\rtimes}}U_{\mathcal{G}}(\xi\otimes\eta))=\alpha_{X}(x)\otimes_{B^{\rtimes}}U_{\mathcal{G}}(\xi\otimes(\operatorname{\mathrm{id}}\otimes\omega)(\widetilde{W})\eta),\qquad\omega\in
B(L^{2}(\mathbb{G}))_{*}.$

Hence
$TU_{\boxtimes\blacktriangleleft}(\omega)=U_{\rtimes\boxtimes}(\omega)T$, and
so $T$ intertwines the two $\check{\mathbb{G}}$-representations.

The intertwining property for right $C$-representations can be obtained by
observing first the identification

$(\mathcal{H}\boxtimes\mathcal{G})\otimes
L^{2}(\mathbb{G})\cong(\mathcal{H}\otimes
L^{2}(\mathbb{G}))\underset{B\bar{\otimes}M}{\boxtimes}(\mathcal{G}\otimes
L^{2}(\mathbb{G})),\qquad(x\otimes_{B}\xi)\otimes\eta\mapsto(x\otimes
1)\otimes_{B\bar{\otimes}M}(\xi\otimes\eta).$

Under this identification, the map $T$ can be written as

$T:(\mathcal{H}\otimes L^{2}(\mathbb{G}))\boxtimes(\mathcal{G}\otimes
L^{2}(\mathbb{G}))\rightarrow\mathcal{H}^{\rtimes}\boxtimes\mathcal{G}^{\rtimes},\qquad(x\otimes
1_{M})\otimes_{B\bar{\otimes}M}\xi\mapsto\alpha_{X}(x)\otimes_{B^{\rtimes}}U_{\mathcal{G}}\xi,$

where $x\in X_{B}(\mathcal{H})$ and $\xi\in\mathcal{G}\otimes
L^{2}(\mathbb{G})$. With this description, it is immediate that $T$
intertwines the right $C$-representations as well.

Finally, we need to check that $T$ is an intertwiner of left
$\check{M}$-representations. For this, we note that we can identify

$(\mathcal{H}\boxtimes\mathcal{G})\otimes L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G})\cong(\mathcal{H}\otimes L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G}))\underset{B\bar{\otimes}\check{M}\bar{\otimes}M}{\boxtimes}(\mathcal{G}\otimes
L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G})),$
$(x\otimes\xi)\otimes\eta\otimes\zeta\mapsto(x\otimes 1\otimes
1)\otimes_{B\bar{\otimes}\check{M}\bar{\otimes}M}(\xi\otimes\eta\otimes\zeta),$

and similarly we can identify

$(\mathcal{H}^{\rtimes}\underset{B^{\rtimes}}{\boxtimes}\mathcal{G}^{\rtimes})\otimes
L^{2}(\mathbb{G})\cong(\mathcal{H}^{\rtimes}\otimes
L^{2}(\mathbb{G}))\underset{B^{\rtimes}\bar{\otimes}M}{\boxtimes}(\mathcal{G}^{\rtimes}\otimes
L^{2}(\mathbb{G})),$ $(x\otimes_{B^{\rtimes}}\xi)\otimes\eta\mapsto(x\otimes
1)\otimes_{B^{\rtimes}\bar{\otimes}M}(\xi\otimes\eta).$

Then $T\otimes\operatorname{\mathrm{id}}_{L^{2}(\mathbb{G})}$ can be viewed as
the unitary

$(\mathcal{H}\otimes L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G}))\underset{B\bar{\otimes}\check{M}\bar{\otimes}M}{\boxtimes}(\mathcal{G}\otimes
L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G}))\rightarrow(\mathcal{H}^{\rtimes}\otimes
L^{2}(\mathbb{G}))\underset{B^{\rtimes}\bar{\otimes}M}{\boxtimes}(\mathcal{G}^{\rtimes}\otimes
L^{2}(\mathbb{G})),$
$z_{13}\otimes_{B\bar{\otimes}\check{M}\bar{\otimes}M}\xi\mapsto(\alpha\otimes\operatorname{\mathrm{id}})(z)\bar{\otimes}_{B^{\rtimes}\bar{\otimes}M}U_{\mathcal{G},12}\xi,$

where $z\in X_{B}(\mathcal{H})\bar{\otimes}M$ and $\xi\in\mathcal{G}\otimes
L^{2}(\mathbb{G})\otimes L^{2}(\mathbb{G})$.

It is now sufficient to prove that, under these models,

$(T\otimes
1)(\pi^{\boxtimes\blacktriangleleft}\otimes\operatorname{\mathrm{id}})(V)=(\pi^{\rtimes\boxtimes}\otimes\operatorname{\mathrm{id}})(V)(T\otimes
1).$ (5.14)

But since by (5.11) we have

$(\pi^{\boxtimes\blacktriangleleft}\otimes\operatorname{\mathrm{id}})(V)=U_{\boxtimes,13}V_{23},$

we see that (5.14) follows from the following computation for $x\in
X_{B}(\mathcal{H})$ and $\xi\in\mathcal{G}\otimes L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G})$:

$\displaystyle(T\otimes
1)(\pi^{\boxtimes\blacktriangleleft}\otimes\operatorname{\mathrm{id}})(V)((x\otimes
1\otimes 1)\otimes_{B\bar{\otimes}\check{M}\bar{\otimes}M}\xi)$
$\displaystyle=$ $\displaystyle(T\otimes 1)U_{\boxtimes,13}((x\otimes 1\otimes
1)\otimes_{B\bar{\otimes}\check{M}\bar{\otimes}M}V_{23}\xi)$ $\displaystyle=$
$\displaystyle(T\otimes
1)(\alpha_{X}(x)_{13}\otimes_{B\bar{\otimes}\check{M}\bar{\otimes}M}U_{\mathcal{G},13}V_{23}\xi)$
$\displaystyle=$
$\displaystyle(\alpha\otimes\operatorname{\mathrm{id}})\alpha(x)\otimes_{B^{\rtimes}\bar{\otimes}M}U_{\mathcal{G},12}U_{\mathcal{G},13}V_{23}\xi$
$\displaystyle=$ $\displaystyle
V_{23}\alpha(x)_{12}\otimes_{B^{\rtimes}\bar{\otimes}M}V_{23}^{*}U_{\mathcal{G},12}U_{\mathcal{G},13}V_{23}\xi$
$\displaystyle=$ $\displaystyle
V_{23}\alpha(x)_{12}\otimes_{B^{\rtimes}\bar{\otimes}M}U_{\mathcal{G},12}\xi$
$\displaystyle=$
$\displaystyle(\pi^{\rtimes\boxtimes}\otimes\operatorname{\mathrm{id}})(V)(\alpha(x)_{12}\otimes_{B^{\rtimes}\bar{\otimes}M}U_{\mathcal{G},12}\xi)$
$\displaystyle=$
$\displaystyle(\pi^{\rtimes\boxtimes}\otimes\operatorname{\mathrm{id}})(V)(T\otimes
1)((x\otimes 1\otimes 1)\otimes_{B\bar{\otimes}\check{M}\bar{\otimes}M}\xi).$

∎

###  5.4. Takesaki-Takai duality

Recall that $M^{\vee\vee}\cong M$ via
$\operatorname{\mathrm{Ad}}(u_{\mathbb{G}})$. Hence any
$\mathbb{G}$-W∗-algebra $A$ can be seen as a
$\mathbb{G}^{\vee\vee}$-W∗-algebra by a coaction that we write

$\widetilde{\alpha}:A\rightarrow A\bar{\otimes}M^{\vee\vee},\qquad
a\mapsto(1\otimes u_{\mathbb{G}})\alpha(a)(1\otimes u_{\mathbb{G}}).$

The _Takesaki-Takai duality_ for actions of locally compact quantum groups on
von Neumann algebras [ES80, Vae01] can now be stated as follows.

######  Theorem 5.22 (Takesaki-Takai duality).

Endow $A\bar{\otimes}B(L^{2}(\mathbb{G}))$ with the $\mathbb{G}$-action

$\alpha_{+}=\operatorname{\mathrm{Ad}}(\Sigma_{23}W_{23})\circ(\alpha\otimes\operatorname{\mathrm{id}}).$

Then there is a $\mathbb{G}^{\vee\vee}$-equivariant $*$-isomorphism

$\Phi:(A^{\rtimes\rtimes},\alpha^{\rtimes\rtimes})\cong(A\bar{\otimes}B(L^{2}(\mathbb{G})),\widetilde{\alpha_{+}}),$

whose inverse is concretely implemented by

$\Phi^{-1}:A\bar{\otimes}B(L^{2}(\mathbb{G}))\rightarrow
A^{\rtimes\rtimes}\subseteq B(L^{2}(A)\otimes L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G})),\quad z\mapsto
V_{23}^{*}(\alpha\otimes\operatorname{\mathrm{id}})(z)V_{23}.$

On generating elements, $\Phi$ is determined by

$\alpha(a)\otimes 1\mapsto\alpha(a),\quad 1\otimes\check{\Delta}(x)\mapsto
1\otimes x,\quad 1\otimes 1\otimes y\mapsto 1\otimes y,\qquad a\in
A,x\in\check{M},y\in M^{\prime},$

We reformulate this result as follows.

######  Theorem 5.23.

Endow the Hilbert space
${}_{A^{\rtimes\rtimes}}\mathscr{M}_{A}:=L^{2}(A)\otimes L^{2}(\mathbb{G})$
with

  * •

the normal unital $*$-representation
$(\pi_{A}\otimes\operatorname{\mathrm{id}})\circ\Phi$ of $A^{\rtimes\rtimes}$,

  * •

the normal unital anti-$*$-representation $\rho_{A}\otimes 1$ of $A$, and

  * •

the $\mathbb{G}^{\vee\vee}$-representation
$u_{\mathbb{G},3}W_{32}U_{\alpha,13}u_{\mathbb{G},3}$.

Then ${}_{A^{\rtimes\rtimes}}\mathscr{M}_{A}$ becomes a
$\mathbb{G}^{\vee\vee}$-W∗-Morita correspondence between
$(A^{\rtimes\rtimes},\alpha^{\rtimes\rtimes})$ and $(A,\widetilde{\alpha})$.

Note now that if $(A,\alpha),(B,\beta)$ are $\mathbb{G}$-W∗-algebras, any
$\mathbb{G}$-$A$-$B$-correspondence $(\mathcal{H},\pi,\rho,U)$ becomes a
$\mathbb{G}^{\vee\vee}$-equivariant
$(A,\widetilde{\alpha})$-$(B,\widetilde{\beta})$-correspondence by means of

$\widetilde{\mathcal{H}}=(\mathcal{H},\pi,\rho,(1\otimes
u_{\mathbb{G}})U(1\otimes u_{\mathbb{G}})).$

We can then extend the Takesaki-Takai duality to general
$\mathbb{G}$-equivariant correspondences.

######  Theorem 5.24.

Let $(\mathcal{H},U_{\mathcal{H}},\pi_{\mathcal{H}},\rho_{\mathcal{H}})$ be a
$\mathbb{G}$-equivariant $A$-$B$-correspondence. Then we have a unitary
intertwiner of $\mathbb{G}^{\vee\vee}$-$A$-$B$-correspondences

$\mathcal{H}^{\rtimes\rtimes}\cong{}_{A^{\rtimes\rtimes}}\mathscr{M}_{A}\boxtimes\widetilde{\mathcal{H}}\boxtimes{}_{B}\overline{\mathscr{M}}_{B^{\rtimes\rtimes}}.$

###### Proof.

Since $\mathscr{M}$ is a $\mathbb{G}^{\vee\vee}$-W∗-Morita correspondence, it
is (by associativity of $\boxtimes$ and Proposition 5.9) enough to show that

${}_{A^{\rtimes\rtimes}}\mathcal{H}^{\rtimes\rtimes}_{B^{\rtimes\rtimes}}\boxtimes{}_{B^{\rtimes\rtimes}}\mathscr{M}_{B}\cong{}_{A^{\rtimes\rtimes}}\mathscr{M}_{A}\boxtimes{}_{A}\widetilde{\mathcal{H}}_{B}.$
(5.15)

Note now that the representation of
$A\rtimes\mathbb{G}\rtimes\check{\mathbb{G}}$ and anti-representation of
$B\rtimes\mathbb{G}\rtimes\check{\mathbb{G}}$ on
$\mathcal{H}^{\rtimes}=\mathcal{H}\otimes L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{G})$ are given by

$\pi^{\rtimes\rtimes}(\alpha(a)\otimes
1)=(\pi_{\mathcal{H}}\otimes\pi_{M})(\alpha(a))_{12},\quad\rho^{\rtimes\rtimes}(\beta(b)\otimes
1)=\rho_{\mathcal{H}}(b)_{1},\qquad a\in A,b\in B,$
$\pi^{\rtimes\rtimes}(1\otimes\check{\Delta}(y))=\check{\Delta}(y)_{23},\quad\rho^{\rtimes\rtimes}(1\otimes\check{\Delta}(y))=U_{\mathcal{H},12}\rho_{\check{M}}(y)_{2}U_{\mathcal{H},12}^{*},\qquad
y\in\check{M},$ $\pi^{\rtimes\rtimes}(1\otimes 1\otimes
z)=\pi_{M^{\vee\vee}}(z)_{3},\quad\rho^{\rtimes\rtimes}(1\otimes 1\otimes
z)=\widetilde{W}_{23}\rho_{M^{\vee\vee}}(z)_{3}\widetilde{W}_{23}^{*},\qquad
z\in M^{\vee\vee},$

with associated unitary $\mathbb{G}^{\vee\vee}$-representation given by
$\widetilde{V}_{34}$, where we recall that
$\widetilde{V}=\operatorname{\mathrm{Ad}}(u_{\mathbb{G}}\otimes
u_{\mathbb{G}})(V)$.

It is then easily verified that

$X_{B^{\rtimes\rtimes}}(\mathcal{H}^{\rtimes\rtimes})=\sigma\textrm{-closure
of the span of