# Quantum direct products and the Künneth class

Rubén Martos Department of Mathematical Sciences, University of Copenhagen
(Denmark). R.M. is supported by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
895141.

Abstract. We introduce a Künneth class in the quantum equivariant setting
inspired by the pioneer work by J. Chabert, H. Oyono-Oyono and S. Echterhoff,
which allows to relate the quantum Baum-Connes property with the Künneth
formula by generalising some key results of Chabert-Oyono-Oyono-Echterhoff to
discrete quantum groups. Finally, we make the observation that the C∗-algebra
defining a compact quantum group with dual satisfying the strong quantum Baum-
Connes property belongs to the Künneth class. This allows to obtain some
K-theory computations for quantum direct products based on earlier work by
Voigt and Vergnioux-Voigt.

Keywords. Baum-Connes conjecture, compact/Discrete quantum groups, K-theory,
Künneth formula, quantum direct product, tensor product of C∗-algebras,
torsion, triangulated categories, universal coefficient theorem.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notations and conventions
    2. 2.2 Actions of quantum groups
    3. 2.3 Quantum direct products
    4. 2.4 The equivariant Kasparov category and the quantum Baum-Connes assembly map
  3. 3 The Künneth classes
    1. 3.1 Preparatory observations
    2. 3.2 The Künneth class $\mathcal{N}_{\widehat{\mathbb{G}}}$
  4. 4 Relating the quantum Baum-Connes property with the Künneth formula
  5. 5 Some K-theory computations

##  Introduction

Universal Coefficient Theorems in algebraic topology establish a connection
between ordinary homology (resp. cohomology) with homology (resp. cohomology)
with coefficients. In noncommutative geometry, C∗-algebras are viewed as
noncommutative analogues of topological spaces and as such it is reasonable to
extend ideas from algebraic topology to a noncommutative framework. In this
sense, K-theory of C∗-algebras is viewed as a homology theory for C∗-algebras,
which provides important invariants for C∗-algebras (e.g. AF-algebras are
completely classified through K-theory [9]). Dually, K-homology for
C∗-algebras is a cohomology theory for C∗-algebras, which agrees with the Ext-
functor in the commutative unital case. K-theory and K-homology can be related
to each other by means of the index theory of elliptic pseudo-differential
operators (e.g. the Atiyah-Singer’s theorem). Kasparov KK-theory is in turn a
bivariant K-theory, which gathers together both K-theory and K-homology of
C∗-algebras. KK-theory plays a central role in the classification program of
C∗-algebras, but it is also relevant to approach diverse problems outside
noncommutative geometry, e.g. the Novikov conjecture.

The Universal Coefficient Theorem (_UCT_ for short) of J. Rosenberg and C.
Schochet [26] approximates the bivariant K-theory of Kasparov in terms of
ordinary K-theory. More precisely, given two (separable) C∗-algebras $A$ and
$B$, there exists a short exact sequence:

$\text{Ext}^{1}_{\mathbb{Z}}(K_{*}(A),K_{*}(B))\rightarrowtail
KK(A,B)\twoheadrightarrow\text{Hom}(K_{*}(A),K_{*}(B)),$

provided that $A$ belongs to certain bootstrap class. A remarkable consequence
of this is that K-equivalences between C∗-algebras satisfying the UCT lift to
KK-equivalences. A classical companion to the UCT is the _Künneth theorem_
(cf. [26]), which asserts that given two (separable) C∗-algebras $A$ and $B$,
there exists a short exact sequence:

$K_{*}(A)\otimes K_{*}(B)\rightarrowtail K_{*}(A\otimes
B)\twoheadrightarrow\text{Tor}^{\mathbb{Z}}_{1}(K_{*}(A),K_{*}(B)),$

provided that $A$ belongs to certain bootstrap class. Roughly speaking,
Künneth theorem allows to compute the K-theory of a tensor product of two
C∗-algebras in terms of the K-theory of the corresponding factors (e.g. when
$K(A)$ or $K(B)$ is torsion-free). When the the first arrow in the above
diagram is an isomorphism, we refer to $K_{*}(A)\otimes K_{*}(B)\cong
K_{*}(A\otimes B)$ as the _Künneth formula_. Classically, the Künneth formula
identifies the (co-)homology of a product of topological spaces with the
tensor product of (co-)homologies.

In [5], J. Chabert, S. Echterhoff and H. Oyono-Oyono establish a connection
between the Baum-Connes conjecture for a locally compact group $G$ with
coefficients in a C∗-algebra $A$ and the Künneth formula for the K-theory of
tensor products by the corresponding crossed product
$A\underset{r}{\rtimes}G$. One remarkable application of the techniques
developed by J. Chabert, S. Echterhoff and H. Oyono-Oyono is a permanence
property of the Baum-Connes conjecture for a direct product of locally compact
groups with trivial coefficients.

In this article, we establish such a connection in analogy to [5] in the realm
of discrete quantum groups and prove a similar permanence property for the
quantum counterpart of the Baum-Connes conjecture. It is important to say that
such a study was partially initiated by the author in [19], but only under
torsion-freeness assumption. This was due mainly to two technical
difficulties: the classification of torsion actions for quantum direct
products and the formulation of a quantum assembly map for arbitrary discrete
quantum groups. Let us give more details about these issues.

The Baum-Connes conjecture has been formulated in 1982 by P. Baum and A.
Connes. We still do not know any counter example to the original conjecture
but it is known that the one with coefficients is false. For this reason we
refer to the Baum-Connes conjecture with coefficients as the _Baum-Connes
property_ (_BC property_ for short). The conjecture aims to understand the
link between two operator K-groups of different nature, which would establish
a strong connection between geometry and topology in a more abstract and
general index-theory context. More precisely, if $G$ is a (second countable)
locally compact group and $A$ is a (separable) $G$-$C^{*}$-algebra, then the
BC property for $G$ with coefficients in $A$ claims that the assembly map
$\mu^{G}_{A}:K^{top}_{*}(G;A)\longrightarrow K_{*}(A\underset{r}{\rtimes}G)$
is an isomorphism, where $K^{top}_{*}(G;A)$ is the equivariant K-homology with
compact support of $G$ with coefficients in $A$ and
$K_{*}(A\underset{r}{\rtimes}G)$ is the K-theory of the reduced crossed
product $A\underset{r}{\rtimes}G$. This property has been proved for a large
class of groups (e.g. a-T-menable groups [14] or hyperbolic groups [16]).

On the one hand, a major problem when studying the quantum counterpart of the
BC property for a discrete quantum group $\widehat{\mathbb{G}}$ is the torsion
structure of $\widehat{\mathbb{G}}$. Indeed, if $G$ is a discrete group, its
torsion phenomenon is completely described in terms of the finite subgroups of
$G$ and encoded in the localizing subcategory of $\mathscr{KK}^{G}$ of
_compactly induced $G$-$C^{*}$-algebras_, denoted by $\mathscr{L}_{G}$,
according to the Meyer-Nest reformulation [22]. We say that $G$ satisfies the
_strong_ BC property if $\mathscr{L}_{G}=\mathscr{KK}^{G}$, which corresponds,
in usual terms, to the existence of a $\gamma$-element that equals
$\mathbbold{1}_{\mathbb{C}}$ (cf. [22]). This approach yields as well a
characterization of the BC property for a discrete group $G$ _only_ in terms
of finite subgroups, K-theory and crossed products. The notion of torsion for
a genuine discrete quantum group, $\widehat{\mathbb{G}}$, has been introduced
firstly by R. Meyer and R. Nest [23], [21] in terms of ergodic actions of
$\mathbb{G}$. It has been re-interpreted later by Y. Arano and K. De Commer in
terms of fusion rings and module $C^{*}$-categories [1].

Given a compact quantum group $\mathbb{G}$, an important question in this
context is to classify all torsion actions of $\mathbb{G}$ (at least up to
equivariant Morita equivalence). Such a classification is known for the most
common examples of compact quantum groups, e.g for the q-deformation of
$SU(2)$, the free unitary quantum group $U^{+}(n)$, the free orthogonal
quantum group $O^{+}(n)$ or the quantum permutation group $S^{+}_{N}$ with
$n,N\in\mathbb{N}$ (cf. [30, 31, 1]). In fact, $\widehat{SU_{q}(2)}$,
$\widehat{U^{+}(n)}$ and $\widehat{O^{+}(n)}$ are all torsion-free and the
only, up to equivariant Morita equivalence, non-trivial torsion action of
$S^{+}_{N}$ is its structural action as quantum automorphism group of
$\mathbb{C}^{N}$. However, little is known about the classification of torsion
actions for general constructions of quantum groups. In this direction, the
only, to the best knowledge to the author, such a general result was given in
[11] by the author in collaboration with A. Freslon. It is shown in [11] that
torsion actions of a free product of compact quantum groups are in one-to-one
correspondence, up to equivariant Morita equivalence, with torsion actions of
each of the factors.

It is important to mention that by virtue of the work [1] by Y. Arano and K.
De Commer, we know that if both $\widehat{\mathbb{G}}$ and
$\widehat{\mathbb{H}}$ are torsion-free, then
$\widehat{\mathbb{G}\times\mathbb{H}}$ is torsion-free too. The converse is
also true because both $\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ can
be viewed as _divisible_ discrete quantum subgroups of
$\widehat{\mathbb{G}\times\mathbb{H}}$ and torsion-freeness is preserved under
divisible discrete quantum subgroups as shown in [20]. However, it is an open
problem to classify all torsion actions of $\mathbb{G}\times\mathbb{H}$. For
example, if $\mathbb{G}=:G$ and $\mathbb{H}=:H$ are classical discrete groups,
then it is well-known that the subgroups of their direct product $G\times H$
are _not_ always of the form $G^{\prime}\times H^{\prime}$ with
$G^{\prime}\leq G$ and $H^{\prime}\leq H$ subgroups. It suffices to take $G=H$
and consider $G$ as a subgroup of $G\times G$ embedded diagonally. The general
classification of the subgroups of a direct product of groups is given by the
celebrated _Goursat’s lemma_ (cf. [12]). In particular, not all finite
subgroups of $G\times H$ are classified as direct product of finite subgroups
of each factor. This already indicates that the classification of torsion
actions of a quantum direct product $\mathbb{G}\times\mathbb{H}$ in terms of
torsion actions of $\mathbb{G}$ and torsion actions of $\mathbb{H}$ would be a
non trivial task.

One could restrict attention to _projective torsion_. A (non-trivial)
projective torsion action of a compact quantum group $\mathbb{G}$ is given by
a simple finite dimensional C∗-algebra equipped with an ergodic action of
$\mathbb{G}$ (not equivariantly Morita equivalent to $\mathbb{C}$). It is
shown in [8] by the author in collaboration with K. De Commer and R. Nest that
these actions by $\mathbb{G}$ are in bijective correspondence with
(measurable) $2$-cocycles on $\mathbb{G}$, i.e. with twisted representations
of $\mathbb{G}$. This is in complete analogy with the situation of classical
compact groups. Accordingly, the classification of projective torsion of a
quantum direct product $\mathbb{G}\times\mathbb{H}$ turns into the
classification of (measurable) $2$-cocycles on $\mathbb{G}\times\mathbb{H}$.
In the classical setting of discrete groups, such a classification goes back
to work by K. Tahara [27]. The author has initiated an investigation in this
direction in order to classify projective torsion of a quantum direct product,
which will be the subject of a subsequent paper.

On the other hand, in order to apply the Meyer-Nest strategy in the quantum
setting, one needs a _complementary pair_ of localizing subcategories,
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$,
where $\mathscr{L}_{\widehat{\mathbb{G}}}$ must encode the torsion phenomenon
of $\widehat{\mathbb{G}}$. A candidate was proposed in [23] and [31] for
specific examples (see also [18, Section 4.1.2] for a description for general
discrete quantum groups), but it has been an open question whether the
corresponding pair is complementary in $\mathscr{KK}^{\widehat{\mathbb{G}}}$,
which prevented from having a definition of a quantum assembly map whenever
$\widehat{\mathbb{G}}$ is not torsion-free. Recently, Y. Arano and A. Skalski
[2] have observed that the candidates for $\mathscr{L}_{\widehat{\mathbb{G}}}$
and $\mathscr{N}_{\widehat{\mathbb{G}}}$ form indeed a complementary pair of
subcategories in $\mathscr{KK}^{\widehat{\mathbb{G}}}$, which allows to define
a quantum assembly map for every discrete quantum group $\widehat{\mathbb{G}}$
(torsion-free or not). Moreover, following a different approach by studying
the projective representation theory of a compact quantum group, the same
conclusion is reached for permutation torsion-free discrete quantum groups by
the author in collaboration with K. De Commer and R. Nest [8]. More details
about this formulation can be found in Section 2.4.

In the present paper we are able to improve several results in this matter
appearing already in [19]. For instance, the functor
$\mathcal{Z}:\mathscr{KK}^{\widehat{\mathbb{G}}}\times\mathscr{KK}^{\widehat{\mathbb{H}}}\rightarrow\mathscr{KK}^{\widehat{\mathbb{G}\times\mathbb{H}}}$
given by the exterior tensor product of Kasparov triples allows to describe
appropriately, _and without any torsion-freeness assumption_ , the quantum BC
property for $\widehat{\mathbb{G}\times\mathbb{H}}$ in terms of the quantum BC
property for $\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$. Namely, if
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ satisfy the strong quantum
BC property, then we show that $\widehat{\mathbb{G}\times\mathbb{H}}$
satisfies the BC property with coefficients in $A\otimes B$, for all
$\widehat{\mathbb{G}}$-C∗-algebra $A$ and all
$\widehat{\mathbb{H}}$-C∗-algebra $B$ (cf. Theorem 3.1.5). Accordingly, an
analogous assertion about the _usual_ quantum BC property needs further
hypothesis related to the Künneth formula in order to compute the K-theory of
a tensor product. Therefore, we are lead to consider a (quantum) equivariant
analogue of the Künneth class $\mathcal{N}$, say
$\mathcal{N}_{\widehat{\mathbb{G}}}$, containing those
$\widehat{\mathbb{G}}$-C∗-algebras which make possible such a K-theory
computation. This is done in analogy to the work by J. Chabert, S. Echterhoff
and H. Oyono-Oyono in [5]. If $\widehat{\mathbb{G}}$ is a classical locally
compact group $G$, then $\mathcal{N}_{\widehat{\mathbb{G}}}=\mathcal{N}_{G}$
as defined in [5]. Furthermore, our approach, as based in the Meyer-Nest
categorical framework, yields a characterisation of the objects in the
equivariant Künneth class in terms of the non-equivariant one, up to replacing
$A$ by a $\mathscr{L}_{G}$-simplicial approximation of $A$. This study is
contained in Section 3.

In Section 4 we generalise some key results appearing in [5] about the
connection between the BC property with the Künneth formula. Namely, let $A$
be a $\widehat{\mathbb{G}}$-C∗-algebra and $B$ a C∗-algebra. Then we show that
$A\in\mathcal{N}_{\widehat{\mathbb{G}}}$ $\Leftrightarrow$
$A\underset{r}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$ provided that
$\widehat{\mathbb{G}}$ satisfies the BC property with coefficients in
$A\otimes B$ (cf. Proposition 4.1). One remarkable result in [5] is the
following permanence property of the BC property for a direct product of
locally compact groups. Let $G$ and $H$ be two locally compact groups
satisfying the BC property with trivial coefficients. If $C^{*}(G)$ or
$C^{*}(H)$ belongs to $\mathcal{N}$, then $G\times H$ satisfies the BC
property with trivial coefficients (cf. [5, Theorem 5.3]). The analogue
statement for quantum groups is stated and proven in Theorem 4.5. In the
quantum setting further hypotheses are needed concerning the behaviour of
$\mathbb{C}$ with respect to the _equivariant_ Künneth formula. Namely, we
have to assume that $\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$. This
supplementary condition is automatically fulfilled in the classical setting
because $\mathbb{C}$ is a type I C∗-algebra and $\mathcal{N}_{G}$ contains all
type I $G$-C∗-algebras by virtue of [5, Theorem 0.1]. In the quantum setting,
a similar related result is Theorem 5.1. However, to the best knowledge of the
author, it is not known for instance whether
$\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$ for every discrete quantum
group $\widehat{\mathbb{G}}$. One reason for this is that in our approach the
objects in $\mathcal{N}_{\widehat{\mathbb{G}}}$ are characterised in terms of
objects in $\mathcal{N}$ up to a _$\mathscr{L}_{\widehat{\mathbb{G}}}$
-simplicial approximation_, which entails to study the localisation functor
$L$ in relation with crossed products and the _equivariant_ Künneth class (cf.
Remark 5.2 for an extended discussion). One possibility to do so might be to
adapt the _Going-Down technique_ from [5] based on Theorem 5.1.

Finally, we make the observation that [2, Theorem 5.2] and [2, Corollary 5.5]
can be also obtained for the Künneth class instead of the bootstrap class (cf.
Theorem 5.1). In particular, one obtains that $C(\mathbb{G})\in\mathcal{N}$ as
soon as $\widehat{\mathbb{G}}$ satisfies the _strong_ quantum BC property. In
the classical setting, one can argue as follows. If $G$ satisfies the BC
property with coefficients (_a fortriori_ when $G$ satisfies the _strong_ BC
property), then we can apply [5, Proposition 4.9] (cf. Proposition 4.1 for the
quantum counterpart). In particular, since we always have
$\mathbb{C}\in\mathcal{N}_{G}$ as explained above, then [5, Proposition 4.9]
implies that $C^{*}(G)=\mathbb{C}\underset{r}{\rtimes}G\in\mathcal{N}$. This
observation allows to put the Künneth formula to work by computing K-theory
groups of the C∗-algebras defining quantum direct products in relevant
examples based, for instance, on works by Voigt and Vergnioux-Voigt (cf. [30],
[29], [31]). See Section 5.

Acknowledgements. The author is grateful to the anonymous referee for their
valuable comments.

##  Preliminaries

###  Notations and conventions

Let us fix the notations and the conventions that we use throughout the whole
article.

Whenever $\mathscr{C}$ denotes a category, we shall assume that $\mathscr{C}$
is essentially small, so morphisms $Hom_{\mathscr{C}}(\cdot,\cdot)$ form sets.
Given a category $\mathscr{C}$, we denote by $\mathscr{C}^{op}$ its opposite
category. We say that $\mathscr{C}$ is _countable additive_ if it is additive
and it admits _countable direct sums_. If $F$ is an _additive_ functor on an
additive category, it is, by definition, compatible with _finite_ direct sums.
The categories considered in the present paper are assumed to be countable
additive. Whenever we require an additive functor to be compatible with
_infinite (countable)_ direct sums, it will be explicitly indicated. We denote
by $\mathscr{A}b$ the abelian category of abelian groups and by
$\mathscr{A}b^{\mathbb{Z}/2}$ the abelian category of $\mathbb{Z}/2$-graded
abelian groups. We denote by $\text{C}^{*}\text{-Alg}$ (resp.
$\widehat{\mathbb{G}}\text{-C}^{*}\text{-Alg}$, where $\mathbb{G}$ is a
compact quantum group) the category of separable C∗-algebras (resp.
$\widehat{\mathbb{G}}\text{-C}^{*}$-algebras) with $*$-homomorphisms (resp.
$\widehat{\mathbb{G}}$-equivariant $*$-homomorphisms) as morphisms. We write
$\mathscr{D}\subset\mathscr{C}$ whenever $\mathscr{D}$ is a (full) subcategory
of $\mathscr{C}$. We use the symbol $0$ to denote either the trivial (abelian)
group, the trivial C∗-algebra, the trivial category or the zero object of an
additive category. The context will distinguish the specific situation.

If $E$ is a $\mathbb{C}$-vector space and $\mathcal{S}$ is a subset of vectors
of $E$, then we write $span\,\mathcal{S}$ for the corresponding
$\mathbb{C}$-vector subspace generated by $\mathcal{S}$. If $(E,||\cdot||)$ is
a normed $\mathbb{C}$-vector space and $\mathcal{S}\subset E$ we write
$[span\,\mathcal{S}]:=\overline{span}\,\mathcal{S}$.

Let $H$ be a Hilbert space. We denote by $\mathcal{B}(H)$ (resp.
$\mathcal{K}(H)$) the space of all linear bounded (resp. compact) operators on
$H$. If $A$ is a C∗-algebra and $H$ a Hilbert $A$-module, we denote by
$\mathcal{L}_{A}(H)$ (resp. $\mathcal{K}_{A}(H)$) the space of all (resp.
compact) adjointable operators on $H$. Hilbert $A$-modules are considered to
be _right $A$-modules_, so that the corresponding inner products are
considered to be conjugate-linear on the left and linear on the right.

All our C∗-algebras (except for obvious exceptions such as multiplier
C∗-algebras and von Neumann algebras) are supposed to be _separable_ and all
our Hilbert modules are supposed to be _countably generated_. If $A$ is a
C∗-algebra and $\mathcal{S}$ is a subset of elements in $A$, we write
$C^{*}\langle\mathcal{S}\rangle:=C^{*}\langle\mathcal{S}\cup\mathcal{S}^{*}\rangle$
for the corresponding C∗-subalgebra of $A$ generated by $\mathcal{S}$, that
is, the intersection of all C∗-subalgebras of $A$ containing $\mathcal{S}$.
The symbol $\otimes$ stands for the minimal tensor product of C∗-algebras and
the exterior tensor product of Hilbert modules depending on the context. The
symbol $\underset{\max}{\otimes}$ stands for the maximal tensor product of
C∗-algebras. In any of the previous cases, the _elementary tensors_ in the
corresponding tensor product are denoted simply by $\otimes$ and the context
will distinguish the specific situation. If $A$ and $B$ are two C∗-algebras,
$\Sigma:A\otimes B\longrightarrow B\otimes A$ denotes the flip map. The symbol
$\Sigma$ is used as well for the suspension functor in the framework of
triangulated categories. The context will distinguish the specific situation.
We use systematically the leg numbering notation, so if $H$ is a Hilbert space
then $X_{12}=X\otimes 1\in\mathcal{B}(H^{\otimes 3})$ for
$X\in\mathcal{B}(H^{\otimes 2})$, etc.

If $S,A$ are C∗-algebras, we denote by $M(A)=\mathcal{L}_{A}(A)$ the
multiplier algebra of $A$ and we put $\widetilde{M}(A\otimes S):=\\{x\in
M(A\otimes S)\ |\ x(id_{A}\otimes S)\subset A\otimes S\mbox{ and
}(id_{A}\otimes S)x\subset A\otimes S\\}$, which contains $A\otimes M(S)$. If
$H$ is a Hilbert $A$-module, we put $M(H):=\mathcal{L}_{A}(A,H)$, which
contains canonically $H\cong\mathcal{K}_{A}(A,H)$. We put
$\widetilde{M}(H\otimes S):=\\{X\in M(H\otimes S)\ |\ X(id_{A}\otimes
S)\subset H\otimes S\mbox{ and }(id_{H}\otimes S)X\subset H\otimes S\\}$,
which contains $H\otimes M(S)$.

If $\mathbb{G}=(C(\mathbb{G}),\Delta)$ is a compact quantum group, the set of
all unitary equivalence classes of irreducible unitary finite dimensional
representations of $\mathbb{G}$ is denoted by $\text{Irr}(\mathbb{G})$. The
trivial representation of $\mathbb{G}$ is denoted by $\epsilon$. If
$x\in\text{Irr}(\mathbb{G})$ is such a class, we write
$u^{x}\in\mathcal{B}(H_{x})\otimes C(\mathbb{G})$ for a representative of $x$
and $H_{x}$ for the finite dimensional Hilbert space on which $u^{x}$ acts (we
write $dim(x):=n_{x}$ for the dimension of $H_{x}$). The matrix coefficients
of $u^{x}$ with respect to an orthonormal basis
$\\{\xi^{x}_{1},\ldots,\xi^{x}_{n_{x}}\\}$ of $H_{x}$ are defined by
$u^{x}_{ij}:=(\omega_{ij}\otimes id)(u^{x})$ for all $i,j=1,\ldots,n_{x}$. The
linear span of matrix coefficients of all finite dimensional representations
of $\mathbb{G}$ is denoted by $\text{Pol}(\mathbb{G})$, which is a Hopf
$*$-algebra with co-multipliction $\Delta$ and co-unit and antipode denoted by
$\varepsilon_{\mathbb{G}}$ and $S_{\mathbb{G}}$, respectively. Given
$x,y\in\text{Irr}(\mathbb{G})$, the tensor product of $x$ and $y$ is denoted
by $x\otimes y$.

The Haar state of $\mathbb{G}$ is denoted by $h_{\mathbb{G}}$. The GNS
construction corresponding to $h_{\mathbb{G}}$ is denoted by
$(L^{2}(\mathbb{G}),\lambda,\xi_{\mathbb{G}})$. We also write
$\Lambda(x)=\lambda(x)\xi_{\mathbb{G}}$ for $x\in C(\mathbb{G})$. We adopt the
standard convention for the inner product on $L^{2}(\mathbb{G})$, which means
that $\langle\Lambda(x),\Lambda(y)\rangle:=h_{\mathbb{G}}(x^{*}y)$ for all
$x,y\in C(\mathbb{G})$. We suppress the notation $\lambda$ in computations so
that we simply write $x\Lambda(y)=\Lambda(xy)$ for all $x,y\in C(\mathbb{G})$.
We will make the standing assumption that $h_{\mathbb{G}}$ is faithful, so we
only work with the reduced form of a compact quantum group, hence
$C_{r}(\mathbb{G})=C(\mathbb{G})$ unless the contrary is specified. The
maximal form of $\mathbb{G}$ is given by the C∗-envelopping algebra of
$\text{Pol}(\mathbb{G})$, denoted by $C_{m}(\mathbb{G})$.

The Haar state extends uniquely to a normal faithful state on
$L^{\infty}(\mathbb{G})$, and we denote by $J_{\mathbb{G}}$ the associated
modular conjugation on $L^{2}(\mathbb{G})$. Let $I_{0}$ be the anti-linear
involutive map $\Lambda(\text{Pol}(\mathbb{G}))\rightarrow L^{2}(\mathbb{G})$
defined by $\Lambda(x)\mapsto\Lambda(S(x)^{*})$ for
$x\in\text{Pol}(\mathbb{G})$. Then $I_{0}$ is closeable, and we denote
$I=\widehat{J}_{\mathbb{G}}|I|$ for the polar decomposition of its closure.
The map $R(x)=\widehat{J}_{\mathbb{G}}x^{*}\widehat{J}_{\mathbb{G}}$, for all
$x\in C(\mathbb{G})$, is a well-defined anti-multiplicative and anti-co-
multiplicative map on $C(\mathbb{G})$ preserving $\text{Pol}(\mathbb{G})$,
called _unitary antipode_. We put
$U_{\mathbb{G}}=J_{\mathbb{G}}\widehat{J}_{\mathbb{G}}=\widehat{J}_{\mathbb{G}}J_{\mathbb{G}}\in\mathcal{B}(L^{2}(\mathbb{G}))$
for the symmetry of the standard Kac system associated to $\mathbb{G}$. The we
put $\rho(a):=U_{\mathbb{G}}\lambda(a)U_{\mathbb{G}}$, for all $a\in
C(\mathbb{G})$.

Given a compact quantum group $\mathbb{G}$, we put
$c_{0}(\widehat{\mathbb{G}}):=[\\{(id\otimes\eta)(V_{\mathbb{G}})\ |\
\eta\in\mathcal{B}(L^{2}(\mathbb{G}))_{*}\\}]\subset\mathcal{B}(L^{2}(\mathbb{G}))$,
where $V_{\mathbb{G}}$ denotes the right regular representation of
$\mathbb{G}$ on $L^{2}(\mathbb{G})$. Recall that $c_{0}(\widehat{\mathbb{G}})$
is a C∗-algebra which defines a locally compact quantum group with co-
multiplication $\widehat{\Delta}(x):=\Sigma V_{\mathbb{G}}^{*}(1\otimes
x)V_{\mathbb{G}}\Sigma$, for all $x\in c_{0}(\widehat{\mathbb{G}})$. There
exists a natural isomorphism $c_{0}(\widehat{\mathbb{G}})\cong\underset{x\in
Irr(\mathbb{G})}{\bigoplus^{c_{0}}}\mathcal{B}(H_{x})$. We denote the identity
map by
$\widehat{\lambda}:c_{0}(\widehat{\mathbb{G}})\rightarrow\mathcal{B}(L^{2}(\mathbb{G}))$.

If $\mathbb{H}$ is another compact quantum group, we say that
$\widehat{\mathbb{H}}$ is a discrete quantum subgroup of
$\widehat{\mathbb{G}}$ if one (hence all) of the following conditions hold:
$i)$ $Pol(\mathbb{H})$ is a Hopf $*$-subalgebra of $\text{Pol}(\mathbb{G})$,
$ii)$ $C_{r}(\mathbb{H})\overset{\iota}{\subset}C_{r}(\mathbb{G})$ such that
$\iota$ intertwines the co-multiplications, $iii)$
$C_{m}(\mathbb{H})\overset{\iota}{\subset}C_{m}(\mathbb{G})$ such that $\iota$
intertwines the co-multiplications; $iv)$ $\mathscr{R}\text{ep}(\mathbb{H})$
is a full subcategory of $\mathscr{R}\text{ep}(\mathbb{G})$ containing the
trivial representation and stable by direct sums, tensor product and adjoint
operations. See [6] for more details. In this case we write
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$. Note that in this case we have
$\epsilon:=\epsilon_{\mathbb{G}}=\epsilon_{\mathbb{H}}$. The trivial quantum
subgroup of $\widehat{\mathbb{G}}$ is denoted by $\mathbb{E}$.

###  Actions of quantum groups

In this section we recall some terminology about actions of quantum groups. We
refer to [7] for more details.

######  2.2.1 Definition.

Let $\mathbb{G}=(C(\mathbb{G}),\Delta)$ be a compact quantum group. A right
$\mathbb{G}$-C∗-algebra is a C∗-algebra $A$ together with an injective non-
degenerate $*$-homomorphism $\alpha:A\longrightarrow A\otimes C(\mathbb{G})$
such that: $i)$ $(\alpha\otimes
id_{C(\mathbb{G})})\circ\alpha=(id_{A}\otimes\Delta)\circ\alpha$ and $ii)$
$[\alpha(A)(1\otimes C(\mathbb{G}))]=A\otimes C(\mathbb{G})$. Such
homomorphism is called a _right action of $\mathbb{G}$ on $A$_ or a _right co-
action of $C(\mathbb{G})$ on $A$_.

Similarly, we can define a _left_ action of $\mathbb{G}$ on $A$ (or a _left_
co-action of $C(\mathbb{G})$ on $A$) as a non-degenerate $*$-homomorphism
$\alpha:A\longrightarrow C(\mathbb{G})\otimes A$ satisfying the analogous
properties of the preceding definition. In the present article, an action of a
compact quantum group $\mathbb{G}$ is supposed to be a _right_ one unless the
contrary is explicitly indicated. Hence, we refer to such actions simply as
_actions of $\mathbb{G}$_. Observe however that if $(A,\alpha)$ is a _right_
$\mathbb{G}$-C∗-algebra, then $(A^{op},\overline{\alpha})$ is a _left_
$\mathbb{G}$-C∗-algebra where $A^{op}$ denotes the opposite C∗-algebra of $A$
and $\overline{\alpha}:A^{op}\longrightarrow C(\mathbb{G})\otimes A^{op}$ is
defined by $\overline{\alpha}:=(R\otimes id)\circ\Sigma\circ\alpha$, where $R$
denotes the unitary antipode of $\mathbb{G}$.

We also recall the notion of action for discrete quantum groups for the sake
of completeness.

######  2.2.2 Definition.

Let $\mathbb{G}=(C(\mathbb{G}),\Delta)$ be a compact quantum group. A right
$\widehat{\mathbb{G}}$-C∗-algebra is a C∗-algebra $A$ together with an
injective non-degenerate $*$-homomorphism
$\alpha:A\longrightarrow\widetilde{M}(A\otimes c_{0}(\widehat{\mathbb{G}}))$
such that: $i)$ $(\alpha\otimes
id_{c_{0}(\widehat{\mathbb{G}})})\circ\alpha=(id_{A}\otimes\widehat{\Delta})\circ\alpha$
and $ii)$ $[\alpha(A)(1\otimes c_{0}(\widehat{\mathbb{G}}))]=A\otimes
c_{0}(\widehat{\mathbb{G}})$. Such homomorphism is called a _right action of
$\widehat{\mathbb{G}}$ on $A$_ or a _right co-action of
$c_{0}(\widehat{\mathbb{G}})$ on $A$_.

Again, one has the analogous notion of a left action of
$\widehat{\mathbb{G}}$. In the following, an action of a discrete quantum
group $\widehat{\mathbb{G}}$ is supposed to be a _right_ one unless the
contrary is explicitly indicated.

We also recall the notion of equivariant Hilbert module with respect to a
compact quantum group for the sake of completeness.

######  2.2.3 Definition.

Let $\mathbb{G}$ be a compact quantum group and $(A,\delta)$ a
$\mathbb{G}$-C∗-algebra. A right $\mathbb{G}$-equivariant Hilbert $A$-module
is a right $A$-module $E$ together with an injective linear map
$\delta_{E}:E\longrightarrow E\otimes C(\mathbb{G})$ such that: $i)$
$\delta_{E}(\xi\cdot a)=\delta_{E}(\xi)\circ\delta(a)$ for all $\xi\in E$ and
$a\in A$; $ii)$
$\delta\big{(}\langle\xi,\eta\rangle\big{)}=\langle\delta_{E}(\xi),\delta_{E}(\eta)\rangle$
for all $\xi,\eta\in E$; $iii)$ $(\delta_{E}\otimes
id)\circ\delta_{E}=(id\otimes\Delta)\circ\delta_{E}$; $iv)$
$[\delta_{E}(E)(A\otimes C(\mathbb{G}))]=E\otimes C(\mathbb{G})$. Such map is
called a _right action of $\mathbb{G}$ on $E$_ or a _right co-action of
$C(\mathbb{G})$ on $E$_.

######  2.2.4 Remark.

The notion of representation of a compact quantum group can be viewed
alternatively as follows. If $u\in\mathcal{B}(H)\otimes C(\mathbb{G})$ is a
(finite dimensional) unitary representation of $\mathbb{G}$ on a (finite
dimensional) Hilbert space $H$, then the map $\delta_{H}:H\rightarrow H\otimes
C(\mathbb{H})$, $\xi\mapsto\delta_{H}(\xi):=u(\xi\otimes id)$, turns $H$ into
a $\mathbb{G}$-equivariant Hilbert space. Conversely, if $(H,\delta_{H})$ is a
(finite dimensional) $\mathbb{G}$-equivariant Hilbert space, then the unitary
$u\in\mathcal{B}(H)\otimes C(\mathbb{G})$ defined by $u(\xi\otimes
1)=\delta_{H}(\xi)$, for all $\xi\in H$ is a (finite dimensional) unitary
representation of $\mathbb{G}$ on $H$.

######  2.2.5 Definition.

Let $\mathbb{G}$ be a compact quantum group and $(A,\delta)$ a
$\mathbb{G}$-C∗-algebra. Let $(E,\delta_{E})$ be a $\mathbb{G}$-equivariant
Hilbert $A$-module. We say that $E$ is irreducible if the space of equivariant
adjointable operators of $E$,
$\mathcal{L}_{\mathbb{G}}(E):=\\{T\in\mathcal{L}_{A}(E)\ |\
\delta_{E}(T(\xi))=(T\otimes 1)\delta_{E}(\xi)\mbox{, for all $\xi\in E$}\\},$
is one-dimensional.

######  2.2.6 Remark.

If $(E,\delta_{E})$ is a $\mathbb{G}$-equivariant Hilbert $A$-module as above,
then $\mathcal{K}_{A}(E)$ is a $\mathbb{G}$-C∗-algebra with action
$\delta_{\mathcal{K}_{A}(E)}$ defined by
$\delta_{\mathcal{K}_{A}(E)}(\theta_{\xi,\eta})=\delta_{E}(\xi)\delta_{E}(\eta)^{*}\in\mathcal{K}_{A}(E)\otimes
C(\mathbb{G})$, for all $\xi,\eta\in E$ where $\theta_{\xi,\eta}$ denotes the
corresponding rank one operator in $E$. By abuse of notation, we still denote
by $\delta_{\mathcal{K}_{A}(E)}$ the extension of this homomorphism to
$\mathcal{L}_{A}(E)=M(\mathcal{K}_{A}(E))\rightarrow
M(\mathcal{K}_{A}(E)\otimes C(\mathbb{G}))$. The latter is however not in
general an action of $\mathbb{G}$ on $\mathcal{L}_{A}(E)$. Recall further that
giving an action $\delta_{E}$ is equivalent to giving a unitary operator
$V_{E}\in\mathcal{L}_{A\otimes
C(\mathbb{G})}\big{(}E\underset{\delta}{\otimes}(A\otimes
C(\mathbb{G})),E\otimes C(\mathbb{G})\big{)}$ such that
$\delta_{E}(\xi)=V_{E}\circ T_{\xi}$ for all $\xi\in E$ where
$T_{\xi}\in\mathcal{L}_{A\otimes C(\mathbb{G})}(A\otimes
C(\mathbb{G}),E\underset{\delta}{\otimes}(A\otimes C(\mathbb{G})))$ is such
that $T_{\xi}(x)=\xi\underset{\delta}{\otimes}x$, for all $x\in A\otimes
C(\mathbb{G})$. One calls $V_{E}$ the _admissible operator for
$(E,\delta_{E})$_. Moreover, we have $\delta_{\mathcal{K}_{A}(E)}=Ad_{V_{E}}$.
We refer to [3] for more details. Note that
$\mathcal{L}_{\mathbb{G}}(E)=\mathcal{L}_{A}(E)^{Ad_{V_{E}}}$. So, if
$(E,\delta_{E})$ is irreducible, then $\mathcal{L}_{A}(E)=\mathcal{K}_{A}(E)$
together with $Ad_{V_{E}}$ defines an ergodic action of $\mathbb{G}$.

######  2.2.7 Definition.

Let $\mathbb{G}$ be a compact quantum group. Let $(A,\alpha)$ and $(B,\beta)$
be two $\mathbb{G}$-C∗-algebras. We say that $A$ and $B$ are
$\mathbb{G}$-equivariantly Morita equivalent if there exists a
$\mathbb{G}$-equivariant Hilbert $A$-module $(E,\delta_{E})$ such that
$B\cong\mathcal{K}_{A}(E)$ as $\mathbb{G}$-C∗-algebras. In this case we write
$A\underset{\mathbb{G}}{\sim}B$.

If $(A,\alpha)$ is a $\mathbb{G}$-C∗-algebra, we put $A^{\alpha}:=\\{a\in A\
|\ \alpha(a)=a\otimes 1_{C(\mathbb{G})}\\}$, which is a sub-C∗-algebra of $A$.
If $A$ is unital, we say that $\alpha$ is ergodic if
$A^{\alpha}=\mathbb{C}\cdot 1_{A}$.

######  2.2.8 Definition.

Let $\mathbb{G}$ be a compact quantum group. A torsion action of $\mathbb{G}$
is a $\mathbb{G}$-C∗-algebra $(T,\tau)$ where $T$ is finite dimensional and
$\tau$ is ergodic. We say that $\widehat{\mathbb{G}}$ is torsion-free if any
torsion action of $\mathbb{G}$ is $\mathbb{G}$-equivariantly Morita equivalent
to the trivial $\mathbb{G}$-C∗-algebra $\mathbb{C}$. The set of all
equivalence classes under equivariant Morita equivalence of non-trivial
torsion actions of $\mathbb{G}$ is denoted by
$\text{Tor}(\widehat{\mathbb{G}})$.

######  2.2.9 Note.

Notice that there is no risk of confusion between the notation
$\text{Tor}(\widehat{\mathbb{G}})$ for torsion actions of $\mathbb{G}$ and the
usual Tor-functor.

Finally, let us recall the _two-sided crossed product_ construction. It
appears already in [25, Section 2.6] in the context of quantum groupoids. It
is used to formulate a quantum Baum-Connes assembly map in [2] and [8], which
we will use later. 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:

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

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 $x\in A$. It is easy to show that
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A=\overline{span}\\{(\beta(B)\otimes
1)(1\otimes c_{0}(\widehat{\mathbb{G}})\otimes 1)(1\otimes\alpha_{U}(A))\\}$.
We use these two descriptions of
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\alpha}{\ltimes}A$
interchangeably.

###  Quantum direct products

Let us recall briefly the construction of quantum direct product in the sense
of S. Wang [32].

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups. The quantum
direct product of $\mathbb{G}$ and $\mathbb{H}$ is a compact quantum group
denoted by $\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ with
$C(\mathbb{F})=C(\mathbb{G})\otimes C(\mathbb{H})$. The co-multiplication on
$\mathbb{F}$ is given by $\Delta_{\mathbb{F}}=(id\otimes\Sigma\otimes
id)(\Delta_{\mathbb{G}}\otimes\Delta_{\mathbb{H}})$. The representation theory
of $\mathbb{F}$ can be described explicitly as follows. For every irreducible
representation $z\in\text{Irr}(\mathbb{F})$, take a representative
$u^{z}\in\mathcal{B}(H_{z})\otimes C(\mathbb{F})$. Then there exist unique
irreducible representations $x\in\text{Irr}(\mathbb{G})$ and
$y\in\text{Irr}(\mathbb{H})$ such that if $u^{x}\in\mathcal{B}(H_{x})\otimes
C(\mathbb{G})$ and $u^{y}\in\mathcal{B}(H_{y})\otimes C(\mathbb{H})$ are
respective representatives of $x$ and $y$, then we have $u^{z}\cong
u^{x}_{13}u^{y}_{24}\in\mathcal{B}(H_{x}\otimes H_{y})\otimes
C(\mathbb{F})\mbox{,}$ where $u^{x}_{13}$ and $u^{y}_{24}$ are the
corresponding legs of $u^{x}$ and $u^{y}$, respectively inside
$\mathcal{B}(H_{x})\otimes\mathcal{B}(H_{y})\otimes C(\mathbb{G})\otimes
C(\mathbb{H})$. In this case we write $u^{(x,y)}:=u^{z}$.

This description of $\text{Irr}(\mathbb{F})$ yields that
$c_{0}(\widehat{\mathbb{F}})\cong c_{0}(\widehat{\mathbb{G}})\otimes
c_{0}(\widehat{\mathbb{H}})$.Therefore, given a
$\widehat{\mathbb{G}}$-C∗-algebra $(A,\alpha)$ and a
$\widehat{\mathbb{H}}$-C∗-algebra $(B,\beta)$, the tensor product $A\otimes B$
is a C∗-algebra equipped with the following action of $\widehat{\mathbb{F}}$:

$\begin{split}A\otimes
B\overset{\alpha\otimes\beta}{\longrightarrow}&\widetilde{M}(c_{0}(\widehat{\mathbb{G}})\otimes
A)\otimes\widetilde{M}(c_{0}(\widehat{\mathbb{H}})\otimes
B)\subset\widetilde{M}(c_{0}(\widehat{\mathbb{G}})\otimes A\otimes
c_{0}(\widehat{\mathbb{H}})\otimes B)\\\
&\overset{\Sigma_{23}}{\cong}\widetilde{M}(c_{0}(\widehat{\mathbb{G}})\otimes
c_{0}(\widehat{\mathbb{H}})\otimes A\otimes
B)\cong\widetilde{M}(c_{0}(\widehat{\mathbb{F}})\otimes A\otimes
B).\end{split}$

By abuse of notation, we denote this composition simply by
$\alpha\otimes\beta$. In a similar way, given a $\mathbb{G}$-C∗-algebra
$(T,\tau)$ and a $\mathbb{H}$-C∗-algebra $(S,\sigma)$ we can still define the
$\mathbb{F}$-C∗-algebra $(T\otimes S,\tau\otimes\sigma)$. In particular, it is
clear that if $(T,\tau)$ is a torsion action of $\mathbb{G}$ and $(S,\sigma)$
is a torsion action of $\mathbb{H}$, then $(R,\rho)$ is a torsion action of
$\mathbb{F}$.

Finally, observe that the quantum direct product construction can be done more
generally for locally compact quantum groups too. In this respect, the
following is an easy observation (see [19, Corollary 2.3.3] for a proof in the
case of discrete quantum groups).

######  2.1 Proposition.

Let $\mathbb{G}$, $\mathbb{H}$ be two locally compact quantum groups and let
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ be the corresponding quantum direct
product of $\mathbb{G}$ and $\mathbb{H}$. If $(A,\alpha)$ is a
$\mathbb{G}$-C∗-algebra and $(B,\beta)$ is a $\mathbb{H}$-C∗-algebra, then
there exists a canonical $*$-isomorphism
$C\underset{\delta,r}{\rtimes}\mathbb{F}\cong
A\underset{\alpha,r}{\rtimes}\mathbb{G}\otimes
B\underset{\beta,r}{\rtimes}\mathbb{H}\mbox{,}$ where $C:=A\otimes B$ is the
$\mathbb{F}$-C∗-algebra with action $\delta:=\alpha\otimes\beta$.

In particular, if $\mathbb{H}=\mathbb{E}$ is the trivial locally compact
quantum group and $B$ is any C∗-algebra equipped with trivial action, then
there exists a canonical $*$-isomorphism $(A\otimes
B)\underset{\delta,r}{\rtimes}\mathbb{G}\cong
A\underset{\alpha,r}{\rtimes}\mathbb{G}\otimes B$, where $A\otimes B$ is the
$\mathbb{G}$-C∗-algebra with action $\delta:=\alpha\otimes id$.

###### Proof.

The isomorphism of the statement is simply induced by the canonical map
$\mathcal{L}_{A}(L^{2}(\mathbb{G})\otimes
A)\otimes\mathcal{L}_{B}(L^{2}(\mathbb{H})\otimes
B)\rightarrow\mathcal{L}_{A\otimes B}\big{(}L^{2}(\mathbb{G})\otimes
L^{2}(\mathbb{H})\otimes A\otimes B\big{)}=\mathcal{L}_{A\otimes
B}(L^{2}(\mathbb{F})\otimes C)$. ∎

In relation with the two-sided crossed product construction recalled in
Section 2.1, the following result (which is a generalisation of Proposition
2.1) is straightforward after a routine computation by applying the
definitions and the fact that $c_{0}(\widehat{\mathbb{F}})\cong
c_{0}(\widehat{\mathbb{G}})\otimes c_{0}(\widehat{\mathbb{H}})$.

######  2.2 Proposition.

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups and let
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ be the corresponding quantum direct
product of $\mathbb{G}$ and $\mathbb{H}$. Let $(A,\alpha)$ and $(B,\beta)$ be
a right $\mathbb{G}$-C∗-algebra and a right $\mathbb{H}$-C∗-algebra,
respectively. Let $(T,\tau)$ and $(S,\sigma)$ be a left
$\mathbb{G}$-C∗-algebra and a left $\mathbb{H}$-C∗-algebra. Then $(A\otimes
B)\underset{r,\alpha\otimes\beta}{\rtimes}\mathbb{F}\underset{r,\tau\otimes\sigma}{\ltimes}(T\otimes
S)\cong(A\underset{r,\alpha}{\rtimes}\mathbb{G}\underset{r,\tau}{\ltimes}T)\otimes(B\underset{r,\beta}{\rtimes}\mathbb{H}\underset{r,\sigma}{\ltimes}S)$.

In particular, if $\mathbb{H}=\mathbb{E}$ is the trivial compact quantum group
and $B$ is any C∗-algebra equipped with trivial action, then $(A\otimes
B)\underset{r,\alpha\otimes
id}{\rtimes}\mathbb{G}\underset{r,\tau}{\ltimes}T\cong(A\underset{r,\alpha}{\rtimes}\mathbb{G}\underset{r,\tau}{\ltimes}T)\otimes
B$.

######  2.3.1 Note.

In the original paper [32] by S. Wang the construction
$\mathbb{G}\times\mathbb{H}$ is called _Cartesian product of $\mathbb{G}$ by
$\mathbb{H}$_. Moreover, this construction is consistent with the construction
of direct product of classical discrete groups, for instance. Hence, the
terminology _direct product of $\mathbb{G}$ by $\mathbb{H}$_ should not cause
any confusion. However, we use the terminology _quantum direct product_ to
emphasise the use of arbitrary compact quantum groups in this construction.

###  The equivariant Kasparov category and the quantum Baum-Connes assembly
map

We refer to [4], [3] for more details about (equivariant) $KK$-theory and to
[15], [22] for more details about triangulated categories and the Meyer-Nest
approach to the BC property, which is fundamental for a quantum counterpart of
the conjecture.

Let $\widehat{\mathbb{G}}$ be a discrete quantum group and consider the
corresponding equivariant Kasparov category,
$\mathscr{KK}^{\widehat{\mathbb{G}}}$, with canonical suspension functor
denoted by $\Sigma$. $\mathscr{KK}^{\widehat{\mathbb{G}}}$ is a triangulated
category whose distinguished triangles are given by mapping cone triangles
(see [22] 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 $\mathscr{KK}^{\mathbb{G}}$.

Exterior tensor product of Kasparov triples is important for the purpose of
the present paper. Namely, given C∗-algebras $A$, $A^{\prime}$, $B$,
$B^{\prime}$ and $D$, the map:

$\begin{array}[]{rccl}\widetilde{\tau}_{D}:&KK(A,A^{\prime}\otimes D)\times
KK(D\otimes B,B^{\prime})&\longrightarrow&KK(A\otimes B,A^{\prime}\otimes
B^{\prime})\\\
&(\mathcal{X},\mathcal{Y})&\longmapsto&\widetilde{\tau}_{D}(\mathcal{X},\mathcal{Y}):=\tau_{B}(\mathcal{X})\underset{A^{\prime}\otimes
D\otimes
B}{\otimes}{{}_{A^{\prime}}}\tau(\mathcal{Y})=:\mathcal{X}\underset{D}{\otimes^{\text{ext}}}\mathcal{Y}\end{array}$

is bilinear, contravariantly functorial in $A$ and $B$ and covariantly
functorial in $A^{\prime}$ and $B^{\prime}$. Here $\underset{A^{\prime}\otimes
D\otimes B}{\otimes}$ denotes the usual Kasparov product over
$A^{\prime}\otimes D\otimes B$; and $\tau_{B}$ and ${{}_{A^{\prime}}}\tau$
denote the right and left exterior tensor products by $B$ and $A^{\prime}$,
respectively (cf. [4, Chapter VIII, Section 18] for details). Notice however
that $\tau_{A}\cong{{}_{A}}\tau$, for all C∗-algebra $A$ because the tensor
product of C∗-algebras is a commutative operation. In particular, taking
$D:=\mathbb{C}$, $\widetilde{\tau}_{\mathbb{C}}$ defines a _tensor product of
Kasparov triples_. Namely, by virtue of [4, Proposition 18.9.1], the operation
$\widetilde{\tau}_{\mathbb{C}}(\ \cdot\ )=(\ \cdot\
)\underset{\mathbb{C}}{\otimes^{\text{ext}}}(\ \cdot\ )$ is associative. For
the sake of completeness, let us check the latter claim by translating the
formulas from [4, Proposition 18.9.1] to the case
$\widetilde{\tau}_{\mathbb{C}}$. Let $A$, $A^{\prime}$, $B$, $B^{\prime}$, $C$
and $C^{\prime}$ be C∗-algebras and $\mathcal{X}\in KK(A,A^{\prime})$,
$\mathcal{Y}\in KK(B,B^{\prime})$, $\mathcal{Z}\in KK(C,C^{\prime})$. Then:

$\begin{split}(\mathcal{X}\underset{\mathbb{C}}{\otimes^{\text{ext}}}\mathcal{Y})\underset{\mathbb{C}}{\otimes^{\text{ext}}}\mathcal{Z}&=\tau_{C}\Big{(}\mathcal{X}\underset{\mathbb{C}}{\otimes^{\text{ext}}}\mathcal{Y}\Big{)}\underset{A^{\prime}\otimes
B^{\prime}\otimes C}{\otimes}{{}_{A^{\prime}\otimes
B^{\prime}}}\tau(\mathcal{Z})\\\
&=\Big{(}\tau_{C}(\mathcal{X})\underset{C}{\otimes^{\text{ext}}}{{}_{C}}\tau(\mathcal{Y})\Big{)}\underset{A^{\prime}\otimes
B^{\prime}\otimes C}{\otimes}{{}_{A^{\prime}\otimes
B^{\prime}}}\tau(\mathcal{Z})\\\
&=\Big{(}\tau_{B}\big{(}\tau_{C}(\mathcal{X})\big{)}\underset{A^{\prime}\otimes
B\otimes
C}{\otimes}{{}_{A^{\prime}}}\tau\big{(}{{}_{C}}\tau(\mathcal{Y})\big{)}\Big{)}\underset{A^{\prime}\otimes
B^{\prime}\otimes C}{\otimes}{{}_{A^{\prime}\otimes
B^{\prime}}}\tau(\mathcal{Z})\\\ &=\Big{(}\tau_{B\otimes
C}(\mathcal{X})\underset{A^{\prime}\otimes B\otimes C}{\otimes}{{}_{C\otimes
A^{\prime}}}\tau(\mathcal{Y})\Big{)}\underset{A^{\prime}\otimes
B^{\prime}\otimes C}{\otimes}{{}_{A^{\prime}\otimes
B^{\prime}}}\tau(\mathcal{Z})\\\ &=\tau_{B\otimes
C}(\mathcal{X})\underset{A^{\prime}\otimes B\otimes
C}{\otimes}\Big{(}{{}_{C\otimes
A^{\prime}}}\tau(\mathcal{Y})\underset{A^{\prime}\otimes B^{\prime}\otimes
C}{\otimes}{{}_{A^{\prime}\otimes B^{\prime}}}\tau(\mathcal{Z})\Big{)}\\\
&=\tau_{B\otimes C}(\mathcal{X})\underset{A^{\prime}\otimes B\otimes
C}{\otimes}\Big{(}\tau_{C}\big{(}\tau_{A^{\prime}}(\mathcal{Y})\big{)}\underset{A^{\prime}\otimes
B^{\prime}\otimes
C}{\otimes}{{}_{B^{\prime}}}\tau\big{(}{{}_{A^{\prime}}}\tau(\mathcal{Z})\big{)}\Big{)}\\\
&=\tau_{B\otimes C}(\mathcal{X})\underset{A^{\prime}\otimes B\otimes
C}{\otimes}\Big{(}\tau_{A^{\prime}}(\mathcal{Y})\underset{A^{\prime}}{\otimes^{\text{ext}}}{{}_{A^{\prime}}}\tau(\mathcal{Z})\Big{)}\\\
&=\tau_{B\otimes C}(\mathcal{X})\underset{A^{\prime}\otimes B\otimes
C}{\otimes}{{}_{A^{\prime}}}\tau\Big{(}\mathcal{Y}\underset{\mathbb{C}}{\otimes^{\text{ext}}}\mathcal{Z}\Big{)}=\mathcal{X}\underset{\mathbb{C}}{\otimes^{\text{ext}}}(\mathcal{Y}\underset{\mathbb{C}}{\otimes^{\text{ext}}}\mathcal{Z}).\end{split}$

Moreover, given a C∗-algebra $A$, the functor $A\otimes(\ \cdot\
):\text{C}^{*}\text{-Alg}\rightarrow\mathscr{KK}$, $B\mapsto A\otimes B$, is a
split-exact, stable and homotopy invariant. By the universal property of the
Kasparov category (cf. [13]), it extends to a functor $A\otimes(\ \cdot\
):\mathscr{KK}\rightarrow\mathscr{KK}$. Similarly, we have a functor $(\
\cdot\ )\otimes A:\mathscr{KK}\rightarrow\mathscr{KK}$. Since these extensions
are natural, we obtain a bifunctor $\widetilde{\tau}_{\mathbb{C}}(\ \cdot\
)=(\ \cdot\ )\underset{\mathbb{C}}{\otimes^{\text{ext}}}(\ \cdot\
):\mathscr{KK}\times\mathscr{KK}\rightarrow\mathscr{KK}$. In other words, the
bifunctor $\widetilde{\tau}_{\mathbb{C}}$ confer a monoidal structure on
$\mathscr{KK}$. Finally, since the Kasparov product plays the role of
composition of morphisms within the category $\mathscr{KK}$, functoriality of
$\widetilde{\tau}$ yields that:

$\begin{split}(y\underset{\mathbb{C}}{\otimes^{\text{ext}}}x)\underset{A\otimes
B}{\otimes}(z\underset{\mathbb{C}}{\otimes^{\text{ext}}}w)=(y\underset{A}{\otimes}z)\underset{\mathbb{C}}{\otimes^{\text{ext}}}(x\underset{B}{\otimes}w),\end{split}$
(2.1)

for all $y\in KK(D,A)$, $x\in KK(D^{\prime},B)$, $z\in KK(A,A^{\prime})$,
$w\in KK(B,B^{\prime})$ and all C∗-algebras $A$, $A^{\prime}$, $B$,
$B^{\prime}$, $D$, $D^{\prime}$.

######  2.4.1 Remark.

If $G$ is a locally compact group, then the equivariant Kasparov category
$\mathscr{KK}^{G}$ is also characterised by a universal property in terms of
split-exactness, stability and homotopy invariance (cf. [28]). The exterior
tensor product can be also defined in the equivariant setting, say
$\widetilde{\tau}^{G}_{\mathbb{C}}$; the action of $G$ on a tensor product of
C∗-algebras being the diagonal action. Again, $\mathscr{KK}^{G}$ is equipped
in this way with a monoidal structure and the analogue to Equation (2.1)
holds. Furthermore, if $\mathbb{G}$ is a regular locally compact quantum
group, then $\mathscr{KK}^{\mathbb{G}}$ is also characterised by a universal
property as in the classical setting (cf. [24]). If $D(\mathbb{G})$ denotes
the Drinfeld double of $\mathbb{G}$, then $\mathscr{KK}^{D(\mathbb{G})}$ is
equipped with a monoidal structure by means of the _braided tensor product_
(cf. [24]). Again, the analogue to Equation (2.1) holds.

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

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 [23] and [31])
is given by the following localizing subcategories of
$\mathscr{KK}^{\widehat{\mathbb{G}}}$:

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

######  2.4.2 Remark.

We put $\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}:=\langle\\{C\otimes T\ |\
C\in\text{Obj}(\mathscr{KK}),\
T\in\mathbb{T}\mbox{or}(\widehat{\mathbb{G}})\\}\rangle$, so that we have
$\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}\rtimes\mathbb{G}=\mathscr{L}_{\widehat{\mathbb{G}}}$
by definition. Similarly we put
$\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}:=\mathscr{N}_{\widehat{\mathbb{G}}}\rtimes\widehat{\mathbb{G}}$.
The pair
$(\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}},\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}})$
is still complementary. We denote by $(\widehat{L},\widehat{N})$ the
corresponding triangulated functors defining the
$(\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}},\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}})$-triangles
in $\mathscr{KK}^{\mathbb{G}}$.

Recently, Y. Arano and A. Skalski [2] have showed that these two subcategories
form indeed a complementary pair of localizing subcategories in
$\mathscr{KK}^{\widehat{\mathbb{G}}}$. Also, the author in collaboration with
K. De Commer and R. Nest has obtained the same conclusion in [8] for
permutation torsion-free discrete quantum groups as an application of the
study of the projective representation theory for compact quantum groups. In
either case, the complementarity of the pair
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$ is
based in a generalisation of the _Green-Julg isomorphism_. More precisely,
given such a torsion action of $\mathbb{G}$, say $(T,\delta)$, we define the
following triangulated functors:

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

The proof of the following result can be found in [2] (see also [8] for a
different approach).

######  2.3 Theorem.

Let $\mathbb{G}$ be a compact quantum group. Let $(T,\delta)$ be a torsion
action of $\mathbb{G}$. Then
$\tau_{T}:\mathscr{KK}\longrightarrow\mathscr{KK}^{\mathbb{G}}$ is a left
adjoint of
$j_{\mathbb{G},T}:\mathscr{KK}^{\mathbb{G}}\longrightarrow\mathscr{KK}$ 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 $C\in\text{Obj}(\mathscr{KK})$ and
$(B,\beta)\in\text{Obj}(\mathscr{KK}^{\mathbb{G}})$.

Then the general Meyer-Nest’s machinery yields in particular that
$\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}=\ker_{\text{Obj}}\big{(}(j_{\mathbb{G},T})_{T\in\text{Tor}(\widehat{\mathbb{G}})}\big{)}$.
This allows to define a quantum assembly map for arbitrary discrete quantum
groups (not necessarily torsion-free) and thus a quantum version of the
(strong) BC property. More precisely, consider the following homological
functor:

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

The quantum assembly map for $\widehat{\mathbb{G}}$ is given by the natural
transformation $\eta^{\widehat{\mathbb{G}}}:\mathbb{L}F_{*}\longrightarrow
F_{*}$, where $\mathbb{L}F_{*}$ denotes the localisation of $F_{*}$ with
respect to the complementary pair
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$.
More precisely, given any $\widehat{\mathbb{G}}$-C∗-algebra $(A,\alpha)$, we
consider a
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$-triangle
associated to $A$, say $\Sigma N(A)\longrightarrow
L(A)\overset{u}{\longrightarrow}A\longrightarrow N(A)$. Then
$\eta^{\widehat{\mathbb{G}}}_{A}=F_{*}(u)$. Let us point out a more precise
picture for $\eta^{\widehat{\mathbb{G}}}_{A}$. On the one hand, given the
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$-triangle
$\Sigma N(A)\longrightarrow L(A)\overset{u}{\longrightarrow}A\longrightarrow
N(A)$, the map
$L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}\overset{u}{\rightarrow}A\underset{r}{\rtimes}\widehat{\mathbb{G}}$
must be viewed as a Kaparov triple
$u\underset{r}{\rtimes}\widehat{\mathbb{G}}\in
KK(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}},A\underset{r}{\rtimes}\widehat{\mathbb{G}})$,
i.e. a homomorphism in $\mathscr{KK}$. On the other hand, the functor $F_{*}(\
\cdot\ )=KK_{*}(\mathbb{C},(\ \cdot\
)\underset{r}{\rtimes}\widehat{\mathbb{G}})$ can be viewed as the covariant
homomorphism functor $KK(\mathbb{C},\ \cdot\ )$ in $\mathscr{KK}$, which
consists in composing, i.e. making a Kasparov product. In other words, when we
apply this to crossed product C∗-algebras we have:

$\begin{split}\eta^{\widehat{\mathbb{G}}}_{A}=F_{*}(u)=KK_{*}(\mathbb{C},u\underset{r}{\rtimes}\widehat{\mathbb{G}})=(\
\cdot\
)\underset{L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}}{\otimes}u\underset{r}{\rtimes}\widehat{\mathbb{G}}:KK(\mathbb{C},L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})\rightarrow
KK(\mathbb{C},A\underset{r}{\rtimes}\widehat{\mathbb{G}}).\end{split}$ (2.2)

######  2.4.3 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_
quantum Baum-Connes property if
$\mathscr{KK}^{\widehat{\mathbb{G}}}=\mathscr{L}_{\widehat{\mathbb{G}}}$. We
abbreviate the predicate _Baum-Connes property_ by _BC property_.

######  2.4.4 Note.

The following nomenclature is useful. Given
$A\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$ consider a
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$-triangle
associated to $A$, say $\Sigma N(A)\longrightarrow
L(A)\overset{u}{\longrightarrow}A\longrightarrow N(A)$. We know that such
triangles are distinguished and unique up to isomorphism (cf. [22] for a
proof). The homomorphism $u:L(A)\longrightarrow A$ is called _Dirac
homomorphism for $A$_. Sometimes we write $D:=u$. 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_.

##  The Künneth classes

We start by recalling some terminology introduced in [5] in the context of the
(abstract) Künneth theorem.

######  3.1 Definition.

A Künneth category is a subcategory $\mathcal{S}$ of $\text{C}^{*}\text{-Alg}$
such that:

  1. i)

$\mathcal{S}$ contains all separable commutative C∗-algebras.

  2. ii)

$\mathcal{S}$ is closed under stabilization, i.e. if
$B\in\text{Obj}(\mathcal{S})$, then $\mathcal{K}\otimes
B\in\text{Obj}(\mathcal{S})$.

  3. iii)

$\mathcal{S}$ is closed under suspension, i.e. if
$B\in\text{Obj}(\mathcal{S})$, then $\Sigma(B)\in\text{Obj}(\mathcal{S})$ are
objects in $\mathcal{S}$.

  4. iv)

If $0\rightarrow J\rightarrow B\rightarrow B/J\rightarrow 0$ is a semi-split
short exact sequence in $\mathcal{S}$ such that two of the C∗-algebras are in
$\mathcal{S}$, the so is the third.

######  3.2 Definition.

Let $\mathcal{S}$ be a Künneth category. A Künneth functor on $\mathcal{S}$ is
an additive convariant functor
$\kappa_{*}:\mathcal{S}\longrightarrow\mathscr{A}\text{b}^{\mathbb{Z}/2}$ such
that:

  1. i)

$\kappa_{*}$ is invariant under stabilization and under homotopy.

  2. ii)

If $0\rightarrow J\rightarrow B\rightarrow B/J\rightarrow 0$ is a semi-split
short exact sequence in $\mathcal{S}$, then the sequence
$\kappa_{*}(J)\rightarrow\kappa_{*}(B)\rightarrow\kappa_{*}(B/J)$ is exact.

  3. iii)

$\kappa_{*}$ is stable, i.e. $\kappa_{*}(\Sigma(B))\cong\kappa_{*+1}(B)$
naturally, for all $B\in\text{Obj}(\mathcal{S})$.

  4. iv)

There exists a natural zero-graded homomorphism
$\alpha:\kappa_{*}(\mathbb{C})\otimes K_{*}(B)\rightarrow\kappa_{*}(B)$, for
all $B\in\text{Obj}(\mathcal{S})$ such that $\alpha$ is an isomorphism
whenever $K_{*}(B)$ is free abelian.

######  3.3 Theorem.

Let $\mathcal{S}$ be a Künneth category. If $\kappa_{*}$ is a Künneth functor
on $\mathcal{S}$, then there exists a natural $1$-graded homomorphism
$\beta:\kappa_{*}(B)\rightarrow\text{Tor}(\kappa_{*}(\mathbb{C}),K_{*}(B))$,
for all $B\in\text{Obj}(\mathcal{S})$ such that the sequence:

$0\longrightarrow\kappa_{*}(\mathbb{C})\otimes
K_{*}(B)\overset{\alpha}{\longrightarrow}\kappa_{*}(B)\overset{\beta}{\longrightarrow}\text{Tor}(\kappa_{*}(\mathbb{C}),K_{*}(B))\longrightarrow
0$

is exact. This sequence is called _Künneth sequence for $\kappa_{*}$_.

######  3.4 Examples.

  1. 1.

Let $A$ and $B$ be two C∗-algebras. The external Kasparov product:

$KK(\mathbb{C},A)\times
KK(\mathbb{C},B)\overset{\widetilde{\tau}_{\mathbb{C}}}{\longrightarrow}KK(\mathbb{C},A\otimes
B)$

yields a group homomorphism $\alpha:K_{*}(A)\otimes K_{*}(B)\rightarrow
K_{*}(A\otimes B)$, which is natural on $A$ and $B$. We denote by
$\mathcal{N}$ the class of all C∗-algebras $A$ in $\text{C}^{*}\text{-Alg}$
such that $\alpha$ is an isomorphism for all separable C∗-algebra $B$ with
$K_{*}(B)$ free abelian. By abuse of notation, we still denote by
$\mathcal{N}$ the full subcategory of $\text{C}^{*}\text{-Alg}$ generated by
the class $\mathcal{N}$. It is well-known that $\mathcal{N}$ is a Künneth
category in the sense of Definition 3.1 (cf. [5]). Moreover, if
$A\in\mathcal{N}$, it is clear that the map $B\mapsto K_{*}(A\otimes B)$
defines a functor on the whole category $\text{C}^{*}\text{-Alg}$ satisfying
properties $(i)-(iv)$ from Definition 3.2 (cf. [5]). In other words, this map
defines a Künneth functor. Consequently, Theorem 3.3 yields that the Künneth
sequence:

$0\longrightarrow K_{*}(A)\otimes
K_{*}(B)\overset{\alpha}{\longrightarrow}K_{*}(A\otimes
B)\overset{\beta}{\longrightarrow}\text{Tor}(K_{*}(A),K_{*}(B))\longrightarrow
0$

is exact for all $A\in\mathcal{N}$ and all
$B\in\text{Obj}(\text{C}^{*}\text{-Alg})$. It is important to mention that the
class $\mathcal{N}$ contains the bootstrap class and it satisfies the
following remarkable stability properties (cf. [5, Lemma 4.4]):

######  3.5 Lemma.

$\mathcal{N}$ contains the bootstrap class and the following properties hold.

    1. i)

$\mathcal{N}$ is stable under $KK$-equivalence hence it is stable under Morita
equivalence.

    2. ii)

If $0\rightarrow I\rightarrow A\rightarrow A/I\rightarrow$ is a semi-split
short exact sequence of C∗-algebras such that two of them are in
$\mathcal{N}$, then so is the third.

    3. iii)

If $A,B\in\mathcal{N}$, then $A\otimes B\in\mathcal{N}$.

    4. iv)

If $A=\underset{\rightarrow}{\lim}\ A_{i}$ such that all structure maps are
injective and $A_{i}\in\mathcal{N}$ for all $i$, then $A\in\mathcal{N}$.

  2. 2.

Let $G$ be a locally compact group. Let $A$ be a $G$-C∗-algebra and $B$ a
C∗-algebra. Observe that one has an obvious map $KK(\mathbb{C},B)\rightarrow
KK^{G}(\mathbb{C},B)$, which consists in equipping with the trivial action of
$G$. Then, for all $G$-compact subspace $X\subset\underline{E}G$, the external
Kasparov product:

$KK^{G}(C_{0}(X),A)\times KK(\mathbb{C},B)\longrightarrow
KK^{G}(C_{0}(X),A)\times
KK^{G}(\mathbb{C},B)\overset{\widetilde{\tau}_{\mathbb{C}}}{\longrightarrow}KK(C_{0}(X),A\otimes
B)$

yields a group homomorphism $\alpha_{X}:KK^{G}(C_{0}(X),A)\otimes
K_{*}(B)\rightarrow KK^{G}(C_{0}(X),A\otimes B)$, for all $G$-compact subspace
$X\subset\underline{E}G$, which is natural on $A$ and $B$. One checks that the
maps
$\\{\alpha_{X}\\}_{\underset{\text{$G$-compact}}{X\subset\underline{E}G}}$ are
compatible with the inclusion of $G$-compact spaces. Hence, one obtains a
group homomorphism $\alpha_{G}:K^{\text{top}}_{*}(G;A)\otimes
K_{*}(B)\rightarrow K^{\text{top}}_{*}(G;A\otimes B)$, which is natural on $A$
and $B$. We denote by $\mathcal{N}_{G}$ the class of all $G$-C∗-algebras $A$
in $G$-$\text{C}^{*}\text{-Alg}$ such that $\alpha_{G}$ is an isomorphism for
all separable C∗-algebra $B$ with $K_{*}(B)$ free abelian. By abuse of
notation, we still denote by $\mathcal{N}_{G}$ the full subcategory of
$G$-$\text{C}^{*}\text{-Alg}$ generated by the class $\mathcal{N}_{G}$. It is
well-known that $\mathcal{N}_{G}$ is a Künneth category in the sense of
Definition 3.1 (cf. [5]). Moreover, if $A\in\mathcal{N}_{G}$, it is clear that
the map $B\mapsto K^{\text{top}}_{*}(G;A\otimes B)$ defines a functor on the
whole category $\text{C}^{*}\text{-Alg}$ satisfying properties $(i)-(iv)$ from
Definition 3.2 (cf. [5]). In other words, this map defines a Künneth functor.
Consequently, Theorem 3.3 yields that the Künneth sequence:

$0\longrightarrow K^{\text{top}}_{*}(G;A)\otimes
K_{*}(B)\overset{\alpha_{G}}{\longrightarrow}K^{\text{top}}_{*}(G;A\otimes
B)\overset{\beta_{G}}{\longrightarrow}\text{Tor}(K^{\text{top}}_{*}(G;A),K_{*}(B))\longrightarrow
0$

is exact for all $A\in\mathcal{N}_{G}$ and all
$B\in\text{Obj}(\text{C}^{*}\text{-Alg})$.

###  Preparatory observations

In order to provide examples of Künneth functors in the framework of quantum
groups, let us recall some constructions appearing in [19]. Let $\mathbb{G}$
and $\mathbb{H}$ be two compact quantum groups. We put
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ for the corresponding quantum direct
product. Consider the complementary pair of localizing subcategories in
$\mathscr{KK}^{\widehat{\mathbb{F}}}$, $\mathscr{KK}^{\widehat{\mathbb{G}}}$
and $\mathscr{KK}^{\widehat{\mathbb{H}}}$:
$(\mathscr{L}_{\widehat{\mathbb{F}}},\mathscr{N}_{\widehat{\mathbb{F}}})$,
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$ and
$(\mathscr{L}_{\widehat{\mathbb{H}}},\mathscr{N}_{\widehat{\mathbb{H}}})$,
respectively; as explained in Section 2.4. Accordingly, the canonical
triangulated functors associated to these complementary pairs are denoted by
$(L,N)$, $(L^{\prime},N^{\prime})$ and $(L^{\prime\prime},N^{\prime\prime})$,
respectively. Consider the homological functors defining the _quantum_ Baum-
Connes assembly maps for $\widehat{\mathbb{F}}$, $\widehat{\mathbb{G}}$ and
$\widehat{\mathbb{H}}$:

$F:\mathscr{KK}^{\widehat{\mathbb{F}}}\rightarrow\mathscr{A}b^{\mathbb{Z}/2}\mbox{,
}(C,\delta)\mapsto
F(C):=K_{*}(C\underset{\delta,r}{\rtimes}\widehat{\mathbb{F}}),$
$F^{\prime}:\mathscr{KK}^{\widehat{\mathbb{G}}}\rightarrow\mathscr{A}b^{\mathbb{Z}/2}\mbox{,
}(A,\alpha)\mapsto
F^{\prime}(A):=K_{*}(A\underset{\alpha,r}{\rtimes}\widehat{\mathbb{G}}),$
$F^{\prime\prime}:\mathscr{KK}^{\widehat{\mathbb{H}}}\rightarrow\mathscr{A}b^{\mathbb{Z}/2}\mbox{,
}(B,\beta)\mapsto
F^{\prime\prime}(B):=K_{*}(B\underset{\beta,r}{\rtimes}\widehat{\mathbb{H}}).$

Therefore, the quantum assembly maps for $\widehat{\mathbb{F}}$,
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are given by the natural
transformations
$\mathbb{L}F\overset{\eta^{\widehat{\mathbb{F}}}}{\longrightarrow}F$,
$\mathbb{L}F^{\prime}\overset{\eta^{\widehat{\mathbb{G}}}}{\longrightarrow}F^{\prime}$
and
$\mathbb{L}F^{\prime\prime}\overset{\eta^{\widehat{\mathbb{H}}}}{\longrightarrow}F^{\prime\prime}$,
respectively. Next, consider the functor:

$\mathcal{Z}:\mathscr{KK}^{\widehat{\mathbb{G}}}\times\mathscr{KK}^{\widehat{\mathbb{H}}}\rightarrow\mathscr{KK}^{\widehat{\mathbb{F}}}$

defined on objects by
$(A,\alpha)\times(B,\beta)\mapsto\mathcal{Z}(A,B):=(C:=A\otimes
B,\delta:=\alpha\otimes\beta)$ and on homomorphisms by
$\mathcal{Z}(\mathcal{X},\mathcal{Y}):=\mathcal{X}\otimes\mathcal{Y}:=\tau_{B}(\mathcal{X})\underset{A^{\prime}\otimes
B}{\otimes}{{}_{A^{\prime}}}\tau(\mathcal{Y})$, for all $\mathcal{X}\in
KK^{\widehat{\mathbb{G}}}(A,A^{\prime})$, $\mathcal{Y}\in
KK^{\widehat{\mathbb{H}}}(B,B^{\prime})$ with
$(A,\alpha)\times(B,\beta),(A^{\prime},\alpha^{\prime})\times(B^{\prime},\beta^{\prime})\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})\times\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{H}}})$.

######  3.1.1 Lemma.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups and let
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ be the corresponding quantum direct
product of $\mathbb{G}$ and $\mathbb{H}$. The functor
$\mathcal{Z}:\mathscr{KK}^{\widehat{\mathbb{G}}}\times\mathscr{KK}^{\widehat{\mathbb{H}}}\rightarrow\mathscr{KK}^{\widehat{\mathbb{F}}}$
is such that
$\mathcal{Z}(\mathscr{L}_{\widehat{\mathbb{G}}}\times\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{F}}}$.
If, moreover, either $\widehat{\mathbb{G}}$ or $\widehat{\mathbb{H}}$
satisfies the strong quantum BC property, then
$\mathcal{Z}(\mathscr{N}_{\widehat{\mathbb{G}}}\times\mathscr{N}_{\widehat{\mathbb{H}}})\subset\mathscr{N}_{\widehat{\mathbb{F}}}$.
Besides, we have the following.

  1. i)

If $(A_{0},\alpha_{0})\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$ is a
given $\widehat{\mathbb{G}}$-C∗-algebra, the functor
${{}_{A_{0}}}\mathcal{Z}:\mathscr{KK}^{\widehat{\mathbb{H}}}\rightarrow\mathscr{KK}^{\widehat{\mathbb{F}}}\mbox{,
}(B,\beta)\mapsto{{}_{A_{0}}}\mathcal{Z}(B):=\mathcal{Z}(A_{0},B)$ is
triangulated. If, moreover, we assume that $\widehat{\mathbb{H}}$ satisfies
the strong quantum BC property, then
${{}_{A_{0}}}\mathcal{Z}(\mathscr{N}_{\widehat{\mathbb{H}}})\subset\mathscr{N}_{\widehat{\mathbb{F}}}$.

  2. ii)

If $(B_{0},\beta_{0})\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{H}}})$ is a
given $\widehat{\mathbb{H}}$-C∗-algebra, the functor
$\mathcal{Z}_{B_{0}}:\mathscr{KK}^{\widehat{\mathbb{G}}}\rightarrow\mathscr{KK}^{\widehat{\mathbb{F}}}\mbox{,
}(A,\alpha)\mapsto\mathcal{Z}_{B_{0}}(A):=\mathcal{Z}(A,B_{0})$ is
triangulated. If, moreover, we assume that $\widehat{\mathbb{G}}$ satisfies
the strong quantum BC property, then
$\mathcal{Z}_{B_{0}}(\mathscr{N}_{\widehat{\mathbb{G}}})\subset\mathscr{N}_{\widehat{\mathbb{F}}}$.

###### Proof.

Most part of the argument used in [19, Lemma 5.2.1] applies. But here we have
removed the torsion-freeness assumption, so we need to modify some steps of
the proof of [19, Lemma 5.2.1]. First, given an object
$(T,\tau)\times(S,\sigma)\in\text{Obj}(\mathscr{KK}^{\mathbb{G}})\times\text{Obj}(\mathscr{KK}^{\mathbb{H}})$
with $(T,\tau)$ a torsion action of $\mathbb{G}$ and $(S,\sigma)$ a torsion
action of $\mathbb{H}$, then $(R:=T\otimes S,\rho:=\tau\otimes\sigma)$ is a
torsion action of $\mathbb{F}$ as observed in Section 2.3. Next, using
Proposition 2.1 we write:

$\begin{split}\mathcal{Z}\big{(}T\underset{\tau,r}{\rtimes}\mathbb{G}\otimes
C_{1},S\underset{\sigma,r}{\rtimes}\mathbb{H}\otimes
C_{2}\big{)}&=T\underset{\tau,r}{\rtimes}\mathbb{G}\otimes C_{1}\otimes
S\underset{\sigma,r}{\rtimes}\mathbb{H}\otimes C_{2}\\\ &\cong
T\underset{\tau,r}{\rtimes}\mathbb{G}\otimes
S\underset{\sigma,r}{\rtimes}\mathbb{H}\otimes C_{1}\otimes C_{2}\cong
R\underset{\rho,r}{\rtimes}\mathbb{F}\otimes C_{3}\mbox{,}\end{split}$

where $C_{3}:=C_{1}\otimes C_{2}\in\text{Obj}(\mathscr{KK})$. Consequently,
$\mathcal{Z}(\mathscr{L}_{\widehat{\mathbb{G}}}\times\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{F}}}$
as in [19, Lemma 5.2.1].

Now, if either $\widehat{\mathbb{G}}$ or $\widehat{\mathbb{H}}$ satisfies the
strong quantum BC property, then we have by definition that either
$\mathscr{KK}^{\widehat{\mathbb{G}}}=\mathscr{L}_{\widehat{\mathbb{G}}}$ or
$\mathscr{KK}^{\widehat{\mathbb{H}}}=\mathscr{L}_{\widehat{\mathbb{H}}}$.
Since
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$ and
$(\mathscr{L}_{\widehat{\mathbb{H}}},\mathscr{N}_{\widehat{\mathbb{H}}})$ are
complementary pairs as explained in Section 2.4, the latter means that we have
either $\mathscr{N}_{\widehat{\mathbb{G}}}=0$ or
$\mathscr{N}_{\widehat{\mathbb{H}}}=0$ hence either
$\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}=0$ or
$\widehat{\mathscr{N}}_{\widehat{\mathbb{H}}}=0$. In either case, given
$(A,\alpha)\in\text{Obj}(\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}})$ and
$(B,\beta)\in\text{Obj}(\widehat{\mathscr{N}}_{\widehat{\mathbb{H}}})$ we have
$j_{\mathbb{F},R}(A\otimes B)=(A\otimes
B)\underset{r,\alpha\otimes\beta}{\rtimes}\mathbb{F}\underset{r,\overline{\rho}}{\ltimes}R^{op}\cong
0$ in $\mathscr{KK}$, for every torsion action $(R,\rho)$ of $\mathbb{F}$.
This means, as explained in Section 2.4, that $(A\otimes
B,\alpha\otimes\beta)\in\text{Obj}(\widehat{\mathscr{N}}_{\widehat{\mathbb{F}}})$.
In other words,
$\mathcal{Z}(\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}\times\widehat{\mathscr{N}}_{\widehat{\mathbb{H}}})\subset\widehat{\mathscr{N}}_{\widehat{\mathbb{F}}}$,
which yields
$\mathcal{Z}(\mathscr{N}_{\widehat{\mathbb{G}}}\times\mathscr{N}_{\widehat{\mathbb{H}}})\subset\mathscr{N}_{\widehat{\mathbb{F}}}$
by virtue of Baaj-Skandalis duality and Proposition 2.1.

To conclude, let us show property $(i)$ of the statement (the proof of
property $(ii)$ is similar). It remains to show that, given a
$\widehat{\mathbb{G}}$-C∗-algebra
$(A_{0},\alpha_{0})\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$, then
the functor ${{}_{A_{0}}}\mathcal{Z}$ is triangulated. For this, the analogous
argument as in [19, Lemma 5.2.1] applies. ∎

The previous lemma allows to improve [19, Lemma 5.2.2] by removing the
torsion-freeness assumption as follows.

######  3.1.2 Lemma.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups and let
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ be the corresponding quantum direct
product of $\mathbb{G}$ and $\mathbb{H}$.

  1. i)

For all $\widehat{\mathbb{G}}$-C∗-algebra $(A,\alpha)$ and all
$\widehat{\mathbb{H}}$-C∗-algebra $(B,\beta)$ there exists a Kasparov triple
$\psi\in KK^{\widehat{\mathbb{F}}}\big{(}L^{\prime}(A)\otimes
L^{\prime\prime}(B),L(A\otimes B)\big{)}$ such that the diagram:

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B)\underset{r}{\rtimes}\widehat{\mathbb{F}}$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
144.30579pt\raise-20.22221pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
144.30579pt\raise-29.94443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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66.5279pt\raise-34.29443pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
113.33995pt\raise-40.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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is commutative where $\Psi:=\psi\underset{r}{\rtimes}\widehat{\mathbb{F}}$ and
$u^{\prime}$, $u^{\prime\prime}$, $u$ are the Dirac homomorphisms for $A$,
$B$, $A\otimes B$, respectively.

  2. ii)

Assume that $\widehat{\mathbb{H}}$ satisfies the strong quantum BC property.
For all $\widehat{\mathbb{G}}$-C∗-algebra
$(A_{0},\alpha_{0})\in\mathscr{L}_{\widehat{\mathbb{G}}}$ and all
$\widehat{\mathbb{H}}$-C∗-algebra $(B,\beta)$ there exists an _invertible_
Kasparov triple ${{}_{A_{0}}}\psi\in
KK^{\widehat{\mathbb{F}}}\big{(}A_{0}\otimes
L^{\prime\prime}(B),L(A_{0}\otimes B)\big{)}$ such that the diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.2222pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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63.3415pt\raise-34.29443pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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is commutative where
${{}_{A_{0}}}\Psi:={{}_{A_{0}}}\psi\underset{r}{\rtimes}\widehat{\mathbb{F}}$
and $u^{\prime\prime}$, $u$ are the Dirac homomorphism for $B$, $A_{0}\otimes
B$, respectively.

  3. iii)

Assume that $\widehat{\mathbb{G}}$ satisfies the strong quantum BC property.
For all $\widehat{\mathbb{H}}$-C∗-algebra
$(B_{0},\beta_{0})\in\mathscr{L}_{\widehat{\mathbb{H}}}$ and all
$\widehat{\mathbb{G}}$-C∗-algebra $(A,\alpha)$ there exists an _invertible_
Kasparov triple $\psi_{B_{0}}\in
KK^{\widehat{\mathbb{F}}}\big{(}L^{\prime}(A)\otimes B_{0},L(A\otimes
B_{0})\big{)}$ such that the diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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63.12704pt\raise-33.85pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 105.13828pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.13828pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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is commutative where
$\Psi_{B_{0}}:=\psi_{B_{0}}\underset{r}{\rtimes}\widehat{\mathbb{F}}$ and
$u^{\prime}$, $u$ are the Dirac homomorphism for $A$, $A\otimes B_{0}$,
respectively

###### Proof.

The proof of item $(i)$ follows the same argument as the one in item $(i)$ of
[19, Lemma 5.2.2]. For item $(ii)$, since we assume that
$\widehat{\mathbb{H}}$ satisfies the strong quantum BC property, one has
${{}_{A_{0}}}\mathcal{Z}(\mathscr{N}_{\widehat{\mathbb{H}}})\subset\mathscr{N}_{\widehat{\mathbb{F}}}$,
for all $(A_{0},\alpha_{0})\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$
thanks to Lemma 3.1.1. Then, the argument appearing in item $(ii)$ of [19,
Lemma 5.2.2] can be applied verbatim. Finally, item $(iii)$ is the symmetric
statement as item $(ii)$ by exchanging $\widehat{\mathbb{G}}$ with
$\widehat{\mathbb{H}}$. ∎

A special case of the previous lemma, which is important for our purpose, is
the situation when $\mathbb{H}:=\mathbb{E}$ is the trivial quantum group. In
this case, one has $\mathbb{F}=\mathbb{G}$ and so
$(L,N)=(L^{\prime},N^{\prime})$. Given a $\widehat{\mathbb{G}}$-C∗-algebra
$(A,\alpha)$, consider its
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$-triangle
in $\mathscr{KK}^{\widehat{\mathbb{G}}}$, say $\Sigma(N(A))\rightarrow
L(A)\rightarrow A\rightarrow N(A)$. Next, given a C∗-algebra
$B\in\text{Obj}(\mathscr{KK}^{\mathbb{E}})$ we apply the triangulated functor
$\mathcal{Z}_{B}$ (cf. Lemma 3.1.1) to this triangle and obtain the
distinguished triangle $\Sigma(N(A)\otimes B)\rightarrow L(A)\otimes
B\rightarrow A\otimes B\rightarrow N(A)\otimes B$ in
$\mathscr{KK}^{\widehat{\mathbb{G}}}$.

On the one hand, $\mathscr{KK}^{\mathbb{E}}=\mathscr{L}_{\mathbb{E}}$ (because
the trivial quantum group satisfies the strong quantum BC property), hence
$B\in\mathscr{L}_{\mathbb{E}}$ and so $L(A)\otimes
B\in\mathscr{L}_{\widehat{\mathbb{G}}}$ (cf. Lemma 3.1.1). On the other hand,
one also has $N(A)\otimes B\in\mathscr{N}_{\widehat{\mathbb{G}}}$. This is
true because if $\mathbb{H}:=\mathbb{E}$ is the trivial quantum group, then
$\text{Tor}(\widehat{\mathbb{F}})=\text{Tor}(\widehat{\mathbb{G}})$ and
$\mathcal{Z}(\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}\times\mathscr{KK})\subset\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}$.
Indeed, if
$(N,\delta)\in\text{Obj}(\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}})$ and
$B\in\text{Obj}(\mathscr{KK})$, then $j_{\mathbb{G},T}(N\otimes B)=(N\otimes
B)\underset{r,\delta\otimes
id}{\rtimes}\mathbb{G}\underset{r,\overline{\tau}}{\ltimes}T^{op}\cong(N\underset{r,\delta}{\rtimes}\mathbb{G}\underset{r,\overline{\tau}}{\ltimes}T^{op})\otimes
B\cong 0$ in $\mathscr{KK}$, for every torsion action $(T,\tau)$ of
$\mathbb{G}$; where we have used Proposition 2.2. In other words,
$\mathcal{Z}(\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}\times\mathscr{KK})\subset\widehat{\mathscr{N}}_{\widehat{\mathbb{G}}}$,
which yields
$\mathcal{Z}(\mathscr{N}_{\widehat{\mathbb{G}}}\times\mathscr{KK})\subset\mathscr{N}_{\widehat{\mathbb{G}}}$
by virtue of Baaj-Skandalis duality. Hence $N(A)\otimes
B\in\mathscr{N}_{\widehat{\mathbb{G}}}$ as claimed above.

In conclusion, we have showed that the distinguished triangle
$\Sigma(N(A)\otimes B)\rightarrow L(A)\otimes B\rightarrow A\otimes
B\rightarrow N(A)\otimes B$ in $\mathscr{KK}^{\widehat{\mathbb{G}}}$ is a
$(\mathscr{L}_{\widehat{\mathbb{G}}},\mathscr{N}_{\widehat{\mathbb{G}}})$-triangle
in $\mathscr{KK}^{\widehat{\mathbb{G}}}$ for $A\otimes B$. Finally, the
uniqueness of this kind of triangles together with Proposition 2.1 yield the
following:

######  3.1.3 Proposition.

Let $\mathbb{G}$ be a compact quantum group. For all
$\widehat{\mathbb{G}}$-C∗-algebra
$(A,\alpha)\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$ and all
C∗-algebra $B\in\text{Obj}(\mathscr{KK})$ there exists an _invertible_
Kasparov triple $\psi_{B}\in KK^{\widehat{\mathbb{G}}}\big{(}L(A)\otimes
B,L(A\otimes B)\big{)}$ such that the diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.77777pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 124.53166pt\raise-29.5pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
93.56583pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mbox{$(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}}$}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(3.1)

is commutative where
$\Psi_{B}:=\psi_{B}\underset{r}{\rtimes}\widehat{\mathbb{G}}$ and
$u^{\prime}$, $u$ are the Dirac homomorphism for $A$, $A\otimes B$,
respectively.

######  3.1.4 Remark.

The previous corollary is a refined version of item $(iii)$ of Lemma 3.1.2.
The main observation in the case of $\mathbb{H}=\mathbb{E}$ is that we do not
need to further assume that $\widehat{\mathbb{G}}$ satisfies the strong
quantum BC property. As the argument above showed, this is possible because
the torsion of $\widehat{\mathbb{F}}$ reduces to the torsion of
$\widehat{\mathbb{G}}$ because $\mathbb{F}=\mathbb{G}$ as soon as $\mathbb{H}$
is the trivial quantum group.

The additional hypothesis about the strong quantum BC property in Lemma 3.1.2
is necessary in order to use that the functor $\mathcal{Z}$ preserves the
subcategories $\mathscr{N}_{*}$ as argued in Lemma 3.1.1. This property holds
automatically for classical locally compact groups (cf. [22, Section 10.4]).
In the quantum setting, the main obstacle is the understanding of the torsion
phenomenon for a quantum direct product. Namely, as explained in Section 2.4,
the subcategory $\widehat{\mathscr{N}}_{\widehat{\mathbb{F}}}$ is completely
described by the vanishing of the functors $j_{\mathbb{F},R}(\ \cdot\ )$ with
$R\in\text{Tor}(\widehat{\mathbb{F}})$. It would be interesting to know the
description of $\text{Tor}(\widehat{\mathbb{F}})$ in terms of
$\text{Tor}(\widehat{\mathbb{G}})$ and $\text{Tor}(\widehat{\mathbb{H}})$ (as
explained in the introduction, it is the subject of a subsequent paper) and
then the behaviour of these C∗-algebras with respect to the two-sided crossed
product construction, which is out of the scope of the present paper. This is
in line with Proposition 3.1.3 above.

In general, the strong quantum BC property assumption allows to overlook this
issue and in practice it is not too restrictive. On the one hand, all the
examples of compact quantum groups studied in this context have duals
satisfying the strong quantum BC property (cf. Section 5). On the other hand,
the permanence property established in Theorem 4.5 (which is analogue to [5,
Theorem 5.3]) requires an additional assumption with respect to the classical
setting due to our approach to the equivariant Künneth class in the quantum
setting (cf. Section 3.2). This additional assumption is automatically
fulfilled as soon as the strong quantum BC property holds (see Remark 4.6 for
more details).

Another consequence of Lemma 3.1.2 is that a quantum direct product satisfies
the quantum BC property with coefficients in tensor products as soon as the
quantum groups involved satisfy the strong quantum BC property. This result is
the content of [19, Theorem 5.2.3-(i)] under torsion-freeness assumption. Here
we are able to improve it by removing this assumption. The argument, as based
on Lemma 3.1.2, remains the same as the one in [19, Theorem 5.2.3-(i)] (see
also [19, Remark 5.2.4]). So, we have:

######  3.1.5 Theorem.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups and let
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ be the corresponding quantum direct
product of $\mathbb{G}$ and $\mathbb{H}$. If $\widehat{\mathbb{G}}$ and
$\widehat{\mathbb{H}}$ satisfy the strong quantum BC property, then
$\widehat{\mathbb{F}}$ satisfies the quantum BC property with coefficients in
$A\otimes B$, for every $A\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$
and $B\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{H}}})$. Moreover,
$L(A\otimes B)\cong A\otimes B$, for every
$A\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$ and
$B\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{H}}})$.

######  3.1.6 Remark.

In the course of the proof of [19, Theorem 5.2.3-(i)] it is shown that
$\eta^{\widehat{\mathbb{G}}}_{A}\otimes\eta^{\widehat{\mathbb{H}}}_{B}=\eta^{\widehat{\mathbb{F}}}_{A\otimes
B}$ as soon as both $\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ satisfy
the strong quantum BC property.

Note that the analogous assertion for the usual quantum BC property needs
further hypothesis, which are related to the Künneth formula in order to
compute the K-theory of a tensor product. This relation will be studied in
Section 4.

###  The Künneth class $\mathcal{N}_{\widehat{\mathbb{G}}}$

The previous preliminary observations allow us to formulate an equivariant
Künneth class in the spirit of Examples 3.4 and [5] in the context of quantum
groups. More precisely, let $\mathbb{G}$ be a compact quantum group,
$(A,\alpha)$ a $\widehat{\mathbb{G}}$-C∗-algebra and $B$ a C∗algebra. The
exterior Kasparov product $\widetilde{\tau}_{\mathbb{C}}$ together with the
invertible Kasparov triple $\psi_{B}$ obtained in Proposition 3.1.3 give:

$KK(\mathbb{C},L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})\times
KK(\mathbb{C},B)\overset{\widetilde{\tau}_{\mathbb{C}}}{\longrightarrow}KK(\mathbb{C},L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}\otimes
B)\cong KK(\mathbb{C},L(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}}),$

which yields a group homomorphism
$\alpha_{\widehat{\mathbb{G}}}:K_{*}(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(B)\rightarrow K_{*}(L(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}})$, which is natural on $A$ and
$B$.

######  3.2.1 Definition.

Let $\mathbb{G}$ be a compact quantum group. We denote by
$\mathcal{N}_{\widehat{\mathbb{G}}}$ the class of all
$\widehat{\mathbb{G}}$-C∗-algebras $(A,\alpha)$ in
$\widehat{\mathbb{G}}\text{-C}^{*}\text{-Alg}$ such that
$\alpha_{\widehat{\mathbb{G}}}$ is an isomorphism for all separable C∗-algebra
$B$ with $K_{*}(B)$ free abelian. By abuse of notation, we still denote by
$\mathcal{N}_{\widehat{\mathbb{G}}}$ the full subcategory of
$\widehat{\mathbb{G}}\text{-C}^{*}\text{-Alg}$ generated by the class
$\mathcal{N}_{\widehat{\mathbb{G}}}$.

Note that the homomorphism $\alpha_{\widehat{\mathbb{G}}}$ is obtained, up to
the identification given by Proposition 3.1.3, as it was obtained the group
homomorphism $\alpha$ from Examples 3.4. Here we use
$L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}$ instead of an arbitrary
C∗-algebra $A$. In this sense, we have that
$(A,\alpha)\in\mathcal{N}_{\widehat{\mathbb{G}}}$ $\Leftrightarrow$
$L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$, by definition.
Therefore, $\mathcal{N}_{\widehat{\mathbb{G}}}$ is a Künneth category in the
sense of Definition 3.1 and, given
$(A,\alpha)\in\mathcal{N}_{\widehat{\mathbb{G}}}$, the map $B\mapsto
K_{*}(L(A\otimes B)\underset{r}{\rtimes}\widehat{\mathbb{G}})$ defines a
functor on the whole category $\text{C}^{*}\text{-Alg}$ satisfying properties
$(i)-(iv)$ from Definition 3.2. In other words, this map defines a Künneth
functor. Consequently, Theorem 3.3 yields that the Künneth sequence:

$0\longrightarrow K_{*}(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(B)\overset{\alpha_{\widehat{\mathbb{G}}}}{\longrightarrow}K_{*}(L(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}})\overset{\beta_{\widehat{\mathbb{G}}}}{\longrightarrow}\text{Tor}(K_{*}(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}),K_{*}(B))\longrightarrow
0$

is exact for all $(A,\alpha)\in\mathcal{N}_{\widehat{\mathbb{G}}}$ and all
$B\in\text{Obj}(\text{C}^{*}\text{-Alg})$.

######  3.2.2 Example.

If $\widehat{\mathbb{G}}$ is a classical locally compact group $G$, then
$\mathcal{N}_{\widehat{\mathbb{G}}}=\mathcal{N}_{G}$ as defined in Examples
3.4. Indeed, from the Meyer-Nest reformulation of the BC property (see [22,
Theorem 5.2]) one knows that $K^{\text{top}}_{*}(G,A)\cong
K_{*}(L(A)\underset{r}{\rtimes}G)$ naturally, for all $G$-C∗-algebra
$(A,\alpha)$. Moreover, this approach to $\mathcal{N}_{G}$ yields a
characterisation of the objects in the equivariant Künneth class in terms of
the non-equivariant one, up to replacing $A$ by a $\mathscr{L}$-simplicial
approximation of $A$. Namely, $(A,\alpha)\in\mathcal{N}_{G}$ $\Leftrightarrow$
$L(A)\underset{r}{\rtimes}G\in\mathcal{N}$.

##  Relating the quantum Baum-Connes property with the Künneth formula

In this section we generalise to the case of discrete quantum groups some key
results appearing in [5]. In particular, we are going to improve the stability
of the quantum BC property for quantum direct products appearing already in
[19].

######  4.1 Proposition.

Let $\mathbb{G}$ be a compact quantum group. The diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
54.15475pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mbox{$K_{*}(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(B)$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.0625pt\hbox{$\scriptstyle{\mbox{$\alpha_{\widehat{\mathbb{G}}}$}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
68.7866pt\raise 6.63194pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.27084pt\hbox{$\scriptstyle{\mbox{$\eta^{\widehat{\mathbb{G}}}_{A}\otimes
id$}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 111.06026pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
111.06026pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mbox{$K_{*}(A\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(B)$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
157.92332pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\mbox{$\alpha$}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 157.92332pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mbox{$K_{*}(L(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}})$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
72.00157pt\raise-33.21562pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.1184pt\hbox{$\scriptstyle{\mbox{$\eta^{\widehat{\mathbb{G}}}_{A\otimes
B}$}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
117.06442pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
117.06442pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mbox{$K_{*}((A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}})$}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(4.1)

commutes for all $\widehat{\mathbb{G}}$-C∗-algebra $A$ and all C∗-algebra $B$.
In particular, if $\widehat{\mathbb{G}}$ satisfies the BC property with
coefficients in $A\otimes B$, for all C∗-algebra $B$ equipped with the trivial
action of $\widehat{\mathbb{G}}$; then
$A\in\mathcal{N}_{\widehat{\mathbb{G}}}$ $\Leftrightarrow$
$A\underset{r}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$.

###### Proof.

Recall that the quantum assembly map $\eta^{\widehat{\mathbb{G}}}$ can be
pictured as a certain Kasparov product (cf. Equation (2.2)). Let $y\in
K_{*}(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})=KK(\mathbb{C},L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})$
and $x\in K_{*}(B)=KK(\mathbb{C},B)$. On the one hand, we have:

$\begin{split}y\otimes x\overset{\alpha_{\widehat{\mathbb{G}}}}{\mapsto}y\
\underset{\mathbb{C}}{\otimes^{\text{ext}}}x\overset{\eta^{\widehat{\mathbb{G}}}_{A\otimes
B}}{\mapsto}(y\
\underset{\mathbb{C}}{\otimes^{\text{ext}}}x)\underset{L(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}}}{\otimes}u\underset{r}{\rtimes}\widehat{\mathbb{G}}&\cong(y\
\underset{\mathbb{C}}{\otimes^{\text{ext}}}x)\underset{L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}\otimes
B}{\otimes}\big{(}u^{\prime}\underset{r}{\rtimes}\widehat{\mathbb{G}}\otimes
id\big{)}\\\
&=\big{(}y\underset{L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}}{\otimes}u^{\prime}\underset{r}{\rtimes}\widehat{\mathbb{G}}\big{)}\
\underset{\mathbb{C}}{\otimes^{\text{ext}}}\big{(}x\underset{B}{\otimes}id),\end{split}$

where the last equality follows from Equation (2.1). On the other hand, we
have:

$\begin{split}y\otimes x&\overset{\eta^{\widehat{\mathbb{G}}}_{A}\otimes
id}{\mapsto}\big{(}y\underset{L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}}{\otimes}u^{\prime}\underset{r}{\rtimes}\widehat{\mathbb{G}}\big{)}\otimes
x\overset{\alpha}{\mapsto}\big{(}y\underset{L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}}{\otimes}u^{\prime}\underset{r}{\rtimes}\widehat{\mathbb{G}}\big{)}\
\underset{\mathbb{C}}{\otimes^{\text{ext}}}x.\end{split}$

These computations show that, indeed, Diagram (4.1) commutes. To conclude,
notice that if $\widehat{\mathbb{G}}$ satisfies the BC property with
coefficients in $A\otimes B$, for all C∗-algebra $B$ equipped with the trivial
action of $\widehat{\mathbb{G}}$; then both $\eta_{A}$ and $\eta_{A\otimes B}$
are isomorphisms, for all C∗-algebra $B$. Therefore, commutativity of Diagram
(4.1) yields that $\alpha_{\widehat{\mathbb{G}}}$ is an isomorphism if and
only if $\alpha$ is an isomorphism. ∎

######  4.2 Proposition.

Let $\mathbb{G}$ be a compact quantum group. Assume that $A$ is a
$\widehat{\mathbb{G}}$-C∗-algebra such that $\widehat{\mathbb{G}}$ satisfies
the quantum BC property with coefficients in $A$.

  1. i)

If $A\in\mathcal{N}_{\widehat{\mathbb{G}}}$ and
$A\underset{r}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$, then
$\widehat{\mathbb{G}}$ satisfies the quantum BC property for $A\otimes B$, for
all C∗-algebra $B$.

  2. ii)

If $A\in\mathcal{N}_{\widehat{\mathbb{G}}}$, then $\widehat{\mathbb{G}}$
satisfies the quantum BC property for $A\otimes B$, for all C∗-algebra
$B\in\mathcal{N}$.

###### Proof.

First, recall that given a $\widehat{\mathbb{G}}$-C∗-algebra $A$, we have
$A\in\mathcal{N}_{\widehat{\mathbb{G}}}$ $\Leftrightarrow$
$L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$ by definition.
Next, consider the following diagram:

$0\longrightarrow K_{*}(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(B)$$\alpha_{\widehat{\mathbb{G}}}$$\eta^{\widehat{\mathbb{G}}}_{A}\otimes
id$$K_{*}(L(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}})$$\beta_{\widehat{\mathbb{G}}}$$\eta^{\widehat{\mathbb{G}}}_{A\otimes
B}$$\text{Tor}(K_{*}(L(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}),K_{*}(B))\longrightarrow
0$$\text{Tor}(\eta^{\widehat{\mathbb{G}}}_{A}\otimes id)$$0\longrightarrow
K_{*}(A\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(B)$$\alpha$$K_{*}((A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{G}})$$\beta$$\text{Tor}(K_{*}(A\underset{r}{\rtimes}\widehat{\mathbb{G}}),K_{*}(B))\longrightarrow
0$

The argument follows then the same lines as the one in [5, Proposition 4.10].
∎

######  4.3 Corollary.

Let $\mathbb{G}$ be a compact quantum group. Assume that
$\widehat{\mathbb{G}}$ satisfies the quantum BC property with coefficients in
$\mathbb{C}$.

  1. i)

If $C(\mathbb{G})\in\mathcal{N}$ and
$\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$, then $\widehat{\mathbb{G}}$
satisfies the quantum BC property for all C∗-algebra $B$ equipped with the
trivial action of $\widehat{\mathbb{G}}$.

  2. ii)

If $\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$, then
$\widehat{\mathbb{G}}$ satisfies the quantum BC property for all C∗-algebra
$B\in\mathcal{N}$.

###### Proof.

Recall that
$C(\mathbb{G})=\mathbb{C}\underset{r}{\rtimes}\widehat{\mathbb{G}}$. So, $(i)$
and $(ii)$ of statement follow from $(i)$ and $(ii)$ of Proposition 4.2,
respectively. ∎

######  4.4 Remark.

In the classical setting when $\widehat{\mathbb{G}}$ is a locally compact
group $G$, the assumption $\mathbb{C}\in\mathcal{N}_{G}$ is automatically
satisfied. Indeed, $\mathbb{C}$ is a type I C∗-algebra and $\mathcal{N}_{G}$
contains all type I $G$-C∗-algebras by virtue of [5, Theorem 0.1]. In the
quantum setting, a similar related result to [5, Theorem 0.1] is Theorem 5.1
in the next section (cf. Remark 5.2 for an explanation to this analogy).
However, to the best knowledge of the author, it is not known for instance
whether $\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$ for every discrete
quantum group $\widehat{\mathbb{G}}$. One reason for this is that in our
approach the objects in $\mathcal{N}_{\widehat{\mathbb{G}}}$ are characterised
in terms of objects in $\mathcal{N}$ up to a
_$\mathscr{L}_{\widehat{\mathbb{G}}}$ -simplicial approximation_, which
entails to study the localisation functor $L$ in relation with crossed
products and the _equivariant_ Künneth class. One possibility to do so might
be to adapt the _Going-Down technique_ from [5] based on Theorem 5.1. But this
is out of the scope of the present article.

The following theorem is an improved version of [19, Corollary 5.2.5].

######  4.5 Theorem.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups and let
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ be the corresponding quantum direct
product of $\mathbb{G}$ and $\mathbb{H}$.

  1. i)

Let $A$ be a $\widehat{\mathbb{G}}$-C∗-algebra and $B$ a
$\widehat{\mathbb{H}}$-C∗-algebra. Assume that $\widehat{\mathbb{G}}$
satisfies the strong quantum BC property and that
$B\in\mathscr{L}_{\widehat{\mathbb{H}}}$. If
$A\underset{r}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$ and
$A\in\mathcal{N}_{\widehat{\mathbb{G}}}$, then $\widehat{\mathbb{F}}$
satisfies the quantum BC property with coefficients in $A\otimes B$.

In particular, if $\widehat{\mathbb{G}}$ satisfies the strong quantum BC
property, $C(\mathbb{G})\in\mathcal{N}$ and
$\mathbb{C}\in\mathscr{L}_{\widehat{\mathbb{H}}}$; then $\widehat{\mathbb{F}}$
satisfies the quantum BC property with coefficients in $\mathbb{C}$.

  2. ii)

Assume that $\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ satisfy the
strong quantum BC property. If $C(\mathbb{G}),C(\mathbb{H})\in\mathcal{N}$ and
$\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{F}}}$, then $\widehat{\mathbb{F}}$
satisfies the quantum BC property for all C∗-algebra $B$ equipped with the
trivial action of $\widehat{\mathbb{F}}$.

###### Proof.

  1. i)

Consider the following diagram:

$0$$K_{*}(L^{\prime}(A)\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(L^{\prime\prime}(B)\underset{r}{\rtimes}\widehat{\mathbb{H}})$$\alpha_{\widehat{\mathbb{G}}}$$\eta^{\widehat{\mathbb{G}}}_{A}\otimes\eta^{\widehat{\mathbb{H}}}_{B}$$K_{*}(L(A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{F}})$$\beta_{\widehat{\mathbb{G}}}$$\eta^{\widehat{\mathbb{F}}}_{A\otimes
B}$$\text{Tor}(K_{*}(L^{\prime}(A)\underset{r}{\rtimes}\widehat{\mathbb{G}}),K_{*}(L^{\prime\prime}(B)\underset{r}{\rtimes}\widehat{\mathbb{H}}))\longrightarrow
0$$\text{Tor}(\eta^{\widehat{\mathbb{G}}}_{A}\otimes\eta^{\widehat{\mathbb{H}}}_{B}$)$0$$K_{*}(A\underset{r}{\rtimes}\widehat{\mathbb{G}})\otimes
K_{*}(B\underset{r}{\rtimes}\widehat{\mathbb{H}})$$\alpha$$K_{*}((A\otimes
B)\underset{r}{\rtimes}\widehat{\mathbb{F}})$$\beta$$\text{Tor}(K_{*}(A\underset{r}{\rtimes}\widehat{\mathbb{G}}),K_{*}(B\underset{r}{\rtimes}\widehat{\mathbb{H}}))\longrightarrow
0$

On the one hand, since $\widehat{\mathbb{G}}$ is assumed to satisfy the strong
quantum BC property and $B\in\mathscr{L}_{\widehat{\mathbb{H}}}$, we can apply
Lemma 3.1.2. In particular, one has that $L^{\prime}(A)\otimes
L^{\prime\prime}(B)\cong L(A\otimes B)$ and the middle arrow is indeed
$\eta^{\widehat{\mathbb{F}}}_{A\otimes B}$. On the other hand,
$\eta^{\widehat{\mathbb{H}}}_{B}$ is an isomorphism (because $B\cong
L^{\prime\prime}(B)$) and $\eta^{\widehat{\mathbb{G}}}_{A}$ is an isomorphism
(because $\widehat{\mathbb{G}}$ satisfies the strong quantum BC property).
Consequently, the left arrow is an isomorphism hence so is the right arrow.
Finally, the assumptions
$A\underset{r}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$ and
$A\in\mathcal{N}_{\widehat{\mathbb{G}}}$ say that the bottom and top lines of
the diagram are exact sequences, respectively. In conclusion, the Five lemma
yields that the middle arrow, i.e. $\eta^{\widehat{\mathbb{F}}}_{A\otimes B}$,
is an isomorphism.

  2. ii)

Since $\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ satisfy the strong
quantum BC property by assumption, then Theorem 3.1.5 yields that
$\widehat{\mathbb{F}}$ satisfies the quantum BC property with coefficients in
$A\otimes B$, for every $A\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$
and $B\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{H}}})$. In particular,
$\widehat{\mathbb{F}}$ satisfies the quantum BC property with coefficients in
$\mathbb{C}$. Next, observe that $C(\mathbb{F})=C(\mathbb{G})\otimes
C(\mathbb{H})\in\mathcal{N}$ because $\mathcal{N}$ is closed under taking
tensor products (cf. Lemma 3.5). Then the conclusion follows from item $(i)$
of Corollary 4.3.

∎

######  4.6 Remark.

Theorem 4.5 above is analogue to [5, Theorem 5.3]. In the quantum setting we
need however further hypothesis. Namely, we need
$\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$ as explained in Remark 4.4
and $\mathbb{C}\in\mathscr{L}_{\widehat{\mathbb{H}}}$. The reason for the
latter is again that we need some control for the
$\mathscr{L}_{\widehat{\mathbb{H}}}$-simplicial approximations in relation
with tensor products. This is made precise by Lemma 3.1.2 (and Remark 3.1.4),
which we use in the proof of Theorem 4.5. In the classical setting, this
control is guaranteed by [5, Corollary 2.10] based on their _Going-Down
technique_.

Note that assuming that $\widehat{\mathbb{G}}$ satisfies the _strong_ quantum
BC property guarantees the property
$\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$. Indeed, in this case one
has in particular that $\mathbb{C}\in\mathscr{L}_{\widehat{\mathbb{G}}}$ hence
$\mathbb{C}\cong L^{\prime}(\mathbb{C})$. Therefore, by definition of our
equivariant Künneth class, one has
$\mathbb{C}\in\mathcal{N}_{\widehat{\mathbb{G}}}$ $\Leftrightarrow$
$C(\mathbb{G})\in\mathcal{N}$. The latter is our assumption in item $(i)$ of
Theorem 4.5, which is the analogous assumption made in item $(i)$ of [5,
Theorem 5.3].

##  Some K-theory computations

Recall that by Theorem 3.1.5, we have that if $\widehat{\mathbb{G}}$ and
$\widehat{\mathbb{H}}$ satisfy the strong quantum BC property, then
$\widehat{\mathbb{G}\times\mathbb{H}}$ satisfies the strong quantum BC
property with coefficients in $A\otimes B$, for all
$A\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{G}}})$ and
$B\in\text{Obj}(\mathscr{KK}^{\widehat{\mathbb{H}}})$. In order to compute the
K-theory groups of $C(\mathbb{G}\times\mathbb{H})$ by applying the homological
techniques from the Meyer-Nest work, we would need further information
concerning the torsion phenomenon of $\widehat{\mathbb{F}}$. Namely, a
classification of torsion actions of a quantum direct product, which seems
hard in general as explained in the introduction. If such a classification is
provided, then it is plausible to construct explicit projective resolutions
for $\mathbb{C}$ in $\mathscr{KK}^{\widehat{\mathbb{G}\times\mathbb{H}}}$ as
tensor products of projective resolutions for $\mathbb{C}$ in
$\mathscr{KK}^{\widehat{\mathbb{G}}}$ and in
$\mathscr{KK}^{\widehat{\mathbb{H}}}$. Note that, since
$C(\mathbb{G}\times\mathbb{H})=C(\mathbb{G})\otimes C(\mathbb{H})$, we need
either $C(\mathbb{G})$ or $C(\mathbb{H})$ to satisfy the Künneth formula to
succeed in such a construction.

However, instead of doing that, we can compute
$K_{*}(C(\mathbb{G}\times\mathbb{H}))$ using simply the Künneth formula. In
order to do so, let us point out that [2, Theorem 5.2] and [2, Corollary 5.5]
can be also obtained for the Künneth class instead of the bootstrap class.
This is true because even if these two classes are not the same, they satisfy
similar stabilising properties in the sense of Lemma 3.5. Namely,
$\mathcal{N}$ contains all finite dimensional C∗-algebras, it is closed under
tensor products, it is closed under semi-split extensions (hence under mapping
cones and homotopy limits) and it contains all type I C∗-algebras. In other
words, the arguments in [2, Theorem 5.2] and [2, Corollary 5.5] can be applied
_verbatim_ by replacing the bootstrap class by $\mathcal{N}$ and we obtain the
following:

######  5.1 Theorem.

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

  1. i)

If $(B,\beta)$ is a $\mathbb{G}$-C∗-algebra such that
$B\underset{r,\beta}{\rtimes}\mathbb{G}\underset{r,\overline{\delta}}{\ltimes}T^{op}\in\mathcal{N}$,
for all torsion action $(T,\delta)\in\text{Tor}(\widehat{\mathbb{G}})$; then
$\widehat{L}(B)\in\mathcal{N}$.

  2. ii)

Assume that $\widehat{\mathbb{G}}$ satisfies the strong quantum BC property.
If $(A,\alpha)$ is a type $I$ $\widehat{\mathbb{G}}$-C∗-algebra, then
$A\underset{r,\alpha}{\rtimes}\widehat{\mathbb{G}}\in\mathcal{N}$. In
particular, $C(\mathbb{G})\in\mathcal{N}$.

######  5.2 Remark.

On the one hand, as we have explained in the introduction, the family of
finite subgroups represents the torsion phenomenon for a classical discrete
group. In the quantum setting it must be replaced by the family of torsion
actions of a compact quantum group. This makes a substantial difference when
it comes to define the quantum assembly map for a discrete quantum group. In
particular, the induction functor must be replaced by the two-sided crossed
product functor as explained in Section 2.4. In particular, we work in the
category $\mathscr{KK}^{\mathbb{G}}$ and not in
$\mathscr{KK}^{\widehat{\mathbb{G}}}$ itself. If $G$ is a discrete group, then
[5, Theorem 0.1] gives a sufficient condition for a $G$-C∗-algebra to be in
$\mathcal{N}_{G}$ in terms of the torsion of $G$. In this sense, item $(i)$ of
Theorem 5.1 gives a sufficient condition for the
_$\widehat{\mathscr{L}}_{\widehat{\mathbb{G}}}$ -simplicial approximation_ of
a $\mathbb{G}$-C∗-algebra to be in $\mathcal{N}$ in terms of the torsion
actions of $\mathbb{G}$. The fact that we work in $\mathscr{KK}^{\mathbb{G}}$
leads to consider _dual_ (in the sense of the Baaj-Skandalis duality)
simplicial approximations, which would lead to a _dual_ equivariant Künneth
class. However, the Künneth class $\mathcal{N}$ is not stable under general
crossed products hence it is not clear whether item $(i)$ of Theorem 5.1
translates into a statement about $\mathcal{N}_{\widehat{\mathbb{G}}}$.

On the other hand, the conclusion of item $(ii)$ of Theorem 5.1, i.e. that
$C(\mathbb{G})\in\mathcal{N}$ as soons as $\widehat{\mathbb{G}}$ satisfies the
strong quantum BC property is also true for classical locally compact groups.
One can argue as follows. Assume that $G$ is a locally compact group
satisfying the BC property with coefficients (_a fortriori_ when $G$ satisfies
the _strong_ BC property). As explained in Remark 4.4, we always have
$\mathbb{C}\in\mathcal{N}_{G}$. Therefore, [5, Proposition 4.9] implies that
$C^{*}(G)=\mathbb{C}\underset{r}{\rtimes}G\in\mathcal{N}$.

By the work of Voigt and Vergnioux-Voigt (e.g. [30], [29], [31]) we have a
number of examples of compact quantum groups with duals satisfying the strong
quantum BC property. Hence the C∗-algebras defining these compact quantum
groups lie in $\mathcal{N}$. Namely:

######  5.3 Corollary.

Let $\mathbb{G}$ be any of the following compact quantum groups: $SU_{q}(2)$,
$O^{+}(F)$, $U^{+}(Q)$, $S^{+}_{N}$, where $N\in\mathbb{N}$, $Q$ is a complex
invertible matrix and $F$ is a complex invertible matrix such that
$F\overline{F}\in\mathbb{R}id$. Then $C(\mathbb{G})\in\mathcal{N}$.

Moreover, a computation of the K-theory groups of the C∗-algebras
$C(\mathbb{G})$ is carried out too in the works [30], [29], [31]. Namely:

######  5.4 Theorem.

Let $N\in\mathbb{N}$, let $Q$ be a complex invertible matrix and let $F$ be a
complex invertible matrix such that $F\overline{F}\in\mathbb{R}id$. Then:

  * -

$K_{0}(C(SU_{q}(2)))=\mathbb{Z}$ and $K_{1}(C(SU_{q}(2)))=\mathbb{Z}$.

  * -

$K_{0}(C(O^{+}(F)))=\mathbb{Z}$ and $K_{1}(C(O^{+}(F)))=\mathbb{Z}$.

  * -

$K_{0}(C(U^{+}(Q)))=\mathbb{Z}$ and
$K_{1}(C(U^{+}(Q)))=\mathbb{Z}\oplus\mathbb{Z}$.

  * -

$K_{0}(C(S^{+}_{N}))=\mathbb{Z}^{N^{2}-2N+2}$ and
$K_{1}(C(S^{+}_{N}))=\mathbb{Z}$.

These computations together with Theorem 5.1 allow to compute the K-theory
groups for the corresponding quantum direct products by means of the Künneth
formula. More precisely, if $\mathbb{G}$ and $\mathbb{H}$ are any of the
compact quantum groups $SU_{q}(2)$, $O^{+}(F)$, $U^{+}(Q)$ or $S^{+}_{N}$ as
above, then it follows from Theorem 5.4 that $K_{*}(C(\mathbb{H}))$ is free
abelian. Moreover, $C(\mathbb{G})\in\mathcal{N}$ by Corollary 5.3. Therefore,
we have $K_{*}(C(\mathbb{F}))=K_{*}(C(\mathbb{G})\otimes
C(\mathbb{H}))=K_{*}(C(\mathbb{G}))\otimes K_{*}(C(\mathbb{H}))$, where
$\mathbb{F}=\mathbb{G}\times\mathbb{H}$. Recall that $K_{*}(\ \cdot\ )$
denotes the $\mathbb{Z}/2$-graded $K$-theory hence
$K_{l}(C(\mathbb{F}))=\underset{\underset{i,j\in\\{0,1\\}}{i+j=l}}{\bigoplus}K_{i}(C(\mathbb{G}))\otimes
K_{j}(C(\mathbb{H}))$, for all $l\in\\{0,1\\}$. Namely, we have the following:

######  5.5 Theorem.

Let $N\in\mathbb{N}$, let $Q$ be a complex invertible matrix and let $F$ be a
complex invertible matrix such that $F\overline{F}\in\mathbb{R}id$. Then:

  * -

For $\mathbb{F}:=O^{+}(F)\times O^{+}(F)$, we have
$K_{0}(C(\mathbb{F}))=\mathbb{Z}^{2}$ and
$K_{1}(C(\mathbb{F}))=\mathbb{Z}^{2}$.

  * -

For $\mathbb{F}:=O^{+}(F)\times U^{+}(Q)$, we have
$K_{0}(C(\mathbb{F}))=\mathbb{Z}^{3}$ and
$K_{1}(C(\mathbb{F}))=\mathbb{Z}^{3}$.

  * -

For $\mathbb{F}:=O^{+}(F)\times S^{+}_{N}$, we have
$K_{0}(C(\mathbb{F}))=\mathbb{Z}^{N^{2}-2N+3}$ and
$K_{1}(C(\mathbb{F}))=\mathbb{Z}^{N^{2}-2N+3}$.

  * -

For $\mathbb{F}:=U^{+}(Q)\times U^{+}(Q)$, we have
$K_{0}(C(\mathbb{F}))=\mathbb{Z}^{5}$ and
$K_{1}(C(\mathbb{F}))=\mathbb{Z}^{4}$.

  * -

For $\mathbb{F}:=U^{+}(Q)\times S^{+}_{N}$, we have
$K_{0}(C(\mathbb{F}))=\mathbb{Z}^{N^{2}-2N+4}$ and
$K_{1}(C(\mathbb{F}))=\mathbb{Z}^{2N^{2}-4N+5}$.

  * -

For $\mathbb{F}:=S^{+}_{N}\times S^{+}_{N}$, we have
$K_{0}(C(\mathbb{F}))=\mathbb{Z}^{(N^{2}-2N+2)^{2}+1}$ and
$K_{1}(C(\mathbb{F}))=\mathbb{Z}^{2(N^{2}-2N+2)}$.

Of course, a similar list of K-theory groups can be obtained with other
combinations of compact quantum groups into a quantum direct product as soon
as the strong quantum BC property is satisfied and the K-groups of the
C∗-algebras defining the quantum groups involved are free abelian. Notice
that, in order to apply the Künneth formula, the computation of the K-groups
of the C∗-algebra defining $\mathbb{G}\times\mathbb{H}$ requires at least
$K_{*}(C(\mathbb{H}))$ to be free abelian.

For instance, let $\mathbb{G}$ be a compact quantum group, $N\geq 4$ and
$\mathbb{G}\wr_{*}S^{+}_{N}$ the corresponding free wreath product of
$\mathbb{G}$ by $S^{+}_{N}$. A result by F. Lemeux and P. Tarrago (cf. [17])
shows that there exists a parameter $q\in[-1,1]$ such that the compact quantum
group $\mathbb{H}_{q}$ is monoidal equivalent to $\mathbb{G}\wr_{*}S^{+}_{N}$,
where $\mathbb{H}_{q}$ is such that $\widehat{\mathbb{H}}_{q}$ is the discrete
quantum subgroup of $\widehat{\mathbb{G}*SU_{q}(2)}$ generated, in the sense
of the Tannaka-Krein duality, by the representations $xux$ (as a word in
$\text{Irr}(\mathbb{G}*SU_{q}(2))$) with $x\in\text{Irr}(\mathbb{G})$ and $u$
being the fundamental representation of $SU_{q}(2)$. It is shown in [11] that
$\widehat{\mathbb{G}\wr_{*}S^{+}_{N}}$ (hence $\widehat{\mathbb{H}}_{q}$)
satisfies the strong quantum BC property as soon as $\widehat{\mathbb{G}}$ is
torsion-free and satisfies the strong quantum BC property.

The K-theory of $C(\mathbb{H}_{q})$ is computed in [11] for concrete and
relevant instances of $\mathbb{G}$. For example, when
$\widehat{\mathbb{G}}:=\mathbb{F}_{n}$ is the classical free group with
$n\in\mathbb{N}$ generators, then
$K_{0}(C(\mathbb{H}_{q}))=\mathbb{Z}\oplus\mathbb{Z}_{2}$ and
$K_{1}(C(\mathbb{H}_{q}))=\mathbb{Z}^{n+1}$. In this case, we see that the
K-groups are _not_ free abelian, so that the Künneth formula cannot be applied
to compute the K-theory of quantum direct products of the form
$\mathbb{X}\times\mathbb{H}_{q}$, but it still applies for quantum direct
products of the form $\mathbb{H}_{q}\times\mathbb{X}$ for $\mathbb{X}$ any
compact quantum group satisfying the strong quantum BC property and such that
$K_{*}(C(\mathbb{X}))$ is free abelian.

The K-theory of $C(\mathbb{G}\wr_{*}S^{+}_{N})$ is computed in the recent
paper [10] by P. Fima and A. Troupel for concrete and relevant instances of
$\mathbb{G}$. For example, when $\widehat{\mathbb{G}}:=\mathbb{F}_{n}$ is the
classical free group with $n\in\mathbb{N}$ generators, then
$K_{0}(C(\widehat{\mathbb{F}}_{n}\wr_{*}S^{+}_{N}))=\mathbb{Z}^{N^{2}-2N+2}$
and $K_{1}(C(\widehat{\mathbb{F}}_{n}\wr_{*}S^{+}_{N}))=\mathbb{Z}^{N^{2}n+1}$
(cf. [10, Corollary 7.2]). In this case, the K-groups are free abelian.

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