bibhow

# Injectivity for algebras and categories with quantum symmetry

Lucas Hataishi  Universitetet i Oslo<EMAIL_ADDRESS>and  Makoto
Yamashita  Universitetet i Oslo<EMAIL_ADDRESS>
(Date: August 18, 2023, minor revision; May 13, 2022)

###### Abstract.

We study completely positive maps and injectivity for Yetter–Drinfeld algebras
over compact quantum groups, and module categories over rigid C∗-tensor
categories. This gives a generalization of Hamana’s theory of injective
envelopes to the framework of dynamical systems over quantum groups. As a
byproduct we also establish a duality between the Yetter–Drinfeld algebras and
certain bimodule categories with central generators.

Supported by the NFR project 300837 “Quantum Symmetry”.

##  1\. Introduction

In this paper, we study completely positive maps and injectivity for
Yetter–Drinfeld algebras over compact quantum groups, and module categories
over rigid C∗-tensor categories.

The study of injectivity of operator systems and operator spaces led to
surprisingly rich applications to the structure theory of operator algebras
old and new. While the foundation of the theory goes back to the 1950s, one
early breakthrough in this context is Arveson’s result on the injectivity of
$\mathcal{B}(H)$ [MR253059]. This, together with subsequent works by Choi and
Effros [MR0376726][ce1] among others, proved the field to be a fruitful
framework at the intersection of abstract functional analysis and more
“applied” fields such as quantum information.

Following this direction, Hamana took on systematic study of injectivity
[h4][h1][h2][h3][MR2985658], defining and proving the existence and uniqueness
of injective envelopes of operator systems, C∗-algebras, and C∗-dynamical
systems with dynamics given by a discrete group, and more generally by a
Hopf–von Neumann algebra. Roughly speaking, his construction builds on two
parts: first is to embed a given object into a bigger injective object, then
next is to construct a ‘smallest’ injective one containing the original object
as the image of minimal idempotent in a convex semigroup of completely
positive maps acting on the injective one from the first step.

In another direction, Furstenberg and his school studied the probabilistic
notion of boundary actions of semisimple Lie groups [furstenberg1][moore], and
more generally of locally compact groups [MR365537]. A central object in their
theory is the notion of a universal (initial) boundary action of $G$, now
known as the Furstenberg boundary $\partial_{F}(G)$. In the case of a
semisimple Lie group $G$, an important application of boundary theory is to
the symmetric space $G/K$ corresponding to $G$, which appears through measure
theoretic considerations on $\partial_{F}(G)$. Furstenberg also considered
another kind of boundary, the Poisson boundary $B(G)$ of $G$, which shows up
through the study of harmonic functions, and studied how $\partial_{F}(G)$
relates to $B(G)$.

While these two theories look completely separate at the outset, it turns out
that the C∗-algebra of continuous functions on the Furstenberg boundary of a
discrete group $G$ agrees with the $G$-injective envelope of the trivial
$G$-C∗-algebra $\mathbb{C}$ [h4][kk]. In recent years, this connection led to
striking implications on the structure of group C∗-algebras: this includes the
equivalence between amenability of a discrete group and triviality of its
Furstenberg–Hamana boundary, results on rigidity and C∗-simplicity
[MR3735864][kennedy].

Probabilistic boundary theory has been brought to the framework of operator
algebraic quantum groups in various flavors in the last 30 years
[biane][izumi1][izumi2][int][neshveyevtuset1][vaesvergnioux][NY1]. More
recently, the analogue of Furstenberg–Hamana boundaries in this framework is
also achieved by the work of Kalantar, Kasprzak, Skalski, and Vergnioux
[kksv1]. They define and prove both the existence and the uniqueness of
Furstenberg–Hamana boundaries for discrete quantum groups, and relate them to
several C∗-algebraic concepts such as simplicity, exactness, and existence of
KMS-states.

This was further brought to the case of Drinfeld doubles $D(G)$ of compact
quantum groups $G$ in a joint work of the first named author with Habbestad
and Neshveyev [HHN1]. When the compact quantum group is the $q$-deformation of
a compact simple group, this Drinfeld double construction can be seen as a
quantization of the corresponding complex simple Lie group, hence coming back
to a setting close to Furstenberg’s original work. This also provides a
conceptual explanation for the equality between the Furstenberg–Hamana
boundary and the Poisson boundary of such quantum groups.

The boundary theory developed in [HHN1] works in the framework of braided
commutative Yetter–Drinfeld $G$-C∗-algebras for a compact quantum group $G$,
and more generally C∗-tensor categories under a rigid tensor category. Such
correspondence follows from a Tannaka–Krein type duality [NY3], but more
general types of duality for module categories and comodule algebras are known
to hold [ydk1][n1], which can be traced back to Ostrik’s result in the purely
algebraic setting [ostrik03].

In this work, we look at a more general setting of module categories, and also
the Yetter–Drinfeld algebras without braided commutativity. Let us explain the
structure of our paper.

In Section 2 we fix our conventions and recall some basic results. Then in
Section 3, we quickly prove our first main result, as follows.

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

For a compact quantum group $G$, every unital Yetter–Drinfeld $G$-C∗-algebra
$A$ admits an injective envelope.

This result, showing that the injective envelopes of continuous $D(G)$-actions
are still continuous, improves a result of Hamana.

In Section 4, we look at the categorical dual of quantum group actions. We
begin with the study of injective envelope of a pointed module category, i.e.,
a module category with a fixed object. Here we make extensive use of the
correspondence between algebra objects in a tensor category and cyclic pointed
module categories, and the concept of multipliers between pointed module
categories introduced in [jp1]. This allows us to bring ideas about completely
positive maps between C∗-algebras to the categorical setting. In fact, special
cases of such maps have already been considered in [PV1][jonesghosh1][NY2] to
study representation theory and approximation properties, like the Haagerup
property and property (T), for rigid C∗-tensor categories, subfactors, and
$\lambda$-lattices.

Given a Hilbert space object $H\in\mathrm{Hilb}(\mathcal{C})$ (equivalently,
an object in the ind-completion of $\mathcal{C}$ in the sense of [NY2]), there
is a C∗-algebra object $\mathcal{B}(H)$ in $\operatorname{Vec}(\mathcal{C})$
playing the role of algebra of bounded operators in $H$. Our first main result
in this setting is the following analogue of Arveson’s theorem.

######  Theorem B (Theorem 4.10).

For any $H\in\mathrm{Hilb}(\mathcal{C})$, the C∗-$\mathcal{C}$-module category
$\mathcal{M}_{\mathcal{B}(H)}$ corresponding to $\mathcal{B}(H)$ is injective.

By a standard argument, this allows us to obtain an analogue of Hamana’s
theorem for module categories (Theorem 4.13). We also show that the
injectivity of a pointed module category $(\mathcal{M},m)$ is actually a
property of the module subcategory of $\mathcal{M}$ generated by $m$
(Proposition 4.9) through a generalization of the Choi matrix construction. In
particular, the injectivity of $G$-C∗-algebra is invariant under Morita
equivalence with respect to finitely generated projective Hilbert modules.

Next in Section 5, we introduce a special type of module categories, which we
call centrally pointed bimodule categories. These are bimodule categories
endowed with an analogue of unitary half-braiding on the distinguished object.
Such bimodule categories and bimodule functors compatible with the half-
braiding on a distinguished object again form a category
$\mathcal{CB}(\mathcal{C})$. In particular, we are interested in its
subcategory $\mathcal{CB}^{\mathrm{c}}(\mathcal{C})$ consisting of the ones in
which the distinguished objects are generating.

In Section 6, we look at the Tannaka–Krein type duality between C∗-module
categories over $\operatorname{Rep}(G)$ and $G$-C∗-algebras for compact
quantum groups $G$.

######  Theorem C (Theorem 6.1).

The category of Yetter–Drinfeld $G$-C∗-algebras is equivalent to
$\mathcal{CB}^{\mathrm{c}}(\operatorname{Rep}(G))$. Under this equivalence,
two morphisms $F,F^{\prime}\colon\mathcal{M}\to\mathcal{M}^{\prime}$ in
$\mathcal{CB}^{\mathrm{c}}(\operatorname{Rep}(G))$ induce the same
homomorphism $B_{\mathcal{M}}\to B_{\mathcal{M}^{\prime}}$ if and only if they
are naturally isomorphic as centrally pointed bimodule functors.

This allows us to drop the braided commutativity assumption in [HHN1], and
work with categorical analogues of general Yetter–Drinfeld $G$-C∗-algebras.

Finally, in Section 7 we investigate injectivity and boundary theory of
centrally pointed bimodule categories based on the schemes of previous
sections. Here our main result is the following analogue of Theorem A.

######  Theorem D (Theorem 7.8).

Every centrally pointed module category has an injective envelope in
$\mathcal{CB}(\mathcal{C})$.

In the boundary theory for $D(G)$ developed in [HHN1], one sees that every
boundary action embeds into the Furstenberg–Hamana boundary, and as a
corollary one sees that every boundary action is braided-commutative.
Following a similar idea, we obtain the following generalization in the
categorical framework, where a _boundary_ of $\mathcal{C}$ is a centrally
pointed module category over $\mathcal{C}$ such that any ucp multiplier to
another category is completely isometric.

######  Theorem E (Theorem 7.10).

Every boundary category $\mathcal{M}$ of $\mathcal{C}$ is a centrally pointed
subcategory of $\partial_{\mathrm{FH}}(\mathcal{C})$.

In Appendix A, we give an intrinsic characterization of C∗-algebra objects in
terms of the associated Hilbert modules over the C∗-algebra at the tensor
unit, which avoids talking directly about the associated module category as
was done in [jp1].

#### Acknowledgments

We would like to thank the anonymous reviewers for their helpful comments
which helped us improve the presentation of the paper.

##  2\. Preliminaries

When $H$ is a Hilbert space, we denote the algebra of bounded operators on $H$
by $\mathcal{B}(H)$, while the algebra of compact operators is denoted by
$\mathcal{K}(H)$. The conjugate Hilbert space is denoted by $\bar{H}$, and we
write $\xi\mapsto\bar{\xi}$ for the anti-linear isomorphism $H\to\bar{H}$.

The multiplier algebra of a C∗-algebra $A$ is denoted by $\mathcal{M}(A)$. We
freely identify $\mathcal{M}(\mathcal{K}(H))$ with $\mathcal{B}(H)$.

As tensor product of C∗-algebras we always take the minimal tensor product,
that we denote by $A\otimes B$. For von Neumann algebras, their von Neumann
algebraic tensor product is denoted by $M\mathbin{\bar{\otimes}}N$.

###  2.1. Operator systems

An _operator system_ is a unital and selfadjoint closed subspace of a unital
C∗-algebra, which we write as $1\in S\subset A$. Such objects form a category
with unital completely positive maps as morphisms. An operator system is
_injective_ if it is injective in this category.

When $S$ is an operator system, its _injective envelope_ is given by an
injective operator system $\mathcal{I}(S)$ and a complete isometry $\phi\colon
S\to\mathcal{I}(S)$ such that $\phi$ is _essential_ in the sense that for any
ucp map $\psi\colon\mathcal{I}(S)\to S^{\prime}$ to another operator system,
$\psi\phi$ is completely isometric if and only if $\psi$ is.

The injective envelope of an operator system always exists and is unique up to
complete order isomorphisms [h1], with various generalizations imposing
additional structures on operator systems. A key technical step in the
construction of injective envelopes is the following proposition.

######  Proposition 2.1 ([HHN1]*Proposition 2.1).

Assume $X$ is a subspace of a dual Banach space $Y^{*}$, and $S$ is a convex
semigroup of contractive linear maps $X\to X$ such that, if we consider $S$ as
a set of maps $X\to Y^{*}$, then $S$ is closed in the topology of pointwise
weak* convergence. Then there is an idempotent $\phi_{0}\in S$ such that

$\phi_{0}\psi\phi_{0}=\phi_{0}\quad(\psi\in S).$

Moreover, $\phi_{0}$ is minimal with respect to the preorder relation
$\preceq$ on $S$ defined as

$\phi\preceq\psi\Leftrightarrow\forall x\in
X\colon\left\|\phi(x)\right\|\leq\left\|\psi(x)\right\|.$

###  2.2. C∗-tensor categories

Here we mostly follow [MR3204665][NY3].

Given a _C ∗-category_ $\mathcal{C}$, we denote the spaces of morphisms from
an object $X$ to another $Y$ as $\mathcal{C}(X,Y)$. The involution
$\mathcal{C}(X,Y)\to\mathcal{C}(Y,X)$ is denoted by $T\mapsto T^{*}$, and the
norm is by $\left\|T\right\|$, so that we have the C∗-identity
$\left\|T^{*}T\right\|=\left\|T\right\|^{2}$. We also assume that the direct
sum $X\oplus Y$ makes sense with structure morphisms given by isometries $X\to
X\oplus Y\leftarrow Y$. Under these assumptions, $\mathcal{C}(X,Y)$ is
naturally a right Hilbert module over the unital C∗-algebra
$\mathcal{C}(X)=\mathcal{C}(X,X)$. We tacitly assume that $\mathcal{C}$ is
closed under taking subobjects, i.e., any projection in $\mathcal{C}(X)$
corresponds to a direct summand of $X$.

A _C ∗-tensor category_ is a C∗-category endowed with monoidal structure given
by: a $*$-bifunctor $\otimes\colon\mathcal{C}\times\mathcal{C}\to\mathcal{C}$,
a unit object $1_{\mathcal{C}}$, and unitary natural isomorphisms

$\displaystyle 1_{\mathcal{C}}$ $\displaystyle\otimes U\to U\leftarrow
U\otimes 1_{\mathcal{C}},$ $\displaystyle\Phi\colon(U\otimes V)\otimes W$
$\displaystyle\to U\otimes(V\otimes W)$

for $U,V,W\in\mathcal{C}$, satisfying a standard set of axioms. Without losing
generality, we may and do assume that $\mathcal{C}$ is _strict_ so that the
above morphisms are identity, and that $1_{\mathcal{C}}$ is simple, unless
explicitly stated otherwise.

A _rigid_ C∗-tensor category is a C∗-tensor category where any object $U$ has
a dual given by: an object $\bar{U}$ and morphisms

$\displaystyle R\colon 1_{\mathcal{C}}$ $\displaystyle\to\bar{U}\otimes U,$
$\displaystyle\bar{R}\colon 1_{\mathcal{C}}$ $\displaystyle\to
U\otimes\bar{U}$

satisfying the conjugate equations for $U$.

When $\mathcal{C}$ is a C∗-tensor category, a _right C ∗-$\mathcal{C}$-module
category_ is given by a C∗-category $\mathcal{M}$, together with a
$*$-bifunctor $\stmryolt\colon\mathcal{M}\times\mathcal{C}\to\mathcal{M}$ and
unitary natural isomorphisms

$\displaystyle X$ $\displaystyle\stmryolt 1_{\mathcal{C}}\to X,$
$\displaystyle\Psi\colon(X\stmryolt U)\stmryolt V$ $\displaystyle\to
X\stmryolt(U\otimes V)$

for $X\in\mathcal{M}$ and $U,V\in\mathcal{C}$, satisfying standard set of
axioms. Again we may and do assume that module category structures are strict
so that the above morphisms are identities.

A _functor of right $\mathcal{C}$-module categories_ is given by a functor
$F\colon\mathcal{M}\to\mathcal{M}^{\prime}$ of the underlying linear
categories, together with natural isomorphisms

$F_{2}=F_{2;m,U}\colon F(m)\stmryolt U\to F(m\stmryolt
U)\quad(m\in\mathcal{M},U\in\mathcal{C})$

satisfying the standard compatibility conditions with structure morphisms of
$\mathcal{M}$ and $\mathcal{M}^{\prime}$. If $\mathcal{M}$ and
$\mathcal{M}^{\prime}$ are C∗-$\mathcal{C}$-module categories, a functor
$(F,F_{2})$ as above is a said to be a _functor of right C
∗-$\mathcal{C}$-module categories_ if it is a $*$-functor and the natural
isomorphism $F_{2}$ is unitary.

###  2.3. Quantum groups

Again we mostly follow [MR3204665][NY3]. A _compact quantum group_ $G$ is
given by a unital C∗-algebra $C(G)$ and a $*$-homomorphism $\Delta\colon
C(G)\to C(G)\otimes C(G)$ satisfying the coassociativity and cancellativity.
We denote its invariant state (_Haar state_) by $h$, and always assume that
$h$ is faithful on $C(G)$. The GNS Hilbert space of $(C(G),h)$ is denoted by
$L^{2}(G)$.

A _finite dimensional unitary representation_ of $G$ is given by a finite
dimensional Hilbert space $H$ and a unitary element $U\in\mathcal{B}(H)\otimes
C(G)$ satisfying

$U_{12}U_{13}=(\mathrm{id}\otimes\Delta)(U).$ (2.1)

We often write $H=H_{U}$. These objects form a rigid C∗-tensor category that
we denote by $\operatorname{Rep}(G)$, with its tensor unit represented by the
multiplicative unit $1\in C(G)\simeq\mathcal{B}(\mathbb{C})\otimes C(G)$. Our
convention of duality is given by $H_{\bar{U}}=\overline{H_{U}}$, with
structure morphisms $R_{U}\colon 1\to\bar{U}\otimes U$ and $\bar{R}_{U}\colon
1\to U\otimes\bar{U}$ given by

$\displaystyle R_{U}$
$\displaystyle=\sum_{i}\bar{\xi}_{i}\otimes\rho_{U}^{-1/2}\xi_{i},$
$\displaystyle\bar{R}_{U}$
$\displaystyle=\sum_{i}\rho_{U}^{1/2}\xi_{i}\otimes\bar{\xi}_{i}$ (2.2)

for any choice of orthonormal basis $(\xi_{i})_{i}\subset H_{U}$. Here
$\rho_{U}$ is the positive operator on $H_{U}$ defined by

$\rho_{U}=(\mathrm{id}\otimes f_{1})(U),$

where $f_{1}$ is a special value of the Woronowicz characters
$(f_{z})_{z\in\mathbb{C}}$.

We denote the set of irreducible classes in $\operatorname{Rep}(G)$ by
$\operatorname{Irr}(G)$, and write $(U_{i})_{i\in\operatorname{Irr}(G)}$ by a
choice of representatives of finite dimensional irreducible unitary
representations, with underlying Hilbert spaces $(H_{i})_{i}$.

As the model of $c_{0}$-sequences on $\hat{G}$, the discrete dual of $G$, we
take the C∗-algebra

$C^{*}_{r}(G)=c_{0}(\hat{G})=c_{0}\mathchar
45\relax\smashoperator[l]{\bigoplus_{i\in\operatorname{Irr}(G)}^{}}\mathcal{B}(H_{i}),$

where $c_{0}\mathchar 45\relax\bigoplus$ denotes the C∗-algebraic direct sum.

As usual, without finite dimensionality assumption on $H$, an unitary
representation of $G$ is given by an unitary element
$U\in\mathcal{M}(\mathcal{K}(H)\otimes C(G))$ satisfying (2.1), where we
extend $\mathrm{id}\otimes\Delta$ to a unital $*$-homomorphism
$\mathcal{M}(\mathcal{K}(H)\otimes C(G))\to\mathcal{M}(\mathcal{K}(H)\otimes
C(G)\otimes C(G))$. In particular, there is a distinguished representation
(the _regular representation_) of $G$ on $L^{2}(G)$, given by the
_multiplicative unitary_ $W_{G}\in\mathcal{M}(c_{0}(\hat{G})\otimes
C(G))\subset\mathcal{M}(\mathcal{K}(L^{2}(G))\otimes C(G))$.

Purely algebraic models of functions on these quantum groups are given as
follows. The _regular subalgebra_ of $C(G)$, or the algebra of _matrix
coefficients_ , is denoted by $\mathcal{O}(G)$, which is spanned by the
elements $(\omega\otimes\mathrm{id})(U)$ for $U\in\operatorname{Rep}(G)$ and
$\omega\in\mathcal{B}(H_{U})_{*}$. On the dual side, we take

$c_{c}(\hat{G})=\bigoplus_{i\in\operatorname{Irr}(G)}\mathcal{B}(H_{i})$

which is the usual algebraic direct sum. There is a nondegenerate linear
duality pairing between $\mathcal{O}(G)$ and $c_{c}(\hat{G})$, so that the
algebra structures and the coalgebra structures on the spaces are related by

$\displaystyle(\phi_{(1)},f_{1})(\phi_{(2)},f_{2})$
$\displaystyle=(\phi,f_{1}f_{2}),$ $\displaystyle(\phi_{1}\phi_{2},f)$
$\displaystyle=(\phi_{1},f_{(1)})(\phi_{2},f_{(2)}).$ (2.3)

The _Drinfeld double_ of $G$ is represented by $\mathcal{O}_{c}(\hat{D}(G))$,
which is modeled on the tensor product coalgebra
$c_{c}(\hat{G})\otimes\mathcal{O}(G)$ endowed with the coproduct
$\Delta(\omega a)=\omega_{(1)}a_{(1)}\otimes\omega_{(2)}a_{(2)}$, and the
$*$-algebra structure induced by those of $\mathcal{O}(G)$ and
$c_{c}(\hat{G})$ together with the exchange rule

$\omega(\cdot a_{(1)})a_{(2)}=a_{(1)}\omega(a_{(2)}\cdot)$

for $\omega\in c_{c}(\hat{G})$ and $a\in\mathcal{O}(G)$. To be precise, this
is to be interpreted as an algebraic model of convolution algebra of functions
on the Drinfeld double quantum group $D(G)$. Similarly, the function algebra
on $D(G)$ is given by the tensor product $*$-algebra
$\mathcal{O}_{c}(D(G))=\mathcal{O}(G)\otimes c_{c}(\hat{G})$ together with the
twisted coproduct induced by the adjoint action of $W_{G}$, see [MR2566309].
These algebras have completion as C∗-algebraic models for locally compact
quantum groups, but we refrain from working with them.

###  2.4. Quantum group actions

A _continuous action_ of $G$ on a C∗-algebra $A$ is given by a nondegenerate
and injective $*$-homomorphism $\alpha\colon A\to C(G)\otimes A$ satisfying
$(\Delta\otimes\mathrm{id})\alpha=(\mathrm{id}\otimes\alpha)\alpha$. We say
that $A$ is a _$G$ -C∗-algebra_.

Given a $G$-C∗-algebra $(A,\alpha)$, its _regular subalgebra_ $\mathcal{A}$ is
defined as the set of elements $a\in A$ such that $\alpha(a)$ belongs to the
algebraic tensor product $\mathcal{O}(G)\otimes A$. This is a left
$\mathcal{O}(G)$-comodule algebra.

A measurable model, _$G$ -von Neumann algebra_, is given by a von Neumann
algebra $M$ and a unital injective normal $*$-homomorphism $\alpha\colon M\to
L^{\infty}(G)\mathbin{\bar{\otimes}}M$ satisfying the coassociativity
condition
$(\Delta\mathbin{\bar{\otimes}}\mathrm{id})\alpha=(\mathrm{id}\mathbin{\bar{\otimes}}\alpha)\alpha$.
The regular subalgebra $\mathcal{M}\subset M$ makes sense as above, and we
denote its C∗-algebraic closure by $\mathcal{R}(M)$. Then $\mathcal{R}(M)$
admits a continuous action of $G$.

######  Example 2.2.

Let $(H,U)$ be a finite dimensional unitary representation of $G$. Under our
convention, $H$ is a right $\mathcal{O}(G)$-comodule, but it can be considered
as a left comodule by the coaction map $\xi\mapsto U_{21}^{*}(1\otimes\xi)$.
This becomes an equivariant right Hilbert $\mathbb{C}$-module by the inner
product

$\langle\xi,\eta\rangle_{\mathbb{C}}=(U_{1}\eta,U_{1^{\prime}}\xi)_{H}h(U_{2^{\prime}}^{*}U_{2})=(\rho_{U}^{-1}\eta,\xi)_{H}$

where $(\cdot,\cdot)_{H}$ denotes the original inner product on $H$, see
[MR3933035]. Then $\mathcal{B}(H)$ admits the induced $G$-C∗-algebra
structure, concretely given by the coaction $T\mapsto U^{*}_{21}T_{2}U_{21}$.
Analogously, when $(H,U)$ is an infinite dimensional unitary representation,
the same formula makes $\mathcal{B}(H)$ a $G$-von Neumann algebra and
$\mathcal{R}(\mathcal{B}(H))$ a $G$-C∗-algebra.

A _Yetter–Drinfeld $G$-C∗-algebra_ is given by a $G$-C∗-algebra $(A,\alpha)$
together with a left action

$\mathcal{O}(G)\otimes\mathcal{A}\to\mathcal{A},\quad f\otimes a\mapsto f\rhd
a$

such that

$\alpha(f\rhd a)=f_{(1)}a_{(1)}S(f_{(3)})\otimes(f_{(2)}\rhd a_{(2)}).$

Equivalently, it can be interpreted as a continuous action of the locally
compact quantum group $D(G)$. Because of this, we also say _$D(G)$
-equivariance_ instead of _Yetter–Drinfeld $G$-equivariance_.

Let $A$ be a left $\mathcal{O}(G)$-module algebra. By duality, we have a
homomorphism

$\beta_{A}\colon
A\to\smashoperator[l]{\prod_{i\in\operatorname{Irr}(G)}^{}}A\otimes\mathcal{B}(H_{i}),\quad
a\mapsto\biggl{(}\sum_{k,l}(u^{(i)}_{kl}\triangleright a)\otimes
m^{(i)}_{kl}\biggr{)}_{i},$

which can be regarded as a right comodule algebra over the multiplier Hopf
algebra representing $\hat{G}$ with the convention of (2.3).

This construction makes sense as a unitary coaction of
$\ell^{\infty}(\hat{G})$ when $A$ is a Yetter–Drinfeld $G$-C∗-algebra.
Moreover, when we consider an action of $\hat{G}$ on a von Neumann algebra
$N$, the target of coaction map becomes

$\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{i\in\operatorname{Irr}(G)}^{}}N\otimes\mathcal{B}(H_{i})\simeq
N\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G}).$

By an analogue of the Tannaka–Krein–Woronowicz duality, the category of unital
$G$-C∗-algebras is equivalent to the category of right
C∗-$(\operatorname{Rep}(G))$-module categories [ydk1][n1].

Concretely, given a unital $G$-C∗-algebra $B$, one takes the category
$\mathcal{D}_{B}$ of finitely generated projective $G$-equivariant right
Hilbert modules over $B$. Thus, an object of $\mathcal{D}_{B}$ is a right
Hilbert module $E_{B}$ with a left coaction $\delta$ of $C(G)$, such that the
action of $B$ is equivariant. Given an object $U$ of $\operatorname{Rep}(G)$,
its right action $E_{B}\stmryolt U$ is represented by the equivariant right
Hilbert module $H_{U}\otimes E_{B}$, where the underlying left comodule is
given as the tensor product of $E_{B}$ and the left comodule $H_{U}$ as
explained in Example 2.2. Explicitly, the new coaction on $E_{B}\stmryolt U$
is given by

$\xi\otimes a\mapsto\left(U_{21}^{*}(1\otimes\xi\otimes
1)\right)\delta(a)_{13}.$

Conversely, given a right C∗-$(\operatorname{Rep}(G))$-module category
$\mathcal{D}$ and an object $X\in\mathcal{D}$, we take the left
$\mathcal{O}(G)$-comodule

$\mathcal{B}_{\mathcal{D},X}=\bigoplus_{i\in\operatorname{Irr}(G)}\bar{H}_{i}\otimes\mathcal{D}(X,X\stmryolt
U_{i}),$

which admits an associative product from irreducible decomposition of monoidal
products. Together with the involution coming from duality of representations,
we obtain a pre C∗-algebra which admits a canonical completion supporting a
coaction of $C(G)$.

######  Remark 2.3.

The formula above explains why left $C(G)$-comodule structures give rise to
right $\operatorname{Rep}(G)$-module categories. Indeed, given two finite
dimensional unitary representations $(H_{U},U)$ and $(H_{V},V)$ of $G$, and
given $\xi\in H_{U}$, $\eta\in H_{V}$ and $a\in E_{B}$, the $C(G)$-coaction on
$\xi\otimes\eta\otimes a\in(H_{V}\otimes E_{B})\stmryolt U$ is given by

$\xi\otimes\eta\otimes a\mapsto
U_{21}^{*}V_{31}^{*}(a_{(1)}\otimes\xi\otimes\eta\otimes a_{(2)}),$

where $a_{(1)}\otimes a_{(2)}=\delta(a)$. Flipping the second and third legs,
we obtain

$U_{21}^{*}V_{31}^{*}(a_{(1)}\otimes\xi\otimes\eta\otimes
a_{(2)})\mapsto(V\otimes U)_{21}^{*}(a_{(1)}\otimes(\eta\otimes\xi)\otimes
a_{(2)}).$

This computation shows that the flip map $H_{U}\otimes H_{V}\to H_{V}\otimes
H_{U}$ induces an equivariant isomorphism $(H_{V}\otimes E_{B})\stmryolt
U\simeq H_{V\otimes U}\otimes E_{B}$.

The generalization of module categories to the nonunital setting is given by
multiplier module categories [AV1].

###  2.5. Vector spaces and algebras over $\mathcal{C}$

Let us review the correspondence between module categories and algebra objects
for linear tensor categories. Further details on the following construction
can be found in [jp1].

Let $\mathcal{C}$ be a rigid tensor category. We denote by
$\operatorname{Vec}(\mathcal{C})$ the category whose objects are the
contravariant linear functors $\mathcal{C}\to\operatorname{Vec}$ and morphisms
are natural transformations. For $V\in\operatorname{Vec}(\mathcal{C})$, we
call its values $V(X)$ on $X\in\mathcal{C}$ the _fibers_ of $V$.

Due to the semisimplicity of $\mathcal{C}$, it is possible to give a concrete
description of $\operatorname{Vec}(\mathcal{C})$; the morphism space between
objects $V$ and $W$ of $\operatorname{Vec}(\mathcal{C})$ can be decomposed
into the algebraic direct product as

$\operatorname{Vec}(\mathcal{C})(V,W)\cong\prod_{i\in\operatorname{Irr}(\mathcal{C})}\mathcal{L}(V(U_{i}),W(U_{i})),$

while the fibers can be written as

$V(Y)\cong\bigoplus_{i\in\operatorname{Irr}(\mathcal{C})}V(U_{i})\otimes\mathcal{C}(Y,U_{i}).$

Under this presentation of $V$, its value on a morphism
$\psi\in\mathcal{C}(X,Y)=\mathcal{C}^{\mathrm{op}}(Y^{\mathrm{op}},X^{\mathrm{op}})$
would then be

$V(\psi)=\bigoplus_{i\in\operatorname{Irr}(\mathcal{C})}\mathrm{id}_{V(U_{i})}\otimes\psi^{\\#}$

with $\psi^{\\#}$ denoting pre-composition with $\psi$.

The category $\operatorname{Vec}(\mathcal{C})$ becomes a monoidal category by
introducing the monoidal product

$(V\otimes
W)(X)=\bigoplus_{i,j\in\operatorname{Irr}(\mathcal{C})}V(U_{i})\otimes\mathcal{C}(X,U_{i}\otimes
U_{j})\otimes W(U_{j})$ (2.4)

together with naturally defined structure morphisms (see [jp1]). Then the
Yoneda embedding $X\to\mathcal{C}(\cdot,X)$ is a fully faithful monoidal
functor from $\mathcal{C}$ to $\operatorname{Vec}(\mathcal{C})$. We regard
$\mathcal{C}$ as a monoidal subcategory of $\operatorname{Vec}(\mathcal{C})$
by means of this embedding.

An _algebra object_ in $\operatorname{Vec}(\mathcal{C})$ is given by
$A\in\operatorname{Vec}(\mathcal{C})$ together with natural transformations
$m\colon A\otimes A\to A$ and $i\colon 1_{\mathcal{C}}\to A$ that make the
algebra diagrams commute. We denote by $\operatorname{Alg}(\mathcal{C})$ the
category whose object are algebras in $\operatorname{Vec}(\mathcal{C})$ and
the morphisms are algebra natural transformations. In other words, the algebra
objects in $\operatorname{Vec}(\mathcal{C})$ are exactly the lax tensor
functors $\mathcal{C}^{\mathrm{op}}\to\operatorname{Vec}$.

######  Definition 2.4.

A _pointed_ right $\mathcal{C}$-module category is a pair $(\mathcal{M},m)$,
where $\mathcal{M}$ is a right $\mathcal{C}$-module category and $m$ is an
object of $\mathcal{M}$. A $\mathcal{C}$-module category is said to be
_cyclic_ if it has a generating object $m$, in the sense that any object
$X\in\mathcal{M}$ is a direct summand of $m\stmryolt U$ for some
$U\in\mathcal{C}$. By a _pointed cyclic $\mathcal{C}$-module category_ we mean
a pointed category $(\mathcal{M},m)$ such that $m$ is a generator of
$\mathcal{M}$.

When the choice of $m$ is implicit, we just write $\mathcal{M}$ instead of
$(\mathcal{M},m)$, and $m_{\mathcal{M}}=m$.

######  Definition 2.5.

A _functor of pointed right $\mathcal{C}$-module categories_ is given by a
pair consisting of a $\mathcal{C}$-module functor
$F\colon\mathcal{M}\to\mathcal{M}^{\prime}$ and an isomorphism $F_{0}\colon
m_{\mathcal{M}^{\prime}}\to F(m_{\mathcal{M}})$. We denote by
$\operatorname{Mod}^{c}_{*}(\mathcal{C})$ the category of pointed cyclic right
$\mathcal{C}$-module categories, with these functors as morphisms.

Let us denote the category of cyclic pointed right $\mathcal{C}$-module
categories and functors of pointed $\mathcal{C}$-module categories by
$\operatorname{Mod}^{c}_{*}(\mathcal{C})$.

######  Theorem 2.6 ([ostrik03][jp1]).

There is an equivalence of categories
$\operatorname{Alg}(\mathcal{C})\simeq\operatorname{Mod}^{c}_{*}(\mathcal{C})$.

Let us sketch this correspondence. Given
$A\in\operatorname{Alg}(\mathcal{C})$, we get a right $\mathcal{C}$-module
category by taking the category $\mathcal{M}_{A}$ of left $A$-module objects
in $\mathcal{C}$. Concretely, we can start with objects
$X_{A}\in\mathcal{M}_{A}$ for $X\in\mathcal{C}$ (corresponding to the left
$A$-modules $A\otimes X$) and morphism sets

$\mathcal{M}_{A}(X_{A},Y_{A})=A(X\otimes\bar{Y}),$

and then take the idempotent completion. In this presentation the
$\mathcal{C}$-module structure is induced by $X_{A}\stmryolt U=(X\otimes
U)_{A}$.

Conversely, given $\mathcal{M}\in\operatorname{Mod}^{c}_{*}(\mathcal{C})$, we
define $A_{\mathcal{M}}\in\operatorname{Vec}(\mathcal{C})$ by setting its
fibers as

$A_{\mathcal{M}}(X)=\mathcal{M}(m_{\mathcal{M}}\stmryolt X,m_{\mathcal{M}}),$

and its action on $\psi\in\mathcal{C}(X,Y)$ as

$A_{\mathcal{M}}(\psi)\colon\mathcal{M}(m_{\mathcal{M}}\stmryolt
Y,m_{\mathcal{M}})\to\mathcal{M}(m_{\mathcal{M}}\stmryolt
X,m_{\mathcal{M}}),\quad f\mapsto f(\mathrm{id}\otimes\psi).$

Now, let us assume that $\mathcal{C}$ is a rigid C∗-tensor category.

######  Definition 2.7 ([jp1]).

An algebra $A\in\operatorname{Alg}(\mathcal{C})$ is said to be a _C ∗-algebra
object_ in $\mathcal{C}$ when the corresponding pointed cyclic
$\mathcal{C}$-module category $\mathcal{M}_{A}$ admits a compatible structure
of C∗-category, and _a $W^{*}$-category_ when $\mathcal{M}_{A}$ admits a
compatible structure of $W^{*}$-category.

See Appendix A for an intrinsic characterization of C∗-algebra objects. When
$A$ is a C∗-algebra object in the above sense, we frequently regard
$A(U\otimes\bar{U})$ as a C∗-algebra up to the isomorphism
$A(U\otimes\bar{U})\cong\mathcal{M}_{A}(U_{A},U_{A})$.

######  Definition 2.8.

The category $\mathrm{Hilb}(\mathcal{C})$ is the subcategory of
$\operatorname{Vec}(\mathcal{C})$ consisting of contravariant $*$-functors
$\mathcal{C}\to\mathrm{Hilb}$, and having uniformly bounded natural
transformations as morphisms: for $H,K\in\mathrm{Hilb}(\mathcal{C})$,

$\mathrm{Hilb}(\mathcal{C})(H,K)\cong\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{i\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{B}(H(U_{i}),K(U_{i})).$

$\mathrm{Hilb}(\mathcal{C})$ becomes monoidal subcategory of
$\operatorname{Vec}(\mathcal{C})$ by giving the following inner product to the
fibers of $H\otimes K$. On the space
$H(U_{i})\otimes\mathcal{C}(X,U_{i}\otimes U_{j})\otimes K(U_{j})$, consider
the Hermitian inner product characterized by

$(\xi_{2}\otimes\alpha_{2}\otimes\eta_{2},\xi_{1}\otimes\alpha_{1}\otimes\eta_{1})1_{X}=\frac{1}{d_{i}d_{j}}(\xi_{2},\xi_{1})(\eta_{2},\eta_{1})\alpha_{1}^{*}\alpha_{2}.$

Then $(H\otimes K)(X)$ has an inner product such that (2.4) gives an
orthogonal decomposition.

######  Remark 2.9.

An object $H\in\mathrm{Hilb}(\mathcal{C})$ can be interpreted as the infinite
direct sum $\bigoplus_{i}H(U_{i})\otimes U_{i}$. This way
$\mathrm{Hilb}(\mathcal{C})$ can be identified with the ind-category
$\operatorname{ind}(\mathcal{C})$ of $\mathcal{C}$ as defined in [NY2]*Section
2.

We will work with the following W∗-algebra object $\mathcal{B}(H)$ for
$H\in\mathrm{Hilb}(\mathcal{C})$ [jp1]*Example 10. The fiber is given by
$\mathcal{B}(H)(U)=\mathrm{Hilb}(\mathcal{C})(H\otimes U,H)$, with a natural
algebra structure such that the associated module category
$\mathcal{M}_{\mathcal{B}(H)}$ is generated by objects of the form
$U_{\mathcal{B}(H)}$ for $U\in\mathcal{C}$ and the morphism spaces are given
by

$\mathcal{M}_{\mathcal{B}(H)}(U_{\mathcal{B}(H)},V_{\mathcal{B}(H)})=\mathrm{Hilb}(\mathcal{C})(H\otimes
U,H\otimes V).$

Later, we will use the following analogue of the Gelfand–Naimark Theorem to
construct injective envelopes of C∗-module categories.

######  Theorem 2.10 ([jp1]*Theorem 4).

Every C∗-algebra object in $\operatorname{Vec}(\mathcal{C})$ admits a faithful
representation into $\mathcal{B}(H)$, for some
$H\in\mathrm{Hilb}(\mathcal{C})$.

Here, a _faithful representation_ means a *-algebra morphism
$A\to\mathcal{B}(H)$ where the associated functor of module categories
$\mathcal{M}_{A}\to\mathcal{M}_{\mathcal{B}(H)}$ is faithful, or equivalently,
the corresponding natural transformation $A(U)\to\mathcal{B}(H)(U)$ is given
by injective maps.

######  Remark 2.11.

When $\mathcal{C}$ is $\operatorname{Rep}(G)$ for a compact quantum group $G$,
the category $\mathrm{Hilb}(\operatorname{Rep}(G))$ is can be identified with
the category of all unitary representations of $G$. Then the above theorem
reduces to the fact that any $G$-C∗-algebra can be equivariantly embedded into
one of the form $\mathcal{R}(\mathcal{B}(H))$ for some infinite dimensional
unitary representation $(H,U)$ of $G$.

Recall that if $A$ and $B$ are C∗-algebra objects in
$\operatorname{Vec}(\mathcal{C})$, then the spaces $A(X\bar{X})$ and
$B(X\bar{X})$ are C∗-algebras for all $X\in\mathcal{C}$.

######  Definition 2.12 ([jp1]).

Let $A$ and $B$ be C∗-algebra objects in $\operatorname{Vec}(\mathcal{C})$. A
_completely positive map_ (or a _cp map_) from $A$ to $B$ is a natural
transformation of contravariant C∗-functors $\theta\colon A\to B$ for which
the induced maps $\theta_{X\bar{X}}\colon A(X\bar{X})\to B(X\bar{X})$, with
$X\in\mathcal{C}$, are positive. If they are also unital, $\theta$ is called a
_unital completely positive map_ , or a _ucp map_.

######  Definition 2.13 ([jp1]).

Let $(\mathcal{M},m)$ and $(\mathcal{M}^{\prime},m^{\prime})$ be pointed
C∗-$\mathcal{C}$-module categories. A _multiplier_
$\Theta\colon(\mathcal{M},m)\to(\mathcal{M}^{\prime},m^{\prime})$ is a
collection of linear maps

$\Theta_{X,Y}\colon\mathcal{M}(m\stmryolt X,m\stmryolt
Y)\to\mathcal{M}^{\prime}(m^{\prime}\stmryolt X,m^{\prime}\stmryolt Y)$

for $X,Y\in\mathcal{C}$ satisfying

$\Theta_{X\otimes U,Z\otimes
U}(((\mathrm{id}_{m}\stmryolt\phi)f(\mathrm{id}_{m}\stmryolt\psi))\stmryolt\mathrm{id}_{U})=((\mathrm{id}_{m^{\prime}}\stmryolt\phi)\Theta_{Y,W}(f)(\mathrm{id}_{m^{\prime}}\stmryolt\psi))\stmryolt\mathrm{id}_{U}$

for all $U,X,Y,W,Z\in\mathcal{C}$, $\psi\in\mathcal{C}(X,Y)$,
$\phi\in\mathcal{C}(W,Z)$, and $f\in\mathcal{M}(m\stmryolt Y,m\stmryolt W)$. A
multiplier $\Theta$ for which $\Theta_{X,X}$ is positive for all $X$ is called
a _cp multiplier_. It is called _ucp multiplier_ if $\Theta_{X,X}$ is ucp for
$X$.

######  Proposition 2.14 ([jp1]*Proposition 7 and Corollary 5).

Given two C∗-algebra objects $A,B$ in $\operatorname{Vec}(\mathcal{C})$, the
space of natural transformations between the contravariant functors $A$ and
$B$ is in bijection with the space of multipliers from $\mathcal{M}_{A}$ to
$\mathcal{M}_{B}$. Under this bijection, the cp maps correspond to the cp
multipliers, as the ucp maps correspond to the ucp multipliers.

Concretely, given a natural transformation $\theta\colon A\to B$, the
corresponding multiplier is given by

$\Theta_{V,W}(T)=\sum_{U,\alpha}((\theta_{U}(\bar{R}_{W}^{*}(T\otimes\mathrm{id}_{\bar{W}})v_{\alpha})v_{\alpha}^{*})\otimes\mathrm{id}_{W})(\mathrm{id}_{V}\otimes
R_{W})$ (2.5)

where $U$ runs over $\operatorname{Irr}(\mathcal{C})$ and
$(v_{\alpha})_{\alpha}$ is an orthonormal basis of
$\mathcal{C}(U,V\otimes\bar{W})$.

##  3\. Injective envelopes for $D(G)$-C∗-algebras

In this section we are going to establish the existence of injective envelopes
for unital Yetter–Drinfeld $G$-C∗-algebras, closely following [HHN1]*Section
2.

Take a unitary representation $(H,U)$ of $G$ on $H$, and as before consider
the coaction

$\alpha_{H}(T)=U_{21}^{*}(1\otimes T)U_{21}$

on $\mathcal{B}(H)$. Then
$\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G})$ becomes a
$G$-von Neumann algebra with coaction

$\beta\colon\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G})\to
L^{\infty}(G)\mathbin{\bar{\otimes}}\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G}),\quad
x\mapsto W_{31}^{*}(\alpha_{H}\otimes\mathrm{id})(x)W_{31},$ (3.1)

where $W=W_{G}$ is the multiplicative unitary. Together with the $\hat{G}$-von
Neumann algebra structure given by $\mathrm{id}\otimes\hat{\Delta}$, we get a
Yetter–Drinfeld von Neumann $G$-algebra structure on
$\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G})$.

######  Proposition 3.1 ( [HHN1], Section 2).

Under the above setting, the $G$-C∗-algebra
$\mathcal{R}(\mathcal{B}(H)\otimes\ell^{\infty}(\hat{G}))$ admits a structure
of Yetter–Drinfeld $G$-C∗-algebra.

Now let us define injective envelopes for unital Yetter–Drinfeld
$G$-C∗-algebras, in the quite standard way. In this section, by equivariance
we always mean equivariance with respect to Yetter–Drinfeld structures.

######  Definition 3.2.

A unital Yetter–Drinfeld $G$-C∗-algebra $A$ is said to be _injective_ if for
any equivariant completely isometric map $\phi\colon B\to C$ of unital
Yetter–Drinfeld $G$-C∗-algebras and any equivariant ucp map $\psi\colon B\to
A$, there is an equivariant ucp map $\tilde{\psi}\colon C\to A$ that extends
$\psi$ along $\phi$, i.e., $\tilde{\psi}\circ\phi=\psi$.

######  Definition 3.3.

An _injective envelope_ of a unital Yetter–Drinfeld $G$-C∗-algebra $A$ is
given by an injective Yetter–Drinfeld $G$-C∗-algebra $\mathcal{I}$ and an
equivariant complete isometry $\phi\colon A\to\mathcal{I}$ which is
_essential_ , i.e., for any equivariant ucp map $\psi\colon\mathcal{I}\to B$,
$\psi$ is completely isometric if and only if $\psi\phi$ is.

The first step in proving the existence of injective envelopes is to show the
existence of enough injective objects. This is accomplished by the following
proposition.

######  Proposition 3.4.

Let $(H,U)$ be a unitary representation of $G$. The Yetter–Drinfeld
$G$-C∗-algebra
$\mathcal{R}(\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G}))$ is
injective.

###### Proof.

The proof is essentially the same as [HHN1]*Corollary 2.6. By
[HHN1]*Proposition 2.5, for any Yetter–Drinfeld $G$-C∗-algebra $A$ there is a
bijective correspondence between the $G$-equivariant completely bounded maps
$\phi\colon A\to\mathcal{B}(H)$ and $D(G)$-equivariant cb maps $P\colon
A\to\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G})$. Moreover,
the image of such map $P$ would be in
$\mathcal{R}(\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G}))$.

By the averaging argument (see [HHN1], Lemma 2.10), the injectivity of
$\mathcal{B}(H)$ as an operator system implies its injectivity as a
$G$-operator system. This implies the injectivity of
$\mathcal{R}(\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G}))$ as
a Yetter–Drinfeld $G$-C∗-algebra. ∎

Let $A$ be a Yetter–Drinfeld $G$-C∗ algebra, and let us take a faithful
representation $\pi\colon A\to\mathcal{B}(H)$ together with a covariant
unitary representation $U$ of $G$ on $H$. For example, we may start from a
faithful nondegenerate representation $\pi_{0}\colon A\to B(H_{0})$, and take

$\displaystyle H$ $\displaystyle=L^{2}(G)\otimes H_{0},$ $\displaystyle\pi$
$\displaystyle=(\lambda\otimes\pi_{0})\alpha,$ $\displaystyle U$
$\displaystyle=W_{13}\in\mathcal{M}(\mathcal{K}(L^{2}(G)\otimes H_{0})\otimes
C(G)).$

######  Definition 3.5.

Under the above setting, the _Poisson integral_ of $\pi$ is defined as the
$D(G)$-equivariant embedding map

$P_{\pi}\colon
A\to\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G}),\quad
a\mapsto(\pi\otimes\mathrm{id})\beta_{A}(a).$

######  Theorem 3.6.

Every unital Yetter–Drinfeld $G$-C∗-algebra $A$ admits an injective envelope.

###### Proof.

Let us fix $(\pi,H)$ as above. Consider the semigroup $S$ of equivariant ucp
maps $f$ from $\mathcal{R}(\mathcal{B}(H)\otimes\ell^{\infty}(\hat{G}))$ to
itself such that $fP_{\pi}=P_{\pi}$. This is closed with respect to the
topology of pointwise convergence for the weak∗-topology. By Proposition 2.1,
there is a unique minimal idempotent in $\phi_{0}\in S$.

The rest is quite standard, as follows.

Put $\mathcal{I}(A)=\operatorname{Im}(\phi_{0})$, and endow it with the
Choi–Effros product coming from $\phi_{0}$. By construction, $\mathcal{I}(A)$
is injective as a Yetter–Drinfeld $G$-C∗-algebra, and we have an embedding
$A\to\mathcal{I}(A)$ given by $P_{\pi}$. We are going to show that
$\mathcal{I}(A)$ is an injective envelope of $A$.

First, we claim that the only ucp $D(G)$-equivariant map
$\psi\colon\mathcal{I}(A)\to\mathcal{I}(A)$ restricting to $\mathrm{id}_{A}$
on $A$ is the identity map. Otherwise $\psi\phi_{0}$ would be an element of
$S$ that is strictly below $\phi_{0}$, contradicting the minimality of
$\phi_{0}$.

Now, let us take a completely isometric $D(G)$-equivariant ucp map
$\psi\colon\mathcal{I}(A)\to B$ such that $\psi P_{\pi}$ is completely
isometric. Without losing generality we may assume that $B$ is injective. By
the injectivity of $\mathcal{I}(A)$, there is an equivariant ucp map
$\eta\colon B\to\mathcal{I}(A)$. By the above claim implies that $\eta\psi$
must be the identity, which implies that $\psi$ is completely isometric. This
shows that $\mathcal{I}(A)$ is an essential extension of $A$. ∎

######  Remark 3.7.

Theorem 3.6 would extend to more general locally compact quantum groups once
we have an analogue of Proposition 3.4. Note that [MR2985658] already
establishes the existence of injective envelopes for the category of operator
spaces with von Neumann algebraic coaction of a Hopf–von Neumann algebra $M$,
such that the predual Banach algebra $M_{*}$ has a bounded approximate unit
with bound $1$. For us, the relevant Hopf–von Neumann algebra is
$M=L^{\infty}(D(G))$, so that $M_{*}$ is a completion of
$\mathcal{O}_{c}(\hat{D}(G))$. Taking an increasing net of finite sets of
$\operatorname{Irr}(G)$, we obtain an approximate unit of $M_{*}$ of the form
$(z_{i}\otimes 1)_{i}$, where $z_{i}$ is a central projection of
$c_{c}(\hat{G})$. In this regard Theorem 3.6 could be regarded as a refinement
of [MR2985658]*Theorem 2.7 to the category of _continuous_ actions of $D(G)$.

##  4\. Injectivity for module categories

As before, let $\mathcal{C}$ be a rigid C∗-tensor category. Our goal here is
to establish that the theory of injectivity work well for $\mathcal{C}$-module
categories, which will form a basis of our later analysis for module
categories with additional structures.

Let us fix an object $U$ of $\mathcal{C}$. Let us denote by $\mathcal{C}_{U}$
the C∗-$\mathcal{C}$-module category defined as (the idempotent completion of)
the category with objects $V\in\mathcal{C}$, and morphism sets

$\mathcal{C}_{U}(V,W)=\mathcal{C}(U\otimes V,U\otimes W).$

In other words, this is simply the category $\mathcal{C}$ as a right module
category over itself, but with shifted base point $U$.

Let us call $H\in\mathrm{Hilb}(\mathcal{C})$ _finite-dimensional_ when it is
isomorphic to the image under the Yoneda embedding of some object
$U\in\mathcal{C}$. A Hilbert space object is finite dimensional if and only if
it has finite dimensional fibers and finite support: $H(V)=0$ for all but
finitely many $V\in\operatorname{Irr}(\mathcal{C})$.

As we show in the next proposition, the categories $\mathcal{C}_{U}$ can be
used as a model of $\mathcal{B}(H)$ for finite dimensional Hilbert space
objects $H$.

######  Proposition 4.1.

Let $H\in\mathrm{Hilb}(\mathcal{C})$ be finite dimensional, corresponding to
an object $U\in\mathcal{C}$. Then the pointed cyclic C∗-$\mathcal{C}$-module
category $\mathcal{M}_{\mathcal{B}(H)}$ is isomorphic to $\mathcal{C}_{U}$.

###### Proof.

Let us write $\tilde{V}$ for the image of Yoneda embedding of
$V\in\mathcal{C}$, so that we have $H\simeq\tilde{U}$. Since
$V\mapsto\tilde{V}$ is a monoidal functor, we have

$(H\otimes\tilde{V})(Z)\simeq\mathcal{C}(Z,U\otimes V).$

Now, recall that we have

$\mathcal{M}_{\mathcal{B}(H)}(V,W)=\mathcal{B}(H)(V\otimes\bar{W})\simeq\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{Z\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{B}\left((H\otimes\tilde{V})(Z),(H\otimes\tilde{W})(Z)\right).$

We thus have

$\mathcal{M}_{\mathcal{B}(H)}(V,W)\simeq\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{Z\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{B}(\mathcal{C}(Z,U\otimes
V),\mathcal{C}(Z,U\otimes W)).$

From this we see that an element $\theta\in\mathcal{M}_{\mathcal{B}(H)}(V,W)$
is the same thing as a bounded family of linear operators

$\theta_{Z}\colon\mathcal{C}(Z,U\otimes V)\to\mathcal{C}(Z,U\otimes
W)\quad(Z\in\operatorname{Irr}(\mathcal{C})).$

This is precisely a natural transformation between the Yoneda embeddings of
$U\otimes V$ and $U\otimes W$, hence given by a morphism $U\otimes V\to
U\otimes W$. We thus obtained an isomorphism
$\mathcal{M}_{\mathcal{B}(H)}(V,W)\simeq\mathcal{C}_{U}(V,W)$. ∎

######  Definition 4.2.

A pointed C∗-$\mathcal{C}$-module category $(\mathcal{M},m)$ is said to be
_injective_ when, for any ucp-multiplier
$\Phi\colon(\mathcal{N},n)\to(\mathcal{M},m)$ and another ucp-multiplier
$\Psi\colon(\mathcal{N},n)\to(\mathcal{N}^{\prime},n^{\prime})$ such that the
maps

$\Psi_{V,W}\colon\mathcal{N}(n\stmryolt V,n\stmryolt
W)\to\mathcal{N}^{\prime}(n^{\prime}\stmryolt V,n^{\prime}\stmryolt W)$

are completely isometric for $V,W\in\mathcal{C}$, there exists a ucp-
multiplier
$\tilde{\Phi}\colon(\mathcal{N}^{\prime},n^{\prime})\to(\mathcal{M},m)$ such
that $\tilde{\Phi}\Psi=\Phi$.

When the generator $m$ is understood from the context we also say
$\mathcal{M}$ is injective. We will later see that this definition is
independent of the choice of $m$ for cyclic module categories.

In the following lemmas, $(\mathcal{M},m)$ is a fixed pointed
C∗-$\mathcal{C}$-module category.

######  Lemma 4.3.

Let $(\mathcal{M}^{\prime},m^{\prime})$ be another pointed
C∗-$\mathcal{C}$-module category. Then there is a bijective correspondence
between the cp-multipliers
$P\colon(\mathcal{M},m)\to(\mathcal{M}^{\prime},m^{\prime}\stmryolt U)$ and
the cp-multipliers
$Q\colon(\mathcal{M},m\stmryolt\bar{U})\to(\mathcal{M}^{\prime},m^{\prime})$.
Under this correspondence, a ucp-multiplier $P$ corresponds to a ucp-
multiplier $Q$ satisfying

$Q_{U\otimes V,U\otimes
V}(\mathrm{id}_{m}\stmryolt(R_{U}R_{U}^{*}\otimes\mathrm{id}_{V}))=d_{U}^{-1}\mathrm{id}_{m^{\prime}\stmryolt
U\otimes V}$ (4.1)

for all $V\in\mathcal{C}$.

###### Proof.

Let $P\colon(\mathcal{M},m)\to(\mathcal{M}^{\prime},m^{\prime}\stmryolt U)$ be
a cp-multiplier, given by the maps

$P_{V,W}\colon\mathcal{M}(m\stmryolt V,m\stmryolt
W)\to\mathcal{M}^{\prime}(m^{\prime}\stmryolt U\otimes V,m^{\prime}\stmryolt
U\otimes W).$

Then we get a multiplier
$Q\colon(\mathcal{M},m\stmryolt\bar{U})\to(\mathcal{M}^{\prime},m^{\prime})$
by setting

$\displaystyle Q_{V,W}\colon\mathcal{M}(m\stmryolt\bar{U}\otimes
V,m\stmryolt\bar{U}\otimes W)$
$\displaystyle\to\mathcal{M}^{\prime}(m^{\prime}\stmryolt
V,m^{\prime}\stmryolt W),$ $\displaystyle T$ $\displaystyle\mapsto
d_{U}^{-1}(\mathrm{id}_{m^{\prime}}\stmryolt\bar{R}_{U}^{*}\otimes\mathrm{id}_{W})P_{\bar{U}\otimes
V,\bar{U}\otimes
W}(T)(\mathrm{id}_{m^{\prime}}\stmryolt\bar{R}_{U}\otimes\mathrm{id}_{V}).$

If we have $V=W$ and $T\geq 0$, then $P_{\bar{U}\otimes V,\bar{U}\otimes
V}(T)$ is positive by assumption, and since $d_{U}^{-1/2}\bar{R}_{U}$ is an
isometry, we conclude that $Q_{V,V}(T)$ is positive. This shows that $Q$ is
completely positive. Moreover, when $P$ is unital, $Q$ is unital by
$\bar{R}_{U}^{*}R_{U}=d_{U}$, and (4.1) follows from $\mathcal{C}$-modularity.

In the other direction, given a cp-multiplier
$Q\colon(\mathcal{M},m\stmryolt\bar{U})\to(\mathcal{M}^{\prime},m^{\prime})$,
we get a multiplier
$P\colon(\mathcal{M},m)\to(\mathcal{M}^{\prime},m^{\prime}\stmryolt U)$ by

$P_{V,W}(T)=d_{U}^{-1}Q_{U\otimes V,U\otimes W}((\mathrm{id}_{m}\stmryolt
R_{U}\otimes\mathrm{id}_{W})T(\mathrm{id}_{m}\stmryolt
R_{U}^{*}\otimes\mathrm{id}_{V})).$

This is again completely positive by the complete positivity of $Q$. Moreover,
when $Q$ satisfies (4.1), the unitality of $P$ is a direct consequence of the
above definition.

It remains to check that these constructions are inverse to each other. Let us
start from a multiplier $P$ as above, and let $Q$ be the corresponding one.
Let
$P^{\prime}\colon(\mathcal{M},m)\to(\mathcal{M}^{\prime},m^{\prime}\stmryolt
U)$ denote the multiplier we obtain from $Q$. We then need to check that
$P^{\prime}$ is equal to $P$. Expanding the definitions, we have

$P^{\prime}(T)=\\\
(\mathrm{id}_{m^{\prime}}\stmryolt\bar{R}_{U}^{*}\otimes\mathrm{id}_{U\otimes
W})P_{\bar{U}\otimes U\otimes V,\bar{U}\otimes U\otimes
W}\bigl{(}(\mathrm{id}_{m}\stmryolt
R_{U}\otimes\mathrm{id}_{W})T(\mathrm{id}_{m}\stmryolt
R_{U}^{*}\otimes\mathrm{id}_{V})\bigr{)}(\mathrm{id}_{m^{\prime}}\stmryolt\bar{R}_{U}\otimes\mathrm{id}_{U\otimes
V}).$

Using

$P_{\bar{U}\otimes U\otimes V,\bar{U}\otimes U\otimes
W}\bigl{(}(\mathrm{id}_{m}\stmryolt
R_{U}\otimes\mathrm{id}_{W})T(\mathrm{id}_{m}\stmryolt
R_{U}^{*}\otimes\mathrm{id}_{V})\bigr{)}=\\\
(\mathrm{id}_{m^{\prime}}\stmryolt\mathrm{id}_{U}\otimes
R_{U}\otimes\mathrm{id}_{W})P_{V,W}(T)(\mathrm{id}_{m^{\prime}}\stmryolt\mathrm{id}_{U}\otimes
R_{U}^{*}\otimes\mathrm{id}_{V})$

and the conjugate equations for $U$, we see that $P^{\prime}(T)=P(T)$ indeed
holds. This can also be seen from the graphical calculus for multipliers
established in [jp1]. ∎

######  Remark 4.4.

In fact, the condition (4.1) for all $V\in\mathcal{C}$ implies that $Q$ is a
unital multiplier. To see this, given $W\in\mathcal{C}$, take
$V=\bar{U}\otimes W$ and consider the operation

$T\mapsto(\mathrm{id}_{m^{\prime}}\stmryolt\bar{R}_{U}^{*}\otimes\mathrm{id}_{W})T(\mathrm{id}_{m^{\prime}}\stmryolt\bar{R}_{U}\otimes\mathrm{id}_{W})$

on both sides of (4.1). The left hand side gives
$Q_{W,W}(\mathrm{id}_{m\stmryolt W})$ by the conjugate equations, while the
right hand side gives $\mathrm{id}_{m^{\prime}\stmryolt W}$ by
$\bar{R}_{U}^{*}\bar{R}_{U}=d_{U}$.

######  Lemma 4.5 (cf. [HHN1]*Proposition 4.11).

There is a bijective correspondence between the ucp-multipliers
$(\mathcal{M},m)\to(\mathcal{C},1_{\mathcal{C}})$ and the states on the
C∗-algebra $\mathcal{M}(m)$.

###### Proof.

Suppose we are given a ucp-multiplier
$P\colon(\mathcal{M},m)\to(\mathcal{C},1_{\mathcal{C}})$.

Given $V\in\mathcal{C}$, define a state $\omega_{V}$ on
$\mathcal{M}(m\stmryolt V)$ by $\omega_{V}(T)=\mathrm{tr}_{V}(P_{V,V}(T))$. By
the multiplier property, we have

$\omega_{V}(T)=d_{V}^{-1}P_{1,1}((\mathrm{id}_{m}\stmryolt\bar{R}_{V}^{*})(T\stmryolt\mathrm{id}_{\bar{V}})(\mathrm{id}_{m}\stmryolt\bar{R}_{V})).$
(4.2)

We claim that the state $\omega_{1}=P_{1,1}$ determines both $\omega_{V}$ and
$P_{V,V}$, which implies that the correspondence from ucp-multipliers to
states is one-to-one.

First, (4.2) shows that $\omega_{V}$ can be written in terms of $\omega_{1}$.

Observe that $\mathcal{C}(V)$ embeds into $\mathcal{M}(m\stmryolt V)$ by
$T\mapsto\mathrm{id}_{m}\stmryolt T$, and the restriction of $\omega_{V}$ to
$\mathcal{C}(V)$ agrees with $\mathrm{tr}_{V}$. Using the spherical structure,
we see that $\mathcal{C}(V)$ is in the centralizer of $\omega_{V}$, implying
the existence of a unique state-preserving conditional expectation

$E_{V}\colon\mathcal{M}(m\stmryolt
V)\to\mathcal{C}(V),\quad\mathrm{tr}_{V}\circ E_{V}=\omega_{V}.$

By the uniqueness, we see that $P_{V,V}=E_{V}$, which shows that $E_{V}$ is
indeed determined by $\omega_{V}$, hence by $\omega_{1}$.

Conversely, let $\omega\colon\mathcal{M}(m)\to\mathbb{C}$ be a state.
Composing the conditional expectation

$\mathcal{M}(m\stmryolt V)\to\mathcal{M}(m),\quad T\mapsto
d_{V}^{-1}(\mathrm{id}_{m}\stmryolt\bar{R}_{V}^{*})(T\stmryolt\mathrm{id}_{\bar{V}})(\mathrm{id}_{m}\stmryolt\bar{R}_{V})$

with $\omega$, we obtain a state $\omega_{V}$ on $\mathcal{M}(m\stmryolt V)$.

Again $\mathcal{C}(V)$ sits in the centralizer of $\omega$. Setting
$P_{V,V}\colon\mathcal{M}(m\stmryolt V)\to\mathcal{C}(V)$ to be the
conditional expectation for $\omega_{V}$, we obtain a ucp-multiplier
satisfying $\omega=P_{1,1}$. ∎

######  Lemma 4.6.

Let $\omega$ be a state on $\mathcal{M}(m\stmryolt\bar{U})$, and
$Q\colon(\mathcal{M},m\stmryolt\bar{U})\to(\mathcal{C},1)$ be the ucp-
multiplier corresponding to $\omega$ given by Lemma 4.5. Then $Q$ satisfies
(4.1) if and only if $\omega$ restricts to the normalized categorical trace on
$\mathcal{C}(\bar{U})$.

###### Proof.

Let $\omega$ be a state on $\mathcal{M}(m\stmryolt\bar{U})$. Recall that from
$\omega$ we can construct state $\omega_{V}$ on the C∗-algebra
$\mathcal{M}(m\stmryolt(\bar{U}\otimes V))$. Let $Q$ be the corresponding ucp-
multiplier. Then, to check (4.1), it is enough to check

$\omega_{U\otimes V}((\mathrm{id}_{m}\stmryolt
R_{U}R_{U}^{*}\otimes\mathrm{id}_{V})(\mathrm{id}_{m\stmryolt\bar{U}}\stmryolt
S))=d_{U}^{-1}\mathrm{tr}_{U\otimes V}(S)\quad(S\in\mathcal{C}(U\otimes V))$

as $Q_{U\otimes V,U\otimes V}$ is the unique conditional expectation
$\mathcal{M}(m\stmryolt(\bar{U}\otimes U\otimes V))\to\mathcal{C}(U\otimes V)$
such that $\mathrm{tr}_{U\otimes V}\circ Q_{U\otimes V,U\otimes
V}=\omega_{U\otimes V,U\otimes V}$. By the definition of $\omega_{U\otimes
V}$, the left hand side of the above is equal to

$d_{U}^{-1}\omega\left(\mathrm{id}_{m}\stmryolt(R_{U}^{*}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{\bar{U}}\otimes
S^{\prime}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{\bar{U}}\otimes\bar{R}_{U})\right)$

for $S^{\prime}=(\mathrm{id}\otimes\mathrm{tr}_{V})(S)$. If $\omega$ restricts
to $\mathrm{tr}_{\bar{U}}$, by a standard sphericity argument we see that this
is indeed equal to $d_{U}^{-1}\mathrm{tr}_{U\otimes V}(S)$.

Conversely, suppose that we know (4.1). We then have

$d_{U}Q_{U,U}(\mathrm{id}_{m}\stmryolt(R_{U}R_{U}^{*}(\mathrm{id}_{\bar{U}}\otimes
S)))=S$

for any $S\in\mathcal{C}(U)$. From the form of state $\omega_{U}$, we obtain

$\omega\left(\mathrm{id}_{m}\stmryolt\left((R_{U}^{*}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{\bar{U}}\otimes
ST\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{\bar{U}}\otimes\bar{R}_{U})\right)\right)=\mathrm{tr}_{U}(ST)\quad(T\in\mathcal{C}(U)).$

Again by standard sphericity argument we obtain
$\omega(S^{\vee})=\mathrm{tr}_{\bar{U}}(S^{\vee})$ for

$S^{\vee}=(R_{U}^{*}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{\bar{U}}\otimes
S\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{\bar{U}}\otimes\bar{R}_{U}).$

As such $S^{\vee}$ exhaust $\mathcal{C}(\bar{U})$, we obtain the claim. ∎

We now move towards proving that the category
$\operatorname{Mod}^{c}_{*}(\mathcal{C})$ has enough injective objects. First
we show that objects of a particular type are injective.

######  Lemma 4.7.

The object $(\mathcal{C},1_{\mathcal{C}})$ is injective in
$\operatorname{Mod}^{c}_{*}(\mathcal{C})$.

###### Proof.

For any $(\mathcal{M},m)\in\operatorname{Mod}^{c}_{*}(\mathcal{C})$, the ucp-
multipliers $(\mathcal{M},m)\to(\mathcal{C},1_{\mathcal{C}})$ are completely
determined by the induced states on $\mathcal{M}(m)$. Moreover, a completely
isometric multiplier $(\mathcal{M},m)\to(\mathcal{M}^{\prime},m^{\prime})$
induces a (complete) isometric map of unital C∗-algebras
$\mathcal{M}(m)\to\mathcal{M}^{\prime}(m^{\prime})$. Then the claim follows
from the Hahn–Banach theorem. ∎

######  Proposition 4.8.

Let $(\mathcal{M},m)$ be an injective pointed $\mathcal{C}$-module category.
For any $U\in\mathcal{C}$ and a direct summand $m^{\prime}$ of $m\stmryolt U$,
the pointed $\mathcal{C}$-module category $(\mathcal{M},m^{\prime})$ is also
injective.

###### Proof.

Lemma 4.3 shows that $(\mathcal{M},m\stmryolt U)$ is injective. Thus, it is
enough to prove the assertion when $U=1$. Let us take the projection
$p\in\mathcal{M}(m)$ corresponding to $m^{\prime}$, and let $m^{\prime\prime}$
be the summand corresponding to $1-p$.

Suppose that $\Phi\colon(\mathcal{N},n)\to(\mathcal{M},m^{\prime})$ is a ucp-
multiplier, and
$\Psi\colon(\mathcal{N},n)\to(\mathcal{N}^{\prime},n^{\prime})$ is a complete
isometric multiplier as in Definition 4.2. Take a state
$\omega\colon\mathcal{N}(n)\to\mathbb{C}$, and let
$P\colon\mathcal{N}\to\mathcal{C}$ be the corresponding ucp-multiplier.
Composing this with the ucp-multiplier
$Q\colon\mathcal{C}\to(\mathcal{M},m^{\prime\prime})$ given by

$\mathcal{C}(V,W)\to\mathcal{M}(m^{\prime\prime}\stmryolt
V,m^{\prime\prime}\stmryolt W),\quad T\mapsto(1-p)\stmryolt T,$

we obtain a ucp-multiplier
$\Phi^{\prime}\colon(\mathcal{N},n)\to(\mathcal{M},m^{\prime\prime})$. By
taking the direct sum $\Phi\oplus\Phi^{\prime}$, we get a ucp multiplier
$(\mathcal{N},n)\to(\mathcal{M},m)$. Then the injectivity of $(\mathcal{M},m)$
gives an ucp extension
$\tilde{\Psi}^{\prime}\colon(\mathcal{N}^{\prime},n^{\prime})\to(\mathcal{M},m)$.
Then the maps
$\tilde{\Psi}_{V,W}(T)=(p\stmryolt\mathrm{id}_{W})\tilde{\Psi}^{\prime}_{V,W}(T)(p\stmryolt\mathrm{id}_{V})$
give a desired ucp extension
$(\mathcal{N}^{\prime},n^{\prime})\to(\mathcal{M},m^{\prime})$. ∎

######  Proposition 4.9.

Let $H\in\mathrm{Hilb}(\mathcal{C})$ be a finite dimensional object. Then
$\mathcal{M}_{\mathcal{B}(H)}$ is an injective pointed $\mathcal{C}$-module
category.

###### Proof.

Let $H$ be a finite dimensional Hilbert space object. By Proposition 4.1,
there is $U\in\mathcal{C}$ such that
$\mathcal{M}_{\mathcal{B}(H)}\simeq\mathcal{C}_{U}\simeq(\mathcal{C},U)$.
Since $(\mathcal{C},1_{\mathcal{C}})$ is injective by Lemma 4.7,
$(\mathcal{C},U)$ is injective by Proposition 4.8. ∎

Now, following Arveson’s proof for the injectivity of $\mathcal{B}(H)$ for
Hilbert spaces, we obtain the following.

######  Theorem 4.10.

For any $H\in\mathrm{Hilb}(\mathcal{C})$, the C∗-$\mathcal{C}$-module category
$\mathcal{M}_{\mathcal{B}(H)}$ is injective.

###### Proof.

Let $A$ and $B$ be C∗-algebra objects, and suppose that $\iota\colon A\to B$
is a completely isometric multiplier and $\psi\colon A\to\mathcal{B}(H)$ is a
ucp multiplier. We then need to construct an extension $\tilde{\psi}\colon
B\to\mathcal{B}(H)$.

Take an increasing net $(H_{\lambda})_{\lambda\in\Lambda}$ of finite
dimensional Hilbert space objects such that $H_{\lambda}(X)\subset H(X)$ and
$H(X)=\varinjlim_{\lambda\in\Lambda}H_{\lambda}(X)$ for all $X\in\mathcal{C}$.
Let us denote the orthogonal projections $H(X)\to H_{\lambda}(X)$ by
$p_{\lambda,X}$. Then we get a morphism $p_{\lambda}\colon H\to H_{\lambda}$
by

$(p_{\lambda,X})_{X\in\operatorname{Irr}(\mathcal{C})}\in\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{X\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{B}(H(X),H_{\lambda}(X))\simeq\mathrm{Hilb}(\mathcal{C})(H,H_{\lambda}).$

The maps
$\operatorname{Ad}(p_{\lambda})\colon\mathcal{B}(H_{\lambda})\to\mathcal{B}(H)$
and
$\operatorname{Ad}(p_{\lambda}^{*})\colon\mathcal{B}(H)\to\mathcal{B}(H_{\lambda})$
are cp and ucp multipliers, respectively [jp1]*Lemma 4.27.

For fixed $\lambda$, since $\operatorname{Ad}(p_{\lambda}^{*})\circ\psi$ is a
ucp map from $A$ to $\mathcal{B}(H_{\lambda})$, we obtain a ucp extension
$\tilde{\psi}_{\lambda}\colon B\to\mathcal{B}(H_{\lambda})$ by Proposition
4.9. Then
$\psi_{\lambda}=\operatorname{Ad}(p_{\lambda})\circ\tilde{\psi}_{\lambda}$ is
a completely contractive positive map from $B$ to $\mathcal{B}(H)$. We claim
that a limit of the family $(\psi_{\lambda})_{\lambda}$ is a desired ucp
extension of $\psi$.

For each $V\in\mathcal{C}$, the map

$\tilde{\psi}_{\lambda,V}\colon
B(V)\to\mathcal{B}(H)(V)\simeq\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{X\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{B}((H\otimes
V)(X),H(X))$

is completely contractive. By passing to a subnet, we may assume that
$\psi_{V}=\lim_{\lambda}\psi_{\lambda,V}$ exists as a complete contraction for
each irreducible $V$, where we consider the topology of pointwise convergence
with respect to the weak operator topology on $\mathcal{B}(H)(V)$ up to the
above identification.

Now, collecting $\tilde{\psi}_{V}$ as above for irreducible $V$, we get a
natural transformation $\tilde{\psi}\colon B\to\mathcal{B}(H)$, and by
Proposition 2.14 a multiplier
$\tilde{\Psi}\colon\mathcal{M}_{B}\to\mathcal{M}_{\mathcal{B}(H)}$. Let
$(\tilde{\Psi}_{\lambda,V,W})_{V,W}$ be the cp multiplier corresponding to
$\psi_{\lambda}$. We then claim that
$\tilde{\Psi}_{V,W}(T)=\lim_{\lambda}\tilde{\Psi}_{\lambda,V,W}(T)$ for the
weak operator topology. This follows from the correspondence (2.5) and
$\tilde{\psi}_{U}(S)\otimes\mathrm{id}_{W}=\lim_{\lambda}\tilde{\psi}_{\lambda,U}(S)\otimes\mathrm{id}_{W}$
for $S\in B(U)$.

Then, setting $V=W$, we obtain the complete positivity of $\tilde{\Psi}$.
Moreover, $\mathrm{id}_{V_{B}}\in\mathcal{M}_{B}(V_{B},V_{B})$ is the image of
$\mathrm{id}_{V_{A}}\in\mathcal{M}_{A}(V_{A},V_{A})$, so we have

$\tilde{\Psi}_{V,V}(\mathrm{id}_{V_{B}})=\lim_{\lambda}(p_{\lambda}^{*}p_{\lambda})_{V}=\mathrm{id}_{H(V)}.$

This proves that $\tilde{\Psi}$ is unital, hence $\tilde{\psi}$ is a ucp
extension of $\psi$. ∎

######  Definition 4.11.

A functor $F\colon\mathcal{M}\to\mathcal{N}$ between C∗-$\mathcal{C}$-module
is said to be an embedding if it is faithful and norm-closed at the level of
morphism spaces.

######  Definition 4.12.

An _injective envelope_ of a C∗-$\mathcal{C}$-module category
$(\mathcal{M},m)$ is an injective $\mathcal{C}$-module category
$(\mathcal{I},m^{\prime})$ endowed with a faithful module functor
$\iota\colon(\mathcal{M},m)\to(\mathcal{I},m^{\prime})$ such that
$\mathrm{Id}_{\mathcal{I}}$ is the only ucp extension of $\iota$.

######  Theorem 4.13.

Every cyclic C∗-$\mathcal{C}$-module category $(\mathcal{M},m)$ has an
injective envelope.

###### Proof.

As we only deal with cyclic module categories in this proof, for simplicity we
write $\mathcal{M}$ instead for $(\mathcal{M},m)$, and similarly for other
cyclic module categories.

Theorems 2.10 and 4.10 imply that $\mathcal{M}$ embeds into an injective
$\mathcal{C}$-module W∗-category $\mathcal{N}$, namely
$\mathcal{N}=\mathcal{M}_{\mathcal{B}(H)}$ for some
$H\in\mathrm{Hilb}(\mathcal{C})$. Define $i\colon\mathcal{M}\to\mathcal{N}$ to
be this embedding. Consider the semigroup of ucp multipliers

$S=\\{\Phi\colon\mathcal{N}\to\mathcal{N}\mid\Phi\circ i=i\\}.$

This can be identified with a weak∗-closed convex set in the dual Banach space

$\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{U,V\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{N}(U,V)=\Biggl{[}\ell^{1}\mathchar
45\relax\smashoperator[l]{\bigoplus_{U,V\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{N}(U,V)_{*}\Biggr{]}^{*}.$

By Proposition 2.1, $S$ has a minimal idempotent $\Psi$.

Let us make sense of the image of $\Psi$ as a C∗-$\mathcal{C}$-module
category. For each $U$ and $V$ in $\mathcal{C}$, set

$\Psi(\mathcal{N})(U,V)=\Psi(\mathcal{N}(U,V)).$

When we have $U=V$, we get a structure of C∗-algebra on this space by the
Choi–Effros product $S\cdot T=\Psi(ST)$. Generally, by considering $X=U\oplus
V\oplus W$ and $\Psi(\mathcal{N})(X,X)$, we get the composition maps

$\Psi(\mathcal{N})(V,W)\times\Psi(\mathcal{N})(U,V)\to\Psi(U,W)$

that defines a C∗-category $\Psi(\mathcal{N})$. Moreover, using the fact that
$\Psi$ is a multiplier, we obtain maps

$\Psi(\mathcal{N})(V,W)\times\mathcal{C}(V^{\prime},W^{\prime})\to\Psi(\mathcal{N})(V\otimes
V^{\prime},W\otimes W^{\prime})$

that defines a structure of $\mathcal{C}$-module category on
$\Psi(\mathcal{N})$.

We next claim that $\Psi(\mathcal{N})$ is an injective envelope for
$\mathcal{M}$. Again by construction $i$ induces maps

$\mathcal{M}(m\stmryolt V,m\stmryolt W)\to\Psi(\mathcal{N})(V,W),$

and by the multiplicative domain argument this is a functor of C∗-categories.
It is straightforward to check the compatibility with $\mathcal{C}$-module
structures. Moreover, the uniqueness of $\mathrm{Id}_{\Psi(\mathcal{N})}$ as a
ucp multiplier stabilizing $\mathcal{M}$ is obvious from the characterization
of $\Psi$. ∎

Let $(\mathcal{M},m)$ be a cyclic C∗-$\mathcal{C}$-module category, and let
$\Psi\colon(\mathcal{M},m)\to(\mathcal{M},m)$ be a ucp multiplier. The proof
of Theorem 4.13 shows that the image of $\Psi$ has a natural composition rule
which makes it a cyclic C∗-$\mathcal{C}$-module category: this result does not
depend on minimality of $\Psi$, and it will be used later again to prove
existence of injective envelopes for a class of bimodule categories, which can
be equivalently understood as module categories over
$\mathcal{C}\boxtimes\mathcal{C}^{\text{op}}$.

##  5\. Centrally pointed bimodule categories

Fix a rigid C∗-tensor category $\mathcal{C}$. Then a _$\mathcal{C}$ -bimodule
category_ $\mathcal{M}$ is defined in an analogous way to right
$\mathcal{C}$-module categories with three sets of module associativity
morphisms

$\displaystyle\begin{aligned} \Psi^{\prime}\colon(V\otimes W)\stmryogt X&\to
V\stmryogt(W\stmryogt X),&\Psi\colon(X\stmryolt V)\stmryolt W&\to
X\stmryolt(V\otimes W)\end{aligned}$
$\displaystyle\Psi^{\prime\prime}\colon(V\stmryogt X)\stmryolt W\to
V\stmryogt(X\stmryolt W)$

for $X\in\mathcal{M}$ and $V,W\in\mathcal{C}$, satisfying the pentagon-type
equations with the associativity morphism of $\mathcal{C}$. A _$\mathcal{C}$
-bimodule functor_ $F\colon\mathcal{M}\to\mathcal{M}^{\prime}$ is given by a
linear functor of underlying categories, together with

$\displaystyle\tensor[_{2}]F{{}_{V,X}}\colon V\stmryogt F(X)$
$\displaystyle\to F(V\stmryogt X),$ $\displaystyle F_{2;X,V}\colon
F(X)\stmryolt V\to F(X\stmryolt V),$

again compatible with structure morphisms of $\mathcal{M}$ and
$\mathcal{M}^{\prime}$.

######  Remark 5.1.

An equivalent, more succinct way of defining C∗-bimodule categories, is by
means of the max tensor product
$\mathcal{C}\underset{\max}{\boxtimes}\mathcal{C}^{\mathrm{op}}$ defined in
[AV1]. Indeed, given a C∗-category $\mathcal{M}$, there is an isomorphism
between the space of C∗-functors from
$\mathcal{C}\underset{\max}{\boxtimes}\mathcal{C}^{\mathrm{op}}$ to
$\operatorname{End}(\mathcal{M})$ and that of bilinear $*$-functors from
$\mathcal{C}\times\mathcal{C}^{\mathrm{op}}$ to
$\operatorname{End}(\mathcal{M})$ by [AV1]*Proposition 3.3. This can be
promoted to a correspondence between the
C∗-$\mathcal{C}\underset{\max}{\boxtimes}\mathcal{C}^{\mathrm{op}}$-module
category structures on $\mathcal{M}$ and the C∗-$\mathcal{C}$-bimodule
category structures on $\mathcal{M}$.

To simplify our presentation we further assume that $\mathcal{C}$ and
$\mathcal{M}$ are strict.

######  Definition 5.2.

Let $\mathcal{M}$ be a C∗-$\mathcal{C}$-bimodule category, and
$m\in\mathcal{M}$. We say that $m$ is _central_ if there is a family of
unitary morphisms

$\sigma_{U}\colon U\stmryogt m\to m\stmryolt U$

that is natural in $U$ and satisfy the braid relations

$\sigma_{U\otimes
V}=(\sigma_{U}\stmryolt\mathrm{id}_{V})(\mathrm{id}_{U}\stmryogt\sigma_{V}),$

for all $U,V\in\mathcal{C}$. We call $(\mathcal{M},m)$ a _centrally pointed
bimodule category_. We call it a _cyclic centrally pointed bimodule category_
when furthermore $m$ is a generator of $\mathcal{M}$ under the action of
$\mathcal{C}\underset{\max}{\boxtimes}\mathcal{C}^{\mathrm{op}}$.

######  Remark 5.3.

In the above definition, because of centrality, $m$ is a generator of
$\mathcal{M}$ as a bimodule category if and only if it generates $\mathcal{M}$
under either the right or left action of $\mathcal{C}$ on $\mathcal{M}$.

######  Definition 5.4.

Let $(\mathcal{M},m)$ and $(\mathcal{M}^{\prime},m^{\prime})$ be centrally
pointed bimodule categories. A _central functor_ between them is a bimodule
functor $F\colon\mathcal{M}\to\mathcal{M}^{\prime}$ endowed with an
isomorphism $F_{0}\colon m^{\prime}\to F(m)$ which are compatible with central
generators and structure morphisms, in the sense that the diagram

${U\stmryogt m^{\prime}}$${m^{\prime}\stmryolt U}$${U\stmryogt
F(m)}$${F(m)\stmryolt U}$${F(U\stmryogt m)}$${F(m\stmryolt
U)}$$\scriptstyle{\sigma^{\prime}_{U}}$$\scriptstyle{\tensor[_{2}]{F}{}}$$\scriptstyle{F_{2}}$$\scriptstyle{F(\sigma_{U})}$
(5.1)

is commutative.

######  Definition 5.5.

Let $F$ and $F^{\prime}$ be central functors from $(\mathcal{M},m)$ to
$(\mathcal{M},m^{\prime})$. A _natural transformation of bimodule functors_
$\alpha\colon F\to F^{\prime}$ is given by a natural transformation of the
underlying linear functors $\alpha_{X}\colon F(X)\to F^{\prime}(X)$ for
$X\in\mathcal{M}$ such that the diagrams

${U\stmryogt F(X)}$${F(U\stmryogt X)}$${U\stmryogt
F^{\prime}(X)}$${F^{\prime}(U\stmryogt
X)}$$\scriptstyle{\tensor[_{2}]{F}{}}$$\scriptstyle{\mathrm{id}_{U}\otimes\alpha_{X}}$$\scriptstyle{\alpha_{U\stmryogt
X}}$$\scriptstyle{\tensor[_{2}]{F}{{}^{\prime}}}$ ${F(X)\stmryolt
U}$${F(X\stmryolt U)}$${F^{\prime}(X)\stmryolt U}$${F^{\prime}(X\stmryolt
U)}$$\scriptstyle{F_{2}}$$\scriptstyle{\alpha_{X}\otimes\mathrm{id}_{U}}$$\scriptstyle{\alpha_{X\stmryolt
U}}$$\scriptstyle{F^{\prime}_{2}}$
${m^{\prime}}$${F(m)}$${F^{\prime}(m)}$$\scriptstyle{F_{0}}$$\scriptstyle{F^{\prime}_{0}}$$\scriptstyle{\alpha_{m}}$

commute.

We denote by $\mathcal{CB}(\mathcal{C})$ be the category of centrally pointed
C∗-$\mathcal{C}$-bimodule categories and central bimodule functors, and by
$\mathcal{CB}^{\mathrm{c}}(\mathcal{C})$ the subcategory of cyclic centrally
pointed C∗-$\mathcal{C}$-bimodule categories.

Given a central object as above, we often work with the induced maps

$\displaystyle\Sigma_{U;V,W}\colon\mathcal{M}(m\stmryolt V,m\stmryolt W)$
$\displaystyle\to\mathcal{M}(m\stmryolt U\otimes V,m\stmryolt U\otimes W)$
$\displaystyle T$
$\displaystyle\mapsto(\sigma_{U}\stmryolt\mathrm{id}_{W})(\mathrm{id}_{U}\stmryogt
T)(\sigma_{U}^{*}\stmryolt\mathrm{id}_{V}).$ (5.2)

######  Proposition 5.6.

Let $(\mathcal{M},m)$ a centrally pointed bimodule category. There is a
embedding of centrally pointed bimodule categories
$F\colon(\mathcal{C},1_{\mathcal{C}})\to(\mathcal{M},m)$ which sends $U$ to
$m\stmryolt U$.

###### Proof.

For simplicity we assume that the right action of $\mathcal{C}$ on
$\mathcal{M}$ is strict. Then we get a right $\mathcal{C}$-module functor
$F(U)=m\stmryolt U$ with $F_{2}=\mathrm{id}_{m\stmryolt U}$. We extend it to a
bimodule functor by setting

${\tensor[_{2}]{F}{}}=\sigma_{U}\otimes\mathrm{id}_{V}\colon(U\stmryogt
m)\stmryolt V\to m\stmryolt U\stmryolt V=m\stmryolt(U\otimes V).$

Then consistency conditions of ${\tensor[_{2}]{F}{}}$ follow from those of the
half-braiding. ∎

######  Definition 5.7.

Let $\mathcal{M}$ be a right C∗-$\mathcal{C}$-module category, and
$m\in\mathcal{M}$. We denote by $\hat{\mathcal{M}}_{m}$ the idempotent
completion of $\mathcal{C}$ with the enlarged morphism sets

$\hat{\mathcal{M}}_{m}(V,W)=\operatorname{Nat}_{b}(m\stmryolt\iota\otimes
V,m\stmryolt\iota\otimes W)\simeq\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{X\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{M}(m\stmryolt
X\otimes V,m\stmryolt X\otimes W).$ (5.3)

This is a bimodule category, with $1$ (which corresponds to $m\in\mathcal{M}$)
being a central generator. The bimodule structure of $\hat{\mathcal{M}}_{m}$
is as follows. At the level of morphisms, for $T\in\hat{\mathcal{M}}_{m}(V,W)$
and $X\in\mathcal{C}$, we define the monoidal products of $T$ and
$\mathrm{id}_{X}$ by

$\displaystyle(T\stmryolt\mathrm{id}_{X})_{Y}$
$\displaystyle=T_{Y}\stmryolt\mathrm{id}_{X},$
$\displaystyle(\mathrm{id}_{X}\stmryogt T)_{Y}$ $\displaystyle=T_{Y\otimes
X}.$

Then we see that $1\in\hat{\mathcal{M}}_{m}$ is indeed a central object, with
$\sigma_{U}$ given by $\mathrm{id}_{U}$.

This is motivated by the ‘dual category’ $\hat{\mathcal{C}}$ introduced in
[NY1], which corresponds to case of $\mathcal{M}=\mathcal{C}$ and
$m=1_{\mathcal{C}}$. In this case there is a natural C∗-tensor structure on
$\hat{\mathcal{C}}$ (with nonsimple unit) such that $U\mapsto U$ is a
C∗-tensor functor from $\mathcal{C}$ to $\hat{\mathcal{C}}$. If we take $m=U$
instead, because of the centrality, the resulting category can be identified
with $\hat{\mathcal{C}}_{U}$, with morphism sets

$\hat{\mathcal{C}}_{U}(V,W)=\hat{\mathcal{C}}(U\otimes V,U\otimes
W)=\ell^{\infty}\mathchar
45\relax\smashoperator[l]{\prod_{i\in\operatorname{Irr}(\mathcal{C})}^{}}\mathcal{C}(U_{i}\otimes
U\otimes V,U_{i}\otimes U\otimes W)$

for $V,W\in\mathcal{C}$. The evaluation at the unit $1_{\mathcal{C}}$ gives a
multiplier $\hat{\mathcal{C}}_{U}\to\mathcal{C}_{U}$ that is a conditional
expectation (completely positive idempotent onto $\mathcal{C}_{U}$).

Suppose $\mathcal{C}=\operatorname{Rep}(G)$ for some compact quantum group
$G$. Up to the Tannaka–Krein type duality for actions, the
C∗-$\mathcal{C}$-module category $\hat{\mathcal{C}}$ corresponds to the
discrete dual quantum group with adjoint action, see [NY3]*Section 4.1. When
$(H,U)$ is a finite dimensional unitary representation of $G$, the module
category corresponding to
$\mathcal{R}(\mathcal{B}(H)\mathbin{\bar{\otimes}}\ell^{\infty}(\hat{G}))$ is
$\hat{\mathcal{C}}_{U}$, cf. Proposition 4.1. Our constructions leading up to
Theorem 3.6 shows that, with a suitable generalization to infinite dimensional
$H$, the categorical dual of any Yetter–Drinfeld C∗-algebra can be embedded in
a category analogous to $\hat{\mathcal{C}}_{U}$, in a way compatible with
extra bimodule structures as we will see in Section 6.

###  5.1. Trivialization of central structure

Let us consider a construction that we call _trivialization of the central
structure_. Let $(\mathcal{M},m,\sigma)$ be a centrally pointed
C∗-$\mathcal{C}$-bimodule category. Then, as usual, we consider the idempotent
completion of $\mathcal{C}$ with morphism sets
$\operatorname{Mor}(V,W)=\mathcal{M}(m\stmryolt V,m\stmryolt W)$. In other
words, our objects are projections $p$ in the C∗-algebras
$\mathcal{M}(m\stmryolt V)$. Given such $p$, and another $U\in\mathcal{C}$,
consider the projection

$\Sigma_{U;V}(p)=(\sigma_{U}\stmryolt\mathrm{id}_{V})(\mathrm{id}_{U}\stmryogt
p)(\sigma_{U}^{-1}\stmryolt\mathrm{id}_{V})\in\mathcal{M}(m\stmryolt U\otimes
V),$

which defines a subobject of $U\otimes V$ in our category. Of course, this is
isomorphic to the left action of $U$ on the subobject of $m\stmryolt V$
defined by $p$, i.e., isomorphic to the subobject of $U\stmryogt m\stmryolt V$
defined by the projection $\mathrm{id}_{U}\stmryogt p$. This way (assuming
$\mathcal{C}$ is strict) we get a strict bimodule category. Moreover, $1$ is a
central object, with the structure morphisms
$\sigma^{\prime}_{U}=\mathrm{id}_{U}$.

######  Proposition 5.8.

The centrally pointed bimodule category $\tilde{\mathcal{M}}$ obtained by
trivialization of the central structure is isomorphic to the original one.

###### Proof.

Define $F_{\sigma}\colon\tilde{\mathcal{M}}\to\mathcal{M}$ as (the canonical
extension of) the functor $V\mapsto m\stmryolt V$ at the level of objects, and
by the tautological map at the level of morphisms. This is a right module
functor. The compatibility with left module structures,

$\tensor[_{2}]{(F_{\sigma})}{}\colon V\stmryogt m\stmryolt W=V\stmryogt
F_{\sigma}(W)\to m\stmryolt V\stmryolt W=F_{\sigma}(V\otimes W),$

is given by $\sigma_{V}\otimes\mathrm{id}_{W}$. Then the commutativity of
(5.1) holds by construction.

In the other direction, to define $F\colon\mathcal{M}\to\tilde{\mathcal{M}}$,
we choose $V_{X}\in\mathcal{C}$ and an isometry $j_{X}\colon X\to m\stmryolt
V_{X}$ for each $X\in\mathcal{M}$. At the level of C∗-functor, for each
$X\in\mathcal{M}$ the object $F(X)$ is represented by the projection
$j_{X}j_{X}^{*}\in\tilde{\mathcal{M}}(V_{X})$, while for each
$T\in\mathcal{M}(X,Y)$ the morphism $F^{\prime}(T)$ is represented by
$j_{Y}Tj_{X}^{*}\in\tilde{\mathcal{M}}(V_{X},V_{Y})$.

Now, we can define the natural isomorphisms giving compatibility with bimodule
structure as follows. For the right module structure, we take

$F_{2}=j_{X\stmryolt U}(j_{X}^{*}\stmryolt\mathrm{id}_{U})\colon
F^{\prime}(X)\stmryolt U\to F(X\stmryolt U)$

while for the left module structure we take

$\tensor[_{2}]{F}{{}^{\prime}}=j_{U\stmryogt X}(\mathrm{id}_{U}\stmryogt
j_{X}^{*})(\sigma_{U}^{*}\stmryolt\mathrm{id}_{V_{X}})\colon U\stmryogt
F(X)\to F(U\stmryogt X).$

Then the composition $F_{\sigma}F\colon\mathcal{M}\to\mathcal{M}$ is naturally
isomorphic to the identity functor $\mathrm{Id}_{\mathcal{M}}$. Indeed, the
assumption on cyclicity says that $\mathcal{M}\ni X\mapsto(m\stmryolt
V_{X},j_{x}j_{X}^{*})$ extends to an equivalence of right
C∗-$\mathcal{C}$-module categories. Under this equivalence, $F_{\sigma}F$
corresponds to the identity functor on $\mathcal{M}$, and this correspondence
is by construction compatible with the module structures. A similar argument
shows that $FF_{\sigma}$ is naturally isomorphic to the identity functor on
$\tilde{\mathcal{M}}$. ∎

######  Proposition 5.9.

Let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ be $\mathcal{C}$-bimodule
categories with central generators $m_{1}$ and $m_{2}$, respectively. Let
$F\colon\mathcal{M}_{1}\to\mathcal{M}_{2}$ be a functor of right
$\mathcal{C}$-module categories such that $F(m_{1})=m_{2}$. Suppose that the
maps (5.2) satisfy

$F_{2;m_{1},U\otimes W}^{-1}F(\Sigma_{U;V,W}(T))F_{2;m_{1},U\otimes
V}=\Sigma_{U;V,W}^{\prime}(F_{2;m_{1},W}^{-1}F(T)F_{2;m_{1},V})\quad(T\in\mathcal{M}_{1}(m_{1}\stmryolt
V,m_{1}\stmryolt W).$ (5.4)

Then we have a strict bimodule functor
$\tilde{F}\colon\tilde{\mathcal{M}}_{1}\to\tilde{\mathcal{M}}_{2}$
characterized by

$\tilde{F}(T)=F_{2;m_{1},W}^{-1}F(T)F_{2;m_{1},V}$

for morphisms $T\in\tilde{\mathcal{M}}_{1}(V,W)=\mathcal{M}_{1}(m_{1}\stmryolt
V,m_{1}\stmryolt W))$.

###### Proof.

By the above formula, an object of $\tilde{\mathcal{M}}_{1}$ represented by a
projection $p\in\mathcal{M}_{1}(m_{1}\stmryolt V)$ is mapped to the object of
$\tilde{\mathcal{M}}_{2}$ represented by the projection
$F_{2;m_{1},V}^{-1}F(p)F_{2;m_{1},V}$.

Then the above claim says that $\tilde{F}$ is a strict functor of left
$\operatorname{Rep}(G)$-module categories. The compatibility of $F_{2}$ and
the (trivial) associativity morphisms imply that $\tilde{F}$ is also a strict
functor of right module categories. ∎

The above functor $\tilde{F}$ satisfies
$\tilde{F}(\tilde{\sigma}_{U})=\tilde{\sigma}^{\prime}_{U}$ for the half-
braidings of $\tilde{\mathcal{M}}_{1}$ and $\tilde{\mathcal{M}}_{2}$, since
$\tilde{\sigma}_{U}$ is given by $\mathrm{id}_{m_{1}\stmryolt
U}\in\mathcal{M}(m_{1}\stmryolt U)$. Combined with Proposition 5.8, we obtain
a central functor $\mathcal{M}_{1}\to\mathcal{M}_{2}$ from a right module
functor satisfying (5.4). Consequently, $\mathcal{CB}(\mathcal{C})$ is
equivalent to the category with the same objects but right module category
functors satisfying (5.4) as morphisms.

##  6\. Duality for Yetter–Drinfeld C∗-algebras

In this section, we establish the Tannaka–Krein duality for the unital
Yetter–Drinfeld $G$-C∗-algebras. In our presentation, their categorical dual
are cyclic bimodules over $\operatorname{Rep}(G)$ from Section 5. More
precisely, let $\mathcal{YD}(G)$ denote the category of Yetter–Drinfeld
$G$-C∗-algebras and equivariant unital $*$-homomorphisms. We then have the
following correspondence.

######  Theorem 6.1.

The categories $\mathcal{YD}(G)$ and
$\mathcal{CB}^{\mathrm{c}}(\operatorname{Rep}(G))$ are equivalent. Under this
equivalence, two morphisms
$F,F^{\prime}\colon\mathcal{M}\to\mathcal{M}^{\prime}$ in
$\mathcal{CB}^{\mathrm{c}}(\operatorname{Rep}(G))$ induce the same
homomorphism $B_{\mathcal{M}}\to B_{\mathcal{M}^{\prime}}$ if and only if they
are naturally isomorphic as central bimodule functors.

The proof of this theorem will occupy the rest of the section.

######  Remark 6.2.

An alternative presentation of duality as above was recently given in
[FreslonTaipeWang] in terms of weak monoidal functors [pr1][n1].

###  6.1. From Yetter–Drinfeld algebras to centrally pointed bimodules

Let $B$ be a Yetter–Drinfeld C∗-algebra with $G$-action $\alpha$ and
$\hat{G}$-action $\beta$. Let us show that the category $\mathcal{D}_{B}$
admits a structure of centrally pointed bimodule category.

Given a unitary finite dimensional representation $U$ of $G$, we fix a choice
$\\{m^{U}_{ij}\\}_{i,j}$ of matrix units for $\mathcal{B}(H_{U})$ and write

$U=\sum_{i,j}m^{U}_{ij}\otimes u_{ij}.$

As we did in Section 2.4, on the right Hilbert $B$-module $H_{U}\otimes B$ we
consider the coaction of $C(G)$ given by

$\delta_{U}(\xi\otimes b)=U_{21}^{*}(b_{(1)}\otimes\xi\otimes b_{(2)}),$

where $\alpha(b)=b_{(1)}\otimes b_{(2)}$ denotes the coaction of $G$ on $B$.

######  Proposition 6.3.

For a finite dimensional unitary representation $U$ of $G$, consider the
homomorphism $\pi_{U}\colon B\to\operatorname{End}_{B}(H_{U}\otimes B)$ given
by

$\pi_{U}(a)=\sum_{i,j}m^{U}_{ij}\otimes(u_{ij}\rhd a).$

Then we have the equivariance condition

$\delta_{U}(\pi_{U}(a)(\xi\otimes
b))=\left((\mathrm{id}\otimes\pi_{U})\alpha(a)\right)(\delta_{U}(\xi\otimes
b))\quad(a,b\in B,\xi\in H_{U}).$

###### Proof.

Let $\\{\xi_{i}\\}_{i}$ be the basis of $H_{U}$ corresponding to the matrix
units $\\{m^{U}_{ij}\\}_{ij}$. Let us check the claim for $\xi=\xi_{i}$.

The left hand side is

$\displaystyle\delta_{U}(\pi_{U}(a)(\xi_{i}\otimes b))$
$\displaystyle=\delta_{U}\left(\sum_{jk}m^{U}_{jk}(\xi_{i})\otimes(u_{jk}\rhd
a)b\right)$ $\displaystyle=\sum_{j,r,s}u_{rs}^{*}(u_{ji}\rhd
a)_{(1)}b_{(1)}\otimes m^{U}_{rs}(\xi_{j})\otimes(u_{ji}\rhd a)_{(2)}b_{(1)}$
$\displaystyle=\sum_{j,s,k,l}u_{js}^{*}u_{jk}a_{(1)}u_{il}^{*}b_{(1)}\otimes\xi_{s}\otimes(u_{kl}\rhd
a_{(2)})b_{(2)},$

since

$\alpha(u_{ij}\rhd a)=(u_{ij}\rhd a)_{(1)}\otimes(u_{ij}\rhd
a)_{(2)}=\sum_{k,l}u_{jr}a_{(1)}u_{il}^{*}\otimes(u_{kl}\rhd a_{(2)})$

holds by the Yetter–Drinfeld property. As
$\sum_{j}u_{js}^{*}u_{jk}=\delta_{sk}$

$\displaystyle\delta_{U}(\pi_{U}(a)(\xi_{i}\otimes
b))=\sum_{kl}a_{(1)}u_{il}^{*}b_{(1)}\otimes\xi_{k}\otimes(u_{kl}\rhd
a_{(2)})b_{(2)}$ (6.1)

For the right hand side, we have

$\displaystyle\left((\mathrm{id}\otimes\pi_{U})\alpha(a)\right)\delta_{U}(\xi_{i}\otimes
b)$
$\displaystyle=(a_{(1)}\otimes\pi_{U}(a_{(2)}))\left(\sum_{j,k}u_{jk}^{*}b_{(1)}\otimes
m^{U}_{kj}(\xi_{i})\otimes b_{(2)}\right)$
$\displaystyle=\sum_{k}a_{(1)}u_{ik}^{*}b_{(1)}\otimes\pi_{U}(a_{(2)})(\xi_{k}\otimes
b_{(2)})$ $\displaystyle=\sum_{k,r,s}a_{(1)}u_{ik}^{*}b_{(1)}\otimes
m^{U}_{rs}(\xi_{k})\otimes(u_{rs}\rhd a_{(2)})b_{(2)}$
$\displaystyle=\sum_{k,l}a_{(1)}u_{ik}^{*}b_{(1)}\otimes\xi_{l}\otimes(u_{lk}\rhd
a_{(2)})b_{(2)},$

which proves the claim. ∎

Thus, $H_{U}\otimes B$ becomes an equivariant bimodule over $B$. We use this
structure to define balanced tensor products

$\left(H_{U}\otimes B\right)\otimes_{B}\left(H_{V}\otimes B\right),$

which is again a equivariant Hilbert $B$-module. Note that the canonical
isomorphism $B\otimes_{B}(H_{U}\otimes B)\simeq H_{U}\otimes B$ is
$G$-equivariant. Given $U,V\in\operatorname{Rep}(G)$, the map

$S_{U,V}\colon(H_{V}\otimes B)\otimes_{B}(H_{U}\otimes B)\to H_{U\otimes
V}\otimes B,\quad(\zeta\otimes b)\otimes_{B}(\xi\otimes
a)\mapsto\sum_{i,j}m^{U}_{ij}(\xi)\otimes\zeta\otimes(u_{ij}\rhd b)a$

is an equivariant isomorphism of right Hilbert $B$-modules.

######  Proposition 6.4.

The map $S_{U,V}$ is a $B$-bimodule isomorphism.

###### Proof.

We need to show that $\\{S_{U,V}\\}$ respect the left $B$-actions, which is
equivalent to the equality

$\pi_{V}(b)\otimes\mathrm{id}=S_{U,V}^{-1}\pi_{U\otimes V}(b)S_{UV}$

for $b\in B$.

Let $a,b,c\in B$ and $\xi_{i_{0}}\in H_{U},\zeta_{k_{0}}\in H_{V}$ be basis
elements corresponding to the respective matrix units of $U$ and $V$. We then
have

$\displaystyle\pi_{U\otimes V}(c)S_{U,V}\left((\zeta_{k_{0}}\otimes
b)\otimes_{B}(\xi_{i_{0}}\otimes a)\right)$ $\displaystyle=\pi_{U\otimes
V}(c)\left(\sum_{i}\xi_{i}\otimes\zeta_{k_{0}}\otimes(u_{ii_{0}}\rhd
a)\right)$
$\displaystyle=\sum_{ij}\xi_{i}\otimes\zeta_{k}\otimes(u_{ij}v_{kk_{0}}\rhd
c)(u_{ji_{0}}\rhd b)a.$

On the other hand, we also have

$\displaystyle S_{U,V}\left(\pi_{V}(c)(\zeta_{K_{0}}\otimes
b)\otimes_{B}(\xi_{i_{0}}\otimes a)\right)$
$\displaystyle=S_{U,V}\left(\sum_{k}\left(\zeta_{k}\otimes(v_{kk_{0}}\rhd
c)b\right)\otimes_{B}(\xi_{i_{0}}\otimes a)\right)$
$\displaystyle=\sum_{i,k}\xi_{i}\otimes\zeta_{k}\otimes\left(u_{ii_{0}}\rhd((v_{kk_{0}}\rhd
c)b)\right)a$
$\displaystyle=\sum_{i,j,k}\xi_{i}\otimes\zeta_{k}\otimes(u_{ij}v_{kk_{0}}\rhd
c)(u_{ji_{0}}\rhd b)a$ $\displaystyle=\pi_{U\otimes
V}S_{U,V}\left((\zeta_{k_{0}}\otimes b)\otimes_{B}(\xi_{i_{0}}\otimes
a)\right),$

which proves the claim. ∎

######  Proposition 6.5.

We have

$S_{U\otimes V,W}(\mathrm{id}_{H_{W}\otimes B}\otimes S_{U,V})=S_{U,V\otimes
W}S_{V,W}\otimes\mathrm{id}_{H_{U}\otimes B}.$

###### Proof.

Let us compare the action of both sides on vectors of the form
$(\zeta_{k_{0}}\otimes c)\otimes(\eta_{j_{0}}\otimes
b)\otimes(\xi_{i_{0}}\otimes a)$. The left hand side gives

$S_{U\otimes V,W}\left(\sum_{i_{1}}(\zeta_{k_{0}}\otimes
c)\otimes(\xi_{i_{1}}\otimes\eta_{j_{0}}\otimes(u_{i_{1}i_{0}}\rhd
b)a)\right)\\\
=\sum_{i_{1},i_{2},j}\xi_{i_{1}}\otimes\eta_{j}\otimes\zeta_{k_{0}}\otimes(u_{i_{1}i_{2}}v_{jj_{0}}\rhd
c)(u_{i_{2}i_{0}}\rhd
b)a=\sum_{ij}\xi_{i}\otimes\eta_{j}\otimes\zeta_{k_{0}}\otimes\left(u_{ii_{0}}\rhd(v_{jj_{0}}\rhd
c)b)\right)a.$

The right hand side gives

$S_{U,V\otimes
W}\biggl{(}\biggl{(}\sum_{j}(\eta_{j}\otimes\zeta_{k_{0}})\otimes(v_{jj_{0}}\rhd
c)b\biggr{)}b\otimes(\xi_{i_{0}}\otimes
a)\biggr{)}=\sum_{ij}\xi_{i}\otimes\eta_{j}\otimes\zeta_{k_{0}}\otimes\left(u_{ii_{0}}\rhd(v_{jj_{0}}\rhd
c)b)\right)a,$

hence we obtain the claim. ∎

Given $W\in\operatorname{Rep}(G)$, let us denote by $\beta_{W}$ the
composition of the $\hat{G}$-action $\beta\colon B\to\mathcal{M}(B\otimes
c_{0}(\hat{G}))$ with the projection $\mathrm{id}\otimes\pi_{W}$, where
$\pi_{W}\colon\ell^{\infty}(\hat{G})\to\mathcal{B}(H_{W})$ is the
representation homomorphism.

######  Definition 6.6.

Define a left action of $\operatorname{Rep}(G)$ on the category
$\mathcal{D}_{B}$ by

$V\stmryogt X=X\otimes_{B}(H_{V}\otimes B),$

and for $T\in\mathcal{D}_{B}(X,Y)$ and $W\in\operatorname{Rep}(G)$,

$\mathrm{id}_{W}\stmryogt T=T\otimes\mathrm{id}_{H_{W}\otimes B}.$

######  Proposition 6.7.

Together with the left $\operatorname{Rep}(G)$ action given by Definition 6.6,
$\mathcal{C}_{B}$ is a $\operatorname{Rep}(G)$-bimodule C∗-category.

###### Proof.

Proposition 6.5 gives natural isomorphisms

$(U\stmryogt B)\stmryolt V=H_{V}\otimes(B\otimes_{B}B_{U})\simeq
B_{V}\otimes_{B}B_{U}=U\stmryogt(B\stmryolt V).$

We need to show that for
$T\in\operatorname{Mor}_{\operatorname{Rep}(G)}(U,V)$,
$T\otimes\mathrm{id}_{B}\in\mathcal{C}_{B}(B_{U},B_{V})$ is a left $B$-module
map. For $a,b\in B$ and $\xi\in H_{U}$,

$\displaystyle(T\otimes\mathrm{id}_{B})\left(\pi_{U}(b)(\xi\otimes a)\right)$
$\displaystyle=(T\otimes\mathrm{id}_{B})\left(m^{U}_{ij}(\xi)\otimes(u_{ij}\rhd
b)a\right)$
$\displaystyle=(T\otimes\mathrm{id}_{B})\left(m^{U}_{ij}(\xi)\otimes((\mathrm{id}\otimes
u_{ij})\beta(b)\right)a.$

By the intertwiner condition for $T$, we see that this is equal to

$m^{V}_{kl}(T\xi)\otimes\left((\mathrm{id}\otimes
v_{kl})\beta(b)\right)a=\pi_{V}(b)(T\xi\otimes a),$

hence we obtain the claim. ∎

We conclude that the $\operatorname{Rep}(G)$-bimodule category
$\mathcal{D}_{B}$ has $B$ as a central generator, with the half-braiding
$\sigma_{U}\colon U\stmryogt B\to B\stmryolt U$ given by the identity map up
to the standard identification

$U\stmryogt B=B\otimes_{B}(H_{U}\otimes B)\simeq H_{U}\otimes B.$

Thus, a Yetter–Drinfeld $G$-C∗-algebra gives rise to a
$\operatorname{Rep}(G)$-bimodule category with a central generator.

###  6.2. From centrally pointed bimodules to Yetter–Drinfeld algebras

In the other direction, let $\mathcal{M}$ be a
$\operatorname{Rep}(G)$-bimodule C∗-category with a central generator $m$,
with central structure $\sigma_{U}\colon U\stmryogt m\to m\stmryolt U$ for
$U\in\operatorname{Rep}(G)$. Being a cyclic right
$\operatorname{Rep}(G)$-module C∗-category, to $\mathcal{M}$ there is
associated a unital $G$-C∗-algebra $B_{\mathcal{M}}$, which is a completion of
the vector space

$\mathcal{B}_{\mathcal{M}}=\bigoplus_{U\in\operatorname{Irr}(G)}\bar{H}_{U}\otimes\mathcal{M}(m,m\stmryolt
U),$

which we call regular part of $B_{\mathcal{M}}$, see Section 2.4.

Let us briefly recall the $*$-algebra structure on this space. For further
reference, see [n1][NY3]. It is built using the auxiliary universal algebra

$\tilde{\mathcal{B}}_{\mathcal{M}}=\bigoplus_{U\in\mathcal{C}}\bar{H}_{U}\otimes\mathcal{M}(m,m\stmryolt
U),$

called the universal cover of $\mathcal{B}_{\mathcal{M}}$. Its product is
defined by

$(\bar{\xi}\otimes T)\bullet(\bar{\eta}\otimes
S)=(\overline{\xi\otimes\eta})\otimes(T\otimes\mathrm{id}_{V})S$

for $\bar{\xi}\otimes T\in\bar{H}_{U}\otimes\mathcal{M}(m,m\stmryolt U)$ and
$\bar{\eta}\otimes S\in\bar{H}_{V}\otimes\mathcal{M}(m,m\stmryolt V)$.

There is a surjective idempotent linear map
$\pi\colon\tilde{\mathcal{B}}_{\mathcal{M}}\to\mathcal{B}_{\mathcal{M}}$
defined by the decompositions of arbitrary elements of $\operatorname{Rep}(G)$
into irreducible objects, and for $x,y\in\mathcal{B}_{\mathcal{M}}$ the
product $xy$ is $\pi(x\bullet y)$. The specific form of $\pi$ will be given in
the proof of Lemma 6.8. The involution $x\mapsto x^{\dagger}$ on
$\tilde{\mathcal{B}}_{\mathcal{M}}$ is given by

$\bar{H}_{U}\otimes\mathcal{M}(m,m\stmryolt U)\to
H_{U}\otimes\mathcal{M}(m,m\stmryolt\bar{U}),\quad\bar{\xi}\otimes
T\mapsto\rho_{U}^{-1/2}\xi\otimes(T^{*}\otimes\mathrm{id}_{U})\bar{R}_{U},$

and we obtain the involution on $\mathcal{B}_{\mathcal{M}}$ by
$\pi(x)^{*}=\pi(x^{\dagger})$.

The construction above can be applied to the Yetter-Drinfeld algebra $C(G)$.
The regular part of it is $\mathcal{O}(G)$, the algebra of polynomial
functions on $G$, and its universal cover is $\tilde{\mathcal{O}}(G)$. In this
case, we denote by $\pi_{G}$ the canonical projection
$\tilde{\mathcal{O}}(G)\to\mathcal{O}(G)$.

We claim that the $*$-algebra $\mathcal{B}_{\mathcal{M}}$ has a
Yetter–Drinfeld structure. Consider the action
$\mathbin{\tilde{\rhd}}\colon\tilde{\mathcal{O}}(G)\otimes\tilde{\mathcal{B}}_{\mathcal{M}}\to\tilde{\mathcal{B}}_{\mathcal{M}}$
defined by

$(\bar{\xi}\otimes\zeta)\mathbin{\tilde{\rhd}}(\bar{\eta}\otimes T)\\\
=(\overline{\xi\otimes\eta\otimes\overline{\rho^{-1/2}\zeta}})\otimes\left((\sigma_{U}\otimes\mathrm{id}_{V}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{U}\otimes
T\otimes\mathrm{id}_{\bar{U}})(\sigma_{U}^{-1}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\right)$
(6.2)

for $T\in\mathcal{M}(m,m\stmryolt V)$, $\bar{\eta}\in\bar{H}_{V}$, and
$\xi,\zeta\in H_{U}$. For $x\in\mathcal{O}(G)$ and
$a\in\mathcal{B}_{\mathcal{M}}$, we put $x\rhd
a=\pi(x\mathbin{\tilde{\rhd}}a)$.

From now on, $U,V$ and $W$ will always stand for finite dimensional unitary
representations of $G$, and to avoid cumbersome formulas we will often write
$UV$ instead of $U\otimes V$.

######  Lemma 6.8.

For all $x\in\tilde{\mathcal{O}}(G)$ and every
$a\in\tilde{\mathcal{B}}_{\mathcal{M}}$,

$\pi_{G}(x)\rhd\pi(a)=\pi(x\mathbin{\tilde{\rhd}}a).$

###### Proof.

Take $x=\bar{\xi}\otimes\zeta\in\bar{H}_{U}\otimes H_{U}$ and
$a=\bar{\eta}\otimes T\in\bar{H}_{V}\otimes\mathcal{M}(m,m\stmryolt V)$. Take
partial isometries $u_{i}\colon H_{s_{i}}\to H_{U}$ and $v_{j}\colon
H_{s_{j}}\to H_{V}$ defining decompositions of $U$ and $V$ into irreducible
finite dimensional unitary representations. Then we have

$\pi_{G}(x)\rhd\pi(a)=\pi\left(\sum_{i,j}(\overline{u_{i}^{*}\xi}\otimes
u_{i}^{*}\zeta)\mathbin{\tilde{\rhd}}(\overline{v_{j}^{*}\eta}\otimes
v_{j}^{*}T)\right),$

which can be expanded as

$\pi\left(\sum_{i,j}\overline{(u_{i}^{*}\xi\otimes
v_{j}^{*}\eta\otimes\overline{\rho_{U}^{-1/2}(u_{i}^{*}\zeta)})}\otimes(\sigma_{U_{s_{i}}}\otimes\mathrm{id}_{V}\otimes\mathrm{id}_{\bar{U}_{s_{i}}})(\mathrm{id}_{U_{s_{i}}}\otimes
v_{j}^{*}T\otimes\mathrm{id}_{\bar{U}_{s_{i}}})(\sigma_{U_{s_{i}}}^{*}\otimes\mathrm{id}_{\bar{U}_{s_{i}}})\bar{R}_{s_{i}}\right).$

On the other hand, we also have

$\pi(x\tilde{\rhd}a)=\pi\left(\overline{(\xi\otimes\eta\otimes\overline{\rho_{U}^{-1/2}\zeta})}\otimes(\sigma_{U}\otimes\mathrm{id}_{V}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{U}\otimes
T\otimes\mathrm{id}_{\bar{U}})(\sigma_{U}^{*}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\right)$

which can be expanded as

$\pi\Bigl{(}\sum_{i,j,k}\overline{(u_{i}^{*}\xi\otimes
v_{j}^{*}\eta\otimes\bar{u}_{k}^{*}\overline{\rho_{U}^{-1/2}\zeta})}\otimes\\\
(\sigma_{U_{s_{i}}}\otimes\mathrm{id}_{U_{s_{j}}}\otimes\mathrm{id}_{\bar{U}_{s_{k}}})(\mathrm{id}_{U_{s_{i}}}\otimes
v_{j}^{*}T\otimes\mathrm{id}_{\bar{U}_{s_{k}}})(\sigma_{U_{s_{i}}}\otimes\mathrm{id}_{\bar{U}_{s_{k}}})(u_{i}^{*}\otimes
u_{k}^{*})\bar{R}_{U}\Bigr{)}.$

We have $u_{k}^{*}\rho_{U}=\rho_{U_{k}}u_{k}^{*}$ and
$\bar{R}_{U}=\sum_{i}(u_{i}\otimes\bar{u}_{i})\bar{R}_{s_{i}}$. Moreover, the
partial isometries $u_{i}$ have mutually orthogonal ranges. We conclude that

$\pi_{G}(x)\rhd\pi(a)=\pi(x\tilde{\rhd}a),$

establishing the claim. ∎

######  Lemma 6.9.

The map $\rhd$ makes $\mathcal{B}_{\mathcal{M}}$ into a left
$\mathcal{O}(G)$-module.

###### Proof.

Taking $x=\bar{\xi}\otimes\zeta\in\bar{H}_{U}\otimes H_{U}$,
$y=\bar{\mu}\otimes\nu\in\bar{H}_{W}\otimes H_{W}$ and $a=\bar{\eta}\otimes
T\in\bar{H}_{V}\otimes\mathcal{M}(m,m\stmryolt V)$, we must show that

$\pi(x\mathbin{\tilde{\rhd}}(y\mathbin{\tilde{\rhd}}a))=\pi((xy)\mathbin{\tilde{\rhd}}a).$

Expanding the left hand side, we obtain

$\pi\Bigl{(}\overline{(\xi\otimes\mu\otimes\eta\otimes\overline{\rho_{W}^{-1/2}\nu}\otimes\overline{\rho_{U}^{-1/2}\zeta})}\\\
\otimes(\sigma_{UW}\otimes\mathrm{id}_{V\bar{W}\bar{U}})(\mathrm{id}_{UW}\otimes
T\otimes\mathrm{id}_{\bar{W}\bar{U}})(\sigma_{UW}^{*}\otimes\mathrm{id}_{\bar{W}\bar{U}})(\mathrm{id}_{U}\otimes\bar{R}_{W}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\Bigr{)}.$

The right hand side is, in turn,

$\displaystyle\pi\left(\overline{(\xi\otimes\mu\otimes\overline{\rho_{U}^{-1/2}\zeta\otimes\rho_{W}^{-1/2}\nu})}\otimes(\sigma_{UW}\otimes\mathrm{id}_{V\bar{UW}})(\mathrm{id}_{UW}\otimes
T\otimes\mathrm{id}_{\bar{UW}})(\sigma_{UW}^{*}\otimes\mathrm{id}_{\bar{UW}})\bar{R}_{UW}\right)$

The equality between the two expressions comes from the fact that

$\gamma\colon\bar{H}_{W}\otimes\bar{H}_{U}\to\overline{H_{U}\otimes
H_{W}},\quad\gamma(\bar{a}\otimes\bar{b})=\overline{b\otimes a}$

is an equivalence of representations, and that we can take

$\bar{R}_{UW}=(\mathrm{id}_{UW}\otimes\gamma)(\mathrm{id}_{U}\otimes\bar{R}_{W}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U},$

hence the claim. ∎

######  Lemma 6.10.

The map $\rhd$ is an algebra-module map, that is,

$x\rhd(ab)=(x_{(1)}\rhd a)(x_{(2)}\rhd b)$

holds for any $x\in\mathcal{O}(G)$ and $a,b\in\mathcal{B}_{\mathcal{M}}$.

###### Proof.

Take $a=\bar{\eta}\otimes T\in\bar{H}_{V}\otimes\mathcal{M}(m,m\stmryolt V)$,
$b=\bar{\zeta}\otimes S\in\bar{H}_{W}\otimes\mathcal{M}(m,m\stmryolt W)$, and
take $x$ to be $u_{ij}=\pi_{G}(\bar{\xi}_{i}\otimes\xi_{j})$, where
$\\{\xi_{i}\\}_{i}$ is an orthonormal basis of $H_{U}$, so that $\\{u_{ij}\\}$
are the matrix coefficients of $U$ with respect to it.

By $\Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj}$, our claim is equivalent to

$\pi\left((\bar{\xi}_{i}\otimes\xi_{j})\mathbin{\tilde{\rhd}}(ab)\right)=\sum_{k}\pi\left(((\bar{\xi}_{i}\otimes\xi_{k})\mathbin{\tilde{\rhd}}a)((\bar{\xi}_{k}\otimes\xi_{j})\mathbin{\tilde{\rhd}}b)\right).$

The left hand side is

$\pi\left(\overline{(\xi_{i}\otimes\eta\otimes\zeta\otimes\overline{\rho_{U}^{-1/2}\xi_{j}})}\otimes(\sigma_{U}\otimes\mathrm{id}_{VW\bar{U}})(\mathrm{id}_{U}\otimes(T\otimes\mathrm{id}_{W})S\otimes\mathrm{id}_{\bar{U}})(\sigma_{U}^{*}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\right),$

while the right hand side is

$\sum_{k}\pi\left(\overline{(\xi_{i}\otimes\eta\otimes\overline{\rho_{U}^{-1/2}\xi_{k}}\otimes\xi_{k}\otimes\zeta\otimes\overline{\rho_{U}^{-1/2}\xi_{j}})}\otimes
X\right),$

where

$X=(\sigma_{U}\otimes\mathrm{id}_{V\bar{U}UW\bar{U}})(\mathrm{id}_{U}\otimes
T\otimes\mathrm{id}_{\bar{U}UW\bar{U}}))\\\
(\sigma_{U}^{*}\otimes\mathrm{id}_{\bar{U}UW\bar{U}})(\bar{R}_{U}\otimes\mathrm{id}_{UW\bar{U}})(\sigma_{U}\otimes\mathrm{id}_{W\bar{U}})(\mathrm{id}_{U}\otimes
S\otimes\mathrm{id}_{\bar{U}})(\sigma_{U}^{*}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}).$

We have $\sum_{k}\overline{\rho_{U}^{-1/2}\xi_{k}}\otimes\xi_{k}=R_{U}(1)$,
and the operator $d_{U}^{-1/2}R_{U}$ is an isometric embedding of the trivial
representation $1$ into $\bar{U}\otimes U$. Therefore

$\sum_{k}\pi\left(\overline{(\xi_{i}\otimes\eta\otimes\overline{\rho_{U}^{-1/2}\xi_{k}}\otimes\xi_{k}\otimes\zeta\otimes\overline{\rho_{U}^{-1/2}\xi_{j}})}\otimes
X\right)$

is equal to

$\pi\left(\overline{(\xi_{i}\otimes\eta\otimes\zeta\otimes\overline{\rho_{U}^{-1/2}\xi_{j}})}\otimes(\mathrm{id}_{UV}\otimes
R_{U}^{*}\otimes_{W\bar{U}})X\right),$

which is equal to
$\pi\left((\bar{\xi}_{i}\otimes\xi_{j})\mathbin{\tilde{\rhd}}(ab)\right)$
since
$(R_{U}^{*}\otimes\mathrm{id}_{U})(\mathrm{id}_{U}\otimes\bar{R}_{U})=\mathrm{id}_{U}$.
∎

######  Lemma 6.11.

The $\mathcal{O}(G)$-action on $\mathcal{B}_{\mathcal{M}}$ is compatible with
the $*$-structure:

$x\rhd a^{*}=\left(S(x)^{*}\rhd a\right)^{*}.$

###### Proof.

For $x=\bar{\xi}\otimes\zeta\in\bar{H}_{U}\otimes H_{U}$, denote
$\bar{\zeta}\otimes\xi$ by $x^{\dagger}$. We then have
$S(\pi_{G}(x))^{*}=\pi_{G}(x^{\dagger})$, see [NY3]. Thus, the claim is
equivalent to

$\pi(x\mathbin{\tilde{\rhd}}a^{\dagger})=\pi((x^{\dagger}\mathbin{\tilde{\rhd}}a))^{\dagger}.$

The left hand side is

$\pi\left(\overline{\xi\otimes\overline{\rho_{V}^{-1/2}\eta}\otimes\overline{\rho_{U}^{-1/2}\zeta})}\otimes(\sigma_{U}\otimes\mathrm{id}_{V\bar{U}})(\mathrm{id}_{U}\otimes
T^{*}\otimes\mathrm{id}_{\bar{V}\bar{U}})(\mathrm{id}_{U}\otimes\bar{R}_{V}\otimes\mathrm{id}_{\bar{U}})(\sigma_{U}^{*}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\right).$

The right hand side is

$\pi\Bigl{(}\overline{\overline{(\rho_{U}^{-1/2}\zeta\otimes\rho_{V}^{-1/2}\eta\otimes\bar{\xi})}}\otimes(\bar{R}_{U}^{*}\otimes\mathrm{id}_{\overline{UV\bar{U}}})(\sigma_{U}\otimes\mathrm{id}_{\bar{U}\overline{UV\bar{U}}})(\mathrm{id}_{U}\otimes
T^{*}\otimes\mathrm{id}_{\bar{U}\overline{UV\bar{U}}})\\\
(\sigma_{U}^{*}\otimes\mathrm{id}_{V\bar{U}\overline{UV\bar{U}}})\bar{R}_{UV\bar{U}}\Bigr{)}.$

The unitary

$K\colon H_{U}\otimes H_{\bar{V}}\otimes H_{\bar{U}}\to
H_{\overline{UV\bar{U}}},\quad\xi\otimes\bar{\eta}\otimes\bar{\zeta}\mapsto\overline{\zeta\otimes\eta\otimes\bar{\xi}}$

is an equivalence of representations, and

$\bar{R}_{UV\bar{U}}=(\mathrm{id}_{UV\bar{U}}\otimes
K)(\mathrm{id}_{UV}\otimes
R_{U}\otimes\mathrm{id}_{\bar{V}\bar{U}})(\mathrm{id}_{U}\otimes\bar{R}_{V}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}.$

With this and the identity
$(\bar{R}_{U}^{*}\otimes\mathrm{id}_{U})(\mathrm{id}_{U}\otimes
R_{U})=\mathrm{id}_{U}$, we see that the claim holds. ∎

######  Lemma 6.12.

The Yetter–Drinfeld condition for $\rhd$ is satisfied: if $\alpha$ is the
action of $G$ on $B_{\mathcal{M}}$, then

$\alpha(x\rhd a)=x_{(1)}a_{(1)}S(x_{(3)})\otimes(x_{(2)}\rhd a_{(2)})$

for all $x\in\mathcal{O}(G)$ and for all $a\in\mathcal{B}_{\mathcal{M}}$.

###### Proof.

Fix orthonormal basis $\\{\xi_{i}\\}_{i}$ and $\\{\eta_{k}\\}_{k}$ of $H_{U}$
and $H_{V}$, respectively, consisting of eigenvectors of the corresponding
operators $\rho_{U}$ and $\rho_{V}$. Let $\rho_{i}$ be the associated
eigenvalues of $\rho_{U}$, so that we have $\rho_{U}\xi_{i}=\rho_{i}\xi_{i}$.
Take
$x=u_{i_{0}j_{0}}=\pi_{G}(\bar{\xi}_{i_{0}}\otimes\xi_{j_{0}})\in\mathcal{O}(G)$
and $a=\pi(\bar{\eta}_{k_{0}}\otimes
T)\in\pi(\bar{H}_{V}\otimes\mathcal{M}(m,m\stmryolt V))$. Recall that the
$G$-action on $\mathcal{B}_{\mathcal{M}}$ is defined by

$\alpha(\pi(\bar{\eta}_{k_{0}}\otimes
T))=\sum_{k}v_{k_{0}k}\otimes\pi(\bar{\eta}_{k}\otimes T),$

with $v_{k_{1}k_{2}}$ denoting the matrix coefficients of $V$ with respect to
the basis $\\{\eta_{k}\\}_{k}$. From the definitions, we get that
$x_{(1)}a_{(1)}S(x_{(3)})\otimes(x_{(2)}\rhd a_{(2)})$ equals to

$\displaystyle\sum_{i,j,k}\rho_{j}^{-1/2}u_{i_{0}i}v_{k_{0}k}S(u_{jj_{0}})\otimes\pi\left(\overline{(\xi_{i}\otimes\eta_{k}\otimes\bar{\xi}_{j})}\otimes(\sigma_{U}\otimes\mathrm{id}_{V\bar{U}})(\mathrm{id}_{U}\otimes
T\otimes\mathrm{id}_{\bar{U}})(\sigma_{U}^{*}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\right).$

The contragredient representation of $U$ has $\\{u_{ij}^{*}\\}_{i,j}$ as
matrix coefficients. As for the dual representation $\bar{U}$, the matrix
coefficients are
$\\{\bar{u}_{ij}=\rho_{i}^{1/2}\rho_{j}^{-1/2}u_{ij}^{*}=\rho_{i}^{1/2}\rho_{j}^{-1/2}S(u_{ji})\\}_{i,j}$.
Therefore we have
$\rho_{j}^{-1/2}S(u_{jj_{0}})=\rho_{j_{0}}^{-1/2}\bar{u}_{j_{0}j}$, and the
expression above is the same as

$\rho_{j_{0}}^{-1/2}\sum_{i,j,k}u_{i_{0}i}v_{k_{0}k}\bar{u}_{jj_{0}}\otimes\pi\left(\overline{(\xi_{i}\otimes\eta_{k}\otimes\bar{\xi}_{j})}\otimes(\sigma_{U}\otimes\mathrm{id}_{V\bar{U}})(\mathrm{id}_{U}\otimes
T\otimes\mathrm{id}_{\bar{U}})(\sigma_{U}^{*}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\right).$

This is the right hand side of the equation in the Lemma; let us compute the
left hand side.

As $u_{i_{0}j_{0}}\rhd(\bar{\eta}_{k_{0}}\otimes
T)\in\pi(\bar{H}_{UV\bar{U}}\otimes\mathcal{M}(m,m\stmryolt(U\otimes
V\otimes\bar{U}))$,

$\alpha(u_{i_{0}j_{0}}\rhd(\bar{\eta}_{k_{0}}\otimes T))=(U\otimes
V\otimes\bar{U})_{21}^{*}(1\otimes(u_{i_{0}j_{0}}\rhd(\bar{\eta}_{k_{0}}\otimes
T)).$

Expanding on the definitions, we see that it is the same as
$x_{(1)}a_{(1)}S(x_{(3)})\otimes(x_{(2)}\rhd a_{(2)})$. ∎

Lemmas 6.8–6.12 say that the closure $B_{\mathcal{M}}$ of
$\mathcal{B}_{\mathcal{M}}$ becomes then a Yetter–Drinfeld $G$-C∗-algebra.

###  6.3. Duality

Our next goal is to prove that, up to isomorphisms, the above constructions
are inverse to each other. Given a (strict) centrally pointed bimodule
category $\mathcal{M}$, let us write
$\mathcal{D}_{\mathcal{M}}=\mathcal{D}_{B_{\mathcal{M}}}$. Then by the duality
for $G$-C∗-algebras, there is an equivalence of right
$\operatorname{Rep}(G)$-module C∗-categories,
$\Psi\colon\mathcal{M}\to\mathcal{D}_{\mathcal{M}}$ ([n1]).

######  Proposition 6.13.

The right module category equivalence functor $\Psi$ extends to a central
functor.

###### Proof.

By Proposition 5.8, it is enough to construct a corresponding extension from
$\tilde{\mathcal{M}}$ to $\tilde{\mathcal{D}}_{\mathcal{M}}$.

First we note that $\tilde{\mathcal{D}}_{\mathcal{M}}$ is the category
$\mathcal{C}_{\mathcal{M}}=\mathcal{C}_{B_{\mathcal{M}}}$ in [NY3], i.e., the
idempotent completion of $\mathcal{C}$ with respect to the morphism sets

$\mathcal{C}_{B_{\mathcal{M}}}(U,V)=\operatorname{Hom}_{\tilde{\mathcal{D}}_{\mathcal{M}}}(H_{U}\otimes
B_{\mathcal{M}},H_{V}\otimes B_{\mathcal{M}}),$

the right-hand-side meaning $G$-equivariant Hilbert $B_{\mathcal{M}}$-module
maps. Then, the right module category equivalence
$\Psi\colon\mathcal{M}\to\mathcal{C}_{B_{\mathcal{M}}}$ is characterized by
$m\stmryolt U\mapsto U$ at the level of objects, and at the level of morphisms
by

$\Psi(T)=\sum_{i,j}\theta_{\zeta_{j},\xi_{i}}\otimes\pi\left(\overline{(\zeta_{j}\otimes\overline{\rho^{-1/2}\xi_{i}})}\otimes(T\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{m}\stmryolt\bar{R}_{U})\right)$
(6.3)

for $T\in\mathcal{M}(m\stmryolt U,m\stmryolt V)$, where $(\xi_{i})_{i}$ is an
orthonormal basis of $H_{U}$, $(\zeta_{j})_{j}$ is an orthonormal basis of
$H_{V}$, and $\theta_{\zeta,\xi}$ is the linear map
$\eta\mapsto(\eta,\xi)\zeta$, see [NY3].

Now let us denote the central structure on $\mathcal{M}=(\mathcal{M},m)$ by
$\sigma$, and the one on $\mathcal{C}_{\mathcal{M}}$ by $S$. Let us fix
$T\in\mathcal{M}(m_{1}\stmryolt V,m_{1}\stmryolt W)$, and take orthonormal
bases $(\xi_{i})_{i}$ on $H_{U}$, $(\zeta_{l})_{l}$ on $H_{V}$, and
$(\eta_{k})_{k}$ on $H_{W}$.

On the one hand, the morphism
$\Psi\left((\sigma\otimes\mathrm{id}_{W})(\mathrm{id}_{U}\otimes
T)(\sigma^{-1}\otimes\mathrm{id}_{V})\right)$ can be expanded as

$\sum_{i,j,k,l}\theta_{\xi_{i}\otimes\eta_{k},\xi_{j}\otimes\zeta_{l}}\otimes\pi\left(\overline{(\xi_{i}\otimes\eta_{k}\otimes\overline{\rho^{-1/2}(\xi_{j}\otimes\zeta_{l}})}\otimes\left((\sigma\otimes\mathrm{id}_{W})(\mathrm{id}_{U}\otimes
T)(\sigma^{-1}\otimes\mathrm{id}_{V})\otimes\mathrm{id}_{\overline{UV}}\right)\bar{R}_{UV}\right),$

which is equal to

$\sum
m^{U}_{ij}\otimes\theta_{\eta_{k},\zeta_{l}}\otimes\left(u_{ij}\rhd\pi\left(\overline{(\eta_{k}\otimes\overline{\rho^{-1/2}\zeta_{l}}})\otimes(T\otimes\mathrm{id}_{\bar{V}})\bar{R}_{V}\right)\right).$

On the other, writing $\Psi(T)=\sum_{l}\theta_{l}\otimes b_{l}$ with
$\theta_{l}\in B(H_{V},H_{W})$ and $b_{l}\in B_{\mathcal{M}}$, we have

$(S\otimes\mathrm{id}_{W})\left(\mathrm{id}_{U}\otimes\sum_{l}T_{l}\otimes
b_{l}\right)(S^{-1}\otimes\mathrm{id}_{V})=\sum_{l,i,j}m^{U}_{ij}\otimes
T_{l}\otimes(u_{ij}\rhd b_{l}).$

Comparing these we get

$\Psi\left((\sigma\otimes\mathrm{id}_{W})(\mathrm{id}_{U}\otimes
T)(\sigma^{-1}\otimes\mathrm{id}_{V})\right)=(S\otimes\mathrm{id}_{W})(\mathrm{id}_{U}\otimes\Psi(T))(S^{-1}\otimes\mathrm{id}_{V}),$

which gives the desired extension by Proposition 5.9. ∎

Going the other way, let $B$ be a Yetter–Drinfeld $G$-C∗-algebra, and consider
the new algebra $B_{\mathcal{D}_{B}}$ we get by the above procedure. Let us
denote their regular subalgebras by $\mathcal{B}$ and
$\mathcal{B}_{\mathcal{D}}$. Then the isomorphism of $G$-algebras
$\lambda\colon B_{\mathcal{D}_{B}}\to B$ is given by

$\lambda(\bar{\eta}\otimes T)=(\eta^{*}\otimes\mathrm{id}_{B})(T)$

for $\bar{\eta}\in\bar{H}_{V}$ and $T\in(H_{V}\otimes
B)^{G}\simeq\mathcal{D}_{B}(B,B\stmryolt V)$, where $\eta^{*}$ is the
functional $\xi\mapsto(\xi,\eta)$ on $H_{V}$.

######  Proposition 6.14.

$\lambda$ is $\hat{G}$-equivariant.

###### Proof.

Fix an orthonormal basis $(\zeta_{k})_{k}$ in $H_{V}$ and take
$T\in\mathcal{D}_{B}(B,B\stmryolt V)$, represented by
$\sum_{k}\zeta_{k}\otimes b_{k}\in H_{V}\otimes B$ up to the above
correspondence.

The central structure $S$ on $\mathcal{D}_{B}$ satisfies

$\displaystyle(S\otimes\mathrm{id}_{V\bar{U}})(\mathrm{id}_{U}\otimes
T\otimes\mathrm{id}_{\bar{U}})(S^{-1}\otimes\mathrm{id}_{\bar{U}})=\sum_{i,j,k}m^{U}_{ij}\otimes\zeta_{k}\otimes
1\otimes(u_{ij}\rhd b_{k})$

in $\mathcal{D}_{B}(B\stmryolt U\otimes\bar{U},B\stmryolt U\otimes
V\otimes\bar{U})$.

From our definition of the action $\rhd$, we see that
$u_{i_{0}j_{0}}\rhd\pi(\bar{\zeta}_{k_{0}}\otimes T)$ is equal to

$\pi\left((\bar{\xi}_{i_{0}}\otimes\xi_{j_{0}})\mathbin{\tilde{\rhd}}(\bar{\zeta}_{k_{0}}\otimes
T)\right)\\\
=\pi\left(\overline{(\xi_{i_{0}}\otimes\zeta_{k_{0}}\otimes\overline{\rho_{U}^{-1/2}\xi_{j_{0}}})}\otimes(S\otimes\mathrm{id}_{V\bar{U}})(\mathrm{id}_{U}\otimes
T\otimes\mathrm{id}_{\bar{U}})(S^{-1}\otimes\mathrm{id}_{\bar{U}})\bar{R}_{U}\right)\\\
=\pi\left(\overline{(\xi_{i_{0}}\otimes\zeta_{k_{0}}\otimes\overline{\rho_{U}^{-1/2}\xi_{j_{0}}})}\otimes\left(\sum_{i,j,k}m^{U}_{ij}\otimes\zeta_{k}\otimes
1\otimes(u_{ij}\rhd b_{k})\right)\bar{R}_{U}\right).$

From (2.2), this is equal to

$\sum_{i,j,k}\overline{(\xi_{i_{0}}\otimes\zeta_{k_{0}}\otimes\overline{\rho_{U}^{-1/2}\xi_{j_{0}}})}\otimes(\xi_{i}\otimes\zeta_{k}\otimes\overline{\rho_{U}^{1/2}\xi_{j}})\otimes(u_{ij}\rhd
b_{k}).$

We thus obtain

$\lambda\left(u_{i_{0}j_{0}}\rhd\pi(\bar{\zeta}_{k_{0}}\otimes
T)\right)=u_{i_{0}j_{0}}\rhd
b_{k_{0}}=u_{i_{0},j_{0}}\rhd\lambda(\bar{\zeta}_{k_{0}}\otimes T),$

establishing the claim. ∎

###  6.4. Moduli of module functors

Let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ be two centrally pointed
$\operatorname{Rep}(G)$-bimodule categories, with respective generators
$m_{1}$ and $m_{2}$ and half-braidings $\sigma$ and $\sigma^{\prime}$, and
denote their corresponding Yetter–Drinfeld $G$-C∗-algebras by $B_{1}$ and
$B_{2}$, respectively. Consider a right $\operatorname{Rep}(G)$-module functor
$F\colon\mathcal{M}_{1}\to\mathcal{M}_{2}$ such that $F(m_{1})=m_{2}$. Let us
work out the condition on $F$ such that the induced $G$-equivariant
homomorphism $f\colon B_{1}\to B_{2}$ becomes a $D(G)$-equivariant
homomorphism.

Restricting to the regular subalgebras, $f$ is given by the collection of

$\bar{H}_{U}\otimes\mathcal{M}_{1}(m_{1},m_{1}\stmryolt
U)\to\bar{H}_{U}\otimes\mathcal{M}_{2}(m_{2},m_{2}\stmryolt U),\quad
T\mapsto\mathrm{id}_{\bar{H}_{U}}\otimes F_{2}^{-1}F(T)$

for the irreducible representations $U$. In view of (6.2), $f$ will be
$\hat{G}$-equivariant if and only if

$(F_{2;m,UV\bar{U}})^{-1}F\mathopen{}\left((\sigma_{U}\stmryolt\mathrm{id}_{V}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{U}\stmryogt
T\stmryolt\mathrm{id}_{\bar{U}})(\sigma_{U}^{-1}\stmryolt\mathrm{id}_{\bar{U}})(\mathrm{id}_{m_{1}}\stmryolt\bar{R}_{U})\right)\\\
=(\sigma_{U}^{\prime}\stmryolt\mathrm{id}_{V}\otimes\mathrm{id}_{\bar{U}})(\mathrm{id}_{U}\stmryogt
F_{2;m,V}^{-1}F(T)\stmryolt\mathrm{id}_{\bar{U}})(\sigma_{U}^{\prime-1}\stmryolt\mathrm{id}_{\bar{U}})(\mathrm{id}_{m_{2}}\stmryolt\bar{R}_{U})$

As before, consider the braiding maps
$\Sigma_{U;V}\colon\mathcal{M}_{1}(m_{1},m_{1}\stmryolt
V)\to\mathcal{M}_{1}(m_{1}\stmryolt U,m_{1}\stmryolt U\stmryolt V)$ given by
$\Sigma_{U;1,V}$ in (5.2). We will write the analogous braiding maps on
$\mathcal{M}_{2}$ as $\Sigma^{\prime}$. Using

$\Sigma_{U;V}(T)=(\mathrm{id}_{m_{1}\stmryolt U\stmryolt V}\stmryolt
R_{U}^{*})(\sigma_{U}\stmryolt\mathrm{id}_{V\bar{U}U})(\mathrm{id}_{U}\stmryogt
T\stmryolt\mathrm{id}_{U\bar{U}})(\sigma_{U}^{-1}\stmryolt\mathrm{id}_{\bar{U}U})(\mathrm{id}_{m_{1}}\stmryolt\bar{R}_{U}\otimes\mathrm{id}_{U}),$

we see that $f$ will be $\hat{G}$-equivariant if and only if

$F_{2;m_{1},UV}^{-1}F(\Sigma_{U;V}(T))F_{2}=\Sigma_{U;V}^{\prime}(F_{2;m,V}^{-1}F(T))$
(6.4)

for all $U,V\in\operatorname{Rep}(G)$ and
$T\in\mathcal{M}_{1}(m_{1},m_{1}\stmryolt V)$.

######  Proposition 6.15.

Let $F\colon\mathcal{M}_{1}\to\mathcal{M}_{2}$ be a right
C∗-$\operatorname{Rep}(G)$-module functor between centrally pointed bimodule
categories. If $F$ satisfies (6.4), then there is a strict bimodule functor
$\tilde{F}\colon\tilde{\mathcal{M}}_{1}\to\tilde{\mathcal{M}}_{2}$ between the
categories obtained by trivialization of the central structures, that is
naturally isomorphic to $F$ up to the equivalence of Proposition 5.8.

###### Proof.

From the assumption we actually have

$F_{2;m_{1},UW}^{-1}F(\Sigma_{U;V}(T))F_{2;m_{1},UV}=\Sigma_{U;V}^{\prime}(F_{2;m,V}^{-1}F(T))\quad(T\in\mathcal{M}_{1}(m_{1}\stmryolt
V,m_{1}\stmryolt W).$

Indeed, given $T\in\mathcal{M}_{1}(m_{1}\stmryolt V,m_{1}\stmryolt W)$, we can
take
$S=(T\stmryolt\mathrm{id}_{\bar{W}})(\mathrm{id}_{m_{1}}\stmryolt\bar{R}_{W})$
and combine (6.4) with the conjugate equation to get this claim. Then the
claim follows from Proposition 5.9. ∎

Finally, let us compare different choices of bimodule functors leading to the
same homomorphisms. Suppose that
$F,F^{\prime}\colon\mathcal{M}\to\mathcal{M}^{\prime}$ are central functors
that are naturally isomorphic. Then the induced homomorphisms
$B_{\mathcal{M}}\to B_{\mathcal{M}^{\prime}}$ agree. Indeed, we just need to
look at the induced maps

$\mathcal{M}(m,m\stmryolt
U)\to\mathcal{M}^{\prime}(m^{\prime},m^{\prime}\stmryolt U)$

which are given by
$T\mapsto(F_{0}^{-1}\otimes\mathrm{id}_{U})F_{2}^{-1}F(T)F_{0}$ and a similar
formula for $F^{\prime}$. Using the commutativity of diagrams in Definition
5.5, we see that these maps indeed agree.

######  Proposition 6.16.

Suppose that the induced homomorphisms $B_{\mathcal{M}}\to
B_{\mathcal{M}^{\prime}}$ agree. Then $F$ and $F^{\prime}$ are naturally
isomorphic as bimodule functors.

###### Proof.

Let $f_{F}\colon B_{\mathcal{M}}\to B_{\mathcal{M}^{\prime}}$ denote the
homomorphism induced by $F$. By Proposition 6.13, the assertion follows once
we can check that the bimodule functor
$F_{f}\colon\mathcal{D}_{\mathcal{M}}\to\mathcal{D}_{\mathcal{M}^{\prime}}$
induced by $f_{F}$ is naturally isomorphic to $F$ up to the bimodule
equivalence $\mathcal{M}\to\mathcal{D}_{\mathcal{M}}$. Looking at the
corresponding functor $\tilde{\mathcal{M}}\to\tilde{\mathcal{M}}^{\prime}$
induced by $F$, the claim is tautological. Then we get the claim by
Proposition 5.8. ∎

##  7\. Operator system theory for centrally pointed bimodule categories

###  7.1. Multipliers and injectivity

######  Definition 7.1.

Let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ be centrally pointed
$\mathcal{C}$-bimodule categories, with central generators $m_{1}$ and
$m_{2}$. A _central ucp $\mathcal{C}$-linear multiplier_
$F\colon\mathcal{M}_{1}\to\mathcal{M}_{2}$ is a ucp right $\mathcal{C}$-linear
multiplier $F=\\{F_{U,V}\colon\mathcal{M}_{1}(m_{1}\stmryolt U,m_{1}\stmryolt
V)\to\mathcal{M}_{2}(m_{2}\stmryolt U,m_{2}\stmryolt V)\\}_{U,V}$ such that

$F_{UV,UW}\Sigma_{U;V,W}=\Sigma_{U;V,W}^{\prime}F_{V,W}\quad(U,V,W\in\mathcal{C}).$
(7.1)

We denote by $\mathcal{CBOS}(\mathcal{C})$ the category of centrally pointed
$\mathcal{C}$-bimodule categories and central ucp $\mathcal{C}$-linear
multipliers.

######  Remark 7.2.

In Section 5 we showed that the category of centrally pointed bimodule
categories is equivalent, through trivialization of the central structures, to
a category of pointed cyclic bimodules $(\mathcal{M},m)$ on which the left
bimodule structure is an extension of $U\stmryogt(m\stmryolt V)=m\stmryolt
U\stmryolt V$. At the level of morphisms, the left action on such a
trivialized category was given by

$\mathrm{id}_{U}\stmryogt T=\Sigma_{U;V,W}(T)\quad(T\in\mathcal{M}(m\stmryolt
V,m\stmryolt W)).$

Equation (7.1) is saying, therefore, that $F$ is a ucp-multiplier of the
corresponding bimodule categories with trivialized central structures.

Recall that a pointed right $\mathcal{C}$-module category
$\mathcal{M}^{\prime}$ gives rise to a centrally pointed $\mathcal{C}$-module
category $\hat{\mathcal{M}}^{\prime}$, see Definition 5.7.

######  Proposition 7.3.

Given $\mathcal{M}\in\mathcal{CBOS}(\mathcal{C})$ and
$\mathcal{M}^{\prime}\in\operatorname{Mod}^{c}_{*}(\mathcal{C})$, there is a
bijective correspondence between the morphisms
$\mathcal{M}\to\hat{\mathcal{M}}^{\prime}$ in $\mathcal{CBOS}(\mathcal{C})$
and the ucp-multipliers $\mathcal{M}\to\mathcal{M}^{\prime}$ of pointed right
$\mathcal{C}$-modules.

###### Proof.

Let $F$ be a ucp-multiplier $\mathcal{M}\to\mathcal{M}^{\prime}$. Given
$T\in\mathcal{M}(m\stmryolt V,m\stmryolt W)$ and $U\in\mathcal{C}$, we define
$\hat{F}_{V,W}(T)_{U}\in\mathcal{M}^{\prime}(m^{\prime}\stmryolt U\stmryolt
V,m^{\prime}\stmryolt U\stmryolt W)$ by

$\hat{F}_{V,W}(T)_{U}=F_{UV,UW}((\sigma_{U}\stmryolt\mathrm{id}_{W})(\mathrm{id}_{U}\stmryogt
T)(\sigma_{U}^{-1}\stmryolt\mathrm{id}_{V}))=F_{UV,UW}(\Sigma_{U;V,W}(T)).$

This is natural in $U$ by construction, hence we get a multiplier
$\hat{F}\colon\mathcal{M}\to\hat{\mathcal{M}}^{\prime}$. This is completely
positive because $\sigma_{U}$ is unitary.

Moreover, $\Sigma^{\prime}_{U;V,W}$ on $\hat{\mathcal{M}}^{\prime}$ is the
left module structure map $S\mapsto\mathrm{id}_{U}\stmryogt S$. Thus, (7.1)
for $\hat{F}$ becomes

$\hat{F}_{UV,UW}(\Sigma_{U;V,W}(T))=\mathrm{id}_{U}\stmryogt\hat{F}_{V,W}(T)\quad(T\in\mathcal{M}(m\stmryolt
V,m\stmryolt W)).$

This follows from the multiplicativity of $\sigma_{U}$ in $U$.

In the other direction, given a ucp-multiplier
$F\colon\mathcal{M}\to\hat{\mathcal{M}}^{\prime}$ (satisfying (7.1)), we get a
ucp-multiplier $\check{F}\colon\mathcal{M}\to\mathcal{M}^{\prime}$ by
$\check{F}_{V,W}(T)=F_{V,W}(T)_{1}$ for $T\in\mathcal{M}(m\stmryolt
V,m\stmryolt W)$.

These correspondences are inverse to each other: since
$\sigma_{1}=\mathrm{id}_{m}$, if $F\colon\mathcal{M}\to\mathcal{M}^{\prime}$
is a right-module ucp-multiplier, $G=\hat{F}$ satisfies

$\check{G}_{V,W}=[\hat{F}_{V,W}]_{1}=F_{V,W}.$

Conversely, suppose $F\colon\mathcal{M}\to\hat{\mathcal{M}}^{\prime}$ is a
central ucp-multiplier, and write $G=\check{F}$. Then we have

$[\hat{G}_{V,W}(T)]_{U}=\check{F}_{UV,UW}(\Sigma_{UV,UW}(T))=F_{UV,UW}(\Sigma_{U;V,W}(T))_{1}=\Sigma^{\prime}_{U;V,W}(F_{V,W}(T))_{1}\\\
=[\mathrm{id}_{U}\stmryogt F_{V,W}(T)]_{1}=[F_{V,W}(T)]_{U}$

for $T\in\mathcal{M}(m\stmryolt V,m\stmryolt W)$. ∎

Now, consider the notions of _injectivity_ and _injective envelopes_ for
centrally pointed bimodule categories analogously to Definitions 4.2 and 4.12,
by restricting to central ucp-multipliers. In other words, we consider only
central ucp-extensions of central ucp-multipliers.

######  Theorem 7.4.

The categories $\hat{\mathcal{M}}_{\mathcal{B}(H)}$ are injective in
$\mathcal{CBOS}(\mathcal{C})$ for $H\in\mathrm{Hilb}(\mathcal{C})$.

###### Proof.

By Proposition 7.3, the central ucp-multipliers
$\mathcal{M}^{\prime}\to\hat{\mathcal{M}}_{\mathcal{B}(H)}$ bijectively
correspond to the right module ucp-multipliers
$\mathcal{M}^{\prime}\to\mathcal{M}_{\mathcal{B}(H)}$. We then get the claim
by Theorem 4.10. ∎

As a consequence, any object in $\mathcal{CBOS}(\mathcal{C})$ embeds into a
injective object.

######  Remark 7.5.

As the above proof shows, $\hat{\mathcal{M}}$ is injective as a centrally
pointed bimodule category whenever $\mathcal{M}$ is an injective pointed right
$\mathcal{C}$-module category.

######  Remark 7.6.

Consider a right $\mathcal{C}$-module C∗-category $\mathcal{M}$ and an
embedding of $\mathcal{M}$ into $\mathcal{M}_{\mathcal{B}(H)}$, for some
$H\in\mathrm{Hilb}(\mathcal{C})$. The composition

$\mathcal{M}\to\mathcal{M}_{\mathcal{B}(H)}\to\hat{\mathcal{M}}_{\mathcal{B}(H)}$

is an analogue of the _Poisson transform_ considered in [kksv1][HHN1], which
we call a categorical Poisson transform.

######  Definition 7.7.

A central bimodule functor $F\colon\mathcal{M}\to\mathcal{N}$ is said to be
_rigid_ when $\mathrm{Id}_{\mathcal{N}}$ is the only central ucp multiplier
$G\colon\mathcal{N}\to\mathcal{N}$ satisfying $GF=F$.

######  Theorem 7.8.

Every object in $\mathcal{CBOS}(\mathcal{C})$ has an injective envelope
$\mathcal{I}_{\mathcal{CBOS}}(\mathcal{M})$. Moreover the embedding
$i\colon\mathcal{M}\to\mathcal{I}_{\mathcal{CBOS}}(\mathcal{M})$ is rigid.

###### Proof.

The first claim can be proved with a slight modification of the proof of
Theorem 4.13. Start with a centrally pointed bimodule category
$(\mathcal{M},m)$, and embed it into $\hat{\mathcal{M}}_{\mathcal{B}(H)}$ for
some $H\in\mathrm{Hilb}(\mathcal{C})$.

Then consider the semigroup $S$ of central ucp multipliers from
$\hat{\mathcal{M}}_{\mathcal{B}(H)}$ to itself that are identity on the image
of $\mathcal{M}$. This is still a closed convex set for the weak∗ topology
with respect to the presentation (5.3), ensuring then the existence of a
minimal idempotent $\Phi\in S$. Consider the image of $\Phi$, endowed with the
Choi–Effros product. Since $\Phi$ is central, its image inherits the centrally
pointed bimodule structure of $\hat{\mathcal{M}}_{\mathcal{B}(H)}$. This is a
model of $\mathcal{I}=\mathcal{I}_{\mathcal{CBOS}}(\mathcal{M})$.

The rest of the proof is also a close analogue of the usual one for injective
envelope of C∗-algebras. Let $\Psi\colon\mathcal{I}\to\mathcal{I}$ be a
central ucp multiplier with $\Psi i=i$, and put $\Phi^{\prime}=\Psi\Phi$. Then
by the complete contractivity of $\Psi$, we have

$\left\|\Phi^{\prime}(T)\right\|\leq\left\|\Phi(T)\right\|$

for any $U,V\in\mathcal{C}$ and any
$T\in\hat{\mathcal{M}}_{\mathcal{B}(H)}(U,V)$. By the minimality of $\Phi$, we
obtain $\Phi^{\prime}=\Phi$, which means $\Psi=\mathrm{Id}_{\mathcal{I}}$. ∎

###  7.2. Boundary theory for centrally pointed bimodule categories

Let us quickly recall the relevant concepts from [HHN1]: A _$\mathcal{C}$
-tensor category_ is a C∗-tensor category $\mathcal{D}$ endowed with a
dominant faithful C∗-tensor functor $F\colon\mathcal{C}\to\mathcal{D}$. (For
simplicity we assume that $\mathcal{C}$ is a subcategory of $\mathcal{D}$, and
that any object of $\mathcal{D}$ is a subobject of some $U\in\mathcal{C}$.) A
_$\mathcal{C}$ -linear transformation_
$\Theta\colon\mathcal{D}_{1}\to\mathcal{D}_{2}$ between $\mathcal{C}$-tensor
categories is given by a family of linear maps

$\Theta_{U,V}\colon\mathcal{D}_{1}(U,V)\to\mathcal{D}_{2}(U,V)\quad(U,V\in\mathcal{C})$

satisfying

$\displaystyle\Theta_{U_{2},V_{1}}(S_{1}TS_{2})=S_{1}\Theta_{V_{2},U_{1}}(T)S_{2}\quad(T\in\mathcal{D}_{1}(V_{2},U_{1}),S_{i}\in\mathcal{C}(U_{i},V_{i})),$
$\displaystyle\Theta_{Z\otimes U\otimes Y,Z\otimes V\otimes
Y}(\mathrm{id}_{Z}\otimes
T\otimes\mathrm{id}_{Y})=\mathrm{id}_{Z}\otimes\Theta_{U,V}(T)\otimes\mathrm{id}_{Y}\quad(T\in\mathcal{D}_{1}(U,V)).$

A $\mathcal{C}$-linear transformation $\Theta$ is _ucp_ if $\Theta_{U,U}$ is
ucp for all $U\in\mathcal{C}$. The _$\mathcal{C}$ -injectivity_ for
$\mathcal{C}$-tensor categories is defined using this class of maps. Finally,
the _Furstenberg–Hamana boundary_ $\partial_{\mathrm{FH}}(\mathcal{C})$ of
$\mathcal{C}$ is an injective $\mathcal{C}$-tensor category such that any
$\mathcal{C}$-linear transformation
$\partial_{\mathrm{FH}}(\mathcal{C})\to\mathcal{D}$ is completely isometric.
Concretely, $\partial_{\mathrm{FH}}(\mathcal{C})$ can be realized inside
$\hat{\mathcal{C}}$ using Proposition 2.1 applied to the semigroup of
$\mathcal{C}$-linear ucp transformations from $\hat{\mathcal{C}}$ to itself.

Now, observe that any $\mathcal{C}$-tensor category $\mathcal{D}$ has a
canonical structure of centrally pointed $\mathcal{C}$-bimodule category: as
the generating object we take the tensor unit, and the central structure
$\sigma_{U}\colon U\otimes 1_{\mathcal{C}}\to 1_{\mathcal{C}}\otimes U$ is
given by the composition of structure morphisms of the monoidal unit. Under
this correspondence, a ucp $\mathcal{C}$-linear transformations between
$\mathcal{C}$-tensor categories is exactly a ucp central $\mathcal{C}$-linear
transformation. This motivates the following definition of boundary bimodules.

######  Definition 7.9.

A centrally pointed $\mathcal{C}$-bimodule category $\mathcal{M}$ is called a
_boundary_ category when any central ucp multiplier
$\mathcal{M}\to\mathcal{N}$ to any centrally pointed bimodule category
$\mathcal{N}$ is completely isometric.

######  Theorem 7.10.

Every boundary $\mathcal{M}$ is a centrally pointed subcategory of
$\partial_{\mathrm{FH}}(\mathcal{C})$.

###### Proof.

By Proposition 5.6, we have central embedding of $\mathcal{C}$ into
$\mathcal{M}$. Since $\mathcal{M}$ is a boundary and
$\partial_{FH}(\mathcal{C})$ is injective, we get a completely isometric
central ucp multiplier $i\colon\mathcal{M}\to\partial_{FH}(\mathcal{C})$.

Take an embedding $\pi$ of $\mathcal{M}$ to
$\hat{\mathcal{M}}_{\mathcal{B}(H)}$ for some
$H\in\mathrm{Hilb}(\mathcal{C})$, as a centrally pointed bimodule category. By
the injectivity of $\hat{\mathcal{M}}_{\mathcal{B}(H)}$, the multiplier $i$
extends to a ucp central multiplier
$\Phi\colon\partial_{\mathrm{FH}}(\mathcal{C})\to\hat{\mathcal{M}}_{\mathcal{B}(H)}$.
Conversely, by the injectivity of $\partial_{FH}(\mathcal{C})$, we also get a
ucp central map
$\Psi\colon\hat{\mathcal{M}}_{\mathcal{B}(H)}\to\partial_{\mathrm{FH}}(\mathcal{C})$.

The composition $\Psi\Phi$ is a $\mathcal{C}$-linear ucp transformation on
$\partial_{\mathrm{FH}}(\mathcal{C})$. Thus, $\Phi$ must be completely
isometric. Moreover, by the above construction of $\partial_{FH}(\mathcal{C})$
via Proposition 2.1, $\Psi\Phi$ must be given by the identity functor. This
implies that $E=\Phi\Psi$ is a conditional expectation on
$\hat{\mathcal{M}}_{\mathcal{B}(H)}$, with image
$\Phi(\partial_{\mathrm{FH}}(\mathcal{C}))$. As before, the Choi–Effros
product

$\Phi(g)\cdot\Phi(f)=E(\Phi(g)\Phi(f))$

turns this image into a category. Since we worked with central ucp central
multipliers throughout, this category is again centrally pointed.

By construction $\pi(\mathcal{M})$ is contained in
$\Phi(\partial_{FH}(\mathcal{C}))$. Moreover, the Choi–Effros product
coincides with the original product of $\pi(\mathcal{M})$, by

$\pi(g)\cdot\pi(f)=E(\pi(g)\pi(f))=\Phi(\Psi(\pi(gf))=\Phi(i(gf))=\pi(gf)=\pi(g)\pi(f).$

Thus $\pi$ defines a functor
$\mathcal{M}\to\Phi(\partial_{\mathrm{FH}}(\mathcal{C}))$, which can be
upgraded to a central functor of centrally pointed bimodules. By the boundary
property of $\mathcal{M}$, this must be an embedding. ∎

######  Corollary 7.11.

The boundary objects in $\mathcal{CBOS}(\mathcal{C})$ are exactly the
centrally pointed $\mathcal{C}$-bimodule subcategories of
$\partial_{\mathrm{FH}}(\mathcal{C})$.

##  Appendix A Intrinsic characterization of C∗-algebra objects

Definition 2.7 does not give an intrinsic condition on C∗-algebra objects.
Here we give an intrinsic characterization of such structures, without
directly referring to the corresponding module category.

When $A$ is an algebra object in $\operatorname{Vec}(\mathcal{C})$, let us
denote the product of $a\in A(U)$ and $b\in A(V)$ by $a\cdot b\in A(V\otimes
U)$. We also write $A_{0}=A(1_{\mathcal{C}})$.

Now, let $A$ be a _$*$ -algebra object_ in $\operatorname{Vec}(\mathcal{C})$
[jp1]. This means that there is a family of conjugate linear maps

$A(U)\to A(\bar{U}),\quad a\mapsto a^{\natural}$

(denoted by $j_{U}$ in [jp1]) satisfying the following conditions:

  * •

the naturality

$A(T)(a)^{\natural}=A(T^{*\vee})(a^{\natural})\quad(T\in\mathcal{C}(U,V),a\in
A(V)),$ (A.1)

  * •

the involutivity $(a^{\natural})^{\natural}=a$ up to the canonical isomorphism
$\bar{\bar{U}}\simeq U$,

  * •

the unitality***There seems to be a typo in [jp1].
$1_{A_{0}}^{\natural}=1_{A_{0}}$ in $A_{0}$ up to the canonical choice
$\bar{1}_{\mathcal{C}}=1_{\mathcal{C}}$, and

  * •

the antimultiplicativity $a^{\natural}\cdot b^{\natural}=(b\cdot
a)^{\natural}$ up to the natural isomorphism
$\bar{U}\otimes\bar{V}\simeq\overline{V\otimes U}$.