Applying the definition of $\mathscr{C}(V^{\Omega})$ together with the
pentagonal equation satisfied by $V^{\Omega}$ from Theorem 3.2.8, we write the
following:

$\begin{split}\overline{span}\\{\mathscr{C}(V^{\Omega})\mathscr{C}(V^{\Omega})\\}&=\overline{span}\\{(id\otimes\eta)(\Sigma
V^{\Omega})\cdot(id\otimes\eta^{\prime})(\Sigma V^{\Omega})\ |\
\eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(id\otimes\eta)(\Sigma
V^{\Omega})\cdot(\eta^{\prime}\otimes id)(V^{\Omega}\Sigma)\ |\
\eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(\eta^{\prime}\otimes
id\otimes\eta)(\Sigma_{23}V^{\Omega}_{23}V^{\Omega}_{12}\Sigma_{12})\ |\
\eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(\eta^{\prime}\otimes
id\otimes\eta)(\Sigma_{23}V^{\Omega}_{12}V^{\Omega}_{13}(V_{\mathbb{G}})_{23}\Sigma_{12})\
|\ \eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(\eta^{\prime}\otimes\eta\otimes
id)(V^{\Omega}_{12}V^{\Omega}_{13}(V_{\mathbb{G}})_{23}\Sigma_{12}\Sigma_{23})\
|\ \eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(\eta^{\prime}\otimes\eta\otimes
id)(V^{\Omega}_{12}V^{\Omega}_{13}(V_{\mathbb{G}})_{23}\Sigma_{23}\Sigma_{13})\
|\ \eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(\eta^{\prime}\otimes\eta\otimes
id)(V^{\Omega}_{13}(V_{\mathbb{G}})_{23}\Sigma_{23}\Sigma_{13})\ |\
\eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(id\otimes\eta\otimes\eta^{\prime})(\Sigma_{13}V^{\Omega}_{13}(V_{\mathbb{G}})_{23}\Sigma_{23})\
|\ \eta,\eta^{\prime}\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(id\otimes\eta)(\Sigma V^{\Omega})\ |\
\eta\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\mathscr{C}(V^{\Omega}).\end{split}$

To see that $\mathscr{C}(V^{\Omega})$ acts non-degenerately, take
$\zeta\in(\mathscr{C}(V^{\Omega})L^{2}(\mathbb{G}))^{\perp}$. For every
$\xi,\xi^{\prime},x\in L^{2}(\mathbb{G})$ we have
$0=\langle\zeta,(id\otimes\omega_{\xi,\xi^{\prime}}(\Sigma
V^{\Omega})(x))\rangle=\langle\zeta\otimes\xi,\Sigma
V^{\Omega}(x\otimes\xi^{\prime})\rangle$. Since $\Sigma V^{\Omega}$ is a
unitary in $\mathcal{B}(L^{2}(G)\otimes L^{2}(G))$ so surjective, the above
equality implies $(\mathscr{C}(V^{\Omega})L^{2}(\mathbb{G}))^{\perp}=(0)$.

  2. ii)

By Theorem 3.2.13, the set
$\\{(id\otimes\omega_{\Lambda(u^{y}_{kl}),\Lambda(a)})(\Sigma V^{\Omega})\ |\
y\in Irr(\mathbb{G},\Omega),k,l=1,\ldots,n_{y},a\in Pol(\mathbb{G})\\}$ is
dense in $\mathscr{C}(V^{\Omega})$. Given $u^{x}_{ij}\in
Pol(\mathbb{G},\Omega)$ we compute with the help of the twisted orthogonality
relations from Theorem 3.2.13 that for $\zeta\in L^{2}(\mathbb{G})$:

$\begin{split}\langle\zeta,(id\otimes\omega_{\Lambda(u^{y}_{kl}),\Lambda(a)})(\Sigma
V^{\Omega})\Lambda(u^{x}_{ij})\rangle&=\langle\Lambda(u^{y}_{kl})\otimes\zeta,V^{\Omega}(\Lambda(u^{x}_{ij})\otimes\Lambda(a))\rangle\\\
&=\langle\Lambda(u^{y}_{kl})\otimes\zeta,\Delta_{\Omega}(u^{x}_{ij})(\xi_{\mathbb{G}}\otimes\Lambda(a))\rangle\\\
&=\langle\Lambda(u^{y}_{kl})\otimes\zeta,\big{(}\overset{n_{x}}{\underset{r=1}{\sum}}u^{x}_{ir}\otimes
u^{x}_{rj}\big{)}(\xi_{\mathbb{G}}\otimes\Lambda(a))\rangle\\\
&=\overset{n_{x}}{\underset{r=1}{\sum}}h_{\mathbb{G}}\big{(}(u^{y}_{kl})^{*}u^{x}_{ir}\big{)}\langle\zeta,\lambda(u^{x}_{rj})\Lambda(a)\rangle\\\
&=F^{x}_{k}\delta_{x,y}\delta_{k,i}\langle\zeta,\lambda(u^{x}_{lj})\Lambda(a)\rangle,\end{split}$

so with respect to the orthonormal basis of Theorem 3.2.14 we have:

$(F^{x}_{k})^{-1}(id\otimes\omega_{\Lambda(u^{y}_{kl}),\Lambda(a)})(\Sigma
V^{\Omega})(F_{i}^{x})^{-1/2}\Lambda(u_{ij}^{x})=\delta_{x,y}\delta_{k,i}(F_{l}^{x})^{-1/2}\lambda(u^{x}_{lj})\Lambda(a).$
(4.3)

Assume now that $\Omega$ is regular. Taking $a=1$, we see from the above that
$(id\otimes\omega_{\Lambda(u^{y}_{kl}),\xi_{\mathbb{G}}})(\Sigma V^{\Omega})$
is of the form $T^{x}_{rs}\otimes 1$ for some non-zero operator $T^{x}_{rs}$
with respect to the model $L^{2}(\mathbb{G})\cong\oplus_{x\in
Irr(\mathbb{G},\Omega)}H_{x}\otimes\overline{H_{x}}$. As this operator needs
to be compact, we see that all $n_{x}$ need to be finite, hence $\Omega$ is of
finite type.

Conversely, if $\Omega$ is of finite type, we see from the above computation
(and the fact that the operators $\Lambda(b)\mapsto\Lambda(ba)$ are bounded
for $a\in Pol(\mathbb{G})$) that
$\mathscr{C}(V^{\Omega})\subset\mathcal{K}(L^{2}(\mathbb{G}))$. To see that
this is an equality, it is sufficient to show that the commutant
$\mathscr{C}(V^{\Omega})^{\prime}=\mathbb{C}$. Now if
$x\in\mathscr{C}(V^{\Omega})^{\prime}$, then in particular $x$ commutes with
all $(id\otimes\omega_{\Lambda(u^{y}_{kl}),\xi_{\mathbb{G}}})(\Sigma
V^{\Omega})$, and it follows from the above computation and (4.1) that $x\in
l^{\infty}(\widehat{\mathbb{G}},\Omega)$. On the other hand, the computation
above also shows that then
$x\in\rho(L^{\infty}(\mathbb{G}))^{\prime}=L^{\infty}(\mathbb{G})$, hence
$x\in l^{\infty}(\widehat{\mathbb{G}},\Omega)\cap L^{\infty}(\mathbb{G})$. But
by Remark 4.1.6, this implies $(W^{\Omega})^{*}(1\otimes x)W^{\Omega}=1\otimes
x$, hence ${}_{\Omega}\Delta_{\Omega^{*}}(x)=1\otimes x$ and so
$x\in\mathbb{C}$.

  3. iii)

By (4.1) we see that $C^{*}_{r}(\mathbb{G},\Omega)$ is formed by compact
operators if and only if $\Omega$ is of finite type. Hence
$\overline{span}\\{U_{\mathbb{G}}C(\mathbb{G})U_{\mathbb{G}}\cdot
C^{*}_{r}(\mathbb{G},\Omega)\\}\subset\mathcal{K}(L^{2}(\mathbb{G}))$ if and
only if $\Omega$ is of finite type. To see that this is an equality if
$\Omega$ is of finite type, we can follow as similar strategy as in ii).
Alternatively, conjugating with $\widehat{J}$ and taking into account Remark
4.1.6, we see that we have an inclusion
$C_{r}^{*}(\Omega,\mathbb{G})JC(\mathbb{G})J\subseteq\mathcal{K}(L^{2}(\mathbb{G}))$,
and this must be an equality by the discussion following [32, Definition 2.9].
Conjugating back with $\widehat{J}$, we see that
$\overline{span}\\{U_{\mathbb{G}}C(\mathbb{G})U_{\mathbb{G}}\cdot
C^{*}_{r}(\mathbb{G},\Omega)\\}=\mathcal{K}(L^{2}(\mathbb{G}))$.

∎

######  4.1.10 Remarks.

  1. i)

As follows from the end of the above proof, our notion of regularity indeed
coincides with the notion of regularity of a $2$-cocycle as introduced in [32,
Definition 2.9].

  2. ii)

Up to unitary conjugation, our operator $V^{\Omega}$ also coincides with the
operator $\Sigma(V_{21}^{1})^{*}\Sigma$ with $V_{21}^{1}$ as it appears in [3,
Proposition 2.44]. It hence follows from that result that $\Omega$ is regular
if and only if $\mathbb{G}_{\Omega}$ is regular, and hence that
$\mathbb{G}_{\Omega}$ is never regular if $\Omega$ is not of finite type. It
is unclear at the moment if in general $\mathbb{G}_{\Omega}$ is semi-regular
(this holds in all known cases). By [3, Proposition 2.44], this is equivalent
to $\mathcal{K}(L^{2}(\mathbb{G}))\subseteq\mathscr{C}(V^{\Omega})$.

The following proposition is a straightforward adaptation of [32, Proposition
2.3] to our setting.

######  4.1.11 Proposition.

Let $\mathbb{G}$ be a compact quantum group and $\Omega$ a $2$-cocycle. Then
the twisted reduced $C^{*}$-algebra $C^{*}_{r}(\mathbb{G},\Omega)$ is a left
$\widehat{\mathbb{G}}$-$C^{*}$-algebra with action
$\widehat{\mathbb{G}}\overset{\alpha^{\Omega}}{\curvearrowright}C^{*}_{r}(\mathbb{G},\Omega)$
defined by:

$\alpha^{\Omega}(x)=\Sigma(V^{\Omega})^{*}(1\otimes x)V^{\Omega}\Sigma,$

for all $x\in C^{*}_{r}(\mathbb{G},\Omega)$.

###### Proof.

Given $x\in C^{*}_{r}(\mathbb{G},\Omega)$, assume without loss of generality
that $x:=(id\otimes\eta)(V^{\Omega})$ for some
$\eta\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}$. Then, with the help of the
pentagonal equation satisfied by $V^{\Omega}$ (see Theorem-Definition 3.2.8),
we have $\alpha^{\Omega}(x)=(id\otimes
id\otimes\eta)V^{\Omega}_{23}(V_{\mathbb{G}})_{13}\in\widetilde{M}(c_{0}(\widehat{\mathbb{G}})\otimes
C^{*}_{r}(\mathbb{G},\Omega))$, which shows that $\alpha^{\Omega}$ is well-
defined as a (injective) $*$-homomorphism
$C^{*}_{r}(\mathbb{G},\Omega)\longrightarrow\widetilde{M}(c_{0}(\widehat{\mathbb{G}})\otimes
C^{*}_{r}(\mathbb{G},\Omega)$). Next, we are going to show that
$\alpha^{\Omega}$ defines an action of $\widehat{\mathbb{G}}$ on
$C^{*}_{r}(\mathbb{G},\Omega)$. On the one hand, since the elements of the
form $x=(id\otimes\eta)(V^{\Omega})\in C^{*}_{r}(\mathbb{G},\Omega)$ with
$\eta\in C(\mathbb{G})_{*}$ are dense in $C^{*}_{r}(\mathbb{G},\Omega)$, then
the previous computation shows also that the subspace
$\alpha^{\Omega}(C^{*}_{r}(\mathbb{G},\Omega))(c_{0}(\widehat{\mathbb{G}})\otimes
1)$ is dense in $c_{0}(\widehat{\mathbb{G}})\otimes
C^{*}_{r}(\mathbb{G},\Omega)$. On the other hand, applying
$id\otimes\alpha^{\Omega}$ and $\widehat{\Delta}\otimes id$ to the above
expression, we obtain $(id\otimes\alpha^{\Omega})\alpha^{\Omega}(x)=(id\otimes
id\otimes
id\otimes\eta)V^{\Omega}_{34}(V_{\mathbb{G}})_{24}(V_{\mathbb{G}})_{14}=(\widehat{\Delta}\otimes
id)\alpha^{\Omega}(x)$ by a direct computation. ∎

######  4.1.12 Remark.

By the formula $\beta^{\Omega}(x):=(V^{\Omega})^{*}(1\otimes x)V^{\Omega}$ for
$x\in C^{*}_{r}(\mathbb{G},\Omega)$, we can also view
$(C^{*}_{r}(\mathbb{G},\Omega),\beta^{\Omega})$ as a (right)
$\widehat{\mathbb{G}}^{cop}$-$C^{*}$-algebra.

###  Twisted crossed products

In this section, we consider twisted crossed products, cf. again [32]. We
start however from the twisted side, and work our way back to the original
compact quantum group. As to spare the reader a battle with conventions, we
spell out some of the details particular to our setting.

######  4.2.1 Definition.

A (measurable) right twisted dynamical system is the data
$(\mathbb{G},A,\delta,\Omega)$ where $\mathbb{G}$ is a compact quantum group,
$\Omega$ is a $2$-cocycle of finite type on $\mathbb{G}$, $A$ is a
$C^{*}$-algebra and $\delta:A\longrightarrow A\otimes C(\mathbb{G}_{\Omega})$
is right action of $\mathbb{G}_{\Omega}$.

We write $A\overset{(\delta,\Omega)}{\curvearrowleft}\mathbb{G}$, and say that
$\delta$ is a twisted action of $\mathbb{G}$ on $A$ with respect to $\Omega$
or simply that $\delta$ is an $\Omega$-action of $\mathbb{G}$ on $A$. We say
that $(A,\delta)$ is a right $\Omega$-$\mathbb{G}$-$C^{*}$-algebra if moreover
$\delta$ is injective.

######  4.2.2 Definition.

Let $(\mathbb{G},A,\delta,\Omega)$ be a twisted dynamical system. The twisted
reduced crossed product of $A$ by $\mathbb{G}$ with respect to
$(\delta,\Omega)$, denoted by
$A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$, is the $C^{*}$-algebra
defined by:

$A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}:=C^{*}\langle(id\otimes\lambda)\delta(A)(1\otimes
C^{*}_{r}(\mathbb{G},\Omega))\rangle\subset\mathcal{L}_{A}(A\otimes
L^{2}(\mathbb{G})).$

######  4.2.3 Note.

To lighten the notation, we will omit the representation $\lambda$ appearing
in the definition of $A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$.

######  4.2.4 Lemma.

We have
$A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}=\overline{span}\\{\delta(A)(1\otimes
C^{*}_{r}(\mathbb{G},\Omega))\\}$.

###### Proof.

It is enough to show that

$[(1\otimes C^{*}_{r}(\mathbb{G},\Omega))\delta(A)]\subset[\delta(A)(1\otimes
C^{*}_{r}(\mathbb{G},\Omega))]$ (4.4)

But, using the implementation of ${}_{\Omega}\Delta=\Omega\cdot\Delta$ in
terms of $V_{\mathbb{G}}$ and $V^{\Omega}$, the compatibility of $\delta$ with
$\Delta$ as a _twisted_ action of $\mathbb{G}$ on $A$ yields that
$(1\otimes(id\otimes\eta)(V^{\Omega}))\delta(a)=(id\otimes
id\otimes\eta)\big{(}(\delta\otimes id)\delta(a)(1\otimes
V^{\Omega})\big{)}=\lim\sum_{i}\delta(a_{i})(1\otimes(id\otimes\eta\cdot
u_{i})(V^{\Omega}))$, for all $a\in A$ and all
$\eta\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}$, where
$\delta(a)=\lim\sum_{i}a_{i}\otimes u_{i}$. This proves (4.4). ∎

As a consequence, the maps $A\longrightarrow\mathcal{L}_{A}(A\otimes
L^{2}(\mathbb{G}))$ and
$C^{*}_{r}(\mathbb{G},\Omega)\longrightarrow\mathcal{L}_{A}(A\otimes
L^{2}(\mathbb{G}))$ given by $a\mapsto\delta(a)$ and $x\mapsto 1\otimes x$,
send $A$ and $C^{*}_{r}(\mathbb{G},\Omega)$ respectively onto non-degenerate
$C^{*}$-subalgebras of $M(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G})$.

######  4.2.5 Example.

We note that both Definition 4.2.1 and Definition 4.2.2 are natural dual
versions of the classical framework. Let $G$ be a classical compact group and
$A$ a unital $C^{*}$-algebra. Given a map $\omega:G\times
G\longrightarrow\mathcal{U}(A)$, a $\omega$-action of $G$ on $A$ is a map
$\alpha:G\longrightarrow Aut(A)$ such that
$\alpha_{g_{1}}\circ\alpha_{g_{2}}=Ad_{\omega(g_{1},g_{2})}\circ\alpha_{g_{1}g_{2}}$
for all $g_{1},g_{2}\in G$;
$\omega(x,y)\omega(xy,z)=\alpha_{x}\big{(}\omega(y,z)\big{)}\omega(x,yz)$ for
all $x,y,z\in G$ and $\omega(x,e)=1=\omega(e,x)$ for all $x\in G$. Consider
the vector space of continuous functions on $G$ with values in $A$ equipped
with the usual point-wise operations, $C(G,A)$. We define the _twisted
convolution product on $C(G,A)$ with respect to $\omega$_ by:

$(f\underset{\omega}{\ast}g)(x):=\int_{G}f(y)\alpha_{y}\big{(}g(y^{-1}x)\big{)}\omega(y,y^{-1}x)dy,$

for all $f,g\in C(G,A)$ and $x\in G$. We define the _twisted involution on
$C(G,A)$ with respect to $\omega$_ by:

$f^{\underset{\omega}{*}}(x):=\omega(x,x^{-1})\alpha_{x}\big{(}f(x^{-1})^{*}\big{)},$

for all $f\in C(G,A)$ and $x\in G$. Straightforward computations show that
$C(G,A)$ is a $*$-algebra with the product and involution above. Next, by
applying standard arguments (analogous to the untwisted case) we find that
$(L^{2}(G)\otimes H_{0},\pi,\lambda^{\omega})$ is a _faithful_ covariant
$\omega$-representation of $(A,\alpha)$, where
$\pi:A\longrightarrow\mathcal{B}(L^{2}(G)\otimes H_{0})$ is such that
$\pi(a)(f\otimes\xi)(x):=\Big{(}f\otimes\pi_{0}(\alpha_{x^{-1}}(a))(\xi)\Big{)}(x)$,
for all $a\in A$, $f\in L^{2}(G)$, $\xi\in H_{0}$, $x\in G$; and
$(\pi_{0},H_{0})$ is any faithful representation of $A$. Thus we define the
_reduced twisted crossed product_ by
$A\underset{r,(\alpha,\omega)}{\rtimes}G:=\overline{(\pi,\lambda^{\omega})(C(G,A))}^{||\cdot||_{\mathcal{B}(L^{2}(G,H_{0}))}}$
and one shows that this definition does not depend on the faithful
representation $\pi_{0}$. Alternatively, we have
$G\underset{r,(\alpha,\omega)}{\ltimes}A:=\overline{span}\\{\alpha(A)(1\otimes
C^{*}_{r}(G,\omega))\\}$, where the action
$G\overset{\alpha}{\curvearrowright}A$ is viewed as a map
$\alpha:A\longrightarrow M(A\otimes
C(G))\hookrightarrow\mathcal{L}_{A}(A\otimes L^{2}(G))$.

Specially interesting is the case when $\omega$ is scalar valued (that is,
$\omega$ takes its values on $S^{1}\cong S^{1}\cdot 1_{A}$), then $\omega$ is
a usual (normalized) $2$-cocycle on $G$ and $\alpha$ a group homomorphism. In
this case, we observe that these constructions yield the usual
$C^{*}$-algebras: if $\omega=1$, we have
$A\underset{r,(\alpha,\omega)}{\rtimes}G=A\underset{r,\alpha}{\rtimes}G$; if
$\alpha$ is trivial, we have $A\underset{r,(\alpha,\omega)}{\rtimes}G=A\otimes
C^{*}_{r}(G,\omega)$.

######  4.2.6 Proposition-Definition.

Let $(\mathbb{G},A,\delta,\Omega)$ be a twisted dynamical system. The twisted
reduced crossed product, $A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$,
is a $\widehat{\mathbb{G}}^{cop}$-$C^{*}$-algebra with action
$A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\overset{\widehat{\delta}}{\curvearrowleft}\widehat{\mathbb{G}}^{cop}$
such that:

$\widehat{\delta}\big{(}\delta(a)(1\otimes x)\big{)}=(\delta(a)\otimes
1)\big{(}1\otimes(V^{\Omega})^{*}(1\otimes x)V^{\Omega}\big{)},$

for all $a\in A$ and all $x\in C^{*}_{r}(\mathbb{G},\Omega)$. The action
$A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\overset{\widehat{\delta}}{\curvearrowleft}\widehat{\mathbb{G}}^{cop}$
is called _twisted dual action of $(\delta,\Omega)$_ or _$\Omega$ -dual action
of $\delta$_.

###### Proof.

Let us consider the unitary $\widetilde{V}_{\mathbb{G}}$ as in Theorem-
Definition 2.2.6. We are going to show that $\widehat{\delta}$ can be written
as a conjugation by $1\otimes\widetilde{V}_{\mathbb{G}}$. Given $x\in
C^{*}_{r}(\mathbb{G},\Omega)$, assume without loss of generality that
$x:=(id\otimes\eta)(V^{\Omega})$ for some $\eta\in C(\mathbb{G})_{*}$. On the
one hand, using the pentagonal equation satisfied by $V^{\Omega}$ (see
Theorem-Definition 3.2.8) and Lemma 3.2.10, a direct computation shows that
$(V^{\Omega})^{*}(1\otimes x)V^{\Omega}=\widetilde{V}_{\mathbb{G}}(x\otimes
1)\widetilde{V}_{\mathbb{G}}^{*}$. On the other hand, Theorem-Definition 2.2.6
guarantees that $\widetilde{V}_{\mathbb{G}}(y\otimes
1)\widetilde{V}_{\mathbb{G}}^{*}=y\otimes 1$, for all $y\in
C(\mathbb{G})^{\prime\prime}=L^{\infty}(\mathbb{G})\supseteq
C(\mathbb{G}_{\Omega})$. Combining these two expressions, it is easy to see
that $(\delta(a)\otimes 1)\big{(}1\otimes(V^{\Omega})^{*}(1\otimes
x)V^{\Omega}\big{)}=(1\otimes\widetilde{V}_{\mathbb{G}})\big{(}\delta(a)(1\otimes
x)\otimes 1\big{)}(1\otimes\widetilde{V}_{\mathbb{G}}^{*})$, for all $a\in A$
and all $x\in C^{*}_{r}(\mathbb{G},\Omega)$. In other words, these expressions
show that the formula of the statement defines a (injective) $*$-homomorphism
$A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\longrightarrow\widetilde{M}((A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G})\otimes
c_{0}(\widehat{\mathbb{G}}))$ given precisely by
$\widehat{\delta}(z)=(1\otimes\widetilde{V}_{\mathbb{G}})(z\otimes
1)(1\otimes\widetilde{V}_{\mathbb{G}}^{*})$, for all $z\in
A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$. It remains to show that
$\widehat{\delta}$ defines an action of $\widehat{\mathbb{G}}$ on
$A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$. On the one hand, the
density condition for $\widehat{\delta}$ is obtained as follows:

$\begin{split}\Big{[}\widehat{\delta}(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G})(1\otimes
c_{0}(\widehat{\mathbb{G}}))\Big{]}&=\Big{[}(\delta(A)\otimes
1)\big{(}1\otimes(V^{\Omega})^{*}(1\otimes
C^{*}_{r}(\mathbb{G},\Omega))V^{\Omega}\big{)}(1\otimes 1\otimes
c_{0}(\widehat{\mathbb{G}}))\Big{]}\\\
&\overset{(*)}{=}\Big{[}(\delta(A)\otimes 1)\big{(}1\otimes
C^{*}_{r}(\mathbb{G},\Omega)\otimes
c_{0}(\widehat{\mathbb{G}})\big{)}\Big{]}\\\
&=(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G})\otimes
c_{0}(\widehat{\mathbb{G}})\mbox{,}\end{split}$

where in $(*)$ we use Remark 4.1.12. On the other hand, the compatibility of
$\widehat{\delta}$ with $\widehat{\Delta}^{cop}$ is obtained by a direct
computation using again Remark 4.1.12. ∎

To end this section, let us introduce the following nomenclature for a special
type of quantum dynamical systems.

######  4.2.7 Definition.

Let $\mathbb{G}$ be a compact quantum group and $\Omega$ a $2$-cocycle of
finite type on $\mathbb{G}$. Let $H$ be a Hilbert space. A right twisted
dynamical system $(\mathbb{G},\mathcal{K}(H),\delta,\Omega)$ is called
$\Omega$-inner if there exists an $\Omega$-representation
$u\in\mathcal{B}(H)\overline{\otimes}L^{\infty}(\mathbb{G})$ such that
$\delta(a)=\delta_{u}(a)=u(a\otimes 1)u^{*}$, for all $a\in\mathcal{K}(H)$. In
this case, the data $(\mathbb{G},\mathcal{K}(H),\delta,\Omega,u)$ is called a
_right twisted inner dynamical system_ or _right $\Omega$-inner dynamical
system_.

So a right twisted dynamical system
$(\mathbb{G},\mathcal{K}(H),\delta,\Omega)$ is nothing but a projective left
$\mathbb{G}_{\Omega}$-representation induced from an $\Omega$-representation
of $\mathbb{G}$.

It is well-known that an inner action is exterior equivalent to the trivial
one, so that the corresponding crossed products are isomorphic (see for
instance [19] for more details). The following proposition shows that a
similar phenomenon occurs in the quantum group setting when the action is
$\Omega$-inner.

######  4.2.8 Proposition.

Let $\mathbb{G}$ be a compact quantum group and $\Omega$ a $2$-cocycle of
finite type on $\mathbb{G}$. Let $H$ be a Hilbert space. If
$(\mathbb{G},\mathcal{K}(H),\delta,\Omega,v)$ is a right $\Omega$-inner
dynamical system, then

$\mathcal{K}(H)\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\cong\mathcal{K}(H)\otimes
c_{0}(\widehat{\mathbb{G}}).$

###### Proof.

We can represent
$\mathcal{K}(H)\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$ as the
normclosure of $(1\otimes
C_{r}^{*}(\mathbb{G},\Omega))\delta_{u}(\mathcal{K}(H))$ on $H\otimes
L^{2}(\mathbb{G})$. It is then sufficient to show that $u^{*}[(1\otimes
C_{r}^{*}(\mathbb{G},\Omega))\delta_{u}(\mathcal{K}(H))]u=\mathcal{K}(H)\otimes
c_{0}(\widehat{\mathbb{G}})$. But since
$u_{12}^{*}(V^{\Omega})_{23}u_{12}=u_{13}(V_{\mathbb{G}})_{23}$ as $u$ is an
$\Omega$-representation, we have that $u^{*}[(1\otimes
C_{r}^{*}(\mathbb{G},\Omega))\delta_{u}(\mathcal{K}(H))]u=[\\{(id\otimes
id\otimes\omega)(u_{13}(x\otimes V_{\mathbb{G}}))\mid
x\in\mathcal{K}(H),\omega\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}]$. As $u$ is
the direct sum of finite-dimensional $\Omega$-representations, it follows that
this last set equals $[\\{(id\otimes id\otimes\omega)(x\otimes
V_{\mathbb{G}})\mid
x\in\mathcal{K}(H),\omega\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}]=\mathcal{K}(H)\otimes
c_{0}(\widehat{\mathbb{G}})$. ∎

######  4.2.9 Remark.

Note that if $v$ is a unitary $\Omega^{*}$-representation, we also have an
ordinary right action
$\mathcal{K}(H)\overset{\delta}{\curvearrowleft}\mathbb{G}$ by putting
$\delta(a)=\delta_{v}(a)=v(a\otimes 1)v^{*}$, for all $a\in\mathcal{B}(H)$. We
then say that the action
$\mathcal{K}(H)\overset{\delta}{\curvearrowleft}\mathbb{G}$ is $\Omega$-inner.
An analogous computation as above yields that in this case we have for the
untwisted crossed product that
$\mathcal{K}(H)\underset{r,\delta}{\rtimes}\mathbb{G}\underset{\mathrm{Ad}(v^{*})}{\cong}\mathcal{K}(H)\otimes
C^{*}_{r}(\mathbb{G},\Omega)$. Analogously, from an $\Omega$-representation
$u$ of $\mathbb{G}$, one could consider an ordinary left action
$\mathbb{G}\overset{\delta}{\curvearrowright}\mathcal{K}(H)$ by putting
$\delta(a)=\Sigma(u^{*}(a\otimes 1)u)$, for all $a\in\mathcal{B}(H)$. In this
case, the (untwisted) crossed product is defined as
$\mathbb{G}\underset{r,\delta}{\ltimes}\mathcal{K}(H)=C^{*}\langle\widehat{\lambda}(c_{0}(\widehat{\mathbb{G}})\otimes
1)(\rho\otimes id)\delta(A)\rangle$ where we recall that
$\rho(x)=U_{\mathbb{G}}\lambda(x)U_{\mathbb{G}}$ for $x\in C(\mathbb{G})$.
Then we now have, upon using the regular representation
$W_{\mathbb{G}}=\check{V}_{\mathbb{G}}$ that
$\mathbb{G}\underset{r,\delta}{\ltimes}\mathcal{K}(H)\underset{\mathrm{Ad}(u_{21}U_{\mathbb{G},1})}{\cong}C^{*}_{r}(\Omega,\mathbb{G})\otimes\mathcal{K}(H)$,
where we use the notation of Remark 4.1.6.

###  Twisted Takesaki-Takai duality, twisted descent map and twisted Baaj-
Skandalis duality

######  4.3.1 Definition.

Let $(\mathbb{G},A,\delta,\Omega)$ be a twisted dynamical system. The double
reduced crossed product of $A$ by $\widehat{\mathbb{G}}^{cop}$ with respect to
$\widehat{\delta}$, denoted by
$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}$,
is the $C^{*}$-algebra defined by:

$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}:=C^{*}\langle\widehat{\delta}_{U}(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G})(1\otimes
C(\mathbb{G}))\rangle\subset\mathcal{L}_{A}((A\otimes
L^{2}(\mathbb{G}))\otimes L^{2}(\mathbb{G})),$

where $\widehat{\delta}_{U}(x)=(1\otimes
U_{\mathbb{G}})\widehat{\delta}(x)(1\otimes U_{\mathbb{G}})$ for $x\in
A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$.

######  4.3.2 Remark.

Observe that the crossed product
$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}$
defined above is the usual one as, say, given in [5, Section 7] (we do not
consider any deformation in this definition, contrary to Definition 4.2.2). In
particular, we have automatically that
$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}=\overline{span}\\{\widehat{\delta}_{U}(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G})(1\otimes
C(\mathbb{G}))\\}$.

Since
$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}$
defined above is a usual crossed product, we can define the corresponding dual
action $\widehat{\widehat{\delta}}$. Hence, the following proposition follows
from the standard theory of crossed products.

######  4.3.3 Proposition-Definition.

Let $(\mathbb{G},A,\delta,\Omega)$ be a twisted dynamical system. The double
reduced crossed product
$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}$
is a $\mathbb{G}$-$C^{*}$-algebra with action
$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}\overset{\widehat{\widehat{\delta}}}{\curvearrowleft}\mathbb{G}$
such that:

$\widehat{\widehat{\delta}}\big{(}\widehat{\delta}_{U}(z)(1\otimes
y)\big{)}=(\widehat{\delta}_{U}(z)\otimes 1)(1\otimes V_{\mathbb{G}}(y\otimes
1)V_{\mathbb{G}}^{*})\mbox{,}$

for all $z\in A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$ and all $y\in
C(\mathbb{G})$. The action
$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}\overset{\widehat{\widehat{\delta}}}{\curvearrowleft}\mathbb{G}$
is called _twisted double dual action of $(\delta,\Omega)$_ or _$\Omega$
-double dual action of $\delta$_.

The following theorem is a special case of [32, Theorem 3.6] by means of [32,
Proposition 3.5], see also [42, Section 1] in the von Neumann algebraic
setting (where no regularity assumption is needed).

######  4.3.4 Theorem (Twisted Takesaki-Takai duality).

Let $(\mathbb{G},A,\delta,\Omega)$ be a twisted dynamical system with $\Omega$
of finite type.

  1. i)

There is a canonical isomorphism of $C^{*}$-algebras,
$A\otimes\mathcal{K}(L^{2}(\mathbb{G}))\cong\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}$,
given by the map
$x\mapsto(U_{\mathbb{G}})_{3}(V^{\Omega})_{23}^{*}(\delta\otimes
id)(x)(V^{\Omega})_{23}(U_{\mathbb{G}})_{3}$.

  2. ii)

Under the above isomorphism, the twisted double dual action
$\widehat{\widehat{\delta}}$ of $(\delta,\Omega)$ is conjugate to the action
$A\otimes\mathcal{K}(L^{2}(\mathbb{G}))\overset{\widetilde{\delta}}{\curvearrowleft}\mathbb{G}$
defined by
$\widetilde{\delta}:=Ad_{(W^{\Omega})_{32}}\circ\delta_{13}\mbox{,}$ where
$\delta_{13}$ denotes the amplified twisted action of $\mathbb{G}$ on
$A\otimes\mathcal{K}(L^{2}(\mathbb{G}))$ such that:

$\delta_{13}(a\otimes T)=(1\otimes\Sigma)(\delta(a)\otimes
id)(1\otimes\Sigma)(1\otimes T\otimes 1)\in
A\otimes\mathcal{K}(L^{2}(\mathbb{G}))\otimes C(\mathbb{G}_{\Omega})\mbox{,}$

for all $a\in A$ and $T\in\mathcal{K}(L^{2}(\mathbb{G}))$.

###### Proof.

By Theorem 4.1.9, we have
$\mathcal{K}(L^{2}(\mathbb{G}))=[C^{*}_{r}(\mathbb{G})U_{\mathbb{G}}C(\mathbb{G})U_{\mathbb{G}}]$.
Since $\delta$ is continuous, we can hence write
$A\otimes\mathcal{K}(L^{2}(\mathbb{G}))=[\delta(A)(1\otimes\mathcal{K}(L^{2}(\mathbb{G})))]=[\delta(A)(1\otimes
C_{r}^{*}(\mathbb{G},\Omega)U_{\mathbb{G}}C(\mathbb{G})U_{\mathbb{G}})]$, and
the first item then follows by a straightforward computation. The second item
then follows from Remark 4.1.6 together with the fact that
$W_{\mathbb{G}}(1\otimes UyU)(W_{\mathbb{G}})^{*}=(1\otimes
U)\Delta^{op}(y)(1\otimes U)$, which follows from the identity
$W=\check{V}_{\mathbb{G}}$ in Theorem-Definition 2.2.6. ∎

As a corollary of the twisted Takesaki-Takai duality established in Theorem
4.3.4 we obtain the following generalization of the well known _Packer-
Raeburn’s untwisting trick_ or _Packer-Raeburn’s stabilisation trick_ [35] to
compact quantum groups.

######  4.3.5 Proposition (Quantum Packer-Raeburn’s untwisting trick).

Let $(\mathbb{G},A,\delta,\Omega)$ be a twisted dynamical system with $\Omega$
of finite type. Then

$\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\otimes\mathcal{K}(L^{2}(\mathbb{G}))\cong\big{(}A\otimes\mathcal{K}(L^{2}(\mathbb{G})\big{)}\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G}.$

###### Proof.

Put $B:=A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$, which is a
$\widehat{\mathbb{G}}^{cop}$-$C^{*}$-algebra with action
$B\overset{\widehat{\delta}}{\curvearrowleft}\widehat{\mathbb{G}}^{cop}$.
Using the usual version of Takesaki-Takai duality for quantum groups, we can
write
$B\otimes\mathcal{K}(L^{2}(\mathbb{G}))\cong\big{(}B\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}\big{)}\underset{r,\widehat{\widehat{\delta}}}{\rtimes}\mathbb{G}$.
Next, using the twisted version of Takesaki-Takai duality from Theorem 4.3.4
we have the following:

$\big{(}B\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}\big{)}\underset{r,\widehat{\widehat{\delta}}}{\rtimes}\mathbb{G}=\big{(}\big{(}A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}\big{)}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}\big{)}\underset{r,\widehat{\widehat{\delta}}}{\rtimes}\mathbb{G}\cong\big{(}A\otimes\mathcal{K}(L^{2}(\mathbb{G})\big{)}\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G}.$

∎

This trick helps to establish a twisted version of the well known _Baaj-
Skandalis duality_ [5, 43].

######  4.3.6 Note.

“Takesaki-Takai duality” for quantum groups is also referred as “Baaj-
Skandalis duality” in the literature. We prefer to reserve the latter
terminology for such duality at the level of $KK$-theory.

######  4.3.7 Theorem (Twisted Baaj-Skandalis duality).

Let $(\mathbb{G},A,\delta,\Omega)$ and $(\mathbb{G},B,\vartheta,\Omega)$ be
two twisted dynamical systems with respect to a given $2$-cocycle $\Omega$ of
finite type on $\mathbb{G}$. Then there exists a canonical group isomorphism:

$J^{\Omega}_{\mathbb{G}}:KK^{\mathbb{G}_{\Omega}}(A,B)\overset{\sim}{\longrightarrow}KK^{\widehat{\mathbb{G}}^{cop}}(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G},B\underset{r,(\vartheta,\Omega)}{\rtimes}\mathbb{G})\mbox{,}$

which is compatible with the Kasparov product, that is, if
$(\mathbb{G},C,\nu,\Omega)$ is another twisted dynamical system with respect
to $\Omega$, then we have:

$J^{\Omega}_{\mathbb{G}}(\mathcal{X}\underset{C}{\otimes}\mathcal{Y})=J^{\Omega}_{\mathbb{G}}(\mathcal{X})\underset{C\underset{r,(\nu,\Omega)}{\rtimes}\mathbb{G}}{\otimes}J^{\Omega}_{\mathbb{G}}(\mathcal{Y})\mbox{
and
}J^{\Omega}_{\mathbb{G}}(\mathbbold{1}_{A})=\mathbbold{1}_{A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}}\mbox{,}$

for all $\mathcal{X}\in KK^{\mathbb{G}_{\Omega}}(A,C)$ and $\mathcal{Y}\in
KK^{\mathbb{G}_{\Omega}}(C,B)$. Moreover, we have
$j^{\Omega}_{\mathbb{G}}=\mathcal{O}_{\mathbb{G}}\circ
J^{\Omega}_{\mathbb{G}}$, where $\mathcal{O}_{\mathbb{G}}$ is the obvious
forgetful functor.

###### Proof.

Given the twisted dynamical system $(\mathbb{G},A,\delta,\Omega)$, put
$A_{1}:=A\underset{r,\delta}{\rtimes}\mathbb{G}_{\Omega}$. Thus
$(A_{1},\widehat{\delta})$ is an object in
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}_{\Omega}}^{cop}}$. It is known
that $\widehat{\mathbb{G}_{\Omega}}$ and $\widehat{\mathbb{G}}$ are monoidally
co-Morita equivalent in the sense of [12], [15]. The corresponding co-linking
quantum groupoid takes the form $C^{*}_{r}(\mathbb{G}_{\Omega})\oplus
C^{*}_{r}(\mathbb{G}_{\Omega},\Omega^{*})\oplus
C^{*}_{r}(\mathbb{G},\Omega)\oplus C^{*}_{r}(\mathbb{G})$. Next, following the
notations from [3, Section 2.4], we consider the _exterior comultiplication_
$\widehat{\Delta}_{ext}\equiv({}_{\Omega}\widehat{\Delta}_{\Omega^{*}})_{11}^{2}:C^{*}_{r}(\mathbb{G}_{\Omega})\longrightarrow\widetilde{M}(C^{*}_{r}(\mathbb{G}_{\Omega},\Omega^{*})\otimes
C^{*}_{r}(\mathbb{G},\Omega))$. Following [3, Proposition 4.1], we consider
the $C^{*}$-algebra:

$\begin{split}A_{2}:&=\overline{span}\\{(id\otimes
id\otimes\eta)(id_{A_{1}}\otimes\widehat{\Delta}_{ext}^{cop})\widehat{\delta}(a^{\prime})\
|\ a^{\prime}\in A_{1},\eta\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}\\\
&=\overline{span}\\{(id\otimes id\otimes\eta)(\delta(a)\otimes
1)(1\otimes\widehat{\Delta}_{ext}^{cop}(x))\ |\ a^{\prime}\in A,x\in
C^{*}_{r}(\mathbb{G}_{\Omega}),\eta\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}.\end{split}$

Since $\\{(id\otimes\eta)\widehat{\Delta}_{ext}^{cop}(x)\mid x\in
C^{*}_{r}(\mathbb{G}_{\Omega}),\eta\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}$
is norm-dense in $C_{r}^{*}(\mathbb{G},\Omega)$, we have by construction that
$A_{2}=A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$. Now, $A_{2}$ is an
object in $\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}^{cop}}$ with
$\widehat{\mathbb{G}}^{cop}$-action given by Proposition 4.2.6, which we still
denote by $\widehat{\delta}$.

In other words, the equivalence of triangulated categories
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}_{\Omega}}^{cop}}\overset{J_{\widehat{\mathbb{G}_{\Omega}},\
\widehat{\mathbb{G}}^{cop}}}{\cong}\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}^{cop}}$
from [3, Section 4.5] sends $A\underset{r,\delta}{\rtimes}\mathbb{G}_{\Omega}$
to $A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}$. An application of the
equivariant Morita equivalence of [3, Section 4.4] yields that
$(A\otimes\mathcal{K}(L^{2}(\mathbb{G})))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G}_{\Omega}\cong
A_{1}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}_{\Omega}}^{cop}\underset{r,\widehat{\widehat{\delta}}}{\rtimes}\mathbb{G}_{\Omega}$
in $\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}_{\Omega}}^{cop}}$ (here
$\widetilde{\delta}$ is defined by Takesaki-Takai duality for
$\mathbb{G}_{\Omega}$) is sent to
$(A\otimes\mathcal{K}(L^{2}(\mathbb{G})))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G}\cong
A_{2}\underset{r,\widehat{\delta}}{\rtimes}\widehat{\mathbb{G}}^{cop}\underset{r,\widehat{\widehat{\delta}}}{\rtimes}\mathbb{G}$
in $\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}^{cop}}$ (here
$\widetilde{\delta}$ is defined through the twisted Takesaki-Takai from
Theorem 4.3.4) through $J_{\widehat{\mathbb{G}_{\Omega}},\
\widehat{\mathbb{G}}^{cop}}$.

Therefore, the twisted Baaj-Skandalis is obtained as follows:

$\begin{split}KK^{\mathbb{G}_{\Omega}}(A,B)&\cong
KK^{\mathbb{G}_{\Omega}}(A\otimes\mathcal{K}(L^{2}(\mathbb{G}),B\otimes\mathcal{K}(L^{2}(\mathbb{G}))\\\
&\overset{J_{\mathbb{G}_{\Omega}}}{\cong}KK^{\widehat{\mathbb{G}_{\Omega}}^{cop}}((A\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G}_{\Omega},(B\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G}_{\Omega})\\\
&\overset{J_{\widehat{\mathbb{G}_{\Omega}},\
\widehat{\mathbb{G}}^{cop}}}{\cong}KK^{\widehat{\mathbb{G}}^{cop}}((A\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G},(B\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G})\\\
&\cong
KK^{\widehat{\mathbb{G}}^{cop}}(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G},B\underset{r,(\vartheta,\Omega)}{\rtimes}\mathbb{G}).\end{split}$

∎

As a consequence, we obtain a twisted version of the well-known _descent map_
for quantum groups [43].

######  4.3.8 Corollary (Twisted descent map).

Let $(\mathbb{G},A,\delta,\Omega)$ and $(\mathbb{G},B,\vartheta,\Omega)$ be
two twisted dynamical systems with respect to a given $2$-cocycle $\Omega$ of
finite type on $\mathbb{G}$. Then there exists a canonical group homomorphism:

$j^{\Omega}_{\mathbb{G}}:KK^{\mathbb{G}_{\Omega}}(A,B)\longrightarrow
KK(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G},B\underset{r,(\vartheta,\Omega)}{\rtimes}\mathbb{G})$

called _twisted descent map (with respect to $\mathbb{G}$)_. Moreover,
$j^{\Omega}_{\mathbb{G}}$ is compatible with the Kasparov product, that is, if
$(\mathbb{G},C,\nu,\Omega)$ is another twisted dynamical system with respect
to $\Omega$, then we have:

$j^{\Omega}_{\mathbb{G}}(\mathcal{X}\underset{C}{\otimes}\mathcal{Y})=j^{\Omega}_{\mathbb{G}}(\mathcal{X})\underset{C\underset{r,(\nu,\Omega)}{\rtimes}\mathbb{G}}{\otimes}j^{\Omega}_{\mathbb{G}}(\mathcal{Y})\mbox{
and
}j^{\Omega}_{\mathbb{G}}(\mathbbold{1}_{A})=\mathbbold{1}_{A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G}}\mbox{,}$

for all $\mathcal{X}\in KK^{\mathbb{G}_{\Omega}}(A,C)$ and $\mathcal{Y}\in
KK^{\mathbb{G}_{\Omega}}(C,B)$.

###### Proof.

Similarly as in Theorem 4.3.7 we have $KK^{\mathbb{G}_{\Omega}}(A,B)\cong
KK^{\widehat{\mathbb{G}}^{cop}}((A\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G},(B\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G})$.
Applying Baaj-Skandalis duality for $\mathbb{G}$, the latter is isomorphic to
$KK^{\mathbb{G}}(A\otimes\mathcal{K}(L^{2}(\mathbb{G}),B\otimes\mathcal{K}(L^{2}(\mathbb{G}))$.
Applying the ordinary descent map $j_{\mathbb{G}}$, we get
$KK((A\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G},(B\otimes\mathcal{K}(L^{2}(\mathbb{G}))\underset{r,\widetilde{\delta}}{\rtimes}\mathbb{G})$,
which is isomophic to
$KK(A\underset{r,(\delta,\Omega)}{\rtimes}\mathbb{G},B\underset{r,(\vartheta,\Omega)}{\rtimes}\mathbb{G})$
thanks to the untwisting trick from Proposition 4.3.5. The map
$j^{\Omega}_{\mathbb{G}}$ of the statement is obtained as result of these
compositions. ∎

##  Application: quantum assembly map for permutation torsion-free discrete
quantum groups

###  The Baum-Connes property for discrete quantum groups

We collect here the main category framework for the formulation of the Baum-
Connes property for discrete quantum groups. We refer to [27] or [22] for a
complete presentation of the subject.

Let $\widehat{\mathbb{G}}$ be a discrete quantum group and consider the
corresponding equivariant Kasparov category,
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$, with canonical suspension
functor denoted by $\Sigma$. $\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$
is a triangulated category whose distinguished triangles are given by mapping
cone triangles (see [27] for more details). The word _homomorphism (resp.
isomorphism)_ will mean _homomorphism (resp. isomorphism) in the corresponding
Kasparov category_ ; it will be a true homomorphism (resp. isomorphism)
between $\widehat{\mathbb{G}}$-$C^{*}$-algebras or any Kasparov triple between
$\widehat{\mathbb{G}}$-$C^{*}$-algebras (resp. any equivariant
$KK$-equivalence between $\widehat{\mathbb{G}}$-$C^{*}$-algebras).
Analogously, we can consider the equivariant Kasparov category
$\mathscr{K}\mathscr{K}^{\mathbb{G}}$.

Assume for the moment that $\widehat{\mathbb{G}}$ is _torsion-free_. In that
case, consider the usual complementary pair of localizing subcategories in
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$,
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$.
Denote by $(L,N)$ the canonical triangulated functors associated to this
complementary pair. More precisely we have that
$\mathscr{L}_{\widehat{\mathbb{G}}}$ is defined as the _localizing subcategory
of $\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$ generated by the objects of
the form $Ind^{\widehat{\mathbb{G}}}_{\mathbb{E}}(C)=C\otimes
c_{0}(\widehat{\mathbb{G}})$ with $C$ any $C^{*}$-algebra in the Kasparov
category $\mathscr{K}\mathscr{K}$_ and $\mathscr{N}_{\widehat{\mathbb{G}}}$ is
defined as the _localizing subcategory of objects which are isomorphic to $0$
in $\mathscr{K}\mathscr{K}$_:
$\mathscr{L}_{\widehat{\mathbb{G}}}:=\langle\\{Ind^{\widehat{\mathbb{G}}}_{\mathbb{E}}(C)=C\otimes
c_{0}(\widehat{\mathbb{G}})\ |\ C\in Obj.(\mathscr{K}\mathscr{K})\\}\rangle$
and $\mathscr{N}_{\widehat{\mathbb{G}}}=\\{A\in
Obj.(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}})\ |\
Res^{\widehat{\mathbb{G}}}_{\mathbb{E}}(A)=0\\}$.

If $\widehat{\mathbb{G}}$ is _not_ torsion-free, then a technical property
lacked in the literature in order to define a suitable complementary pair. The
natural candidate used in the related works (see for instance [28] and [45])
is given by the following localizing subcategories of
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$:

$\mathscr{L}_{\widehat{\mathbb{G}}}:=\langle\\{C\otimes\mathbb{G}\underset{r}{\ltimes}T\
|\ C\in Obj.(\mathscr{K}\mathscr{K})\mbox{,
}T\in\mathbb{T}\mbox{or}(\widehat{\mathbb{G}})\\}\rangle,$
$\mathscr{N}_{\widehat{\mathbb{G}}}:=\mathscr{L}^{\dashv}_{\widehat{\mathbb{G}}}=\\{A\in
Obj(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}})\ |\
KK^{\widehat{\mathbb{G}}}(L,A)=(0)\mbox{, $\forall$ $L\in
Obj(\mathscr{L}_{\widehat{\mathbb{G}}})$}\\}.$

######  5.1.1 Remark.

We put $\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}:=\langle\\{T\otimes C\ |\
C\in Obj.(\mathscr{K}\mathscr{K})\mbox{,
}T\in\mathbb{T}\mbox{or}(\widehat{\mathbb{G}})\\}\rangle$, so that we have
$\mathbb{G}\ltimes\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}=\mathscr{L}_{\widehat{\mathbb{G}}}$
by definition. Similarly we put
$\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}:=\widehat{\mathbb{G}}\ltimes\mathscr{N}_{\widehat{\mathbb{G}}}$.

Recently, Y. Arano and A. Skalski [2] have showed that these two subcategories
form indeed a complementary pair of localizing subcategories in
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$. As an application of our
cleftness property obtained in Theorem 3.1.5, we are going to prove in Section
5.3 that
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$ is
indeed complementary when $\mathscr{L}_{\widehat{\mathbb{G}}}$ is defined only
in terms of torsion actions of $\mathbb{G}$ of projective type. It consists in
generalizing the _Green-Julg isomorphism_.

######  5.1.2 Note.

The following nomenclature is useful. Given $A\in
Obj.(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}})$ consider a
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$-triangle
associated to $A$, say $\Sigma N(A)\longrightarrow
L(A)\overset{D}{\longrightarrow}A\longrightarrow N(A)$. We know that such
triangles are distinguished and unique up to isomorphism (recall Theorem
2.5.11). The homomorphism $D:L(A)\longrightarrow A$ is called _Dirac
homomorphism for $A$_. In particular, we consider the Dirac homomorphism for
$\mathbb{C}$ (as trivial $\widehat{\mathbb{G}}$-$C^{*}$-algebra),
$D_{\mathbb{C}}:L(\mathbb{C})\longrightarrow\mathbb{C}$. We refer to
$D_{\mathbb{C}}$ simply as _Dirac homomorphism_.

###  Two-sided crossed products

In this section we give some technical tools necessary for the next section.
In order to define an assembly map for discrete quantum groups, we need to
define a suitable pair of adjoint functors on
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$ taking into account the
torsion phenomena of $\widehat{\mathbb{G}}$. For classical discrete groups,
the torsion phenomena are completely described in terms of finite subgroups.
Hence, the induction and restriction functors will provide such an adjunction.
In the quantum case, the torsion is described in terms of
$\mathbb{G}$-$C^{*}$-algebras, so the induction-restriction approach is no
longer valid since finite discrete quantum groups do not exhaust the torsion
phenomena for $\widehat{\mathbb{G}}$. Moreover, the torsion objects in the
Kasparov category are also $\mathbb{G}$-$C^{*}$-algebras. In this sense, we
need a construction encoding both the induction process from the classical
setting and the _diagonal_ action with respect to $\mathbb{G}$. We call it a
_two-sided crossed product_ and it is already used in [2], see also [33,
Section 2.6].

######  5.2.1 Definition.

Let $\mathbb{G}$ be a compact quantum group. If $(B,\beta)$ is a right
$\mathbb{G}$-$C^{*}$-algebra and $(A,\alpha)$ is a left
$\mathbb{G}$-$C^{*}$-algebra, then the two-sided crossed product of $B$ and
$A$ by $\mathbb{G}$, denoted by
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$, is the
$C^{*}$-algebra defined by:

$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A:=C^{*}\langle((id\otimes\lambda)\beta(B)\otimes
1)(1\otimes\widehat{\lambda}(c_{0}(\widehat{\mathbb{G}}))\otimes
1)(1\otimes(\rho\otimes id)(\alpha(A)))\rangle\subset\mathcal{L}_{B\otimes
A}(B\otimes L^{2}(\mathbb{G})\otimes A).$

######  5.2.2 Remark.

First, to lighten the notations we will omit the representations $\lambda$,
$\widehat{\lambda}$ and $\rho$ in the definition of
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$, and
note that then $\rho(x)=U_{\mathbb{G}}xU_{\mathbb{G}}$ for $x\in
C(\mathbb{G})$. We also write $\alpha_{U}(x)=(U_{\mathbb{G}}\otimes
id)\alpha(x)(U_{\mathbb{G}}\otimes id)$ for $x\in A$. Next, it is easy to show
that
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A=\overline{span}\\{(\beta(B)\otimes
1)(1\otimes c_{0}(\widehat{\mathbb{G}})\otimes 1)(1\otimes\alpha_{U}(A))\\}$,
cf. Lemma 4.2.4. From now on we will use these two descriptions of
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$
interchangeably. As a consequence, we see that the maps
$B\longrightarrow\mathcal{L}_{B\otimes A}(B\otimes L^{2}(\mathbb{G})\otimes
A)$, $A\longrightarrow\mathcal{L}_{B\otimes A}(B\otimes
L^{2}(\mathbb{G})\otimes A)$ and
$c_{0}(\widehat{\mathbb{G}})\longrightarrow\mathcal{L}_{B\otimes A}(B\otimes
L^{2}(\mathbb{G})\otimes A)$ given by $b\mapsto\beta(b)\otimes 1$, $a\mapsto
1\otimes\alpha_{U}(a)$ and $x\mapsto 1\otimes\widehat{\lambda}(x)\otimes 1$
respectively, send $B$, $A$ and $c_{0}(\widehat{\mathbb{G}})$ respectively
onto non-degenerate $C^{*}$-subalgebras of
$M(B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A)$.

As for usual crossed products, we can show that the two-sided crossed product
construction is functorial. More precisely, we have the following:

######  5.2.3 Proposition.

Let $\mathbb{G}$ be a compact quantum group. Let $(B,\beta)$,
$(B^{\prime},\beta^{\prime})$ be right $\mathbb{G}$-$C^{*}$-algebras and
$(A,\alpha)$, $(A^{\prime},\alpha^{\prime})$ left
$\mathbb{G}$-$C^{*}$-algebras.

  1. i)

If $\phi:B\longrightarrow M(B^{\prime})$ is a non-degenerate
$\mathbb{G}$-equivariant $*$-homomorphism, then there exists a non-degenerate
$*$-homomorphism:

$\phi\rtimes id\ltimes
id:B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A\longrightarrow
M(B^{\prime}\underset{r,\beta^{\prime}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A)$

such that $\phi\rtimes id\ltimes id\big{(}(\beta(b)\otimes 1)(1\otimes
x\otimes 1)(1\otimes\alpha_{U}(a))\big{)}=(\beta^{\prime}(\phi(b))\otimes
1)(1\otimes x\otimes 1)(1\otimes\alpha_{U}(a))$, for all $b\in B$, $a\in A$
and $x\in c_{0}(\widehat{\mathbb{G}})$.

  2. ii)

If $\psi:A\longrightarrow M(A^{\prime})$ is a non-degenerate
$\mathbb{G}$-equivariant $*$-homomorphism, then there exists a non-degenerate
$*$-homomorphism:

$id\rtimes
id\ltimes\psi:B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A\longrightarrow
M(B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha^{\prime}}{\ltimes}A^{\prime})$

such that $id\rtimes id\ltimes\psi\big{(}(\beta(b)\otimes 1)(1\otimes x\otimes
1)(1\otimes\alpha_{U}(a))\big{)}=(\beta(b)\otimes 1)(1\otimes x\otimes
1)(1\otimes\alpha_{U}^{\prime}(\psi(a)))$, for all $b\in B$, $a\in A$ and
$x\in c_{0}(\widehat{\mathbb{G}})$.

If a torsion action of projective type of $\mathbb{G}$ is involved in a two-
sided crossed product, then we can given an alternative description of the
latter, which is useful for our purpose. Recall first from Lemma 3.2.3 that,
given a torsion action of projective type $\delta$ on $T=\mathcal{B}(H)$ with
implementing $\Omega^{*}$-representation $u$ for some $\Omega$, we can
construct the $\Omega$-representation $u^{\circ}$ on $\overline{H}$ with
associated coaction $\overline{\delta}$ on
$T^{op}\cong\mathcal{B}(\overline{H})$. By Remark 4.1.4 we can view
$u^{\circ}$ as a $\Omega=(\Omega^{*})^{*}$-representation of
$\mathbb{G}_{\Omega}$, and correspondingly $\delta_{u^{\circ}}$ as a right
action of $\mathbb{G}_{\Omega}$ on $T^{op}$.

######  5.2.4 Proposition.

Let $\mathbb{G}$ be a compact quantum group. Let $(T,\delta)$ be a torsion
action of projective type of $\mathbb{G}$. Let $u$ be an
$\Omega^{*}$-representation of $\mathbb{G}$ implementing $\delta$ for some
$2$-cocycle $\Omega$ (necessarily of finite type). Let $(B,\beta)$ be a
$\mathbb{G}$-$C^{*}$-algebra. Then
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\cong(B\otimes
T^{op})\underset{\widetilde{\beta}}{\rtimes}\mathbb{G}_{\Omega}$, where
$\widetilde{\beta}:=Ad_{u^{\circ}_{23}}\circ\beta_{13}$.

We use here that the twisted quantum group $\mathbb{G}_{\Omega}$ is again a
compact quantum group by Theorem 3.3.7.

###### Proof.

First of all, it is straightforward to check that
$\widetilde{\beta}:=Ad_{u^{\circ}_{23}}\circ\beta_{13}$ is an action of
$\mathbb{G}_{\Omega}$ on $B\otimes T^{op}$. For given $b\in B$ and $x\in
T^{op}$ we have:

$\begin{split}(id\otimes{}_{\Omega}\Delta_{\Omega^{*}})\widetilde{\beta}(b\otimes
x)&=(id\otimes{}_{\Omega}\Delta_{\Omega^{*}})\big{(}u^{\circ}_{23}(b_{(0)}\otimes
x\otimes b_{(1)})(u^{\circ}_{23})^{*}\big{)}\\\
&=\Omega_{34}\Delta_{3}(u^{\circ}_{23})(b_{(0)}\otimes
x\otimes\Delta(b_{(1)}))\Delta_{3}(u^{\circ}_{23})^{*}\Omega^{*}_{34}\\\
&\overset{(*)}{=}u^{\circ}_{23}u^{\circ}_{24}(b_{(0)}\otimes
x\otimes\Delta(b_{(1)}))(u^{\circ}_{24})^{*}(u^{\circ}_{23})^{*}\\\
&\overset{(**)}{=}u^{\circ}_{23}u^{\circ}_{24}(b_{(0)(0)}\otimes x\otimes
b_{(0)(1)}\otimes b_{(1)})(u^{\circ}_{24})^{*}(u^{\circ}_{23})^{*}\\\
&=(\widetilde{\beta}\otimes id)\big{(}u^{\circ}_{23}(b_{(0)}\otimes x\otimes
b_{(1)})(u^{\circ}_{23})^{*}\big{)}=(\widetilde{\beta}\otimes
id)\widetilde{\beta}(b\otimes x),\end{split}$

where in $(*)$ we used the fact that $u^{\circ}$ is a $\Omega$-representation
of $\mathbb{G}$ and in $(**)$ we use the fact that $\beta$ is a
$\mathbb{G}$-action on $B$.

Next, recall that $u^{\circ}$ implements $\overline{\delta}$. Then on the one
hand, by Remark 4.2.9 we have
$\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\cong
C^{*}_{r}(\Omega,\mathbb{G})\otimes T^{op}$ (here the identification is given
by conjugating with $u^{\circ}_{21}(U_{\mathbb{G}}\otimes 1)$). On the other
hand, by Remark 4.1.4 we view $u^{\circ}$ as a
$\Theta^{*}:=(\Omega^{*})^{*}$-representation of $\mathbb{G}_{\Omega}$ so
that, by applying again Remark 4.2.9, we obtain that
$T^{op}\underset{r,Ad_{u^{\circ}}}{\rtimes}\mathbb{G}_{\Omega}\cong
T^{op}\otimes C^{*}_{r}(\mathbb{G}_{\Omega},\Theta)=T^{op}\otimes
C^{*}_{r}(\mathbb{G}_{\Omega},\Omega^{*})$ (here the identification is given
by conjugating with $(u^{\circ})^{*}$). Recall from Remark 4.1.5 that
$C^{*}_{r}(\mathbb{G}_{\Omega},\Omega^{*})=JC^{*}_{r}(\mathbb{G},\Omega)J$.
Therefore, by Remark 4.1.6 we see that
$T^{op}\underset{r,Ad_{u^{\circ}}}{\rtimes}\mathbb{G}_{\Omega}$ is
$*$-isomorphic to $\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}$ by
introducing an extra conjugation with $X_{\Omega}U_{\mathbb{G}}$ (and a flip
map). This allows to conclude the isomorphism of the statement. Indeed, since
$U_{\mathbb{G}}X_{\Omega}U_{\mathbb{G}}\in L^{\infty}(\mathbb{G})^{\prime}$,
we compute

$\begin{split}(B&\otimes
T^{op})\underset{\widetilde{\beta}}{\rtimes}\mathbb{G}_{\Omega}=\overline{span}\\{(id\otimes
id\otimes\lambda)\widetilde{\beta}(B\otimes T^{op})(1\otimes 1\otimes
c_{0}(\widehat{\mathbb{G}_{\Omega}}))\\}\\\ &=\overline{span}\\{(id\otimes
id\otimes\lambda)u^{\circ}_{23}(B_{(0)}\otimes T^{op}\otimes
B_{(1)})(u^{\circ}_{23})^{*}(1\otimes 1\otimes
c_{0}(\widehat{\mathbb{G}_{\Omega}}))\\}\\\
&\overset{Ad_{(u^{\circ}_{23})^{*}}}{\cong}\overline{span}\\{(id\otimes
id\otimes\lambda)(B_{(0)}\otimes T^{op}\otimes
B_{(1)})(u^{\circ}_{23})^{*}(1\otimes 1\otimes
c_{0}(\widehat{\mathbb{G}_{\Omega}}))u^{\circ}_{23}\\}\\\
&=\overline{span}\\{(id\otimes id\otimes\lambda)(B_{(0)}\otimes 1\otimes
B_{(1)})(1\otimes\underbracket{T^{op}\otimes
1})(1\otimes\underbracket{(u^{\circ})^{*}(1\otimes
c_{0}(\widehat{\mathbb{G}_{\Omega}}))u^{\circ}})\\}\\\
&\underset{(*)}{=}\overline{span}\\{(id\otimes
id\otimes\lambda)(B_{(0)}\otimes 1\otimes
B_{(1)})(1\otimes\underbracket{T^{op}\otimes
C^{*}_{r}(\mathbb{G}_{\Omega},\Omega^{*})})\\}\\\
&\cong\overline{span}\\{(id\otimes id\otimes\lambda)(B_{(0)}\otimes 1\otimes
B_{(1)})(1\otimes T^{op}\otimes JC^{*}_{r}(\mathbb{G},\Omega)J)\\}\\\
&\overset{Ad_{\Sigma_{23}(U_{\mathbb{G}}X_{\Omega}U_{\mathbb{G}})_{3}}}{\cong}\overline{span}\\{(id\otimes\lambda\otimes
id)(B_{(0)}\otimes B_{(1)}\otimes
1)(1\otimes\underbracket{U_{\mathbb{G}}C^{*}_{r}(\Omega,\mathbb{G})U_{\mathbb{G}}\otimes
T^{op}})\\}\\\
&\overset{Ad_{u_{21}^{\circ}}}{\underset{(**)}{\cong}}\overline{span}\\{(id\otimes\lambda\otimes
id)(B_{(0)}\otimes B_{(1)}\otimes
1)(1\otimes\underbracket{c_{0}(\widehat{\mathbb{G}})\otimes
1})(1\otimes\underbracket{\rho\otimes
1})(1\otimes\underbracket{\Sigma(u^{\circ})^{*}(1\otimes
T^{op})u^{\circ}\Sigma})\\}\\\
&=\overline{span}\\{((id\otimes\lambda)\beta(B)\otimes 1)(1\otimes
c_{0}(\widehat{\mathbb{G}})\otimes 1)(1\otimes(\rho\otimes
1)\overline{\delta}(T^{op}))\\}=B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op},\end{split}$

where in $(*)$ and $(**)$ we have used the identifications
$T^{op}\underset{r,Ad_{u^{\circ}}}{\rtimes}\mathbb{G}_{\Omega}\cong
T^{op}\otimes C^{*}_{r}(\mathbb{G}_{\Omega},\Omega^{*})$ and
$\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\cong
C^{*}_{r}(\Omega,\mathbb{G})\otimes T^{op}$ explained above, respectively. ∎

The two-sided crossed product construction can also be defined for Hilbert
modules in a similar way as we do for usual crossed products.

######  5.2.5 Definition.

Let $\mathbb{G}$ be a compact quantum group. Let $(B,\beta)$ be a right
$\mathbb{G}$-$C^{*}$-algebra and $(A,\alpha)$ a left
$\mathbb{G}$-$C^{*}$-algebra. If $(E,\delta_{E})$ is a
$\mathbb{G}$-equivariant Hilbert $B$-module, we define the two-sided crossed
product of $E$ and $A$ by $\mathbb{G}$, denoted by
$E\underset{r,\delta_{E}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$, as
the following Hilbert
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$-module
$E\underset{r,\delta_{E}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A:=E\underset{B}{\otimes}B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$.

As for the usual crossed products, the embeddings of
$E\cong\mathcal{K}_{B}(B,E)$ and $B\cong\mathcal{K}_{B}(B)$ into
$\mathcal{K}_{B}(B\oplus E)$ induce an embedding of
$E\underset{r,\delta_{E}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$
into $\mathcal{K}_{B}(B\oplus
E)\underset{r}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$. In this way
we have the following (see for instance [43, Lemme 5.2] for a proof):

######  5.2.6 Proposition.

Let $\mathbb{G}$ be a compact quantum group. Let $(B,\beta)$ be a right
$\mathbb{G}$-$C^{*}$-algebra and $(A,\alpha)$ a left
$\mathbb{G}$-$C^{*}$-algebra. If $(E,\delta_{E})$ is a
$\mathbb{G}$-equivariant Hilbert $B$-module, then
$\mathcal{K}_{B}(E)\underset{r}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A\cong\mathcal{K}_{B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A}\big{(}E\underset{r,\delta_{E}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A\big{)}$.

Following similar arguments as for usual crossed products (see for instance
[43, Proposition 5.3] for more details), it is easy to show that Definition
5.2.5 above passes also at the level of Kasparov triples. More precisely, we
have the following:

######  5.2.7 Proposition.

Let $\mathbb{G}$ be a compact quantum group. Let $(B,\beta)$,
$(B^{\prime},\beta^{\prime})$ be right $\mathbb{G}$-$C^{*}$-algebras and
$(A,\alpha)$ a left $\mathbb{G}$-$C^{*}$-algebra. If $((E,\delta_{E}),\pi,F)$
is a $\mathbb{G}$-equivariant Kasparov $(B^{\prime},B)$-module in
$KK^{\mathbb{G}}(B^{\prime},B)$, then the triple
$\big{(}E\underset{r,\delta_{E}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A,\pi\rtimes
id\ltimes id,F\underset{B}{\otimes}id\big{)}$ defines a Kasparov
$\big{(}B^{\prime}\underset{r,\beta^{\prime}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A,B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A\big{)}$-module
in
$KK\big{(}B^{\prime}\underset{r,\beta^{\prime}}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A,B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A\big{)}$.

Finally, as for usual crossed products, the two-sided crossed product functor
intertwines the suspension of $\mathbb{G}$-$C^{*}$-algebras and transforms
mapping cone triangles into mapping cone triangles. In other words, the
functor $(\ \cdot\
)\underset{r}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$ preserves
_semi-split extensions_ i.e. extensions of $\mathbb{G}$-equivariant
$C^{*}$-algebras that split through a $\mathbb{G}$-equivariant completely
positive contractive linear section, see for instance [25]; and the class of
all triangles in $\mathscr{K}\mathscr{K}^{\mathbb{G}}$ isomorphic to mapping
cone triangles is the same as the class of all triangles in
$\mathscr{K}\mathscr{K}^{\mathbb{G}}$ isomorphic to extension triangles (see
for instance [24, Lemma 1.2.3.7]). In conclusion, we have obtained that, for a
fixed left $\mathbb{G}$-$C^{*}$-algebra $(A_{0},\alpha_{0})$, the association
$(B,\beta)\mapsto
B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha_{0}}{\ltimes}A_{0}$,
for all right $\mathbb{G}$-$C^{*}$-algebra $(B,\beta)$ defines a _triangulated
functor_ $j_{\mathbb{G},A_{0}}:=(\ \cdot\
)\underset{r}{\rtimes}\mathbb{G}\underset{r,\alpha_{0}}{\ltimes}A_{0}:\mathscr{K}\mathscr{K}^{\mathbb{G}}\longrightarrow\mathscr{K}\mathscr{K}$.

###  Twisted Green-Julg isomorphism

First of all, let us recall briefly the Green-Julg isomorphism for compact
quantum groups (see [43] for more details). If $C$ is a $C^{*}$-algebra
equipped with the trivial action of $\mathbb{G}$, then we have that
$\Psi:KK^{\mathbb{G}}(C,B)\overset{\sim}{\longrightarrow}KK(C,B\underset{r,\beta}{\rtimes}\mathbb{G})$,
for all $\mathbb{G}$-$C^{*}$-algebra $(B,\beta)$. Since $C$ is equipped with
the trivial action of $\mathbb{G}$, then
$C\underset{r}{\rtimes}\mathbb{G}\cong C\otimes c_{0}(\widehat{\mathbb{G}})$
and we have a natural $*$-homomorphism:

$\begin{array}[]{rcclccl}\phi_{C}:&C&\longrightarrow&C\otimes
c_{0}(\widehat{\mathbb{G}}),&c&\longmapsto&\phi(c):=c\otimes
p_{0}\mbox{,}\end{array}$

where $p_{0}:=(id\otimes h_{\mathbb{G}})(V_{\mathbb{G}})\in
c_{0}(\widehat{\mathbb{G}})$ is the canonical projection onto the subspace of
invariant vectors of $(V_{\mathbb{G}},L^{2}(\mathbb{G}))$. In this way we
obtain a Kasparov triple $[\phi_{C}]\in KK(C,C\otimes
c_{0}(\widehat{\mathbb{G}}))$. The Green-Julg isomorphism is given precisely
by
$\Psi(\mathcal{X}):=\phi_{C}^{*}(j_{\mathbb{G}}(\mathcal{X}))=[\phi_{C}]\underset{C\otimes
c_{0}(\widehat{\mathbb{G}})}{\otimes}j_{\mathbb{G}}(\mathcal{X})$, for all
$\mathcal{X}\in KK^{\mathbb{G}}(C,B)$. It is also possible to give an explicit
expression of its inverse. Given any $C^{*}$-algebra $C$ in
$\mathscr{K}\mathscr{K}$, we denote by $\tau(C)$ the same $C^{*}$-algebra $C$
equipped with the trivial action of $\mathbb{G}$ and so we regard it as an
object in $\mathscr{K}\mathscr{K}^{\mathbb{G}}$. In this way, we define the
Kasparov triple $\mathcal{E}_{B}:=[(B\otimes L^{2}(\mathbb{G}),\pi_{r},0)]\in
KK^{\mathbb{G}}\Big{(}\tau\big{(}B\underset{r,\beta}{\rtimes}\mathbb{G}\big{)},B\Big{)}$,
where $\pi_{r}$ denotes the canonical representation of
$B\underset{r,\beta}{\rtimes}\mathbb{G}$ in $B\otimes L^{2}(\mathbb{G})$. The
action of $\mathbb{G}$ on $B\otimes L^{2}(\mathbb{G})$ is defined as the
tensor product action of $\beta$ with the action of $\mathbb{G}$ on
$L^{2}(\mathbb{G})$ induced by the unitary
$\Sigma\check{V}_{\mathbb{G}}\Sigma=(U_{\mathbb{G}}\otimes
1)V_{\mathbb{G}}(U_{\mathbb{G}}\otimes 1)$ (see [43] for the precise
definitions). Then we have
$\Psi^{-1}(\mathcal{Y})=\tau(\mathcal{Y})\underset{\tau\big{(}B\underset{r,\beta}{\rtimes}\mathbb{G}\big{)}}{\otimes}\mathcal{E}_{B}$,
for all $\mathcal{Y}\in KK(C,B\underset{r,\beta}{\rtimes}\mathbb{G})$. In
other words, the Green-Julg isomorphism can be rephrased by saying that the
functors
$\mathscr{K}\mathscr{K}^{\mathbb{G}}\overset{j_{\mathbb{G}}}{\longrightarrow}\mathscr{K}\mathscr{K}\mbox{
and
}\mathscr{K}\mathscr{K}\overset{\tau}{\longrightarrow}\mathscr{K}\mathscr{K}^{\mathbb{G}}$
are adjoint: $\tau$ is a left adjoint of $j_{\mathbb{G}}$. Precisely, the unit
of the adjunction is given by $\eta_{C}:=[\phi_{C}]$ and the counit by
$\varepsilon_{B}:=\mathcal{E}_{B}$, for all $C\in Obj(\mathscr{K}\mathscr{K})$
and all $B\in Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$.

The goal of this section is to generalise these constructions when $C$ is
replaced by an object of the form $C\otimes
T\in\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}$, where $(T,\delta)$ is a
torsion action of $\mathbb{G}$ of projective type. Recall that a torsion
action of projective type of $\mathbb{G}$, $(T,\delta)$, means simply that
$T=\mathcal{M}_{k}(\mathbb{C})$ for some $k\in\mathbb{N}$ and that $\delta$ is
ergodic such that $T$ is not $\mathbb{G}$-Morita equivalent to $\mathbb{C}$.
We fix a state $\varphi_{T}=\text{Tr}(\varrho\ \cdot)$ on $T$ (recall Section
2.1). Recall as well that, by virtue of Theorem 3.1.5, $\delta$ is implemented
by a $\Omega^{*}$-representation of $\mathbb{G}$, say $u$, for some
(measurable) $2$-cocycle $\Omega$ on $\mathbb{G}$. The $2$-cocycle $\Omega$ is
necessarily of finite type (recall Definition 3.3.1). Hence
$\mathbb{G}_{\Omega}$ is again a compact quantum group by Theorem 3.3.7.
Following Equation (2.1), we denote by $(T^{op},\overline{\delta})$ the
corresponding opposite twisted dynamical system. In this case,
$\overline{\delta}$ is implemented (in the sense of (3.2)) by a
$\Omega$-representation of $\mathbb{G}$ that we denote by $u^{\circ}$. The
unitary representation of $\mathbb{G}$ on $L^{2}(T)$ implementing $\delta$
according to Proposition 2.3.6 is denoted by $V_{T}$. Given such a projective
torsion action of $\mathbb{G}$, we define the following triangulated functors:

$\begin{array}[]{rcclccl}j_{\mathbb{G},T}:&\mathscr{K}\mathscr{K}^{\mathbb{G}}&\longrightarrow&\mathscr{K}\mathscr{K},&(B,\beta)&\longmapsto&j_{\mathbb{G},T}(B,\beta):=B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op},\end{array}$
$\begin{array}[]{rcclccl}\tau_{T}:&\mathscr{K}\mathscr{K}&\longrightarrow&\mathscr{K}\mathscr{K}^{\mathbb{G}},&C&\longmapsto&\tau_{T}(C):=(C\otimes
T,id\otimes\delta).\end{array}$

We are going to show that $\tau_{T}$ is a left adjoint of $j_{\mathbb{G},T}$
for every torsion action of projective type $(T,\delta)$ of $\mathbb{G}$. To
do so we start by showing an appropriate equivalence of triangulated
categories between $\mathscr{K}\mathscr{K}^{\mathbb{G}}$ and
$\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}}$. Then the adjunction between
$\tau_{T}$ and $j_{\mathbb{G},T}$ will result from the usual Green-Julg
isomorphism applied to $\mathbb{G}_{\Omega}$.

Let us consider the following triangulated functors:

$\begin{array}[]{rcclccl}\Pi_{T^{op}}:&\mathscr{K}\mathscr{K}^{\mathbb{G}}&\longrightarrow&\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}},&(B,\beta)&\longmapsto&\Pi_{T^{op}}(B,\beta):=(B\otimes
T^{op},\widetilde{\beta}:=Ad_{u^{\circ}_{23}}\circ\beta_{13}),\end{array}$
$\begin{array}[]{rcclccl}\Pi_{T}:&\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}}&\longrightarrow&\mathscr{K}\mathscr{K}^{\mathbb{G}},&(C,\gamma)&\longmapsto&\Pi_{T}(C,\gamma):=(C\otimes
T,\widetilde{\gamma}:=Ad_{u_{23}}\circ\gamma_{13}).\end{array}$

First of all, observe that these functors are well defined. On the one hand,
given $(B,\beta)\in Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$, we have proved
in Proposition 5.2.4 that
$\widetilde{\beta}:=Ad_{u^{\circ}_{23}}\circ\beta_{13}$ is an action of
$\mathbb{G}_{\Omega}$ on $B\otimes T^{op}$. So $\Pi_{T^{op}}(B,\beta)\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}})$. A similar computation
yields that if $(C,\gamma)\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}})$, then
$\widetilde{\gamma}:=Ad_{u_{23}}\circ\gamma_{13}$ is an action of $\mathbb{G}$
on $C\otimes T$. So $\Pi_{T}(C,\gamma)\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$.

On the other hand, given two objects $(B_{1},\beta_{1}),(B_{2},\beta_{2})\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$ and a Kasparov triple
$\mathcal{X}\in KK^{\mathbb{G}}(B_{1},B_{2})$, then
$\Pi_{T^{op}}(\mathcal{X})$ is given by the right exterior tensor product of
Kasparov triples with respect to $T^{op}$ i.e.
$\Pi_{T^{op}}(\mathcal{X})=\mathcal{X}\otimes T^{op}\in
KK^{\mathbb{G}_{\Omega}}(B_{1}\otimes T^{op},B_{2}\otimes T^{op})$ (if
$\mathcal{X}$ is represented by the $\mathbb{G}$-equivariant Hilbert
$B_{2}$-module $E$ with action $\delta_{E}$, then $\Pi_{T^{op}}(\mathcal{X})$
is represented by the Hilbert $B_{2}\otimes T^{op}$-module $E\otimes T^{op}$
with action of $\mathbb{G}_{\Omega}$ given by
$Ad_{v_{23}}\circ(\delta_{E})_{13}$). Similarly,
$\Pi_{T}(\mathcal{Y})=\mathcal{Y}\otimes T$, for all $\mathcal{Y}\in
KK^{\mathbb{G}_{\Omega}}(C_{1},C_{2})$ with
$(C_{1},\gamma_{1}),(C_{2},\gamma_{2})\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}})$. Clearly, both
$\Pi_{T^{op}}$ and $\Pi_{T}$ intertwine the suspensions of each category.
Moreover, they transform mapping cone triangles into mapping cone triangles.
This is true by the following general fact: if $\phi:A\longrightarrow B$ is a
homomorphism between $C^{*}$-algebras and $D$ is any other $C^{*}$-algebra,
then we have that $C_{\phi}\otimes D\cong C_{\phi\otimes id}$ induced by the
canonical identification $C_{0}\big{(}(0,1],B\big{)}\otimes D\cong
C_{0}\big{(}(0,1],B\otimes D\big{)}$, $f\otimes d\mapsto\Big{(}t\mapsto
f(t)\otimes d\Big{)}$. If in addition $\phi$ is a $\mathbb{G}$-equivariant
homomorphism between the $\mathbb{G}$-$C^{*}$-algebras $(A,\alpha)$ and
$(B,\beta)$, then $C_{\phi}$ is a $\mathbb{G}$-$C^{*}$-algebra with action
$\gamma\big{(}(a,h)\big{)}:=(\alpha(a),\beta\circ h)$, for all $(a,h)\in
C_{\phi}$. In this way, it is straightforward to check that given a
$\mathbb{G}$-equivariant homomorphism between two
$\mathbb{G}$-$C^{*}$-algebras $(B_{1},\beta_{1})$ and $(B_{2},\beta_{2})$, say
$\phi:B_{1}\longrightarrow B_{2}$, then the isomorphism $C_{\phi}\otimes
T^{op}\cong C_{\phi\otimes id}$ is $\mathbb{G}_{\Omega}$-equivariant. As a
consequence, the functor $\Pi_{T^{op}}$ preserves mapping cone triangles; and
similarly for $\Pi_{T}$. In conclusion, both $\Pi_{T^{op}}$ and $\Pi_{T}$ are
well defined triangulated functors.

######  5.3.1 Lemma.

Following the previous notations, the pair of functors
$(\Pi_{T^{op}},\Pi_{T})$ defines an equivalence of triangulated categories
between $\mathscr{K}\mathscr{K}^{\mathbb{G}}$ and
$\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}}$.

###### Proof.

It only remains to show that $\Pi_{T}\circ\Pi_{T^{op}}\cong
id_{\mathscr{K}\mathscr{K}^{\mathbb{G}}}$ and $\Pi_{T^{op}}\circ\Pi_{T}\cong
id_{\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}}}$.

On the one hand, given an object $(B,\beta)\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$, we have
$\Pi_{T}(\Pi_{T^{op}}(B,\beta))=B\otimes T^{op}\otimes
T\overset{id\otimes\Sigma}{\cong}B\otimes T\otimes T^{op}$ equipped with the
$\mathbb{G}$-action $\widetilde{\widetilde{\beta}}=Ad_{u_{24}}\circ
Ad_{u^{\circ}_{34}}\circ\beta_{14}$. By identifying $T\otimes
T^{op}\cong\mathcal{B}(L^{2}(T))=\mathcal{K}(L^{2}(T))$ along with the
$\mathbb{G}$-action $Ad_{V_{T}}$, where $V_{T}=u_{13}u^{\circ}_{23}$ as in
Lemma 3.2.3, we obtain that $\Pi_{T}(\Pi_{T^{op}}(B,\beta))$ is
$\mathbb{G}$-equivariantly Morita equivalent to $(B,\beta)$; and this
identification is natural. So $\Pi_{T}\circ\Pi_{T^{op}}\cong
id_{\mathscr{K}\mathscr{K}^{\mathbb{G}}}$. On the other hand, given an object
$(C,\gamma)\in Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}})$, we have
$\Pi_{T^{op}}(\Pi_{T}(C,\gamma))=C\otimes T\otimes
T^{op}\overset{id\otimes\Sigma}{\cong}B\otimes T^{op}\otimes T$ equipped with
the $\mathbb{G}$-action
$\widetilde{\widetilde{\gamma}}=Ad_{u^{\circ}_{24}}\circ
Ad_{u_{34}}\circ\gamma_{14}$. By identifying $T^{op}\otimes T=(T\otimes
T^{op})^{op}\cong\mathcal{B}(L^{2}(T))^{op}\cong\mathcal{B}(L^{2}(T^{op}))=\mathcal{K}(L^{2}(T^{op}))$
along with the $\mathbb{G}_{\Omega}$-action $Ad_{V_{T^{op}}}$, where we define
the (ordinary) $\mathbb{G}_{\Omega}$-representation
$V_{T^{op}}=u^{\circ}_{13}u_{23}$, we obtain that
$\Pi_{T^{op}}(\Pi_{T}(C,\gamma))$ is $\mathbb{G}_{\Omega}$-equivariantly
Morita equivalent to $(C,\gamma)$; and this identification is natural. So
$\Pi_{T^{op}}\circ\Pi_{T}\cong
id_{\mathscr{K}\mathscr{K}^{\mathbb{G}_{\Omega}}}$. ∎

######  5.3.2 Theorem (Twisted Green-Julg isomorphism).

Let $\mathbb{G}$ be a compact quantum group. Let $(T,\delta)$ be a torsion
action of projective type of $\mathbb{G}$. Let $u$ be an
$\Omega^{*}$-representation of $\mathbb{G}$ implementing $\delta$ for some
$2$-cocycle $\Omega$ (necessarily of finite type). Then
$\tau_{T}:\mathscr{K}\mathscr{K}\longrightarrow\mathscr{K}\mathscr{K}^{\mathbb{G}}$
is a left adjoint of
$j_{\mathbb{G},T}:\mathscr{K}\mathscr{K}^{\mathbb{G}}\longrightarrow\mathscr{K}\mathscr{K}$
as triangulated functors. More precisely,

$\Psi_{T}:KK^{\mathbb{G}}(C\otimes
T,B)\overset{\sim}{\longrightarrow}KK\big{(}C,B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\big{)},$

for all $C\in Obj(\mathscr{K}\mathscr{K})$ and $(B,\beta)\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$.

###### Proof.

Since the $2$-cocycle $\Omega$ is necessarily of finite type, the twisted
quantum group $\mathbb{G}_{\Omega}$ is compact by Theorem 3.3.7. Given a
$C^{*}$-algebra $C\in Obj(\mathscr{K}\mathscr{K})$ and a
$\mathbb{G}$-$C^{*}$-algebra $(B,\beta)\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$, the previous lemma allows to write
the following:

$KK^{\mathbb{G}}(C\otimes T,B)\cong KK^{\mathbb{G}_{\Omega}}(C\otimes T\otimes
T^{op},B\otimes T^{op})\cong KK^{\mathbb{G}_{\Omega}}(C,B\otimes T^{op}).$

Next, by applying the usual Green-Julg isomorphism we obtain that:
$KK^{\mathbb{G}_{\Omega}}(C,B\otimes
T^{op})\overset{\Psi}{\cong}KK(C,(B\otimes
T^{op})\underset{\widetilde{\beta}}{\rtimes}\mathbb{G}_{\Omega}).$ To
conclude, we observe that $(B\otimes
T^{op})\underset{\widetilde{\beta}}{\rtimes}\mathbb{G}_{\Omega}\cong
B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}$
by virtue of Proposition 5.2.4. Therefore $\Psi_{T}=\Psi\circ\Pi_{T^{op}}$. ∎

###  Compact objects in $\mathscr{K}\mathscr{K}^{\mathbb{G}}$

First of all, as a corollary of the twisted Green-Julg isomorphism established
in Theorem 5.3.2 we obtain a _finiteness presentation property type_ for the
_projective_ torsion objects within the Kasparov category
$\mathscr{K}\mathscr{K}^{\mathbb{G}}$.

######  5.4.1 Theorem.

Let $\mathbb{G}$ be a compact quantum group. Let $(T,\delta)\in
Obj(\mathscr{K}\mathscr{K}^{\mathbb{G}})$ be a torsion action of projective
type of $\mathbb{G}$. Then $(T,\delta)$ is a compact object in
$\mathscr{K}\mathscr{K}^{\mathbb{G}}$, that is, the functor
$KK^{\mathbb{G}}(T,\ \cdot\ )$ is compatible with countable direct sums.

###### Proof.

Let $\\{(B_{n},\beta_{n})\\}_{n\in\mathbb{N}}$ be a countable family of
$\mathbb{G}$-$C^{*}$-algebras. By virtue of the twisted Green-Julg isomorphism
of Theorem 5.3.2 with $C:=\mathbb{C}$ we have

$\begin{split}KK^{\mathbb{G}}(T,\underset{n\in\mathbb{N}}{\bigoplus}B_{n})&\cong
KK\big{(}\mathbb{C},\big{(}\underset{n\in\mathbb{N}}{\bigoplus}B_{n}\big{)}\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\big{)}\cong
K_{0}\big{(}\big{(}\underset{n\in\mathbb{N}}{\bigoplus}B_{n}\big{)}\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\big{)}\\\
&=K_{0}\circ(\cdot)\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\
\Big{(}\underset{n\in\mathbb{N}}{\bigoplus}B^{op}_{n}\Big{)}.\end{split}$

To conclude it is enough to note that the $K_{0}$ functor and the two-sided
crossed product functor
$(\cdot)\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}$
are compatible with countable direct sums. ∎

######  5.4.2 Remark.

Thanks to the more general result in [2], _any_ torsion object in
$\mathscr{K}\mathscr{K}^{\mathbb{G}}$ (not necessarily of projective type) is
a compact object in $\mathscr{K}\mathscr{K}^{\mathbb{G}}$. This is because an
analogous isomorphism as the one from Theorem 5.3.2 is obtained in [2] for any
torsion action of $\mathbb{G}$ (not necessarily of projective type) allowing
to apply the same argument as above.

We believe that this property will be useful to study the equivariant Kasparov
category $\mathscr{K}\mathscr{K}^{\mathbb{G}}$ from a geometrical and
topological perspective according to works by I. Dell’Ambrogio and his
collaborators (see for example [16], [17]). For instance, the above theorem
(and remark) yields that the subcategory
$\mathscr{P}_{D(\mathbb{G})}:=\langle\\{T\ |\
T\in\mathbb{T}\mbox{or}(\widehat{D(\mathbb{G})})\\}\rangle$ is a compactly
generated tensor triangular subcategory of
$\mathscr{K}\mathscr{K}^{D(\mathbb{G})}$ when $\mathbb{G}$ is finite (note
that we need to consider the Drinfeld double construction to provide a tensor
structure on the Kasparov category). In particular,
$(\mathscr{P}_{D(\mathbb{G})},\mathscr{P}_{D(\mathbb{G})}^{\dashv})$ is a
complementary pair of localizing subcategories in
$\mathscr{K}\mathscr{K}^{D(\mathbb{G})}$ (recall Corollary 2.5.16). Hence it
will be interesting to compute its spectrum,
$Spc(\mathscr{P}_{D(\mathbb{G})})$, in the sense of Balmer [6] and to make a
connection with the Baum-Connes property for $\widehat{\mathbb{G}}$.

###  The quantum assembly map

######  5.5.1 Theorem.

Let $\mathbb{G}$ be a compact quantum group. If $\widehat{\mathbb{G}}$ is
permutation torsion-free, then the subcategories
$\mathscr{L}_{\widehat{\mathbb{G}}}$ and $\mathscr{N}_{\widehat{\mathbb{G}}}$
form a complementary pair of localizing subcategories in
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$.

###### Proof.

Since $\widehat{\mathbb{G}}$ is permutation torsion-free, any torsion action
of $\mathbb{G}$ is of projective type. Moreover, by virtue of Theorem 3.1.5
(and Remark 3.2.7), the projective torsion of $\widehat{\mathbb{G}}$ is
parametrised by the family of all (measurable) $2$-cocycles on $\mathbb{G}$,
which is a countable one. We put the following triangulated functor
$F:=(j_{\mathbb{G},T})_{T\in\mathbb{T}or(\widehat{\mathbb{G}})}:\mathscr{K}\mathscr{K}^{\mathbb{G}}\longrightarrow\underset{T\in\mathbb{T}or(\widehat{\mathbb{G}})}{\prod}\mathscr{K}\mathscr{K}$
defined in the obvious way. Then we put
$\mathcal{C}\mathcal{C}_{\mathbb{G}}:=\ker_{Obj}(F)$ and
$\mathcal{J}_{\mathbb{G}}:=\ker_{Hom}(F)$. By virtue of Theorem 5.3.2 the
triangulated functor
$F^{\vdash}:=\underset{T\in\mathbb{T}or(\widehat{\mathbb{G}})}{\bigoplus}\tau_{T}:\underset{T\in\mathbb{T}or(\widehat{\mathbb{G}})}{\prod}\mathscr{K}\mathscr{K}\longrightarrow\mathscr{K}\mathscr{K}^{\mathbb{G}}$
is the left adjoint of $F$ (note that the equivariant KK-theory is additive on
the first variable). Consequently, $\mathscr{K}\mathscr{K}^{\mathbb{G}}$ has
enough $\mathcal{J}$-projective objects and so the pair
$(\langle\mathfrak{p}_{\mathcal{J}}\rangle,\mathcal{C}\mathcal{C}_{\mathbb{G}})$
is complementary in $\mathscr{K}\mathscr{K}^{\mathbb{G}}$ with
$\langle\mathfrak{p}_{\mathcal{J}}\rangle=\langle
F^{\vdash}\big{(}Obj\big{(}\underset{T\in\mathbb{T}or(\widehat{\mathbb{G}})}{\prod}\mathscr{K}\mathscr{K}\big{)}\big{)}\rangle$
(recall Theorem 2.5.13). Note that, by using again the twisted Green-Julg
isomorphism, we have
$\mathcal{C}\mathcal{C}_{\mathbb{G}}=\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}$.
To conclude, recall that $\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}$ is, by
definition, the minimal localising subcategory containing the objects of the
form $C\otimes T$ with $T\in\mathbb{T}or(\widehat{\mathbb{G}})$ and $C\in
Obj(\mathscr{K}\mathscr{K})$. So we have $\langle
F^{\vdash}\big{(}Obj\big{(}\underset{T\in\mathbb{T}or(\widehat{\mathbb{G}})}{\prod}\mathscr{K}\mathscr{K}\big{)}\big{)}\rangle\subset\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}$
and by minimality this inclusion must be an equality. By applying Baaj-
Skandalis duality, we obtain the complementary pair as in the statement. ∎

We are now ready to apply the general Meyer-Nest’s machinery to define a
quantum assembly map for permutation torsion-free discrete quantum groups and
thus a quantum version of the (strong) Baum-Connes property for this class of
discrete quantum groups. More precisely, consider the following homological
functor:

$\begin{array}[]{rcclccl}F:&\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}&\longrightarrow&\mathscr{A}b^{\mathbb{Z}/2},&(A,\delta)&\longmapsto&F(A):=K_{*}(\widehat{\mathbb{G}}\underset{\delta,r}{\ltimes}A).\end{array}$

The quantum assembly map for $\widehat{\mathbb{G}}$ is given by the natural
transformation $\eta^{\widehat{\mathbb{G}}}:\mathbb{L}F\longrightarrow F$
(recall Note 2.5.12).

######  5.5.2 Definition.

Let $\widehat{\mathbb{G}}$ be a discrete quantum group. We say that
$\widehat{\mathbb{G}}$ satisfies the quantum Baum-Connes property (with
coefficients) if the natural transformation
$\eta^{\widehat{\mathbb{G}}}:\mathbb{L}F\longrightarrow F$ is a natural
equivalence. We say that $\widehat{\mathbb{G}}$ satisfies the _strong_ Baum-
Connes property if
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}=\mathscr{L}_{\widehat{\mathbb{G}}}$.

######  5.5.3 Remark.

If $\widehat{\mathbb{G}}:=\Gamma$ is a classical discrete group, then the
Meyer-Nest approach to the Baum-Connes conjecture leads to consider the
following complementary pair of localizing subcategories in
$\mathscr{K}\mathscr{K}^{\Gamma}$:

$\mathscr{L}_{\Gamma}=\langle\\{A\in Obj(\mathscr{K}\mathscr{K}^{\Gamma})\ |\
A\cong Ind^{\Gamma}_{\Lambda}(B)\mbox{ with }\Lambda\in\mathcal{F},B\in
Obj(\mathscr{K}\mathscr{K}^{\Lambda})\rangle,$ $\mathscr{N}_{\Gamma}=\\{A\in
Obj(\mathscr{K}\mathscr{K}^{\Gamma})\ |\ Res^{\Gamma}_{\Lambda}(A)\cong 0\
\forall\Lambda\in\mathcal{F}\\}=\underset{\Lambda\in\mathcal{F}}{\bigcap}ker_{Obj}(Res^{\Gamma}_{\Lambda})\mbox{,}$

where $\mathcal{F}$ denotes the family of all finite subgroups of $\Gamma$.
Observe that every generator of $\mathscr{L}_{\Gamma}$ is a proper
$\Gamma$-$C^{*}$-algebra over $\Gamma/\Lambda$. Conversely, it is well-known
that any proper $\Gamma$-$C^{*}$-algebra is built up out of induced
$C^{*}$-algebras from finite subgroups, so they belong to
$\mathscr{L}_{\Gamma}$ (see [20] for more details). However, one open question
is to decide whether every object of $\mathscr{L}_{\Gamma}$ is a proper
$\Gamma$-$C^{*}$-algebra in $\mathscr{K}\mathscr{K}^{\Gamma}$. It is well-
known that any torsion action of $\widehat{\Gamma}$ is of the form
$C^{*}_{r}(\Lambda,\omega)$ for some finite subgroup $\Lambda\leq\Gamma$ and
some $2$-cocycle $\omega\in H^{2}(\Lambda,S^{1})$. In addition, it can be
shown that $\widehat{\Gamma}\ltimes C^{*}_{r}(\Lambda,\omega)\in
Obj(\mathscr{K}\mathscr{K}^{\Gamma})$ is a proper $\Gamma$-$C^{*}$-algebra
over $\Gamma/\Lambda$, so it is an object in $\mathscr{L}_{\Gamma}$.
Therefore, the previous discussion yields that $\mathscr{L}_{\Gamma}$ contains
the subcategory $\mathscr{L}_{\widehat{\mathbbold{\Gamma}}}$ where
$\mathbbold{\Gamma}:=(C^{*}_{r}(\Gamma),\Delta_{\Gamma})$ as defined
previously in the more general setting of quantum groups. In other words, the
condition
$\mathscr{K}\mathscr{K}^{\widehat{\mathbbold{\Gamma}}}=\mathscr{L}_{\widehat{\mathbbold{\Gamma}}}$
is stronger than the condition
$\mathscr{K}\mathscr{K}^{\Gamma}=\mathscr{L}_{\Gamma}$. The latter translates
into the usual strong Baum-Connes conjecture for $\Gamma$, that is, the
existence of a $\gamma$-element that equals $\mathbbold{1}_{\mathbb{C}}$ (see
[27, Section 8] for a proof). For this reason, it is convenient to refer to
the property
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}=\mathscr{L}_{\widehat{\mathbb{G}}}$
as _$\mathscr{L}_{\widehat{\mathbb{G}}}$ -strong_ Baum-Connes property instead
of simply _strong_ Baum-Connes property.

## References

  * [1] Arano, Y., and De Commer, K. Torsion-freeness for fusion rings and tensor ${C}^{*}$-categories.  J. Noncommut. Geom. 13, 1 (2019), 35–58.
  * [2] Arano, Y., and Skalski, A. On the Baum-Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture.  Proc. Amer. Math. Soc. 149 (2021), 5237–5254.
  * [3] Baaj, S., and Crespo, J. Equivalence monoidale de groupes quantiques et ${K}$-théorie bivariante.  Bulletin de la société mathématique de France 4, 145 (2017), 711–802.
  * [4] Baaj, S., and Skandalis, G. ${C}^{*}$-algèbres de Hopf et théorie de Kasparov équivariante.  K-theory 2, 6 (1989), 683–721.
  * [5] Baaj, S., and Skandalis, G. Unitaires multiplicatifs et dualité pour les produits croisés de ${C}^{*}$-algèbres.  Annales Scientifiques de l’Ecole Normale Supérieure 26, 4 (1993), 425–488.
  * [6] Balmer, P. The spectrum of prime ideals in tensor triangulated categories.  J. Reine Angew. Math. 588 (2005), 149–168.
  * [7] Bedos, E. C., and Conti, R. On discrete twisted ${C}^{*}$-dynamical systems, hilbert ${C}^{*}$-modules and regularity.  Münster Journal of Mathematics, 5 (2012), 183–208.
  * [8] Bichon, J., De Rijdt, A., and Vaes, S. Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups.  Comm. Math. Phys 262 (2006), 703–728.
  * [9] Boca, F. Ergodic actions of compact matrix pseudogroups on ${C}^{*}$-algebras.  Recent Advances in Operator Algebras, Astérisque 232 (1995), 93–110.
  * [10] Cheng, C. A character theory for projective representations of finite groups.  Linear Algebra and its applications 469 (2015), 230–242.
  * [11] De Commer, K. On cocycle twisting of compact quantum groups.  Journal of Functional Analysis 258 (2010), 3362–3375.
  * [12] De Commer, K. Comonoidal ${W}^{*}$-Morita equivalence for von Neumann bialgebras.  Journal of Noncommutative Geometry 5 (2011), 547–571.
  * [13] De Commer, K. Galois objects and cocycle twisting for locally compact quantum groups.  Journal of Operator Theory 66, 1 (2011), 59–106.
  * [14] De Commer, K. Projective representations for compact quantum groups.  Journal of Functional Analysis 260, 12 (2011), 3596–3644.
  * [15] De Commer, K. On a Morita equivalence between the duals of quantum ${S}{U}(2)$ and quantum ${E}(2)$.  Adv. Math. 2, 229 (2012), 1047–1079.
  * [16] Dell’Ambrogio, I. Tensor triangular geometry and KK-theory.  J. Homotopy Relat. Struct. 5, 1 (2010), 319–358.
  * [17] Dell’Ambrogio, I., and Meyer, R. The spectrum of equivariant Kasparov theory for cyclic groups of prime order.  Ann. ${K}$-Theory 6, 3 (2021), 543–558.
  * [18] Freslon, A., and Martos, R. Torsion and ${K}$-theory for some free wreath products.  Int. Math. Res. Not. 2020, 6 (2018), 1639–1670.
  * [19] Giordano, T., Kerr, D., Phillips, N. C., and Toms, A. Crossed Products of ${C}*$-Algebras, Topological Dynamics, and Classification.  Birkhäuser, 2018.
  * [20] Guentner, E., Higson, N., and Trout, J. Equivariant E-theory for ${C}^{*}$-algebras.  Memoirs of the American Mathematical Society 703 (2000).
  * [21] Higson, N., and Kasparov, G. E-theory and KK-theory for groups which act properly and isometrically on hilbert space.  Inventiones mathematicae 144, 1 (2001), 23–74.
  * [22] Holm, T., Jorgensen, P., and Rouquier, R. Triangulated Categories.  Cambridge University Press, 2010.
  * [23] Lafforgue, V. La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques.  J. Noncommutative Geometry (2012), 1–197.
  * [24] Martos, R. The Baum-Connes conjecture for Quantum Groups. Stability properties and ${K}$${K}$-theory computations.  PhD thesis, Université Paris VII, 2018.
  * [25] Meyer, R. Categorical aspects of bivariant ${K}$-theory.  ${K}$-theory and Noncommutative Geometry, EMS Ser. Congr. Rep (2008), 1–39.
  * [26] Meyer, R. Homological algebra in bivariant K-theory and other triangulated categories II.  Tbilisi Mathematical Journal 1 (2008), 165–210.
  * [27] Meyer, R., and Nest, R. The Baum–Connes conjecture via localisation of categories.  Topology 45, 2 (2006), 209–259.
  * [28] Meyer, R., and Nest, R. An analogue of the Baum-Connes isomorphism for coactions of compact groups.  Mathematica Scandinavica 100, 2 (2007), 301–316.
  * [29] Meyer, R., and Nest, R. Homological algebra in bivariant K-theory and other triangulated categories I.  London Mathematical Society Lecture Notes 375 (2010), 236–290.
  * [30] Neeman, A. Triangulated Categories.  Princeton University Press, 2014.
  * [31] Neshveyev, S., and Tuset, L. Compact quantum groups and their representation categories.  Société mathématique de France, 2013.
  * [32] Neshveyev, S., and Tuset, L. Deformation of ${C}^{*}$-algebras by cocycles on locally compact quantum groups.  Advances in Mathematics 254 (2014), 454–496.
  * [33] Nikshych, D., and Vainerman, L. Finite quantum groupoids and their applications.  In New directions in Hopf algebras, vol. 43 of Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 2002, pp. 211–262.
  * [34] Packer, J. Transformation group ${C}^{*}$-algebras: a selective survey.  Contemporary Mathematics 167 (1994), 183–183.
  * [35] Packer, J., and Raeburn, I. Twisted crossed products of ${C}^{*}$-algebras.  Math. Proc. Camb. Phil. Soc. 106 (1989), 293–311.
  * [36] Quigg, J. C. Duality for Reduced Twisted Crossed Products of ${C}^{*}$-algebras.  Indiana University Mathematics Journal 35, 3 (1986), 549–571.
  * [37] Soł tan, P. M. Quantum Bohr compactification.  Illinois J. Math. 49, 4 (2005), 1245–1270.
  * [38] Soltan, P. M. Quantum ${S}{O}(3)$ groups and quantum group actions on ${M}_{2}$.  J. Noncommut. Geom. 4, 1 (2010), 1–28.
  * [39] Takesaki, M. Theory of operator algebras II, vol. 125.  Encyclopaedia Math. Sci. Springer-Verlag, 2003.
  * [40] Timmermann, T. An invitation to quantum groups and duality: From Hopf algebras to multiplicative unitaries and beyond.  European Mathematical Society, 2008.
  * [41] Vaes, S. The unitary implementation of a locally compact quantum group action.  Journal of Functional Analysis 180, 2 (2001), 426–480.
  * [42] Vaes, S., and Vainerman, L. Extensions of locally compact quantum groups and the bicrossed product construction.  Advances in Mathematics 175 (2003), 1–101.
  * [43] Vergnioux, R. KK-théorie équivariante et opérateur de Julg-Valette pour les groupes quantiques.  PhD thesis, Université Paris VII, 2002.
  * [44] Vergnioux, R., and Voigt, C. The ${K}$-theory of free quantum groups.  Mathematische Annalen 357, 1 (2013), 355–400.
  * [45] Voigt, C. On the structure of quantum automorphism groups.  Journal für die reine und angewandte Mathematik (Crelles Journal) (2015).
  * [46] Woronowicz, S. L. Compact quantum groups.  Symétries quantiques (1998), 845–884.
  * [47] Yamashita, M. Deformation of algebras associated to group cocycles.  arXiv (2011).